src/HOL/Filter.thy
author paulson <lp15@cam.ac.uk>
Mon, 22 Feb 2016 14:37:56 +0000
changeset 62379 340738057c8c
parent 62378 85ed00c1fe7c
child 63343 fb5d8a50c641
permissions -rw-r--r--
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
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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
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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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  "\<lbrakk>eventually P F; \<And>x. P x \<Longrightarrow> Q x\<rbrakk> \<Longrightarrow> eventually Q F"
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  unfolding eventually_def
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  by (blast intro: 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 -
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  have "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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  then show ?thesis
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    by (blast intro: eventually_mono)
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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_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_mono)
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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_mono)
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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_mono)
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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
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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 facts =
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    CONTEXT_SUBGOAL (fn (goal, i) => fn (ctxt, st) =>
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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 CONTEXT_CASES cases (resolve_tac ctxt [raw_elim_thm] i) (ctxt, st) end)
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\<close>
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method_setup eventually_elim = \<open>
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  Scan.succeed (fn _ => CONTEXT_METHOD (fn facts => eventually_elim_tac facts 1))
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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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   274
  "sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"
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   275
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definition
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   277
  "inf F F' = Abs_filter
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   278
      (\<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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   279
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definition
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   281
  "Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)"
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   282
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   283
definition
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   284
  "Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}"
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   285
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   286
lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
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   287
  unfolding top_filter_def
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   288
  by (rule eventually_Abs_filter, rule is_filter.intro, auto)
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   289
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   290
lemma eventually_bot [simp]: "eventually P bot"
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   291
  unfolding bot_filter_def
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   292
  by (subst eventually_Abs_filter, rule is_filter.intro, auto)
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   293
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   294
lemma eventually_sup:
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   295
  "eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'"
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   296
  unfolding sup_filter_def
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parents:
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   297
  by (rule eventually_Abs_filter, rule is_filter.intro)
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   298
     (auto elim!: eventually_rev_mp)
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   299
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lemma eventually_inf:
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   301
  "eventually P (inf F F') \<longleftrightarrow>
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   302
   (\<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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   303
  unfolding inf_filter_def
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   304
  apply (rule eventually_Abs_filter, rule is_filter.intro)
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   305
  apply (fast intro: eventually_True)
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parents:
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   306
  apply clarify
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   307
  apply (intro exI conjI)
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parents:
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   308
  apply (erule (1) eventually_conj)
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   309
  apply (erule (1) eventually_conj)
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   310
  apply simp
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   311
  apply auto
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   312
  done
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   313
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   314
lemma eventually_Sup:
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   315
  "eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)"
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   316
  unfolding Sup_filter_def
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   317
  apply (rule eventually_Abs_filter, rule is_filter.intro)
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parents:
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   318
  apply (auto intro: eventually_conj elim!: eventually_rev_mp)
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parents:
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   319
  done
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   320
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   321
instance proof
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   322
  fix F F' F'' :: "'a filter" and S :: "'a filter set"
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   323
  { show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
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parents:
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   324
    by (rule less_filter_def) }
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   325
  { show "F \<le> F"
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   326
    unfolding le_filter_def by simp }
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   327
  { assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"
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   328
    unfolding le_filter_def by simp }
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   329
  { assume "F \<le> F'" and "F' \<le> F" thus "F = F'"
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   330
    unfolding le_filter_def filter_eq_iff by fast }
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   331
  { show "inf F F' \<le> F" and "inf F F' \<le> F'"
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   332
    unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
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   333
  { assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"
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parents:
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   334
    unfolding le_filter_def eventually_inf
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61806
diff changeset
   335
    by (auto intro: eventually_mono [OF eventually_conj]) }
60036
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   336
  { show "F \<le> sup F F'" and "F' \<le> sup F F'"
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   337
    unfolding le_filter_def eventually_sup by simp_all }
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   338
  { assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"
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   339
    unfolding le_filter_def eventually_sup by simp }
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   340
  { assume "F'' \<in> S" thus "Inf S \<le> F''"
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   341
    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
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   342
  { assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"
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   343
    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
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   344
  { assume "F \<in> S" thus "F \<le> Sup S"
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   345
    unfolding le_filter_def eventually_Sup by simp }
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   346
  { assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'"
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   347
    unfolding le_filter_def eventually_Sup by simp }
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   348
  { show "Inf {} = (top::'a filter)"
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   349
    by (auto simp: top_filter_def Inf_filter_def Sup_filter_def)
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   350
      (metis (full_types) top_filter_def always_eventually eventually_top) }
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   351
  { show "Sup {} = (bot::'a filter)"
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   352
    by (auto simp: bot_filter_def Sup_filter_def) }
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   353
qed
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   354
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   355
end
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   356
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   357
lemma filter_leD:
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   358
  "F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"
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parents:
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   359
  unfolding le_filter_def by simp
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   360
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   361
lemma filter_leI:
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   362
  "(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"
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parents:
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   363
  unfolding le_filter_def by simp
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diff changeset
   364
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   365
lemma eventually_False:
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   366
  "eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot"
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   367
  unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
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parents:
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   368
60040
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   369
lemma eventually_frequently: "F \<noteq> bot \<Longrightarrow> eventually P F \<Longrightarrow> frequently P F"
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   370
  using eventually_conj[of P F "\<lambda>x. \<not> P x"]
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   371
  by (auto simp add: frequently_def eventually_False)
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   372
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
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   373
lemma eventually_const_iff: "eventually (\<lambda>x. P) F \<longleftrightarrow> P \<or> F = bot"
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   374
  by (cases P) (auto simp: eventually_False)
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   375
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   376
lemma eventually_const[simp]: "F \<noteq> bot \<Longrightarrow> eventually (\<lambda>x. P) F \<longleftrightarrow> P"
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   377
  by (simp add: eventually_const_iff)
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   378
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   379
lemma frequently_const_iff: "frequently (\<lambda>x. P) F \<longleftrightarrow> P \<and> F \<noteq> bot"
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   380
  by (simp add: frequently_def eventually_const_iff)
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   381
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   382
lemma frequently_const[simp]: "F \<noteq> bot \<Longrightarrow> frequently (\<lambda>x. P) F \<longleftrightarrow> P"
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   383
  by (simp add: frequently_const_iff)
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   384
61245
b77bf45efe21 prove Liminf_inverse_ereal
hoelzl
parents: 61233
diff changeset
   385
lemma eventually_happens: "eventually P net \<Longrightarrow> net = bot \<or> (\<exists>x. P x)"
b77bf45efe21 prove Liminf_inverse_ereal
hoelzl
parents: 61233
diff changeset
   386
  by (metis frequentlyE eventually_frequently)
b77bf45efe21 prove Liminf_inverse_ereal
hoelzl
parents: 61233
diff changeset
   387
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   388
lemma eventually_happens':
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   389
  assumes "F \<noteq> bot" "eventually P F"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   390
  shows   "\<exists>x. P x"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   391
  using assms eventually_frequently frequentlyE by blast
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   392
60036
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parents:
diff changeset
   393
abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"
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hoelzl
parents:
diff changeset
   394
  where "trivial_limit F \<equiv> F = bot"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   395
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   396
lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   397
  by (rule eventually_False [symmetric])
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hoelzl
parents:
diff changeset
   398
61806
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   399
lemma False_imp_not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> \<not> trivial_limit net \<Longrightarrow> \<not> eventually (\<lambda>x. P x) net"
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   400
  by (simp add: eventually_False)
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   401
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   402
lemma eventually_Inf: "eventually P (Inf B) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X))"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   403
proof -
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   404
  let ?F = "\<lambda>P. \<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X)"
61806
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   405
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   406
  { fix P have "eventually P (Abs_filter ?F) \<longleftrightarrow> ?F P"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   407
    proof (rule eventually_Abs_filter is_filter.intro)+
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   408
      show "?F (\<lambda>x. True)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   409
        by (rule exI[of _ "{}"]) (simp add: le_fun_def)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   410
    next
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   411
      fix P Q
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   412
      assume "?F P" then guess X ..
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   413
      moreover
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   414
      assume "?F Q" then guess Y ..
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   415
      ultimately show "?F (\<lambda>x. P x \<and> Q x)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   416
        by (intro exI[of _ "X \<union> Y"])
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   417
           (auto simp: Inf_union_distrib eventually_inf)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   418
    next
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   419
      fix P Q
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   420
      assume "?F P" then guess X ..
