src/HOL/Filter.thy
author wenzelm
Mon, 12 Apr 2021 14:14:47 +0200
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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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   219
  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_facts = facts RL @{thms eventually_rev_mp}
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        val rule =
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          @{thm eventuallyI}
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          |> fold (fn mp_fact => fn th => th RS mp_fact) mp_facts
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          |> funpow (length facts) (fn th => @{thm impI} RS th)
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        val cases_prop =
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          Thm.prop_of (Rule_Cases.internalize_params (rule 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 [rule] 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>\<open>F \<le> F'\<close> means that filter \<^term>\<open>F\<close> is finer than
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filter \<^term>\<open>F'\<close>.\<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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218fcc645d22 move filters to their own theory
hoelzl
parents:
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   272
definition
218fcc645d22 move filters to their own theory
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parents:
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   273
  "sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"
218fcc645d22 move filters to their own theory
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parents:
diff changeset
   274
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   275
definition
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   276
  "inf F F' = Abs_filter
218fcc645d22 move filters to their own theory
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parents:
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   277
      (\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   278
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   279
definition
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   280
  "Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)"
218fcc645d22 move filters to their own theory
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parents:
diff changeset
   281
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   282
definition
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   283
  "Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   284
218fcc645d22 move filters to their own theory
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parents:
diff changeset
   285
lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   286
  unfolding top_filter_def
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   287
  by (rule eventually_Abs_filter, rule is_filter.intro, auto)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   288
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   289
lemma eventually_bot [simp]: "eventually P bot"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   290
  unfolding bot_filter_def
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   291
  by (subst eventually_Abs_filter, rule is_filter.intro, auto)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   292
218fcc645d22 move filters to their own theory
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parents:
diff changeset
   293
lemma eventually_sup:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   294
  "eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   295
  unfolding sup_filter_def
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   296
  by (rule eventually_Abs_filter, rule is_filter.intro)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   297
     (auto elim!: eventually_rev_mp)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   298
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   299
lemma eventually_inf:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   300
  "eventually P (inf F F') \<longleftrightarrow>
218fcc645d22 move filters to their own theory
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parents:
diff changeset
   301
   (\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   302
  unfolding inf_filter_def
71743
0239bee6bffd more removal of applys
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   303
  apply (rule eventually_Abs_filter [OF is_filter.intro])
0239bee6bffd more removal of applys
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   304
  apply (blast intro: eventually_True)
0239bee6bffd more removal of applys
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   305
   apply (force elim!: eventually_conj)+
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   306
  done
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   307
218fcc645d22 move filters to their own theory
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parents:
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   308
lemma eventually_Sup:
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hoelzl
parents:
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   309
  "eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   310
  unfolding Sup_filter_def
71743
0239bee6bffd more removal of applys
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   311
  apply (rule eventually_Abs_filter [OF is_filter.intro])
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   312
  apply (auto intro: eventually_conj elim!: eventually_rev_mp)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   313
  done
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   314
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   315
instance proof
218fcc645d22 move filters to their own theory
hoelzl
parents:
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   316
  fix F F' F'' :: "'a filter" and S :: "'a filter set"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   317
  { show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   318
    by (rule less_filter_def) }
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   319
  { show "F \<le> F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   320
    unfolding le_filter_def by simp }
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   321
  { assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   322
    unfolding le_filter_def by simp }
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   323
  { assume "F \<le> F'" and "F' \<le> F" thus "F = F'"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   324
    unfolding le_filter_def filter_eq_iff by fast }
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   325
  { show "inf F F' \<le> F" and "inf F F' \<le> F'"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   326
    unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   327
  { assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   328
    unfolding le_filter_def eventually_inf
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61806
diff changeset
   329
    by (auto intro: eventually_mono [OF eventually_conj]) }
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   330
  { show "F \<le> sup F F'" and "F' \<le> sup F F'"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   331
    unfolding le_filter_def eventually_sup by simp_all }
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   332
  { assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   333
    unfolding le_filter_def eventually_sup by simp }
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   334
  { assume "F'' \<in> S" thus "Inf S \<le> F''"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   335
    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   336
  { assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   337
    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   338
  { assume "F \<in> S" thus "F \<le> Sup S"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   339
    unfolding le_filter_def eventually_Sup by simp }
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   340
  { assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   341
    unfolding le_filter_def eventually_Sup by simp }
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   342
  { show "Inf {} = (top::'a filter)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   343
    by (auto simp: top_filter_def Inf_filter_def Sup_filter_def)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   344
      (metis (full_types) top_filter_def always_eventually eventually_top) }
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   345
  { show "Sup {} = (bot::'a filter)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   346
    by (auto simp: bot_filter_def Sup_filter_def) }
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   347
qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   348
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   349
end
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   350
66171
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   351
instance filter :: (type) distrib_lattice
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   352
proof
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   353
  fix F G H :: "'a filter"
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   354
  show "sup F (inf G H) = inf (sup F G) (sup F H)"
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   355
  proof (rule order.antisym)
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   356
    show "inf (sup F G) (sup F H) \<le> sup F (inf G H)" 
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   357
      unfolding le_filter_def eventually_sup
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   358
    proof safe
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   359
      fix P assume 1: "eventually P F" and 2: "eventually P (inf G H)"
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   360
      from 2 obtain Q R 
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   361
        where QR: "eventually Q G" "eventually R H" "\<And>x. Q x \<Longrightarrow> R x \<Longrightarrow> P x"
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   362
        by (auto simp: eventually_inf)
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   363
      define Q' where "Q' = (\<lambda>x. Q x \<or> P x)"
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   364
      define R' where "R' = (\<lambda>x. R x \<or> P x)"
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   365
      from 1 have "eventually Q' F" 
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   366
        by (elim eventually_mono) (auto simp: Q'_def)
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   367
      moreover from 1 have "eventually R' F" 
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   368
        by (elim eventually_mono) (auto simp: R'_def)
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   369
      moreover from QR(1) have "eventually Q' G" 
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   370
        by (elim eventually_mono) (auto simp: Q'_def)
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   371
      moreover from QR(2) have "eventually R' H" 
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   372
        by (elim eventually_mono)(auto simp: R'_def)
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   373
      moreover from QR have "P x" if "Q' x" "R' x" for x 
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   374
        using that by (auto simp: Q'_def R'_def)
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   375
      ultimately show "eventually P (inf (sup F G) (sup F H))"
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   376
        by (auto simp: eventually_inf eventually_sup)
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   377
    qed
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   378
  qed (auto intro: inf.coboundedI1 inf.coboundedI2)
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   379
qed
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   380
454abfe923fe distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents: 66162
diff changeset
   381
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   382
lemma filter_leD:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   383
  "F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   384
  unfolding le_filter_def by simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   385
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   386
lemma filter_leI:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   387
  "(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   388
  unfolding le_filter_def by simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   389
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   390
lemma eventually_False:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   391
  "eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   392
  unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   393
60040
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   394
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
   395
  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
   396
  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
   397
67706
4ddc49205f5d Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents: 67616
diff changeset
   398
lemma eventually_frequentlyE:
4ddc49205f5d Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents: 67616
diff changeset
   399
  assumes "eventually P F"
4ddc49205f5d Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents: 67616
diff changeset
   400
  assumes "eventually (\<lambda>x. \<not> P x \<or> Q x) F" "F\<noteq>bot"
4ddc49205f5d Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents: 67616
diff changeset
   401
  shows "frequently Q F"
4ddc49205f5d Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents: 67616
diff changeset
   402
proof -
4ddc49205f5d Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents: 67616
diff changeset
   403
  have "eventually Q F"
4ddc49205f5d Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents: 67616
diff changeset
   404
    using eventually_conj[OF assms(1,2),simplified] by (auto elim:eventually_mono)
4ddc49205f5d Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents: 67616
diff changeset
   405
  then show ?thesis using eventually_frequently[OF \<open>F\<noteq>bot\<close>] by auto
4ddc49205f5d Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents: 67616
diff changeset
   406
qed
4ddc49205f5d Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents: 67616
diff changeset
   407
60040
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   408
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
   409
  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
   410
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   411
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
   412
  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
   413
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   414
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
   415
  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
   416
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   417
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
   418
  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
   419
61245
b77bf45efe21 prove Liminf_inverse_ereal
hoelzl
parents: 61233
diff changeset
   420
lemma eventually_happens: "eventually P net \<Longrightarrow> net = bot \<or> (\<exists>x. P x)"
b77bf45efe21 prove Liminf_inverse_ereal
hoelzl
parents: 61233
diff changeset
   421
  by (metis frequentlyE eventually_frequently)
b77bf45efe21 prove Liminf_inverse_ereal
hoelzl
parents: 61233
diff changeset
   422
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   423
lemma eventually_happens':
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   424
  assumes "F \<noteq> bot" "eventually P F"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   425
  shows   "\<exists>x. P x"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   426
  using assms eventually_frequently frequentlyE by blast
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   427
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   428
abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   429
  where "trivial_limit F \<equiv> F = bot"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   430
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   431
lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   432
  by (rule eventually_False [symmetric])
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   433
61806
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   434
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
   435
  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
   436
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   437
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
   438
proof -
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   439
  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
   440
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   441
  { fix P have "eventually P (Abs_filter ?F) \<longleftrightarrow> ?F P"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   442
    proof (rule eventually_Abs_filter is_filter.intro)+
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   443
      show "?F (\<lambda>x. True)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   444
        by (rule exI[of _ "{}"]) (simp add: le_fun_def)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   445
    next
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   446
      fix P Q
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   447
      assume "?F P" then guess X ..
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   448
      moreover
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   449
      assume "?F Q" then guess Y ..
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   450
      ultimately show "?F (\<lambda>x. P x \<and> Q x)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   451
        by (intro exI[of _ "X \<union> Y"])
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   452
           (auto simp: Inf_union_distrib eventually_inf)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   453
    next
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   454
      fix P Q
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   455
      assume "?F P" then guess X ..
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   456
      moreover assume "\<forall>x. P x \<longrightarrow> Q x"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   457
      ultimately show "?F Q"
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61806
diff changeset
   458
        by (intro exI[of _ X]) (auto elim: eventually_mono)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   459
    qed }
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   460
  note eventually_F = this
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   461
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   462
  have "Inf B = Abs_filter ?F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   463
  proof (intro antisym Inf_greatest)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   464
    show "Inf B \<le> Abs_filter ?F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   465
      by (auto simp: le_filter_def eventually_F dest: Inf_superset_mono)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   466
  next
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   467
    fix F assume "F \<in> B" then show "Abs_filter ?F \<le> F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   468
      by (auto simp add: le_filter_def eventually_F intro!: exI[of _ "{F}"])
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   469
  qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   470
  then show ?thesis
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   471
    by (simp add: eventually_F)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   472
qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   473
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
   474
lemma eventually_INF: "eventually P (\<Sqinter>b\<in>B. F b) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (\<Sqinter>b\<in>X. F b))"
62343
24106dc44def prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents: 62123
diff changeset
   475
  unfolding eventually_Inf [of P "F`B"]
24106dc44def prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents: 62123
diff changeset
   476
  by (metis finite_imageI image_mono finite_subset_image)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   477
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   478
lemma Inf_filter_not_bot:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   479
  fixes B :: "'a filter set"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   480
  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
   481
  unfolding trivial_limit_def eventually_Inf[of _ B]
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   482
    bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   483
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   484
lemma INF_filter_not_bot:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   485
  fixes F :: "'i \<Rightarrow> 'a filter"
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
   486
  shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> (\<Sqinter>b\<in>X. F b) \<noteq> bot) \<Longrightarrow> (\<Sqinter>b\<in>B. F b) \<noteq> bot"
62343
24106dc44def prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents: 62123
diff changeset
   487
  unfolding trivial_limit_def eventually_INF [of _ _ B]
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   488
    bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   489
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   490
lemma eventually_Inf_base:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   491
  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
   492
  shows "eventually P (Inf B) \<longleftrightarrow> (\<exists>b\<in>B. eventually P b)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   493
proof (subst eventually_Inf, safe)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   494
  fix X assume "finite X" "X \<subseteq> B"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   495
  then have "\<exists>b\<in>B. \<forall>x\<in>X. b \<le> x"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   496
  proof induct
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   497
    case empty then show ?case
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
   498
      using \<open>B \<noteq> {}\<close> by auto
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   499
  next
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   500
    case (insert x X)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   501
    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
   502
      by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
   503
    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
   504
      by (auto intro: order_trans)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   505
  qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   506
  then obtain b where "b \<in> B" "b \<le> Inf X"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   507
    by (auto simp: le_Inf_iff)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   508
  then show "eventually P (Inf X) \<Longrightarrow> Bex B (eventually P)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   509
    by (intro bexI[of _ b]) (auto simp: le_filter_def)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   510
qed (auto intro!: exI[of _ "{x}" for x])
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   511
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   512
lemma eventually_INF_base:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   513
  "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>
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
   514
    eventually P (\<Sqinter>b\<in>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
   515
  by (subst eventually_Inf_base) auto
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   516
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
   517
lemma eventually_INF1: "i \<in> I \<Longrightarrow> eventually P (F i) \<Longrightarrow> eventually P (\<Sqinter>i\<in>I. F i)"
62369
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62367
diff changeset
   518
  using filter_leD[OF INF_lower] .
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62367
diff changeset
   519
68860
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   520
lemma eventually_INF_finite:
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   521
  assumes "finite A"
69260
0a9688695a1b removed relics of ASCII syntax for indexed big operators
haftmann
parents: 68860
diff changeset
   522
  shows   "eventually P (\<Sqinter> x\<in>A. F x) \<longleftrightarrow>
68860
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   523
             (\<exists>Q. (\<forall>x\<in>A. eventually (Q x) (F x)) \<and> (\<forall>y. (\<forall>x\<in>A. Q x y) \<longrightarrow> P y))" 
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   524
  using assms
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   525
proof (induction arbitrary: P rule: finite_induct)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   526
  case (insert a A P)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   527
  from insert.hyps have [simp]: "x \<noteq> a" if "x \<in> A" for x
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   528
    using that by auto
69260
0a9688695a1b removed relics of ASCII syntax for indexed big operators
haftmann
parents: 68860
diff changeset
   529
  have "eventually P (\<Sqinter> x\<in>insert a A. F x) \<longleftrightarrow>
68860
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   530
          (\<exists>Q R S. eventually Q (F a) \<and> (( (\<forall>x\<in>A. eventually (S x) (F x)) \<and>
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   531
            (\<forall>y. (\<forall>x\<in>A. S x y) \<longrightarrow> R y)) \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x)))"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   532
    unfolding ex_simps by (simp add: eventually_inf insert.IH)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   533
  also have "\<dots> \<longleftrightarrow> (\<exists>Q. (\<forall>x\<in>insert a A. eventually (Q x) (F x)) \<and>
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   534
                           (\<forall>y. (\<forall>x\<in>insert a A. Q x y) \<longrightarrow> P y))"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   535
  proof (safe, goal_cases)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   536
    case (1 Q R S)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   537
    thus ?case using 1 by (intro exI[of _ "S(a := Q)"]) auto
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   538
  next
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   539
    case (2 Q)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   540
    show ?case
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   541
      by (rule exI[of _ "Q a"], rule exI[of _ "\<lambda>y. \<forall>x\<in>A. Q x y"],
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   542
          rule exI[of _ "Q(a := (\<lambda>_. True))"]) (use 2 in auto)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   543
  qed
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   544
  finally show ?case .
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   545
qed auto
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   546
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
   547
subsubsection \<open>Map function for filters\<close>
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   548
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   549
definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   550
  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
   551
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   552
lemma eventually_filtermap:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   553
  "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   554
  unfolding filtermap_def
71743
0239bee6bffd more removal of applys
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   555
  apply (rule eventually_Abs_filter [OF is_filter.intro])
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   556
  apply (auto elim!: eventually_rev_mp)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   557
  done
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   558
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   559
lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   560
  by (simp add: filter_eq_iff eventually_filtermap)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   561
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   562
lemma filtermap_filtermap:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   563
  "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   564
  by (simp add: filter_eq_iff eventually_filtermap)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   565
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   566
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
   567
  unfolding le_filter_def eventually_filtermap by simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   568
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   569
lemma filtermap_bot [simp]: "filtermap f bot = bot"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   570
  by (simp add: filter_eq_iff eventually_filtermap)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   571
67956
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
   572
lemma filtermap_bot_iff: "filtermap f F = bot \<longleftrightarrow> F = bot"
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
   573
  by (simp add: trivial_limit_def eventually_filtermap)
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
   574
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   575
lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)"
67956
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
   576
  by (simp add: filter_eq_iff eventually_filtermap eventually_sup)
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
   577
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
   578
lemma filtermap_SUP: "filtermap f (\<Squnion>b\<in>B. F b) = (\<Squnion>b\<in>B. filtermap f (F b))"
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
   579
  by (simp add: filter_eq_iff eventually_Sup eventually_filtermap)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   580
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   581
lemma filtermap_inf: "filtermap f (inf F1 F2) \<le> inf (filtermap f F1) (filtermap f F2)"
67956
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
   582
  by (intro inf_greatest filtermap_mono inf_sup_ord)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   583
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
   584
lemma filtermap_INF: "filtermap f (\<Sqinter>b\<in>B. F b) \<le> (\<Sqinter>b\<in>B. filtermap f (F b))"
67956
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
   585
  by (rule INF_greatest, rule filtermap_mono, erule INF_lower)
62101
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
   586
66162
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   587
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   588
subsubsection \<open>Contravariant map function for filters\<close>
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   589
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   590
definition filtercomap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter" where
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   591
  "filtercomap f F = Abs_filter (\<lambda>P. \<exists>Q. eventually Q F \<and> (\<forall>x. Q (f x) \<longrightarrow> P x))"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   592
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   593
lemma eventually_filtercomap:
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   594
  "eventually P (filtercomap f F) \<longleftrightarrow> (\<exists>Q. eventually Q F \<and> (\<forall>x. Q (f x) \<longrightarrow> P x))"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   595
  unfolding filtercomap_def
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   596
proof (intro eventually_Abs_filter, unfold_locales, goal_cases)
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   597
  case 1
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   598
  show ?case by (auto intro!: exI[of _ "\<lambda>_. True"])
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   599
next
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   600
  case (2 P Q)
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   601
  from 2(1) guess P' by (elim exE conjE) note P' = this
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   602
  from 2(2) guess Q' by (elim exE conjE) note Q' = this
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   603
  show ?case
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   604
    by (rule exI[of _ "\<lambda>x. P' x \<and> Q' x"])
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   605
       (insert P' Q', auto intro!: eventually_conj)
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   606
next
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   607
  case (3 P Q)
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   608
  thus ?case by blast
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   609
qed
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   610
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   611
lemma filtercomap_ident: "filtercomap (\<lambda>x. x) F = F"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   612
  by (auto simp: filter_eq_iff eventually_filtercomap elim!: eventually_mono)
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   613
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   614
lemma filtercomap_filtercomap: "filtercomap f (filtercomap g F) = filtercomap (\<lambda>x. g (f x)) F"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   615
  unfolding filter_eq_iff by (auto simp: eventually_filtercomap)
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   616
  
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   617
lemma filtercomap_mono: "F \<le> F' \<Longrightarrow> filtercomap f F \<le> filtercomap f F'"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   618
  by (auto simp: eventually_filtercomap le_filter_def)
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   619
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   620
lemma filtercomap_bot [simp]: "filtercomap f bot = bot"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   621
  by (auto simp: filter_eq_iff eventually_filtercomap)
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   622
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   623
lemma filtercomap_top [simp]: "filtercomap f top = top"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   624
  by (auto simp: filter_eq_iff eventually_filtercomap)
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   625
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   626
lemma filtercomap_inf: "filtercomap f (inf F1 F2) = inf (filtercomap f F1) (filtercomap f F2)"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   627
  unfolding filter_eq_iff
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   628
proof safe
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   629
  fix P
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   630
  assume "eventually P (filtercomap f (F1 \<sqinter> F2))"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   631
  then obtain Q R S where *:
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   632
    "eventually Q F1" "eventually R F2" "\<And>x. Q x \<Longrightarrow> R x \<Longrightarrow> S x" "\<And>x. S (f x) \<Longrightarrow> P x"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   633
    unfolding eventually_filtercomap eventually_inf by blast
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   634
  from * have "eventually (\<lambda>x. Q (f x)) (filtercomap f F1)" 
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   635
              "eventually (\<lambda>x. R (f x)) (filtercomap f F2)"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   636
    by (auto simp: eventually_filtercomap)
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   637
  with * show "eventually P (filtercomap f F1 \<sqinter> filtercomap f F2)"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   638
    unfolding eventually_inf by blast
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   639
next
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   640
  fix P
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   641
  assume "eventually P (inf (filtercomap f F1) (filtercomap f F2))"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   642
  then obtain Q Q' R R' where *:
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   643
    "eventually Q F1" "eventually R F2" "\<And>x. Q (f x) \<Longrightarrow> Q' x" "\<And>x. R (f x) \<Longrightarrow> R' x" 
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   644
    "\<And>x. Q' x \<Longrightarrow> R' x \<Longrightarrow> P x"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   645
    unfolding eventually_filtercomap eventually_inf by blast
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   646
  from * have "eventually (\<lambda>x. Q x \<and> R x) (F1 \<sqinter> F2)" by (auto simp: eventually_inf)
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   647
  with * show "eventually P (filtercomap f (F1 \<sqinter> F2))"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   648
    by (auto simp: eventually_filtercomap)
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   649
qed
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   650
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   651
lemma filtercomap_sup: "filtercomap f (sup F1 F2) \<ge> sup (filtercomap f F1) (filtercomap f F2)"
67956
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
   652
  by (intro sup_least filtercomap_mono inf_sup_ord)
66162
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   653
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
   654
lemma filtercomap_INF: "filtercomap f (\<Sqinter>b\<in>B. F b) = (\<Sqinter>b\<in>B. filtercomap f (F b))"
66162
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   655
proof -
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
   656
  have *: "filtercomap f (\<Sqinter>b\<in>B. F b) = (\<Sqinter>b\<in>B. filtercomap f (F b))" if "finite B" for B
66162
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   657
    using that by induction (simp_all add: filtercomap_inf)
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   658
  show ?thesis unfolding filter_eq_iff
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   659
  proof
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   660
    fix P
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
   661
    have "eventually P (\<Sqinter>b\<in>B. filtercomap f (F b)) \<longleftrightarrow> 
66162
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   662
            (\<exists>X. (X \<subseteq> B \<and> finite X) \<and> eventually P (\<Sqinter>b\<in>X. filtercomap f (F b)))"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   663
      by (subst eventually_INF) blast
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
   664
    also have "\<dots> \<longleftrightarrow> (\<exists>X. (X \<subseteq> B \<and> finite X) \<and> eventually P (filtercomap f (\<Sqinter>b\<in>X. F b)))"
66162
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   665
      by (rule ex_cong) (simp add: *)
69275
9bbd5497befd clarified status of legacy input abbreviations
haftmann
parents: 69272
diff changeset
   666
    also have "\<dots> \<longleftrightarrow> eventually P (filtercomap f (\<Sqinter>(F ` B)))"
66162
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   667
      unfolding eventually_filtercomap by (subst eventually_INF) blast
69275
9bbd5497befd clarified status of legacy input abbreviations
haftmann
parents: 69272
diff changeset
   668
    finally show "eventually P (filtercomap f (\<Sqinter>(F ` B))) = 
66162
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   669
                    eventually P (\<Sqinter>b\<in>B. filtercomap f (F b))" ..
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   670
  qed
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   671
qed
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   672
67956
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
   673
lemma filtercomap_SUP:
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
   674
  "filtercomap f (\<Squnion>b\<in>B. F b) \<ge> (\<Squnion>b\<in>B. filtercomap f (F b))"
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
   675
  by (intro SUP_least filtercomap_mono SUP_upper)
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
   676
68860
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   677
lemma filtermap_le_iff_le_filtercomap: "filtermap f F \<le> G \<longleftrightarrow> F \<le> filtercomap f G"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   678
  unfolding le_filter_def eventually_filtermap eventually_filtercomap
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   679
  using eventually_mono by auto
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   680
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   681
lemma filtercomap_neq_bot:
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   682
  assumes "\<And>P. eventually P F \<Longrightarrow> \<exists>x. P (f x)"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   683
  shows   "filtercomap f F \<noteq> bot"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   684
  using assms by (auto simp: trivial_limit_def eventually_filtercomap)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   685
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   686
lemma filtercomap_neq_bot_surj:
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   687
  assumes "F \<noteq> bot" and "surj f"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   688
  shows   "filtercomap f F \<noteq> bot"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   689
proof (rule filtercomap_neq_bot)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   690
  fix P assume *: "eventually P F"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   691
  show "\<exists>x. P (f x)"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   692
  proof (rule ccontr)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   693
    assume **: "\<not>(\<exists>x. P (f x))"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   694
    from * have "eventually (\<lambda>_. False) F"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   695
    proof eventually_elim
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   696
      case (elim x)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   697
      from \<open>surj f\<close> obtain y where "x = f y" by auto
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   698
      with elim and ** show False by auto
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   699
    qed
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   700
    with assms show False by (simp add: trivial_limit_def)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   701
  qed
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   702
qed
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   703
66162
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   704
lemma eventually_filtercomapI [intro]:
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   705
  assumes "eventually P F"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   706
  shows   "eventually (\<lambda>x. P (f x)) (filtercomap f F)"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   707
  using assms by (auto simp: eventually_filtercomap)
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   708
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   709
lemma filtermap_filtercomap: "filtermap f (filtercomap f F) \<le> F"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   710
  by (auto simp: le_filter_def eventually_filtermap eventually_filtercomap)
67956
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
   711
66162
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   712
lemma filtercomap_filtermap: "filtercomap f (filtermap f F) \<ge> F"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   713
  unfolding le_filter_def eventually_filtermap eventually_filtercomap
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   714
  by (auto elim!: eventually_mono)
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   715
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   716
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
   717
subsubsection \<open>Standard filters\<close>
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   718
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   719
definition principal :: "'a set \<Rightarrow> 'a filter" where
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   720
  "principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   721
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   722
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
   723
  unfolding principal_def
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   724
  by (rule eventually_Abs_filter, rule is_filter.intro) auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   725
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   726
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
   727
  unfolding eventually_inf eventually_principal by (auto elim: eventually_mono)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   728
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   729
lemma principal_UNIV[simp]: "principal UNIV = top"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   730
  by (auto simp: filter_eq_iff eventually_principal)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   731
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   732
lemma principal_empty[simp]: "principal {} = bot"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   733
  by (auto simp: filter_eq_iff eventually_principal)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   734
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   735
lemma principal_eq_bot_iff: "principal X = bot \<longleftrightarrow> X = {}"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   736
  by (auto simp add: filter_eq_iff eventually_principal)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   737
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   738
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
   739
  by (auto simp: le_filter_def eventually_principal)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   740
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   741
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
   742
  unfolding le_filter_def eventually_principal
71743
0239bee6bffd more removal of applys
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   743
  by (force elim: eventually_mono)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   744
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   745
lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   746
  unfolding eq_iff by simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   747
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   748
lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   749
  unfolding filter_eq_iff eventually_sup eventually_principal by auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   750
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   751
lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   752
  unfolding filter_eq_iff eventually_inf eventually_principal
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   753
  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
   754
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
   755
lemma SUP_principal[simp]: "(\<Squnion>i\<in>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
   756
  unfolding filter_eq_iff eventually_Sup by (auto simp: eventually_principal)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   757
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
   758
lemma INF_principal_finite: "finite X \<Longrightarrow> (\<Sqinter>x\<in>X. principal (f x)) = principal (\<Inter>x\<in>X. f x)"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   759
  by (induct X rule: finite_induct) auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   760
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   761
lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   762
  unfolding filter_eq_iff eventually_filtermap eventually_principal by simp
66162
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   763
    
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   764
lemma filtercomap_principal[simp]: "filtercomap f (principal A) = principal (f -` A)"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   765
  unfolding filter_eq_iff eventually_filtercomap eventually_principal by fast
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   766
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
   767
subsubsection \<open>Order filters\<close>
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   768
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   769
definition at_top :: "('a::order) filter"
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
   770
  where "at_top = (\<Sqinter>k. principal {k ..})"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   771
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
   772
lemma at_top_sub: "at_top = (\<Sqinter>k\<in>{c::'a::linorder..}. principal {k ..})"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   773
  by (auto intro!: INF_eq max.cobounded1 max.cobounded2 simp: at_top_def)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   774
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   775
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
   776
  unfolding at_top_def
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   777
  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
   778
66162
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   779
lemma eventually_filtercomap_at_top_linorder: 
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   780
  "eventually P (filtercomap f at_top) \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>x. f x \<ge> N \<longrightarrow> P x)"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   781
  by (auto simp: eventually_filtercomap eventually_at_top_linorder)
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   782
63556
36e9732988ce numerical bounds on pi
immler
parents: 63540
diff changeset
   783
lemma eventually_at_top_linorderI:
36e9732988ce numerical bounds on pi
immler
parents: 63540
diff changeset
   784
  fixes c::"'a::linorder"
36e9732988ce numerical bounds on pi
immler
parents: 63540
diff changeset
   785
  assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
36e9732988ce numerical bounds on pi
immler
parents: 63540
diff changeset
   786
  shows "eventually P at_top"
36e9732988ce numerical bounds on pi
immler
parents: 63540
diff changeset
   787
  using assms by (auto simp: eventually_at_top_linorder)
36e9732988ce numerical bounds on pi
immler
parents: 63540
diff changeset
   788
65578
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 63967
diff changeset
   789
lemma eventually_ge_at_top [simp]:
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   790
  "eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   791
  unfolding eventually_at_top_linorder by auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   792
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   793
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
   794
proof -
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
   795
  have "eventually P (\<Sqinter>k. principal {k <..}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n>N. P n)"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   796
    by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
   797
  also have "(\<Sqinter>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
   798
    unfolding at_top_def
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   799
    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
   800
  finally show ?thesis .
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   801
qed
66162
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   802
  
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   803
lemma eventually_filtercomap_at_top_dense: 
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   804
  "eventually P (filtercomap f at_top) \<longleftrightarrow> (\<exists>N::'a::{no_top, linorder}. \<forall>x. f x > N \<longrightarrow> P x)"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   805
  by (auto simp: eventually_filtercomap eventually_at_top_dense)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   806
65578
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 63967
diff changeset
   807
lemma eventually_at_top_not_equal [simp]: "eventually (\<lambda>x::'a::{no_top, linorder}. x \<noteq> c) at_top"
60721
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
   808
  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
   809
65578
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 63967
diff changeset
   810
lemma eventually_gt_at_top [simp]: "eventually (\<lambda>x. (c::_::{no_top, linorder}) < x) at_top"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   811
  unfolding eventually_at_top_dense by auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   812
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   813
lemma eventually_all_ge_at_top:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   814
  assumes "eventually P (at_top :: ('a :: linorder) filter)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   815
  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
   816
proof -
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   817
  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
   818
  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
   819
  thus ?thesis by (auto simp: eventually_at_top_linorder)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   820
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61378
diff changeset
   821
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   822
definition at_bot :: "('a::order) filter"
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
   823
  where "at_bot = (\<Sqinter>k. principal {.. k})"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   824
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
   825
lemma at_bot_sub: "at_bot = (\<Sqinter>k\<in>{.. c::'a::linorder}. principal {.. k})"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   826
  by (auto intro!: INF_eq min.cobounded1 min.cobounded2 simp: at_bot_def)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   827
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   828
lemma eventually_at_bot_linorder:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   829
  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
   830
  unfolding at_bot_def
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   831
  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
   832
66162
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   833
lemma eventually_filtercomap_at_bot_linorder: 
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   834
  "eventually P (filtercomap f at_bot) \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>x. f x \<le> N \<longrightarrow> P x)"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   835
  by (auto simp: eventually_filtercomap eventually_at_bot_linorder)
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   836
65578
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 63967
diff changeset
   837
lemma eventually_le_at_bot [simp]:
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   838
  "eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   839
  unfolding eventually_at_bot_linorder by auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   840
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   841
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
   842
proof -
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
   843
  have "eventually P (\<Sqinter>k. principal {..< k}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n<N. P n)"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   844
    by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
   845
  also have "(\<Sqinter>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
   846
    unfolding at_bot_def
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   847
    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
   848
  finally show ?thesis .
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   849
qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   850
66162
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   851
lemma eventually_filtercomap_at_bot_dense: 
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   852
  "eventually P (filtercomap f at_bot) \<longleftrightarrow> (\<exists>N::'a::{no_bot, linorder}. \<forall>x. f x < N \<longrightarrow> P x)"
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   853
  by (auto simp: eventually_filtercomap eventually_at_bot_dense)
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
   854
65578
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 63967
diff changeset
   855
lemma eventually_at_bot_not_equal [simp]: "eventually (\<lambda>x::'a::{no_bot, linorder}. x \<noteq> c) at_bot"
60721
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
   856
  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
   857
65578
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 63967
diff changeset
   858
lemma eventually_gt_at_bot [simp]:
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   859
  "eventually (\<lambda>x. x < (c::_::unbounded_dense_linorder)) at_bot"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   860
  unfolding eventually_at_bot_dense by auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   861
63967
2aa42596edc3 new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents: 63556
diff changeset
   862
lemma trivial_limit_at_bot_linorder [simp]: "\<not> trivial_limit (at_bot ::('a::linorder) filter)"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   863
  unfolding trivial_limit_def
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   864
  by (metis eventually_at_bot_linorder order_refl)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   865
63967
2aa42596edc3 new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents: 63556
diff changeset
   866
lemma trivial_limit_at_top_linorder [simp]: "\<not> trivial_limit (at_top ::('a::linorder) filter)"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   867
  unfolding trivial_limit_def
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   868
  by (metis eventually_at_top_linorder order_refl)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   869
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
   870
subsection \<open>Sequentially\<close>
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   871
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   872
abbreviation sequentially :: "nat filter"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   873
  where "sequentially \<equiv> at_top"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   874
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   875
lemma eventually_sequentially:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   876
  "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   877
  by (rule eventually_at_top_linorder)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   878
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   879
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   880
  unfolding filter_eq_iff eventually_sequentially by auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   881
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   882
lemmas trivial_limit_sequentially = sequentially_bot
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   883
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   884
lemma eventually_False_sequentially [simp]:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   885
  "\<not> eventually (\<lambda>n. False) sequentially"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   886
  by (simp add: eventually_False)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   887
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   888
lemma le_sequentially:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   889
  "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   890
  by (simp add: at_top_def le_INF_iff le_principal)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   891
60974
6a6f15d8fbc4 New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents: 60758
diff changeset
   892
lemma eventually_sequentiallyI [intro?]:
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   893
  assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   894
  shows "eventually P sequentially"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   895
using assms by (auto simp: eventually_sequentially)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
   896
63967
2aa42596edc3 new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents: 63556
diff changeset
   897
lemma eventually_sequentially_Suc [simp]: "eventually (\<lambda>i. P (Suc i)) sequentially \<longleftrightarrow> eventually P sequentially"
60040
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   898
  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
   899
63967
2aa42596edc3 new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents: 63556
diff changeset
   900
lemma eventually_sequentially_seg [simp]: "eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially"
60040
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
   901
  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
   902
67956
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
   903
lemma filtermap_sequentually_ne_bot: "filtermap f sequentially \<noteq> bot"
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
   904
  by (simp add: filtermap_bot_iff)
61955
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61953
diff changeset
   905
68860
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   906
subsection \<open>Increasing finite subsets\<close>
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   907
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   908
definition finite_subsets_at_top where
69260
0a9688695a1b removed relics of ASCII syntax for indexed big operators
haftmann
parents: 68860
diff changeset
   909
  "finite_subsets_at_top A = (\<Sqinter> X\<in>{X. finite X \<and> X \<subseteq> A}. principal {Y. finite Y \<and> X \<subseteq> Y \<and> Y \<subseteq> A})"
68860
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   910
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   911
lemma eventually_finite_subsets_at_top:
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   912
  "eventually P (finite_subsets_at_top A) \<longleftrightarrow>
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   913
     (\<exists>X. finite X \<and> X \<subseteq> A \<and> (\<forall>Y. finite Y \<and> X \<subseteq> Y \<and> Y \<subseteq> A \<longrightarrow> P Y))"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   914
  unfolding finite_subsets_at_top_def
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   915
proof (subst eventually_INF_base, goal_cases)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   916
  show "{X. finite X \<and> X \<subseteq> A} \<noteq> {}" by auto
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   917
next
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   918
  case (2 B C)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   919
  thus ?case by (intro bexI[of _ "B \<union> C"]) auto
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   920
qed (simp_all add: eventually_principal)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   921
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   922
lemma eventually_finite_subsets_at_top_weakI [intro]:
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   923
  assumes "\<And>X. finite X \<Longrightarrow> X \<subseteq> A \<Longrightarrow> P X"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   924
  shows   "eventually P (finite_subsets_at_top A)"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   925
proof -
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   926
  have "eventually (\<lambda>X. finite X \<and> X \<subseteq> A) (finite_subsets_at_top A)"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   927
    by (auto simp: eventually_finite_subsets_at_top)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   928
  thus ?thesis by eventually_elim (use assms in auto)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   929
qed
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   930
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   931
lemma finite_subsets_at_top_neq_bot [simp]: "finite_subsets_at_top A \<noteq> bot"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   932
proof -
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   933
  have "\<not>eventually (\<lambda>x. False) (finite_subsets_at_top A)"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   934
    by (auto simp: eventually_finite_subsets_at_top)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   935
  thus ?thesis by auto
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   936
qed
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   937
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   938
lemma filtermap_image_finite_subsets_at_top:
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   939
  assumes "inj_on f A"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   940
  shows   "filtermap ((`) f) (finite_subsets_at_top A) = finite_subsets_at_top (f ` A)"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   941
  unfolding filter_eq_iff eventually_filtermap
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   942
proof (safe, goal_cases)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   943
  case (1 P)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   944
  then obtain X where X: "finite X" "X \<subseteq> A" "\<And>Y. finite Y \<Longrightarrow> X \<subseteq> Y \<Longrightarrow> Y \<subseteq> A \<Longrightarrow> P (f ` Y)"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   945
    unfolding eventually_finite_subsets_at_top by force
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   946
  show ?case unfolding eventually_finite_subsets_at_top eventually_filtermap
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   947
  proof (rule exI[of _ "f ` X"], intro conjI allI impI, goal_cases)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   948
    case (3 Y)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   949
    with assms and X(1,2) have "P (f ` (f -` Y \<inter> A))" using X(1,2)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   950
      by (intro X(3) finite_vimage_IntI) auto
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   951
    also have "f ` (f -` Y \<inter> A) = Y" using assms 3 by blast
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   952
    finally show ?case .
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   953
  qed (insert assms X(1,2), auto intro!: finite_vimage_IntI)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   954
next
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   955
  case (2 P)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   956
  then obtain X where X: "finite X" "X \<subseteq> f ` A" "\<And>Y. finite Y \<Longrightarrow> X \<subseteq> Y \<Longrightarrow> Y \<subseteq> f ` A \<Longrightarrow> P Y"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   957
    unfolding eventually_finite_subsets_at_top by force
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   958
  show ?case unfolding eventually_finite_subsets_at_top eventually_filtermap
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   959
  proof (rule exI[of _ "f -` X \<inter> A"], intro conjI allI impI, goal_cases)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   960
    case (3 Y)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   961
    with X(1,2) and assms show ?case by (intro X(3)) force+
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   962
  qed (insert assms X(1), auto intro!: finite_vimage_IntI)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   963
qed
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   964
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   965
lemma eventually_finite_subsets_at_top_finite:
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   966
  assumes "finite A"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   967
  shows   "eventually P (finite_subsets_at_top A) \<longleftrightarrow> P A"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   968
  unfolding eventually_finite_subsets_at_top using assms by force
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   969
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   970
lemma finite_subsets_at_top_finite: "finite A \<Longrightarrow> finite_subsets_at_top A = principal {A}"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   971
  by (auto simp: filter_eq_iff eventually_finite_subsets_at_top_finite eventually_principal)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   972
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
   973
61955
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61953
diff changeset
   974
subsection \<open>The cofinite filter\<close>
60039
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   975
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   976
definition "cofinite = Abs_filter (\<lambda>P. finite {x. \<not> P x})"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   977
61955
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61953
diff changeset
   978
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
   979
  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
   980
61955
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61953
diff changeset
   981
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
   982
  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
   983
61955
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61953
diff changeset
   984
notation (ASCII)
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61953
diff changeset
   985
  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
   986
  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
   987
60039
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   988
lemma eventually_cofinite: "eventually P cofinite \<longleftrightarrow> finite {x. \<not> P x}"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   989
  unfolding cofinite_def
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   990
proof (rule eventually_Abs_filter, rule is_filter.intro)
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   991
  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
   992
  from finite_UnI[OF this] show "finite {x. \<not> (P x \<and> Q x)}"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   993
    by (rule rev_finite_subset) auto
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   994
next
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   995
  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
   996
  from * show "finite {x. \<not> Q x}"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   997
    by (intro finite_subset[OF _ P]) auto
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   998
qed simp
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
   999
60040
1fa1023b13b9 move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents: 60039
diff changeset
  1000
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
  1001
  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
  1002
60039
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
  1003
lemma cofinite_bot[simp]: "cofinite = (bot::'a filter) \<longleftrightarrow> finite (UNIV :: 'a set)"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
  1004
  unfolding trivial_limit_def eventually_cofinite by simp
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
  1005
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
  1006
lemma cofinite_eq_sequentially: "cofinite = sequentially"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
  1007
  unfolding filter_eq_iff eventually_sequentially eventually_cofinite
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
  1008
proof safe
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
  1009
  fix P :: "nat \<Rightarrow> bool" assume [simp]: "finite {x. \<not> P x}"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
  1010
  show "\<exists>N. \<forall>n\<ge>N. P n"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
  1011
  proof cases
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
  1012
    assume "{x. \<not> P x} \<noteq> {}" then show ?thesis
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
  1013
      by (intro exI[of _ "Suc (Max {x. \<not> P x})"]) (auto simp: Suc_le_eq)
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
  1014
  qed auto
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
  1015
next
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
  1016
  fix P :: "nat \<Rightarrow> bool" and N :: nat assume "\<forall>n\<ge>N. P n"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
  1017
  then have "{x. \<not> P x} \<subseteq> {..< N}"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
  1018
    by (auto simp: not_le)
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
  1019
  then show "finite {x. \<not> P x}"
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
  1020
    by (blast intro: finite_subset)
d55937a8f97e add cofinite filter
hoelzl
parents: 60038
diff changeset
  1021
qed
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1022
62101
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1023
subsubsection \<open>Product of filters\<close>
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1024
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1025
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
  1026
  "prod_filter F G =
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
  1027
    (\<Sqinter>(P, Q)\<in>{(P, Q). eventually P F \<and> eventually Q G}. principal {(x, y). P x \<and> Q y})"
62101
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1028
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1029
lemma eventually_prod_filter: "eventually P (F \<times>\<^sub>F G) \<longleftrightarrow>
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1030
  (\<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
  1031
  unfolding prod_filter_def
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1032
proof (subst eventually_INF_base, goal_cases)
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1033
  case 2
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1034
  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
  1035
    \<exists>P Q. eventually P F \<and> eventually Q G \<and>
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1036
      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
  1037
    by (intro conjI exI[of _ "inf Pf Qf"] exI[of _ "inf Pg Qg"])
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1038
       (auto simp: inf_fun_def eventually_conj)
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1039
  ultimately show ?case
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1040
    by auto
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1041
qed (auto simp: eventually_principal intro: eventually_True)
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1042
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1043
lemma eventually_prod1:
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1044
  assumes "B \<noteq> bot"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1045
  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
  1046
  unfolding eventually_prod_filter
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1047
proof safe
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1048
  fix R Q
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1049
  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"
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1050
  with \<open>B \<noteq> bot\<close> obtain y where "Q y" by (auto dest: eventually_happens)
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1051
  with * show "eventually P A"
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1052
    by (force elim: eventually_mono)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1053
next
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1054
  assume "eventually P A"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1055
  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
  1056
    by (intro exI[of _ P] exI[of _ "\<lambda>x. True"]) auto
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1057
qed
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1058
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1059
lemma eventually_prod2:
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1060
  assumes "A \<noteq> bot"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1061
  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
  1062
  unfolding eventually_prod_filter
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1063
proof safe
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1064
  fix R Q
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1065
  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"
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1066
  with \<open>A \<noteq> bot\<close> obtain x where "R x" by (auto dest: eventually_happens)
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1067
  with * show "eventually P B"
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1068
    by (force elim: eventually_mono)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1069
next
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1070
  assume "eventually P B"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1071
  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
  1072
    by (intro exI[of _ P] exI[of _ "\<lambda>x. True"]) auto
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1073
qed
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1074
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1075
lemma INF_filter_bot_base:
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1076
  fixes F :: "'a \<Rightarrow> 'b filter"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1077
  assumes *: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> \<exists>k\<in>I. F k \<le> F i \<sqinter> F j"
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
  1078
  shows "(\<Sqinter>i\<in>I. F i) = bot \<longleftrightarrow> (\<exists>i\<in>I. F i = bot)"
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1079
proof (cases "\<exists>i\<in>I. F i = bot")
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1080
  case True
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
  1081
  then have "(\<Sqinter>i\<in>I. F i) \<le> bot"
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1082
    by (auto intro: INF_lower2)
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1083
  with True show ?thesis
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1084
    by (auto simp: bot_unique)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1085
next
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1086
  case False
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
  1087
  moreover have "(\<Sqinter>i\<in>I. F i) \<noteq> bot"
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1088
  proof (cases "I = {}")
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1089
    case True
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1090
    then show ?thesis
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1091
      by (auto simp add: filter_eq_iff)
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1092
  next
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1093
    case False': False
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1094
    show ?thesis
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1095
    proof (rule INF_filter_not_bot)
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1096
      fix J
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1097
      assume "finite J" "J \<subseteq> I"
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1098
      then have "\<exists>k\<in>I. F k \<le> (\<Sqinter>i\<in>J. F i)"
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1099
      proof (induct J)
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1100
        case empty
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1101
        then show ?case
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1102
          using \<open>I \<noteq> {}\<close> by auto
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1103
      next
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1104
        case (insert i J)
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1105
        then obtain k where "k \<in> I" "F k \<le> (\<Sqinter>i\<in>J. F i)" by auto
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1106
        with insert *[of i k] show ?case
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1107
          by auto
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1108
      qed
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1109
      with False show "(\<Sqinter>i\<in>J. F i) \<noteq> \<bottom>"
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1110
        by (auto simp: bot_unique)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1111
    qed
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1112
  qed
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1113
  ultimately show ?thesis
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1114
    by auto
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1115
qed
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1116
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1117
lemma Collect_empty_eq_bot: "Collect P = {} \<longleftrightarrow> P = \<bottom>"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1118
  by auto
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1119
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1120
lemma prod_filter_eq_bot: "A \<times>\<^sub>F B = bot \<longleftrightarrow> A = bot \<or> B = bot"
67956
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1121
  unfolding trivial_limit_def
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1122
proof
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1123
  assume "\<forall>\<^sub>F x in A \<times>\<^sub>F B. False"
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1124
  then obtain Pf Pg
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1125
    where Pf: "eventually (\<lambda>x. Pf x) A" and Pg: "eventually (\<lambda>y. Pg y) B"
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1126
    and *: "\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> False"
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1127
    unfolding eventually_prod_filter by fast
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1128
  from * have "(\<forall>x. \<not> Pf x) \<or> (\<forall>y. \<not> Pg y)" by fast
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1129
  with Pf Pg show "(\<forall>\<^sub>F x in A. False) \<or> (\<forall>\<^sub>F x in B. False)" by auto
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1130
next
67956
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1131
  assume "(\<forall>\<^sub>F x in A. False) \<or> (\<forall>\<^sub>F x in B. False)"
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1132
  then show "\<forall>\<^sub>F x in A \<times>\<^sub>F B. False"
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1133
    unfolding eventually_prod_filter by (force intro: eventually_True)
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1134
qed
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1135
62101
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1136
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
  1137
  by (auto simp: le_filter_def eventually_prod_filter)
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1138
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1139
lemma prod_filter_mono_iff:
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1140
  assumes nAB: "A \<noteq> bot" "B \<noteq> bot"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1141
  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
  1142
proof safe
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1143
  assume *: "A \<times>\<^sub>F B \<le> C \<times>\<^sub>F D"
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1144
  with assms have "A \<times>\<^sub>F B \<noteq> bot"
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1145
    by (auto simp: bot_unique prod_filter_eq_bot)
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1146
  with * have "C \<times>\<^sub>F D \<noteq> bot"
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1147
    by (auto simp: bot_unique)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1148
  then have nCD: "C \<noteq> bot" "D \<noteq> bot"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1149
    by (auto simp: prod_filter_eq_bot)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1150
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1151
  show "A \<le> C"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1152
  proof (rule filter_leI)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1153
    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
  1154
      using nAB nCD by (simp add: eventually_prod1 eventually_prod2)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1155
  qed
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1156
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1157
  show "B \<le> D"
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1158
  proof (rule filter_leI)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1159
    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
  1160
      using nAB nCD by (simp add: eventually_prod1 eventually_prod2)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1161
  qed
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1162
qed (intro prod_filter_mono)
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1163
62101
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1164
lemma eventually_prod_same: "eventually P (F \<times>\<^sub>F F) \<longleftrightarrow>
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1165
    (\<exists>Q. eventually Q F \<and> (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y)))"
71743
0239bee6bffd more removal of applys
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1166
  unfolding eventually_prod_filter by (blast intro!: eventually_conj)
62101
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1167
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1168
lemma eventually_prod_sequentially:
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1169
  "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
  1170
  unfolding eventually_prod_same eventually_sequentially by auto
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1171
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1172
lemma principal_prod_principal: "principal A \<times>\<^sub>F principal B = principal (A \<times> B)"
67956
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1173
  unfolding filter_eq_iff eventually_prod_filter eventually_principal
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1174
  by (fast intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"])
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1175
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1176
lemma le_prod_filterI:
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1177
  "filtermap fst F \<le> A \<Longrightarrow> filtermap snd F \<le> B \<Longrightarrow> F \<le> A \<times>\<^sub>F B"
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1178
  unfolding le_filter_def eventually_filtermap eventually_prod_filter
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1179
  by (force elim: eventually_elim2)
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1180
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1181
lemma filtermap_fst_prod_filter: "filtermap fst (A \<times>\<^sub>F B) \<le> A"
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1182
  unfolding le_filter_def eventually_filtermap eventually_prod_filter
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1183
  by (force intro: eventually_True)
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1184
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1185
lemma filtermap_snd_prod_filter: "filtermap snd (A \<times>\<^sub>F B) \<le> B"
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1186
  unfolding le_filter_def eventually_filtermap eventually_prod_filter
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1187
  by (force intro: eventually_True)
62101
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61955
diff changeset
  1188
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1189
lemma prod_filter_INF:
67956
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1190
  assumes "I \<noteq> {}" and "J \<noteq> {}"
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
  1191
  shows "(\<Sqinter>i\<in>I. A i) \<times>\<^sub>F (\<Sqinter>j\<in>J. B j) = (\<Sqinter>i\<in>I. \<Sqinter>j\<in>J. A i \<times>\<^sub>F B j)"
67956
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1192
proof (rule antisym)
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1193
  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
  1194
  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
  1195
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1196
  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)"
67956
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1197
    by (fast intro: le_prod_filterI INF_greatest INF_lower2
69272
15e9ed5b28fb isabelle update_cartouches -t;
wenzelm
parents: 69260
diff changeset
  1198
      order_trans[OF filtermap_INF] \<open>i \<in> I\<close> \<open>j \<in> J\<close>
67956
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1199
      filtermap_fst_prod_filter filtermap_snd_prod_filter)
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1200
  show "(\<Sqinter>i\<in>I. A i) \<times>\<^sub>F (\<Sqinter>j\<in>J. B j) \<le> (\<Sqinter>i\<in>I. \<Sqinter>j\<in>J. A i \<times>\<^sub>F B j)"
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1201
    by (intro INF_greatest prod_filter_mono INF_lower)
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1202
qed
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1203
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1204
lemma filtermap_Pair: "filtermap (\<lambda>x. (f x, g x)) F \<le> filtermap f F \<times>\<^sub>F filtermap g F"
67956
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1205
  by (rule le_prod_filterI, simp_all add: filtermap_filtermap)
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1206
62369
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62367
diff changeset
  1207
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)"
67956
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1208
  unfolding eventually_prod_filter by auto
62369
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62367
diff changeset
  1209
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
  1210
lemma prod_filter_INF1: "I \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>I. A i) \<times>\<^sub>F B = (\<Sqinter>i\<in>I. A i \<times>\<^sub>F B)"
62369
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62367
diff changeset
  1211
  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
  1212
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
  1213
lemma prod_filter_INF2: "J \<noteq> {} \<Longrightarrow> A \<times>\<^sub>F (\<Sqinter>i\<in>J. B i) = (\<Sqinter>i\<in>J. A \<times>\<^sub>F B i)"
62369
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62367
diff changeset
  1214
  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
  1215
68667
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1216
lemma prod_filtermap1: "prod_filter (filtermap f F) G = filtermap (apfst f) (prod_filter F G)"
71743
0239bee6bffd more removal of applys
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1217
  unfolding filter_eq_iff eventually_filtermap eventually_prod_filter
0239bee6bffd more removal of applys
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1218
  apply safe
68667
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1219
  subgoal by auto
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1220
  subgoal for P Q R by(rule exI[where x="\<lambda>y. \<exists>x. y = f x \<and> Q x"])(auto intro: eventually_mono)
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1221
  done
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1222
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1223
lemma prod_filtermap2: "prod_filter F (filtermap g G) = filtermap (apsnd g) (prod_filter F G)"
71743
0239bee6bffd more removal of applys
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1224
  unfolding filter_eq_iff eventually_filtermap eventually_prod_filter
0239bee6bffd more removal of applys
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1225
  apply safe
68667
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1226
  subgoal by auto
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1227
  subgoal for P Q R  by(auto intro: exI[where x="\<lambda>y. \<exists>x. y = g x \<and> R x"] eventually_mono)
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1228
  done
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1229
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1230
lemma prod_filter_assoc:
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1231
  "prod_filter (prod_filter F G) H = filtermap (\<lambda>(x, y, z). ((x, y), z)) (prod_filter F (prod_filter G H))"
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1232
  apply(clarsimp simp add: filter_eq_iff eventually_filtermap eventually_prod_filter; safe)
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1233
  subgoal for P Q R S T by(auto 4 4 intro: exI[where x="\<lambda>(a, b). T a \<and> S b"])
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1234
  subgoal for P Q R S T by(auto 4 3 intro: exI[where x="\<lambda>(a, b). Q a \<and> S b"])
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1235
  done
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1236
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1237
lemma prod_filter_principal_singleton: "prod_filter (principal {x}) F = filtermap (Pair x) F"
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1238
  by(fastforce simp add: filter_eq_iff eventually_prod_filter eventually_principal eventually_filtermap elim: eventually_mono intro: exI[where x="\<lambda>a. a = x"])
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1239
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1240
lemma prod_filter_principal_singleton2: "prod_filter F (principal {x}) = filtermap (\<lambda>a. (a, x)) F"
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1241
  by(fastforce simp add: filter_eq_iff eventually_prod_filter eventually_principal eventually_filtermap elim: eventually_mono intro: exI[where x="\<lambda>a. a = x"])
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1242
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1243
lemma prod_filter_commute: "prod_filter F G = filtermap prod.swap (prod_filter G F)"
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1244
  by(auto simp add: filter_eq_iff eventually_prod_filter eventually_filtermap)
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1245
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
  1246
subsection \<open>Limits\<close>
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1247
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1248
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
  1249
  "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1250
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1251
syntax
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1252
  "_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
  1253
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1254
translations
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1255
  "LIM x F1. f :> F2" == "CONST filterlim (\<lambda>x. f) F2 F1"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1256
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
  1257
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
  1258
  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
  1259
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1260
lemma filterlim_iff:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1261
  "(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
  1262
  unfolding filterlim_def le_filter_def eventually_filtermap ..
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1263
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1264
lemma filterlim_compose:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1265
  "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
  1266
  unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1267
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1268
lemma filterlim_mono:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1269
  "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
  1270
  unfolding filterlim_def by (metis filtermap_mono order_trans)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1271
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1272
lemma filterlim_ident: "LIM x F. x :> F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1273
  by (simp add: filterlim_def filtermap_ident)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1274
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1275
lemma filterlim_cong:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1276
  "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
  1277
  by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1278
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1279
lemma filterlim_mono_eventually:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1280
  assumes "filterlim f F G" and ord: "F \<le> F'" "G' \<le> G"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1281
  assumes eq: "eventually (\<lambda>x. f x = f' x) G'"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1282
  shows "filterlim f' F' G'"
71743
0239bee6bffd more removal of applys
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1283
proof -
0239bee6bffd more removal of applys
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1284
  have "filterlim f F' G'"
0239bee6bffd more removal of applys
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1285
    by (simp add: filterlim_mono[OF _ ord] assms)
0239bee6bffd more removal of applys
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1286
  then show ?thesis
0239bee6bffd more removal of applys
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1287
    by (rule filterlim_cong[OF refl refl eq, THEN iffD1])
0239bee6bffd more removal of applys
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1288
qed
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1289
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1290
lemma filtermap_mono_strong: "inj f \<Longrightarrow> filtermap f F \<le> filtermap f G \<longleftrightarrow> F \<le> G"
67956
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1291
  apply (safe intro!: filtermap_mono)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1292
  apply (auto simp: le_filter_def eventually_filtermap)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1293
  apply (erule_tac x="\<lambda>x. P (inv f x)" in allE)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1294
  apply auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1295
  done
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1296
67950
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67855
diff changeset
  1297
lemma eventually_compose_filterlim:
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67855
diff changeset
  1298
  assumes "eventually P F" "filterlim f F G"
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67855
diff changeset
  1299
  shows "eventually (\<lambda>x. P (f x)) G"
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67855
diff changeset
  1300
  using assms by (simp add: filterlim_iff)
99eaa5cedbb7 Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents: 67855
diff changeset
  1301
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1302
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
  1303
  by (simp add: filtermap_mono_strong eq_iff)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1304
60721
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
  1305
lemma filtermap_fun_inverse:
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
  1306
  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
  1307
  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
  1308
  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
  1309
  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
  1310
proof (rule antisym)
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
  1311
  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
  1312
    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
  1313
  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
  1314
    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
  1315
  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
  1316
    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
  1317
  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
  1318
qed
c1b7793c23a3 generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents: 60589
diff changeset
  1319
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1320
lemma filterlim_principal:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1321
  "(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
  1322
  unfolding filterlim_def eventually_filtermap le_principal ..
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1323
68860
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1324
lemma filterlim_filtercomap [intro]: "filterlim f F (filtercomap f F)"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1325
  unfolding filterlim_def by (rule filtermap_filtercomap)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1326
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1327
lemma filterlim_inf:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1328
  "(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
  1329
  unfolding filterlim_def by simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1330
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1331
lemma filterlim_INF:
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
  1332
  "(LIM x F. f x :> (\<Sqinter>b\<in>B. G b)) \<longleftrightarrow> (\<forall>b\<in>B. LIM x F. f x :> G b)"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1333
  unfolding filterlim_def le_INF_iff ..
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1334
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1335
lemma filterlim_INF_INF:
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
  1336
  "(\<And>m. m \<in> J \<Longrightarrow> \<exists>i\<in>I. filtermap f (F i) \<le> G m) \<Longrightarrow> LIM x (\<Sqinter>i\<in>I. F i). f x :> (\<Sqinter>j\<in>J. G j)"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1337
  unfolding filterlim_def by (rule order_trans[OF filtermap_INF INF_mono])
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1338
69260
0a9688695a1b removed relics of ASCII syntax for indexed big operators
haftmann
parents: 68860
diff changeset
  1339
lemma filterlim_INF': "x \<in> A \<Longrightarrow> filterlim f F (G x) \<Longrightarrow> filterlim f F (\<Sqinter> x\<in>A. G x)"
68860
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1340
  unfolding filterlim_def by (rule order.trans[OF filtermap_mono[OF INF_lower]])
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1341
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1342
lemma filterlim_filtercomap_iff: "filterlim f (filtercomap g G) F \<longleftrightarrow> filterlim (g \<circ> f) G F"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1343
  by (simp add: filterlim_def filtermap_le_iff_le_filtercomap filtercomap_filtercomap o_def)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1344
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1345
lemma filterlim_iff_le_filtercomap: "filterlim f F G \<longleftrightarrow> G \<le> filtercomap f F"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1346
  by (simp add: filterlim_def filtermap_le_iff_le_filtercomap)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1347
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1348
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
  1349
  "(\<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>
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
  1350
    LIM x (\<Sqinter>i\<in>I. principal (F i)). f x :> (\<Sqinter>j\<in>J. principal (G j))"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1351
  by (force intro!: filterlim_INF_INF simp: image_subset_iff)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1352
61806
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
  1353
lemma filterlim_base_iff:
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1354
  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"
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67399
diff changeset
  1355
  shows "(LIM x (\<Sqinter>i\<in>I. principal (F i)). f x :> \<Sqinter>j\<in>J. principal (G j)) \<longleftrightarrow>
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1356
    (\<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
  1357
  unfolding filterlim_INF filterlim_principal
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1358
proof (subst eventually_INF_base)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1359
  fix i j assume "i \<in> I" "j \<in> I"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1360
  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
  1361
    by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
  1362
qed (auto simp: eventually_principal \<open>I \<noteq> {}\<close>)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1363
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1364
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
  1365
  unfolding filterlim_def filtermap_filtermap ..
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1366
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1367
lemma filterlim_sup:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1368
  "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
  1369
  unfolding filterlim_def filtermap_sup by auto
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 filterlim_sequentially_Suc:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1372
  "(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
  1373
  unfolding filterlim_iff by (subst eventually_sequentially_Suc) simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1374
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1375
lemma filterlim_Suc: "filterlim Suc sequentially sequentially"
63967
2aa42596edc3 new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents: 63556
diff changeset
  1376
  by (simp add: filterlim_iff eventually_sequentially)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1377
60182
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60040
diff changeset
  1378
lemma filterlim_If:
e1ea5a6379c9 generalized tends over powr; added DERIV rule for powr
hoelzl
parents: 60040
diff changeset
  1379
  "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
  1380
    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
  1381
    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
  1382
  unfolding filterlim_iff eventually_inf_principal by (auto simp: eventually_conj_iff)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1383
62367
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1384
lemma filterlim_Pair:
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1385
  "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
  1386
  unfolding filterlim_def
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1387
  by (rule order_trans[OF filtermap_Pair prod_filter_mono])
d2bc8a7e5fec move product topology to HOL-Complex_Main
hoelzl
parents: 62343
diff changeset
  1388
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69275
diff changeset
  1389
subsection \<open>Limits to \<^const>\<open>at_top\<close> and \<^const>\<open>at_bot\<close>\<close>
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1390
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1391
lemma filterlim_at_top:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1392
  fixes f :: "'a \<Rightarrow> ('b::linorder)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1393
  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
  1394
  by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_mono)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1395
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1396
lemma filterlim_at_top_mono:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1397
  "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
  1398
    LIM x F. g x :> at_top"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1399
  by (auto simp: filterlim_at_top elim: eventually_elim2 intro: order_trans)
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 filterlim_at_top_dense:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1402
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1403
  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
  1404
  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
  1405
            filterlim_at_top[of f F] filterlim_iff[of f at_top F])
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1406
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1407
lemma filterlim_at_top_ge:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1408
  fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1409
  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
  1410
  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
  1411
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1412
lemma filterlim_at_top_at_top:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1413
  fixes f :: "'a::linorder \<Rightarrow> 'b::linorder"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1414
  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
  1415
  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
  1416
  assumes Q: "eventually Q at_top"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1417
  assumes P: "eventually P at_top"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1418
  shows "filterlim f at_top at_top"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1419
proof -
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1420
  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
  1421
    unfolding eventually_at_top_linorder by auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1422
  show ?thesis
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1423
  proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1424
    fix z assume "x \<le> z"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1425
    with x have "P z" by auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1426
    have "eventually (\<lambda>x. g z \<le> x) at_top"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1427
      by (rule eventually_ge_at_top)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1428
    with Q show "eventually (\<lambda>x. z \<le> f x) at_top"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60752
diff changeset
  1429
      by eventually_elim (metis mono bij \<open>P z\<close>)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1430
  qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1431
qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1432
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1433
lemma filterlim_at_top_gt:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1434
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1435
  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
  1436
  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
  1437
61806
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
  1438
lemma filterlim_at_bot:
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1439
  fixes f :: "'a \<Rightarrow> ('b::linorder)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1440
  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
  1441
  by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_mono)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1442
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1443
lemma filterlim_at_bot_dense:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1444
  fixes f :: "'a \<Rightarrow> ('b::{dense_linorder, no_bot})"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1445
  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
  1446
proof (auto simp add: filterlim_at_bot[of f F])
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1447
  fix Z :: 'b
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1448
  from lt_ex [of Z] obtain Z' where 1: "Z' < Z" ..
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1449
  assume "\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1450
  hence "eventually (\<lambda>x. f x \<le> Z') F" by auto
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1451
  thus "eventually (\<lambda>x. f x < Z) F"
71743
0239bee6bffd more removal of applys
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1452
    by (rule eventually_mono) (use 1 in auto)
61806
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
  1453
  next
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
  1454
    fix Z :: 'b
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1455
    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
  1456
      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
  1457
qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1458
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1459
lemma filterlim_at_bot_le:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1460
  fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1461
  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
  1462
  unfolding filterlim_at_bot
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1463
proof safe
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1464
  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
  1465
  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
  1466
    by (auto elim!: eventually_mono)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1467
qed simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1468
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1469
lemma filterlim_at_bot_lt:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1470
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1471
  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
  1472
  by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans)
66162
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
  1473
    
68860
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1474
lemma filterlim_finite_subsets_at_top:
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1475
  "filterlim f (finite_subsets_at_top A) F \<longleftrightarrow>
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1476
     (\<forall>X. finite X \<and> X \<subseteq> A \<longrightarrow> eventually (\<lambda>y. finite (f y) \<and> X \<subseteq> f y \<and> f y \<subseteq> A) F)"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1477
  (is "?lhs = ?rhs")
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1478
proof 
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1479
  assume ?lhs
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1480
  thus ?rhs
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1481
  proof (safe, goal_cases)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1482
    case (1 X)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1483
    hence *: "(\<forall>\<^sub>F x in F. P (f x))" if "eventually P (finite_subsets_at_top A)" for P
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1484
      using that by (auto simp: filterlim_def le_filter_def eventually_filtermap)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1485
    have "\<forall>\<^sub>F Y in finite_subsets_at_top A. finite Y \<and> X \<subseteq> Y \<and> Y \<subseteq> A"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1486
      using 1 unfolding eventually_finite_subsets_at_top by force
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1487
    thus ?case by (intro *) auto
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1488
  qed
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1489
next
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1490
  assume rhs: ?rhs
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1491
  show ?lhs unfolding filterlim_def le_filter_def eventually_finite_subsets_at_top
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1492
  proof (safe, goal_cases)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1493
    case (1 P X)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1494
    with rhs have "\<forall>\<^sub>F y in F. finite (f y) \<and> X \<subseteq> f y \<and> f y \<subseteq> A" by auto
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1495
    thus "eventually P (filtermap f F)" unfolding eventually_filtermap
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1496
      by eventually_elim (insert 1, auto)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1497
  qed
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1498
qed
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1499
68860
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1500
lemma filterlim_atMost_at_top:
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1501
  "filterlim (\<lambda>n. {..n}) (finite_subsets_at_top (UNIV :: nat set)) at_top"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1502
  unfolding filterlim_finite_subsets_at_top
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1503
proof (safe, goal_cases)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1504
  case (1 X)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1505
  then obtain n where n: "X \<subseteq> {..n}" by (auto simp: finite_nat_set_iff_bounded_le)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1506
  show ?case using eventually_ge_at_top[of n]
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1507
    by eventually_elim (insert n, auto)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1508
qed
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1509
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1510
lemma filterlim_lessThan_at_top:
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1511
  "filterlim (\<lambda>n. {..<n}) (finite_subsets_at_top (UNIV :: nat set)) at_top"
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1512
  unfolding filterlim_finite_subsets_at_top
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1513
proof (safe, goal_cases)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1514
  case (1 X)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1515
  then obtain n where n: "X \<subseteq> {..<n}" by (auto simp: finite_nat_set_iff_bounded)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1516
  show ?case using eventually_ge_at_top[of n]
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1517
    by eventually_elim (insert n, auto)
f443ec10447d Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents: 68667
diff changeset
  1518
qed
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1519
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69275
diff changeset
  1520
subsection \<open>Setup \<^typ>\<open>'a filter\<close> for lifting and transfer\<close>
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1521
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1522
lemma filtermap_id [simp, id_simps]: "filtermap id = id"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1523
by(simp add: fun_eq_iff id_def filtermap_ident)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1524
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1525
lemma filtermap_id' [simp]: "filtermap (\<lambda>x. x) = (\<lambda>F. F)"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1526
using filtermap_id unfolding id_def .
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1527
67616
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1528
context includes lifting_syntax
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1529
begin
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1530
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1531
definition map_filter_on :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter" where
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1532
  "map_filter_on X f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x) \<and> x \<in> X) F)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1533
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1534
lemma is_filter_map_filter_on:
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1535
  "is_filter (\<lambda>P. \<forall>\<^sub>F x in F. P (f x) \<and> x \<in> X) \<longleftrightarrow> eventually (\<lambda>x. x \<in> X) F"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1536
proof(rule iffI; unfold_locales)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1537
  show "\<forall>\<^sub>F x in F. True \<and> x \<in> X" if "eventually (\<lambda>x. x \<in> X) F" using that by simp
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1538
  show "\<forall>\<^sub>F x in F. (P (f x) \<and> Q (f x)) \<and> x \<in> X" if "\<forall>\<^sub>F x in F. P (f x) \<and> x \<in> X" "\<forall>\<^sub>F x in F. Q (f x) \<and> x \<in> X" for P Q
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1539
    using eventually_conj[OF that] by(auto simp add: conj_ac cong: conj_cong)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1540
  show "\<forall>\<^sub>F x in F. Q (f x) \<and> x \<in> X" if "\<forall>x. P x \<longrightarrow> Q x" "\<forall>\<^sub>F x in F. P (f x) \<and> x \<in> X" for P Q
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1541
    using that(2) by(rule eventually_mono)(use that(1) in auto)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1542
  show "eventually (\<lambda>x. x \<in> X) F" if "is_filter (\<lambda>P. \<forall>\<^sub>F x in F. P (f x) \<and> x \<in> X)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1543
    using is_filter.True[OF that] by simp
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1544
qed
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1545
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1546
lemma eventually_map_filter_on: "eventually P (map_filter_on X f F) = (\<forall>\<^sub>F x in F. P (f x) \<and> x \<in> X)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1547
  if "eventually (\<lambda>x. x \<in> X) F"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1548
  by(simp add: is_filter_map_filter_on map_filter_on_def eventually_Abs_filter that)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1549
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1550
lemma map_filter_on_UNIV: "map_filter_on UNIV = filtermap"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1551
  by(simp add: map_filter_on_def filtermap_def fun_eq_iff)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1552
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1553
lemma map_filter_on_comp: "map_filter_on X f (map_filter_on Y g F) = map_filter_on Y (f \<circ> g) F"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1554
  if "g ` Y \<subseteq> X" and "eventually (\<lambda>x. x \<in> Y) F"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1555
  unfolding map_filter_on_def using that(1)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1556
  by(auto simp add: eventually_Abs_filter that(2) is_filter_map_filter_on intro!: arg_cong[where f=Abs_filter] arg_cong2[where f=eventually])
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1557
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1558
inductive rel_filter :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> 'b filter \<Rightarrow> bool" for R F G where
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1559
  "rel_filter R F G" if "eventually (case_prod R) Z" "map_filter_on {(x, y). R x y} fst Z = F" "map_filter_on {(x, y). R x y} snd Z = G"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1560
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1561
lemma rel_filter_eq [relator_eq]: "rel_filter (=) = (=)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1562
proof(intro ext iffI)+
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1563
  show "F = G" if "rel_filter (=) F G" for F G using that
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1564
    by cases(clarsimp simp add: filter_eq_iff eventually_map_filter_on split_def cong: rev_conj_cong)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1565
  show "rel_filter (=) F G" if "F = G" for F G unfolding \<open>F = G\<close>
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1566
  proof
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1567
    let ?Z = "map_filter_on UNIV (\<lambda>x. (x, x)) G"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1568
    have [simp]: "range (\<lambda>x. (x, x)) \<subseteq> {(x, y). x = y}" by auto
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1569
    show "map_filter_on {(x, y). x = y} fst ?Z = G" and "map_filter_on {(x, y). x = y} snd ?Z = G"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1570
      by(simp_all add: map_filter_on_comp)(simp_all add: map_filter_on_UNIV o_def)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1571
    show "\<forall>\<^sub>F (x, y) in ?Z. x = y" by(simp add: eventually_map_filter_on)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1572
  qed
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1573
qed
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1574
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1575
lemma rel_filter_mono [relator_mono]: "rel_filter A \<le> rel_filter B" if le: "A \<le> B"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1576
proof(clarify elim!: rel_filter.cases)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1577
  show "rel_filter B (map_filter_on {(x, y). A x y} fst Z) (map_filter_on {(x, y). A x y} snd Z)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1578
    (is "rel_filter _ ?X ?Y") if "\<forall>\<^sub>F (x, y) in Z. A x y" for Z
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1579
  proof
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1580
    let ?Z = "map_filter_on {(x, y). A x y} id Z"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1581
    show "\<forall>\<^sub>F (x, y) in ?Z. B x y" using le that
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1582
      by(simp add: eventually_map_filter_on le_fun_def split_def conj_commute cong: conj_cong)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1583
    have [simp]: "{(x, y). A x y} \<subseteq> {(x, y). B x y}" using le by auto
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1584
    show "map_filter_on {(x, y). B x y} fst ?Z = ?X" "map_filter_on {(x, y). B x y} snd ?Z = ?Y"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1585
      using le that by(simp_all add: le_fun_def map_filter_on_comp)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1586
  qed
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1587
qed
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1588
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1589
lemma rel_filter_conversep: "rel_filter A\<inverse>\<inverse> = (rel_filter A)\<inverse>\<inverse>"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1590
proof(safe intro!: ext elim!: rel_filter.cases)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1591
  show *: "rel_filter A (map_filter_on {(x, y). A\<inverse>\<inverse> x y} snd Z) (map_filter_on {(x, y). A\<inverse>\<inverse> x y} fst Z)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1592
    (is "rel_filter _ ?X ?Y") if "\<forall>\<^sub>F (x, y) in Z. A\<inverse>\<inverse> x y" for A Z
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1593
  proof
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1594
    let ?Z = "map_filter_on {(x, y). A y x} prod.swap Z"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1595
    show "\<forall>\<^sub>F (x, y) in ?Z. A x y" using that by(simp add: eventually_map_filter_on)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1596
    have [simp]: "prod.swap ` {(x, y). A y x} \<subseteq> {(x, y). A x y}" by auto
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1597
    show "map_filter_on {(x, y). A x y} fst ?Z = ?X" "map_filter_on {(x, y). A x y} snd ?Z = ?Y"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1598
      using that by(simp_all add: map_filter_on_comp o_def)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1599
  qed
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1600
  show "rel_filter A\<inverse>\<inverse> (map_filter_on {(x, y). A x y} snd Z) (map_filter_on {(x, y). A x y} fst Z)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1601
    if "\<forall>\<^sub>F (x, y) in Z. A x y" for Z using *[of "A\<inverse>\<inverse>" Z] that by simp
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1602
qed
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1603
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1604
lemma rel_filter_distr [relator_distr]:
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1605
  "rel_filter A OO rel_filter B = rel_filter (A OO B)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1606
proof(safe intro!: ext elim!: rel_filter.cases)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1607
  let ?AB = "{(x, y). (A OO B) x y}"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1608
  show "(rel_filter A OO rel_filter B)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1609
     (map_filter_on {(x, y). (A OO B) x y} fst Z) (map_filter_on {(x, y). (A OO B) x y} snd Z)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1610
    (is "(_ OO _) ?F ?H") if "\<forall>\<^sub>F (x, y) in Z. (A OO B) x y" for Z
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1611
  proof
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1612
    let ?G = "map_filter_on ?AB (\<lambda>(x, y). SOME z. A x z \<and> B z y) Z"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1613
    show "rel_filter A ?F ?G"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1614
    proof
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1615
      let ?Z = "map_filter_on ?AB (\<lambda>(x, y). (x, SOME z. A x z \<and> B z y)) Z"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1616
      show "\<forall>\<^sub>F (x, y) in ?Z. A x y" using that
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1617
        by(auto simp add: eventually_map_filter_on split_def elim!: eventually_mono intro: someI2)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1618
      have [simp]: "(\<lambda>p. (fst p, SOME z. A (fst p) z \<and> B z (snd p))) ` {p. (A OO B) (fst p) (snd p)} \<subseteq> {p. A (fst p) (snd p)}" by(auto intro: someI2)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1619
      show "map_filter_on {(x, y). A x y} fst ?Z = ?F" "map_filter_on {(x, y). A x y} snd ?Z = ?G"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1620
        using that by(simp_all add: map_filter_on_comp split_def o_def)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1621
    qed
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1622
    show "rel_filter B ?G ?H"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1623
    proof
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1624
      let ?Z = "map_filter_on ?AB (\<lambda>(x, y). (SOME z. A x z \<and> B z y, y)) Z"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1625
      show "\<forall>\<^sub>F (x, y) in ?Z. B x y" using that
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1626
        by(auto simp add: eventually_map_filter_on split_def elim!: eventually_mono intro: someI2)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1627
      have [simp]: "(\<lambda>p. (SOME z. A (fst p) z \<and> B z (snd p), snd p)) ` {p. (A OO B) (fst p) (snd p)} \<subseteq> {p. B (fst p) (snd p)}" by(auto intro: someI2)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1628
      show "map_filter_on {(x, y). B x y} fst ?Z = ?G" "map_filter_on {(x, y). B x y} snd ?Z = ?H"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1629
        using that by(simp_all add: map_filter_on_comp split_def o_def)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1630
