author  paulson <lp15@cam.ac.uk> 
Mon, 22 Feb 2016 14:37:56 +0000  
changeset 62379  340738057c8c 
parent 62378  85ed00c1fe7c 
child 63343  fb5d8a50c641 
permissions  rwrr 
60036  1 
(* 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 nonproper filters. 
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\<close> 
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locale is_filter = 

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fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool" 

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assumes True: "F (\<lambda>x. True)" 

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assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)" 

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assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)" 

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typedef 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}" 

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proof 

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show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro) 

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qed 

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lemma is_filter_Rep_filter: "is_filter (Rep_filter F)" 

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using Rep_filter [of F] by simp 

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lemma Abs_filter_inverse': 

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assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F" 

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using assms by (simp add: Abs_filter_inverse) 

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subsubsection \<open>Eventually\<close> 
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definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool" 

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where "eventually P F \<longleftrightarrow> Rep_filter F P" 

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syntax 
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"_eventually" :: "pttrn => 'a filter => bool => bool" ("(3\<forall>\<^sub>F _ in _./ _)" [0, 0, 10] 10) 

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translations 
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"\<forall>\<^sub>Fx in F. P" == "CONST eventually (\<lambda>x. P) F" 
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lemma eventually_Abs_filter: 
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assumes "is_filter F" shows "eventually P (Abs_filter F) = F P" 

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unfolding eventually_def using assms by (simp add: Abs_filter_inverse) 

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lemma filter_eq_iff: 

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shows "F = F' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')" 

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unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def .. 

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lemma eventually_True [simp]: "eventually (\<lambda>x. True) F" 

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unfolding eventually_def 

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by (rule is_filter.True [OF is_filter_Rep_filter]) 

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lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P F" 

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proof  

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assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext) 

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thus "eventually P F" by simp 

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qed 

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lemma eventuallyI: "(\<And>x. P x) \<Longrightarrow> eventually P F" 
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by (auto intro: always_eventually) 
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lemma eventually_mono: 
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"\<lbrakk>eventually P F; \<And>x. P x \<Longrightarrow> Q x\<rbrakk> \<Longrightarrow> eventually Q F" 
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unfolding eventually_def 
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by (blast intro: is_filter.mono [OF is_filter_Rep_filter]) 
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lemma eventually_conj: 
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assumes P: "eventually (\<lambda>x. P x) F" 

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assumes Q: "eventually (\<lambda>x. Q x) F" 

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shows "eventually (\<lambda>x. P x \<and> Q x) F" 

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using assms unfolding eventually_def 

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by (rule is_filter.conj [OF is_filter_Rep_filter]) 

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lemma eventually_mp: 

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assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F" 

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assumes "eventually (\<lambda>x. P x) F" 

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shows "eventually (\<lambda>x. Q x) F" 

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proof  
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have "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F" 
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using assms by (rule eventually_conj) 
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then show ?thesis 
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by (blast intro: eventually_mono) 
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qed 
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lemma eventually_rev_mp: 

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assumes "eventually (\<lambda>x. P x) F" 

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assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F" 

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shows "eventually (\<lambda>x. Q x) F" 

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using assms(2) assms(1) by (rule eventually_mp) 

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lemma eventually_conj_iff: 

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"eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F" 

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by (auto intro: eventually_conj elim: eventually_rev_mp) 

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lemma eventually_elim2: 

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assumes "eventually (\<lambda>i. P i) F" 

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assumes "eventually (\<lambda>i. Q i) F" 

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assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i" 

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shows "eventually (\<lambda>i. R i) F" 

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using assms by (auto elim!: eventually_rev_mp) 

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lemma eventually_ball_finite_distrib: 
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"finite A \<Longrightarrow> (eventually (\<lambda>x. \<forall>y\<in>A. P x y) net) \<longleftrightarrow> (\<forall>y\<in>A. eventually (\<lambda>x. P x y) net)" 
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by (induction A rule: finite_induct) (auto simp: eventually_conj_iff) 
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lemma eventually_ball_finite: 
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"finite A \<Longrightarrow> \<forall>y\<in>A. eventually (\<lambda>x. P x y) net \<Longrightarrow> eventually (\<lambda>x. \<forall>y\<in>A. P x y) net" 
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by (auto simp: eventually_ball_finite_distrib) 
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lemma eventually_all_finite: 
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fixes P :: "'a \<Rightarrow> 'b::finite \<Rightarrow> bool" 
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assumes "\<And>y. eventually (\<lambda>x. P x y) net" 
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shows "eventually (\<lambda>x. \<forall>y. P x y) net" 
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using eventually_ball_finite [of UNIV P] assms by simp 
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lemma eventually_ex: "(\<forall>\<^sub>Fx in F. \<exists>y. P x y) \<longleftrightarrow> (\<exists>Y. \<forall>\<^sub>Fx in F. P x (Y x))" 
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proof 
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assume "\<forall>\<^sub>Fx in F. \<exists>y. P x y" 
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then have "\<forall>\<^sub>Fx in F. P x (SOME y. P x y)" 
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by (auto intro: someI_ex eventually_mono) 
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then show "\<exists>Y. \<forall>\<^sub>Fx in F. P x (Y x)" 
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by auto 
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qed (auto intro: eventually_mono) 
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lemma not_eventually_impI: "eventually P F \<Longrightarrow> \<not> eventually Q F \<Longrightarrow> \<not> eventually (\<lambda>x. P x \<longrightarrow> Q x) F" 
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by (auto intro: eventually_mp) 

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lemma not_eventuallyD: "\<not> eventually P F \<Longrightarrow> \<exists>x. \<not> P x" 

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by (metis always_eventually) 

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lemma eventually_subst: 

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assumes "eventually (\<lambda>n. P n = Q n) F" 

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shows "eventually P F = eventually Q F" (is "?L = ?R") 

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proof  

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from assms have "eventually (\<lambda>x. P x \<longrightarrow> Q x) F" 

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and "eventually (\<lambda>x. Q x \<longrightarrow> P x) F" 

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by (auto elim: eventually_mono) 
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then show ?thesis by (auto elim: eventually_elim2) 
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qed 

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subsection \<open> Frequently as dual to eventually \<close> 
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definition frequently :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool" 
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where "frequently P F \<longleftrightarrow> \<not> eventually (\<lambda>x. \<not> P x) F" 
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syntax 
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"_frequently" :: "pttrn \<Rightarrow> 'a filter \<Rightarrow> bool \<Rightarrow> bool" ("(3\<exists>\<^sub>F _ in _./ _)" [0, 0, 10] 10) 

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translations 
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"\<exists>\<^sub>Fx in F. P" == "CONST frequently (\<lambda>x. P) F" 
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lemma not_frequently_False [simp]: "\<not> (\<exists>\<^sub>Fx in F. False)" 
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by (simp add: frequently_def) 
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lemma frequently_ex: "\<exists>\<^sub>Fx in F. P x \<Longrightarrow> \<exists>x. P x" 
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by (auto simp: frequently_def dest: not_eventuallyD) 
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lemma frequentlyE: assumes "frequently P F" obtains x where "P x" 
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using frequently_ex[OF assms] by auto 
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lemma frequently_mp: 
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assumes ev: "\<forall>\<^sub>Fx in F. P x \<longrightarrow> Q x" and P: "\<exists>\<^sub>Fx in F. P x" shows "\<exists>\<^sub>Fx in F. Q x" 
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proof  
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from ev have "eventually (\<lambda>x. \<not> Q x \<longrightarrow> \<not> P x) F" 
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by (rule eventually_rev_mp) (auto intro!: always_eventually) 
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from eventually_mp[OF this] P show ?thesis 
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by (auto simp: frequently_def) 
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qed 
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lemma frequently_rev_mp: 
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assumes "\<exists>\<^sub>Fx in F. P x" 
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assumes "\<forall>\<^sub>Fx in F. P x \<longrightarrow> Q x" 
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shows "\<exists>\<^sub>Fx in F. Q x" 
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using assms(2) assms(1) by (rule frequently_mp) 
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lemma frequently_mono: "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> frequently P F \<Longrightarrow> frequently Q F" 
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using frequently_mp[of P Q] by (simp add: always_eventually) 
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lemma frequently_elim1: "\<exists>\<^sub>Fx in F. P x \<Longrightarrow> (\<And>i. P i \<Longrightarrow> Q i) \<Longrightarrow> \<exists>\<^sub>Fx in F. Q x" 
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by (metis frequently_mono) 
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lemma frequently_disj_iff: "(\<exists>\<^sub>Fx in F. P x \<or> Q x) \<longleftrightarrow> (\<exists>\<^sub>Fx in F. P x) \<or> (\<exists>\<^sub>Fx in F. Q x)" 
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by (simp add: frequently_def eventually_conj_iff) 
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lemma frequently_disj: "\<exists>\<^sub>Fx in F. P x \<Longrightarrow> \<exists>\<^sub>Fx in F. Q x \<Longrightarrow> \<exists>\<^sub>Fx in F. P x \<or> Q x" 
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by (simp add: frequently_disj_iff) 
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lemma frequently_bex_finite_distrib: 
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assumes "finite A" shows "(\<exists>\<^sub>Fx in F. \<exists>y\<in>A. P x y) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>\<^sub>Fx in F. P x y)" 
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using assms by induction (auto simp: frequently_disj_iff) 
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lemma frequently_bex_finite: "finite A \<Longrightarrow> \<exists>\<^sub>Fx in F. \<exists>y\<in>A. P x y \<Longrightarrow> \<exists>y\<in>A. \<exists>\<^sub>Fx in F. P x y" 
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by (simp add: frequently_bex_finite_distrib) 
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lemma frequently_all: "(\<exists>\<^sub>Fx in F. \<forall>y. P x y) \<longleftrightarrow> (\<forall>Y. \<exists>\<^sub>Fx in F. P x (Y x))" 
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using eventually_ex[of "\<lambda>x y. \<not> P x y" F] by (simp add: frequently_def) 
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lemma 
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shows not_eventually: "\<not> eventually P F \<longleftrightarrow> (\<exists>\<^sub>Fx in F. \<not> P x)" 
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and not_frequently: "\<not> frequently P F \<longleftrightarrow> (\<forall>\<^sub>Fx in F. \<not> P x)" 
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by (auto simp: frequently_def) 
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lemma frequently_imp_iff: 
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"(\<exists>\<^sub>Fx in F. P x \<longrightarrow> Q x) \<longleftrightarrow> (eventually P F \<longrightarrow> frequently Q F)" 
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unfolding imp_conv_disj frequently_disj_iff not_eventually[symmetric] .. 
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lemma eventually_frequently_const_simps: 
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"(\<exists>\<^sub>Fx in F. P x \<and> C) \<longleftrightarrow> (\<exists>\<^sub>Fx in F. P x) \<and> C" 
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"(\<exists>\<^sub>Fx in F. C \<and> P x) \<longleftrightarrow> C \<and> (\<exists>\<^sub>Fx in F. P x)" 
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"(\<forall>\<^sub>Fx in F. P x \<or> C) \<longleftrightarrow> (\<forall>\<^sub>Fx in F. P x) \<or> C" 
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"(\<forall>\<^sub>Fx in F. C \<or> P x) \<longleftrightarrow> C \<or> (\<forall>\<^sub>Fx in F. P x)" 
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"(\<forall>\<^sub>Fx in F. P x \<longrightarrow> C) \<longleftrightarrow> ((\<exists>\<^sub>Fx in F. P x) \<longrightarrow> C)" 
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"(\<forall>\<^sub>Fx in F. C \<longrightarrow> P x) \<longleftrightarrow> (C \<longrightarrow> (\<forall>\<^sub>Fx in F. P x))" 
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by (cases C; simp add: not_frequently)+ 
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lemmas eventually_frequently_simps = 
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eventually_frequently_const_simps 
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not_eventually 
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eventually_conj_iff 
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eventually_ball_finite_distrib 
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eventually_ex 
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not_frequently 
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frequently_disj_iff 
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frequently_bex_finite_distrib 
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frequently_all 
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frequently_imp_iff 
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60758  233 
ML \<open> 
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fun eventually_elim_tac facts = 
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CONTEXT_SUBGOAL (fn (goal, i) => fn (ctxt, st) => 
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let 
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val mp_thms = facts RL @{thms eventually_rev_mp} 
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val raw_elim_thm = 
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(@{thm allI} RS @{thm always_eventually}) 
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> fold (fn thm1 => fn thm2 => thm2 RS thm1) mp_thms 
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> fold (fn _ => fn thm => @{thm impI} RS thm) facts 
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val cases_prop = 
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Thm.prop_of 
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(Rule_Cases.internalize_params (raw_elim_thm RS Goal.init (Thm.cterm_of ctxt goal))) 
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val cases = Rule_Cases.make_common ctxt cases_prop [(("elim", []), [])] 
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in CONTEXT_CASES cases (resolve_tac ctxt [raw_elim_thm] i) (ctxt, st) end) 
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\<close> 
60036  248 