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   421
      moreover assume "\<forall>x. P x \<longrightarrow> Q x"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   422
      ultimately show "?F Q"
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61806
diff changeset
   423
        by (intro exI[of _ X]) (auto elim: eventually_mono)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   424
    qed }
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   425
  note eventually_F = this
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   426
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   427
  have "Inf B = Abs_filter ?F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   428
  proof (intro antisym Inf_greatest)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   429
    show "Inf B \<le> Abs_filter ?F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   430
      by (auto simp: le_filter_def eventually_F dest: Inf_superset_mono)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   431
  next
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   432
    fix F assume "F \<in> B" then show "Abs_filter ?F \<le> F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   433
      by (auto simp add: le_filter_def eventually_F intro!: exI[of _ "{F}"])
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   434
  qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   435
  then show ?thesis
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   436
    by (simp add: eventually_F)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   437
qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   438
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   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))"
62343
24106dc44def prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents: 62123
diff changeset
   440
  unfolding eventually_Inf [of P "F`B"]
24106dc44def prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents: 62123
diff changeset
   441
  by (metis finite_imageI image_mono finite_subset_image)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   442
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   443
lemma Inf_filter_not_bot:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   444
  fixes B :: "'a filter set"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   445
  shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> Inf X \<noteq> bot) \<Longrightarrow> Inf B \<noteq> bot"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   446
  unfolding trivial_limit_def eventually_Inf[of _ B]
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   447
    bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   448
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   449
lemma INF_filter_not_bot:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   450
  fixes F :: "'i \<Rightarrow> 'a filter"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   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"
62343
24106dc44def prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents: 62123
diff changeset
   452
  unfolding trivial_limit_def eventually_INF [of _ _ B]
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   453
    bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   454
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   455
lemma eventually_Inf_base:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   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"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   457
  shows "eventually P (Inf B) \<longleftrightarrow> (\<exists>b\<in>B. eventually P b)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   458
proof (subst eventually_Inf, safe)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   459
  fix X assume "finite X" "X \<subseteq> B"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   460
  then have "\<exists>b\<in>B. \<forall>x\<in>X. b \<le> x"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   461
  proof induct
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   462
    case empty then show ?case
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
   463
      using \<open>B \<noteq> {}\<close> by auto
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   464
  next
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   465
    case (insert x X)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   466
    then obtain b where "b \<in> B" "\<And>x. x \<in> X \<Longrightarrow> b \<le> x"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   467
      by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
   468
    with \<open>insert x X \<subseteq> B\<close> base[of b x] show ?case
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   469
      by (auto intro: order_trans)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   470
  qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   471
  then obtain b where "b \<in> B" "b \<le> Inf X"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   472
    by (auto simp: le_Inf_iff)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   473
  then show "eventually P (Inf X) \<Longrightarrow> Bex B (eventually P)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   474
    by (intro bexI[of _ b]) (auto simp: le_filter_def)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   475
qed (auto intro!: exI[of _ "{x}" for x])
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   476
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   477
lemma eventually_INF_base:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   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>
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   479
    eventually P (INF b:B. F b) \<longleftrightarrow> (\<exists>b\<in>B. eventually P (F b))"
62343
24106dc44def prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents: 62123
diff changeset
   480
  by (subst eventually_Inf_base) auto
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   481
62369
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62367
diff changeset
   482
lemma eventually_INF1: "i \<in> I \<Longrightarrow> eventually P (F i) \<Longrightarrow> eventually P (INF i:I. F i)"
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62367
diff changeset
   483
  using filter_leD[OF INF_lower] .
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62367
diff changeset
   484
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   485
lemma eventually_INF_mono:
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   486
  assumes *: "\<forall>\<^sub>F x in \<Sqinter>i\<in>I. F i. P x"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   487
  assumes T1: "\<And>Q R P. (\<And>x. Q x \<and> R x \<longrightarrow> P x) \<Longrightarrow> (\<And>x. T Q x \<Longrightarrow> T R x \<Longrightarrow> T P x)"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   488
  assumes T2: "\<And>P. (\<And>x. P x) \<Longrightarrow> (\<And>x. T P x)"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   489
  assumes **: "\<And>i P. i \<in> I \<Longrightarrow> \<forall>\<^sub>F x in F i. P x \<Longrightarrow> \<forall>\<^sub>F x in F' i. T P x"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   490
  shows "\<forall>\<^sub>F x in \<Sqinter>i\<in>I. F' i. T P x"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   491
proof -
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   492
  from * obtain X where "finite X" "X \<subseteq> I" "\<forall>\<^sub>F x in \<Sqinter>i\<in>X. F i. P x"
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62369
diff changeset
   493
    unfolding eventually_INF[of _ _ I] by auto
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   494
  moreover then have "eventually (T P) (INFIMUM X F')"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   495
    apply (induction X arbitrary: P)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   496
    apply (auto simp: eventually_inf T2)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   497
    subgoal for x S P Q R
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   498
      apply (intro exI[of _ "T Q"])
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   499
      apply (auto intro!: **) []
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   500
      apply (intro exI[of _ "T R"])
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   501
      apply (auto intro: T1) []
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   502
      done
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   503
    done
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   504
  ultimately show "\<forall>\<^sub>F x in \<Sqinter>i\<in>I. F' i. T P x"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   505
    by (subst eventually_INF) auto
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   506
qed
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   507
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   508
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
   509
subsubsection \<open>Map function for filters\<close>
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   510
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   511
definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   512
  where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   513
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   514
lemma eventually_filtermap:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   515
  "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   516
  unfolding filtermap_def
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   517
  apply (rule eventually_Abs_filter)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   518
  apply (rule is_filter.intro)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   519
  apply (auto elim!: eventually_rev_mp)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   520
  done
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   521
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   522
lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   523
  by (simp add: filter_eq_iff eventually_filtermap)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   524
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   525
lemma filtermap_filtermap:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   526
  "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   527
  by (simp add: filter_eq_iff eventually_filtermap)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   528
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   529
lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   530
  unfolding le_filter_def eventually_filtermap by simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   531
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   532
lemma filtermap_bot [simp]: "filtermap f bot = bot"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   533
  by (simp add: filter_eq_iff eventually_filtermap)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   534
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   535
lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   536
  by (auto simp: filter_eq_iff eventually_filtermap eventually_sup)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   537
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   538
lemma filtermap_inf: "filtermap f (inf F1 F2) \<le> inf (filtermap f F1) (filtermap f F2)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   539
  by (auto simp: le_filter_def eventually_filtermap eventually_inf)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   540
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   541
lemma filtermap_INF: "filtermap f (INF b:B. F b) \<le> (INF b:B. filtermap f (F b))"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   542
proof -
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   543
  { fix X :: "'c set" assume "finite X"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   544
    then have "filtermap f (INFIMUM X F) \<le> (INF b:X. filtermap f (F b))"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   545
    proof induct
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   546
      case (insert x X)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   547
      have "filtermap f (INF a:insert x X. F a) \<le> inf (filtermap f (F x)) (filtermap f (INF a:X. F a))"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   548
        by (rule order_trans[OF _ filtermap_inf]) simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   549
      also have "\<dots> \<le> inf (filtermap f (F x)) (INF a:X. filtermap f (F a))"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   550
        by (intro inf_mono insert order_refl)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   551
      finally show ?case
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   552
        by simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   553
    qed simp }
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   554
  then show ?thesis
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   555
    unfolding le_filter_def eventually_filtermap
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   556
    by (subst (1 2) eventually_INF) auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   557
qed
62101
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   558
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
   559
subsubsection \<open>Standard filters\<close>
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   560
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   561
definition principal :: "'a set \<Rightarrow> 'a filter" where
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   562
  "principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   563
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   564
lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   565
  unfolding principal_def
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   566
  by (rule eventually_Abs_filter, rule is_filter.intro) auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   567
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   568
lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F"
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61806
diff changeset
   569
  unfolding eventually_inf eventually_principal by (auto elim: eventually_mono)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   570
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   571
lemma principal_UNIV[simp]: "principal UNIV = top"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   572
  by (auto simp: filter_eq_iff eventually_principal)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   573
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   574
lemma principal_empty[simp]: "principal {} = bot"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   575
  by (auto simp: filter_eq_iff eventually_principal)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   576
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   577
lemma principal_eq_bot_iff: "principal X = bot \<longleftrightarrow> X = {}"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   578
  by (auto simp add: filter_eq_iff eventually_principal)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   579
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   580
lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   581
  by (auto simp: le_filter_def eventually_principal)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   582
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   583
lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   584
  unfolding le_filter_def eventually_principal
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   585
  apply safe
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   586
  apply (erule_tac x="\<lambda>x. x \<in> A" in allE)
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61806
diff changeset
   587
  apply (auto elim: eventually_mono)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   588
  done
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   589
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   590
lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   591
  unfolding eq_iff by simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   592
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   593
lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   594
  unfolding filter_eq_iff eventually_sup eventually_principal by auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   595
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   596
lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   597
  unfolding filter_eq_iff eventually_inf eventually_principal
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   598
  by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"])
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   599
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   600
lemma SUP_principal[simp]: "(SUP i : I. principal (A i)) = principal (\<Union>i\<in>I. A i)"
62343
24106dc44def prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents: 62123
diff changeset
   601
  unfolding filter_eq_iff eventually_Sup by (auto simp: eventually_principal)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   602
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   603
lemma INF_principal_finite: "finite X \<Longrightarrow> (INF x:X. principal (f x)) = principal (\<Inter>x\<in>X. f x)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   604
  by (induct X rule: finite_induct) auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   605
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   606
lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   607
  unfolding filter_eq_iff eventually_filtermap eventually_principal by simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   608
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
   609
subsubsection \<open>Order filters\<close>
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   610
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   611
definition at_top :: "('a::order) filter"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   612
  where "at_top = (INF k. principal {k ..})"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   613
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   614
lemma at_top_sub: "at_top = (INF k:{c::'a::linorder..}. principal {k ..})"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   615
  by (auto intro!: INF_eq max.cobounded1 max.cobounded2 simp: at_top_def)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   616
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   617
lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   618
  unfolding at_top_def
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   619
  by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   620
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   621
lemma eventually_ge_at_top:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   622
  "eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   623
  unfolding eventually_at_top_linorder by auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   624
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   625
lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::{no_top, linorder}. \<forall>n>N. P n)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   626
proof -
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   627
  have "eventually P (INF k. principal {k <..}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n>N. P n)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   628
    by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   629
  also have "(INF k. principal {k::'a <..}) = at_top"
61806
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   630
    unfolding at_top_def
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   631
    by (intro INF_eq) (auto intro: less_imp_le simp: Ici_subset_Ioi_iff gt_ex)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   632
  finally show ?thesis .