    qed
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1631
  qed
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1632
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1633
  fix F G
67616
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1634
  assume F: "\<forall>\<^sub>F (x, y) in F. A x y" and G: "\<forall>\<^sub>F (x, y) in G. B x y"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1635
    and eq: "map_filter_on {(x, y). B x y} fst G = map_filter_on {(x, y). A x y} snd F" (is "?Y2 = ?Y1")
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1636
  let ?X = "map_filter_on {(x, y). A x y} fst F"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1637
    and ?Z = "(map_filter_on {(x, y). B x y} snd G)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1638
  have step: "\<exists>P'\<le>P. \<exists>Q' \<le> Q. eventually P' F \<and> eventually Q' G \<and> {y. \<exists>x. P' (x, y)} = {y. \<exists>z. Q' (y, z)}"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1639
    if P: "eventually P F" and Q: "eventually Q G" for P Q
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1640
  proof -
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1641
    let ?P = "\<lambda>(x, y). P (x, y) \<and> A x y" and ?Q = "\<lambda>(y, z). Q (y, z) \<and> B y z"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1642
    define P' where "P' \<equiv> \<lambda>(x, y). ?P (x, y) \<and> (\<exists>z. ?Q (y, z))"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1643
    define Q' where "Q' \<equiv> \<lambda>(y, z). ?Q (y, z) \<and> (\<exists>x. ?P (x, y))"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1644
    have "P' \<le> P" "Q' \<le> Q" "{y. \<exists>x. P' (x, y)} = {y. \<exists>z. Q' (y, z)}"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1645
      by(auto simp add: P'_def Q'_def)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1646
    moreover
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1647
    from P Q F G have P': "eventually ?P F" and Q': "eventually ?Q G" 
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1648
      by(simp_all add: eventually_conj_iff split_def)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1649
    from P' F have "\<forall>\<^sub>F y in ?Y1. \<exists>x. P (x, y) \<and> A x y"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1650
      by(auto simp add: eventually_map_filter_on elim!: eventually_mono)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1651
    from this[folded eq] obtain Q'' where Q'': "eventually Q'' G"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1652
      and Q''P: "{y. \<exists>z. Q'' (y, z)} \<subseteq> {y. \<exists>x. ?P (x, y)}"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1653
      using G by(fastforce simp add: eventually_map_filter_on)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1654
    have "eventually (inf Q'' ?Q) G" using Q'' Q' by(auto intro: eventually_conj simp add: inf_fun_def)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1655
    then have "eventually Q' G" using Q''P  by(auto elim!: eventually_mono simp add: Q'_def)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1656
    moreover
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1657
    from Q' G have "\<forall>\<^sub>F y in ?Y2. \<exists>z. Q (y, z) \<and> B y z"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1658
      by(auto simp add: eventually_map_filter_on elim!: eventually_mono)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1659
    from this[unfolded eq] obtain P'' where P'': "eventually P'' F"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1660
      and P''Q: "{y. \<exists>x. P'' (x, y)} \<subseteq> {y. \<exists>z. ?Q (y, z)}"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1661
      using F by(fastforce simp add: eventually_map_filter_on)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1662
    have "eventually (inf P'' ?P) F" using P'' P' by(auto intro: eventually_conj simp add: inf_fun_def)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1663
    then have "eventually P' F" using P''Q  by(auto elim!: eventually_mono simp add: P'_def)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1664
    ultimately show ?thesis by blast
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1665
  qed
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1666
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1667
  show "rel_filter (A OO B) ?X ?Z"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1668
  proof
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1669
    let ?Y = "\<lambda>Y. \<exists>X Z. eventually X ?X \<and> eventually Z ?Z \<and> (\<lambda>(x, z). X x \<and> Z z \<and> (A OO B) x z) \<le> Y"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1670
    have Y: "is_filter ?Y"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1671
    proof
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1672
      show "?Y (\<lambda>_. True)" by(auto simp add: le_fun_def intro: eventually_True)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1673
      show "?Y (\<lambda>x. P x \<and> Q x)" if "?Y P" "?Y Q" for P Q using that
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1674
        apply clarify
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1675
        apply(intro exI conjI; (elim eventually_rev_mp; fold imp_conjL; intro always_eventually allI; rule imp_refl)?)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1676
        apply auto
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1677
        done
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1678
      show "?Y Q" if "?Y P" "\<forall>x. P x \<longrightarrow> Q x" for P Q using that by blast
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1679
    qed
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1680
    define Y where "Y = Abs_filter ?Y"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1681
    have eventually_Y: "eventually P Y \<longleftrightarrow> ?Y P" for P
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1682
      using eventually_Abs_filter[OF Y, of P] by(simp add: Y_def)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1683
    show YY: "\<forall>\<^sub>F (x, y) in Y. (A OO B) x y" using F G
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1684
      by(auto simp add: eventually_Y eventually_map_filter_on eventually_conj_iff intro!: eventually_True)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1685
    have "?Y (\<lambda>(x, z). P x \<and> (A OO B) x z) \<longleftrightarrow> (\<forall>\<^sub>F (x, y) in F. P x \<and> A x y)" (is "?lhs = ?rhs") for P
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1686
    proof
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1687
      show ?lhs if ?rhs using G F that
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1688
        by(auto 4 3 intro: exI[where x="\<lambda>_. True"] simp add: eventually_map_filter_on split_def)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1689
      assume ?lhs
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1690
      then obtain X Z where "\<forall>\<^sub>F (x, y) in F. X x \<and> A x y"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1691
        and "\<forall>\<^sub>F (x, y) in G. Z y \<and> B x y"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1692
        and "(\<lambda>(x, z). X x \<and> Z z \<and> (A OO B) x z) \<le> (\<lambda>(x, z). P x \<and> (A OO B) x z)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1693
        using F G by(auto simp add: eventually_map_filter_on split_def)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1694
      from step[OF this(1, 2)] this(3)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1695
      show ?rhs by(clarsimp elim!: eventually_rev_mp simp add: le_fun_def)(fastforce intro: always_eventually)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1696
    qed
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1697
    then show "map_filter_on ?AB fst Y = ?X"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1698
      by(simp add: filter_eq_iff YY eventually_map_filter_on)(simp add: eventually_Y eventually_map_filter_on F G; simp add: split_def)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1699
67616
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1700
    have "?Y (\<lambda>(x, z). P z \<and> (A OO B) x z) \<longleftrightarrow> (\<forall>\<^sub>F (x, y) in G. P y \<and> B x y)" (is "?lhs = ?rhs") for P
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1701
    proof
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1702
      show ?lhs if ?rhs using G F that
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1703
        by(auto 4 3 intro: exI[where x="\<lambda>_. True"] simp add: eventually_map_filter_on split_def)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1704
      assume ?lhs
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1705
      then obtain X Z where "\<forall>\<^sub>F (x, y) in F. X x \<and> A x y"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1706
        and "\<forall>\<^sub>F (x, y) in G. Z y \<and> B x y"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1707
        and "(\<lambda>(x, z). X x \<and> Z z \<and> (A OO B) x z) \<le> (\<lambda>(x, z). P z \<and> (A OO B) x z)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1708
        using F G by(auto simp add: eventually_map_filter_on split_def)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1709
      from step[OF this(1, 2)] this(3)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1710
      show ?rhs by(clarsimp elim!: eventually_rev_mp simp add: le_fun_def)(fastforce intro: always_eventually)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1711
    qed
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1712
    then show "map_filter_on ?AB snd Y = ?Z"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1713
      by(simp add: filter_eq_iff YY eventually_map_filter_on)(simp add: eventually_Y eventually_map_filter_on F G; simp add: split_def)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1714
  qed
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1715
qed
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1716
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1717
lemma filtermap_parametric: "((A ===> B) ===> rel_filter A ===> rel_filter B) filtermap filtermap"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1718
proof(intro rel_funI; erule rel_filter.cases; hypsubst)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1719
  fix f g Z
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1720
  assume fg: "(A ===> B) f g" and Z: "\<forall>\<^sub>F (x, y) in Z. A x y"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1721
  have "rel_filter B (map_filter_on {(x, y). A x y} (f \<circ> fst) Z) (map_filter_on {(x, y). A x y} (g \<circ> snd) Z)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1722
    (is "rel_filter _ ?F ?G")
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1723
  proof
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1724
    let ?Z = "map_filter_on {(x, y). A x y} (map_prod f g) Z"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1725
    show "\<forall>\<^sub>F (x, y) in ?Z. B x y" using fg Z
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1726
      by(auto simp add: eventually_map_filter_on split_def elim!: eventually_mono rel_funD)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1727
    have [simp]: "map_prod f g ` {p. A (fst p) (snd p)} \<subseteq> {p. B (fst p) (snd p)}"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1728
      using fg by(auto dest: rel_funD)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1729
    show "map_filter_on {(x, y). B x y} fst ?Z = ?F" "map_filter_on {(x, y). B x y} snd ?Z = ?G"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1730
      using Z by(auto simp add: map_filter_on_comp split_def)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1731
  qed
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1732
  thus "rel_filter B (filtermap f (map_filter_on {(x, y). A x y} fst Z)) (filtermap g (map_filter_on {(x, y). A x y} snd Z))"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1733
    using Z by(simp add: map_filter_on_UNIV[symmetric] map_filter_on_comp)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1734
qed
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1735
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1736
lemma rel_filter_Grp: "rel_filter (Grp UNIV f) = Grp UNIV (filtermap f)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1737
proof((intro antisym predicate2I; (elim GrpE; hypsubst)?), rule GrpI[OF _ UNIV_I])
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1738
  fix F G
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1739
  assume "rel_filter (Grp UNIV f) F G"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1740
  hence "rel_filter (=) (filtermap f F) (filtermap id G)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1741
    by(rule filtermap_parametric[THEN rel_funD, THEN rel_funD, rotated])(simp add: Grp_def rel_fun_def)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1742
  thus "filtermap f F = G" by(simp add: rel_filter_eq)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1743
next
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1744
  fix F :: "'a filter"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1745
  have "rel_filter (=) F F" by(simp add: rel_filter_eq)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1746
  hence "rel_filter (Grp UNIV f) (filtermap id F) (filtermap f F)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1747
    by(rule filtermap_parametric[THEN rel_funD, THEN rel_funD, rotated])(simp add: Grp_def rel_fun_def)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1748
  thus "rel_filter (Grp UNIV f) F (filtermap f F)" by simp
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1749
qed
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1750
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1751
lemma Quotient_filter [quot_map]:
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1752
  "Quotient R Abs Rep T \<Longrightarrow> Quotient (rel_filter R) (filtermap Abs) (filtermap Rep) (rel_filter T)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1753
  unfolding Quotient_alt_def5 rel_filter_eq[symmetric] rel_filter_Grp[symmetric]
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1754
  by(simp add: rel_filter_distr[symmetric] rel_filter_conversep[symmetric] rel_filter_mono)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1755
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1756
lemma left_total_rel_filter [transfer_rule]: "left_total A \<Longrightarrow> left_total (rel_filter A)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1757
unfolding left_total_alt_def rel_filter_eq[symmetric] rel_filter_conversep[symmetric] rel_filter_distr
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1758
by(rule rel_filter_mono)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1759
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1760
lemma right_total_rel_filter [transfer_rule]: "right_total A \<Longrightarrow> right_total (rel_filter A)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1761
using left_total_rel_filter[of "A\<inverse>\<inverse>"] by(simp add: rel_filter_conversep)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1762
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1763
lemma bi_total_rel_filter [transfer_rule]: "bi_total A \<Longrightarrow> bi_total (rel_filter A)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1764
unfolding bi_total_alt_def by(simp add: left_total_rel_filter right_total_rel_filter)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1765
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1766
lemma left_unique_rel_filter [transfer_rule]: "left_unique A \<Longrightarrow> left_unique (rel_filter A)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1767
unfolding left_unique_alt_def rel_filter_eq[symmetric] rel_filter_conversep[symmetric] rel_filter_distr
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1768
by(rule rel_filter_mono)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1769
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1770
lemma right_unique_rel_filter [transfer_rule]:
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1771
  "right_unique A \<Longrightarrow> right_unique (rel_filter A)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1772
using left_unique_rel_filter[of "A\<inverse>\<inverse>"] by(simp add: rel_filter_conversep)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1773
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1774
lemma bi_unique_rel_filter [transfer_rule]: "bi_unique A \<Longrightarrow> bi_unique (rel_filter A)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1775
by(simp add: bi_unique_alt_def left_unique_rel_filter right_unique_rel_filter)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1776
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1777
lemma eventually_parametric [transfer_rule]:
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66171
diff changeset
  1778
  "((A ===> (=)) ===> rel_filter A ===> (=)) eventually eventually"
67616
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1779
by(auto 4 4 intro!: rel_funI elim!: rel_filter.cases simp add: eventually_map_filter_on dest: rel_funD intro: always_eventually elim!: eventually_rev_mp)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1780
67616
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1781
lemma frequently_parametric [transfer_rule]: "((A ===> (=)) ===> rel_filter A ===> (=)) frequently frequently"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1782
  unfolding frequently_def[abs_def] by transfer_prover
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1783
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1784
lemma is_filter_parametric [transfer_rule]:
67956
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1785
  assumes [transfer_rule]: "bi_total A"
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1786
  assumes [transfer_rule]: "bi_unique A"
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1787
  shows "(((A ===> (=)) ===> (=)) ===> (=)) is_filter is_filter"
79dbb9dccc99 tuned some proofs about filters
huffman
parents: 67950
diff changeset
  1788
  unfolding is_filter_def by transfer_prover
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1789
67616
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1790
lemma top_filter_parametric [transfer_rule]: "rel_filter A top top" if "bi_total A"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1791
proof
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1792
  let ?Z = "principal {(x, y). A x y}"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1793
  show "\<forall>\<^sub>F (x, y) in ?Z. A x y" by(simp add: eventually_principal)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1794
  show "map_filter_on {(x, y). A x y} fst ?Z = top" "map_filter_on {(x, y). A x y} snd ?Z = top"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1795
    using that by(auto simp add: filter_eq_iff eventually_map_filter_on eventually_principal bi_total_def)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1796
qed
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1797
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1798
lemma bot_filter_parametric [transfer_rule]: "rel_filter A bot bot"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1799
proof
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1800
  show "\<forall>\<^sub>F (x, y) in bot. A x y" by simp
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1801
  show "map_filter_on {(x, y). A x y} fst bot = bot" "map_filter_on {(x, y). A x y} snd bot = bot"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1802
    by(simp_all add: filter_eq_iff eventually_map_filter_on)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1803
qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1804
67616
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1805
lemma principal_parametric [transfer_rule]: "(rel_set A ===> rel_filter A) principal principal"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1806
proof(rule rel_funI rel_filter.intros)+
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1807
  fix S S'
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1808
  assume *: "rel_set A S S'"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1809
  define SS' where "SS' = S \<times> S' \<inter> {(x, y). A x y}"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1810
  have SS': "SS' \<subseteq> {(x, y). A x y}" and [simp]: "S = fst ` SS'" "S' = snd ` SS'"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1811
    using * by(auto 4 3 dest: rel_setD1 rel_setD2 intro: rev_image_eqI simp add: SS'_def)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1812
  let ?Z = "principal SS'"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1813
  show "\<forall>\<^sub>F (x, y) in ?Z. A x y" using SS' by(auto simp add: eventually_principal)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1814
  then show "map_filter_on {(x, y). A x y} fst ?Z = principal S"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1815
    and "map_filter_on {(x, y). A x y} snd ?Z = principal S'"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1816
    by(auto simp add: filter_eq_iff eventually_map_filter_on eventually_principal)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1817
qed
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1818
67616
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1819
lemma sup_filter_parametric [transfer_rule]:
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1820
  "(rel_filter A ===> rel_filter A ===> rel_filter A) sup sup"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1821
proof(intro rel_funI; elim rel_filter.cases; hypsubst)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1822
  show "rel_filter A
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1823
    (map_filter_on {(x, y). A x y} fst FG \<squnion> map_filter_on {(x, y). A x y} fst FG')
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1824
    (map_filter_on {(x, y). A x y} snd FG \<squnion> map_filter_on {(x, y). A x y} snd FG')"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1825
    (is "rel_filter _ (sup ?F ?G) (sup ?F' ?G')")
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1826
    if "\<forall>\<^sub>F (x, y) in FG. A x y" "\<forall>\<^sub>F (x, y) in FG'. A x y" for FG FG'
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1827
  proof
67616
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1828
    let ?Z = "sup FG FG'"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1829
    show "\<forall>\<^sub>F (x, y) in ?Z. A x y" by(simp add: eventually_sup that)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1830
    then show "map_filter_on {(x, y). A x y} fst ?Z = sup ?F ?G" 
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1831
      and "map_filter_on {(x, y). A x y} snd ?Z = sup ?F' ?G'"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1832
      by(simp_all add: filter_eq_iff eventually_map_filter_on eventually_sup)
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1833
  qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1834
qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1835
67616
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1836
lemma Sup_filter_parametric [transfer_rule]: "(rel_set (rel_filter A) ===> rel_filter A) Sup Sup"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1837
proof(rule rel_funI)
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1838
  fix S S'
67616
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1839
  define SS' where "SS' = S \<times> S' \<inter> {(F, G). rel_filter A F G}"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1840
  assume "rel_set (rel_filter A) S S'"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1841
  then have SS': "SS' \<subseteq> {(F, G). rel_filter A F G}" and [simp]: "S = fst ` SS'" "S' = snd ` SS'"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1842
    by(auto 4 3 dest: rel_setD1 rel_setD2 intro: rev_image_eqI simp add: SS'_def)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1843
  from SS' obtain Z where Z: "\<And>F G. (F, G) \<in> SS' \<Longrightarrow>
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1844
    (\<forall>\<^sub>F (x, y) in Z F G. A x y) \<and>
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1845
    id F = map_filter_on {(x, y). A x y} fst (Z F G) \<and>
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1846
    id G = map_filter_on {(x, y). A x y} snd (Z F G)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1847
    unfolding rel_filter.simps by atomize_elim((rule choice allI)+; auto)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1848
  have id: "eventually P F = eventually P (id F)" "eventually Q G = eventually Q (id G)"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1849
    if "(F, G) \<in> SS'" for P Q F G by simp_all
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1850
  show "rel_filter A (Sup S) (Sup S')"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1851
  proof
69260
0a9688695a1b removed relics of ASCII syntax for indexed big operators
haftmann
parents: 68860
diff changeset
  1852
    let ?Z = "\<Squnion>(F, G)\<in>SS'. Z F G"
67616
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1853
    show *: "\<forall>\<^sub>F (x, y) in ?Z. A x y" using Z by(auto simp add: eventually_Sup)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1854
    show "map_filter_on {(x, y). A x y} fst ?Z = Sup S" "map_filter_on {(x, y). A x y} snd ?Z = Sup S'"
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1855
      unfolding filter_eq_iff
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1856
      by(auto 4 4 simp add: id eventually_Sup eventually_map_filter_on *[simplified eventually_Sup] simp del: id_apply dest: Z)
1d005f514417 strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents: 67613
diff changeset
  1857
  qed
66162
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
  1858
qed
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
  1859
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1860
context
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1861
  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
  1862
  assumes [transfer_rule]: "bi_unique A"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1863
begin
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1864
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1865
lemma le_filter_parametric [transfer_rule]:
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66171
diff changeset
  1866
  "(rel_filter A ===> rel_filter A ===> (=)) (\<le>) (\<le>)"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1867
unfolding le_filter_def[abs_def] by transfer_prover
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1868
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1869
lemma less_filter_parametric [transfer_rule]:
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66171
diff changeset
  1870
  "(rel_filter A ===> rel_filter A ===> (=)) (<) (<)"
60036
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1871
unfolding less_filter_def[abs_def] by transfer_prover
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1872
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1873
context
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1874
  assumes [transfer_rule]: "bi_total A"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1875
begin
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1876
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1877
lemma Inf_filter_parametric [transfer_rule]:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1878
  "(rel_set (rel_filter A) ===> rel_filter A) Inf Inf"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1879
unfolding Inf_filter_def[abs_def] by transfer_prover
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1880
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1881
lemma inf_filter_parametric [transfer_rule]:
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1882
  "(rel_filter A ===> rel_filter A ===> rel_filter A) inf inf"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1883
proof(intro rel_funI)+
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1884
  fix F F' G G'
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1885
  assume [transfer_rule]: "rel_filter A F F'" "rel_filter A G G'"
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1886
  have "rel_filter A (Inf {F, G}) (Inf {F', G'})" by transfer_prover
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1887
  thus "rel_filter A (inf F G) (inf F' G')" by simp
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1888
qed
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1889
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1890
end
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1891
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1892
end
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1893
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1894
end
218fcc645d22 move filters to their own theory
hoelzl
parents:
diff changeset
  1895
70927
cc204e10385c tuned syntax
haftmann
parents: 69593
diff changeset
  1896
context
cc204e10385c tuned syntax
haftmann
parents: 69593
diff changeset
  1897
  includes lifting_syntax
cc204e10385c tuned syntax
haftmann
parents: 69593
diff changeset
  1898
begin
cc204e10385c tuned syntax
haftmann
parents: 69593
diff changeset
  1899
cc204e10385c tuned syntax
haftmann
parents: 69593
diff changeset
  1900
lemma prod_filter_parametric [transfer_rule]:
68667
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1901
   "(rel_filter R ===> rel_filter S ===> rel_filter (rel_prod R S)) prod_filter prod_filter"
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1902
proof(intro rel_funI; elim rel_filter.cases; hypsubst)
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1903
  fix F G
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1904
  assume F: "\<forall>\<^sub>F (x, y) in F. R x y" and G: "\<forall>\<^sub>F (x, y) in G. S x y"
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1905
  show "rel_filter (rel_prod R S)
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1906
    (map_filter_on {(x, y). R x y} fst F \<times>\<^sub>F map_filter_on {(x, y). S x y} fst G)
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1907
    (map_filter_on {(x, y). R x y} snd F \<times>\<^sub>F map_filter_on {(x, y). S x y} snd G)"
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1908
    (is "rel_filter ?RS ?F ?G")
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1909
  proof
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1910
    let ?Z = "filtermap (\<lambda>((a, b), (a', b')). ((a, a'), (b, b'))) (prod_filter F G)"
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1911
    show *: "\<forall>\<^sub>F (x, y) in ?Z. rel_prod R S x y" using F G
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1912
      by(auto simp add: eventually_filtermap split_beta eventually_prod_filter)
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1913
    show "map_filter_on {(x, y). ?RS x y} fst ?Z = ?F"
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1914
      using F G
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1915
      apply(clarsimp simp add: filter_eq_iff eventually_map_filter_on *)
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1916
      apply(simp add: eventually_filtermap split_beta eventually_prod_filter)
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1917
      apply(subst eventually_map_filter_on; simp)+
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1918
      apply(rule iffI; clarsimp)
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1919
      subgoal for P P' P''
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1920
        apply(rule exI[where x="\<lambda>a. \<exists>b. P' (a, b) \<and> R a b"]; rule conjI)
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1921
        subgoal by(fastforce elim: eventually_rev_mp eventually_mono)
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1922
        subgoal
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1923
          by(rule exI[where x="\<lambda>a. \<exists>b. P'' (a, b) \<and> S a b"])(fastforce elim: eventually_rev_mp eventually_mono)
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1924
        done
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1925
      subgoal by fastforce
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1926
      done
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1927
    show "map_filter_on {(x, y). ?RS x y} snd ?Z = ?G"
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1928
      using F G
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1929
      apply(clarsimp simp add: filter_eq_iff eventually_map_filter_on *)
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1930
      apply(simp add: eventually_filtermap split_beta eventually_prod_filter)
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1931
      apply(subst eventually_map_filter_on; simp)+
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1932
      apply(rule iffI; clarsimp)
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1933
      subgoal for P P' P''
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1934
        apply(rule exI[where x="\<lambda>b. \<exists>a. P' (a, b) \<and> R a b"]; rule conjI)
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1935
        subgoal by(fastforce elim: eventually_rev_mp eventually_mono)
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1936
        subgoal
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1937
          by(rule exI[where x="\<lambda>b. \<exists>a. P'' (a, b) \<and> S a b"])(fastforce elim: eventually_rev_mp eventually_mono)
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1938
        done
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1939
      subgoal by fastforce
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1940
      done
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1941
  qed
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1942
qed
96aae7c44bb2 add lemmas about prod_filter
Andreas Lochbihler
parents: 67956
diff changeset
  1943
70927
cc204e10385c tuned syntax
haftmann
parents: 69593
diff changeset
  1944
end
cc204e10385c tuned syntax
haftmann
parents: 69593
diff changeset
  1945
cc204e10385c tuned syntax
haftmann
parents: 69593
diff changeset
  1946
62123
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1947
text \<open>Code generation for filters\<close>
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1948
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1949
definition abstract_filter :: "(unit \<Rightarrow> 'a filter) \<Rightarrow> 'a filter"
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1950
  where [simp]: "abstract_filter f = f ()"
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1951
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1952
code_datatype principal abstract_filter
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1953
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1954
hide_const (open) abstract_filter
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1955
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1956
declare [[code drop: filterlim prod_filter filtermap eventually
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1957
  "inf :: _ filter \<Rightarrow> _" "sup :: _ filter \<Rightarrow> _" "less_eq :: _ filter \<Rightarrow> _"
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1958
  Abs_filter]]
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1959
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1960
declare filterlim_principal [code]
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1961
declare principal_prod_principal [code]
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1962
declare filtermap_principal [code]
66162
65cd285f6b9c Contravariant map on filters
eberlm <eberlm@in.tum.de>
parents: 65578
diff changeset
  1963
declare filtercomap_principal [code]
62123
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1964
declare eventually_principal [code]
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1965
declare inf_principal [code]
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1966
declare sup_principal [code]
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1967
declare principal_le_iff [code]
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1968
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1969
lemma Rep_filter_iff_eventually [simp, code]:
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1970
  "Rep_filter F P \<longleftrightarrow> eventually P F"
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1971
  by (simp add: eventually_def)
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1972
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1973
lemma bot_eq_principal_empty [code]:
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1974
  "bot = principal {}"
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1975
  by simp
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1976
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1977
lemma top_eq_principal_UNIV [code]:
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1978
  "top = principal UNIV"
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1979
  by simp
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1980
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1981
instantiation filter :: (equal) equal
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1982
begin
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1983
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1984
definition equal_filter :: "'a filter \<Rightarrow> 'a filter \<Rightarrow> bool"
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1985
  where "equal_filter F F' \<longleftrightarrow> F = F'"
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1986
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1987
lemma equal_filter [code]:
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1988
  "HOL.equal (principal A) (principal B) \<longleftrightarrow> A = B"
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1989
  by (simp add: equal_filter_def)
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1990
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1991
instance
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1992
  by standard (simp add: equal_filter_def)
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1993
61806
d2e62ae01cd8 Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
  1994
end
62123
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1995
df65f5c27c15 setup code generation for filters as suggested by Florian
hoelzl
parents: 62102
diff changeset
  1996
end