60758  249 
method_setup eventually_elim = \<open> 
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Scan.succeed (fn _ => CONTEXT_METHOD (fn facts => eventually_elim_tac facts 1)) 
60758  251 
\<close> "elimination of eventually quantifiers" 
60036  252 

60758  253 
subsubsection \<open>Finerthan relation\<close> 
60036  254 

60758  255 
text \<open>@{term "F \<le> F'"} means that filter @{term F} is finer than 
256 
filter @{term F'}.\<close> 

60036  257 

258 
instantiation filter :: (type) complete_lattice 

259 
begin 

260 

261 
definition le_filter_def: 

262 
"F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)" 

263 

264 
definition 

265 
"(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F" 

266 

267 
definition 

268 
"top = Abs_filter (\<lambda>P. \<forall>x. P x)" 

269 

270 
definition 

271 
"bot = Abs_filter (\<lambda>P. True)" 

272 

273 
definition 

274 
"sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')" 

275 

276 
definition 

277 
"inf F F' = Abs_filter 

278 
(\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))" 

279 

280 
definition 

281 
"Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)" 

282 

283 
definition 

284 
"Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}" 

285 

286 
lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)" 

287 
unfolding top_filter_def 

288 
by (rule eventually_Abs_filter, rule is_filter.intro, auto) 

289 

290 
lemma eventually_bot [simp]: "eventually P bot" 

291 
unfolding bot_filter_def 

292 
by (subst eventually_Abs_filter, rule is_filter.intro, auto) 

293 

294 
lemma eventually_sup: 

295 
"eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'" 

296 
unfolding sup_filter_def 

297 
by (rule eventually_Abs_filter, rule is_filter.intro) 

298 
(auto elim!: eventually_rev_mp) 

299 

300 
lemma eventually_inf: 

301 
"eventually P (inf F F') \<longleftrightarrow> 

302 
(\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))" 

303 
unfolding inf_filter_def 

304 
apply (rule eventually_Abs_filter, rule is_filter.intro) 

305 
apply (fast intro: eventually_True) 

306 
apply clarify 

307 
apply (intro exI conjI) 

308 
apply (erule (1) eventually_conj) 

309 
apply (erule (1) eventually_conj) 

310 
apply simp 

311 
apply auto 

312 
done 

313 

314 
lemma eventually_Sup: 

315 
"eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)" 

316 
unfolding Sup_filter_def 

317 
apply (rule eventually_Abs_filter, rule is_filter.intro) 

318 
apply (auto intro: eventually_conj elim!: eventually_rev_mp) 

319 
done 

320 

321 
instance proof 

322 
fix F F' F'' :: "'a filter" and S :: "'a filter set" 

323 
{ show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F" 

324 
by (rule less_filter_def) } 

325 
{ show "F \<le> F" 

326 
unfolding le_filter_def by simp } 

327 
{ assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''" 

328 
unfolding le_filter_def by simp } 

329 
{ assume "F \<le> F'" and "F' \<le> F" thus "F = F'" 

330 
unfolding le_filter_def filter_eq_iff by fast } 

331 
{ show "inf F F' \<le> F" and "inf F F' \<le> F'" 

332 
unfolding le_filter_def eventually_inf by (auto intro: eventually_True) } 

333 
{ assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''" 

334 
unfolding le_filter_def eventually_inf 

61810  335 
by (auto intro: eventually_mono [OF eventually_conj]) } 
60036  336 
{ show "F \<le> sup F F'" and "F' \<le> sup F F'" 
337 
unfolding le_filter_def eventually_sup by simp_all } 

338 
{ assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''" 

339 
unfolding le_filter_def eventually_sup by simp } 

340 
{ assume "F'' \<in> S" thus "Inf S \<le> F''" 

341 
unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp } 

342 
{ assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S" 

343 
unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp } 

344 
{ assume "F \<in> S" thus "F \<le> Sup S" 

345 
unfolding le_filter_def eventually_Sup by simp } 

346 
{ assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'" 

347 
unfolding le_filter_def eventually_Sup by simp } 

348 
{ show "Inf {} = (top::'a filter)" 

349 
by (auto simp: top_filter_def Inf_filter_def Sup_filter_def) 

350 
(metis (full_types) top_filter_def always_eventually eventually_top) } 

351 
{ show "Sup {} = (bot::'a filter)" 

352 
by (auto simp: bot_filter_def Sup_filter_def) } 

353 
qed 

354 

355 
end 

356 

357 
lemma filter_leD: 

358 
"F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F" 

359 
unfolding le_filter_def by simp 

360 

361 
lemma filter_leI: 

362 
"(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'" 

363 
unfolding le_filter_def by simp 

364 

365 
lemma eventually_False: 

366 
"eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot" 

367 
unfolding filter_eq_iff by (auto elim: eventually_rev_mp) 

368 

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lemma eventually_frequently: "F \<noteq> bot \<Longrightarrow> eventually P F \<Longrightarrow> frequently P F" 
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using eventually_conj[of P F "\<lambda>x. \<not> P x"] 
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by (auto simp add: frequently_def eventually_False) 
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lemma eventually_const_iff: "eventually (\<lambda>x. P) F \<longleftrightarrow> P \<or> F = bot" 
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by (cases P) (auto simp: eventually_False) 
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lemma eventually_const[simp]: "F \<noteq> bot \<Longrightarrow> eventually (\<lambda>x. P) F \<longleftrightarrow> P" 
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by (simp add: eventually_const_iff) 
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378 

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lemma frequently_const_iff: "frequently (\<lambda>x. P) F \<longleftrightarrow> P \<and> F \<noteq> bot" 
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by (simp add: frequently_def eventually_const_iff) 
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381 

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lemma frequently_const[simp]: "F \<noteq> bot \<Longrightarrow> frequently (\<lambda>x. P) F \<longleftrightarrow> P" 
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by (simp add: frequently_const_iff) 
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384 

61245  385 
lemma eventually_happens: "eventually P net \<Longrightarrow> net = bot \<or> (\<exists>x. P x)" 
386 
by (metis frequentlyE eventually_frequently) 

387 

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lemma eventually_happens': 
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assumes "F \<noteq> bot" "eventually P F" 
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390 
shows "\<exists>x. P x" 
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using assms eventually_frequently frequentlyE by blast 
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60036  393 
abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool" 
394 
where "trivial_limit F \<equiv> F = bot" 

395 

396 
lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F" 

397 
by (rule eventually_False [symmetric]) 

398 

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lemma False_imp_not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> \<not> trivial_limit net \<Longrightarrow> \<not> eventually (\<lambda>x. P x) net" 
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by (simp add: eventually_False) 
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401 

60036  402 
lemma eventually_Inf: "eventually P (Inf B) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X))" 
403 
proof  

404 
let ?F = "\<lambda>P. \<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X)" 

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405 

60036  406 
{ fix P have "eventually P (Abs_filter ?F) \<longleftrightarrow> ?F P" 
407 
proof (rule eventually_Abs_filter is_filter.intro)+ 

408 
show "?F (\<lambda>x. True)" 

409 
by (rule exI[of _ "{}"]) (simp add: le_fun_def) 

410 
next 

411 
fix P Q 

412 
assume "?F P" then guess X .. 

413 
moreover 

414 
assume "?F Q" then guess Y .. 

415 
ultimately show "?F (\<lambda>x. P x \<and> Q x)" 

416 
by (intro exI[of _ "X \<union> Y"]) 

417 
(auto simp: Inf_union_distrib eventually_inf) 

418 
next 

419 
fix P Q 

420 
assume "?F P" then guess X .. 

421 
moreover assume "\<forall>x. P x \<longrightarrow> Q x" 

422 
ultimately show "?F Q" 

61810  423 
by (intro exI[of _ X]) (auto elim: eventually_mono) 
60036  424 
qed } 
425 
note eventually_F = this 

426 

427 
have "Inf B = Abs_filter ?F" 

428 
proof (intro antisym Inf_greatest) 

429 
show "Inf B \<le> Abs_filter ?F" 

430 
by (auto simp: le_filter_def eventually_F dest: Inf_superset_mono) 

431 
next 

432 
fix F assume "F \<in> B" then show "Abs_filter ?F \<le> F" 

433 
by (auto simp add: le_filter_def eventually_F intro!: exI[of _ "{F}"]) 

434 
qed 

435 
then show ?thesis 

436 
by (simp add: eventually_F) 

437 
qed 

438 

439 
lemma eventually_INF: "eventually P (INF b:B. F b) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (INF b:X. F b))" 

62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62123
diff
changeset

440 
unfolding eventually_Inf [of P "F`B"] 
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62123
diff
changeset

441 
by (metis finite_imageI image_mono finite_subset_image) 
60036  442 

443 
lemma Inf_filter_not_bot: 

444 
fixes B :: "'a filter set" 

445 
shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> Inf X \<noteq> bot) \<Longrightarrow> Inf B \<noteq> bot" 

446 
unfolding trivial_limit_def eventually_Inf[of _ B] 

447 
bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp 

448 

449 
lemma INF_filter_not_bot: 

450 
fixes F :: "'i \<Rightarrow> 'a filter" 

451 
shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> (INF b:X. F b) \<noteq> bot) \<Longrightarrow> (INF b:B. F b) \<noteq> bot" 

62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62123
diff
changeset

452 
unfolding trivial_limit_def eventually_INF [of _ _ B] 
60036  453 
bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp 
454 

455 
lemma eventually_Inf_base: 

456 
assumes "B \<noteq> {}" and base: "\<And>F G. F \<in> B \<Longrightarrow> G \<in> B \<Longrightarrow> \<exists>x\<in>B. x \<le> inf F G" 

457 
shows "eventually P (Inf B) \<longleftrightarrow> (\<exists>b\<in>B. eventually P b)" 

458 
proof (subst eventually_Inf, safe) 

459 
fix X assume "finite X" "X \<subseteq> B" 

460 
then have "\<exists>b\<in>B. \<forall>x\<in>X. b \<le> x" 

461 
proof induct 

462 
case empty then show ?case 

60758  463 
using \<open>B \<noteq> {}\<close> by auto 
60036  464 
next 
465 
case (insert x X) 

466 
then obtain b where "b \<in> B" "\<And>x. x \<in> X \<Longrightarrow> b \<le> x" 

467 
by auto 

60758  468 
with \<open>insert x X \<subseteq> B\<close> base[of b x] show ?case 
60036  469 
by (auto intro: order_trans) 
470 
qed 

471 
then obtain b where "b \<in> B" "b \<le> Inf X" 

472 
by (auto simp: le_Inf_iff) 

473 
then show "eventually P (Inf X) \<Longrightarrow> Bex B (eventually P)" 

474 
by (intro bexI[of _ b]) (auto simp: le_filter_def) 

475 
qed (auto intro!: exI[of _ "{x}" for x]) 

476 

477 
lemma eventually_INF_base: 

478 
"B \<noteq> {} \<Longrightarrow> (\<And>a b. a \<in> B \<Longrightarrow> b \<in> B \<Longrightarrow> \<exists>x\<in>B. F x \<le> inf (F a) (F b)) \<Longrightarrow> 

479 
eventually P (INF b:B. F b) \<longleftrightarrow> (\<exists>b\<in>B. eventually P (F b))" 

62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62123
diff
changeset

480 
by (subst eventually_Inf_base) auto 
60036  481 

62369  482 
lemma eventually_INF1: "i \<in> I \<Longrightarrow> eventually P (F i) \<Longrightarrow> eventually P (INF i:I. F i)" 
483 
using filter_leD[OF INF_lower] . 

484 

62367  485 
lemma eventually_INF_mono: 
486 
assumes *: "\<forall>\<^sub>F x in \<Sqinter>i\<in>I. F i. P x" 

487 
assumes T1: "\<And>Q R P. (\<And>x. Q x \<and> R x \<longrightarrow> P x) \<Longrightarrow> (\<And>x. T Q x \<Longrightarrow> T R x \<Longrightarrow> T P x)" 

488 
assumes T2: "\<And>P. (\<And>x. P x) \<Longrightarrow> (\<And>x. T P x)" 

489 
assumes **: "\<And>i P. i \<in> I \<Longrightarrow> \<forall>\<^sub>F x in F i. P x \<Longrightarrow> \<forall>\<^sub>F x in F' i. T P x" 

490 
shows "\<forall>\<^sub>F x in \<Sqinter>i\<in>I. F' i. T P x" 

491 
proof  

492 
from * obtain X where "finite X" "X \<subseteq> I" "\<forall>\<^sub>F x in \<Sqinter>i\<in>X. F i. P x" 

62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62369
diff
changeset

493 
unfolding eventually_INF[of _ _ I] by auto 
62367  494 
moreover then have "eventually (T P) (INFIMUM X F')" 
495 
apply (induction X arbitrary: P) 

496 
apply (auto simp: eventually_inf T2) 

497 
subgoal for x S P Q R 

498 
apply (intro exI[of _ "T Q"]) 

499 
apply (auto intro!: **) [] 

500 
apply (intro exI[of _ "T R"]) 

501 
apply (auto intro: T1) [] 

502 
done 

503 
done 

504 
ultimately show "\<forall>\<^sub>F x in \<Sqinter>i\<in>I. F' i. T P x" 

505 
by (subst eventually_INF) auto 

506 
qed 

507 

60036  508 

60758  509 
subsubsection \<open>Map function for filters\<close> 
60036  510 

511 
definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter" 

512 
where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)" 

513 

514 
lemma eventually_filtermap: 

515 
"eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F" 

516 
unfolding filtermap_def 

517 
apply (rule eventually_Abs_filter) 

518 
apply (rule is_filter.intro) 

519 
apply (auto elim!: eventually_rev_mp) 

520 
done 

521 

522 
lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F" 

523 
by (simp add: filter_eq_iff eventually_filtermap) 

524 

525 
lemma filtermap_filtermap: 

526 
"filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F" 

527 
by (simp add: filter_eq_iff eventually_filtermap) 

528 

529 
lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'" 

530 
unfolding le_filter_def eventually_filtermap by simp 

531 

532 
lemma filtermap_bot [simp]: "filtermap f bot = bot" 

533 
by (simp add: filter_eq_iff eventually_filtermap) 

534 

535 
lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)" 

536 
by (auto simp: filter_eq_iff eventually_filtermap eventually_sup) 

537 

538 
lemma filtermap_inf: "filtermap f (inf F1 F2) \<le> inf (filtermap f F1) (filtermap f F2)" 

539 
by (auto simp: le_filter_def eventually_filtermap eventually_inf) 

540 

541 
lemma filtermap_INF: "filtermap f (INF b:B. F b) \<le> (INF b:B. filtermap f (F b))" 

542 
proof  

543 
{ fix X :: "'c set" assume "finite X" 

544 
then have "filtermap f (INFIMUM X F) \<le> (INF b:X. filtermap f (F b))" 

545 
proof induct 

546 
case (insert x X) 

547 
have "filtermap f (INF a:insert x X. F a) \<le> inf (filtermap f (F x)) (filtermap f (INF a:X. F a))" 

548 
by (rule order_trans[OF _ filtermap_inf]) simp 

549 
also have "\<dots> \<le> inf (filtermap f (F x)) (INF a:X. filtermap f (F a))" 

550 
by (intro inf_mono insert order_refl) 

551 
finally show ?case 

552 
by simp 

553 
qed simp } 

554 
then show ?thesis 

555 
unfolding le_filter_def eventually_filtermap 

556 
by (subst (1 2) eventually_INF) auto 

557 
qed 

62101  558 

60758  559 
subsubsection \<open>Standard filters\<close> 
60036  560 

561 
definition principal :: "'a set \<Rightarrow> 'a filter" where 

562 
"principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)" 

563 

564 
lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)" 

565 
unfolding principal_def 

566 
by (rule eventually_Abs_filter, rule is_filter.intro) auto 

567 

568 
lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F" 

61810  569 
unfolding eventually_inf eventually_principal by (auto elim: eventually_mono) 
60036  570 

571 
lemma principal_UNIV[simp]: "principal UNIV = top" 

572 
by (auto simp: filter_eq_iff eventually_principal) 

573 

574 
lemma principal_empty[simp]: "principal {} = bot" 

575 
by (auto simp: filter_eq_iff eventually_principal) 

576 

577 
lemma principal_eq_bot_iff: "principal X = bot \<longleftrightarrow> X = {}" 

578 
by (auto simp add: filter_eq_iff eventually_principal) 

579 

580 
lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B" 

581 
by (auto simp: le_filter_def eventually_principal) 

582 

583 
lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F" 

584 
unfolding le_filter_def eventually_principal 

585 
apply safe 

586 
apply (erule_tac x="\<lambda>x. x \<in> A" in allE) 

61810  587 
apply (auto elim: eventually_mono) 
60036  588 
done 
589 

590 
lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B" 

591 
unfolding eq_iff by simp 

592 

593 
lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)" 

594 
unfolding filter_eq_iff eventually_sup eventually_principal by auto 

595 

596 
lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)" 

597 
unfolding filter_eq_iff eventually_inf eventually_principal 

598 
by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"]) 

599 

600 
lemma SUP_principal[simp]: "(SUP i : I. principal (A i)) = principal (\<Union>i\<in>I. A i)" 

62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62123
diff
changeset

601 
unfolding filter_eq_iff eventually_Sup by (auto simp: eventually_principal) 
60036  602 

603 
lemma INF_principal_finite: "finite X \<Longrightarrow> (INF x:X. principal (f x)) = principal (\<Inter>x\<in>X. f x)" 

604 
by (induct X rule: finite_induct) auto 

605 

606 
lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)" 

607 
unfolding filter_eq_iff eventually_filtermap eventually_principal by simp 

608 

60758  609 
subsubsection \<open>Order filters\<close> 
60036  610 

611 
definition at_top :: "('a::order) filter" 

612 
where "at_top = (INF k. principal {k ..})" 

613 

614 
lemma at_top_sub: "at_top = (INF k:{c::'a::linorder..}. principal {k ..})" 

615 
by (auto intro!: INF_eq max.cobounded1 max.cobounded2 simp: at_top_def) 

616 

617 
lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)" 

618 
unfolding at_top_def 

619 
by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2) 

620 

621 
lemma eventually_ge_at_top: 

622 
"eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top" 

623 
unfolding eventually_at_top_linorder by auto 

624 

625 
lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::{no_top, linorder}. \<forall>n>N. P n)" 

626 
proof  

627 
have "eventually P (INF k. principal {k <..}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n>N. P n)" 

628 
by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2) 

629 
also have "(INF k. principal {k::'a <..}) = at_top" 

61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset

630 
unfolding at_top_def 
60036  631 
by (intro INF_eq) (auto intro: less_imp_le simp: Ici_subset_Ioi_iff gt_ex) 
632 
finally show ?thesis . 

633 
qed 

634 

60721
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset

635 
lemma eventually_at_top_not_equal: "eventually (\<lambda>x::'a::{no_top, linorder}. x \<noteq> c) at_top" 
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset

636 
unfolding eventually_at_top_dense by auto 
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset

637 

c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset

638 
lemma eventually_gt_at_top: "eventually (\<lambda>x. (c::_::{no_top, linorder}) < x) at_top" 
60036  639 
unfolding eventually_at_top_dense by auto 
640 

61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset

641 
lemma eventually_all_ge_at_top: 
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset

642 
assumes "eventually P (at_top :: ('a :: linorder) filter)" 
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset

643 
shows "eventually (\<lambda>x. \<forall>y\<ge>x. P y) at_top" 
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset

644 
proof  
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset

645 
from assms obtain x where "\<And>y. y \<ge> x \<Longrightarrow> P y" by (auto simp: eventually_at_top_linorder) 
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset

646 
hence "\<forall>z\<ge>y. P z" if "y \<ge> x" for y using that by simp 
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset

647 
thus ?thesis by (auto simp: eventually_at_top_linorder) 
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset

648 
qed 
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset

649 

60036  650 
definition at_bot :: "('a::order) filter" 
651 
where "at_bot = (INF k. principal {.. k})" 

652 

653 
lemma at_bot_sub: "at_bot = (INF k:{.. c::'a::linorder}. principal {.. k})" 

654 
by (auto intro!: INF_eq min.cobounded1 min.cobounded2 simp: at_bot_def) 

655 

656 
lemma eventually_at_bot_linorder: 

657 
fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)" 

658 
unfolding at_bot_def 

659 
by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2) 

660 

661 
lemma eventually_le_at_bot: 

662 
"eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot" 

663 
unfolding eventually_at_bot_linorder by auto 

664 

665 
lemma eventually_at_bot_dense: "eventually P at_bot \<longleftrightarrow> (\<exists>N::'a::{no_bot, linorder}. \<forall>n<N. P n)" 

666 
proof  

667 
have "eventually P (INF k. principal {..< k}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n<N. P n)" 

668 
by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2) 

669 
also have "(INF k. principal {..< k::'a}) = at_bot" 

61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset

670 
unfolding at_bot_def 
60036  671 
by (intro INF_eq) (auto intro: less_imp_le simp: Iic_subset_Iio_iff lt_ex) 
672 
finally show ?thesis . 

673 
qed 

674 

60721
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset

675 
lemma eventually_at_bot_not_equal: "eventually (\<lambda>x::'a::{no_bot, linorder}. x \<noteq> c) at_bot" 
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset

676 
unfolding eventually_at_bot_dense by auto 
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset

677 

60036  678 
lemma eventually_gt_at_bot: 
679 
"eventually (\<lambda>x. x < (c::_::unbounded_dense_linorder)) at_bot" 

680 
unfolding eventually_at_bot_dense by auto 

681 

682 
lemma trivial_limit_at_bot_linorder: "\<not> trivial_limit (at_bot ::('a::linorder) filter)" 

683 
unfolding trivial_limit_def 

684 
by (metis eventually_at_bot_linorder order_refl) 

685 

686 
lemma trivial_limit_at_top_linorder: "\<not> trivial_limit (at_top ::('a::linorder) filter)" 

687 
unfolding trivial_limit_def 

688 
by (metis eventually_at_top_linorder order_refl) 

689 

60758  690 
subsection \<open>Sequentially\<close> 
60036  691 

692 
abbreviation sequentially :: "nat filter" 

693 
where "sequentially \<equiv> at_top" 

694 

695 
lemma eventually_sequentially: 

696 
"eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)" 

697 
by (rule eventually_at_top_linorder) 

698 

699 
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot" 

700 
unfolding filter_eq_iff eventually_sequentially by auto 

701 

702 
lemmas trivial_limit_sequentially = sequentially_bot 

703 

704 
lemma eventually_False_sequentially [simp]: 

705 
"\<not> eventually (\<lambda>n. False) sequentially" 

706 
by (simp add: eventually_False) 

707 

708 
lemma le_sequentially: 

709 
"F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)" 

710 
by (simp add: at_top_def le_INF_iff le_principal) 

711 

60974
6a6f15d8fbc4
New material and fixes related to the forthcoming StoneWeierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset

712 
lemma eventually_sequentiallyI [intro?]: 
60036  713 
assumes "\<And>x. c \<le> x \<Longrightarrow> P x" 
714 
shows "eventually P sequentially" 

715 
using assms by (auto simp: eventually_sequentially) 

716 

60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset

717 
lemma eventually_sequentially_Suc: "eventually (\<lambda>i. P (Suc i)) sequentially \<longleftrightarrow> eventually P sequentially" 
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset

718 
unfolding eventually_sequentially by (metis Suc_le_D Suc_le_mono le_Suc_eq) 
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset

719 

1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset

720 
lemma eventually_sequentially_seg: "eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially" 
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset

721 
using eventually_sequentially_Suc[of "\<lambda>n. P (n + k)" for k] by (induction k) auto 
60036  722 

61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset

723 

e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset

724 
subsection \<open>The cofinite filter\<close> 
60039  725 

726 
definition "cofinite = Abs_filter (\<lambda>P. finite {x. \<not> P x})" 

727 

61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset

728 
abbreviation Inf_many :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "\<exists>\<^sub>\<infinity>" 10) 
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset

729 
where "Inf_many P \<equiv> frequently P cofinite" 
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset

730 

61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset

731 
abbreviation Alm_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "\<forall>\<^sub>\<infinity>" 10) 
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset

732 
where "Alm_all P \<equiv> eventually P cofinite" 
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset

733 

61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset

734 
notation (ASCII) 
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset

735 
Inf_many (binder "INFM " 10) and 
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset

736 
Alm_all (binder "MOST " 10) 
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset

737 

60039  738 
lemma eventually_cofinite: "eventually P cofinite \<longleftrightarrow> finite {x. \<not> P x}" 
739 
unfolding cofinite_def 

740 
proof (rule eventually_Abs_filter, rule is_filter.intro) 

741 
fix P Q :: "'a \<Rightarrow> bool" assume "finite {x. \<not> P x}" "finite {x. \<not> Q x}" 

742 
from finite_UnI[OF this] show "finite {x. \<not> (P x \<and> Q x)}" 

743 
by (rule rev_finite_subset) auto 

744 
next 

745 
fix P Q :: "'a \<Rightarrow> bool" assume P: "finite {x. \<not> P x}" and *: "\<forall>x. P x \<longrightarrow> Q x" 

746 
from * show "finite {x. \<not> Q x}" 

747 
by (intro finite_subset[OF _ P]) auto 

748 
qed simp 

749 

60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset

750 
lemma frequently_cofinite: "frequently P cofinite \<longleftrightarrow> \<not> finite {x. P x}" 
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset

751 
by (simp add: frequently_def eventually_cofinite) 
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset

752 

60039  753 
lemma cofinite_bot[simp]: "cofinite = (bot::'a filter) \<longleftrightarrow> finite (UNIV :: 'a set)" 
754 
unfolding trivial_limit_def eventually_cofinite by simp 

755 

756 
lemma cofinite_eq_sequentially: "cofinite = sequentially" 

757 
unfolding filter_eq_iff eventually_sequentially eventually_cofinite 

758 
proof safe 

759 
fix P :: "nat \<Rightarrow> bool" assume [simp]: "finite {x. \<not> P x}" 

760 
show "\<exists>N. \<forall>n\<ge>N. P n" 

761 
proof cases 

762 
assume "{x. \<not> P x} \<noteq> {}" then show ?thesis 

763 
by (intro exI[of _ "Suc (Max {x. \<not> P x})"]) (auto simp: Suc_le_eq) 

764 
qed auto 

765 
next 

766 
fix P :: "nat \<Rightarrow> bool" and N :: nat assume "\<forall>n\<ge>N. P n" 

767 
then have "{x. \<not> P x} \<subseteq> {..< N}" 

768 
by (auto simp: not_le) 

769 
then show "finite {x. \<not> P x}" 

770 
by (blast intro: finite_subset) 

771 
qed 

60036  772 

62101  773 
subsubsection \<open>Product of filters\<close> 
774 

775 
lemma filtermap_sequentually_ne_bot: "filtermap f sequentially \<noteq> bot" 

776 
by (auto simp add: filter_eq_iff eventually_filtermap eventually_sequentially) 

777 

778 
definition prod_filter :: "'a filter \<Rightarrow> 'b filter \<Rightarrow> ('a \<times> 'b) filter" (infixr "\<times>\<^sub>F" 80) where 

779 
"prod_filter F G = 

780 
(INF (P, Q):{(P, Q). eventually P F \<and> eventually Q G}. principal {(x, y). P x \<and> Q y})" 

781 

782 
lemma eventually_prod_filter: "eventually P (F \<times>\<^sub>F G) \<longleftrightarrow> 

783 
(\<exists>Pf Pg. eventually Pf F \<and> eventually Pg G \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P (x, y)))" 

784 
unfolding prod_filter_def 

785 
proof (subst eventually_INF_base, goal_cases) 

786 
case 2 

787 
moreover have "eventually Pf F \<Longrightarrow> eventually Qf F \<Longrightarrow> eventually Pg G \<Longrightarrow> eventually Qg G \<Longrightarrow> 

788 
\<exists>P Q. eventually P F \<and> eventually Q G \<and> 

789 
Collect P \<times> Collect Q \<subseteq> Collect Pf \<times> Collect Pg \<inter> Collect Qf \<times> Collect Qg" for Pf Pg Qf Qg 

790 
by (intro conjI exI[of _ "inf Pf Qf"] exI[of _ "inf Pg Qg"]) 

791 
(auto simp: inf_fun_def eventually_conj) 

792 
ultimately show ?case 

793 
by auto 

794 
qed (auto simp: eventually_principal intro: eventually_True) 

795 

62367  796 
lemma eventually_prod1: 
797 
assumes "B \<noteq> bot" 

798 
shows "(\<forall>\<^sub>F (x, y) in A \<times>\<^sub>F B. P x) \<longleftrightarrow> (\<forall>\<^sub>F x in A. P x)" 

799 
unfolding eventually_prod_filter 

800 
proof safe 

801 
fix R Q assume "\<forall>\<^sub>F x in A. R x" "\<forall>\<^sub>F x in B. Q x" "\<forall>x y. R x \<longrightarrow> Q y \<longrightarrow> P x" 

802 
moreover with \<open>B \<noteq> bot\<close> obtain y where "Q y" by (auto dest: eventually_happens) 

803 
ultimately show "eventually P A" 

804 
by (force elim: eventually_mono) 

805 
next 

806 
assume "eventually P A" 

807 
then show "\<exists>Pf Pg. eventually Pf A \<and> eventually Pg B \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P x)" 

808 
by (intro exI[of _ P] exI[of _ "\<lambda>x. True"]) auto 

809 
qed 

810 

811 
lemma eventually_prod2: 

812 
assumes "A \<noteq> bot" 

813 
shows "(\<forall>\<^sub>F (x, y) in A \<times>\<^sub>F B. P y) \<longleftrightarrow> (\<forall>\<^sub>F y in B. P y)" 

814 
unfolding eventually_prod_filter 

815 
proof safe 

816 
fix R Q assume "\<forall>\<^sub>F x in A. R x" "\<forall>\<^sub>F x in B. Q x" "\<forall>x y. R x \<longrightarrow> Q y \<longrightarrow> P y" 

817 
moreover with \<open>A \<noteq> bot\<close> obtain x where "R x" by (auto dest: eventually_happens) 

818 
ultimately show "eventually P B" 

819 
by (force elim: eventually_mono) 

820 
next 

821 
assume "eventually P B" 

822 
then show "\<exists>Pf Pg. eventually Pf A \<and> eventually Pg B \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P y)" 

823 
by (intro exI[of _ P] exI[of _ "\<lambda>x. True"]) auto 

824 
qed 

825 

826 
lemma INF_filter_bot_base: 

827 
fixes F :: "'a \<Rightarrow> 'b filter" 

828 
assumes *: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> \<exists>k\<in>I. F k \<le> F i \<sqinter> F j" 

829 
shows "(INF i:I. F i) = bot \<longleftrightarrow> (\<exists>i\<in>I. F i = bot)" 

830 
proof cases 

831 
assume "\<exists>i\<in>I. F i = bot" 

832 
moreover then have "(INF i:I. F i) \<le> bot" 

833 
by (auto intro: INF_lower2) 

834 
ultimately show ?thesis 

835 
by (auto simp: bot_unique) 

836 
next 

837 
assume **: "\<not> (\<exists>i\<in>I. F i = bot)" 

838 
moreover have "(INF i:I. F i) \<noteq> bot" 

839 
proof cases 

840 
assume "I \<noteq> {}" 

841 
show ?thesis 

842 
proof (rule INF_filter_not_bot) 

843 
fix J assume "finite J" "J \<subseteq> I" 

844 
then have "\<exists>k\<in>I. F k \<le> (\<Sqinter>i\<in>J. F i)" 

845 
proof (induction J) 

846 
case empty then show ?case 

847 
using \<open>I \<noteq> {}\<close> by auto 

848 
next 

849 
case (insert i J) 

850 
moreover then obtain k where "k \<in> I" "F k \<le> (\<Sqinter>i\<in>J. F i)" by auto 

851 
moreover note *[of i k] 

852 
ultimately show ?case 

853 
by auto 

854 
qed 

855 
with ** show "(\<Sqinter>i\<in>J. F i) \<noteq> \<bottom>" 

856 
by (auto simp: bot_unique) 

857 
qed 

858 
qed (auto simp add: filter_eq_iff) 

859 
ultimately show ?thesis 

860 
by auto 

861 
qed 

862 

863 
lemma Collect_empty_eq_bot: "Collect P = {} \<longleftrightarrow> P = \<bottom>" 

864 
by auto 

865 

866 
lemma prod_filter_eq_bot: "A \<times>\<^sub>F B = bot \<longleftrightarrow> A = bot \<or> B = bot" 

867 
unfolding prod_filter_def 

868 
proof (subst INF_filter_bot_base; clarsimp simp: principal_eq_bot_iff Collect_empty_eq_bot bot_fun_def simp del: Collect_empty_eq) 

869 
fix A1 A2 B1 B2 assume "\<forall>\<^sub>F x in A. A1 x" "\<forall>\<^sub>F x in A. A2 x" "\<forall>\<^sub>F x in B. B1 x" "\<forall>\<^sub>F x in B. B2 x" 

870 
then show "\<exists>x. eventually x A \<and> (\<exists>y. eventually y B \<and> Collect x \<times> Collect y \<subseteq> Collect A1 \<times> Collect B1 \<and> Collect x \<times> Collect y \<subseteq> Collect A2 \<times> Collect B2)" 

871 
by (intro exI[of _ "\<lambda>x. A1 x \<and> A2 x"] exI[of _ "\<lambda>x. B1 x \<and> B2 x"] conjI) 

872 
(auto simp: eventually_conj_iff) 

873 
next 

874 
show "(\<exists>x. eventually x A \<and> (\<exists>y. eventually y B \<and> (x = (\<lambda>x. False) \<or> y = (\<lambda>x. False)))) = (A = \<bottom> \<or> B = \<bottom>)" 

875 
by (auto simp: trivial_limit_def intro: eventually_True) 

876 
qed 

877 

62101  878 
lemma prod_filter_mono: "F \<le> F' \<Longrightarrow> G \<le> G' \<Longrightarrow> F \<times>\<^sub>F G \<le> F' \<times>\<^sub>F G'" 
879 
by (auto simp: le_filter_def eventually_prod_filter) 

880 

62367  881 
lemma prod_filter_mono_iff: 
882 
assumes nAB: "A \<noteq> bot" "B \<noteq> bot" 

883 
shows "A \<times>\<^sub>F B \<le> C \<times>\<^sub>F D \<longleftrightarrow> A \<le> C \<and> B \<le> D" 

884 
proof safe 

885 
assume *: "A \<times>\<^sub>F B \<le> C \<times>\<^sub>F D" 

886 
moreover with assms have "A \<times>\<^sub>F B \<noteq> bot" 

887 
by (auto simp: bot_unique prod_filter_eq_bot) 

888 
ultimately have "C \<times>\<^sub>F D \<noteq> bot" 

889 
by (auto simp: bot_unique) 

890 
then have nCD: "C \<noteq> bot" "D \<noteq> bot" 

891 
by (auto simp: prod_filter_eq_bot) 

892 

893 
show "A \<le> C" 

894 
proof (rule filter_leI) 

895 
fix P assume "eventually P C" with *[THEN filter_leD, of "\<lambda>(x, y). P x"] show "eventually P A" 

896 
using nAB nCD by (simp add: eventually_prod1 eventually_prod2) 

897 
qed 

898 

899 
show "B \<le> D" 

900 
proof (rule filter_leI) 

901 
fix P assume "eventually P D" with *[THEN filter_leD, of "\<lambda>(x, y). P y"] show "eventually P B" 

902 
using nAB nCD by (simp add: eventually_prod1 eventually_prod2) 

903 
qed 

904 
qed (intro prod_filter_mono) 

905 

62101  906 
lemma eventually_prod_same: "eventually P (F \<times>\<^sub>F F) \<longleftrightarrow> 
907 
(\<exists>Q. eventually Q F \<and> (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y)))" 

908 
unfolding eventually_prod_filter 

909 
apply safe 

910 
apply (rule_tac x="inf Pf Pg" in exI) 

911 
apply (auto simp: inf_fun_def intro!: eventually_conj) 

912 
done 

913 

914 
lemma eventually_prod_sequentially: 

915 
"eventually P (sequentially \<times>\<^sub>F sequentially) \<longleftrightarrow> (\<exists>N. \<forall>m \<ge> N. \<forall>n \<ge> N. P (n, m))" 

916 
unfolding eventually_prod_same eventually_sequentially by auto 

917 

918 
lemma principal_prod_principal: "principal A \<times>\<^sub>F principal B = principal (A \<times> B)" 

919 
apply (simp add: filter_eq_iff eventually_prod_filter eventually_principal) 

920 
apply safe 

921 
apply blast 

922 
apply (intro conjI exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"]) 

923 
apply auto 

924 
done 

925 

62367  926 
lemma prod_filter_INF: 
927 
assumes "I \<noteq> {}" "J \<noteq> {}" 

928 
shows "(INF i:I. A i) \<times>\<^sub>F (INF j:J. B j) = (INF i:I. INF j:J. A i \<times>\<^sub>F B j)" 

929 
proof (safe intro!: antisym INF_greatest) 