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   633
qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   634
60721
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
   635
lemma eventually_at_top_not_equal: "eventually (\<lambda>x::'a::{no_top, linorder}. x \<noteq> c) at_top"
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
   636
  unfolding eventually_at_top_dense by auto
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
   637
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
   638
lemma eventually_gt_at_top: "eventually (\<lambda>x. (c::_::{no_top, linorder}) < x) at_top"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   639
  unfolding eventually_at_top_dense by auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   640
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   641
lemma eventually_all_ge_at_top:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   642
  assumes "eventually P (at_top :: ('a :: linorder) filter)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   643
  shows   "eventually (\<lambda>x. \<forall>y\<ge>x. P y) at_top"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   644
proof -
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   645
  from assms obtain x where "\<And>y. y \<ge> x \<Longrightarrow> P y" by (auto simp: eventually_at_top_linorder)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   646
  hence "\<forall>z\<ge>y. P z" if "y \<ge> x" for y using that by simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   647
  thus ?thesis by (auto simp: eventually_at_top_linorder)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   648
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   649
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   650
definition at_bot :: "('a::order) filter"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   651
  where "at_bot = (INF k. principal {.. k})"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   652
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   653
lemma at_bot_sub: "at_bot = (INF k:{.. c::'a::linorder}. principal {.. k})"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   654
  by (auto intro!: INF_eq min.cobounded1 min.cobounded2 simp: at_bot_def)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   655
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   656
lemma eventually_at_bot_linorder:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   657
  fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   658
  unfolding at_bot_def
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   659
  by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   660
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   661
lemma eventually_le_at_bot:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   662
  "eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   663
  unfolding eventually_at_bot_linorder by auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   664
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   665
lemma eventually_at_bot_dense: "eventually P at_bot \<longleftrightarrow> (\<exists>N::'a::{no_bot, linorder}. \<forall>n<N. P n)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   666
proof -
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   667
  have "eventually P (INF k. principal {..< k}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n<N. P n)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   668
    by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   669
  also have "(INF k. principal {..< k::'a}) = at_bot"
61806
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   670
    unfolding at_bot_def
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   671
    by (intro INF_eq) (auto intro: less_imp_le simp: Iic_subset_Iio_iff lt_ex)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   672
  finally show ?thesis .
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   673
qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   674
60721
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
   675
lemma eventually_at_bot_not_equal: "eventually (\<lambda>x::'a::{no_bot, linorder}. x \<noteq> c) at_bot"
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
   676
  unfolding eventually_at_bot_dense by auto
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
   677
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   678
lemma eventually_gt_at_bot:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   679
  "eventually (\<lambda>x. x < (c::_::unbounded_dense_linorder)) at_bot"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   680
  unfolding eventually_at_bot_dense by auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   681
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   682
lemma trivial_limit_at_bot_linorder: "\<not> trivial_limit (at_bot ::('a::linorder) filter)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   683
  unfolding trivial_limit_def
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   684
  by (metis eventually_at_bot_linorder order_refl)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   685
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   686
lemma trivial_limit_at_top_linorder: "\<not> trivial_limit (at_top ::('a::linorder) filter)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   687
  unfolding trivial_limit_def
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   688
  by (metis eventually_at_top_linorder order_refl)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   689
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
   690
subsection \<open>Sequentially\<close>
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   691
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   692
abbreviation sequentially :: "nat filter"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   693
  where "sequentially \<equiv> at_top"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   694
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   695
lemma eventually_sequentially:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   696
  "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   697
  by (rule eventually_at_top_linorder)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   698
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   699
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   700
  unfolding filter_eq_iff eventually_sequentially by auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   701
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   702
lemmas trivial_limit_sequentially = sequentially_bot
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   703
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   704
lemma eventually_False_sequentially [simp]:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   705
  "\<not> eventually (\<lambda>n. False) sequentially"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   706
  by (simp add: eventually_False)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   707
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   708
lemma le_sequentially:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   709
  "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   710
  by (simp add: at_top_def le_INF_iff le_principal)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   711
60974
6a6f15d8fbc4 New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents: 60758
diff changeset
   712
lemma eventually_sequentiallyI [intro?]:
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   713
  assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   714
  shows "eventually P sequentially"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   715
using assms by (auto simp: eventually_sequentially)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   716
60040
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   717
lemma eventually_sequentially_Suc: "eventually (\<lambda>i. P (Suc i)) sequentially \<longleftrightarrow> eventually P sequentially"
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   718
  unfolding eventually_sequentially by (metis Suc_le_D Suc_le_mono le_Suc_eq)
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   719
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   720
lemma eventually_sequentially_seg: "eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially"
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   721
  using eventually_sequentially_Suc[of "\<lambda>n. P (n + k)" for k] by (induction k) auto
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   722
61955
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61953
diff changeset
   723
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61953
diff changeset
   724
subsection \<open>The cofinite filter\<close>
60039
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   725
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   726
definition "cofinite = Abs_filter (\<lambda>P. finite {x. \<not> P x})"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   727
61955
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61953
diff changeset
   728
abbreviation Inf_many :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "\<exists>\<^sub>\<infinity>" 10)
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61953
diff changeset
   729
  where "Inf_many P \<equiv> frequently P cofinite"
60040
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   730
61955
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61953
diff changeset
   731
abbreviation Alm_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "\<forall>\<^sub>\<infinity>" 10)
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61953
diff changeset
   732
  where "Alm_all P \<equiv> eventually P cofinite"
60040
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   733
61955
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61953
diff changeset
   734
notation (ASCII)
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61953
diff changeset
   735
  Inf_many  (binder "INFM " 10) and
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61953
diff changeset
   736
  Alm_all  (binder "MOST " 10)
60040
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   737
60039
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   738
lemma eventually_cofinite: "eventually P cofinite \<longleftrightarrow> finite {x. \<not> P x}"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   739
  unfolding cofinite_def
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   740
proof (rule eventually_Abs_filter, rule is_filter.intro)
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   741
  fix P Q :: "'a \<Rightarrow> bool" assume "finite {x. \<not> P x}" "finite {x. \<not> Q x}"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   742
  from finite_UnI[OF this] show "finite {x. \<not> (P x \<and> Q x)}"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   743
    by (rule rev_finite_subset) auto
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   744
next
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   745
  fix P Q :: "'a \<Rightarrow> bool" assume P: "finite {x. \<not> P x}" and *: "\<forall>x. P x \<longrightarrow> Q x"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   746
  from * show "finite {x. \<not> Q x}"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   747
    by (intro finite_subset[OF _ P]) auto
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   748
qed simp
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   749
60040
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   750
lemma frequently_cofinite: "frequently P cofinite \<longleftrightarrow> \<not> finite {x. P x}"
