src/HOL/Filter.thy
author hoelzl
Sun Apr 12 11:33:19 2015 +0200 (2015-04-12)
changeset 60036 218fcc645d22
child 60037 071a99649dde
permissions -rw-r--r--
move filters to their own theory
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(*  Title:      HOL/Filter.thy
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    Author:     Brian Huffman
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    Author:     Johannes Hölzl
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*)
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section {* Filters on predicates *}
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theory Filter
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imports Set_Interval Lifting_Set
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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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subsubsection {* 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_Ball_finite:
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  assumes "finite A" and "\<forall>y\<in>A. eventually (\<lambda>x. P x y) net"
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  shows "eventually (\<lambda>x. \<forall>y\<in>A. P x y) net"
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using assms by (induct set: finite, simp, simp add: eventually_conj)
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lemma eventually_all_finite:
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  fixes P :: "'a \<Rightarrow> 'b::finite \<Rightarrow> bool"
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  assumes "\<And>y. eventually (\<lambda>x. P x y) net"
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  shows "eventually (\<lambda>x. \<forall>y. P x y) net"
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using eventually_Ball_finite [of UNIV P] assms by simp
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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 not_eventually_impI: "eventually P F \<Longrightarrow> \<not> eventually Q F \<Longrightarrow> \<not> eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
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  by (auto intro: eventually_mp)
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lemma not_eventuallyD: "\<not> eventually P F \<Longrightarrow> \<exists>x. \<not> P x"
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  by (metis always_eventually)
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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 facts = SUBGOAL_CASES (fn (goal, i) =>
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    let
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      val mp_thms = facts RL @{thms 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) facts
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      val cases_prop = Thm.prop_of (raw_elim_thm RS Goal.init (Thm.cterm_of ctxt goal))
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      val cases = Rule_Cases.make_common ctxt cases_prop [(("elim", []), [])]
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    in
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      CASES cases (rtac raw_elim_thm i)
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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 (HEADGOAL o eventually_elim_tac ctxt))
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*} "elimination of eventually quantifiers"
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subsubsection {* 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 "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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  { 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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  { 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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  { 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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  { show "Inf {} = (top::'a filter)"
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    by (auto simp: top_filter_def Inf_filter_def Sup_filter_def)
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      (metis (full_types) top_filter_def always_eventually eventually_top) }
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  { show "Sup {} = (bot::'a filter)"
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    by (auto simp: bot_filter_def Sup_filter_def) }
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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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lemma eventually_const: "\<not> trivial_limit net \<Longrightarrow> eventually (\<lambda>x. P) net \<longleftrightarrow> P"
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  by (cases P) (simp_all add: eventually_False)
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lemma eventually_Inf: "eventually P (Inf B) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X))"
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proof -
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  let ?F = "\<lambda>P. \<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X)"
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  { fix P have "eventually P (Abs_filter ?F) \<longleftrightarrow> ?F P"
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    proof (rule eventually_Abs_filter is_filter.intro)+
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      show "?F (\<lambda>x. True)"
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        by (rule exI[of _ "{}"]) (simp add: le_fun_def)
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    next
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      fix P Q
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      assume "?F P" then guess X ..
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      moreover
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      assume "?F Q" then guess Y ..
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      ultimately show "?F (\<lambda>x. P x \<and> Q x)"
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        by (intro exI[of _ "X \<union> Y"])
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           (auto simp: Inf_union_distrib eventually_inf)
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    next
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      fix P Q
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      assume "?F P" then guess X ..
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      moreover assume "\<forall>x. P x \<longrightarrow> Q x"
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      ultimately show "?F Q"
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        by (intro exI[of _ X]) (auto elim: eventually_elim1)
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    qed }
hoelzl@60036
   300
  note eventually_F = this
hoelzl@60036
   301
hoelzl@60036
   302
  have "Inf B = Abs_filter ?F"
hoelzl@60036
   303
  proof (intro antisym Inf_greatest)
hoelzl@60036
   304
    show "Inf B \<le> Abs_filter ?F"
hoelzl@60036
   305
      by (auto simp: le_filter_def eventually_F dest: Inf_superset_mono)
hoelzl@60036
   306
  next
hoelzl@60036
   307
    fix F assume "F \<in> B" then show "Abs_filter ?F \<le> F"
hoelzl@60036
   308
      by (auto simp add: le_filter_def eventually_F intro!: exI[of _ "{F}"])
hoelzl@60036
   309
  qed
hoelzl@60036
   310
  then show ?thesis
hoelzl@60036
   311
    by (simp add: eventually_F)
hoelzl@60036
   312
qed
hoelzl@60036
   313
hoelzl@60036
   314
lemma eventually_INF: "eventually P (INF b:B. F b) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (INF b:X. F b))"
hoelzl@60036
   315
  unfolding INF_def[of B] eventually_Inf[of P "F`B"]
hoelzl@60036
   316
  by (metis Inf_image_eq finite_imageI image_mono finite_subset_image)
hoelzl@60036
   317
hoelzl@60036
   318
lemma Inf_filter_not_bot:
hoelzl@60036
   319
  fixes B :: "'a filter set"
hoelzl@60036
   320
  shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> Inf X \<noteq> bot) \<Longrightarrow> Inf B \<noteq> bot"
hoelzl@60036
   321
  unfolding trivial_limit_def eventually_Inf[of _ B]
hoelzl@60036
   322
    bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
hoelzl@60036
   323
hoelzl@60036
   324
lemma INF_filter_not_bot:
hoelzl@60036
   325
  fixes F :: "'i \<Rightarrow> 'a filter"
hoelzl@60036
   326
  shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> (INF b:X. F b) \<noteq> bot) \<Longrightarrow> (INF b:B. F b) \<noteq> bot"
hoelzl@60036
   327
  unfolding trivial_limit_def eventually_INF[of _ B]
hoelzl@60036
   328
    bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
hoelzl@60036
   329
hoelzl@60036
   330
lemma eventually_Inf_base:
hoelzl@60036
   331
  assumes "B \<noteq> {}" and base: "\<And>F G. F \<in> B \<Longrightarrow> G \<in> B \<Longrightarrow> \<exists>x\<in>B. x \<le> inf F G"
hoelzl@60036
   332
  shows "eventually P (Inf B) \<longleftrightarrow> (\<exists>b\<in>B. eventually P b)"
hoelzl@60036
   333
proof (subst eventually_Inf, safe)
hoelzl@60036
   334
  fix X assume "finite X" "X \<subseteq> B"
hoelzl@60036
   335
  then have "\<exists>b\<in>B. \<forall>x\<in>X. b \<le> x"
hoelzl@60036
   336
  proof induct
hoelzl@60036
   337
    case empty then show ?case
hoelzl@60036
   338
      using `B \<noteq> {}` by auto
hoelzl@60036
   339
  next
hoelzl@60036
   340
    case (insert x X)
hoelzl@60036
   341
    then obtain b where "b \<in> B" "\<And>x. x \<in> X \<Longrightarrow> b \<le> x"
hoelzl@60036
   342
      by auto
hoelzl@60036
   343
    with `insert x X \<subseteq> B` base[of b x] show ?case
hoelzl@60036
   344
      by (auto intro: order_trans)
hoelzl@60036
   345
  qed
hoelzl@60036
   346
  then obtain b where "b \<in> B" "b \<le> Inf X"
hoelzl@60036
   347
    by (auto simp: le_Inf_iff)
hoelzl@60036
   348
  then show "eventually P (Inf X) \<Longrightarrow> Bex B (eventually P)"
hoelzl@60036
   349
    by (intro bexI[of _ b]) (auto simp: le_filter_def)
hoelzl@60036
   350
qed (auto intro!: exI[of _ "{x}" for x])
hoelzl@60036
   351
hoelzl@60036
   352
lemma eventually_INF_base:
hoelzl@60036
   353
  "B \<noteq> {} \<Longrightarrow> (\<And>a b. a \<in> B \<Longrightarrow> b \<in> B \<Longrightarrow> \<exists>x\<in>B. F x \<le> inf (F a) (F b)) \<Longrightarrow>
hoelzl@60036
   354
    eventually P (INF b:B. F b) \<longleftrightarrow> (\<exists>b\<in>B. eventually P (F b))"
hoelzl@60036
   355
  unfolding INF_def by (subst eventually_Inf_base) auto
hoelzl@60036
   356
hoelzl@60036
   357
hoelzl@60036
   358
subsubsection {* Map function for filters *}
hoelzl@60036
   359
hoelzl@60036
   360
definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
hoelzl@60036
   361
  where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
hoelzl@60036
   362
hoelzl@60036
   363
lemma eventually_filtermap:
hoelzl@60036
   364
  "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
hoelzl@60036
   365
  unfolding filtermap_def
hoelzl@60036
   366
  apply (rule eventually_Abs_filter)
hoelzl@60036
   367
  apply (rule is_filter.intro)
hoelzl@60036
   368
  apply (auto elim!: eventually_rev_mp)
hoelzl@60036
   369
  done
hoelzl@60036
   370
hoelzl@60036
   371
lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
hoelzl@60036
   372
  by (simp add: filter_eq_iff eventually_filtermap)
hoelzl@60036
   373
hoelzl@60036
   374
lemma filtermap_filtermap:
hoelzl@60036
   375
  "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
hoelzl@60036
   376
  by (simp add: filter_eq_iff eventually_filtermap)
hoelzl@60036
   377
hoelzl@60036
   378
lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"
hoelzl@60036
   379
  unfolding le_filter_def eventually_filtermap by simp
hoelzl@60036
   380
hoelzl@60036
   381
lemma filtermap_bot [simp]: "filtermap f bot = bot"
hoelzl@60036
   382
  by (simp add: filter_eq_iff eventually_filtermap)
hoelzl@60036
   383
hoelzl@60036
   384
lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)"
hoelzl@60036
   385
  by (auto simp: filter_eq_iff eventually_filtermap eventually_sup)
hoelzl@60036
   386
hoelzl@60036
   387
lemma filtermap_inf: "filtermap f (inf F1 F2) \<le> inf (filtermap f F1) (filtermap f F2)"
hoelzl@60036
   388
  by (auto simp: le_filter_def eventually_filtermap eventually_inf)
hoelzl@60036
   389
hoelzl@60036
   390
lemma filtermap_INF: "filtermap f (INF b:B. F b) \<le> (INF b:B. filtermap f (F b))"
hoelzl@60036
   391
proof -
hoelzl@60036
   392
  { fix X :: "'c set" assume "finite X"
hoelzl@60036
   393
    then have "filtermap f (INFIMUM X F) \<le> (INF b:X. filtermap f (F b))"
hoelzl@60036
   394
    proof induct
hoelzl@60036
   395
      case (insert x X)
hoelzl@60036
   396
      have "filtermap f (INF a:insert x X. F a) \<le> inf (filtermap f (F x)) (filtermap f (INF a:X. F a))"
hoelzl@60036
   397
        by (rule order_trans[OF _ filtermap_inf]) simp
hoelzl@60036
   398
      also have "\<dots> \<le> inf (filtermap f (F x)) (INF a:X. filtermap f (F a))"
hoelzl@60036
   399
        by (intro inf_mono insert order_refl)
hoelzl@60036
   400
      finally show ?case
hoelzl@60036
   401
        by simp
hoelzl@60036
   402
    qed simp }
hoelzl@60036
   403
  then show ?thesis
hoelzl@60036
   404
    unfolding le_filter_def eventually_filtermap
hoelzl@60036
   405
    by (subst (1 2) eventually_INF) auto
hoelzl@60036
   406
qed
hoelzl@60036
   407
subsubsection {* Standard filters *}
hoelzl@60036
   408
hoelzl@60036
   409
definition principal :: "'a set \<Rightarrow> 'a filter" where
hoelzl@60036
   410
  "principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)"
hoelzl@60036
   411
hoelzl@60036
   412
lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)"
hoelzl@60036
   413
  unfolding principal_def
hoelzl@60036
   414
  by (rule eventually_Abs_filter, rule is_filter.intro) auto
hoelzl@60036
   415
hoelzl@60036
   416
lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F"
hoelzl@60036
   417
  unfolding eventually_inf eventually_principal by (auto elim: eventually_elim1)
hoelzl@60036
   418
hoelzl@60036
   419
lemma principal_UNIV[simp]: "principal UNIV = top"
hoelzl@60036
   420
  by (auto simp: filter_eq_iff eventually_principal)
hoelzl@60036
   421
hoelzl@60036
   422
lemma principal_empty[simp]: "principal {} = bot"
hoelzl@60036
   423
  by (auto simp: filter_eq_iff eventually_principal)
hoelzl@60036
   424
hoelzl@60036
   425
lemma principal_eq_bot_iff: "principal X = bot \<longleftrightarrow> X = {}"
hoelzl@60036
   426
  by (auto simp add: filter_eq_iff eventually_principal)
hoelzl@60036
   427
hoelzl@60036
   428
lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B"
hoelzl@60036
   429
  by (auto simp: le_filter_def eventually_principal)
hoelzl@60036
   430
hoelzl@60036
   431
lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F"
hoelzl@60036
   432
  unfolding le_filter_def eventually_principal
hoelzl@60036
   433
  apply safe
hoelzl@60036
   434
  apply (erule_tac x="\<lambda>x. x \<in> A" in allE)
hoelzl@60036
   435
  apply (auto elim: eventually_elim1)
hoelzl@60036
   436
  done
hoelzl@60036
   437
hoelzl@60036
   438
lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B"
hoelzl@60036
   439
  unfolding eq_iff by simp
hoelzl@60036
   440
hoelzl@60036
   441
lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)"
hoelzl@60036
   442
  unfolding filter_eq_iff eventually_sup eventually_principal by auto
hoelzl@60036
   443
hoelzl@60036
   444
lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)"
hoelzl@60036
   445
  unfolding filter_eq_iff eventually_inf eventually_principal
hoelzl@60036
   446
  by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"])
hoelzl@60036
   447
hoelzl@60036
   448
lemma SUP_principal[simp]: "(SUP i : I. principal (A i)) = principal (\<Union>i\<in>I. A i)"
hoelzl@60036
   449
  unfolding filter_eq_iff eventually_Sup SUP_def by (auto simp: eventually_principal)
hoelzl@60036
   450
hoelzl@60036
   451
lemma INF_principal_finite: "finite X \<Longrightarrow> (INF x:X. principal (f x)) = principal (\<Inter>x\<in>X. f x)"
hoelzl@60036
   452
  by (induct X rule: finite_induct) auto
hoelzl@60036
   453
hoelzl@60036
   454
lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)"
hoelzl@60036
   455
  unfolding filter_eq_iff eventually_filtermap eventually_principal by simp
hoelzl@60036
   456
hoelzl@60036
   457
subsubsection {* Order filters *}
hoelzl@60036
   458
hoelzl@60036
   459
definition at_top :: "('a::order) filter"
hoelzl@60036
   460
  where "at_top = (INF k. principal {k ..})"
hoelzl@60036
   461
hoelzl@60036
   462
lemma at_top_sub: "at_top = (INF k:{c::'a::linorder..}. principal {k ..})"
hoelzl@60036
   463
  by (auto intro!: INF_eq max.cobounded1 max.cobounded2 simp: at_top_def)
hoelzl@60036
   464
hoelzl@60036
   465
lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)"
hoelzl@60036
   466
  unfolding at_top_def
hoelzl@60036
   467
  by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
hoelzl@60036
   468
hoelzl@60036
   469
lemma eventually_ge_at_top:
hoelzl@60036
   470
  "eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top"
hoelzl@60036
   471
  unfolding eventually_at_top_linorder by auto
hoelzl@60036
   472
hoelzl@60036
   473
lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::{no_top, linorder}. \<forall>n>N. P n)"
hoelzl@60036
   474
proof -
hoelzl@60036
   475
  have "eventually P (INF k. principal {k <..}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n>N. P n)"
hoelzl@60036
   476
    by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
hoelzl@60036
   477
  also have "(INF k. principal {k::'a <..}) = at_top"
hoelzl@60036
   478
    unfolding at_top_def 
hoelzl@60036
   479
    by (intro INF_eq) (auto intro: less_imp_le simp: Ici_subset_Ioi_iff gt_ex)
hoelzl@60036
   480
  finally show ?thesis .
hoelzl@60036
   481
qed
hoelzl@60036
   482
hoelzl@60036
   483
lemma eventually_gt_at_top:
hoelzl@60036
   484
  "eventually (\<lambda>x. (c::_::unbounded_dense_linorder) < x) at_top"
hoelzl@60036
   485
  unfolding eventually_at_top_dense by auto
hoelzl@60036
   486
hoelzl@60036
   487
definition at_bot :: "('a::order) filter"
hoelzl@60036
   488
  where "at_bot = (INF k. principal {.. k})"
hoelzl@60036
   489
hoelzl@60036
   490
lemma at_bot_sub: "at_bot = (INF k:{.. c::'a::linorder}. principal {.. k})"
hoelzl@60036
   491
  by (auto intro!: INF_eq min.cobounded1 min.cobounded2 simp: at_bot_def)
hoelzl@60036
   492
hoelzl@60036
   493
lemma eventually_at_bot_linorder:
hoelzl@60036
   494
  fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"
hoelzl@60036
   495
  unfolding at_bot_def
hoelzl@60036
   496
  by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
hoelzl@60036
   497
hoelzl@60036
   498
lemma eventually_le_at_bot:
hoelzl@60036
   499
  "eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot"
hoelzl@60036
   500
  unfolding eventually_at_bot_linorder by auto
hoelzl@60036
   501
hoelzl@60036
   502
lemma eventually_at_bot_dense: "eventually P at_bot \<longleftrightarrow> (\<exists>N::'a::{no_bot, linorder}. \<forall>n<N. P n)"
hoelzl@60036
   503
proof -
hoelzl@60036
   504
  have "eventually P (INF k. principal {..< k}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n<N. P n)"
hoelzl@60036
   505
    by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
hoelzl@60036
   506
  also have "(INF k. principal {..< k::'a}) = at_bot"
hoelzl@60036
   507
    unfolding at_bot_def 
hoelzl@60036
   508
    by (intro INF_eq) (auto intro: less_imp_le simp: Iic_subset_Iio_iff lt_ex)
hoelzl@60036
   509
  finally show ?thesis .
hoelzl@60036
   510
qed
hoelzl@60036
   511
hoelzl@60036
   512
lemma eventually_gt_at_bot:
hoelzl@60036
   513
  "eventually (\<lambda>x. x < (c::_::unbounded_dense_linorder)) at_bot"
hoelzl@60036
   514
  unfolding eventually_at_bot_dense by auto
hoelzl@60036
   515
hoelzl@60036
   516
lemma trivial_limit_at_bot_linorder: "\<not> trivial_limit (at_bot ::('a::linorder) filter)"
hoelzl@60036
   517
  unfolding trivial_limit_def
hoelzl@60036
   518
  by (metis eventually_at_bot_linorder order_refl)
hoelzl@60036
   519
hoelzl@60036
   520
lemma trivial_limit_at_top_linorder: "\<not> trivial_limit (at_top ::('a::linorder) filter)"
hoelzl@60036
   521
  unfolding trivial_limit_def
hoelzl@60036
   522
  by (metis eventually_at_top_linorder order_refl)
hoelzl@60036
   523
hoelzl@60036
   524
subsection {* Sequentially *}
hoelzl@60036
   525
hoelzl@60036
   526
abbreviation sequentially :: "nat filter"
hoelzl@60036
   527
  where "sequentially \<equiv> at_top"
hoelzl@60036
   528
hoelzl@60036
   529
lemma eventually_sequentially:
hoelzl@60036
   530
  "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
hoelzl@60036
   531
  by (rule eventually_at_top_linorder)
hoelzl@60036
   532
hoelzl@60036
   533
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
hoelzl@60036
   534
  unfolding filter_eq_iff eventually_sequentially by auto
hoelzl@60036
   535
hoelzl@60036
   536
lemmas trivial_limit_sequentially = sequentially_bot
hoelzl@60036
   537
hoelzl@60036
   538
lemma eventually_False_sequentially [simp]:
hoelzl@60036
   539
  "\<not> eventually (\<lambda>n. False) sequentially"
hoelzl@60036
   540
  by (simp add: eventually_False)
hoelzl@60036
   541
hoelzl@60036
   542
lemma le_sequentially:
hoelzl@60036
   543
  "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
hoelzl@60036
   544
  by (simp add: at_top_def le_INF_iff le_principal)
hoelzl@60036
   545
hoelzl@60036
   546
lemma eventually_sequentiallyI:
hoelzl@60036
   547
  assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
hoelzl@60036
   548
  shows "eventually P sequentially"
hoelzl@60036
   549
using assms by (auto simp: eventually_sequentially)
hoelzl@60036
   550
hoelzl@60036
   551
lemma eventually_sequentially_seg:
hoelzl@60036
   552
  "eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially"
hoelzl@60036
   553
  unfolding eventually_sequentially
hoelzl@60036
   554
  apply safe
hoelzl@60036
   555
   apply (rule_tac x="N + k" in exI)
hoelzl@60036
   556
   apply rule
hoelzl@60036
   557
   apply (erule_tac x="n - k" in allE)
hoelzl@60036
   558
   apply auto []
hoelzl@60036
   559
  apply (rule_tac x=N in exI)
hoelzl@60036
   560
  apply auto []
hoelzl@60036
   561
  done
hoelzl@60036
   562
hoelzl@60036
   563
hoelzl@60036
   564
subsection {* Limits *}
hoelzl@60036
   565
hoelzl@60036
   566
definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where
hoelzl@60036
   567
  "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
hoelzl@60036
   568
hoelzl@60036
   569
syntax
hoelzl@60036
   570
  "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)
hoelzl@60036
   571
hoelzl@60036
   572
translations
hoelzl@60036
   573
  "LIM x F1. f :> F2"   == "CONST filterlim (%x. f) F2 F1"
hoelzl@60036
   574
hoelzl@60036
   575
lemma filterlim_iff:
hoelzl@60036
   576
  "(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)"
hoelzl@60036
   577
  unfolding filterlim_def le_filter_def eventually_filtermap ..
hoelzl@60036
   578
hoelzl@60036
   579
lemma filterlim_compose:
hoelzl@60036
   580
  "filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1"
hoelzl@60036
   581
  unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)
hoelzl@60036
   582
hoelzl@60036
   583
lemma filterlim_mono:
hoelzl@60036
   584
  "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'"
hoelzl@60036
   585
  unfolding filterlim_def by (metis filtermap_mono order_trans)
hoelzl@60036
   586
hoelzl@60036
   587
lemma filterlim_ident: "LIM x F. x :> F"
hoelzl@60036
   588
  by (simp add: filterlim_def filtermap_ident)
hoelzl@60036
   589
hoelzl@60036
   590
lemma filterlim_cong:
hoelzl@60036
   591
  "F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'"
hoelzl@60036
   592
  by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2)
hoelzl@60036
   593
hoelzl@60036
   594
lemma filterlim_mono_eventually:
hoelzl@60036
   595
  assumes "filterlim f F G" and ord: "F \<le> F'" "G' \<le> G"
hoelzl@60036
   596
  assumes eq: "eventually (\<lambda>x. f x = f' x) G'"
hoelzl@60036
   597
  shows "filterlim f' F' G'"
hoelzl@60036
   598
  apply (rule filterlim_cong[OF refl refl eq, THEN iffD1])
hoelzl@60036
   599
  apply (rule filterlim_mono[OF _ ord])
hoelzl@60036
   600
  apply fact
hoelzl@60036
   601
  done
hoelzl@60036
   602
hoelzl@60036
   603
lemma filtermap_mono_strong: "inj f \<Longrightarrow> filtermap f F \<le> filtermap f G \<longleftrightarrow> F \<le> G"
hoelzl@60036
   604
  apply (auto intro!: filtermap_mono) []
hoelzl@60036
   605
  apply (auto simp: le_filter_def eventually_filtermap)
hoelzl@60036
   606
  apply (erule_tac x="\<lambda>x. P (inv f x)" in allE)
hoelzl@60036
   607
  apply auto
hoelzl@60036
   608
  done
hoelzl@60036
   609
hoelzl@60036
   610
lemma filtermap_eq_strong: "inj f \<Longrightarrow> filtermap f F = filtermap f G \<longleftrightarrow> F = G"
hoelzl@60036
   611
  by (simp add: filtermap_mono_strong eq_iff)
hoelzl@60036
   612
hoelzl@60036
   613
lemma filterlim_principal:
hoelzl@60036
   614
  "(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)"
hoelzl@60036
   615
  unfolding filterlim_def eventually_filtermap le_principal ..
hoelzl@60036
   616
hoelzl@60036
   617
lemma filterlim_inf:
hoelzl@60036
   618
  "(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))"
hoelzl@60036
   619
  unfolding filterlim_def by simp
hoelzl@60036
   620
hoelzl@60036
   621
lemma filterlim_INF:
hoelzl@60036
   622
  "(LIM x F. f x :> (INF b:B. G b)) \<longleftrightarrow> (\<forall>b\<in>B. LIM x F. f x :> G b)"
hoelzl@60036
   623
  unfolding filterlim_def le_INF_iff ..
hoelzl@60036
   624
hoelzl@60036
   625
lemma filterlim_INF_INF:
hoelzl@60036
   626
  "(\<And>m. m \<in> J \<Longrightarrow> \<exists>i\<in>I. filtermap f (F i) \<le> G m) \<Longrightarrow> LIM x (INF i:I. F i). f x :> (INF j:J. G j)"
hoelzl@60036
   627
  unfolding filterlim_def by (rule order_trans[OF filtermap_INF INF_mono])
hoelzl@60036
   628
hoelzl@60036
   629
lemma filterlim_base:
hoelzl@60036
   630
  "(\<And>m x. m \<in> J \<Longrightarrow> i m \<in> I) \<Longrightarrow> (\<And>m x. m \<in> J \<Longrightarrow> x \<in> F (i m) \<Longrightarrow> f x \<in> G m) \<Longrightarrow> 
hoelzl@60036
   631
    LIM x (INF i:I. principal (F i)). f x :> (INF j:J. principal (G j))"
hoelzl@60036
   632
  by (force intro!: filterlim_INF_INF simp: image_subset_iff)
hoelzl@60036
   633
hoelzl@60036
   634
lemma filterlim_base_iff: 
hoelzl@60036
   635
  assumes "I \<noteq> {}" and chain: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> F i \<subseteq> F j \<or> F j \<subseteq> F i"
hoelzl@60036
   636
  shows "(LIM x (INF i:I. principal (F i)). f x :> INF j:J. principal (G j)) \<longleftrightarrow>
hoelzl@60036
   637
    (\<forall>j\<in>J. \<exists>i\<in>I. \<forall>x\<in>F i. f x \<in> G j)"
hoelzl@60036
   638
  unfolding filterlim_INF filterlim_principal
hoelzl@60036
   639
proof (subst eventually_INF_base)
hoelzl@60036
   640
  fix i j assume "i \<in> I" "j \<in> I"
hoelzl@60036
   641
  with chain[OF this] show "\<exists>x\<in>I. principal (F x) \<le> inf (principal (F i)) (principal (F j))"
hoelzl@60036
   642
    by auto
hoelzl@60036
   643
qed (auto simp: eventually_principal `I \<noteq> {}`)
hoelzl@60036
   644
hoelzl@60036
   645
lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2"
hoelzl@60036
   646
  unfolding filterlim_def filtermap_filtermap ..
hoelzl@60036
   647
hoelzl@60036
   648
lemma filterlim_sup:
hoelzl@60036
   649
  "filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)"
hoelzl@60036
   650
  unfolding filterlim_def filtermap_sup by auto
hoelzl@60036
   651
hoelzl@60036
   652
lemma eventually_sequentially_Suc: "eventually (\<lambda>i. P (Suc i)) sequentially \<longleftrightarrow> eventually P sequentially"
hoelzl@60036
   653
  unfolding eventually_sequentially by (metis Suc_le_D Suc_le_mono le_Suc_eq)
hoelzl@60036
   654
hoelzl@60036
   655
lemma filterlim_sequentially_Suc:
hoelzl@60036
   656
  "(LIM x sequentially. f (Suc x) :> F) \<longleftrightarrow> (LIM x sequentially. f x :> F)"
hoelzl@60036
   657
  unfolding filterlim_iff by (subst eventually_sequentially_Suc) simp
hoelzl@60036
   658
hoelzl@60036
   659
lemma filterlim_Suc: "filterlim Suc sequentially sequentially"
hoelzl@60036
   660
  by (simp add: filterlim_iff eventually_sequentially) (metis le_Suc_eq)
hoelzl@60036
   661
hoelzl@60036
   662
hoelzl@60036
   663
subsection {* Limits to @{const at_top} and @{const at_bot} *}
hoelzl@60036
   664
hoelzl@60036
   665
lemma filterlim_at_top:
hoelzl@60036
   666
  fixes f :: "'a \<Rightarrow> ('b::linorder)"
hoelzl@60036
   667
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)"
hoelzl@60036
   668
  by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_elim1)
hoelzl@60036
   669
hoelzl@60036
   670
lemma filterlim_at_top_mono:
hoelzl@60036
   671
  "LIM x F. f x :> at_top \<Longrightarrow> eventually (\<lambda>x. f x \<le> (g x::'a::linorder)) F \<Longrightarrow>
hoelzl@60036
   672
    LIM x F. g x :> at_top"
hoelzl@60036
   673
  by (auto simp: filterlim_at_top elim: eventually_elim2 intro: order_trans)
hoelzl@60036
   674
hoelzl@60036
   675
lemma filterlim_at_top_dense:
hoelzl@60036
   676
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)"
hoelzl@60036
   677
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)"
hoelzl@60036
   678
  by (metis eventually_elim1[of _ F] eventually_gt_at_top order_less_imp_le
hoelzl@60036
   679
            filterlim_at_top[of f F] filterlim_iff[of f at_top F])
hoelzl@60036
   680
hoelzl@60036
   681
lemma filterlim_at_top_ge:
hoelzl@60036
   682
  fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
hoelzl@60036
   683
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)"
hoelzl@60036
   684
  unfolding at_top_sub[of c] filterlim_INF by (auto simp add: filterlim_principal)
hoelzl@60036
   685
hoelzl@60036
   686
lemma filterlim_at_top_at_top:
hoelzl@60036
   687
  fixes f :: "'a::linorder \<Rightarrow> 'b::linorder"
hoelzl@60036
   688
  assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
hoelzl@60036
   689
  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
hoelzl@60036
   690
  assumes Q: "eventually Q at_top"
hoelzl@60036
   691
  assumes P: "eventually P at_top"
hoelzl@60036
   692
  shows "filterlim f at_top at_top"
hoelzl@60036
   693
proof -
hoelzl@60036
   694
  from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
hoelzl@60036
   695
    unfolding eventually_at_top_linorder by auto
hoelzl@60036
   696
  show ?thesis
hoelzl@60036
   697
  proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
hoelzl@60036
   698
    fix z assume "x \<le> z"
hoelzl@60036
   699
    with x have "P z" by auto
hoelzl@60036
   700
    have "eventually (\<lambda>x. g z \<le> x) at_top"
hoelzl@60036
   701
      by (rule eventually_ge_at_top)
hoelzl@60036
   702
    with Q show "eventually (\<lambda>x. z \<le> f x) at_top"
hoelzl@60036
   703
      by eventually_elim (metis mono bij `P z`)
hoelzl@60036
   704
  qed
hoelzl@60036
   705
qed
hoelzl@60036
   706
hoelzl@60036
   707
lemma filterlim_at_top_gt:
hoelzl@60036
   708
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
hoelzl@60036
   709
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)"
hoelzl@60036
   710
  by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge)
hoelzl@60036
   711
hoelzl@60036
   712
lemma filterlim_at_bot: 
hoelzl@60036
   713
  fixes f :: "'a \<Rightarrow> ('b::linorder)"
hoelzl@60036
   714
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)"
hoelzl@60036
   715
  by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_elim1)
hoelzl@60036
   716
hoelzl@60036
   717
lemma filterlim_at_bot_dense:
hoelzl@60036
   718
  fixes f :: "'a \<Rightarrow> ('b::{dense_linorder, no_bot})"
hoelzl@60036
   719
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x < Z) F)"
hoelzl@60036
   720
proof (auto simp add: filterlim_at_bot[of f F])
hoelzl@60036
   721
  fix Z :: 'b
hoelzl@60036
   722
  from lt_ex [of Z] obtain Z' where 1: "Z' < Z" ..
hoelzl@60036
   723
  assume "\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F"
hoelzl@60036
   724
  hence "eventually (\<lambda>x. f x \<le> Z') F" by auto
hoelzl@60036
   725
  thus "eventually (\<lambda>x. f x < Z) F"
hoelzl@60036
   726
    apply (rule eventually_mono[rotated])
hoelzl@60036
   727
    using 1 by auto
hoelzl@60036
   728
  next 
hoelzl@60036
   729
    fix Z :: 'b 
hoelzl@60036
   730
    show "\<forall>Z. eventually (\<lambda>x. f x < Z) F \<Longrightarrow> eventually (\<lambda>x. f x \<le> Z) F"
hoelzl@60036
   731
      by (drule spec [of _ Z], erule eventually_mono[rotated], auto simp add: less_imp_le)
hoelzl@60036
   732
qed
hoelzl@60036
   733
hoelzl@60036
   734
lemma filterlim_at_bot_le:
hoelzl@60036
   735
  fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
hoelzl@60036
   736
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)"
hoelzl@60036
   737
  unfolding filterlim_at_bot
hoelzl@60036
   738
proof safe
hoelzl@60036
   739
  fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F"
hoelzl@60036
   740
  with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F"
hoelzl@60036
   741
    by (auto elim!: eventually_elim1)
hoelzl@60036
   742
qed simp
hoelzl@60036
   743
hoelzl@60036
   744
lemma filterlim_at_bot_lt:
hoelzl@60036
   745
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
hoelzl@60036
   746
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)"
hoelzl@60036
   747
  by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans)
hoelzl@60036
   748
hoelzl@60036
   749
hoelzl@60036
   750
subsection {* Setup @{typ "'a filter"} for lifting and transfer *}
hoelzl@60036
   751
hoelzl@60036
   752
context begin interpretation lifting_syntax .
hoelzl@60036
   753
hoelzl@60036
   754
definition rel_filter :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> 'b filter \<Rightarrow> bool"
hoelzl@60036
   755
where "rel_filter R F G = ((R ===> op =) ===> op =) (Rep_filter F) (Rep_filter G)"
hoelzl@60036
   756
hoelzl@60036
   757
lemma rel_filter_eventually:
hoelzl@60036
   758
  "rel_filter R F G \<longleftrightarrow> 
hoelzl@60036
   759
  ((R ===> op =) ===> op =) (\<lambda>P. eventually P F) (\<lambda>P. eventually P G)"
hoelzl@60036
   760
by(simp add: rel_filter_def eventually_def)
hoelzl@60036
   761
hoelzl@60036
   762
lemma filtermap_id [simp, id_simps]: "filtermap id = id"
hoelzl@60036
   763
by(simp add: fun_eq_iff id_def filtermap_ident)
hoelzl@60036
   764
hoelzl@60036
   765
lemma filtermap_id' [simp]: "filtermap (\<lambda>x. x) = (\<lambda>F. F)"
hoelzl@60036
   766
using filtermap_id unfolding id_def .
hoelzl@60036
   767
hoelzl@60036
   768
lemma Quotient_filter [quot_map]:
hoelzl@60036
   769
  assumes Q: "Quotient R Abs Rep T"
hoelzl@60036
   770
  shows "Quotient (rel_filter R) (filtermap Abs) (filtermap Rep) (rel_filter T)"
hoelzl@60036
   771
unfolding Quotient_alt_def
hoelzl@60036
   772
proof(intro conjI strip)
hoelzl@60036
   773
  from Q have *: "\<And>x y. T x y \<Longrightarrow> Abs x = y"
hoelzl@60036
   774
    unfolding Quotient_alt_def by blast
hoelzl@60036
   775
hoelzl@60036
   776
  fix F G
hoelzl@60036
   777
  assume "rel_filter T F G"
hoelzl@60036
   778
  thus "filtermap Abs F = G" unfolding filter_eq_iff
hoelzl@60036
   779
    by(auto simp add: eventually_filtermap rel_filter_eventually * rel_funI del: iffI elim!: rel_funD)
hoelzl@60036
   780
next
hoelzl@60036
   781
  from Q have *: "\<And>x. T (Rep x) x" unfolding Quotient_alt_def by blast
hoelzl@60036
   782
hoelzl@60036
   783
  fix F
hoelzl@60036
   784
  show "rel_filter T (filtermap Rep F) F" 
hoelzl@60036
   785
    by(auto elim: rel_funD intro: * intro!: ext arg_cong[where f="\<lambda>P. eventually P F"] rel_funI
hoelzl@60036
   786
            del: iffI simp add: eventually_filtermap rel_filter_eventually)
hoelzl@60036
   787
qed(auto simp add: map_fun_def o_def eventually_filtermap filter_eq_iff fun_eq_iff rel_filter_eventually
hoelzl@60036
   788
         fun_quotient[OF fun_quotient[OF Q identity_quotient] identity_quotient, unfolded Quotient_alt_def])
hoelzl@60036
   789
hoelzl@60036
   790
lemma eventually_parametric [transfer_rule]:
hoelzl@60036
   791
  "((A ===> op =) ===> rel_filter A ===> op =) eventually eventually"
hoelzl@60036
   792
by(simp add: rel_fun_def rel_filter_eventually)
hoelzl@60036
   793
hoelzl@60036
   794
lemma rel_filter_eq [relator_eq]: "rel_filter op = = op ="
hoelzl@60036
   795
by(auto simp add: rel_filter_eventually rel_fun_eq fun_eq_iff filter_eq_iff)
hoelzl@60036
   796
hoelzl@60036
   797
lemma rel_filter_mono [relator_mono]:
hoelzl@60036
   798
  "A \<le> B \<Longrightarrow> rel_filter A \<le> rel_filter B"
hoelzl@60036
   799
unfolding rel_filter_eventually[abs_def]
hoelzl@60036
   800
by(rule le_funI)+(intro fun_mono fun_mono[THEN le_funD, THEN le_funD] order.refl)
hoelzl@60036
   801
hoelzl@60036
   802
lemma rel_filter_conversep [simp]: "rel_filter A\<inverse>\<inverse> = (rel_filter A)\<inverse>\<inverse>"
hoelzl@60036
   803
by(auto simp add: rel_filter_eventually fun_eq_iff rel_fun_def)
hoelzl@60036
   804
hoelzl@60036
   805
lemma is_filter_parametric_aux:
hoelzl@60036
   806
  assumes "is_filter F"
hoelzl@60036
   807
  assumes [transfer_rule]: "bi_total A" "bi_unique A"
hoelzl@60036
   808
  and [transfer_rule]: "((A ===> op =) ===> op =) F G"
hoelzl@60036
   809
  shows "is_filter G"
hoelzl@60036
   810
proof -
hoelzl@60036
   811
  interpret is_filter F by fact
hoelzl@60036
   812
  show ?thesis
hoelzl@60036
   813
  proof
hoelzl@60036
   814
    have "F (\<lambda>_. True) = G (\<lambda>x. True)" by transfer_prover
hoelzl@60036
   815
    thus "G (\<lambda>x. True)" by(simp add: True)
hoelzl@60036
   816
  next
hoelzl@60036
   817
    fix P' Q'
hoelzl@60036
   818
    assume "G P'" "G Q'"
hoelzl@60036
   819
    moreover
hoelzl@60036
   820
    from bi_total_fun[OF `bi_unique A` bi_total_eq, unfolded bi_total_def]
hoelzl@60036
   821
    obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast
hoelzl@60036
   822
    have "F P = G P'" "F Q = G Q'" by transfer_prover+
hoelzl@60036
   823
    ultimately have "F (\<lambda>x. P x \<and> Q x)" by(simp add: conj)
hoelzl@60036
   824
    moreover have "F (\<lambda>x. P x \<and> Q x) = G (\<lambda>x. P' x \<and> Q' x)" by transfer_prover
hoelzl@60036
   825
    ultimately show "G (\<lambda>x. P' x \<and> Q' x)" by simp
hoelzl@60036
   826
  next
hoelzl@60036
   827
    fix P' Q'
hoelzl@60036
   828
    assume "\<forall>x. P' x \<longrightarrow> Q' x" "G P'"
hoelzl@60036
   829
    moreover
hoelzl@60036
   830
    from bi_total_fun[OF `bi_unique A` bi_total_eq, unfolded bi_total_def]
hoelzl@60036
   831
    obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast
hoelzl@60036
   832
    have "F P = G P'" by transfer_prover
hoelzl@60036
   833
    moreover have "(\<forall>x. P x \<longrightarrow> Q x) \<longleftrightarrow> (\<forall>x. P' x \<longrightarrow> Q' x)" by transfer_prover
hoelzl@60036
   834
    ultimately have "F Q" by(simp add: mono)
hoelzl@60036
   835
    moreover have "F Q = G Q'" by transfer_prover
hoelzl@60036
   836
    ultimately show "G Q'" by simp
hoelzl@60036
   837
  qed
hoelzl@60036
   838
qed
hoelzl@60036
   839
hoelzl@60036
   840
lemma is_filter_parametric [transfer_rule]:
hoelzl@60036
   841
  "\<lbrakk> bi_total A; bi_unique A \<rbrakk>
hoelzl@60036
   842
  \<Longrightarrow> (((A ===> op =) ===> op =) ===> op =) is_filter is_filter"
hoelzl@60036
   843
apply(rule rel_funI)
hoelzl@60036
   844
apply(rule iffI)
hoelzl@60036
   845
 apply(erule (3) is_filter_parametric_aux)
hoelzl@60036
   846
apply(erule is_filter_parametric_aux[where A="conversep A"])
hoelzl@60036
   847
apply(auto simp add: rel_fun_def)
hoelzl@60036
   848
done
hoelzl@60036
   849
hoelzl@60036
   850
lemma left_total_rel_filter [transfer_rule]:
hoelzl@60036
   851
  assumes [transfer_rule]: "bi_total A" "bi_unique A"
hoelzl@60036
   852
  shows "left_total (rel_filter A)"
hoelzl@60036
   853
proof(rule left_totalI)
hoelzl@60036
   854
  fix F :: "'a filter"
hoelzl@60036
   855
  from bi_total_fun[OF bi_unique_fun[OF `bi_total A` bi_unique_eq] bi_total_eq]
hoelzl@60036
   856
  obtain G where [transfer_rule]: "((A ===> op =) ===> op =) (\<lambda>P. eventually P F) G" 
hoelzl@60036
   857
    unfolding  bi_total_def by blast
hoelzl@60036
   858
  moreover have "is_filter (\<lambda>P. eventually P F) \<longleftrightarrow> is_filter G" by transfer_prover
hoelzl@60036
   859
  hence "is_filter G" by(simp add: eventually_def is_filter_Rep_filter)
hoelzl@60036
   860
  ultimately have "rel_filter A F (Abs_filter G)"
hoelzl@60036
   861
    by(simp add: rel_filter_eventually eventually_Abs_filter)
hoelzl@60036
   862
  thus "\<exists>G. rel_filter A F G" ..
hoelzl@60036
   863
qed
hoelzl@60036
   864
hoelzl@60036
   865
lemma right_total_rel_filter [transfer_rule]:
hoelzl@60036
   866
  "\<lbrakk> bi_total A; bi_unique A \<rbrakk> \<Longrightarrow> right_total (rel_filter A)"
hoelzl@60036
   867
using left_total_rel_filter[of "A\<inverse>\<inverse>"] by simp
hoelzl@60036
   868
hoelzl@60036
   869
lemma bi_total_rel_filter [transfer_rule]:
hoelzl@60036
   870
  assumes "bi_total A" "bi_unique A"
hoelzl@60036
   871
  shows "bi_total (rel_filter A)"
hoelzl@60036
   872
unfolding bi_total_alt_def using assms
hoelzl@60036
   873
by(simp add: left_total_rel_filter right_total_rel_filter)
hoelzl@60036
   874
hoelzl@60036
   875
lemma left_unique_rel_filter [transfer_rule]:
hoelzl@60036
   876
  assumes "left_unique A"
hoelzl@60036
   877
  shows "left_unique (rel_filter A)"
hoelzl@60036
   878
proof(rule left_uniqueI)
hoelzl@60036
   879
  fix F F' G
hoelzl@60036
   880
  assume [transfer_rule]: "rel_filter A F G" "rel_filter A F' G"
hoelzl@60036
   881
  show "F = F'"
hoelzl@60036
   882
    unfolding filter_eq_iff
hoelzl@60036
   883
  proof
hoelzl@60036
   884
    fix P :: "'a \<Rightarrow> bool"
hoelzl@60036
   885
    obtain P' where [transfer_rule]: "(A ===> op =) P P'"
hoelzl@60036
   886
      using left_total_fun[OF assms left_total_eq] unfolding left_total_def by blast
hoelzl@60036
   887
    have "eventually P F = eventually P' G" 
hoelzl@60036
   888
      and "eventually P F' = eventually P' G" by transfer_prover+
hoelzl@60036
   889
    thus "eventually P F = eventually P F'" by simp
hoelzl@60036
   890
  qed
hoelzl@60036
   891
qed
hoelzl@60036
   892
hoelzl@60036
   893
lemma right_unique_rel_filter [transfer_rule]:
hoelzl@60036
   894
  "right_unique A \<Longrightarrow> right_unique (rel_filter A)"
hoelzl@60036
   895
using left_unique_rel_filter[of "A\<inverse>\<inverse>"] by simp
hoelzl@60036
   896
hoelzl@60036
   897
lemma bi_unique_rel_filter [transfer_rule]:
hoelzl@60036
   898
  "bi_unique A \<Longrightarrow> bi_unique (rel_filter A)"
hoelzl@60036
   899
by(simp add: bi_unique_alt_def left_unique_rel_filter right_unique_rel_filter)
hoelzl@60036
   900
hoelzl@60036
   901
lemma top_filter_parametric [transfer_rule]:
hoelzl@60036
   902
  "bi_total A \<Longrightarrow> (rel_filter A) top top"
hoelzl@60036
   903
by(simp add: rel_filter_eventually All_transfer)
hoelzl@60036
   904
hoelzl@60036
   905
lemma bot_filter_parametric [transfer_rule]: "(rel_filter A) bot bot"
hoelzl@60036
   906
by(simp add: rel_filter_eventually rel_fun_def)
hoelzl@60036
   907
hoelzl@60036
   908
lemma sup_filter_parametric [transfer_rule]:
hoelzl@60036
   909
  "(rel_filter A ===> rel_filter A ===> rel_filter A) sup sup"
hoelzl@60036
   910
by(fastforce simp add: rel_filter_eventually[abs_def] eventually_sup dest: rel_funD)
hoelzl@60036
   911
hoelzl@60036
   912
lemma Sup_filter_parametric [transfer_rule]:
hoelzl@60036
   913
  "(rel_set (rel_filter A) ===> rel_filter A) Sup Sup"
hoelzl@60036
   914
proof(rule rel_funI)
hoelzl@60036
   915
  fix S T
hoelzl@60036
   916
  assume [transfer_rule]: "rel_set (rel_filter A) S T"
hoelzl@60036
   917
  show "rel_filter A (Sup S) (Sup T)"
hoelzl@60036
   918
    by(simp add: rel_filter_eventually eventually_Sup) transfer_prover
hoelzl@60036
   919
qed
hoelzl@60036
   920
hoelzl@60036
   921
lemma principal_parametric [transfer_rule]:
hoelzl@60036
   922
  "(rel_set A ===> rel_filter A) principal principal"
hoelzl@60036
   923
proof(rule rel_funI)
hoelzl@60036
   924
  fix S S'
hoelzl@60036
   925
  assume [transfer_rule]: "rel_set A S S'"
hoelzl@60036
   926
  show "rel_filter A (principal S) (principal S')"
hoelzl@60036
   927
    by(simp add: rel_filter_eventually eventually_principal) transfer_prover
hoelzl@60036
   928
qed
hoelzl@60036
   929
hoelzl@60036
   930
context
hoelzl@60036
   931
  fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
hoelzl@60036
   932
  assumes [transfer_rule]: "bi_unique A" 
hoelzl@60036
   933
begin
hoelzl@60036
   934
hoelzl@60036
   935
lemma le_filter_parametric [transfer_rule]:
hoelzl@60036
   936
  "(rel_filter A ===> rel_filter A ===> op =) op \<le> op \<le>"
hoelzl@60036
   937
unfolding le_filter_def[abs_def] by transfer_prover
hoelzl@60036
   938
hoelzl@60036
   939
lemma less_filter_parametric [transfer_rule]:
hoelzl@60036
   940
  "(rel_filter A ===> rel_filter A ===> op =) op < op <"
hoelzl@60036
   941
unfolding less_filter_def[abs_def] by transfer_prover
hoelzl@60036
   942
hoelzl@60036
   943
context
hoelzl@60036
   944
  assumes [transfer_rule]: "bi_total A"
hoelzl@60036
   945
begin
hoelzl@60036
   946
hoelzl@60036
   947
lemma Inf_filter_parametric [transfer_rule]:
hoelzl@60036
   948
  "(rel_set (rel_filter A) ===> rel_filter A) Inf Inf"
hoelzl@60036
   949
unfolding Inf_filter_def[abs_def] by transfer_prover
hoelzl@60036
   950
hoelzl@60036
   951
lemma inf_filter_parametric [transfer_rule]:
hoelzl@60036
   952
  "(rel_filter A ===> rel_filter A ===> rel_filter A) inf inf"
hoelzl@60036
   953
proof(intro rel_funI)+
hoelzl@60036
   954
  fix F F' G G'
hoelzl@60036
   955
  assume [transfer_rule]: "rel_filter A F F'" "rel_filter A G G'"
hoelzl@60036
   956
  have "rel_filter A (Inf {F, G}) (Inf {F', G'})" by transfer_prover
hoelzl@60036
   957
  thus "rel_filter A (inf F G) (inf F' G')" by simp
hoelzl@60036
   958
qed
hoelzl@60036
   959
hoelzl@60036
   960
end
hoelzl@60036
   961
hoelzl@60036
   962
end
hoelzl@60036
   963
hoelzl@60036
   964
end
hoelzl@60036
   965
hoelzl@60036
   966
end