src/HOL/Limits.thy
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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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lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)"
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  by (auto simp: filter_eq_iff eventually_filtermap eventually_sup)
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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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   297
  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_ge_at_top:
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  "eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top"
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  unfolding eventually_at_top_linorder by auto
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   304
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lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::dense_linorder. \<forall>n>N. P n)"
50247
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   306
  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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lemma eventually_gt_at_top:
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
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   316
  "eventually (\<lambda>x. (c::_::dense_linorder) < x) at_top"
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  unfolding eventually_at_top_dense by auto
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definition at_bot :: "('a::order) filter"
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  where "at_bot = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<le>k. P n)"
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   321
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   322
lemma eventually_at_bot_linorder:
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   323
  fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"
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   324
  unfolding at_bot_def
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   325
proof (rule eventually_Abs_filter, rule is_filter.intro)
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   326
  fix P Q :: "'a \<Rightarrow> bool"
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   327
  assume "\<exists>i. \<forall>n\<le>i. P n" and "\<exists>j. \<forall>n\<le>j. Q n"
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   328
  then obtain i j where "\<forall>n\<le>i. P n" and "\<forall>n\<le>j. Q n" by auto
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   329
  then have "\<forall>n\<le>min i j. P n \<and> Q n" by simp
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   330
  then show "\<exists>k. \<forall>n\<le>k. P n \<and> Q n" ..
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
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   331
qed auto
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   332
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   333
lemma eventually_le_at_bot:
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
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   334
  "eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
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   335
  unfolding eventually_at_bot_linorder by auto
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
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diff changeset
   336
50247
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lemma eventually_at_bot_dense:
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   338
  fixes P :: "'a::dense_linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n<N. P n)"
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   339
  unfolding eventually_at_bot_linorder
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   340
proof safe
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   341
  fix N assume "\<forall>n\<le>N. P n" then show "\<exists>N. \<forall>n<N. P n" by (auto intro!: exI[of _ N])
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   342
next
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
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   343
  fix N assume "\<forall>n<N. P n" 
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
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   344
  moreover from lt_ex[of N] guess y ..
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
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   345
  ultimately show "\<exists>N. \<forall>n\<le>N. P n" by (auto intro!: exI[of _ y])
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   346
qed
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   347
50346
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   348
lemma eventually_gt_at_bot:
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   349
  "eventually (\<lambda>x. x < (c::_::dense_linorder)) at_bot"
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   350
  unfolding eventually_at_bot_dense by auto
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   351
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   352
subsection {* Sequentially *}
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   353
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   354
abbreviation sequentially :: "nat filter"
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   355
  where "sequentially == at_top"
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   356
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   357
lemma sequentially_def: "sequentially = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
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   358
  unfolding at_top_def by simp
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diff changeset
   359
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
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   360
lemma eventually_sequentially:
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   361
  "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
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diff changeset
   362
  by (rule eventually_at_top_linorder)
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diff changeset
   363
44342
8321948340ea redefine constant 'trivial_limit' as an abbreviation
huffman
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diff changeset
   364
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
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diff changeset
   365
  unfolding filter_eq_iff eventually_sequentially by auto
36662
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diff changeset
   366
44342
8321948340ea redefine constant 'trivial_limit' as an abbreviation
huffman
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   367
lemmas trivial_limit_sequentially = sequentially_bot
8321948340ea redefine constant 'trivial_limit' as an abbreviation
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diff changeset
   368
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
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diff changeset
   369
lemma eventually_False_sequentially [simp]:
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
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diff changeset
   370
  "\<not> eventually (\<lambda>n. False) sequentially"
44081
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huffman
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diff changeset
   371
  by (simp add: eventually_False)
36662
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diff changeset
   372
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
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diff changeset
   373
lemma le_sequentially:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
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diff changeset
   374
  "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
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diff changeset
   375
  unfolding le_filter_def eventually_sequentially
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
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diff changeset
   376
  by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
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diff changeset
   377
45892
8dcf6692433f add lemmas about limits
noschinl
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diff changeset
   378
lemma eventually_sequentiallyI:
8dcf6692433f add lemmas about limits
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diff changeset
   379
  assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
8dcf6692433f add lemmas about limits
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diff changeset
   380
  shows "eventually P sequentially"
8dcf6692433f add lemmas about limits
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diff changeset
   381
using assms by (auto simp: eventually_sequentially)
8dcf6692433f add lemmas about limits
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diff changeset
   382
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
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diff changeset
   383
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
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diff changeset
   384
subsection {* Standard filters *}
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
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diff changeset
   385
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
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diff changeset
   386
definition within :: "'a filter \<Rightarrow> 'a set \<Rightarrow> 'a filter" (infixr "within" 70)
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
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diff changeset
   387
  where "F within S = Abs_filter (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F)"
31392
69570155ddf8 replace filters with filter bases
huffman
parents: 31357
diff changeset
   388
44206
5e4a1664106e locale-ize some constant definitions, so complete_space can inherit from metric_space
huffman
parents: 44205
diff changeset
   389
definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   390
  where "nhds a = Abs_filter (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
36654
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   391
44206
5e4a1664106e locale-ize some constant definitions, so complete_space can inherit from metric_space
huffman
parents: 44205
diff changeset
   392
definition (in topological_space) at :: "'a \<Rightarrow> 'a filter"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   393
  where "at a = nhds a within - {a}"
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   394
50326
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hoelzl
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diff changeset
   395
abbreviation at_right :: "'a\<Colon>{topological_space, order} \<Rightarrow> 'a filter" where
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 50325
diff changeset
   396
  "at_right x \<equiv> at x within {x <..}"
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
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diff changeset
   397
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
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diff changeset
   398
abbreviation at_left :: "'a\<Colon>{topological_space, order} \<Rightarrow> 'a filter" where
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
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diff changeset
   399