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   751
  by (simp add: frequently_def eventually_cofinite)
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   752
60039
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   753
lemma cofinite_bot[simp]: "cofinite = (bot::'a filter) \<longleftrightarrow> finite (UNIV :: 'a set)"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   754
  unfolding trivial_limit_def eventually_cofinite by simp
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   755
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   756
lemma cofinite_eq_sequentially: "cofinite = sequentially"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   757
  unfolding filter_eq_iff eventually_sequentially eventually_cofinite
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   758
proof safe
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   759
  fix P :: "nat \<Rightarrow> bool" assume [simp]: "finite {x. \<not> P x}"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   760
  show "\<exists>N. \<forall>n\<ge>N. P n"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   761
  proof cases
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   762
    assume "{x. \<not> P x} \<noteq> {}" then show ?thesis
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   763
      by (intro exI[of _ "Suc (Max {x. \<not> P x})"]) (auto simp: Suc_le_eq)
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   764
  qed auto
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   765
next
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   766
  fix P :: "nat \<Rightarrow> bool" and N :: nat assume "\<forall>n\<ge>N. P n"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   767
  then have "{x. \<not> P x} \<subseteq> {..< N}"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   768
    by (auto simp: not_le)
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   769
  then show "finite {x. \<not> P x}"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   770
    by (blast intro: finite_subset)
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   771
qed
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   772
62101
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   773
subsubsection \<open>Product of filters\<close>
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   774
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   775
lemma filtermap_sequentually_ne_bot: "filtermap f sequentially \<noteq> bot"
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   776
  by (auto simp add: filter_eq_iff eventually_filtermap eventually_sequentially)
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   777
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   778
definition prod_filter :: "'a filter \<Rightarrow> 'b filter \<Rightarrow> ('a \<times> 'b) filter" (infixr "\<times>\<^sub>F" 80) where
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   779
  "prod_filter F G =
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   780
    (INF (P, Q):{(P, Q). eventually P F \<and> eventually Q G}. principal {(x, y). P x \<and> Q y})"
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   781
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   782
lemma eventually_prod_filter: "eventually P (F \<times>\<^sub>F G) \<longleftrightarrow>
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   783
  (\<exists>Pf Pg. eventually Pf F \<and> eventually Pg G \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P (x, y)))"
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   784
  unfolding prod_filter_def
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   785
proof (subst eventually_INF_base, goal_cases)
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   786
  case 2
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   787
  moreover have "eventually Pf F \<Longrightarrow> eventually Qf F \<Longrightarrow> eventually Pg G \<Longrightarrow> eventually Qg G \<Longrightarrow>
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   788
    \<exists>P Q. eventually P F \<and> eventually Q G \<and>
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   789
      Collect P \<times> Collect Q \<subseteq> Collect Pf \<times> Collect Pg \<inter> Collect Qf \<times> Collect Qg" for Pf Pg Qf Qg
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   790
    by (intro conjI exI[of _ "inf Pf Qf"] exI[of _ "inf Pg Qg"])
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   791
       (auto simp: inf_fun_def eventually_conj)
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   792
  ultimately show ?case
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   793
    by auto
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   794
qed (auto simp: eventually_principal intro: eventually_True)
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   795
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   796
lemma eventually_prod1:
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   797
  assumes "B \<noteq> bot"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   798
  shows "(\<forall>\<^sub>F (x, y) in A \<times>\<^sub>F B. P x) \<longleftrightarrow> (\<forall>\<^sub>F x in A. P x)"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   799
  unfolding eventually_prod_filter
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   800
proof safe
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   801
  fix R Q assume "\<forall>\<^sub>F x in A. R x" "\<forall>\<^sub>F x in B. Q x" "\<forall>x y. R x \<longrightarrow> Q y \<longrightarrow> P x"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   802
  moreover with \<open>B \<noteq> bot\<close> obtain y where "Q y" by (auto dest: eventually_happens)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   803
  ultimately show "eventually P A"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   804
    by (force elim: eventually_mono)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   805
next
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   806
  assume "eventually P A"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   807
  then show "\<exists>Pf Pg. eventually Pf A \<and> eventually Pg B \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P x)"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   808
    by (intro exI[of _ P] exI[of _ "\<lambda>x. True"]) auto
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   809
qed
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   810
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   811
lemma eventually_prod2:
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   812
  assumes "A \<noteq> bot"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   813
  shows "(\<forall>\<^sub>F (x, y) in A \<times>\<^sub>F B. P y) \<longleftrightarrow> (\<forall>\<^sub>F y in B. P y)"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   814
  unfolding eventually_prod_filter
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   815
proof safe
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   816
  fix R Q assume "\<forall>\<^sub>F x in A. R x" "\<forall>\<^sub>F x in B. Q x" "\<forall>x y. R x \<longrightarrow> Q y \<longrightarrow> P y"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   817
  moreover with \<open>A \<noteq> bot\<close> obtain x where "R x" by (auto dest: eventually_happens)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   818
  ultimately show "eventually P B"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   819
    by (force elim: eventually_mono)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   820
next
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   821
  assume "eventually P B"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   822
  then show "\<exists>Pf Pg. eventually Pf A \<and> eventually Pg B \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P y)"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   823
    by (intro exI[of _ P] exI[of _ "\<lambda>x. True"]) auto
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   824
qed
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   825
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   826
lemma INF_filter_bot_base:
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   827
  fixes F :: "'a \<Rightarrow> 'b filter"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   828
  assumes *: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> \<exists>k\<in>I. F k \<le> F i \<sqinter> F j"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   829
  shows "(INF i:I. F i) = bot \<longleftrightarrow> (\<exists>i\<in>I. F i = bot)"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   830
proof cases
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   831
  assume "\<exists>i\<in>I. F i = bot"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   832
  moreover then have "(INF i:I. F i) \<le> bot"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   833
    by (auto intro: INF_lower2)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   834
  ultimately show ?thesis
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   835
    by (auto simp: bot_unique)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   836
next
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   837
  assume **: "\<not> (\<exists>i\<in>I. F i = bot)"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   838
  moreover have "(INF i:I. F i) \<noteq> bot"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   839
  proof cases
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   840
    assume "I \<noteq> {}"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   841
    show ?thesis
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   842
    proof (rule INF_filter_not_bot)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   843
      fix J assume "finite J" "J \<subseteq> I"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   844
      then have "\<exists>k\<in>I. F k \<le> (\<Sqinter>i\<in>J. F i)"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   845
      proof (induction J)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   846
        case empty then show ?case
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   847
          using \<open>I \<noteq> {}\<close> by auto
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   848
      next
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   849
        case (insert i J)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   850
        moreover then obtain k where "k \<in> I" "F k \<le> (\<Sqinter>i\<in>J. F i)" by auto
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   851
        moreover note *[of i k]
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   852
        ultimately show ?case
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   853
          by auto
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   854
      qed
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   855
      with ** show "(\<Sqinter>i\<in>J. F i) \<noteq> \<bottom>"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   856
        by (auto simp: bot_unique)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   857
    qed
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   858
  qed (auto simp add: filter_eq_iff)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   859
  ultimately show ?thesis
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   860
    by auto
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   861
qed
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   862
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   863