930 
from \<open>I \<noteq> {}\<close> obtain i where "i \<in> I" by auto 

931 
from \<open>J \<noteq> {}\<close> obtain j where "j \<in> J" by auto 

932 

933 
show "(\<Sqinter>i\<in>I. \<Sqinter>j\<in>J. A i \<times>\<^sub>F B j) \<le> (\<Sqinter>i\<in>I. A i) \<times>\<^sub>F (\<Sqinter>j\<in>J. B j)" 

934 
unfolding prod_filter_def 

935 
proof (safe intro!: INF_greatest) 

936 
fix P Q assume P: "\<forall>\<^sub>F x in \<Sqinter>i\<in>I. A i. P x" and Q: "\<forall>\<^sub>F x in \<Sqinter>j\<in>J. B j. Q x" 

937 
let ?X = "(\<Sqinter>i\<in>I. \<Sqinter>j\<in>J. \<Sqinter>(P, Q)\<in>{(P, Q). (\<forall>\<^sub>F x in A i. P x) \<and> (\<forall>\<^sub>F x in B j. Q x)}. principal {(x, y). P x \<and> Q y})" 

938 
have "?X \<le> principal {x. P (fst x)} \<sqinter> principal {x. Q (snd x)}" 

939 
proof (intro inf_greatest) 

940 
have "?X \<le> (\<Sqinter>i\<in>I. \<Sqinter>P\<in>{P. eventually P (A i)}. principal {x. P (fst x)})" 

941 
by (auto intro!: INF_greatest INF_lower2[of j] INF_lower2 \<open>j\<in>J\<close> INF_lower2[of "(_, \<lambda>x. True)"]) 

942 
also have "\<dots> \<le> principal {x. P (fst x)}" 

943 
unfolding le_principal 

944 
proof (rule eventually_INF_mono[OF P]) 

945 
fix i P assume "i \<in> I" "eventually P (A i)" 

946 
then show "\<forall>\<^sub>F x in \<Sqinter>P\<in>{P. eventually P (A i)}. principal {x. P (fst x)}. x \<in> {x. P (fst x)}" 

947 
unfolding le_principal[symmetric] by (auto intro!: INF_lower) 

948 
qed auto 

949 
finally show "?X \<le> principal {x. P (fst x)}" . 

950 

951 
have "?X \<le> (\<Sqinter>i\<in>J. \<Sqinter>P\<in>{P. eventually P (B i)}. principal {x. P (snd x)})" 

952 
by (auto intro!: INF_greatest INF_lower2[of i] INF_lower2 \<open>i\<in>I\<close> INF_lower2[of "(\<lambda>x. True, _)"]) 

953 
also have "\<dots> \<le> principal {x. Q (snd x)}" 

954 
unfolding le_principal 

955 
proof (rule eventually_INF_mono[OF Q]) 

956 
fix j Q assume "j \<in> J" "eventually Q (B j)" 

957 
then show "\<forall>\<^sub>F x in \<Sqinter>P\<in>{P. eventually P (B j)}. principal {x. P (snd x)}. x \<in> {x. Q (snd x)}" 

958 
unfolding le_principal[symmetric] by (auto intro!: INF_lower) 

959 
qed auto 

960 
finally show "?X \<le> principal {x. Q (snd x)}" . 

961 
qed 

962 
also have "\<dots> = principal {(x, y). P x \<and> Q y}" 

963 
by auto 

964 
finally show "?X \<le> principal {(x, y). P x \<and> Q y}" . 

965 
qed 

966 
qed (intro prod_filter_mono INF_lower) 

967 

968 
lemma filtermap_Pair: "filtermap (\<lambda>x. (f x, g x)) F \<le> filtermap f F \<times>\<^sub>F filtermap g F" 

969 
by (simp add: le_filter_def eventually_filtermap eventually_prod_filter) 

970 
(auto elim: eventually_elim2) 

971 

62369  972 
lemma eventually_prodI: "eventually P F \<Longrightarrow> eventually Q G \<Longrightarrow> eventually (\<lambda>x. P (fst x) \<and> Q (snd x)) (F \<times>\<^sub>F G)" 
973 
unfolding prod_filter_def 

974 
by (intro eventually_INF1[of "(P, Q)"]) (auto simp: eventually_principal) 

975 

976 
lemma prod_filter_INF1: "I \<noteq> {} \<Longrightarrow> (INF i:I. A i) \<times>\<^sub>F B = (INF i:I. A i \<times>\<^sub>F B)" 

977 
using prod_filter_INF[of I "{B}" A "\<lambda>x. x"] by simp 

978 

979 
lemma prod_filter_INF2: "J \<noteq> {} \<Longrightarrow> A \<times>\<^sub>F (INF i:J. B i) = (INF i:J. A \<times>\<^sub>F B i)" 

980 
using prod_filter_INF[of "{A}" J "\<lambda>x. x" B] by simp 

981 

60758  982 
subsection \<open>Limits\<close> 
60036  983 

984 
definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where 

985 
"filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2" 

986 

987 
syntax 

988 
"_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10) 

989 

990 
translations 

62367  991 
"LIM x F1. f :> F2" == "CONST filterlim (\<lambda>x. f) F2 F1" 
60036  992 

62379
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62378
diff
changeset

993 
lemma filterlim_top [simp]: "filterlim f top F" 
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62378
diff
changeset

994 
by (simp add: filterlim_def) 
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62378
diff
changeset

995 

60036  996 
lemma filterlim_iff: 
997 
"(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)" 

998 
unfolding filterlim_def le_filter_def eventually_filtermap .. 

999 

1000 
lemma filterlim_compose: 

1001 
"filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1" 

1002 
unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans) 

1003 

1004 
lemma filterlim_mono: 

1005 
"filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'" 

1006 
unfolding filterlim_def by (metis filtermap_mono order_trans) 

1007 

1008 
lemma filterlim_ident: "LIM x F. x :> F" 

1009 
by (simp add: filterlim_def filtermap_ident) 

1010 

1011 
lemma filterlim_cong: 

1012 
"F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'" 

1013 
by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2) 

1014 

1015 
lemma filterlim_mono_eventually: 

1016 
assumes "filterlim f F G" and ord: "F \<le> F'" "G' \<le> G" 

1017 
assumes eq: "eventually (\<lambda>x. f x = f' x) G'" 

1018 
shows "filterlim f' F' G'" 

1019 
apply (rule filterlim_cong[OF refl refl eq, THEN iffD1]) 

1020 
apply (rule filterlim_mono[OF _ ord]) 

1021 
apply fact 

1022 
done 

1023 

1024 
lemma filtermap_mono_strong: "inj f \<Longrightarrow> filtermap f F \<le> filtermap f G \<longleftrightarrow> F \<le> G" 

1025 
apply (auto intro!: filtermap_mono) [] 

1026 
apply (auto simp: le_filter_def eventually_filtermap) 

1027 
apply (erule_tac x="\<lambda>x. P (inv f x)" in allE) 

1028 
apply auto 

1029 
done 

1030 

1031 
lemma filtermap_eq_strong: "inj f \<Longrightarrow> filtermap f F = filtermap f G \<longleftrightarrow> F = G" 

1032 
by (simp add: filtermap_mono_strong eq_iff) 

1033 

60721
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset

1034 
lemma filtermap_fun_inverse: 
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset

1035 
assumes g: "filterlim g F G" 
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset

1036 
assumes f: "filterlim f G F" 
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset

1037 
assumes ev: "eventually (\<lambda>x. f (g x) = x) G" 
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset

1038 
shows "filtermap f F = G" 
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset

1039 
proof (rule antisym) 
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset

1040 
show "filtermap f F \<le> G" 
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset

1041 
using f unfolding filterlim_def . 
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset

1042 
have "G = filtermap f (filtermap g G)" 
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset

1043 
using ev by (auto elim: eventually_elim2 simp: filter_eq_iff eventually_filtermap) 
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset

1044 
also have "\<dots> \<le> filtermap f F" 
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset

1045 
using g by (intro filtermap_mono) (simp add: filterlim_def) 
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset

1046 
finally show "G \<le> filtermap f F" . 
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset

1047 
qed 
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset

1048 

60036  1049 
lemma filterlim_principal: 
1050 
"(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)" 

1051 
unfolding filterlim_def eventually_filtermap le_principal .. 

1052 

1053 
lemma filterlim_inf: 

1054 
"(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))" 

1055 
unfolding filterlim_def by simp 

1056 

1057 
lemma filterlim_INF: 

1058 
"(LIM x F. f x :> (INF b:B. G b)) \<longleftrightarrow> (\<forall>b\<in>B. LIM x F. f x :> G b)" 

1059 
unfolding filterlim_def le_INF_iff .. 

1060 

1061 
lemma filterlim_INF_INF: 

1062 
"(\<And>m. m \<in> J \<Longrightarrow> \<exists>i\<in>I. filtermap f (F i) \<le> G m) \<Longrightarrow> LIM x (INF i:I. F i). f x :> (INF j:J. G j)" 

1063 
unfolding filterlim_def by (rule order_trans[OF filtermap_INF INF_mono]) 

1064 

1065 
lemma filterlim_base: 

61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset

1066 
"(\<And>m x. m \<in> J \<Longrightarrow> i m \<in> I) \<Longrightarrow> (\<And>m x. m \<in> J \<Longrightarrow> x \<in> F (i m) \<Longrightarrow> f x \<in> G m) \<Longrightarrow> 
60036  1067 
LIM x (INF i:I. principal (F i)). f x :> (INF j:J. principal (G j))" 
1068 
by (force intro!: filterlim_INF_INF simp: image_subset_iff) 

1069 

61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset

1070 
lemma filterlim_base_iff: 
60036  1071 
assumes "I \<noteq> {}" and chain: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> F i \<subseteq> F j \<or> F j \<subseteq> F i" 
1072 
shows "(LIM x (INF i:I. principal (F i)). f x :> INF j:J. principal (G j)) \<longleftrightarrow> 

1073 
(\<forall>j\<in>J. \<exists>i\<in>I. \<forall>x\<in>F i. f x \<in> G j)" 

1074 
unfolding filterlim_INF filterlim_principal 

1075 
proof (subst eventually_INF_base) 

1076 
fix i j assume "i \<in> I" "j \<in> I" 

1077 
with chain[OF this] show "\<exists>x\<in>I. principal (F x) \<le> inf (principal (F i)) (principal (F j))" 

1078 
by auto 

60758  1079 
qed (auto simp: eventually_principal \<open>I \<noteq> {}\<close>) 
60036  1080 

1081 
lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2" 

1082 
unfolding filterlim_def filtermap_filtermap .. 

1083 

1084 
lemma filterlim_sup: 

1085 
"filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)" 

1086 
unfolding filterlim_def filtermap_sup by auto 

1087 

1088 
lemma filterlim_sequentially_Suc: 

1089 
"(LIM x sequentially. f (Suc x) :> F) \<longleftrightarrow> (LIM x sequentially. f x :> F)" 

1090 
unfolding filterlim_iff by (subst eventually_sequentially_Suc) simp 

1091 

1092 
lemma filterlim_Suc: "filterlim Suc sequentially sequentially" 

1093 
by (simp add: filterlim_iff eventually_sequentially) (metis le_Suc_eq) 

1094 

60182
e1ea5a6379c9
generalized tends over powr; added DERIV rule for powr
hoelzl
parents:
60040
diff
changeset

1095 
lemma filterlim_If: 
e1ea5a6379c9
generalized tends over powr; added DERIV rule for powr
hoelzl
parents:
60040
diff
changeset

1096 
"LIM x inf F (principal {x. P x}). f x :> G \<Longrightarrow> 
e1ea5a6379c9
generalized tends over powr; added DERIV rule for powr
hoelzl
parents:
60040
diff
changeset

1097 
LIM x inf F (principal {x. \<not> P x}). g x :> G \<Longrightarrow> 
e1ea5a6379c9
generalized tends over powr; added DERIV rule for powr
hoelzl
parents:
60040
diff
changeset

1098 
LIM x F. if P x then f x else g x :> G" 
e1ea5a6379c9
generalized tends over powr; added DERIV rule for powr
hoelzl
parents:
60040
diff
changeset

1099 
unfolding filterlim_iff eventually_inf_principal by (auto simp: eventually_conj_iff) 
60036  1100 

62367  1101 
lemma filterlim_Pair: 
1102 
"LIM x F. f x :> G \<Longrightarrow> LIM x F. g x :> H \<Longrightarrow> LIM x F. (f x, g x) :> G \<times>\<^sub>F H" 

1103 
unfolding filterlim_def 

1104 
by (rule order_trans[OF filtermap_Pair prod_filter_mono]) 

1105 

60758  1106 
subsection \<open>Limits to @{const at_top} and @{const at_bot}\<close> 
60036  1107 

1108 
lemma filterlim_at_top: 

1109 
fixes f :: "'a \<Rightarrow> ('b::linorder)" 

1110 
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)" 

61810  1111 
by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_mono) 
60036  1112 

1113 
lemma filterlim_at_top_mono: 

1114 
"LIM x F. f x :> at_top \<Longrightarrow> eventually (\<lambda>x. f x \<le> (g x::'a::linorder)) F \<Longrightarrow> 

1115 
LIM x F. g x :> at_top" 

1116 
by (auto simp: filterlim_at_top elim: eventually_elim2 intro: order_trans) 

1117 

1118 
lemma filterlim_at_top_dense: 

1119 
fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" 

1120 
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)" 

61810  1121 
by (metis eventually_mono[of _ F] eventually_gt_at_top order_less_imp_le 
60036  1122 
filterlim_at_top[of f F] filterlim_iff[of f at_top F]) 
1123 

1124 
lemma filterlim_at_top_ge: 

1125 
fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b" 

1126 
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)" 

1127 
unfolding at_top_sub[of c] filterlim_INF by (auto simp add: filterlim_principal) 

1128 

1129 
lemma filterlim_at_top_at_top: 

1130 
fixes f :: "'a::linorder \<Rightarrow> 'b::linorder" 

1131 
assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y" 

1132 
assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)" 

1133 
assumes Q: "eventually Q at_top" 

1134 
assumes P: "eventually P at_top" 

1135 
shows "filterlim f at_top at_top" 

1136 
proof  

1137 
from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y" 

1138 
unfolding eventually_at_top_linorder by auto 

1139 
show ?thesis 

1140 
proof (intro filterlim_at_top_ge[THEN iffD2] allI impI) 

1141 
fix z assume "x \<le> z" 

1142 
with x have "P z" by auto 

1143 
have "eventually (\<lambda>x. g z \<le> x) at_top" 

1144 
by (rule eventually_ge_at_top) 

1145 
with Q show "eventually (\<lambda>x. z \<le> f x) at_top" 

60758  1146 
by eventually_elim (metis mono bij \<open>P z\<close>) 
60036  1147 
qed 
1148 
qed 

1149 

1150 
lemma filterlim_at_top_gt: 

1151 
fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b" 

1152 
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)" 

1153 
by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge) 

1154 

61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset

1155 
lemma filterlim_at_bot: 
60036  1156 
fixes f :: "'a \<Rightarrow> ('b::linorder)" 
1157 
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)" 

61810  1158 
by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_mono) 
60036  1159 

1160 
lemma filterlim_at_bot_dense: 

1161 
fixes f :: "'a \<Rightarrow> ('b::{dense_linorder, no_bot})" 

1162 
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x < Z) F)" 

1163 
proof (auto simp add: filterlim_at_bot[of f F]) 

1164 
fix Z :: 'b 

1165 
from lt_ex [of Z] obtain Z' where 1: "Z' < Z" .. 

1166 
assume "\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F" 

1167 
hence "eventually (\<lambda>x. f x \<le> Z') F" by auto 

1168 
thus "eventually (\<lambda>x. f x < Z) F" 

61810  1169 
apply (rule eventually_mono) 
60036  1170 
using 1 by auto 
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset

1171 
next 
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset

1172 
fix Z :: 'b 
60036  1173 
show "\<forall>Z. eventually (\<lambda>x. f x < Z) F \<Longrightarrow> eventually (\<lambda>x. f x \<le> Z) F" 
61810  1174 
by (drule spec [of _ Z], erule eventually_mono, auto simp add: less_imp_le) 
60036  1175 
qed 
1176 

1177 
lemma filterlim_at_bot_le: 

1178 
fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b" 

1179 
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)" 

1180 
unfolding filterlim_at_bot 

1181 
proof safe 

1182 
fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F" 

1183 
with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F" 

61810  1184 
by (auto elim!: eventually_mono) 
60036  1185 
qed simp 
1186 

1187 
lemma filterlim_at_bot_lt: 

1188 
fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b" 

1189 
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)" 

1190 
by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans) 

1191 

1192 

60758  1193 
subsection \<open>Setup @{typ "'a filter"} for lifting and transfer\<close> 
60036  1194 

1195 
context begin interpretation lifting_syntax . 

1196 

1197 
definition rel_filter :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> 'b filter \<Rightarrow> bool" 

1198 
where "rel_filter R F G = ((R ===> op =) ===> op =) (Rep_filter F) (Rep_filter G)" 

1199 

1200 
lemma rel_filter_eventually: 

61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset

1201 
"rel_filter R F G \<longleftrightarrow> 
60036  1202 
((R ===> op =) ===> op =) (\<lambda>P. eventually P F) (\<lambda>P. eventually P G)" 
1203 
by(simp add: rel_filter_def eventually_def) 

1204 

1205 
lemma filtermap_id [simp, id_simps]: "filtermap id = id" 

1206 
by(simp add: fun_eq_iff id_def filtermap_ident) 

1207 

1208 
lemma filtermap_id' [simp]: "filtermap (\<lambda>x. x) = (\<lambda>F. F)" 

1209 
using filtermap_id unfolding id_def . 

1210 

1211 
lemma Quotient_filter [quot_map]: 

1212 
assumes Q: "Quotient R Abs Rep T" 

1213 
shows "Quotient (rel_filter R) (filtermap Abs) (filtermap Rep) (rel_filter T)" 

1214 
unfolding Quotient_alt_def 

1215 
proof(intro conjI strip) 

1216 
from Q have *: "\<And>x y. T x y \<Longrightarrow> Abs x = y" 

1217 
unfolding Quotient_alt_def by blast 

1218 

1219 
fix F G 

1220 
assume "rel_filter T F G" 

1221 
thus "filtermap Abs F = G" unfolding filter_eq_iff 

1222 
by(auto simp add: eventually_filtermap rel_filter_eventually * rel_funI del: iffI elim!: rel_funD) 

1223 
next 

1224 
from Q have *: "\<And>x. T (Rep x) x" unfolding Quotient_alt_def by blast 

1225 

1226 
fix F 

61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset

1227 
show "rel_filter T (filtermap Rep F) F" 
60036  1228 
by(auto elim: rel_funD intro: * intro!: ext arg_cong[where f="\<lambda>P. eventually P F"] rel_funI 
1229 
del: iffI simp add: eventually_filtermap rel_filter_eventually) 

1230 
qed(auto simp add: map_fun_def o_def eventually_filtermap filter_eq_iff fun_eq_iff rel_filter_eventually 

1231 
fun_quotient[OF fun_quotient[OF Q identity_quotient] identity_quotient, unfolded Quotient_alt_def]) 

1232 

1233 
lemma eventually_parametric [transfer_rule]: 

1234 
"((A ===> op =) ===> rel_filter A ===> op =) eventually eventually" 

1235 
by(simp add: rel_fun_def rel_filter_eventually) 

1236 

60038  1237 
lemma frequently_parametric [transfer_rule]: 
1238 
"((A ===> op =) ===> rel_filter A ===> op =) frequently frequently" 

1239 
unfolding frequently_def[abs_def] by transfer_prover 

1240 

60036  1241 
lemma rel_filter_eq [relator_eq]: "rel_filter op = = op =" 
1242 
by(auto simp add: rel_filter_eventually rel_fun_eq fun_eq_iff filter_eq_iff) 

1243 

1244 
lemma rel_filter_mono [relator_mono]: 

1245 
"A \<le> B \<Longrightarrow> rel_filter A \<le> rel_filter B" 

1246 
unfolding rel_filter_eventually[abs_def] 

1247 
by(rule le_funI)+(intro fun_mono fun_mono[THEN le_funD, THEN le_funD] order.refl) 

