src/HOL/Filter.thy
author wenzelm
Fri Oct 09 20:26:03 2015 +0200 (2015-10-09)
changeset 61378 3e04c9ca001a
parent 61245 b77bf45efe21
child 61531 ab2e862263e7
permissions -rw-r--r--
discontinued specific HTML syntax;
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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 \<open>Filters on predicates\<close>
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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 \<open>Filters\<close>
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text \<open>
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  This definition also allows non-proper filters.
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\<close>
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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 \<open>Eventually\<close>
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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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syntax (xsymbols)
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  "_eventually"  :: "pttrn => 'a filter => bool => bool"      ("(3\<forall>\<^sub>F _ in _./ _)" [0, 0, 10] 10)
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translations
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  "\<forall>\<^sub>Fx in F. P" == "CONST eventually (\<lambda>x. P) F"
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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 eventuallyI: "(\<And>x. P x) \<Longrightarrow> eventually P F"
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  by (auto intro: always_eventually)
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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_ball_finite_distrib:
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  "finite A \<Longrightarrow> (eventually (\<lambda>x. \<forall>y\<in>A. P x y) net) \<longleftrightarrow> (\<forall>y\<in>A. eventually (\<lambda>x. P x y) net)"
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  by (induction A rule: finite_induct) (auto simp: eventually_conj_iff)
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lemma eventually_ball_finite:
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  "finite A \<Longrightarrow> \<forall>y\<in>A. eventually (\<lambda>x. P x y) net \<Longrightarrow> eventually (\<lambda>x. \<forall>y\<in>A. P x y) net"
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  by (auto simp: eventually_ball_finite_distrib)
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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_ex: "(\<forall>\<^sub>Fx in F. \<exists>y. P x y) \<longleftrightarrow> (\<exists>Y. \<forall>\<^sub>Fx in F. P x (Y x))"
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proof
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  assume "\<forall>\<^sub>Fx in F. \<exists>y. P x y"
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  then have "\<forall>\<^sub>Fx in F. P x (SOME y. P x y)"
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    by (auto intro: someI_ex eventually_elim1)
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  then show "\<exists>Y. \<forall>\<^sub>Fx in F. P x (Y x)"
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    by auto
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qed (auto intro: eventually_elim1)
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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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subsection \<open> Frequently as dual to eventually \<close>
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definition frequently :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
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  where "frequently P F \<longleftrightarrow> \<not> eventually (\<lambda>x. \<not> P x) F"
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syntax (xsymbols)
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  "_frequently"  :: "pttrn \<Rightarrow> 'a filter \<Rightarrow> bool \<Rightarrow> bool"      ("(3\<exists>\<^sub>F _ in _./ _)" [0, 0, 10] 10)
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translations
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  "\<exists>\<^sub>Fx in F. P" == "CONST frequently (\<lambda>x. P) F"
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lemma not_frequently_False [simp]: "\<not> (\<exists>\<^sub>Fx in F. False)"
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  by (simp add: frequently_def)
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lemma frequently_ex: "\<exists>\<^sub>Fx in F. P x \<Longrightarrow> \<exists>x. P x"
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  by (auto simp: frequently_def dest: not_eventuallyD)
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lemma frequentlyE: assumes "frequently P F" obtains x where "P x"
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  using frequently_ex[OF assms] by auto
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lemma frequently_mp:
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  assumes ev: "\<forall>\<^sub>Fx in F. P x \<longrightarrow> Q x" and P: "\<exists>\<^sub>Fx in F. P x" shows "\<exists>\<^sub>Fx in F. Q x"
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proof - 
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  from ev have "eventually (\<lambda>x. \<not> Q x \<longrightarrow> \<not> P x) F"
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    by (rule eventually_rev_mp) (auto intro!: always_eventually)
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  from eventually_mp[OF this] P show ?thesis
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    by (auto simp: frequently_def)
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qed
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lemma frequently_rev_mp:
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  assumes "\<exists>\<^sub>Fx in F. P x"
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  assumes "\<forall>\<^sub>Fx in F. P x \<longrightarrow> Q x"
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  shows "\<exists>\<^sub>Fx in F. Q x"
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using assms(2) assms(1) by (rule frequently_mp)
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lemma frequently_mono: "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> frequently P F \<Longrightarrow> frequently Q F"
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  using frequently_mp[of P Q] by (simp add: always_eventually)
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lemma frequently_elim1: "\<exists>\<^sub>Fx in F. P x \<Longrightarrow> (\<And>i. P i \<Longrightarrow> Q i) \<Longrightarrow> \<exists>\<^sub>Fx in F. Q x"
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  by (metis frequently_mono)
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lemma frequently_disj_iff: "(\<exists>\<^sub>Fx in F. P x \<or> Q x) \<longleftrightarrow> (\<exists>\<^sub>Fx in F. P x) \<or> (\<exists>\<^sub>Fx in F. Q x)"
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  by (simp add: frequently_def eventually_conj_iff)
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lemma frequently_disj: "\<exists>\<^sub>Fx in F. P x \<Longrightarrow> \<exists>\<^sub>Fx in F. Q x \<Longrightarrow> \<exists>\<^sub>Fx in F. P x \<or> Q x"
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  by (simp add: frequently_disj_iff)
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lemma frequently_bex_finite_distrib:
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  assumes "finite A" shows "(\<exists>\<^sub>Fx in F. \<exists>y\<in>A. P x y) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>\<^sub>Fx in F. P x y)"
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  using assms by induction (auto simp: frequently_disj_iff)
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lemma frequently_bex_finite: "finite A \<Longrightarrow> \<exists>\<^sub>Fx in F. \<exists>y\<in>A. P x y \<Longrightarrow> \<exists>y\<in>A. \<exists>\<^sub>Fx in F. P x y"
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  by (simp add: frequently_bex_finite_distrib)
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lemma frequently_all: "(\<exists>\<^sub>Fx in F. \<forall>y. P x y) \<longleftrightarrow> (\<forall>Y. \<exists>\<^sub>Fx in F. P x (Y x))"
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  using eventually_ex[of "\<lambda>x y. \<not> P x y" F] by (simp add: frequently_def)
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lemma
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  shows not_eventually: "\<not> eventually P F \<longleftrightarrow> (\<exists>\<^sub>Fx in F. \<not> P x)"
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    and not_frequently: "\<not> frequently P F \<longleftrightarrow> (\<forall>\<^sub>Fx in F. \<not> P x)"
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  by (auto simp: frequently_def)
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lemma frequently_imp_iff:
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  "(\<exists>\<^sub>Fx in F. P x \<longrightarrow> Q x) \<longleftrightarrow> (eventually P F \<longrightarrow> frequently Q F)"
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  unfolding imp_conv_disj frequently_disj_iff not_eventually[symmetric] ..