lemma Collect_empty_eq_bot: "Collect P = {} \<longleftrightarrow> P = \<bottom>"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   864
  by auto
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   865
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   866
lemma prod_filter_eq_bot: "A \<times>\<^sub>F B = bot \<longleftrightarrow> A = bot \<or> B = bot"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   867
  unfolding prod_filter_def
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   868
proof (subst INF_filter_bot_base; clarsimp simp: principal_eq_bot_iff Collect_empty_eq_bot bot_fun_def simp del: Collect_empty_eq)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   869
  fix A1 A2 B1 B2 assume "\<forall>\<^sub>F x in A. A1 x" "\<forall>\<^sub>F x in A. A2 x" "\<forall>\<^sub>F x in B. B1 x" "\<forall>\<^sub>F x in B. B2 x"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   870
  then show "\<exists>x. eventually x A \<and> (\<exists>y. eventually y B \<and> Collect x \<times> Collect y \<subseteq> Collect A1 \<times> Collect B1 \<and> Collect x \<times> Collect y \<subseteq> Collect A2 \<times> Collect B2)"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   871
    by (intro exI[of _ "\<lambda>x. A1 x \<and> A2 x"] exI[of _ "\<lambda>x. B1 x \<and> B2 x"] conjI)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   872
       (auto simp: eventually_conj_iff)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   873
next
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   874
  show "(\<exists>x. eventually x A \<and> (\<exists>y. eventually y B \<and> (x = (\<lambda>x. False) \<or> y = (\<lambda>x. False)))) = (A = \<bottom> \<or> B = \<bottom>)"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   875
    by (auto simp: trivial_limit_def intro: eventually_True)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   876
qed
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   877
62101
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   878
lemma prod_filter_mono: "F \<le> F' \<Longrightarrow> G \<le> G' \<Longrightarrow> F \<times>\<^sub>F G \<le> F' \<times>\<^sub>F G'"
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   879
  by (auto simp: le_filter_def eventually_prod_filter)
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   880
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   881
lemma prod_filter_mono_iff:
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   882
  assumes nAB: "A \<noteq> bot" "B \<noteq> bot"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   883
  shows "A \<times>\<^sub>F B \<le> C \<times>\<^sub>F D \<longleftrightarrow> A \<le> C \<and> B \<le> D"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   884
proof safe
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   885
  assume *: "A \<times>\<^sub>F B \<le> C \<times>\<^sub>F D"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   886
  moreover with assms have "A \<times>\<^sub>F B \<noteq> bot"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   887
    by (auto simp: bot_unique prod_filter_eq_bot)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   888
  ultimately have "C \<times>\<^sub>F D \<noteq> bot"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   889
    by (auto simp: bot_unique)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   890
  then have nCD: "C \<noteq> bot" "D \<noteq> bot"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   891
    by (auto simp: prod_filter_eq_bot)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   892
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   893
  show "A \<le> C"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   894
  proof (rule filter_leI)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   895
    fix P assume "eventually P C" with *[THEN filter_leD, of "\<lambda>(x, y). P x"] show "eventually P A"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   896
      using nAB nCD by (simp add: eventually_prod1 eventually_prod2)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   897
  qed
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   898
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   899
  show "B \<le> D"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   900
  proof (rule filter_leI)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   901
    fix P assume "eventually P D" with *[THEN filter_leD, of "\<lambda>(x, y). P y"] show "eventually P B"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   902
      using nAB nCD by (simp add: eventually_prod1 eventually_prod2)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   903
  qed
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   904
qed (intro prod_filter_mono)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   905
62101
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   906
lemma eventually_prod_same: "eventually P (F \<times>\<^sub>F F) \<longleftrightarrow>
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   907
    (\<exists>Q. eventually Q F \<and> (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y)))"
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   908
  unfolding eventually_prod_filter
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   909
  apply safe
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   910
  apply (rule_tac x="inf Pf Pg" in exI)
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   911
  apply (auto simp: inf_fun_def intro!: eventually_conj)
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   912
  done
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   913
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   914
lemma eventually_prod_sequentially:
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   915
  "eventually P (sequentially \<times>\<^sub>F sequentially) \<longleftrightarrow> (\<exists>N. \<forall>m \<ge> N. \<forall>n \<ge> N. P (n, m))"
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   916
  unfolding eventually_prod_same eventually_sequentially by auto
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   917
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   918
lemma principal_prod_principal: "principal A \<times>\<^sub>F principal B = principal (A \<times> B)"
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   919
  apply (simp add: filter_eq_iff eventually_prod_filter eventually_principal)
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   920
  apply safe
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   921
  apply blast
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   922
  apply (intro conjI exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"])
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   923
  apply auto
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   924
  done
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   925
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   926
lemma prod_filter_INF:
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   927
  assumes "I \<noteq> {}" "J \<noteq> {}"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   928
  shows "(INF i:I. A i) \<times>\<^sub>F (INF j:J. B j) = (INF i:I. INF j:J. A i \<times>\<^sub>F B j)"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   929
proof (safe intro!: antisym INF_greatest)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   930
  from \<open>I \<noteq> {}\<close> obtain i where "i \<in> I" by auto
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   931
  from \<open>J \<noteq> {}\<close> obtain j where "j \<in> J" by auto
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   932
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   933
  show "(\<Sqinter>i\<in>I. \<Sqinter>j\<in>J. A i \<times>\<^sub>F B j) \<le> (\<Sqinter>i\<in>I. A i) \<times>\<^sub>F (\<Sqinter>j\<in>J. B j)"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   934
    unfolding prod_filter_def
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   935
  proof (safe intro!: INF_greatest)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   936
    fix P Q assume P: "\<forall>\<^sub>F x in \<Sqinter>i\<in>I. A i. P x" and Q: "\<forall>\<^sub>F x in \<Sqinter>j\<in>J. B j. Q x"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   937
    let ?X = "(\<Sqinter>i\<in>I. \<Sqinter>j\<in>J. \<Sqinter>(P, Q)\<in>{(P, Q). (\<forall>\<^sub>F x in A i. P x) \<and> (\<forall>\<^sub>F x in B j. Q x)}. principal {(x, y). P x \<and> Q y})"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   938
    have "?X \<le> principal {x. P (fst x)} \<sqinter> principal {x. Q (snd x)}"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   939
    proof (intro inf_greatest)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   940
      have "?X \<le> (\<Sqinter>i\<in>I. \<Sqinter>P\<in>{P. eventually P (A i)}. principal {x. P (fst x)})"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   941
        by (auto intro!: INF_greatest INF_lower2[of j] INF_lower2 \<open>j\<in>J\<close> INF_lower2[of "(_, \<lambda>x. True)"])
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   942
      also have "\<dots> \<le> principal {x. P (fst x)}"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   943
        unfolding le_principal
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   944
      proof (rule eventually_INF_mono[OF P])
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   945
        fix i P assume "i \<in> I" "eventually P (A i)"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   946
        then show "\<forall>\<^sub>F x in \<Sqinter>P\<in>{P. eventually P (A i)}. principal {x. P (fst x)}. x \<in> {x. P (fst x)}"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   947
          unfolding le_principal[symmetric] by (auto intro!: INF_lower)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   948
      qed auto
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   949
      finally show "?X \<le> principal {x. P (fst x)}" .
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   950
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   951
      have "?X \<le> (\<Sqinter>i\<in>J. \<Sqinter>P\<in>{P. eventually P (B i)}. principal {x. P (snd x)})"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   952
        by (auto intro!: INF_greatest INF_lower2[of i] INF_lower2 \<open>i\<in>I\<close> INF_lower2[of "(\<lambda>x. True, _)"])
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   953
      also have "\<dots> \<le> principal {x. Q (snd x)}"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   954
        unfolding le_principal
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   955
      proof (rule eventually_INF_mono[OF Q])
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   956
        fix j Q assume "j \<in> J" "eventually Q (B j)"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   957
        then show "\<forall>\<^sub>F x in \<Sqinter>P\<in>{P. eventually P (B j)}. principal {x. P (snd x)}. x \<in> {x. Q (snd x)}"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   958
          unfolding le_principal[symmetric] by (auto intro!: INF_lower)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   959
      qed auto
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   960
      finally show "?X \<le> principal {x. Q (snd x)}" .
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   961
    qed
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   962
    also have "\<dots> = principal {(x, y). P x \<and> Q y}"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   963
      by auto
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   964
    finally show "?X \<le> principal {(x, y). P x \<and> Q y}" .