1248 

1249 
lemma rel_filter_conversep [simp]: "rel_filter A\<inverse>\<inverse> = (rel_filter A)\<inverse>\<inverse>" 

61233
1da01148d4b1
Prepared two nonterminating proofs; no obvious link with my changes
paulson <lp15@cam.ac.uk>
parents:
60974
diff
changeset

1250 
apply (simp add: rel_filter_eventually fun_eq_iff rel_fun_def) 
1da01148d4b1
Prepared two nonterminating proofs; no obvious link with my changes
paulson <lp15@cam.ac.uk>
parents:
60974
diff
changeset

1251 
apply (safe; metis) 
1da01148d4b1
Prepared two nonterminating proofs; no obvious link with my changes
paulson <lp15@cam.ac.uk>
parents:
60974
diff
changeset

1252 
done 
60036  1253 

1254 
lemma is_filter_parametric_aux: 

1255 
assumes "is_filter F" 

1256 
assumes [transfer_rule]: "bi_total A" "bi_unique A" 

1257 
and [transfer_rule]: "((A ===> op =) ===> op =) F G" 

1258 
shows "is_filter G" 

1259 
proof  

1260 
interpret is_filter F by fact 

1261 
show ?thesis 

1262 
proof 

1263 
have "F (\<lambda>_. True) = G (\<lambda>x. True)" by transfer_prover 

1264 
thus "G (\<lambda>x. True)" by(simp add: True) 

1265 
next 

1266 
fix P' Q' 

1267 
assume "G P'" "G Q'" 

1268 
moreover 

60758  1269 
from bi_total_fun[OF \<open>bi_unique A\<close> bi_total_eq, unfolded bi_total_def] 
60036  1270 
obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast 
1271 
have "F P = G P'" "F Q = G Q'" by transfer_prover+ 

1272 
ultimately have "F (\<lambda>x. P x \<and> Q x)" by(simp add: conj) 

1273 
moreover have "F (\<lambda>x. P x \<and> Q x) = G (\<lambda>x. P' x \<and> Q' x)" by transfer_prover 

1274 
ultimately show "G (\<lambda>x. P' x \<and> Q' x)" by simp 

1275 
next 

1276 
fix P' Q' 

1277 
assume "\<forall>x. P' x \<longrightarrow> Q' x" "G P'" 

1278 
moreover 

60758  1279 
from bi_total_fun[OF \<open>bi_unique A\<close> bi_total_eq, unfolded bi_total_def] 
60036  1280 
obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast 
1281 
have "F P = G P'" by transfer_prover 

1282 
moreover have "(\<forall>x. P x \<longrightarrow> Q x) \<longleftrightarrow> (\<forall>x. P' x \<longrightarrow> Q' x)" by transfer_prover 

1283 
ultimately have "F Q" by(simp add: mono) 

1284 
moreover have "F Q = G Q'" by transfer_prover 

1285 
ultimately show "G Q'" by simp 

1286 
qed 

1287 
qed 

1288 

1289 
lemma is_filter_parametric [transfer_rule]: 

1290 
"\<lbrakk> bi_total A; bi_unique A \<rbrakk> 

1291 
\<Longrightarrow> (((A ===> op =) ===> op =) ===> op =) is_filter is_filter" 

1292 
apply(rule rel_funI) 

1293 
apply(rule iffI) 

1294 
apply(erule (3) is_filter_parametric_aux) 

1295 
apply(erule is_filter_parametric_aux[where A="conversep A"]) 

61233
1da01148d4b1
Prepared two nonterminating proofs; no obvious link with my changes
paulson <lp15@cam.ac.uk>
parents:
60974
diff
changeset

1296 
apply (simp_all add: rel_fun_def) 
1da01148d4b1
Prepared two nonterminating proofs; no obvious link with my changes
paulson <lp15@cam.ac.uk>
parents:
60974
diff
changeset

1297 
apply metis 
60036  1298 
done 
1299 

1300 
lemma left_total_rel_filter [transfer_rule]: 

1301 
assumes [transfer_rule]: "bi_total A" "bi_unique A" 

1302 
shows "left_total (rel_filter A)" 

1303 
proof(rule left_totalI) 

1304 
fix F :: "'a filter" 

60758  1305 
from bi_total_fun[OF bi_unique_fun[OF \<open>bi_total A\<close> bi_unique_eq] bi_total_eq] 
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset

1306 
obtain G where [transfer_rule]: "((A ===> op =) ===> op =) (\<lambda>P. eventually P F) G" 
60036  1307 
unfolding bi_total_def by blast 
1308 
moreover have "is_filter (\<lambda>P. eventually P F) \<longleftrightarrow> is_filter G" by transfer_prover 

1309 
hence "is_filter G" by(simp add: eventually_def is_filter_Rep_filter) 

1310 
ultimately have "rel_filter A F (Abs_filter G)" 

1311 
by(simp add: rel_filter_eventually eventually_Abs_filter) 

1312 
thus "\<exists>G. rel_filter A F G" .. 

1313 
qed 

1314 

1315 
lemma right_total_rel_filter [transfer_rule]: 

1316 
"\<lbrakk> bi_total A; bi_unique A \<rbrakk> \<Longrightarrow> right_total (rel_filter A)" 

1317 
using left_total_rel_filter[of "A\<inverse>\<inverse>"] by simp 

1318 

1319 
lemma bi_total_rel_filter [transfer_rule]: 

1320 
assumes "bi_total A" "bi_unique A" 

1321 
shows "bi_total (rel_filter A)" 

1322 
unfolding bi_total_alt_def using assms 

1323 
by(simp add: left_total_rel_filter right_total_rel_filter) 

1324 

1325 
lemma left_unique_rel_filter [transfer_rule]: 

1326 
assumes "left_unique A" 

1327 
shows "left_unique (rel_filter A)" 

1328 
proof(rule left_uniqueI) 

1329 
fix F F' G 

1330 
assume [transfer_rule]: "rel_filter A F G" "rel_filter A F' G" 

1331 
show "F = F'" 

1332 
unfolding filter_eq_iff 

1333 
proof 

1334 
fix P :: "'a \<Rightarrow> bool" 

1335 
obtain P' where [transfer_rule]: "(A ===> op =) P P'" 

1336 
using left_total_fun[OF assms left_total_eq] unfolding left_total_def by blast 

61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset

1337 
have "eventually P F = eventually P' G" 
60036  1338 
and "eventually P F' = eventually P' G" by transfer_prover+ 
1339 
thus "eventually P F = eventually P F'" by simp 

1340 
qed 

1341 
qed 

1342 

1343 
lemma right_unique_rel_filter [transfer_rule]: 

1344 
"right_unique A \<Longrightarrow> right_unique (rel_filter A)" 

1345 
using left_unique_rel_filter[of "A\<inverse>\<inverse>"] by simp 

1346 

1347 
lemma bi_unique_rel_filter [transfer_rule]: 

1348 
"bi_unique A \<Longrightarrow> bi_unique (rel_filter A)" 

1349 
by(simp add: bi_unique_alt_def left_unique_rel_filter right_unique_rel_filter) 

1350 

1351 
lemma top_filter_parametric [transfer_rule]: 

1352 
"bi_total A \<Longrightarrow> (rel_filter A) top top" 

1353 
by(simp add: rel_filter_eventually All_transfer) 

1354 

1355 
lemma bot_filter_parametric [transfer_rule]: "(rel_filter A) bot bot" 

1356 
by(simp add: rel_filter_eventually rel_fun_def) 

1357 

1358 
lemma sup_filter_parametric [transfer_rule]: 

1359 
"(rel_filter A ===> rel_filter A ===> rel_filter A) sup sup" 

1360 
by(fastforce simp add: rel_filter_eventually[abs_def] eventually_sup dest: rel_funD) 

1361 

1362 
lemma Sup_filter_parametric [transfer_rule]: 

1363 
"(rel_set (rel_filter A) ===> rel_filter A) Sup Sup" 

1364 
proof(rule rel_funI) 

1365 
fix S T 

1366 
assume [transfer_rule]: "rel_set (rel_filter A) S T" 

1367 
show "rel_filter A (Sup S) (Sup T)" 

1368 
by(simp add: rel_filter_eventually eventually_Sup) transfer_prover 

1369 
qed 

1370 

1371 
lemma principal_parametric [transfer_rule]: 

1372 
"(rel_set A ===> rel_filter A) principal principal" 

1373 
proof(rule rel_funI) 

1374 
fix S S' 

1375 
assume [transfer_rule]: "rel_set A S S'" 

1376 
show "rel_filter A (principal S) (principal S')" 

1377 
by(simp add: rel_filter_eventually eventually_principal) transfer_prover 

1378 
qed 

1379 

1380 
context 

1381 
fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" 

61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset

1382 
assumes [transfer_rule]: "bi_unique A" 
60036  1383 
begin 
1384 

1385 
lemma le_filter_parametric [transfer_rule]: 

1386 
"(rel_filter A ===> rel_filter A ===> op =) op \<le> op \<le>" 

1387 
unfolding le_filter_def[abs_def] by transfer_prover 

1388 

1389 
lemma less_filter_parametric [transfer_rule]: 

1390 
"(rel_filter A ===> rel_filter A ===> op =) op < op <" 

1391 
unfolding less_filter_def[abs_def] by transfer_prover 

1392 

1393 
context 

1394 
assumes [transfer_rule]: "bi_total A" 

1395 
begin 

1396 

1397 
lemma Inf_filter_parametric [transfer_rule]: 

1398 
"(rel_set (rel_filter A) ===> rel_filter A) Inf Inf" 

1399 
unfolding Inf_filter_def[abs_def] by transfer_prover 

1400 

1401 
lemma inf_filter_parametric [transfer_rule]: 

1402 
"(rel_filter A ===> rel_filter A ===> rel_filter A) inf inf" 

1403 
proof(intro rel_funI)+ 

1404 
fix F F' G G' 

1405 
assume [transfer_rule]: "rel_filter A F F'" "rel_filter A G G'" 

1406 
have "rel_filter A (Inf {F, G}) (Inf {F', G'})" by transfer_prover 

1407 
thus "rel_filter A (inf F G) (inf F' G')" by simp 

1408 
qed 

1409 

1410 
end 