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lemma eventually_frequently_const_simps:
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  "(\<exists>\<^sub>Fx in F. P x \<and> C) \<longleftrightarrow> (\<exists>\<^sub>Fx in F. P x) \<and> C"
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  "(\<exists>\<^sub>Fx in F. C \<and> P x) \<longleftrightarrow> C \<and> (\<exists>\<^sub>Fx in F. P x)"
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  "(\<forall>\<^sub>Fx in F. P x \<or> C) \<longleftrightarrow> (\<forall>\<^sub>Fx in F. P x) \<or> C"
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  "(\<forall>\<^sub>Fx in F. C \<or> P x) \<longleftrightarrow> C \<or> (\<forall>\<^sub>Fx in F. P x)"
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  "(\<forall>\<^sub>Fx in F. P x \<longrightarrow> C) \<longleftrightarrow> ((\<exists>\<^sub>Fx in F. P x) \<longrightarrow> C)"
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  "(\<forall>\<^sub>Fx in F. C \<longrightarrow> P x) \<longleftrightarrow> (C \<longrightarrow> (\<forall>\<^sub>Fx in F. P x))"
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  by (cases C; simp add: not_frequently)+
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lemmas eventually_frequently_simps = 
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  eventually_frequently_const_simps
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  not_eventually
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  eventually_conj_iff
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  eventually_ball_finite_distrib
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  eventually_ex
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  not_frequently
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  frequently_disj_iff
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  frequently_bex_finite_distrib
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  frequently_all
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  frequently_imp_iff
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ML \<open>
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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 =
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        Thm.prop_of
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          (Rule_Cases.internalize_params (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 (resolve_tac ctxt [raw_elim_thm] i)
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    end)
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\<close>
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method_setup eventually_elim = \<open>
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  Scan.succeed (fn ctxt => METHOD_CASES (HEADGOAL o eventually_elim_tac ctxt))
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\<close> "elimination of eventually quantifiers"
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subsubsection \<open>Finer-than relation\<close>
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text \<open>@{term "F \<le> F'"} means that filter @{term F} is finer than
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filter @{term F'}.\<close>
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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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   282
  "sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"
hoelzl@60036
   283
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   284
definition
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   285
  "inf F F' = Abs_filter
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   286
      (\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
hoelzl@60036
   287
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   288
definition
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   289
  "Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)"
hoelzl@60036
   290
hoelzl@60036
   291
definition
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   292
  "Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}"
hoelzl@60036
   293
hoelzl@60036
   294
lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
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   295
  unfolding top_filter_def
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   296
  by (rule eventually_Abs_filter, rule is_filter.intro, auto)
hoelzl@60036
   297
hoelzl@60036
   298
lemma eventually_bot [simp]: "eventually P bot"
hoelzl@60036
   299
  unfolding bot_filter_def
hoelzl@60036
   300
  by (subst eventually_Abs_filter, rule is_filter.intro, auto)
hoelzl@60036
   301
hoelzl@60036
   302
lemma eventually_sup:
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   303
  "eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'"
hoelzl@60036
   304
  unfolding sup_filter_def
hoelzl@60036
   305
  by (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@60036
   306
     (auto elim!: eventually_rev_mp)
hoelzl@60036
   307
hoelzl@60036
   308
lemma eventually_inf:
hoelzl@60036
   309
  "eventually P (inf F F') \<longleftrightarrow>
hoelzl@60036
   310
   (\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
hoelzl@60036
   311
  unfolding inf_filter_def
hoelzl@60036
   312
  apply (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@60036
   313
  apply (fast intro: eventually_True)
hoelzl@60036
   314
  apply clarify
hoelzl@60036
   315
  apply (intro exI conjI)
hoelzl@60036
   316
  apply (erule (1) eventually_conj)
hoelzl@60036
   317
  apply (erule (1) eventually_conj)
hoelzl@60036
   318
  apply simp
hoelzl@60036
   319
  apply auto
hoelzl@60036
   320
  done
hoelzl@60036
   321
hoelzl@60036
   322
lemma eventually_Sup:
hoelzl@60036
   323
  "eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)"
hoelzl@60036
   324
  unfolding Sup_filter_def
hoelzl@60036
   325
  apply (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@60036
   326
  apply (auto intro: eventually_conj elim!: eventually_rev_mp)
hoelzl@60036
   327
  done
hoelzl@60036
   328
hoelzl@60036
   329
instance proof
hoelzl@60036
   330
  fix F F' F'' :: "'a filter" and S :: "'a filter set"
hoelzl@60036
   331
  { show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
hoelzl@60036
   332
    by (rule less_filter_def) }
hoelzl@60036
   333
  { show "F \<le> F"
hoelzl@60036
   334
    unfolding le_filter_def by simp }
hoelzl@60036
   335
  { assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"
hoelzl@60036
   336
    unfolding le_filter_def by simp }
hoelzl@60036
   337
  { assume "F \<le> F'" and "F' \<le> F" thus "F = F'"
hoelzl@60036
   338
    unfolding le_filter_def filter_eq_iff by fast }
hoelzl@60036
   339
  { show "inf F F' \<le> F" and "inf F F' \<le> F'"
hoelzl@60036
   340
    unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
hoelzl@60036
   341
  { assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"
hoelzl@60036
   342
    unfolding le_filter_def eventually_inf
hoelzl@60036
   343
    by (auto elim!: eventually_mono intro: eventually_conj) }
hoelzl@60036
   344
  { show "F \<le> sup F F'" and "F' \<le> sup F F'"
hoelzl@60036
   345
    unfolding le_filter_def eventually_sup by simp_all }
hoelzl@60036
   346
  { assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"
hoelzl@60036
   347
    unfolding le_filter_def eventually_sup by simp }
hoelzl@60036
   348
  { assume "F'' \<in> S" thus "Inf S \<le> F''"
hoelzl@60036
   349
    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
hoelzl@60036
   350
  { assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"
hoelzl@60036
   351
    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
hoelzl@60036
   352
  { assume "F \<in> S" thus "F \<le> Sup S"
hoelzl@60036
   353
    unfolding le_filter_def eventually_Sup by simp }
hoelzl@60036
   354
  { assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'"
hoelzl@60036
   355
    unfolding le_filter_def eventually_Sup by simp }
hoelzl@60036
   356
  { show "Inf {} = (top::'a filter)"
hoelzl@60036
   357
    by (auto simp: top_filter_def Inf_filter_def Sup_filter_def)
hoelzl@60036
   358
      (metis (full_types) top_filter_def always_eventually eventually_top) }
hoelzl@60036
   359
  { show "Sup {} = (bot::'a filter)"
hoelzl@60036
   360
    by (auto simp: bot_filter_def Sup_filter_def) }
hoelzl@60036
   361
qed
hoelzl@60036
   362
hoelzl@60036
   363
end
hoelzl@60036
   364
hoelzl@60036
   365
lemma filter_leD:
hoelzl@60036
   366
  "F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"
hoelzl@60036
   367
  unfolding le_filter_def by simp
hoelzl@60036
   368
hoelzl@60036
   369
lemma filter_leI:
hoelzl@60036
   370
  "(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"
hoelzl@60036
   371
  unfolding le_filter_def by simp
hoelzl@60036
   372
hoelzl@60036
   373
lemma eventually_False:
hoelzl@60036
   374
  "eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot"
hoelzl@60036
   375
  unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
hoelzl@60036
   376
hoelzl@60040
   377
lemma eventually_frequently: "F \<noteq> bot \<Longrightarrow> eventually P F \<Longrightarrow> frequently P F"
hoelzl@60040
   378
  using eventually_conj[of P F "\<lambda>x. \<not> P x"]
hoelzl@60040
   379
  by (auto simp add: frequently_def eventually_False)
hoelzl@60040
   380
hoelzl@60040
   381
lemma eventually_const_iff: "eventually (\<lambda>x. P) F \<longleftrightarrow> P \<or> F = bot"
hoelzl@60040
   382
  by (cases P) (auto simp: eventually_False)
hoelzl@60040
   383
hoelzl@60040
   384
lemma eventually_const[simp]: "F \<noteq> bot \<Longrightarrow> eventually (\<lambda>x. P) F \<longleftrightarrow> P"
hoelzl@60040
   385
  by (simp add: eventually_const_iff)
hoelzl@60040
   386
hoelzl@60040
   387
lemma frequently_const_iff: "frequently (\<lambda>x. P) F \<longleftrightarrow> P \<and> F \<noteq> bot"
hoelzl@60040
   388
  by (simp add: frequently_def eventually_const_iff)
hoelzl@60040
   389
hoelzl@60040
   390
lemma frequently_const[simp]: "F \<noteq> bot \<Longrightarrow> frequently (\<lambda>x. P) F \<longleftrightarrow> P"
hoelzl@60040
   391
  by (simp add: frequently_const_iff)
hoelzl@60040
   392
hoelzl@61245
   393
lemma eventually_happens: "eventually P net \<Longrightarrow> net = bot \<or> (\<exists>x. P x)"
hoelzl@61245
   394
  by (metis frequentlyE eventually_frequently)
hoelzl@61245
   395
hoelzl@60036
   396
abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"
hoelzl@60036
   397
  where "trivial_limit F \<equiv> F = bot"
hoelzl@60036
   398
hoelzl@60036
   399
lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"
hoelzl@60036
   400
  by (rule eventually_False [symmetric])
hoelzl@60036
   401
hoelzl@60036
   402
lemma eventually_Inf: "eventually P (Inf B) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X))"
hoelzl@60036
   403
proof -
hoelzl@60036
   404
  let ?F = "\<lambda>P. \<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X)"
hoelzl@60036
   405
  
hoelzl@60036
   406
  { fix P have "eventually P (Abs_filter ?F) \<longleftrightarrow> ?F P"
hoelzl@60036
   407
    proof (rule eventually_Abs_filter is_filter.intro)+
hoelzl@60036
   408
      show "?F (\<lambda>x. True)"
hoelzl@60036
   409
        by (rule exI[of _ "{}"]) (simp add: le_fun_def)
hoelzl@60036
   410
    next
hoelzl@60036
   411
      fix P Q
hoelzl@60036
   412
      assume "?F P" then guess X ..
hoelzl@60036
   413
      moreover
hoelzl@60036
   414
      assume "?F Q" then guess Y ..
hoelzl@60036
   415
      ultimately show "?F (\<lambda>x. P x \<and> Q x)"
hoelzl@60036
   416
        by (intro exI[of _ "X \<union> Y"])
hoelzl@60036
   417
           (auto simp: Inf_union_distrib eventually_inf)
hoelzl@60036
   418
    next
hoelzl@60036
   419
      fix P Q
hoelzl@60036
   420
      assume "?F P" then guess X ..
hoelzl@60036
   421
      moreover assume "\<forall>x. P x \<longrightarrow> Q x"
hoelzl@60036
   422
      ultimately show "?F Q"
hoelzl@60036
   423
        by (intro exI[of _ X]) (auto elim: eventually_elim1)
hoelzl@60036
   424
    qed }
hoelzl@60036
   425
  note eventually_F = this
hoelzl@60036
   426
hoelzl@60036
   427
  have "Inf B = Abs_filter ?F"
hoelzl@60036
   428
  proof (intro antisym Inf_greatest)
hoelzl@60036
   429
    show "Inf B \<le> Abs_filter ?F"
hoelzl@60036
   430
      by (auto simp: le_filter_def eventually_F dest: Inf_superset_mono)
hoelzl@60036
   431
  next
hoelzl@60036
   432
    fix F assume "F \<in> B" then show "Abs_filter ?F \<le> F"
hoelzl@60036
   433
      by (auto simp add: le_filter_def eventually_F intro!: exI[of _ "{F}"])
hoelzl@60036
   434
  qed
hoelzl@60036
   435
  then show ?thesis
hoelzl@60036
   436
    by (simp add: eventually_F)
hoelzl@60036
   437
qed
hoelzl@60036
   438
hoelzl@60036
   439
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
   440
  unfolding INF_def[of B] eventually_Inf[of P "F`B"]
hoelzl@60036
   441
  by (metis Inf_image_eq finite_imageI image_mono finite_subset_image)
hoelzl@60036
   442
hoelzl@60036
   443
lemma Inf_filter_not_bot:
hoelzl@60036
   444
  fixes B :: "'a filter set"
hoelzl@60036
   445
  shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> Inf X \<noteq> bot) \<Longrightarrow> Inf B \<noteq> bot"
hoelzl@60036
   446
  unfolding trivial_limit_def eventually_Inf[of _ B]
hoelzl@60036
   447
    bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
hoelzl@60036
   448
hoelzl@60036
   449
lemma INF_filter_not_bot:
hoelzl@60036
   450
  fixes F :: "'i \<Rightarrow> 'a filter"
hoelzl@60036
   451
  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
   452
  unfolding trivial_limit_def eventually_INF[of _ B]
hoelzl@60036
   453
    bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
hoelzl@60036
   454
hoelzl@60036
   455
lemma eventually_Inf_base:
hoelzl@60036
   456
  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
   457
  shows "eventually P (Inf B) \<longleftrightarrow> (\<exists>b\<in>B. eventually P b)"
hoelzl@60036
   458
proof (subst eventually_Inf, safe)
hoelzl@60036
   459
  fix X assume "finite X" "X \<subseteq> B"
hoelzl@60036
   460
  then have "\<exists>b\<in>B. \<forall>x\<in>X. b \<le> x"
hoelzl@60036
   461
  proof induct
hoelzl@60036
   462
    case empty then show ?case
wenzelm@60758
   463
      using \<open>B \<noteq> {}\<close> by auto
hoelzl@60036
   464
  next
hoelzl@60036
   465
    case (insert x X)
hoelzl@60036
   466
    then obtain b where "b \<in> B" "\<And>x. x \<in> X \<Longrightarrow> b \<le> x"
hoelzl@60036
   467
      by auto
wenzelm@60758
   468
    with \<open>insert x X \<subseteq> B\<close> base[of b x] show ?case
hoelzl@60036
   469
      by (auto intro: order_trans)
hoelzl@60036
   470
  qed
hoelzl@60036
   471
  then obtain b where "b \<in> B" "b \<le> Inf X"
hoelzl@60036
   472
    by (auto simp: le_Inf_iff)
hoelzl@60036
   473
  then show "eventually P (Inf X) \<Longrightarrow> Bex B (eventually P)"
hoelzl@60036
   474
    by (intro bexI[of _ b]) (auto simp: le_filter_def)
hoelzl@60036
   475
qed (auto intro!: exI[of _ "{x}" for x])
hoelzl@60036
   476
hoelzl@60036
   477
lemma eventually_INF_base:
hoelzl@60036
   478
  "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
   479
    eventually P (INF b:B. F b) \<longleftrightarrow> (\<exists>b\<in>B. eventually P (F b))"
hoelzl@60036
   480
  unfolding INF_def by (subst eventually_Inf_base) auto
hoelzl@60036
   481
hoelzl@60036
   482
wenzelm@60758
   483
subsubsection \<open>Map function for filters\<close>
hoelzl@60036
   484
hoelzl@60036
   485
definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
hoelzl@60036
   486
  where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
hoelzl@60036
   487
hoelzl@60036
   488
lemma eventually_filtermap:
hoelzl@60036
   489
  "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
hoelzl@60036
   490
  unfolding filtermap_def
hoelzl@60036
   491
  apply (rule eventually_Abs_filter)
hoelzl@60036
   492
  apply (rule is_filter.intro)
hoelzl@60036
   493
  apply (auto elim!: eventually_rev_mp)
hoelzl@60036
   494
  done
hoelzl@60036
   495
hoelzl@60036
   496
lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
hoelzl@60036
   497
  by (simp add: filter_eq_iff eventually_filtermap)
hoelzl@60036
   498
hoelzl@60036
   499
lemma filtermap_filtermap:
hoelzl@60036
   500
  "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
hoelzl@60036
   501
  by (simp add: filter_eq_iff eventually_filtermap)
hoelzl@60036
   502
hoelzl@60036
   503
lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"
hoelzl@60036
   504
  unfolding le_filter_def eventually_filtermap by simp
hoelzl@60036
   505
hoelzl@60036
   506
lemma filtermap_bot [simp]: "filtermap f bot = bot"
hoelzl@60036
   507
  by (simp add: filter_eq_iff eventually_filtermap)
hoelzl@60036
   508
hoelzl@60036
   509
lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)"
hoelzl@60036
   510
  by (auto simp: filter_eq_iff eventually_filtermap eventually_sup)
hoelzl@60036
   511
hoelzl@60036
   512
lemma filtermap_inf: "filtermap f (inf F1 F2) \<le> inf (filtermap f F1) (filtermap f F2)"
hoelzl@60036
   513
  by (auto simp: le_filter_def eventually_filtermap eventually_inf)
hoelzl@60036
   514
hoelzl@60036
   515
lemma filtermap_INF: "filtermap f (INF b:B. F b) \<le> (INF b:B. filtermap f (F b))"
hoelzl@60036
   516
proof -
hoelzl@60036
   517
  { fix X :: "'c set" assume "finite X"
hoelzl@60036
   518
    then have "filtermap f (INFIMUM X F) \<le> (INF b:X. filtermap f (F b))"
hoelzl@60036
   519
    proof induct
hoelzl@60036
   520
      case (insert x X)
hoelzl@60036
   521
      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
   522
        by (rule order_trans[OF _ filtermap_inf]) simp
hoelzl@60036
   523
      also have "\<dots> \<le> inf (filtermap f (F x)) (INF a:X. filtermap f (F a))"
hoelzl@60036
   524
        by (intro inf_mono insert order_refl)
hoelzl@60036
   525
      finally show ?case
hoelzl@60036
   526
        by simp
hoelzl@60036
   527
    qed simp }
hoelzl@60036
   528
  then show ?thesis
hoelzl@60036
   529
    unfolding le_filter_def eventually_filtermap
hoelzl@60036
   530
    by (subst (1 2) eventually_INF) auto
hoelzl@60036
   531
qed
wenzelm@60758
   532
subsubsection \<open>Standard filters\<close>
hoelzl@60036
   533
hoelzl@60036
   534
definition principal :: "'a set \<Rightarrow> 'a filter" where
hoelzl@60036
   535
  "principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)"
hoelzl@60036
   536
hoelzl@60036
   537
lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)"
hoelzl@60036
   538
  unfolding principal_def
hoelzl@60036
   539
  by (rule eventually_Abs_filter, rule is_filter.intro) auto
hoelzl@60036
   540
hoelzl@60036
   541
lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F"
hoelzl@60036
   542
  unfolding eventually_inf eventually_principal by (auto elim: eventually_elim1)
hoelzl@60036
   543
hoelzl@60036
   544
lemma principal_UNIV[simp]: "principal UNIV = top"
hoelzl@60036
   545
  by (auto simp: filter_eq_iff eventually_principal)
hoelzl@60036
   546
hoelzl@60036
   547
lemma principal_empty[simp]: "principal {} = bot"
hoelzl@60036
   548
  by (auto simp: filter_eq_iff eventually_principal)
hoelzl@60036
   549
hoelzl@60036
   550
lemma principal_eq_bot_iff: "principal X = bot \<longleftrightarrow> X = {}"
hoelzl@60036
   551
  by (auto simp add: filter_eq_iff eventually_principal)
hoelzl@60036
   552
hoelzl@60036
   553
lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B"
hoelzl@60036
   554
  by (auto simp: le_filter_def eventually_principal)
hoelzl@60036
   555
hoelzl@60036
   556
lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F"
hoelzl@60036
   557
  unfolding le_filter_def eventually_principal
hoelzl@60036
   558
  apply safe
hoelzl@60036
   559
  apply (erule_tac x="\<lambda>x. x \<in> A" in allE)
hoelzl@60036
   560
  apply (auto elim: eventually_elim1)
hoelzl@60036
   561
  done
hoelzl@60036
   562
hoelzl@60036
   563
lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B"
hoelzl@60036
   564
  unfolding eq_iff by simp
hoelzl@60036
   565
hoelzl@60036
   566
lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)"
hoelzl@60036
   567
  unfolding filter_eq_iff eventually_sup eventually_principal by auto
hoelzl@60036
   568
hoelzl@60036
   569
lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)"
hoelzl@60036
   570
  unfolding filter_eq_iff eventually_inf eventually_principal
hoelzl@60036
   571
  by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"])
hoelzl@60036
   572
hoelzl@60036
   573
lemma SUP_principal[simp]: "(SUP i : I. principal (A i)) = principal (\<Union>i\<in>I. A i)"
hoelzl@60036
   574
  unfolding filter_eq_iff eventually_Sup SUP_def by (auto simp: eventually_principal)
hoelzl@60036
   575
hoelzl@60036
   576
lemma INF_principal_finite: "finite X \<Longrightarrow> (INF x:X. principal (f x)) = principal (\<Inter>x\<in>X. f x)"
hoelzl@60036
   577
  by (induct X rule: finite_induct) auto
hoelzl@60036
   578
hoelzl@60036
   579
lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)"
hoelzl@60036
   580
  unfolding filter_eq_iff eventually_filtermap eventually_principal by simp
hoelzl@60036
   581
wenzelm@60758
   582
subsubsection \<open>Order filters\<close>
hoelzl@60036
   583
hoelzl@60036
   584
definition at_top :: "('a::order) filter"
hoelzl@60036
   585
  where "at_top = (INF k. principal {k ..})"
hoelzl@60036
   586
hoelzl@60036
   587
lemma at_top_sub: "at_top = (INF k:{c::'a::linorder..}. principal {k ..})"
hoelzl@60036
   588
  by (auto intro!: INF_eq max.cobounded1 max.cobounded2 simp: at_top_def)
hoelzl@60036
   589
hoelzl@60036
   590
lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)"
hoelzl@60036
   591
  unfolding at_top_def
hoelzl@60036
   592
  by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
hoelzl@60036
   593
hoelzl@60036
   594
lemma eventually_ge_at_top:
hoelzl@60036
   595
  "eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top"
hoelzl@60036
   596
  unfolding eventually_at_top_linorder by auto
hoelzl@60036
   597
hoelzl@60036
   598
lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::{no_top, linorder}. \<forall>n>N. P n)"
hoelzl@60036
   599
proof -
hoelzl@60036
   600
  have "eventually P (INF k. principal {k <..}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n>N. P n)"
hoelzl@60036
   601
    by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
hoelzl@60036
   602
  also have "(INF k. principal {k::'a <..}) = at_top"
hoelzl@60036
   603
    unfolding at_top_def 
hoelzl@60036
   604
    by (intro INF_eq) (auto intro: less_imp_le simp: Ici_subset_Ioi_iff gt_ex)
hoelzl@60036
   605
  finally show ?thesis .
hoelzl@60036
   606
qed
hoelzl@60036
   607
hoelzl@60721
   608
lemma eventually_at_top_not_equal: "eventually (\<lambda>x::'a::{no_top, linorder}. x \<noteq> c) at_top"
hoelzl@60721
   609
  unfolding eventually_at_top_dense by auto
hoelzl@60721
   610
hoelzl@60721
   611
lemma eventually_gt_at_top: "eventually (\<lambda>x. (c::_::{no_top, linorder}) < x) at_top"
hoelzl@60036
   612
  unfolding eventually_at_top_dense by auto
hoelzl@60036
   613
hoelzl@60036
   614
definition at_bot :: "('a::order) filter"
hoelzl@60036
   615
  where "at_bot = (INF k. principal {.. k})"
hoelzl@60036
   616
hoelzl@60036
   617
lemma at_bot_sub: "at_bot = (INF k:{.. c::'a::linorder}. principal {.. k})"
hoelzl@60036
   618
  by (auto intro!: INF_eq min.cobounded1 min.cobounded2 simp: at_bot_def)
hoelzl@60036
   619
hoelzl@60036
   620
lemma eventually_at_bot_linorder:
hoelzl@60036
   621
  fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"
hoelzl@60036
   622
  unfolding at_bot_def
hoelzl@60036
   623
  by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
hoelzl@60036
   624
hoelzl@60036
   625
lemma eventually_le_at_bot:
hoelzl@60036
   626
  "eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot"
hoelzl@60036
   627
  unfolding eventually_at_bot_linorder by auto
hoelzl@60036
   628
hoelzl@60036
   629
lemma eventually_at_bot_dense: "eventually P at_bot \<longleftrightarrow> (\<exists>N::'a::{no_bot, linorder}. \<forall>n<N. P n)"
hoelzl@60036
   630
proof -
hoelzl@60036
   631
  have "eventually P (INF k. principal {..< k}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n<N. P n)"
hoelzl@60036
   632
    by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
hoelzl@60036
   633
  also have "(INF k. principal {..< k::'a}) = at_bot"
hoelzl@60036
   634
    unfolding at_bot_def 
hoelzl@60036
   635
    by (intro INF_eq) (auto intro: less_imp_le simp: Iic_subset_Iio_iff lt_ex)
hoelzl@60036
   636
  finally show ?thesis .
hoelzl@60036
   637
qed
hoelzl@60036
   638
hoelzl@60721
   639
lemma eventually_at_bot_not_equal: "eventually (\<lambda>x::'a::{no_bot, linorder}. x \<noteq> c) at_bot"
hoelzl@60721
   640
  unfolding eventually_at_bot_dense by auto
hoelzl@60721
   641
hoelzl@60036
   642
lemma eventually_gt_at_bot:
hoelzl@60036
   643
  "eventually (\<lambda>x. x < (c::_::unbounded_dense_linorder)) at_bot"
hoelzl@60036
   644
  unfolding eventually_at_bot_dense by auto
hoelzl@60036
   645
hoelzl@60036
   646
lemma trivial_limit_at_bot_linorder: "\<not> trivial_limit (at_bot ::('a::linorder) filter)"
hoelzl@60036
   647
  unfolding trivial_limit_def
hoelzl@60036
   648
  by (metis eventually_at_bot_linorder order_refl)
hoelzl@60036
   649
hoelzl@60036
   650
lemma trivial_limit_at_top_linorder: "\<not> trivial_limit (at_top ::('a::linorder) filter)"
hoelzl@60036
   651
  unfolding trivial_limit_def
hoelzl@60036
   652
  by (metis eventually_at_top_linorder order_refl)
hoelzl@60036
   653
wenzelm@60758
   654
subsection \<open>Sequentially\<close>
hoelzl@60036
   655
hoelzl@60036
   656
abbreviation sequentially :: "nat filter"
hoelzl@60036
   657
  where "sequentially \<equiv> at_top"
hoelzl@60036
   658
hoelzl@60036
   659
lemma eventually_sequentially:
hoelzl@60036
   660
  "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
hoelzl@60036
   661
  by (rule eventually_at_top_linorder)
hoelzl@60036
   662
hoelzl@60036
   663
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
hoelzl@60036
   664
  unfolding filter_eq_iff eventually_sequentially by auto
hoelzl@60036
   665
hoelzl@60036
   666
lemmas trivial_limit_sequentially = sequentially_bot
hoelzl@60036
   667
hoelzl@60036
   668
lemma eventually_False_sequentially [simp]:
hoelzl@60036
   669
  "\<not> eventually (\<lambda>n. False) sequentially"
hoelzl@60036
   670
  by (simp add: eventually_False)
hoelzl@60036
   671
hoelzl@60036
   672
lemma le_sequentially:
hoelzl@60036
   673
  "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
hoelzl@60036
   674
  by (simp add: at_top_def le_INF_iff le_principal)
hoelzl@60036
   675
lp15@60974
   676
lemma eventually_sequentiallyI [intro?]:
hoelzl@60036
   677
  assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
hoelzl@60036
   678
  shows "eventually P sequentially"
hoelzl@60036
   679
using assms by (auto simp: eventually_sequentially)
hoelzl@60036
   680
hoelzl@60040
   681
lemma eventually_sequentially_Suc: "eventually (\<lambda>i. P (Suc i)) sequentially \<longleftrightarrow> eventually P sequentially"
hoelzl@60040
   682
  unfolding eventually_sequentially by (metis Suc_le_D Suc_le_mono le_Suc_eq)
hoelzl@60040
   683
hoelzl@60040
   684
lemma eventually_sequentially_seg: "eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially"
hoelzl@60040
   685
  using eventually_sequentially_Suc[of "\<lambda>n. P (n + k)" for k] by (induction k) auto
hoelzl@60036
   686
hoelzl@60039
   687
subsection \<open> The cofinite filter \<close>
hoelzl@60039
   688
hoelzl@60039
   689
definition "cofinite = Abs_filter (\<lambda>P. finite {x. \<not> P x})"
hoelzl@60039
   690
hoelzl@60040
   691
abbreviation Inf_many :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "INFM " 10) where
hoelzl@60040
   692
  "Inf_many P \<equiv> frequently P cofinite"
hoelzl@60040
   693
hoelzl@60040
   694
abbreviation Alm_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "MOST " 10) where
hoelzl@60040
   695
  "Alm_all P \<equiv> eventually P cofinite"
hoelzl@60040
   696
hoelzl@60040
   697
notation (xsymbols)
hoelzl@60040
   698
  Inf_many  (binder "\<exists>\<^sub>\<infinity>" 10) and
hoelzl@60040
   699
  Alm_all  (binder "\<forall>\<^sub>\<infinity>" 10)
hoelzl@60040
   700
hoelzl@60039
   701
lemma eventually_cofinite: "eventually P cofinite \<longleftrightarrow> finite {x. \<not> P x}"
hoelzl@60039
   702
  unfolding cofinite_def
hoelzl@60039
   703
proof (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@60039
   704
  fix P Q :: "'a \<Rightarrow> bool" assume "finite {x. \<not> P x}" "finite {x. \<not> Q x}"
hoelzl@60039
   705
  from finite_UnI[OF this] show "finite {x. \<not> (P x \<and> Q x)}"
hoelzl@60039
   706
    by (rule rev_finite_subset) auto
hoelzl@60039
   707
next
hoelzl@60039
   708
  fix P Q :: "'a \<Rightarrow> bool" assume P: "finite {x. \<not> P x}" and *: "\<forall>x. P x \<longrightarrow> Q x"
hoelzl@60039
   709
  from * show "finite {x. \<not> Q x}"
hoelzl@60039
   710
    by (intro finite_subset[OF _ P]) auto
hoelzl@60039
   711
qed simp
hoelzl@60039
   712
hoelzl@60040
   713
lemma frequently_cofinite: "frequently P cofinite \<longleftrightarrow> \<not> finite {x. P x}"
hoelzl@60040
   714
  by (simp add: frequently_def eventually_cofinite)
hoelzl@60040
   715
hoelzl@60039
   716
lemma cofinite_bot[simp]: "cofinite = (bot::'a filter) \<longleftrightarrow> finite (UNIV :: 'a set)"
hoelzl@60039
   717
  unfolding trivial_limit_def eventually_cofinite by simp
hoelzl@60039
   718
hoelzl@60039
   719
lemma cofinite_eq_sequentially: "cofinite = sequentially"
hoelzl@60039
   720
  unfolding filter_eq_iff eventually_sequentially eventually_cofinite
hoelzl@60039
   721
proof safe
hoelzl@60039
   722
  fix P :: "nat \<Rightarrow> bool" assume [simp]: "finite {x. \<not> P x}"
hoelzl@60039
   723
  show "\<exists>N. \<forall>n\<ge>N. P n"
hoelzl@60039
   724
  proof cases
hoelzl@60039
   725
    assume "{x. \<not> P x} \<noteq> {}" then show ?thesis
hoelzl@60039
   726
      by (intro exI[of _ "Suc (Max {x. \<not> P x})"]) (auto simp: Suc_le_eq)
hoelzl@60039
   727
  qed auto
hoelzl@60039
   728
next
hoelzl@60039
   729
  fix P :: "nat \<Rightarrow> bool" and N :: nat assume "\<forall>n\<ge>N. P n"
hoelzl@60039
   730
  then have "{x. \<not> P x} \<subseteq> {..< N}"
hoelzl@60039
   731
    by (auto simp: not_le)
hoelzl@60039
   732
  then show "finite {x. \<not> P x}"
hoelzl@60039
   733
    by (blast intro: finite_subset)
hoelzl@60039
   734
qed
hoelzl@60036
   735
wenzelm@60758
   736
subsection \<open>Limits\<close>
hoelzl@60036
   737
hoelzl@60036
   738
definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where
hoelzl@60036
   739
  "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
hoelzl@60036
   740
hoelzl@60036
   741
syntax
hoelzl@60036
   742
  "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)
hoelzl@60036
   743
hoelzl@60036
   744
translations
hoelzl@60036
   745
  "LIM x F1. f :> F2"   == "CONST filterlim (%x. f) F2 F1"
hoelzl@60036
   746
hoelzl@60036
   747
lemma filterlim_iff:
hoelzl@60036
   748
  "(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)"
hoelzl@60036
   749
  unfolding filterlim_def le_filter_def eventually_filtermap ..
hoelzl@60036
   750
hoelzl@60036
   751
lemma filterlim_compose:
hoelzl@60036
   752
  "filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1"
hoelzl@60036
   753
  unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)
hoelzl@60036
   754
hoelzl@60036
   755
lemma filterlim_mono:
hoelzl@60036
   756
  "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'"
hoelzl@60036
   757
  unfolding filterlim_def by (metis filtermap_mono order_trans)
hoelzl@60036
   758
hoelzl@60036
   759
lemma filterlim_ident: "LIM x F. x :> F"
hoelzl@60036
   760
  by (simp add: filterlim_def filtermap_ident)
hoelzl@60036
   761
hoelzl@60036
   762
lemma filterlim_cong:
hoelzl@60036
   763
  "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
   764
  by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2)
hoelzl@60036
   765
hoelzl@60036
   766
lemma filterlim_mono_eventually:
hoelzl@60036
   767
  assumes "filterlim f F G" and ord: "F \<le> F'" "G' \<le> G"
hoelzl@60036
   768
  assumes eq: "eventually (\<lambda>x. f x = f' x) G'"
hoelzl@60036
   769
  shows "filterlim f' F' G'"
hoelzl@60036
   770
  apply (rule filterlim_cong[OF refl refl eq, THEN iffD1])
hoelzl@60036
   771
  apply (rule filterlim_mono[OF _ ord])
hoelzl@60036
   772
  apply fact
hoelzl@60036
   773
  done
hoelzl@60036
   774
hoelzl@60036
   775
lemma filtermap_mono_strong: "inj f \<Longrightarrow> filtermap f F \<le> filtermap f G \<longleftrightarrow> F \<le> G"
hoelzl@60036
   776
  apply (auto intro!: filtermap_mono) []
hoelzl@60036
   777
  apply (auto simp: le_filter_def eventually_filtermap)
hoelzl@60036
   778
  apply (erule_tac x="\<lambda>x. P (inv f x)" in allE)
hoelzl@60036
   779
  apply auto
hoelzl@60036
   780
  done
hoelzl@60036
   781
hoelzl@60036
   782
lemma filtermap_eq_strong: "inj f \<Longrightarrow> filtermap f F = filtermap f G \<longleftrightarrow> F = G"
hoelzl@60036
   783
  by (simp add: filtermap_mono_strong eq_iff)
hoelzl@60036
   784
hoelzl@60721
   785
lemma filtermap_fun_inverse:
hoelzl@60721
   786
  assumes g: "filterlim g F G"
hoelzl@60721
   787
  assumes f: "filterlim f G F"
hoelzl@60721
   788
  assumes ev: "eventually (\<lambda>x. f (g x) = x) G"
hoelzl@60721
   789
  shows "filtermap f F = G"
hoelzl@60721
   790
proof (rule antisym)
hoelzl@60721
   791
  show "filtermap f F \<le> G"
hoelzl@60721
   792
    using f unfolding filterlim_def .
hoelzl@60721
   793
  have "G = filtermap f (filtermap g G)"
hoelzl@60721
   794
    using ev by (auto elim: eventually_elim2 simp: filter_eq_iff eventually_filtermap)
hoelzl@60721
   795
  also have "\<dots> \<le> filtermap f F"
hoelzl@60721
   796
    using g by (intro filtermap_mono) (simp add: filterlim_def)
hoelzl@60721
   797
  finally show "G \<le> filtermap f F" .
hoelzl@60721
   798
qed
hoelzl@60721
   799
hoelzl@60036
   800
lemma filterlim_principal:
hoelzl@60036
   801
  "(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)"
hoelzl@60036
   802
  unfolding filterlim_def eventually_filtermap le_principal ..
hoelzl@60036
   803
hoelzl@60036
   804
lemma filterlim_inf:
hoelzl@60036
   805
  "(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))"
hoelzl@60036
   806
  unfolding filterlim_def by simp
hoelzl@60036
   807
hoelzl@60036
   808
lemma filterlim_INF:
hoelzl@60036
   809
  "(LIM x F. f x :> (INF b:B. G b)) \<longleftrightarrow> (\<forall>b\<in>B. LIM x F. f x :> G b)"
hoelzl@60036
   810
  unfolding filterlim_def le_INF_iff ..
hoelzl@60036
   811
hoelzl@60036
   812
lemma filterlim_INF_INF:
hoelzl@60036
   813
  "(\<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
   814
  unfolding filterlim_def by (rule order_trans[OF filtermap_INF INF_mono])
hoelzl@60036
   815
hoelzl@60036
   816
lemma filterlim_base:
hoelzl@60036
   817
  "(\<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
   818
    LIM x (INF i:I. principal (F i)). f x :> (INF j:J. principal (G j))"
hoelzl@60036
   819
  by (force intro!: filterlim_INF_INF simp: image_subset_iff)
hoelzl@60036
   820
hoelzl@60036
   821
lemma filterlim_base_iff: 
hoelzl@60036
   822
  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
   823
  shows "(LIM x (INF i:I. principal (F i)). f x :> INF j:J. principal (G j)) \<longleftrightarrow>
hoelzl@60036
   824
    (\<forall>j\<in>J. \<exists>i\<in>I. \<forall>x\<in>F i. f x \<in> G j)"
hoelzl@60036
   825
  unfolding filterlim_INF filterlim_principal
hoelzl@60036
   826
proof (subst eventually_INF_base)
hoelzl@60036
   827
  fix i j assume "i \<in> I" "j \<in> I"
hoelzl@60036
   828
  with chain[OF this] show "\<exists>x\<in>I. principal (F x) \<le> inf (principal (F i)) (principal (F j))"
hoelzl@60036
   829
    by auto
wenzelm@60758
   830
qed (auto simp: eventually_principal \<open>I \<noteq> {}\<close>)
hoelzl@60036
   831
hoelzl@60036
   832
lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2"
hoelzl@60036
   833
  unfolding filterlim_def filtermap_filtermap ..
hoelzl@60036
   834
hoelzl@60036
   835
lemma filterlim_sup:
hoelzl@60036
   836
  "filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)"
hoelzl@60036
   837
  unfolding filterlim_def filtermap_sup by auto
hoelzl@60036
   838
hoelzl@60036
   839
lemma filterlim_sequentially_Suc:
hoelzl@60036
   840
  "(LIM x sequentially. f (Suc x) :> F) \<longleftrightarrow> (LIM x sequentially. f x :> F)"
hoelzl@60036
   841
  unfolding filterlim_iff by (subst eventually_sequentially_Suc) simp
hoelzl@60036
   842
hoelzl@60036
   843
lemma filterlim_Suc: "filterlim Suc sequentially sequentially"
hoelzl@60036
   844
  by (simp add: filterlim_iff eventually_sequentially) (metis le_Suc_eq)
hoelzl@60036
   845
hoelzl@60182
   846
lemma filterlim_If:
hoelzl@60182
   847
  "LIM x inf F (principal {x. P x}). f x :> G \<Longrightarrow>
hoelzl@60182
   848
    LIM x inf F (principal {x. \<not> P x}). g x :> G \<Longrightarrow>
hoelzl@60182
   849
    LIM x F. if P x then f x else g x :> G"
hoelzl@60182
   850
  unfolding filterlim_iff eventually_inf_principal by (auto simp: eventually_conj_iff)
hoelzl@60036
   851
wenzelm@60758
   852
subsection \<open>Limits to @{const at_top} and @{const at_bot}\<close>
hoelzl@60036
   853
hoelzl@60036
   854
lemma filterlim_at_top:
hoelzl@60036
   855
  fixes f :: "'a \<Rightarrow> ('b::linorder)"
hoelzl@60036
   856
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)"
hoelzl@60036
   857
  by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_elim1)
hoelzl@60036
   858
hoelzl@60036
   859
lemma filterlim_at_top_mono:
hoelzl@60036
   860
  "LIM x F. f x :> at_top \<Longrightarrow> eventually (\<lambda>x. f x \<le> (g x::'a::linorder)) F \<Longrightarrow>
hoelzl@60036
   861
    LIM x F. g x :> at_top"
hoelzl@60036
   862
  by (auto simp: filterlim_at_top elim: eventually_elim2 intro: order_trans)
hoelzl@60036
   863
hoelzl@60036
   864
lemma filterlim_at_top_dense:
hoelzl@60036
   865
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)"
hoelzl@60036
   866
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)"
hoelzl@60036
   867
  by (metis eventually_elim1[of _ F] eventually_gt_at_top order_less_imp_le
hoelzl@60036
   868
            filterlim_at_top[of f F] filterlim_iff[of f at_top F])
hoelzl@60036
   869
hoelzl@60036
   870
lemma filterlim_at_top_ge:
hoelzl@60036
   871
  fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
hoelzl@60036
   872
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)"
hoelzl@60036
   873
  unfolding at_top_sub[of c] filterlim_INF by (auto simp add: filterlim_principal)
hoelzl@60036
   874
hoelzl@60036
   875
lemma filterlim_at_top_at_top:
hoelzl@60036
   876
  fixes f :: "'a::linorder \<Rightarrow> 'b::linorder"
hoelzl@60036
   877
  assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
hoelzl@60036
   878
  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
hoelzl@60036
   879
  assumes Q: "eventually Q at_top"
hoelzl@60036
   880
  assumes P: "eventually P at_top"
hoelzl@60036
   881
  shows "filterlim f at_top at_top"
hoelzl@60036
   882
proof -
hoelzl@60036
   883
  from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
hoelzl@60036
   884
    unfolding eventually_at_top_linorder by auto
hoelzl@60036
   885
  show ?thesis
hoelzl@60036
   886
  proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
hoelzl@60036
   887
    fix z assume "x \<le> z"
hoelzl@60036
   888
    with x have "P z" by auto
hoelzl@60036
   889
    have "eventually (\<lambda>x. g z \<le> x) at_top"
hoelzl@60036
   890
      by (rule eventually_ge_at_top)
hoelzl@60036
   891
    with Q show "eventually (\<lambda>x. z \<le> f x) at_top"
wenzelm@60758
   892
      by eventually_elim (metis mono bij \<open>P z\<close>)
hoelzl@60036
   893
  qed
hoelzl@60036
   894
qed
hoelzl@60036
   895
hoelzl@60036
   896
lemma filterlim_at_top_gt:
hoelzl@60036
   897
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
hoelzl@60036
   898
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)"
hoelzl@60036
   899
  by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge)
hoelzl@60036
   900
hoelzl@60036
   901
lemma filterlim_at_bot: 
hoelzl@60036
   902
  fixes f :: "'a \<Rightarrow> ('b::linorder)"
hoelzl@60036
   903
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)"
hoelzl@60036
   904
  by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_elim1)
hoelzl@60036
   905
hoelzl@60036
   906
lemma filterlim_at_bot_dense:
hoelzl@60036
   907
  fixes f :: "'a \<Rightarrow> ('b::{dense_linorder, no_bot})"
hoelzl@60036
   908
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x < Z) F)"
hoelzl@60036
   909
proof (auto simp add: filterlim_at_bot[of f F])
hoelzl@60036
   910
  fix Z :: 'b
hoelzl@60036
   911
  from lt_ex [of Z] obtain Z' where 1: "Z' < Z" ..
hoelzl@60036
   912
  assume "\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F"
hoelzl@60036
   913
  hence "eventually (\<lambda>x. f x \<le> Z') F" by auto
hoelzl@60036
   914
  thus "eventually (\<lambda>x. f x < Z) F"
hoelzl@60036
   915
    apply (rule eventually_mono[rotated])
hoelzl@60036
   916
    using 1 by auto
hoelzl@60036
   917
  next 
hoelzl@60036
   918
    fix Z :: 'b 
hoelzl@60036
   919
    show "\<forall>Z. eventually (\<lambda>x. f x < Z) F \<Longrightarrow> eventually (\<lambda>x. f x \<le> Z) F"
hoelzl@60036
   920
      by (drule spec [of _ Z], erule eventually_mono[rotated], auto simp add: less_imp_le)
hoelzl@60036
   921
qed
hoelzl@60036
   922
hoelzl@60036
   923
lemma filterlim_at_bot_le:
hoelzl@60036
   924
  fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
hoelzl@60036
   925
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)"
hoelzl@60036
   926
  unfolding filterlim_at_bot
hoelzl@60036
   927
proof safe
hoelzl@60036
   928
  fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F"
hoelzl@60036
   929
  with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F"
hoelzl@60036
   930
    by (auto elim!: eventually_elim1)
hoelzl@60036
   931
qed simp
hoelzl@60036
   932
hoelzl@60036
   933
lemma filterlim_at_bot_lt:
hoelzl@60036
   934
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
hoelzl@60036
   935
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)"
hoelzl@60036
   936
  by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans)
hoelzl@60036
   937
hoelzl@60036
   938
wenzelm@60758
   939
subsection \<open>Setup @{typ "'a filter"} for lifting and transfer\<close>
hoelzl@60036
   940
hoelzl@60036
   941
context begin interpretation lifting_syntax .
hoelzl@60036
   942
hoelzl@60036
   943
definition rel_filter :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> 'b filter \<Rightarrow> bool"
hoelzl@60036
   944
where "rel_filter R F G = ((R ===> op =) ===> op =) (Rep_filter F) (Rep_filter G)"
hoelzl@60036
   945
hoelzl@60036
   946
lemma rel_filter_eventually:
hoelzl@60036
   947
  "rel_filter R F G \<longleftrightarrow> 
hoelzl@60036
   948
  ((R ===> op =) ===> op =) (\<lambda>P. eventually P F) (\<lambda>P. eventually P G)"
hoelzl@60036
   949
by(simp add: rel_filter_def eventually_def)
hoelzl@60036
   950
hoelzl@60036
   951
lemma filtermap_id [simp, id_simps]: "filtermap id = id"
hoelzl@60036
   952
by(simp add: fun_eq_iff id_def filtermap_ident)
hoelzl@60036
   953
hoelzl@60036
   954
lemma filtermap_id' [simp]: "filtermap (\<lambda>x. x) = (\<lambda>F. F)"
hoelzl@60036
   955
using filtermap_id unfolding id_def .
hoelzl@60036
   956
hoelzl@60036
   957
lemma Quotient_filter [quot_map]:
hoelzl@60036
   958
  assumes Q: "Quotient R Abs Rep T"
hoelzl@60036
   959
  shows "Quotient (rel_filter R) (filtermap Abs) (filtermap Rep) (rel_filter T)"
hoelzl@60036
   960
unfolding Quotient_alt_def
hoelzl@60036
   961
proof(intro conjI strip)
hoelzl@60036
   962
  from Q have *: "\<And>x y. T x y \<Longrightarrow> Abs x = y"
hoelzl@60036
   963
    unfolding Quotient_alt_def by blast
hoelzl@60036
   964
hoelzl@60036
   965
  fix F G
hoelzl@60036
   966
  assume "rel_filter T F G"
hoelzl@60036
   967
  thus "filtermap Abs F = G" unfolding filter_eq_iff
hoelzl@60036
   968
    by(auto simp add: eventually_filtermap rel_filter_eventually * rel_funI del: iffI elim!: rel_funD)
hoelzl@60036
   969
next
hoelzl@60036
   970
  from Q have *: "\<And>x. T (Rep x) x" unfolding Quotient_alt_def by blast
hoelzl@60036
   971
hoelzl@60036
   972
  fix F
hoelzl@60036
   973
  show "rel_filter T (filtermap Rep F) F" 
hoelzl@60036
   974
    by(auto elim: rel_funD intro: * intro!: ext arg_cong[where f="\<lambda>P. eventually P F"] rel_funI
hoelzl@60036
   975
            del: iffI simp add: eventually_filtermap rel_filter_eventually)
hoelzl@60036
   976
qed(auto simp add: map_fun_def o_def eventually_filtermap filter_eq_iff fun_eq_iff rel_filter_eventually
hoelzl@60036
   977
         fun_quotient[OF fun_quotient[OF Q identity_quotient] identity_quotient, unfolded Quotient_alt_def])
hoelzl@60036
   978
hoelzl@60036
   979
lemma eventually_parametric [transfer_rule]:
hoelzl@60036
   980
  "((A ===> op =) ===> rel_filter A ===> op =) eventually eventually"
hoelzl@60036
   981
by(simp add: rel_fun_def rel_filter_eventually)
hoelzl@60036
   982
hoelzl@60038
   983
lemma frequently_parametric [transfer_rule]:
hoelzl@60038
   984
  "((A ===> op =) ===> rel_filter A ===> op =) frequently frequently"
hoelzl@60038
   985
  unfolding frequently_def[abs_def] by transfer_prover
hoelzl@60038
   986
hoelzl@60036
   987
lemma rel_filter_eq [relator_eq]: "rel_filter op = = op ="
hoelzl@60036
   988
by(auto simp add: rel_filter_eventually rel_fun_eq fun_eq_iff filter_eq_iff)
hoelzl@60036
   989
hoelzl@60036
   990
lemma rel_filter_mono [relator_mono]:
hoelzl@60036
   991
  "A \<le> B \<Longrightarrow> rel_filter A \<le> rel_filter B"
hoelzl@60036
   992
unfolding rel_filter_eventually[abs_def]
hoelzl@60036
   993
by(rule le_funI)+(intro fun_mono fun_mono[THEN le_funD, THEN le_funD] order.refl)
hoelzl@60036
   994
hoelzl@60036
   995
lemma rel_filter_conversep [simp]: "rel_filter A\<inverse>\<inverse> = (rel_filter A)\<inverse>\<inverse>"
lp15@61233
   996
apply (simp add: rel_filter_eventually fun_eq_iff rel_fun_def)
lp15@61233
   997
apply (safe; metis)
lp15@61233
   998
done
hoelzl@60036
   999
hoelzl@60036
  1000
lemma is_filter_parametric_aux:
hoelzl@60036
  1001
  assumes "is_filter F"
hoelzl@60036
  1002
  assumes [transfer_rule]: "bi_total A" "bi_unique A"
hoelzl@60036
  1003
  and [transfer_rule]: "((A ===> op =) ===> op =) F G"
hoelzl@60036
  1004
  shows "is_filter G"
hoelzl@60036
  1005
proof -
hoelzl@60036
  1006
  interpret is_filter F by fact
hoelzl@60036
  1007
  show ?thesis
hoelzl@60036
  1008
  proof
hoelzl@60036
  1009
    have "F (\<lambda>_. True) = G (\<lambda>x. True)" by transfer_prover
hoelzl@60036
  1010
    thus "G (\<lambda>x. True)" by(simp add: True)
hoelzl@60036
  1011
  next
hoelzl@60036
  1012
    fix P' Q'
hoelzl@60036
  1013
    assume "G P'" "G Q'"
hoelzl@60036
  1014
    moreover
wenzelm@60758
  1015
    from bi_total_fun[OF \<open>bi_unique A\<close> bi_total_eq, unfolded bi_total_def]
hoelzl@60036
  1016
    obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast
hoelzl@60036
  1017
    have "F P = G P'" "F Q = G Q'" by transfer_prover+
hoelzl@60036
  1018
    ultimately have "F (\<lambda>x. P x \<and> Q x)" by(simp add: conj)
hoelzl@60036
  1019
    moreover have "F (\<lambda>x. P x \<and> Q x) = G (\<lambda>x. P' x \<and> Q' x)" by transfer_prover
hoelzl@60036
  1020
    ultimately show "G (\<lambda>x. P' x \<and> Q' x)" by simp
hoelzl@60036
  1021
  next
hoelzl@60036
  1022
    fix P' Q'
hoelzl@60036
  1023
    assume "\<forall>x. P' x \<longrightarrow> Q' x" "G P'"
hoelzl@60036
  1024
    moreover
wenzelm@60758
  1025
    from bi_total_fun[OF \<open>bi_unique A\<close> bi_total_eq, unfolded bi_total_def]
hoelzl@60036
  1026
    obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast
hoelzl@60036
  1027
    have "F P = G P'" by transfer_prover
hoelzl@60036
  1028
    moreover have "(\<forall>x. P x \<longrightarrow> Q x) \<longleftrightarrow> (\<forall>x. P' x \<longrightarrow> Q' x)" by transfer_prover
hoelzl@60036
  1029
    ultimately have "F Q" by(simp add: mono)
hoelzl@60036
  1030
    moreover have "F Q = G Q'" by transfer_prover
hoelzl@60036
  1031
    ultimately show "G Q'" by simp
hoelzl@60036
  1032
  qed
hoelzl@60036
  1033
qed
hoelzl@60036
  1034
hoelzl@60036
  1035
lemma is_filter_parametric [transfer_rule]:
hoelzl@60036
  1036
  "\<lbrakk> bi_total A; bi_unique A \<rbrakk>
hoelzl@60036
  1037
  \<Longrightarrow> (((A ===> op =) ===> op =) ===> op =) is_filter is_filter"
hoelzl@60036
  1038
apply(rule rel_funI)
hoelzl@60036
  1039
apply(rule iffI)
hoelzl@60036
  1040
 apply(erule (3) is_filter_parametric_aux)
hoelzl@60036
  1041
apply(erule is_filter_parametric_aux[where A="conversep A"])
lp15@61233
  1042
apply (simp_all add: rel_fun_def)
lp15@61233
  1043
apply metis
hoelzl@60036
  1044
done
hoelzl@60036
  1045
hoelzl@60036
  1046
lemma left_total_rel_filter [transfer_rule]:
hoelzl@60036
  1047
  assumes [transfer_rule]: "bi_total A" "bi_unique A"
hoelzl@60036
  1048
  shows "left_total (rel_filter A)"
hoelzl@60036
  1049
proof(rule left_totalI)
hoelzl@60036
  1050
  fix F :: "'a filter"
wenzelm@60758
  1051
  from bi_total_fun[OF bi_unique_fun[OF \<open>bi_total A\<close> bi_unique_eq] bi_total_eq]
hoelzl@60036
  1052
  obtain G where [transfer_rule]: "((A ===> op =) ===> op =) (\<lambda>P. eventually P F) G" 
hoelzl@60036
  1053
    unfolding  bi_total_def by blast
hoelzl@60036
  1054
  moreover have "is_filter (\<lambda>P. eventually P F) \<longleftrightarrow> is_filter G" by transfer_prover
hoelzl@60036
  1055
  hence "is_filter G" by(simp add: eventually_def is_filter_Rep_filter)
hoelzl@60036
  1056
  ultimately have "rel_filter A F (Abs_filter G)"
hoelzl@60036
  1057
    by(simp add: rel_filter_eventually eventually_Abs_filter)
hoelzl@60036
  1058
  thus "\<exists>G. rel_filter A F G" ..
hoelzl@60036
  1059
qed
hoelzl@60036
  1060
hoelzl@60036
  1061
lemma right_total_rel_filter [transfer_rule]:
hoelzl@60036
  1062
  "\<lbrakk> bi_total A; bi_unique A \<rbrakk> \<Longrightarrow> right_total (rel_filter A)"
hoelzl@60036
  1063
using left_total_rel_filter[of "A\<inverse>\<inverse>"] by simp
hoelzl@60036
  1064
hoelzl@60036
  1065
lemma bi_total_rel_filter [transfer_rule]:
hoelzl@60036
  1066
  assumes "bi_total A" "bi_unique A"
hoelzl@60036
  1067
  shows "bi_total (rel_filter A)"
hoelzl@60036
  1068
unfolding bi_total_alt_def using assms
hoelzl@60036
  1069
by(simp add: left_total_rel_filter right_total_rel_filter)
hoelzl@60036
  1070
hoelzl@60036
  1071
lemma left_unique_rel_filter [transfer_rule]:
hoelzl@60036
  1072
  assumes "left_unique A"
hoelzl@60036
  1073
  shows "left_unique (rel_filter A)"
hoelzl@60036
  1074
proof(rule left_uniqueI)
hoelzl@60036
  1075
  fix F F' G
hoelzl@60036
  1076
  assume [transfer_rule]: "rel_filter A F G" "rel_filter A F' G"
hoelzl@60036
  1077
  show "F = F'"
hoelzl@60036
  1078
    unfolding filter_eq_iff
hoelzl@60036
  1079
  proof
hoelzl@60036
  1080
    fix P :: "'a \<Rightarrow> bool"
hoelzl@60036
  1081
    obtain P' where [transfer_rule]: "(A ===> op =) P P'"
hoelzl@60036
  1082
      using left_total_fun[OF assms left_total_eq] unfolding left_total_def by blast
hoelzl@60036
  1083
    have "eventually P F = eventually P' G" 
hoelzl@60036
  1084
      and "eventually P F' = eventually P' G" by transfer_prover+
hoelzl@60036
  1085
    thus "eventually P F = eventually P F'" by simp
hoelzl@60036
  1086
  qed
hoelzl@60036
  1087
qed
hoelzl@60036
  1088
hoelzl@60036
  1089
lemma right_unique_rel_filter [transfer_rule]:
hoelzl@60036
  1090
  "right_unique A \<Longrightarrow> right_unique (rel_filter A)"
hoelzl@60036
  1091
using left_unique_rel_filter[of "A\<inverse>\<inverse>"] by simp
hoelzl@60036
  1092
hoelzl@60036
  1093
lemma bi_unique_rel_filter [transfer_rule]:
hoelzl@60036
  1094
  "bi_unique A \<Longrightarrow> bi_unique (rel_filter A)"
hoelzl@60036
  1095
by(simp add: bi_unique_alt_def left_unique_rel_filter right_unique_rel_filter)
hoelzl@60036
  1096
hoelzl@60036
  1097
lemma top_filter_parametric [transfer_rule]:
hoelzl@60036
  1098
  "bi_total A \<Longrightarrow> (rel_filter A) top top"
hoelzl@60036
  1099
by(simp add: rel_filter_eventually All_transfer)
hoelzl@60036
  1100
hoelzl@60036
  1101
lemma bot_filter_parametric [transfer_rule]: "(rel_filter A) bot bot"
hoelzl@60036
  1102
by(simp add: rel_filter_eventually rel_fun_def)
hoelzl@60036
  1103
hoelzl@60036
  1104
lemma sup_filter_parametric [transfer_rule]:
hoelzl@60036
  1105
  "(rel_filter A ===> rel_filter A ===> rel_filter A) sup sup"
hoelzl@60036
  1106
by(fastforce simp add: rel_filter_eventually[abs_def] eventually_sup dest: rel_funD)
hoelzl@60036
  1107
hoelzl@60036
  1108
lemma Sup_filter_parametric [transfer_rule]:
hoelzl@60036
  1109
  "(rel_set (rel_filter A) ===> rel_filter A) Sup Sup"
hoelzl@60036
  1110
proof(rule rel_funI)
hoelzl@60036
  1111
  fix S T
hoelzl@60036
  1112
  assume [transfer_rule]: "rel_set (rel_filter A) S T"
hoelzl@60036
  1113
  show "rel_filter A (Sup S) (Sup T)"
hoelzl@60036
  1114
    by(simp add: rel_filter_eventually eventually_Sup) transfer_prover
hoelzl@60036
  1115
qed
hoelzl@60036
  1116
hoelzl@60036
  1117
lemma principal_parametric [transfer_rule]:
hoelzl@60036
  1118
  "(rel_set A ===> rel_filter A) principal principal"
hoelzl@60036
  1119
proof(rule rel_funI)
hoelzl@60036
  1120
  fix S S'
hoelzl@60036
  1121
  assume [transfer_rule]: "rel_set A S S'"
hoelzl@60036
  1122
  show "rel_filter A (principal S) (principal S')"
hoelzl@60036
  1123
    by(simp add: rel_filter_eventually eventually_principal) transfer_prover
hoelzl@60036
  1124
qed
hoelzl@60036
  1125
hoelzl@60036
  1126
context
hoelzl@60036
  1127
  fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
hoelzl@60036
  1128
  assumes [transfer_rule]: "bi_unique A" 
hoelzl@60036
  1129
begin
hoelzl@60036
  1130
hoelzl@60036
  1131
lemma le_filter_parametric [transfer_rule]:
hoelzl@60036
  1132
  "(rel_filter A ===> rel_filter A ===> op =) op \<le> op \<le>"
hoelzl@60036
  1133
unfolding le_filter_def[abs_def] by transfer_prover
hoelzl@60036
  1134
hoelzl@60036
  1135
lemma less_filter_parametric [transfer_rule]:
hoelzl@60036
  1136
  "(rel_filter A ===> rel_filter A ===> op =) op < op <"
hoelzl@60036
  1137
unfolding less_filter_def[abs_def] by transfer_prover
hoelzl@60036
  1138
hoelzl@60036
  1139
context
hoelzl@60036
  1140
  assumes [transfer_rule]: "bi_total A"
hoelzl@60036
  1141
begin
hoelzl@60036
  1142
hoelzl@60036
  1143
lemma Inf_filter_parametric [transfer_rule]:
hoelzl@60036
  1144
  "(rel_set (rel_filter A) ===> rel_filter A) Inf Inf"
hoelzl@60036
  1145
unfolding Inf_filter_def[abs_def] by transfer_prover
hoelzl@60036
  1146
hoelzl@60036
  1147
lemma inf_filter_parametric [transfer_rule]:
hoelzl@60036
  1148
  "(rel_filter A ===> rel_filter A ===> rel_filter A) inf inf"
hoelzl@60036
  1149
proof(intro rel_funI)+
hoelzl@60036
  1150
  fix F F' G G'
hoelzl@60036
  1151
  assume [transfer_rule]: "rel_filter A F F'" "rel_filter A G G'"
hoelzl@60036
  1152
  have "rel_filter A (Inf {F, G}) (Inf {F', G'})" by transfer_prover
hoelzl@60036
  1153
  thus "rel_filter A (inf F G) (inf F' G')" by simp
hoelzl@60036
  1154
qed
hoelzl@60036
  1155
hoelzl@60036
  1156
end
hoelzl@60036
  1157
hoelzl@60036
  1158
end
hoelzl@60036
  1159
hoelzl@60036
  1160
end
hoelzl@60036
  1161
hoelzl@60036
  1162
end