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   965
  qed
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   966
qed (intro prod_filter_mono INF_lower)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   967
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   968
lemma filtermap_Pair: "filtermap (\<lambda>x. (f x, g x)) F \<le> filtermap f F \<times>\<^sub>F filtermap g F"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   969
  by (simp add: le_filter_def eventually_filtermap eventually_prod_filter)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   970
     (auto elim: eventually_elim2)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   971
62369
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62367
diff changeset
   972
lemma eventually_prodI: "eventually P F \<Longrightarrow> eventually Q G \<Longrightarrow> eventually (\<lambda>x. P (fst x) \<and> Q (snd x)) (F \<times>\<^sub>F G)"
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62367
diff changeset
   973
  unfolding prod_filter_def
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62367
diff changeset
   974
  by (intro eventually_INF1[of "(P, Q)"]) (auto simp: eventually_principal)
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62367
diff changeset
   975
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62367
diff changeset
   976
lemma prod_filter_INF1: "I \<noteq> {} \<Longrightarrow> (INF i:I. A i) \<times>\<^sub>F B = (INF i:I. A i \<times>\<^sub>F B)"
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62367
diff changeset
   977
  using prod_filter_INF[of I "{B}" A "\<lambda>x. x"] by simp
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62367
diff changeset
   978
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62367
diff changeset
   979
lemma prod_filter_INF2: "J \<noteq> {} \<Longrightarrow> A \<times>\<^sub>F (INF i:J. B i) = (INF i:J. A \<times>\<^sub>F B i)"
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62367
diff changeset
   980
  using prod_filter_INF[of "{A}" J "\<lambda>x. x" B] by simp
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62367
diff changeset
   981
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
   982
subsection \<open>Limits\<close>
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   983
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   984
definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   985
  "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   986
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   987
syntax
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   988
  "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   989
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   990
translations
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
   991
  "LIM x F1. f :> F2" == "CONST filterlim (\<lambda>x. f) F2 F1"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   992
62379
340738057c8c An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents: 62378
diff changeset
   993
lemma filterlim_top [simp]: "filterlim f top F"
340738057c8c An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents: 62378
diff changeset
   994
  by (simp add: filterlim_def)
340738057c8c An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents: 62378
diff changeset
   995
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   996
lemma filterlim_iff:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   997
  "(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   998
  unfolding filterlim_def le_filter_def eventually_filtermap ..
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   999
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1000
lemma filterlim_compose:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1001
  "filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1002
  unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1003
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1004
lemma filterlim_mono:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1005
  "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1006
  unfolding filterlim_def by (metis filtermap_mono order_trans)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1007
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1008
lemma filterlim_ident: "LIM x F. x :> F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1009
  by (simp add: filterlim_def filtermap_ident)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1010
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1011
lemma filterlim_cong:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1012
  "F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1013
  by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1014
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1015
lemma filterlim_mono_eventually:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1016
  assumes "filterlim f F G" and ord: "F \<le> F'" "G' \<le> G"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1017
  assumes eq: "eventually (\<lambda>x. f x = f' x) G'"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1018
  shows "filterlim f' F' G'"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1019
  apply (rule filterlim_cong[OF refl refl eq, THEN iffD1])
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1020
  apply (rule filterlim_mono[OF _ ord])
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1021
  apply fact
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1022
  done
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1023
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1024
lemma filtermap_mono_strong: "inj f \<Longrightarrow> filtermap f F \<le> filtermap f G \<longleftrightarrow> F \<le> G"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1025
  apply (auto intro!: filtermap_mono) []
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1026
  apply (auto simp: le_filter_def eventually_filtermap)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1027
  apply (erule_tac x="\<lambda>x. P (inv f x)" in allE)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1028
  apply auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1029
  done
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1030
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1031
lemma filtermap_eq_strong: "inj f \<Longrightarrow> filtermap f F = filtermap f G \<longleftrightarrow> F = G"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1032
  by (simp add: filtermap_mono_strong eq_iff)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1033
60721
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
  1034
lemma filtermap_fun_inverse:
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
  1035
  assumes g: "filterlim g F G"
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
  1036
  assumes f: "filterlim f G F"
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
  1037
  assumes ev: "eventually (\<lambda>x. f (g x) = x) G"
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
  1038
  shows "filtermap f F = G"
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
  1039
proof (rule antisym)
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
  1040
  show "filtermap f F \<le> G"
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
  1041
    using f unfolding filterlim_def .
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
  1042
  have "G = filtermap f (filtermap g G)"
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
  1043
    using ev by (auto elim: eventually_elim2 simp: filter_eq_iff eventually_filtermap)
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
  1044
  also have "\<dots> \<le> filtermap f F"
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
  1045
    using g by (intro filtermap_mono) (simp add: filterlim_def)
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
  1046
  finally show "G \<le> filtermap f F" .
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
  1047
qed
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
  1048
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1049
lemma filterlim_principal:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1050
  "(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1051
  unfolding filterlim_def eventually_filtermap le_principal ..
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1052
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1053
lemma filterlim_inf:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1054
  "(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1055
  unfolding filterlim_def by simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1056
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1057
lemma filterlim_INF:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1058
  "(LIM x F. f x :> (INF b:B. G b)) \<longleftrightarrow> (\<forall>b\<in>B. LIM x F. f x :> G b)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1059
  unfolding filterlim_def le_INF_iff ..
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1060
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1061
lemma filterlim_INF_INF:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1062
  "(\<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)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1063
  unfolding filterlim_def by (rule order_trans[OF filtermap_INF INF_mono])
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1064
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1065
lemma filterlim_base:
61806
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
  1066
  "(\<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>
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1067
    LIM x (INF i:I. principal (F i)). f x :> (INF j:J. principal (G j))"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1068
  by (force intro!: filterlim_INF_INF simp: image_subset_iff)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1069
61806
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
  1070
lemma filterlim_base_iff:
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1071
  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"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1072
  shows "(LIM x (INF i:I. principal (F i)). f x :> INF j:J. principal (G j)) \<longleftrightarrow>
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1073
    (\<forall>j\<in>J. \<exists>i\<in>I. \<forall>x\<in>F i. f x \<in> G j)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1074
  unfolding filterlim_INF filterlim_principal
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1075
proof (subst eventually_INF_base)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1076
  fix i j assume "i \<in> I" "j \<in> I"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1077
  with chain[OF this] show "\<exists>x\<in>I. principal (F x) \<le> inf (principal (F i)) (principal (F j))"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1078
    by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
  1079
qed (auto simp: eventually_principal \<open>I \<noteq> {}\<close>)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1080
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1081
lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1082
  unfolding filterlim_def filtermap_filtermap ..
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1083
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1084
lemma filterlim_sup:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1085
  "filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1086
  unfolding filterlim_def filtermap_sup by auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1087
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1088
lemma filterlim_sequentially_Suc:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1089
  "(LIM x sequentially. f (Suc x) :> F) \<longleftrightarrow> (LIM x sequentially. f x :> F)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1090
  unfolding filterlim_iff by (subst eventually_sequentially_Suc) simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1091
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1092
lemma filterlim_Suc: "filterlim Suc sequentially sequentially"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1093
  by (simp add: filterlim_iff eventually_sequentially) (metis le_Suc_eq)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1094
60182
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60040
diff changeset
  1095
lemma filterlim_If:
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60040
diff changeset
  1096
  "LIM x inf F (principal {x. P x}). f x :> G \<Longrightarrow>
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60040
diff changeset
  1097
    LIM x inf F (principal {x. \<not> P x}). g x :> G \<Longrightarrow>
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60040
diff changeset
  1098
    LIM x F. if P x then f x else g x :> G"
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60040
diff changeset
  1099
  unfolding filterlim_iff eventually_inf_principal by (auto simp: eventually_conj_iff)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1100
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1101
lemma filterlim_Pair:
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1102
  "LIM x F. f x :> G \<Longrightarrow> LIM x F. g x :> H \<Longrightarrow> LIM x F. (f x, g x) :> G \<times>\<^sub>F H"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1103
  unfolding filterlim_def
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1104
  by (rule order_trans[OF filtermap_Pair prod_filter_mono])
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1105
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
  1106
subsection \<open>Limits to @{const at_top} and @{const at_bot}\<close>
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1107
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1108
lemma filterlim_at_top:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1109
  fixes f :: "'a \<Rightarrow> ('b::linorder)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1110
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)"
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61806
diff changeset
  1111
  by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_mono)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1112
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1113
lemma filterlim_at_top_mono:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1114
  "LIM x F. f x :> at_top \<Longrightarrow> eventually (\<lambda>x. f x \<le> (g x::'a::linorder)) F \<Longrightarrow>
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1115
    LIM x F. g x :> at_top"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1116
  by (auto simp: filterlim_at_top elim: eventually_elim2 intro: order_trans)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1117
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1118
lemma filterlim_at_top_dense:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1119
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1120
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)"
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61806
diff changeset
  1121
  by (metis eventually_mono[of _ F] eventually_gt_at_top order_less_imp_le
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1122
            filterlim_at_top[of f F] filterlim_iff[of f at_top F])
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1123
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1124
lemma filterlim_at_top_ge:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1125
  fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1126
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1127
  unfolding at_top_sub[of c] filterlim_INF by (auto simp add: filterlim_principal)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1128
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1129
lemma filterlim_at_top_at_top:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1130
  fixes f :: "'a::linorder \<Rightarrow> 'b::linorder"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1131
  assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1132
  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1133
  assumes Q: "eventually Q at_top"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1134
  assumes P: "eventually P at_top"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1135
  shows "filterlim f at_top at_top"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1136
proof -
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1137
  from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1138
    unfolding eventually_at_top_linorder by auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1139
  show ?thesis
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1140
  proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1141
    fix z assume "x \<le> z"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1142
    with x have "P z" by auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1143
    have "eventually (\<lambda>x. g z \<le> x) at_top"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1144
      by (rule eventually_ge_at_top)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1145
    with Q show "eventually (\<lambda>x. z \<le> f x) at_top"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
  1146
      by eventually_elim (metis mono bij \<open>P z\<close>)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1147
  qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1148
qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1149
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1150
lemma filterlim_at_top_gt:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1151
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1152
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1153
  by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1154
61806
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
  1155
lemma filterlim_at_bot:
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1156
  fixes f :: "'a \<Rightarrow> ('b::linorder)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1157
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)"
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61806
diff changeset
  1158
  by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_mono)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1159
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1160
lemma filterlim_at_bot_dense:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1161
  fixes f :: "'a \<Rightarrow> ('b::{dense_linorder, no_bot})"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1162
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x < Z) F)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1163
proof (auto simp add: filterlim_at_bot[of f F])
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1164
  fix Z :: 'b
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1165
  from lt_ex [of Z] obtain Z' where 1: "Z' < Z" ..