  "at_left x \<equiv> at x within {..< x}"
b5afeccab2db add filterlim rules for exp and ln to infinity
hoelzl
parents: 50325
diff changeset
   400
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
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diff changeset
   401
definition at_infinity :: "'a::real_normed_vector filter" where
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
   402
  "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
   403
31392
69570155ddf8 replace filters with filter bases
huffman
parents: 31357
diff changeset
   404
lemma eventually_within:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   405
  "eventually P (F within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   406
  unfolding within_def
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   407
  by (rule eventually_Abs_filter, rule is_filter.intro)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   408
     (auto elim!: eventually_rev_mp)
31392
69570155ddf8 replace filters with filter bases
huffman
parents: 31357
diff changeset
   409
45031
9583f2b56f85 add lemmas within_empty and tendsto_bot;
huffman
parents: 44627
diff changeset
   410
lemma within_UNIV [simp]: "F within UNIV = F"
9583f2b56f85 add lemmas within_empty and tendsto_bot;
huffman
parents: 44627
diff changeset
   411
  unfolding filter_eq_iff eventually_within by simp
9583f2b56f85 add lemmas within_empty and tendsto_bot;
huffman
parents: 44627
diff changeset
   412
9583f2b56f85 add lemmas within_empty and tendsto_bot;
huffman
parents: 44627
diff changeset
   413
lemma within_empty [simp]: "F within {} = bot"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   414
  unfolding filter_eq_iff eventually_within by simp
36360
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   415
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
   416
lemma within_within_eq: "(F within S) within T = F within (S \<inter> T)"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
   417
  by (auto simp: filter_eq_iff eventually_within elim: eventually_elim1)
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
   418
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
   419
lemma at_within_eq: "at x within T = nhds x within (T - {x})"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
   420
  unfolding at_def within_within_eq by (simp add: ac_simps Diff_eq)
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
   421
50247
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
   422
lemma within_le: "F within S \<le> F"
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
   423
  unfolding le_filter_def eventually_within by (auto elim: eventually_elim1)
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
   424
50323
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
   425
lemma le_withinI: "F \<le> F' \<Longrightarrow> eventually (\<lambda>x. x \<in> S) F \<Longrightarrow> F \<le> F' within S"
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
   426
  unfolding le_filter_def eventually_within by (auto elim: eventually_elim2)
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
   427
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
   428
lemma le_within_iff: "eventually (\<lambda>x. x \<in> S) F \<Longrightarrow> F \<le> F' within S \<longleftrightarrow> F \<le> F'"
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
   429
  by (blast intro: within_le le_withinI order_trans)
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
   430
36654
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   431
lemma eventually_nhds:
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   432
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   433
unfolding nhds_def
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   434
proof (rule eventually_Abs_filter, rule is_filter.intro)
36654
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   435
  have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
   436
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" ..
36358
246493d61204 define nets directly as filters, instead of as filter bases
huffman
parents: 31902
diff changeset
   437
next
246493d61204 define nets directly as filters, instead of as filter bases
huffman
parents: 31902
diff changeset
   438
  fix P Q
36654
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   439
  assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   440
     and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"
36358
246493d61204 define nets directly as filters, instead of as filter bases
huffman
parents: 31902
diff changeset
   441
  then obtain S T where
36654
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   442
    "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   443
    "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   444
  hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). P x \<and> Q x)"
36358
246493d61204 define nets directly as filters, instead of as filter bases
huffman
parents: 31902
diff changeset
   445
    by (simp add: open_Int)
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
   446
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" ..
36358
246493d61204 define nets directly as filters, instead of as filter bases
huffman
parents: 31902
diff changeset
   447
qed auto
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   448
36656
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   449
lemma eventually_nhds_metric:
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   450
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   451
unfolding eventually_nhds open_dist
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   452
apply safe
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   453
apply fast
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31488
diff changeset
   454
apply (rule_tac x="{x. dist x a < d}" in exI, simp)
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   455
apply clarsimp
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   456
apply (rule_tac x="d - dist x a" in exI, clarsimp)
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   457
apply (simp only: less_diff_eq)
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   458
apply (erule le_less_trans [OF dist_triangle])
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   459
done
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   460
44571
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44568
diff changeset
   461
lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot"
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44568
diff changeset
   462
  unfolding trivial_limit_def eventually_nhds by simp
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44568
diff changeset
   463
36656
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   464
lemma eventually_at_topological:
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   465
  "eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   466
unfolding at_def eventually_within eventually_nhds by simp
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   467
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   468
lemma eventually_at:
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   469
  fixes a :: "'a::metric_space"
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   470
  shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   471
unfolding at_def eventually_within eventually_nhds_metric by auto
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   472
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 50326
diff changeset
   473
lemma eventually_within_less: (* COPY FROM Topo/eventually_within *)
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 50326
diff changeset
   474
  "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 50326
diff changeset
   475
  unfolding eventually_within eventually_at dist_nz by auto
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 50326
diff changeset
   476
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 50326
diff changeset
   477
lemma eventually_within_le: (* COPY FROM Topo/eventually_within_le *)
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 50326
diff changeset
   478
  "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 50326
diff changeset
   479
  unfolding eventually_within_less by auto (metis dense order_le_less_trans)
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 50326
diff changeset
   480
44571
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44568
diff changeset
   481
lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}"
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44568
diff changeset
   482
  unfolding trivial_limit_def eventually_at_topological
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44568
diff changeset
   483
  by (safe, case_tac "S = {a}", simp, fast, fast)
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44568
diff changeset
   484
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44568
diff changeset
   485
lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \<noteq> bot"
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44568
diff changeset
   486
  by (simp add: at_eq_bot_iff not_open_singleton)
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44568
diff changeset
   487
50331
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
   488
lemma trivial_limit_at_left_real [simp]: (* maybe generalize type *)
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
   489
  "\<not> trivial_limit (at_left (x::real))"
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
   490
  unfolding trivial_limit_def eventually_within_le
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
   491
  apply clarsimp
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
   492
  apply (rule_tac x="x - d/2" in bexI)
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
   493
  apply (auto simp: dist_real_def)
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
   494
  done
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
   495
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
   496
lemma trivial_limit_at_right_real [simp]: (* maybe generalize type *)
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
   497
  "\<not> trivial_limit (at_right (x::real))"
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
   498
  unfolding trivial_limit_def eventually_within_le
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
   499
  apply clarsimp
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
   500
  apply (rule_tac x="x + d/2" in bexI)
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
   501
  apply (auto simp: dist_real_def)
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
   502
  done
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
   503
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
   504
lemma eventually_at_infinity:
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   505
  "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"
50324
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
   506
unfolding at_infinity_def
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
   507
proof (rule eventually_Abs_filter, rule is_filter.intro)
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
   508
  fix P Q :: "'a \<Rightarrow> bool"
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
   509
  assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
   510
  then obtain r s where
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
   511
    "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
   512
  then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
   513
  then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..
0a1242d5e7d4 add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents: 50323
diff changeset
   514
qed auto
31392
69570155ddf8 replace filters with filter bases
huffman
parents: 31357
diff changeset
   515
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   516
lemma at_infinity_eq_at_top_bot:
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   517
  "(at_infinity \<Colon> real filter) = sup at_top at_bot"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   518
  unfolding sup_filter_def at_infinity_def eventually_at_top_linorder eventually_at_bot_linorder
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   519
proof (intro arg_cong[where f=Abs_filter] ext iffI)
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   520
  fix P :: "real \<Rightarrow> bool" assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   521
  then guess r ..
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   522
  then have "(\<forall>x\<ge>r. P x) \<and> (\<forall>x\<le>-r. P x)" by auto
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   523
  then show "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)" by auto
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   524
next
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   525
  fix P :: "real \<Rightarrow> bool" assume "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   526
  then obtain p q where "\<forall>x\<ge>p. P x" "\<forall>x\<le>q. P x" by auto
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   527
  then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   528
    by (intro exI[of _ "max p (-q)"])
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   529
       (auto simp: abs_real_def)
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   530
qed
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   531
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   532
lemma at_top_le_at_infinity:
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   533
  "at_top \<le> (at_infinity :: real filter)"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   534
  unfolding at_infinity_eq_at_top_bot by simp
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   535
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   536
lemma at_bot_le_at_infinity:
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   537
  "at_bot \<le> (at_infinity :: real filter)"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   538
  unfolding at_infinity_eq_at_top_bot by simp
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   539
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   540
subsection {* Boundedness *}
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   541
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   542
definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   543
  where "Bfun f F = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   544
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   545
lemma BfunI:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   546
  assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   547
unfolding Bfun_def
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   548
proof (intro exI conjI allI)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   549
  show "0 < max K 1" by simp
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   550
next
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   551
  show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   552
    using K by (rule eventually_elim1, simp)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   553
qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   554
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   555
lemma BfunE:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   556
  assumes "Bfun f F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   557
  obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   558
using assms unfolding Bfun_def by fast
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   559
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   560
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   561
subsection {* Convergence to Zero *}
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   562
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   563
definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   564
  where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   565
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   566
lemma ZfunI:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   567
  "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   568
  unfolding Zfun_def by simp
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   569
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   570
lemma ZfunD:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   571
  "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   572
  unfolding Zfun_def by simp
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   573
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   574
lemma Zfun_ssubst:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   575
  "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   576
  unfolding Zfun_def by (auto elim!: eventually_rev_mp)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   577
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   578
lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   579
  unfolding Zfun_def by simp
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   580
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   581
lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   582
  unfolding Zfun_def by simp
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   583
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   584
lemma Zfun_imp_Zfun:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   585
  assumes f: "Zfun f F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   586
  assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   587
  shows "Zfun (\<lambda>x. g x) F"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   588
proof (cases)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   589
  assume K: "0 < K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   590
  show ?thesis
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   591
  proof (rule ZfunI)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   592
    fix r::real assume "0 < r"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   593
    hence "0 < r / K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   594
      using K by (rule divide_pos_pos)
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   595
    then have "eventually (\<lambda>x. norm (f x) < r / K) F"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   596
      using ZfunD [OF f] by fast
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   597
    with g show "eventually (\<lambda>x. norm (g x) < r) F"
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   598
    proof eventually_elim
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   599
      case (elim x)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   600
      hence "norm (f x) * K < r"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   601
        by (simp add: pos_less_divide_eq K)
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   602
      thus ?case
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   603
        by (simp add: order_le_less_trans [OF elim(1)])
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   604
    qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   605
  qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   606
next
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   607
  assume "\<not> 0 < K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   608
  hence K: "K \<le> 0" by (simp only: not_less)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   609
  show ?thesis
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   610
  proof (rule ZfunI)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   611
    fix r :: real
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   612
    assume "0 < r"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   613
    from g show "eventually (\<lambda>x. norm (g x) < r) F"
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   614
    proof eventually_elim
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   615
      case (elim x)
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   616
      also have "norm (f x) * K \<le> norm (f x) * 0"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   617
        using K norm_ge_zero by (rule mult_left_mono)
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   618
      finally show ?case
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   619
        using `0 < r` by simp
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   620
    qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   621
  qed
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   622
qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   623
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   624
lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   625
  by (erule_tac K="1" in Zfun_imp_Zfun, simp)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   626
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   627
lemma Zfun_add:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   628
  assumes f: "Zfun f F" and g: "Zfun g F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   629
  shows "Zfun (\<lambda>x. f x + g x) F"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   630
proof (rule ZfunI)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   631
  fix r::real assume "0 < r"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   632
  hence r: "0 < r / 2" by simp
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   633
  have "eventually (\<lambda>x. norm (f x) < r/2) F"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   634
    using f r by (rule ZfunD)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   635
  moreover
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   636
  have "eventually (\<lambda>x. norm (g x) < r/2) F"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   637
    using g r by (rule ZfunD)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   638
  ultimately
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   639
  show "eventually (\<lambda>x. norm (f x + g x) < r) F"
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   640
  proof eventually_elim
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   641
    case (elim x)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   642
    have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   643
      by (rule norm_triangle_ineq)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   644
    also have "\<dots> < r/2 + r/2"
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   645
      using elim by (rule add_strict_mono)
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   646
    finally show ?case
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   647
      by simp
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   648
  qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   649
qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   650
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   651
lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   652
  unfolding Zfun_def by simp
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   653
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   654
lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   655
  by (simp only: diff_minus Zfun_add Zfun_minus)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   656
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   657
lemma (in bounded_linear) Zfun:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   658
  assumes g: "Zfun g F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   659
  shows "Zfun (\<lambda>x. f (g x)) F"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   660
proof -
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   661
  obtain K where "\<And>x. norm (f x) \<le> norm x * K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   662
    using bounded by fast
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   663
  then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   664
    by simp
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   665
  with g show ?thesis
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   666
    by (rule Zfun_imp_Zfun)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   667
qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   668
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   669
lemma (in bounded_bilinear) Zfun:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   670
  assumes f: "Zfun f F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   671
  assumes g: "Zfun g F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   672
  shows "Zfun (\<lambda>x. f x ** g x) F"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   673
proof (rule ZfunI)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   674
  fix r::real assume r: "0 < r"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   675
  obtain K where K: "0 < K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   676
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   677
    using pos_bounded by fast
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   678
  from K have K': "0 < inverse K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   679
    by (rule positive_imp_inverse_positive)
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   680
  have "eventually (\<lambda>x. norm (f x) < r) F"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   681
    using f r by (rule ZfunD)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   682
  moreover
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   683
  have "eventually (\<lambda>x. norm (g x) < inverse K) F"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   684
    using g K' by (rule ZfunD)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   685
  ultimately
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   686
  show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   687
  proof eventually_elim
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   688
    case (elim x)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   689
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   690
      by (rule norm_le)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   691
    also have "norm (f x) * norm (g x) * K < r * inverse K * K"
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   692
      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   693
    also from K have "r * inverse K * K = r"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   694
      by simp
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   695
    finally show ?case .
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   696
  qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   697
qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   698
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   699
lemma (in bounded_bilinear) Zfun_left:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   700
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   701
  by (rule bounded_linear_left [THEN bounded_linear.Zfun])
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   702
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   703
lemma (in bounded_bilinear) Zfun_right:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   704
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   705
  by (rule bounded_linear_right [THEN bounded_linear.Zfun])
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   706
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   707
lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   708
lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   709
lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   710
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   711
31902
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   712
subsection {* Limits *}
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   713
50322
b06b95a5fda2 rename filter_lim to filterlim to be consistent with filtermap
hoelzl
parents: 50247
diff changeset
   714
definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where
b06b95a5fda2 rename filter_lim to filterlim to be consistent with filtermap
hoelzl
parents: 50247
diff changeset
   715
  "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
50247
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
   716
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
   717
syntax
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
   718
  "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
   719
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
   720
translations
50322
b06b95a5fda2 rename filter_lim to filterlim to be consistent with filtermap
hoelzl
parents: 50247
diff changeset
   721
  "LIM x F1. f :> F2"   == "CONST filterlim (%x. f) F2 F1"
50247
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
   722
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   723
lemma filterlim_iff:
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   724
  "(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   725
  unfolding filterlim_def le_filter_def eventually_filtermap ..
50247
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
   726
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 50326
diff changeset
   727
lemma filterlim_compose:
50323
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
   728
  "filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1"
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
   729
  unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
   730
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 50326
diff changeset
   731
lemma filterlim_mono:
50323
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
   732
  "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'"
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
   733
  unfolding filterlim_def by (metis filtermap_mono order_trans)
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
   734
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 50326
diff changeset
   735
lemma filterlim_cong:
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 50326
diff changeset
   736
  "F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 50326
diff changeset
   737
  by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2)
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 50326
diff changeset
   738
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   739
lemma filterlim_within:
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   740
  "(LIM x F1. f x :> F2 within S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F1 \<and> (LIM x F1. f x :> F2))"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   741
  unfolding filterlim_def
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   742
proof safe
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   743
  assume "filtermap f F1 \<le> F2 within S" then show "eventually (\<lambda>x. f x \<in> S) F1"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   744
    by (auto simp: le_filter_def eventually_filtermap eventually_within elim!: allE[of _ "\<lambda>x. x \<in> S"])
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   745
qed (auto intro: within_le order_trans simp: le_within_iff eventually_filtermap)
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   746
50330
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50327
diff changeset
   747
lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2"
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50327
diff changeset
   748
  unfolding filterlim_def filtermap_filtermap ..
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50327
diff changeset
   749
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50327
diff changeset
   750
lemma filterlim_sup:
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50327
diff changeset
   751
  "filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)"
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50327
diff changeset
   752
  unfolding filterlim_def filtermap_sup by auto
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50327
diff changeset
   753
50331
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
   754
lemma filterlim_Suc: "filterlim Suc sequentially sequentially"
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
   755
  by (simp add: filterlim_iff eventually_sequentially) (metis le_Suc_eq)
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
   756
50247
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
   757
abbreviation (in topological_space)
44206
5e4a1664106e locale-ize some constant definitions, so complete_space can inherit from metric_space
huffman
parents: 44205
diff changeset
   758
  tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "--->" 55) where
50322
b06b95a5fda2 rename filter_lim to filterlim to be consistent with filtermap
hoelzl
parents: 50247
diff changeset
   759
  "(f ---> l) F \<equiv> filterlim f (nhds l) F"
45892
8dcf6692433f add lemmas about limits
noschinl
parents: 45294
diff changeset
   760
31902
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   761
ML {*
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   762
structure Tendsto_Intros = Named_Thms
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   763
(
45294
3c5d3d286055 tuned Named_Thms: proper binding;
wenzelm
parents: 45031
diff changeset
   764
  val name = @{binding tendsto_intros}
31902
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   765
  val description = "introduction rules for tendsto"
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   766
)
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   767
*}
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   768
31902
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   769
setup Tendsto_Intros.setup
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   770
50247
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
   771
lemma tendsto_def: "(f ---> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
50322
b06b95a5fda2 rename filter_lim to filterlim to be consistent with filtermap
hoelzl
parents: 50247
diff changeset
   772
  unfolding filterlim_def
50247
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
   773
proof safe
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
   774
  fix S assume "open S" "l \<in> S" "filtermap f F \<le> nhds l"
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
   775
  then show "eventually (\<lambda>x. f x \<in> S) F"
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
   776
    unfolding eventually_nhds eventually_filtermap le_filter_def
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
   777
    by (auto elim!: allE[of _ "\<lambda>x. x \<in> S"] eventually_rev_mp)
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
   778
qed (auto elim!: eventually_rev_mp simp: eventually_nhds eventually_filtermap le_filter_def)
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
   779
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   780
lemma filterlim_at:
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   781
  "(LIM x F. f x :> at b) \<longleftrightarrow> (eventually (\<lambda>x. f x \<noteq> b) F \<and> (f ---> b) F)"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   782
  by (simp add: at_def filterlim_within)
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   783
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   784
lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f ---> l) F' \<Longrightarrow> (f ---> l) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   785
  unfolding tendsto_def le_filter_def by fast
36656
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   786
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   787
lemma topological_tendstoI:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   788
  "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F)
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   789
    \<Longrightarrow> (f ---> l) F"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   790
  unfolding tendsto_def by auto
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   791
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   792
lemma topological_tendstoD:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   793
  "(f ---> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   794
  unfolding tendsto_def by auto
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   795
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   796
lemma tendstoI:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   797
  assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   798
  shows "(f ---> l) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   799
  apply (rule topological_tendstoI)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   800
  apply (simp add: open_dist)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   801
  apply (drule (1) bspec, clarify)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   802
  apply (drule assms)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   803
  apply (erule eventually_elim1, simp)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   804
  done
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   805
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   806
lemma tendstoD:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   807
  "(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   808
  apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   809
  apply (clarsimp simp add: open_dist)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   810
  apply (rule_tac x="e - dist x l" in exI, clarsimp)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   811
  apply (simp only: less_diff_eq)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   812
  apply (erule le_less_trans [OF dist_triangle])
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   813
  apply simp
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   814
  apply simp
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   815
  done
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   816
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   817
lemma tendsto_iff:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   818
  "(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   819
  using tendstoI tendstoD by fast
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   820
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   821
lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   822
  by (simp only: tendsto_iff Zfun_def dist_norm)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   823
45031
9583f2b56f85 add lemmas within_empty and tendsto_bot;
huffman
parents: 44627
diff changeset
   824
lemma tendsto_bot [simp]: "(f ---> a) bot"
9583f2b56f85 add lemmas within_empty and tendsto_bot;
huffman
parents: 44627
diff changeset
   825
  unfolding tendsto_def by simp
9583f2b56f85 add lemmas within_empty and tendsto_bot;
huffman
parents: 44627
diff changeset
   826
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   827
lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   828
  unfolding tendsto_def eventually_at_topological by auto
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   829
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   830
lemma tendsto_ident_at_within [tendsto_intros]:
36655
88f0125c3bd2 remove unneeded premise
huffman
parents: 36654
diff changeset
   831
  "((\<lambda>x. x) ---> a) (at a within S)"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   832
  unfolding tendsto_def eventually_within eventually_at_topological by auto
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   833
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   834
lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   835
  by (simp add: tendsto_def)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   836
44205
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   837
lemma tendsto_unique:
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   838
  fixes f :: "'a \<Rightarrow> 'b::t2_space"
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   839
  assumes "\<not> trivial_limit F" and "(f ---> a) F" and "(f ---> b) F"
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   840
  shows "a = b"
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   841
proof (rule ccontr)
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   842
  assume "a \<noteq> b"
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   843
  obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   844
    using hausdorff [OF `a \<noteq> b`] by fast
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   845
  have "eventually (\<lambda>x. f x \<in> U) F"
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   846
    using `(f ---> a) F` `open U` `a \<in> U` by (rule topological_tendstoD)
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   847
  moreover
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   848
  have "eventually (\<lambda>x. f x \<in> V) F"
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   849
    using `(f ---> b) F` `open V` `b \<in> V` by (rule topological_tendstoD)
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   850
  ultimately
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   851
  have "eventually (\<lambda>x. False) F"
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   852
  proof eventually_elim
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   853
    case (elim x)
44205
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   854
    hence "f x \<in> U \<inter> V" by simp
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   855
    with `U \<inter> V = {}` show ?case by simp
44205
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   856
  qed
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   857
  with `\<not> trivial_limit F` show "False"
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   858
    by (simp add: trivial_limit_def)
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   859
qed
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   860
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   861
lemma tendsto_const_iff:
44205
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   862
  fixes a b :: "'a::t2_space"
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   863
  assumes "\<not> trivial_limit F" shows "((\<lambda>x. a) ---> b) F \<longleftrightarrow> a = b"
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   864
  by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const])
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   865
50323
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
   866
lemma tendsto_at_iff_tendsto_nhds:
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
   867
  "(g ---> g l) (at l) \<longleftrightarrow> (g ---> g l) (nhds l)"
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
   868
  unfolding tendsto_def at_def eventually_within
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
   869
  by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1)
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
   870
44218
f0e442e24816 add lemma tendsto_compose
huffman
parents: 44206
diff changeset
   871
lemma tendsto_compose:
50323
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
   872
  "(g ---> g l) (at l) \<Longrightarrow> (f ---> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F"
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
   873
  unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g])
44218
f0e442e24816 add lemma tendsto_compose
huffman
parents: 44206
diff changeset
   874
44253
c073a0bd8458 add lemma tendsto_compose_eventually; use it to shorten some proofs
huffman
parents: 44251
diff changeset
   875
lemma tendsto_compose_eventually:
50325
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   876
  "(g ---> m) (at l) \<Longrightarrow> (f ---> l) F \<Longrightarrow> eventually (\<lambda>x. f x \<noteq> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> m) F"
5e40ad9f212a add filterlim rules for inverse and at_infinity
hoelzl
parents: 50324
diff changeset
   877
  by (rule filterlim_compose[of g _ "at l"]) (auto simp add: filterlim_at)
44253
c073a0bd8458 add lemma tendsto_compose_eventually; use it to shorten some proofs
huffman
parents: 44251
diff changeset
   878
44251
d101ed3177b6 add lemma metric_tendsto_imp_tendsto
huffman
parents: 44218
diff changeset
   879
lemma metric_tendsto_imp_tendsto:
d101ed3177b6 add lemma metric_tendsto_imp_tendsto
huffman
parents: 44218
diff changeset
   880
  assumes f: "(f ---> a) F"
d101ed3177b6 add lemma metric_tendsto_imp_tendsto
huffman
parents: 44218
diff changeset
   881
  assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
d101ed3177b6 add lemma metric_tendsto_imp_tendsto
huffman
parents: 44218
diff changeset
   882
  shows "(g ---> b) F"
d101ed3177b6 add lemma metric_tendsto_imp_tendsto
huffman
parents: 44218
diff changeset
   883
proof (rule tendstoI)
d101ed3177b6 add lemma metric_tendsto_imp_tendsto
huffman
parents: 44218
diff changeset
   884
  fix e :: real assume "0 < e"
d101ed3177b6 add lemma metric_tendsto_imp_tendsto
huffman
parents: 44218
diff changeset
   885
  with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
d101ed3177b6 add lemma metric_tendsto_imp_tendsto
huffman
parents: 44218
diff changeset
   886
  with le show "eventually (\<lambda>x. dist (g x) b < e) F"
d101ed3177b6 add lemma metric_tendsto_imp_tendsto
huffman
parents: 44218
diff changeset
   887
    using le_less_trans by (rule eventually_elim2)
d101ed3177b6 add lemma metric_tendsto_imp_tendsto
huffman
parents: 44218
diff changeset
   888
qed
d101ed3177b6 add lemma metric_tendsto_imp_tendsto
huffman
parents: 44218
diff changeset
   889
44205
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   890
subsubsection {* Distance and norms *}
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   891
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   892
lemma tendsto_dist [tendsto_intros]:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   893
  assumes f: "(f ---> l) F" and g: "(g ---> m) F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   894
  shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   895
proof (rule tendstoI)
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   896
  fix e :: real assume "0 < e"
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   897
  hence e2: "0 < e/2" by simp
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   898
  from tendstoD [OF f e2] tendstoD [OF g e2]
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   899
  show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   900
  proof (eventually_elim)
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
   901
    case (elim x)
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   902
    then show "dist (dist (f x) (g x)) (dist l m) < e"
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   903
      unfolding dist_real_def
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   904
      using dist_triangle2 [of "f x" "g x" "l"]
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   905
      using dist_triangle2 [of "g x" "l" "m"]
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   906
      using dist_triangle3 [of "l" "m" "f x"]
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   907
      using dist_triangle [of "f x" "m" "g x"]
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   908
      by arith
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   909
  qed
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   910
qed
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   911
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   912
lemma norm_conv_dist: "norm x = dist x 0"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   913
  unfolding dist_norm by simp
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   914
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   915
lemma tendsto_norm [tendsto_intros]:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   916
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   917
  unfolding norm_conv_dist by (intro tendsto_intros)
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   918
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   919
lemma tendsto_norm_zero:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   920
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   921
  by (drule tendsto_norm, simp)
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   922
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   923
lemma tendsto_norm_zero_cancel:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   924
  "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   925
  unfolding tendsto_iff dist_norm by simp
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   926
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   927
lemma tendsto_norm_zero_iff:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   928
  "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   929
  unfolding tendsto_iff dist_norm by simp
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   930
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   931
lemma tendsto_rabs [tendsto_intros]:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   932
  "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   933
  by (fold real_norm_def, rule tendsto_norm)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   934
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   935
lemma tendsto_rabs_zero:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   936
  "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   937
  by (fold real_norm_def, rule tendsto_norm_zero)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   938
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   939
lemma tendsto_rabs_zero_cancel:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   940
  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   941
  by (fold real_norm_def, rule tendsto_norm_zero_cancel)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   942
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   943
lemma tendsto_rabs_zero_iff:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   944
  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   945
  by (fold real_norm_def, rule tendsto_norm_zero_iff)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   946
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   947
subsubsection {* Addition and subtraction *}
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   948
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   949
lemma tendsto_add [tendsto_intros]:
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   950
  fixes a b :: "'a::real_normed_vector"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   951
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   952
  by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   953
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   954
lemma tendsto_add_zero:
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   955
  fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   956
  shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   957
  by (drule (1) tendsto_add, simp)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   958
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   959
lemma tendsto_minus [tendsto_intros]:
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   960
  fixes a :: "'a::real_normed_vector"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   961
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   962
  by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   963
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   964
lemma tendsto_minus_cancel:
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   965
  fixes a :: "'a::real_normed_vector"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   966
  shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   967
  by (drule tendsto_minus, simp)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   968
50330
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50327
diff changeset
   969
lemma tendsto_minus_cancel_left:
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50327
diff changeset
   970
    "(f ---> - (y::_::real_normed_vector)) F \<longleftrightarrow> ((\<lambda>x. - f x) ---> y) F"
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50327
diff changeset
   971
  using tendsto_minus_cancel[of f "- y" F]  tendsto_minus[of f "- y" F]
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50327
diff changeset
   972
  by auto
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50327
diff changeset
   973
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   974
lemma tendsto_diff [tendsto_intros]:
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   975
  fixes a b :: "'a::real_normed_vector"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   976
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   977
  by (simp add: diff_minus tendsto_add tendsto_minus)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   978
31588
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   979
lemma tendsto_setsum [tendsto_intros]:
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   980
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   981
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   982
  shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
31588
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   983
proof (cases "finite S")
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   984
  assume "finite S" thus ?thesis using assms
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   985
    by (induct, simp add: tendsto_const, simp add: tendsto_add)
31588
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   986
next
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   987
  assume "\<not> finite S" thus ?thesis
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   988
    by (simp add: tendsto_const)
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   989
qed
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   990
45892
8dcf6692433f add lemmas about limits
noschinl
parents: 45294
diff changeset
   991
lemma real_tendsto_sandwich:
8dcf6692433f add lemmas about limits
noschinl
parents: 45294
diff changeset
   992
  fixes f g h :: "'a \<Rightarrow> real"
8dcf6692433f add lemmas about limits
noschinl
parents: 45294
diff changeset
   993
  assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net"
8dcf6692433f add lemmas about limits
noschinl
parents: 45294
diff changeset
   994
  assumes lim: "(f ---> c) net" "(h ---> c) net"
8dcf6692433f add lemmas about limits
noschinl
parents: 45294
diff changeset
   995
  shows "(g ---> c) net"
8dcf6692433f add lemmas about limits
noschinl
parents: 45294
diff changeset
   996
proof -
8dcf6692433f add lemmas about limits
noschinl
parents: 45294
diff changeset
   997
  have "((\<lambda>n. g n - f n) ---> 0) net"
8dcf6692433f add lemmas about limits
noschinl
parents: 45294
diff changeset
   998
  proof (rule metric_tendsto_imp_tendsto)
8dcf6692433f add lemmas about limits
noschinl
parents: 45294
diff changeset
   999
    show "eventually (\<lambda>n. dist (g n - f n) 0 \<le> dist (h n - f n) 0) net"
8dcf6692433f add lemmas about limits
noschinl
parents: 45294
diff changeset
  1000
      using ev by (rule eventually_elim2) (simp add: dist_real_def)
8dcf6692433f add lemmas about limits
noschinl
parents: 45294
diff changeset
  1001
    show "((\<lambda>n. h n - f n) ---> 0) net"
8dcf6692433f add lemmas about limits
noschinl
parents: 45294
diff changeset
  1002
      using tendsto_diff[OF lim(2,1)] by simp
8dcf6692433f add lemmas about limits
noschinl
parents: 45294
diff changeset
  1003
  qed
8dcf6692433f add lemmas about limits
noschinl
parents: 45294
diff changeset
  1004
  from tendsto_add[OF this lim(1)] show ?thesis by simp
8dcf6692433f add lemmas about limits
noschinl
parents: 45294
diff changeset
  1005
qed
8dcf6692433f add lemmas about limits
noschinl
parents: 45294
diff changeset
  1006
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1007
subsubsection {* Linear operators and multiplication *}
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1008
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
  1009
lemma (in bounded_linear) tendsto:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1010
  "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
  1011
  by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
  1012
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1013
lemma (in bounded_linear) tendsto_zero:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1014
  "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1015
  by (drule tendsto, simp only: zero)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1016
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
  1017
lemma (in bounded_bilinear) tendsto:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1018
  "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
  1019
  by (simp only: tendsto_Zfun_iff prod_diff_prod
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
  1020
                 Zfun_add Zfun Zfun_left Zfun_right)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
  1021
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1022
lemma (in bounded_bilinear) tendsto_zero:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1023
  assumes f: "(f ---> 0) F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1024
  assumes g: "(g ---> 0) F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1025
  shows "((\<lambda>x. f x ** g x) ---> 0) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1026
  using tendsto [OF f g] by (simp add: zero_left)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1027
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1028
lemma (in bounded_bilinear) tendsto_left_zero:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1029
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1030
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1031
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1032
lemma (in bounded_bilinear) tendsto_right_zero:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1033
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1034
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1035
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
  1036
lemmas tendsto_of_real [tendsto_intros] =
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
  1037
  bounded_linear.tendsto [OF bounded_linear_of_real]
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
  1038
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
  1039
lemmas tendsto_scaleR [tendsto_intros] =
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
  1040
  bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
  1041
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
  1042
lemmas tendsto_mult [tendsto_intros] =
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
  1043
  bounded_bilinear.tendsto [OF bounded_bilinear_mult]
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1044
44568
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
  1045
lemmas tendsto_mult_zero =
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
  1046
  bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
  1047
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
  1048
lemmas tendsto_mult_left_zero =
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
  1049
  bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
  1050
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
  1051
lemmas tendsto_mult_right_zero =
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
  1052
  bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
  1053
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1054
lemma tendsto_power [tendsto_intros]:
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1055
  fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1056
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1057
  by (induct n) (simp_all add: tendsto_const tendsto_mult)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1058
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1059
lemma tendsto_setprod [tendsto_intros]:
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1060
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1061
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1062
  shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1063
proof (cases "finite S")
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1064
  assume "finite S" thus ?thesis using assms
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1065
    by (induct, simp add: tendsto_const, simp add: tendsto_mult)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1066
next
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1067
  assume "\<not> finite S" thus ?thesis
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1068
    by (simp add: tendsto_const)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1069
qed
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1070
50331
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1071
lemma tendsto_le_const:
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1072
  fixes f :: "_ \<Rightarrow> real" 
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1073
  assumes F: "\<not> trivial_limit F"
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1074
  assumes x: "(f ---> x) F" and a: "eventually (\<lambda>x. a \<le> f x) F"
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1075
  shows "a \<le> x"
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1076
proof (rule ccontr)
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1077
  assume "\<not> a \<le> x"
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1078
  with x have "eventually (\<lambda>x. f x < a) F"
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1079
    by (auto simp add: tendsto_def elim!: allE[of _ "{..< a}"])
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1080
  with a have "eventually (\<lambda>x. False) F"
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1081
    by eventually_elim auto
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1082
  with F show False
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1083
    by (simp add: eventually_False)
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1084
qed
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1085
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1086
lemma tendsto_le:
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1087
  fixes f g :: "_ \<Rightarrow> real" 
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1088
  assumes F: "\<not> trivial_limit F"
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1089
  assumes x: "(f ---> x) F" and y: "(g ---> y) F"
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1090
  assumes ev: "eventually (\<lambda>x. g x \<le> f x) F"
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1091
  shows "y \<le> x"
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1092
  using tendsto_le_const[OF F tendsto_diff[OF x y], of 0] ev
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1093
  by (simp add: sign_simps)
4b6dc5077e98 use filterlim in Lim and SEQ; tuned proofs
hoelzl
parents: 50330
diff changeset
  1094
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1095
subsubsection {* Inverse and division *}
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1096
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1097
lemma (in bounded_bilinear) Zfun_prod_Bfun:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1098
  assumes f: "Zfun f F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1099
  assumes g: "Bfun g F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1100
  shows "Zfun (\<lambda>x. f x ** g x) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1101
proof -
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1102
  obtain K where K: "0 \<le> K"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1103
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1104
    using nonneg_bounded by fast
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1105
  obtain B where B: "0 < B"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1106
    and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1107
    using g by (rule BfunE)
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1108
  have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
  1109
  using norm_g proof eventually_elim
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
  1110
    case (elim x)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1111
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1112
      by (rule norm_le)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1113
    also have "\<dots> \<le> norm (f x) * B * K"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1114
      by (intro mult_mono' order_refl norm_g norm_ge_zero
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
  1115
                mult_nonneg_nonneg K elim)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1116
    also have "\<dots> = norm (f x) * (B * K)"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1117
      by (rule mult_assoc)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1118
    finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1119
  qed
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1120
  with f show ?thesis
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1121
    by (rule Zfun_imp_Zfun)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1122
qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1123
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1124
lemma (in bounded_bilinear) flip:
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1125
  "bounded_bilinear (\<lambda>x y. y ** x)"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
  1126
  apply default
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
  1127
  apply (rule add_right)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
  1128
  apply (rule add_left)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
  1129
  apply (rule scaleR_right)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
  1130
  apply (rule scaleR_left)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
  1131
  apply (subst mult_commute)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
  1132
  using bounded by fast
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1133
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1134
lemma (in bounded_bilinear) Bfun_prod_Zfun:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1135
  assumes f: "Bfun f F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1136
  assumes g: "Zfun g F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1137
  shows "Zfun (\<lambda>x. f x ** g x) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
  1138
  using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1139
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1140
lemma Bfun_inverse_lemma:
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1141
  fixes x :: "'a::real_normed_div_algebra"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1142
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
  1143
  apply (subst nonzero_norm_inverse, clarsimp)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
  1144
  apply (erule (1) le_imp_inverse_le)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
  1145
  done
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1146
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1147
lemma Bfun_inverse:
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1148
  fixes a :: "'a::real_normed_div_algebra"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1149
  assumes f: "(f ---> a) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1150
  assumes a: "a \<noteq> 0"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1151
  shows "Bfun (\<lambda>x. inverse (f x)) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1152
proof -
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1153
  from a have "0 < norm a" by simp
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1154
  hence "\<exists>r>0. r < norm a" by (rule dense)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1155
  then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1156
  have "eventually (\<lambda>x. dist (f x) a < r) F"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1157
    using tendstoD [OF f r1] by fast
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1158
  hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
46887
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
  1159
  proof eventually_elim
cb891d9a23c1 use eventually_elim method
noschinl
parents: 46886
diff changeset
  1160
    case (elim x)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1161
    hence 1: "norm (f x - a) < r"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1162
      by (simp add: dist_norm)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1163
    hence 2: "f x \<noteq> 0" using r2 by auto
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1164
    hence "norm (inverse (f x)) = inverse (norm (f x))"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1165
      by (rule nonzero_norm_inverse)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1166
    also have "\<dots> \<le> inverse (norm a - r)"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1167
    proof (rule le_imp_inverse_le)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1168
      show "0 < norm a - r" using r2 by simp
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1169
    next
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1170
      have "norm a - norm (f x) \<le> norm (a - f x)"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1171
        by (rule norm_triangle_ineq2)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1172
      also have "\<dots> = norm (f x - a)"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1173
        by (rule norm_minus_commute)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1174
      also have "\<dots> < r" using 1 .
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1175
      finally show "norm a - r \<le> norm (f x)" by simp
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1176
    qed
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
  1177
    finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1178
  qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1179
  thus ?thesis by (rule BfunI)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1180
qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1181
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
  1182
lemma tendsto_inverse [tendsto_intros]:
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1183
  fixes a :: "'a::real_normed_div_algebra"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1184
  assumes f: "(f ---> a) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1185
  assumes a: "a \<noteq> 0"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1186
  shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1187
proof -
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1188
  from a have "0 < norm a" by simp
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1189
  with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1190
    by (rule tendstoD)
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1191
  then have "eventually (\<lambda>x. f x \<noteq> 0) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1192
    unfolding dist_norm by (auto elim!: eventually_elim1)
44627
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
  1193
  with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
  1194
    - (inverse (f x) * (f x - a) * inverse a)) F"
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
  1195
    by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
  1196
  moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
  1197
    by (intro Zfun_minus Zfun_mult_left
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
  1198
      bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
  1199
      Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
  1200
  ultimately show ?thesis
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
  1201
    unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1202
qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1203
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
  1204
lemma tendsto_divide [tendsto_intros]:
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1205
  fixes a b :: "'a::real_normed_field"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1206
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1207
    \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
  1208
  by (simp add: tendsto_mult tendsto_inverse divide_inverse)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
  1209
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1210
lemma tendsto_sgn [tendsto_intros]:
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1211
  fixes l :: "'a::real_normed_vector"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
  1212
  shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1213
  unfolding sgn_div_norm by (simp add: tendsto_intros)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
  1214
50247
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
  1215
subsection {* Limits to @{const at_top} and @{const at_bot} *}
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
  1216
50322
b06b95a5fda2 rename filter_lim to filterlim to be consistent with filtermap
hoelzl
parents: 50247
diff changeset
  1217
lemma filterlim_at_top:
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1218
  fixes f :: "'a \<Rightarrow> ('b::linorder)"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1219
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1220
  by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_elim1)
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1221
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1222
lemma filterlim_at_top_dense:
50247
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
  1223
  fixes f :: "'a \<Rightarrow> ('b::dense_linorder)"
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
  1224
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)"
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1225
  by (metis eventually_elim1[of _ F] eventually_gt_at_top order_less_imp_le
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1226
            filterlim_at_top[of f F] filterlim_iff[of f at_top F])
50247
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
  1227
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1228
lemma filterlim_at_top_ge:
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1229
  fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1230
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)"
50323
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
  1231
  unfolding filterlim_at_top
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
  1232
proof safe
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1233
  fix Z assume *: "\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1234
  with *[THEN spec, of "max Z c"] show "eventually (\<lambda>x. Z \<le> f x) F"
50323
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
  1235
    by (auto elim!: eventually_elim1)
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
  1236
qed simp
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
  1237
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1238
lemma filterlim_at_top_at_top:
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1239
  fixes f :: "'a::linorder \<Rightarrow> 'b::linorder"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1240
  assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1241
  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1242
  assumes Q: "eventually Q at_top"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1243
  assumes P: "eventually P at_top"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1244
  shows "filterlim f at_top at_top"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1245
proof -
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1246
  from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1247
    unfolding eventually_at_top_linorder by auto
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1248
  show ?thesis
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1249
  proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1250
    fix z assume "x \<le> z"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1251
    with x have "P z" by auto
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1252
    have "eventually (\<lambda>x. g z \<le> x) at_top"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1253
      by (rule eventually_ge_at_top)
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1254
    with Q show "eventually (\<lambda>x. z \<le> f x) at_top"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1255
      by eventually_elim (metis mono bij `P z`)
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1256
  qed
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1257
qed
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1258
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1259
lemma filterlim_at_top_gt:
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1260
  fixes f :: "'a \<Rightarrow> ('b::dense_linorder)" and c :: "'b"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1261
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1262
  by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge)
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1263
50322
b06b95a5fda2 rename filter_lim to filterlim to be consistent with filtermap
hoelzl
parents: 50247
diff changeset
  1264
lemma filterlim_at_bot: 
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1265
  fixes f :: "'a \<Rightarrow> ('b::linorder)"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1266
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1267
  by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_elim1)
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1268
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1269
lemma filterlim_at_bot_le:
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1270
  fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1271
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1272
  unfolding filterlim_at_bot
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1273
proof safe
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1274
  fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1275
  with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F"
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1276
    by (auto elim!: eventually_elim1)
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1277
qed simp
50247
491c5c81c2e8 introduce filter_lim as a generatlization of tendsto
hoelzl
parents: 49834
diff changeset
  1278
50323
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
  1279
lemma filterlim_at_bot_lt:
3764d4620fb3 add filterlim rules for unary minus and inverse
hoelzl
parents: 50322
diff changeset
  1280
  fixes f :: "'a \<Rightarrow> ('b::dense_linorder)" and c :: "'b"