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1166
  assume "\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1167
  hence "eventually (\<lambda>x. f x \<le> Z') F" by auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1168
  thus "eventually (\<lambda>x. f x < Z) F"
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61806
diff changeset
  1169
    apply (rule eventually_mono)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1170
    using 1 by auto
61806
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
  1171
  next
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
  1172
    fix Z :: 'b
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1173
    show "\<forall>Z. eventually (\<lambda>x. f x < Z) F \<Longrightarrow> eventually (\<lambda>x. f x \<le> Z) F"
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61806
diff changeset
  1174
      by (drule spec [of _ Z], erule eventually_mono, auto simp add: less_imp_le)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1175
qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1176
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1177
lemma filterlim_at_bot_le:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1178
  fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1179
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1180
  unfolding filterlim_at_bot
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1181
proof safe
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1182
  fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1183
  with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F"
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61806
diff changeset
  1184
    by (auto elim!: eventually_mono)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1185
qed simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1186
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1187
lemma filterlim_at_bot_lt:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1188
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1189
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1190
  by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1191
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1192
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
  1193
subsection \<open>Setup @{typ "'a filter"} for lifting and transfer\<close>
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1194
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1195
context begin interpretation lifting_syntax .
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1196
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1197
definition rel_filter :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> 'b filter \<Rightarrow> bool"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1198
where "rel_filter R F G = ((R ===> op =) ===> op =) (Rep_filter F) (Rep_filter G)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1199
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1200
lemma rel_filter_eventually:
61806
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
  1201
  "rel_filter R F G \<longleftrightarrow>
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1202
  ((R ===> op =) ===> op =) (\<lambda>P. eventually P F) (\<lambda>P. eventually P G)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1203
by(simp add: rel_filter_def eventually_def)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1204
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1205
lemma filtermap_id [simp, id_simps]: "filtermap id = id"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1206
by(simp add: fun_eq_iff id_def filtermap_ident)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1207
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1208
lemma filtermap_id' [simp]: "filtermap (\<lambda>x. x) = (\<lambda>F. F)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1209
using filtermap_id unfolding id_def .
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1210
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1211
lemma Quotient_filter [quot_map]:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1212
  assumes Q: "Quotient R Abs Rep T"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1213
  shows "Quotient (rel_filter R) (filtermap Abs) (filtermap Rep) (rel_filter T)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1214
unfolding Quotient_alt_def
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1215
proof(intro conjI strip)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1216
  from Q have *: "\<And>x y. T x y \<Longrightarrow> Abs x = y"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1217
    unfolding Quotient_alt_def by blast
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1218
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1219
  fix F G
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1220
  assume "rel_filter T F G"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1221
  thus "filtermap Abs F = G" unfolding filter_eq_iff
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1222
    by(auto simp add: eventually_filtermap rel_filter_eventually * rel_funI del: iffI elim!: rel_funD)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1223
next
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1224
  from Q have *: "\<And>x. T (Rep x) x" unfolding Quotient_alt_def by blast
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1225
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1226
  fix F
61806
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
  1227
  show "rel_filter T (filtermap Rep F) F"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1228
    by(auto elim: rel_funD intro: * intro!: ext arg_cong[where f="\<lambda>P. eventually P F"] rel_funI
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1229
            del: iffI simp add: eventually_filtermap rel_filter_eventually)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1230
qed(auto simp add: map_fun_def o_def eventually_filtermap filter_eq_iff fun_eq_iff rel_filter_eventually
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1231
         fun_quotient[OF fun_quotient[OF Q identity_quotient] identity_quotient, unfolded Quotient_alt_def])
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1232
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1233
lemma eventually_parametric [transfer_rule]:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1234
  "((A ===> op =) ===> rel_filter A ===> op =) eventually eventually"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1235
by(simp add: rel_fun_def rel_filter_eventually)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1236
60038
ca431cbce2a3 add frequently as dual for eventually
hoelzl
parents: 60037
diff changeset
  1237
lemma frequently_parametric [transfer_rule]:
ca431cbce2a3 add frequently as dual for eventually
hoelzl
parents: 60037
diff changeset
  1238
  "((A ===> op =) ===> rel_filter A ===> op =) frequently frequently"
ca431cbce2a3 add frequently as dual for eventually
hoelzl
parents: 60037
diff changeset
  1239
  unfolding frequently_def[abs_def] by transfer_prover
ca431cbce2a3 add frequently as dual for eventually
hoelzl
parents: 60037
diff changeset
  1240
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1241
lemma rel_filter_eq [relator_eq]: "rel_filter op = = op ="
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1242
by(auto simp add: rel_filter_eventually rel_fun_eq fun_eq_iff filter_eq_iff)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1243
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1244
lemma rel_filter_mono [relator_mono]:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1245
  "A \<le> B \<Longrightarrow> rel_filter A \<le> rel_filter B"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1246
unfolding rel_filter_eventually[abs_def]
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1247
by(rule le_funI)+(intro fun_mono fun_mono[THEN le_funD, THEN le_funD] order.refl)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1248
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1249
lemma rel_filter_conversep [simp]: "rel_filter A\<inverse>\<inverse> = (rel_filter A)\<inverse>\<inverse>"
61233
1da01148d4b1 Prepared two non-terminating proofs; no obvious link with my changes
paulson <lp15@cam.ac.uk>
parents: 60974
diff changeset
  1250
apply (simp add: rel_filter_eventually fun_eq_iff rel_fun_def)
1da01148d4b1 Prepared two non-terminating proofs; no obvious link with my changes
paulson <lp15@cam.ac.uk>
parents: 60974
diff changeset
  1251
apply (safe; metis)
1da01148d4b1 Prepared two non-terminating proofs; no obvious link with my changes
paulson <lp15@cam.ac.uk>
parents: 60974
diff changeset
  1252
done
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1253
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1254
lemma is_filter_parametric_aux:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1255
  assumes "is_filter F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1256
  assumes [transfer_rule]: "bi_total A" "bi_unique A"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1257
  and [transfer_rule]: "((A ===> op =) ===> op =) F G"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1258
  shows "is_filter G"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1259
proof -
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1260
  interpret is_filter F by fact
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1261
  show ?thesis
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1262
  proof
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1263
    have "F (\<lambda>_. True) = G (\<lambda>x. True)" by transfer_prover
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1264
    thus "G (\<lambda>x. True)" by(simp add: True)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1265
  next
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1266
    fix P' Q'
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1267
    assume "G P'" "G Q'"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1268
    moreover
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
  1269
    from bi_total_fun[OF \<open>bi_unique A\<close> bi_total_eq, unfolded bi_total_def]
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1270
    obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1271
    have "F P = G P'" "F Q = G Q'" by transfer_prover+
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1272
    ultimately have "F (\<lambda>x. P x \<and> Q x)" by(simp add: conj)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1273
    moreover have "F (\<lambda>x. P x \<and> Q x) = G (\<lambda>x. P' x \<and> Q' x)" by transfer_prover
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1274
    ultimately show "G (\<lambda>x. P' x \<and> Q' x)" by simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1275
  next
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1276
    fix P' Q'
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1277
    assume "\<forall>x. P' x \<longrightarrow> Q' x" "G P'"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1278
    moreover
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
  1279
    from bi_total_fun[OF \<open>bi_unique A\<close> bi_total_eq, unfolded bi_total_def]
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1280
    obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1281
    have "F P = G P'" by transfer_prover
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1282
    moreover have "(\<forall>x. P x \<longrightarrow> Q x) \<longleftrightarrow> (\<forall>x. P' x \<longrightarrow> Q' x)" by transfer_prover
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1283
    ultimately have "F Q" by(simp add: mono)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1284
    moreover have "F Q = G Q'" by transfer_prover
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1285
    ultimately show "G Q'" by simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1286
  qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1287
qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1288
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1289
lemma is_filter_parametric [transfer_rule]:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1290
  "\<lbrakk> bi_total A; bi_unique A \<rbrakk>
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1291
  \<Longrightarrow> (((A ===> op =) ===> op =) ===> op =) is_filter is_filter"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1292
apply(rule rel_funI)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1293
apply(rule iffI)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1294
 apply(erule (3) is_filter_parametric_aux)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1295
apply(erule is_filter_parametric_aux[where A="conversep A"])
61233
1da01148d4b1 Prepared two non-terminating proofs; no obvious link with my changes
paulson <lp15@cam.ac.uk>
parents: 60974
diff changeset
  1296
apply (simp_all add: rel_fun_def)
1da01148d4b1 Prepared two non-terminating proofs; no obvious link with my changes
paulson <lp15@cam.ac.uk>
parents: 60974
diff changeset
  1297
apply metis
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1298
done
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1299
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1300
lemma left_total_rel_filter [transfer_rule]:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1301
  assumes [transfer_rule]: "bi_total A" "bi_unique A"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1302
  shows "left_total (rel_filter A)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1303
proof(rule left_totalI)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1304
  fix F :: "'a filter"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
  1305
  from bi_total_fun[OF bi_unique_fun[OF \<open>bi_total A\<close> bi_unique_eq] bi_total_eq]
61806
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
  1306
  obtain G where [transfer_rule]: "((A ===> op =) ===> op =) (\<lambda>P. eventually P F) G"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1307
    unfolding  bi_total_def by blast
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1308
  moreover have "is_filter (\<lambda>P. eventually P F) \<longleftrightarrow> is_filter G" by transfer_prover
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1309
  hence "is_filter G" by(simp add: eventually_def is_filter_Rep_filter)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1310
  ultimately have "rel_filter A F (Abs_filter G)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1311
    by(simp add: rel_filter_eventually eventually_Abs_filter)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1312
  thus "\<exists>G. rel_filter A F G" ..
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1313
qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1314
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1315
lemma right_total_rel_filter [transfer_rule]:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1316
  "\<lbrakk> bi_total A; bi_unique A \<rbrakk> \<Longrightarrow> right_total (rel_filter A)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1317
using left_total_rel_filter[of "A\<inverse>\<inverse>"] by simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1318
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1319
lemma bi_total_rel_filter [transfer_rule]:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1320
  assumes "bi_total A" "bi_unique A"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1321
  shows "bi_total (rel_filter A)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1322
unfolding bi_total_alt_def using assms
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1323
by(simp add: left_total_rel_filter right_total_rel_filter)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1324
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1325
lemma left_unique_rel_filter [transfer_rule]:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1326
  assumes "left_unique A"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1327
  shows "left_unique (rel_filter A)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1328
proof(rule left_uniqueI)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1329
  fix F F' G
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1330
  assume [transfer_rule]: "rel_filter A F G" "rel_filter A F' G"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1331
  show "F = F'"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1332
    unfolding filter_eq_iff
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1333
  proof
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1334
    fix P :: "'a \<Rightarrow> bool"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1335
    obtain P' where [transfer_rule]: "(A ===> op =) P P'"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1336
      using left_total_fun[OF assms left_total_eq] unfolding left_total_def by blast
61806
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
  1337
    have "eventually P F = eventually P' G"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1338
      and "eventually P F' = eventually P' G" by transfer_prover+
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1339
    thus "eventually P F = eventually P F'" by simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1340
  qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1341
qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1342
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1343
lemma right_unique_rel_filter [transfer_rule]:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1344
  "right_unique A \<Longrightarrow> right_unique (rel_filter A)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1345
using left_unique_rel_filter[of "A\<inverse>\<inverse>"] by simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1346
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1347
lemma bi_unique_rel_filter [transfer_rule]:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1348
  "bi_unique A \<Longrightarrow> bi_unique (rel_filter A)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1349
by(simp add: bi_unique_alt_def left_unique_rel_filter right_unique_rel_filter)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1350
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1351
lemma top_filter_parametric [transfer_rule]:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1352
  "bi_total A \<Longrightarrow> (rel_filter A) top top"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1353
by(simp add: rel_filter_eventually All_transfer)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1354
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1355
lemma bot_filter_parametric [transfer_rule]: "(rel_filter A) bot bot"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1356
by(simp add: rel_filter_eventually rel_fun_def)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1357
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1358
lemma sup_filter_parametric [transfer_rule]:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1359
  "(rel_filter A ===> rel_filter A ===> rel_filter A) sup sup"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1360
by(fastforce simp add: rel_filter_eventually[abs_def] eventually_sup dest: rel_funD)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1361
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1362
lemma Sup_filter_parametric [transfer_rule]:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1363
  "(rel_set (rel_filter A) ===> rel_filter A) Sup Sup"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1364
proof(rule rel_funI)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1365
  fix S T
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1366
  assume [transfer_rule]: "rel_set (rel_filter A) S T"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1367
  show "rel_filter A (Sup S) (Sup T)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1368
    by(simp add: rel_filter_eventually eventually_Sup) transfer_prover
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1369
qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1370
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1371
lemma principal_parametric [transfer_rule]:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1372
  "(rel_set A ===> rel_filter A) principal principal"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1373
proof(rule rel_funI)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1374
  fix S S'
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1375
  assume [transfer_rule]: "rel_set A S S'"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1376
  show "rel_filter A (principal S) (principal S')"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1377
    by(simp add: rel_filter_eventually eventually_principal) transfer_prover
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1378
qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1379
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1380
context
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1381
  fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
61806
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
  1382
  assumes [transfer_rule]: "bi_unique A"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1383
begin
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1384
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1385
lemma le_filter_parametric [transfer_rule]:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1386
  "(rel_filter A ===> rel_filter A ===> op =) op \<le> op \<le>"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1387
unfolding le_filter_def[abs_def] by transfer_prover
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1388
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1389
lemma less_filter_parametric [transfer_rule]:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1390
  "(rel_filter A ===> rel_filter A ===> op =) op < op <"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1391
unfolding less_filter_def[abs_def] by transfer_prover
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1392
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1393
context
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1394
  assumes [transfer_rule]: "bi_total A"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1395
begin
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1396
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1397
lemma Inf_filter_parametric [transfer_rule]:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1398
  "(rel_set (rel_filter A) ===> rel_filter A) Inf Inf"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1399
unfolding Inf_filter_def[abs_def] by transfer_prover
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1400
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1401
lemma inf_filter_parametric [transfer_rule]:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1402
  "(rel_filter A ===> rel_filter A ===> rel_filter A) inf inf"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1403
proof(intro rel_funI)+
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1404
  fix F F' G G'
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1405
  assume [transfer_rule]: "rel_filter A F F'" "rel_filter A G G'"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1406
  have "rel_filter A (Inf {F, G}) (Inf {F', G'})" by transfer_prover
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1407
  thus "rel_filter A (inf F G) (inf F' G')" by simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1408
qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1409
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1410
end
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff