60036

1 
(* Title: HOL/Filter.thy


2 
Author: Brian Huffman


3 
Author: Johannes Hölzl


4 
*)


5 


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section {* Filters on predicates *}


7 


8 
theory Filter


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imports Set_Interval Lifting_Set


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begin


11 


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subsection {* Filters *}


13 


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text {*


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This definition also allows nonproper filters.


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*}


17 


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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)"


23 


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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


28 


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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


31 


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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)


35 


36 


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subsubsection {* Eventually *}


38 


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definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"


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


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syntax (xsymbols)

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

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translations

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"\<forall>\<^sub>Fx in F. P" == "CONST eventually (\<lambda>x. P) F"

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lemma eventually_Abs_filter:


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assumes "is_filter F" shows "eventually P (Abs_filter F) = F P"


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unfolding eventually_def using assms by (simp add: Abs_filter_inverse)


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lemma filter_eq_iff:


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


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unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..


55 


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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])


59 


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


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proof 


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


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thus "eventually P F" by simp


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qed


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lemma eventually_mono:


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"(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P F \<Longrightarrow> eventually Q F"


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unfolding eventually_def


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by (rule is_filter.mono [OF is_filter_Rep_filter])


70 


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lemma eventually_conj:


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


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


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


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using assms unfolding eventually_def


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by (rule is_filter.conj [OF is_filter_Rep_filter])


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lemma eventually_Ball_finite:


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assumes "finite A" and "\<forall>y\<in>A. eventually (\<lambda>x. P x y) net"


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shows "eventually (\<lambda>x. \<forall>y\<in>A. P x y) net"


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using assms by (induct set: finite, simp, simp add: eventually_conj)


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lemma eventually_all_finite:


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fixes P :: "'a \<Rightarrow> 'b::finite \<Rightarrow> bool"


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assumes "\<And>y. eventually (\<lambda>x. P x y) net"


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shows "eventually (\<lambda>x. \<forall>y. P x y) net"


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using eventually_Ball_finite [of UNIV P] assms by simp


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lemma eventually_mp:


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


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


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


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proof (rule eventually_mono)


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show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp


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


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using assms by (rule eventually_conj)


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qed


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lemma eventually_rev_mp:


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


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


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


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


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lemma eventually_conj_iff:


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


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by (auto intro: eventually_conj elim: eventually_rev_mp)


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lemma eventually_elim1:


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


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


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


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using assms by (auto elim!: eventually_rev_mp)


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lemma eventually_elim2:


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


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


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


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


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using assms by (auto elim!: eventually_rev_mp)


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lemma not_eventually_impI: "eventually P F \<Longrightarrow> \<not> eventually Q F \<Longrightarrow> \<not> eventually (\<lambda>x. P x \<longrightarrow> Q x) F"


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by (auto intro: eventually_mp)


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


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by (metis always_eventually)


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lemma eventually_subst:


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


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


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proof 


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


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


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by (auto elim: eventually_elim1)


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then show ?thesis by (auto elim: eventually_elim2)


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qed


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ML {*


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fun eventually_elim_tac ctxt facts = SUBGOAL_CASES (fn (goal, i) =>


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let


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val mp_thms = facts RL @{thms eventually_rev_mp}


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val raw_elim_thm =


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(@{thm allI} RS @{thm always_eventually})


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> fold (fn thm1 => fn thm2 => thm2 RS thm1) mp_thms


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> fold (fn _ => fn thm => @{thm impI} RS thm) facts


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val cases_prop = Thm.prop_of (raw_elim_thm RS Goal.init (Thm.cterm_of ctxt goal))


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val cases = Rule_Cases.make_common ctxt cases_prop [(("elim", []), [])]


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in


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CASES cases (rtac raw_elim_thm i)


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end)


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*}


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method_setup eventually_elim = {*


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Scan.succeed (fn ctxt => METHOD_CASES (HEADGOAL o eventually_elim_tac ctxt))


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*} "elimination of eventually quantifiers"


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subsection \<open> Frequently as dual to eventually \<close>


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definition frequently :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"


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where "frequently P F \<longleftrightarrow> \<not> eventually (\<lambda>x. \<not> P x) F"


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syntax (xsymbols)


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"_frequently" :: "pttrn \<Rightarrow> 'a filter \<Rightarrow> bool \<Rightarrow> bool" ("(3\<exists>\<^sub>F _ in _./ _)" [0, 0, 10] 10)


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translations


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"\<exists>\<^sub>Fx in F. P" == "CONST frequently (\<lambda>x. P) F"


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lemma not_frequently_False [simp]: "\<not> frequently (\<lambda>x. False) F"


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by (simp add: frequently_def)


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lemma frequently_ex: "frequently P F \<Longrightarrow> \<exists>x. P x"


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by (auto simp: frequently_def dest: not_eventuallyD)


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lemma frequently_mp:


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assumes ev: "eventually (\<lambda>x. P x \<longrightarrow> Q x) F" and P: "frequently (\<lambda>x. P x) F"


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


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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 "frequently (\<lambda>x. P x) F"


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


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


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


189 


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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)


192 


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lemma frequently_disj_iff:


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"frequently (\<lambda>x. P x \<or> Q x) F \<longleftrightarrow> frequently (\<lambda>x. P x) F \<or> frequently (\<lambda>x. Q x) F"


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by (simp add: frequently_def eventually_conj_iff)


196 


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lemma frequently_disj:


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"frequently (\<lambda>x. P x) F \<Longrightarrow> frequently (\<lambda>x. Q x) F \<Longrightarrow> frequently (\<lambda>x. P x \<or> Q x) F"


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by (simp add: frequently_disj_iff)


200 


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lemma frequently_Bex_finite:


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assumes "finite A" shows "frequently (\<lambda>x. \<exists>y\<in>A. P x y) net \<longleftrightarrow> (\<exists>y\<in>A. frequently (\<lambda>x. P x y) net)"


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using assms by induction (auto simp: frequently_disj_iff)

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subsubsection {* Finerthan relation *}


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text {* @{term "F \<le> F'"} means that filter @{term F} is finer than


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filter @{term F'}. *}


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instantiation filter :: (type) complete_lattice


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begin


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definition le_filter_def:


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


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definition


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


218 


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definition


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


221 


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definition


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


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definition


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


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definition


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"inf F F' = Abs_filter


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


231 


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definition


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


234 


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definition


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


237 


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


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unfolding top_filter_def


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


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lemma eventually_bot [simp]: "eventually P bot"


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unfolding bot_filter_def


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


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lemma eventually_sup:


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


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unfolding sup_filter_def


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by (rule eventually_Abs_filter, rule is_filter.intro)


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(auto elim!: eventually_rev_mp)


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lemma eventually_inf:


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


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


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unfolding inf_filter_def


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apply (rule eventually_Abs_filter, rule is_filter.intro)


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apply (fast intro: eventually_True)


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apply clarify


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apply (intro exI conjI)


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apply (erule (1) eventually_conj)


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apply (erule (1) eventually_conj)


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apply simp


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apply auto


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done


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lemma eventually_Sup:


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


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unfolding Sup_filter_def


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apply (rule eventually_Abs_filter, rule is_filter.intro)


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apply (auto intro: eventually_conj elim!: eventually_rev_mp)


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done


272 


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instance proof


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


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


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by (rule less_filter_def) }


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{ show "F \<le> F"


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unfolding le_filter_def by simp }


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


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unfolding le_filter_def by simp }


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


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unfolding le_filter_def filter_eq_iff by fast }


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


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unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }


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


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unfolding le_filter_def eventually_inf


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by (auto elim!: eventually_mono intro: eventually_conj) }


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{ show "F \<le> sup F F'" and "F' \<le> sup F F'"


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unfolding le_filter_def eventually_sup by simp_all }


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


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unfolding le_filter_def eventually_sup by simp }


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


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unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }


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


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unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }


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


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unfolding le_filter_def eventually_Sup by simp }


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


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unfolding le_filter_def eventually_Sup by simp }


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


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by (auto simp: top_filter_def Inf_filter_def Sup_filter_def)


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(metis (full_types) top_filter_def always_eventually eventually_top) }


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


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by (auto simp: bot_filter_def Sup_filter_def) }


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qed


306 


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end


308 


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lemma filter_leD:


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


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unfolding le_filter_def by simp


312 


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lemma filter_leI:


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


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unfolding le_filter_def by simp


316 


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lemma eventually_False:


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


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unfolding filter_eq_iff by (auto elim: eventually_rev_mp)


320 


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abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"


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where "trivial_limit F \<equiv> F = bot"


323 


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


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by (rule eventually_False [symmetric])


326 


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lemma eventually_const: "\<not> trivial_limit net \<Longrightarrow> eventually (\<lambda>x. P) net \<longleftrightarrow> P"


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by (cases P) (simp_all add: eventually_False)


329 


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


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proof 


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


333 


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{ fix P have "eventually P (Abs_filter ?F) \<longleftrightarrow> ?F P"


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proof (rule eventually_Abs_filter is_filter.intro)+


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


337 
by (rule exI[of _ "{}"]) (simp add: le_fun_def)


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next


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fix P Q


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assume "?F P" then guess X ..


341 
moreover


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assume "?F Q" then guess Y ..


343 
ultimately show "?F (\<lambda>x. P x \<and> Q x)"


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


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(auto simp: Inf_union_distrib eventually_inf)


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next


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fix P Q


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assume "?F P" then guess X ..


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


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ultimately show "?F Q"


351 
by (intro exI[of _ X]) (auto elim: eventually_elim1)


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qed }


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note eventually_F = this


354 


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have "Inf B = Abs_filter ?F"


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proof (intro antisym Inf_greatest)


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


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by (auto simp: le_filter_def eventually_F dest: Inf_superset_mono)


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next


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


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


362 
qed


363 
then show ?thesis


364 
by (simp add: eventually_F)


365 
qed


366 


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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))"


368 
unfolding INF_def[of B] eventually_Inf[of P "F`B"]


369 
by (metis Inf_image_eq finite_imageI image_mono finite_subset_image)


370 


371 
lemma Inf_filter_not_bot:


372 
fixes B :: "'a filter set"


373 
shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> Inf X \<noteq> bot) \<Longrightarrow> Inf B \<noteq> bot"


374 
unfolding trivial_limit_def eventually_Inf[of _ B]


375 
bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp


376 


377 
lemma INF_filter_not_bot:


378 
fixes F :: "'i \<Rightarrow> 'a filter"


379 
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"


380 
unfolding trivial_limit_def eventually_INF[of _ B]


381 
bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp


382 


383 
lemma eventually_Inf_base:


384 
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"


385 
shows "eventually P (Inf B) \<longleftrightarrow> (\<exists>b\<in>B. eventually P b)"


386 
proof (subst eventually_Inf, safe)


387 
fix X assume "finite X" "X \<subseteq> B"


388 
then have "\<exists>b\<in>B. \<forall>x\<in>X. b \<le> x"


389 
proof induct


390 
case empty then show ?case


391 
using `B \<noteq> {}` by auto


392 
next


393 
case (insert x X)


394 
then obtain b where "b \<in> B" "\<And>x. x \<in> X \<Longrightarrow> b \<le> x"


395 
by auto


396 
with `insert x X \<subseteq> B` base[of b x] show ?case


397 
by (auto intro: order_trans)


398 
qed


399 
then obtain b where "b \<in> B" "b \<le> Inf X"


400 
by (auto simp: le_Inf_iff)


401 
then show "eventually P (Inf X) \<Longrightarrow> Bex B (eventually P)"


402 
by (intro bexI[of _ b]) (auto simp: le_filter_def)


403 
qed (auto intro!: exI[of _ "{x}" for x])


404 


405 
lemma eventually_INF_base:


406 
"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>


407 
eventually P (INF b:B. F b) \<longleftrightarrow> (\<exists>b\<in>B. eventually P (F b))"


408 
unfolding INF_def by (subst eventually_Inf_base) auto


409 


410 


411 
subsubsection {* Map function for filters *}


412 


413 
definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"


414 
where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"


415 


416 
lemma eventually_filtermap:


417 
"eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"


418 
unfolding filtermap_def


419 
apply (rule eventually_Abs_filter)


420 
apply (rule is_filter.intro)


421 
apply (auto elim!: eventually_rev_mp)


422 
done


423 


424 
lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"


425 
by (simp add: filter_eq_iff eventually_filtermap)


426 


427 
lemma filtermap_filtermap:


428 
"filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"


429 
by (simp add: filter_eq_iff eventually_filtermap)


430 


431 
lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"


432 
unfolding le_filter_def eventually_filtermap by simp


433 


434 
lemma filtermap_bot [simp]: "filtermap f bot = bot"


435 
by (simp add: filter_eq_iff eventually_filtermap)


436 


437 
lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)"


438 
by (auto simp: filter_eq_iff eventually_filtermap eventually_sup)


439 


440 
lemma filtermap_inf: "filtermap f (inf F1 F2) \<le> inf (filtermap f F1) (filtermap f F2)"


441 
by (auto simp: le_filter_def eventually_filtermap eventually_inf)


442 


443 
lemma filtermap_INF: "filtermap f (INF b:B. F b) \<le> (INF b:B. filtermap f (F b))"


444 
proof 


445 
{ fix X :: "'c set" assume "finite X"


446 
then have "filtermap f (INFIMUM X F) \<le> (INF b:X. filtermap f (F b))"


447 
proof induct


448 
case (insert x X)


449 
have "filtermap f (INF a:insert x X. F a) \<le> inf (filtermap f (F x)) (filtermap f (INF a:X. F a))"


450 
by (rule order_trans[OF _ filtermap_inf]) simp


451 
also have "\<dots> \<le> inf (filtermap f (F x)) (INF a:X. filtermap f (F a))"


452 
by (intro inf_mono insert order_refl)


453 
finally show ?case


454 
by simp


455 
qed simp }


456 
then show ?thesis


457 
unfolding le_filter_def eventually_filtermap


458 
by (subst (1 2) eventually_INF) auto


459 
qed


460 
subsubsection {* Standard filters *}


461 


462 
definition principal :: "'a set \<Rightarrow> 'a filter" where


463 
"principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)"


464 


465 
lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)"


466 
unfolding principal_def


467 
by (rule eventually_Abs_filter, rule is_filter.intro) auto


468 


469 
lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F"


470 
unfolding eventually_inf eventually_principal by (auto elim: eventually_elim1)


471 


472 
lemma principal_UNIV[simp]: "principal UNIV = top"


473 
by (auto simp: filter_eq_iff eventually_principal)


474 


475 
lemma principal_empty[simp]: "principal {} = bot"


476 
by (auto simp: filter_eq_iff eventually_principal)


477 


478 
lemma principal_eq_bot_iff: "principal X = bot \<longleftrightarrow> X = {}"


479 
by (auto simp add: filter_eq_iff eventually_principal)


480 


481 
lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B"


482 
by (auto simp: le_filter_def eventually_principal)


483 


484 
lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F"


485 
unfolding le_filter_def eventually_principal


486 
apply safe


487 
apply (erule_tac x="\<lambda>x. x \<in> A" in allE)


488 
apply (auto elim: eventually_elim1)


489 
done


490 


491 
lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B"


492 
unfolding eq_iff by simp


493 


494 
lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)"


495 
unfolding filter_eq_iff eventually_sup eventually_principal by auto


496 


497 
lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)"


498 
unfolding filter_eq_iff eventually_inf eventually_principal


499 
by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"])


500 


501 
lemma SUP_principal[simp]: "(SUP i : I. principal (A i)) = principal (\<Union>i\<in>I. A i)"


502 
unfolding filter_eq_iff eventually_Sup SUP_def by (auto simp: eventually_principal)


503 


504 
lemma INF_principal_finite: "finite X \<Longrightarrow> (INF x:X. principal (f x)) = principal (\<Inter>x\<in>X. f x)"


505 
by (induct X rule: finite_induct) auto


506 


507 
lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)"


508 
unfolding filter_eq_iff eventually_filtermap eventually_principal by simp


509 


510 
subsubsection {* Order filters *}


511 


512 
definition at_top :: "('a::order) filter"


513 
where "at_top = (INF k. principal {k ..})"


514 


515 
lemma at_top_sub: "at_top = (INF k:{c::'a::linorder..}. principal {k ..})"


516 
by (auto intro!: INF_eq max.cobounded1 max.cobounded2 simp: at_top_def)


517 


518 
lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)"


519 
unfolding at_top_def


520 
by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)


521 


522 
lemma eventually_ge_at_top:


523 
"eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top"


524 
unfolding eventually_at_top_linorder by auto


525 


526 
lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::{no_top, linorder}. \<forall>n>N. P n)"


527 
proof 


528 
have "eventually P (INF k. principal {k <..}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n>N. P n)"


529 
by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)


530 
also have "(INF k. principal {k::'a <..}) = at_top"


531 
unfolding at_top_def


532 
by (intro INF_eq) (auto intro: less_imp_le simp: Ici_subset_Ioi_iff gt_ex)


533 
finally show ?thesis .


534 
qed


535 


536 
lemma eventually_gt_at_top:


537 
"eventually (\<lambda>x. (c::_::unbounded_dense_linorder) < x) at_top"


538 
unfolding eventually_at_top_dense by auto


539 


540 
definition at_bot :: "('a::order) filter"


541 
where "at_bot = (INF k. principal {.. k})"


542 


543 
lemma at_bot_sub: "at_bot = (INF k:{.. c::'a::linorder}. principal {.. k})"


544 
by (auto intro!: INF_eq min.cobounded1 min.cobounded2 simp: at_bot_def)


545 


546 
lemma eventually_at_bot_linorder:


547 
fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"


548 
unfolding at_bot_def


549 
by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)


550 


551 
lemma eventually_le_at_bot:


552 
"eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot"


553 
unfolding eventually_at_bot_linorder by auto


554 


555 
lemma eventually_at_bot_dense: "eventually P at_bot \<longleftrightarrow> (\<exists>N::'a::{no_bot, linorder}. \<forall>n<N. P n)"


556 
proof 


557 
have "eventually P (INF k. principal {..< k}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n<N. P n)"


558 
by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)


559 
also have "(INF k. principal {..< k::'a}) = at_bot"


560 
unfolding at_bot_def


561 
by (intro INF_eq) (auto intro: less_imp_le simp: Iic_subset_Iio_iff lt_ex)


562 
finally show ?thesis .


563 
qed


564 


565 
lemma eventually_gt_at_bot:


566 
"eventually (\<lambda>x. x < (c::_::unbounded_dense_linorder)) at_bot"


567 
unfolding eventually_at_bot_dense by auto


568 


569 
lemma trivial_limit_at_bot_linorder: "\<not> trivial_limit (at_bot ::('a::linorder) filter)"


570 
unfolding trivial_limit_def


571 
by (metis eventually_at_bot_linorder order_refl)


572 


573 
lemma trivial_limit_at_top_linorder: "\<not> trivial_limit (at_top ::('a::linorder) filter)"


574 
unfolding trivial_limit_def


575 
by (metis eventually_at_top_linorder order_refl)


576 


577 
subsection {* Sequentially *}


578 


579 
abbreviation sequentially :: "nat filter"


580 
where "sequentially \<equiv> at_top"


581 


582 
lemma eventually_sequentially:


583 
"eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"


584 
by (rule eventually_at_top_linorder)


585 


586 
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"


587 
unfolding filter_eq_iff eventually_sequentially by auto


588 


589 
lemmas trivial_limit_sequentially = sequentially_bot


590 


591 
lemma eventually_False_sequentially [simp]:


592 
"\<not> eventually (\<lambda>n. False) sequentially"


593 
by (simp add: eventually_False)


594 


595 
lemma le_sequentially:


596 
"F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"


597 
by (simp add: at_top_def le_INF_iff le_principal)


598 


599 
lemma eventually_sequentiallyI:


600 
assumes "\<And>x. c \<le> x \<Longrightarrow> P x"


601 
shows "eventually P sequentially"


602 
using assms by (auto simp: eventually_sequentially)


603 


604 
lemma eventually_sequentially_seg:


605 
"eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially"


606 
unfolding eventually_sequentially


607 
apply safe


608 
apply (rule_tac x="N + k" in exI)


609 
apply rule


610 
apply (erule_tac x="n  k" in allE)


611 
apply auto []


612 
apply (rule_tac x=N in exI)


613 
apply auto []


614 
done


615 

60039

616 
subsection \<open> The cofinite filter \<close>


617 


618 
definition "cofinite = Abs_filter (\<lambda>P. finite {x. \<not> P x})"


619 


620 
lemma eventually_cofinite: "eventually P cofinite \<longleftrightarrow> finite {x. \<not> P x}"


621 
unfolding cofinite_def


622 
proof (rule eventually_Abs_filter, rule is_filter.intro)


623 
fix P Q :: "'a \<Rightarrow> bool" assume "finite {x. \<not> P x}" "finite {x. \<not> Q x}"


624 
from finite_UnI[OF this] show "finite {x. \<not> (P x \<and> Q x)}"


625 
by (rule rev_finite_subset) auto


626 
next


627 
fix P Q :: "'a \<Rightarrow> bool" assume P: "finite {x. \<not> P x}" and *: "\<forall>x. P x \<longrightarrow> Q x"


628 
from * show "finite {x. \<not> Q x}"


629 
by (intro finite_subset[OF _ P]) auto


630 
qed simp


631 


632 
lemma cofinite_bot[simp]: "cofinite = (bot::'a filter) \<longleftrightarrow> finite (UNIV :: 'a set)"


633 
unfolding trivial_limit_def eventually_cofinite by simp


634 


635 
lemma cofinite_eq_sequentially: "cofinite = sequentially"


636 
unfolding filter_eq_iff eventually_sequentially eventually_cofinite


637 
proof safe


638 
fix P :: "nat \<Rightarrow> bool" assume [simp]: "finite {x. \<not> P x}"


639 
show "\<exists>N. \<forall>n\<ge>N. P n"


640 
proof cases


641 
assume "{x. \<not> P x} \<noteq> {}" then show ?thesis


642 
by (intro exI[of _ "Suc (Max {x. \<not> P x})"]) (auto simp: Suc_le_eq)


643 
qed auto


644 
next


645 
fix P :: "nat \<Rightarrow> bool" and N :: nat assume "\<forall>n\<ge>N. P n"


646 
then have "{x. \<not> P x} \<subseteq> {..< N}"


647 
by (auto simp: not_le)


648 
then show "finite {x. \<not> P x}"


649 
by (blast intro: finite_subset)


650 
qed

60036

651 


652 
subsection {* Limits *}


653 


654 
definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where


655 
"filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"


656 


657 
syntax


658 
"_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)


659 


660 
translations


661 
"LIM x F1. f :> F2" == "CONST filterlim (%x. f) F2 F1"


662 


663 
lemma filterlim_iff:


664 
"(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)"


665 
unfolding filterlim_def le_filter_def eventually_filtermap ..


666 


667 
lemma filterlim_compose:


668 
"filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1"


669 
unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)


670 


671 
lemma filterlim_mono:


672 
"filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'"


673 
unfolding filterlim_def by (metis filtermap_mono order_trans)


674 


675 
lemma filterlim_ident: "LIM x F. x :> F"


676 
by (simp add: filterlim_def filtermap_ident)


677 


678 
lemma filterlim_cong:


679 
"F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'"


680 
by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2)


681 


682 
lemma filterlim_mono_eventually:


683 
assumes "filterlim f F G" and ord: "F \<le> F'" "G' \<le> G"


684 
assumes eq: "eventually (\<lambda>x. f x = f' x) G'"


685 
shows "filterlim f' F' G'"


686 
apply (rule filterlim_cong[OF refl refl eq, THEN iffD1])


687 
apply (rule filterlim_mono[OF _ ord])


688 
apply fact


689 
done


690 


691 
lemma filtermap_mono_strong: "inj f \<Longrightarrow> filtermap f F \<le> filtermap f G \<longleftrightarrow> F \<le> G"


692 
apply (auto intro!: filtermap_mono) []


693 
apply (auto simp: le_filter_def eventually_filtermap)


694 
apply (erule_tac x="\<lambda>x. P (inv f x)" in allE)


695 
apply auto


696 
done


697 


698 
lemma filtermap_eq_strong: "inj f \<Longrightarrow> filtermap f F = filtermap f G \<longleftrightarrow> F = G"


699 
by (simp add: filtermap_mono_strong eq_iff)


700 


701 
lemma filterlim_principal:


702 
"(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)"


703 
unfolding filterlim_def eventually_filtermap le_principal ..


704 


705 
lemma filterlim_inf:


706 
"(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))"


707 
unfolding filterlim_def by simp


708 


709 
lemma filterlim_INF:


710 
"(LIM x F. f x :> (INF b:B. G b)) \<longleftrightarrow> (\<forall>b\<in>B. LIM x F. f x :> G b)"


711 
unfolding filterlim_def le_INF_iff ..


712 


713 
lemma filterlim_INF_INF:


714 
"(\<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)"


715 
unfolding filterlim_def by (rule order_trans[OF filtermap_INF INF_mono])


716 


717 
lemma filterlim_base:


718 
"(\<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>


719 
LIM x (INF i:I. principal (F i)). f x :> (INF j:J. principal (G j))"


720 
by (force intro!: filterlim_INF_INF simp: image_subset_iff)


721 


722 
lemma filterlim_base_iff:


723 
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"


724 
shows "(LIM x (INF i:I. principal (F i)). f x :> INF j:J. principal (G j)) \<longleftrightarrow>


725 
(\<forall>j\<in>J. \<exists>i\<in>I. \<forall>x\<in>F i. f x \<in> G j)"


726 
unfolding filterlim_INF filterlim_principal


727 
proof (subst eventually_INF_base)


728 
fix i j assume "i \<in> I" "j \<in> I"


729 
with chain[OF this] show "\<exists>x\<in>I. principal (F x) \<le> inf (principal (F i)) (principal (F j))"


730 
by auto


731 
qed (auto simp: eventually_principal `I \<noteq> {}`)


732 


733 
lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2"


734 
unfolding filterlim_def filtermap_filtermap ..


735 


736 
lemma filterlim_sup:


737 
"filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)"


738 
unfolding filterlim_def filtermap_sup by auto


739 


740 
lemma eventually_sequentially_Suc: "eventually (\<lambda>i. P (Suc i)) sequentially \<longleftrightarrow> eventually P sequentially"


741 
unfolding eventually_sequentially by (metis Suc_le_D Suc_le_mono le_Suc_eq)


742 


743 
lemma filterlim_sequentially_Suc:


744 
"(LIM x sequentially. f (Suc x) :> F) \<longleftrightarrow> (LIM x sequentially. f x :> F)"


745 
unfolding filterlim_iff by (subst eventually_sequentially_Suc) simp


746 


747 
lemma filterlim_Suc: "filterlim Suc sequentially sequentially"


748 
by (simp add: filterlim_iff eventually_sequentially) (metis le_Suc_eq)


749 


750 


751 
subsection {* Limits to @{const at_top} and @{const at_bot} *}


752 


753 
lemma filterlim_at_top:


754 
fixes f :: "'a \<Rightarrow> ('b::linorder)"


755 
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)"


756 
by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_elim1)


757 


758 
lemma filterlim_at_top_mono:


759 
"LIM x F. f x :> at_top \<Longrightarrow> eventually (\<lambda>x. f x \<le> (g x::'a::linorder)) F \<Longrightarrow>


760 
LIM x F. g x :> at_top"


761 
by (auto simp: filterlim_at_top elim: eventually_elim2 intro: order_trans)


762 


763 
lemma filterlim_at_top_dense:


764 
fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)"


765 
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)"


766 
by (metis eventually_elim1[of _ F] eventually_gt_at_top order_less_imp_le


767 
filterlim_at_top[of f F] filterlim_iff[of f at_top F])


768 


769 
lemma filterlim_at_top_ge:


770 
fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"


771 
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)"


772 
unfolding at_top_sub[of c] filterlim_INF by (auto simp add: filterlim_principal)


773 


774 
lemma filterlim_at_top_at_top:


775 
fixes f :: "'a::linorder \<Rightarrow> 'b::linorder"


776 
assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"


777 
assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"


778 
assumes Q: "eventually Q at_top"


779 
assumes P: "eventually P at_top"


780 
shows "filterlim f at_top at_top"


781 
proof 


782 
from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"


783 
unfolding eventually_at_top_linorder by auto


784 
show ?thesis


785 
proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)


786 
fix z assume "x \<le> z"


787 
with x have "P z" by auto


788 
have "eventually (\<lambda>x. g z \<le> x) at_top"


789 
by (rule eventually_ge_at_top)


790 
with Q show "eventually (\<lambda>x. z \<le> f x) at_top"


791 
by eventually_elim (metis mono bij `P z`)


792 
qed


793 
qed


794 


795 
lemma filterlim_at_top_gt:


796 
fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"


797 
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)"


798 
by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge)


799 


800 
lemma filterlim_at_bot:


801 
fixes f :: "'a \<Rightarrow> ('b::linorder)"


802 
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)"


803 
by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_elim1)


804 


805 
lemma filterlim_at_bot_dense:


806 
fixes f :: "'a \<Rightarrow> ('b::{dense_linorder, no_bot})"


807 
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x < Z) F)"


808 
proof (auto simp add: filterlim_at_bot[of f F])


809 
fix Z :: 'b


810 
from lt_ex [of Z] obtain Z' where 1: "Z' < Z" ..


811 
assume "\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F"


812 
hence "eventually (\<lambda>x. f x \<le> Z') F" by auto


813 
thus "eventually (\<lambda>x. f x < Z) F"


814 
apply (rule eventually_mono[rotated])


815 
using 1 by auto


816 
next


817 
fix Z :: 'b


818 
show "\<forall>Z. eventually (\<lambda>x. f x < Z) F \<Longrightarrow> eventually (\<lambda>x. f x \<le> Z) F"


819 
by (drule spec [of _ Z], erule eventually_mono[rotated], auto simp add: less_imp_le)


820 
qed


821 


822 
lemma filterlim_at_bot_le:


823 
fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"


824 
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)"


825 
unfolding filterlim_at_bot


826 
proof safe


827 
fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F"


828 
with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F"


829 
by (auto elim!: eventually_elim1)


830 
qed simp


831 


832 
lemma filterlim_at_bot_lt:


833 
fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"


834 
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)"


835 
by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans)


836 


837 


838 
subsection {* Setup @{typ "'a filter"} for lifting and transfer *}


839 


840 
context begin interpretation lifting_syntax .


841 


842 
definition rel_filter :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> 'b filter \<Rightarrow> bool"


843 
where "rel_filter R F G = ((R ===> op =) ===> op =) (Rep_filter F) (Rep_filter G)"


844 


845 
lemma rel_filter_eventually:


846 
"rel_filter R F G \<longleftrightarrow>


847 
((R ===> op =) ===> op =) (\<lambda>P. eventually P F) (\<lambda>P. eventually P G)"


848 
by(simp add: rel_filter_def eventually_def)


849 


850 
lemma filtermap_id [simp, id_simps]: "filtermap id = id"


851 
by(simp add: fun_eq_iff id_def filtermap_ident)


852 


853 
lemma filtermap_id' [simp]: "filtermap (\<lambda>x. x) = (\<lambda>F. F)"


854 
using filtermap_id unfolding id_def .


855 


856 
lemma Quotient_filter [quot_map]:


857 
assumes Q: "Quotient R Abs Rep T"


858 
shows "Quotient (rel_filter R) (filtermap Abs) (filtermap Rep) (rel_filter T)"


859 
unfolding Quotient_alt_def


860 
proof(intro conjI strip)


861 
from Q have *: "\<And>x y. T x y \<Longrightarrow> Abs x = y"


862 
unfolding Quotient_alt_def by blast


863 


864 
fix F G


865 
assume "rel_filter T F G"


866 
thus "filtermap Abs F = G" unfolding filter_eq_iff


867 
by(auto simp add: eventually_filtermap rel_filter_eventually * rel_funI del: iffI elim!: rel_funD)


868 
next


869 
from Q have *: "\<And>x. T (Rep x) x" unfolding Quotient_alt_def by blast


870 


871 
fix F


872 
show "rel_filter T (filtermap Rep F) F"


873 
by(auto elim: rel_funD intro: * intro!: ext arg_cong[where f="\<lambda>P. eventually P F"] rel_funI


874 
del: iffI simp add: eventually_filtermap rel_filter_eventually)


875 
qed(auto simp add: map_fun_def o_def eventually_filtermap filter_eq_iff fun_eq_iff rel_filter_eventually


876 
fun_quotient[OF fun_quotient[OF Q identity_quotient] identity_quotient, unfolded Quotient_alt_def])


877 


878 
lemma eventually_parametric [transfer_rule]:


879 
"((A ===> op =) ===> rel_filter A ===> op =) eventually eventually"


880 
by(simp add: rel_fun_def rel_filter_eventually)


881 

60038

882 
lemma frequently_parametric [transfer_rule]:


883 
"((A ===> op =) ===> rel_filter A ===> op =) frequently frequently"


884 
unfolding frequently_def[abs_def] by transfer_prover


885 

60036

886 
lemma rel_filter_eq [relator_eq]: "rel_filter op = = op ="


887 
by(auto simp add: rel_filter_eventually rel_fun_eq fun_eq_iff filter_eq_iff)


888 


889 
lemma rel_filter_mono [relator_mono]:


890 
"A \<le> B \<Longrightarrow> rel_filter A \<le> rel_filter B"


891 
unfolding rel_filter_eventually[abs_def]


892 
by(rule le_funI)+(intro fun_mono fun_mono[THEN le_funD, THEN le_funD] order.refl)


893 


894 
lemma rel_filter_conversep [simp]: "rel_filter A\<inverse>\<inverse> = (rel_filter A)\<inverse>\<inverse>"


895 
by(auto simp add: rel_filter_eventually fun_eq_iff rel_fun_def)


896 


897 
lemma is_filter_parametric_aux:


898 
assumes "is_filter F"


899 
assumes [transfer_rule]: "bi_total A" "bi_unique A"


900 
and [transfer_rule]: "((A ===> op =) ===> op =) F G"


901 
shows "is_filter G"


902 
proof 


903 
interpret is_filter F by fact


904 
show ?thesis


905 
proof


906 
have "F (\<lambda>_. True) = G (\<lambda>x. True)" by transfer_prover


907 
thus "G (\<lambda>x. True)" by(simp add: True)


908 
next


909 
fix P' Q'


910 
assume "G P'" "G Q'"


911 
moreover


912 
from bi_total_fun[OF `bi_unique A` bi_total_eq, unfolded bi_total_def]


913 
obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast


914 
have "F P = G P'" "F Q = G Q'" by transfer_prover+


915 
ultimately have "F (\<lambda>x. P x \<and> Q x)" by(simp add: conj)


916 
moreover have "F (\<lambda>x. P x \<and> Q x) = G (\<lambda>x. P' x \<and> Q' x)" by transfer_prover


917 
ultimately show "G (\<lambda>x. P' x \<and> Q' x)" by simp


918 
next


919 
fix P' Q'


920 
assume "\<forall>x. P' x \<longrightarrow> Q' x" "G P'"


921 
moreover


922 
from bi_total_fun[OF `bi_unique A` bi_total_eq, unfolded bi_total_def]


923 
obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast


924 
have "F P = G P'" by transfer_prover


925 
moreover have "(\<forall>x. P x \<longrightarrow> Q x) \<longleftrightarrow> (\<forall>x. P' x \<longrightarrow> Q' x)" by transfer_prover


926 
ultimately have "F Q" by(simp add: mono)


927 
moreover have "F Q = G Q'" by transfer_prover


928 
ultimately show "G Q'" by simp


929 
qed


930 
qed


931 


932 
lemma is_filter_parametric [transfer_rule]:


933 
"\<lbrakk> bi_total A; bi_unique A \<rbrakk>


934 
\<Longrightarrow> (((A ===> op =) ===> op =) ===> op =) is_filter is_filter"


935 
apply(rule rel_funI)


936 
apply(rule iffI)


937 
apply(erule (3) is_filter_parametric_aux)


938 
apply(erule is_filter_parametric_aux[where A="conversep A"])


939 
apply(auto simp add: rel_fun_def)


940 
done


941 


942 
lemma left_total_rel_filter [transfer_rule]:


943 
assumes [transfer_rule]: "bi_total A" "bi_unique A"


944 
shows "left_total (rel_filter A)"


945 
proof(rule left_totalI)


946 
fix F :: "'a filter"


947 
from bi_total_fun[OF bi_unique_fun[OF `bi_total A` bi_unique_eq] bi_total_eq]


948 
obtain G where [transfer_rule]: "((A ===> op =) ===> op =) (\<lambda>P. eventually P F) G"


949 
unfolding bi_total_def by blast


950 
moreover have "is_filter (\<lambda>P. eventually P F) \<longleftrightarrow> is_filter G" by transfer_prover


951 
hence "is_filter G" by(simp add: eventually_def is_filter_Rep_filter)


952 
ultimately have "rel_filter A F (Abs_filter G)"


953 
by(simp add: rel_filter_eventually eventually_Abs_filter)


954 
thus "\<exists>G. rel_filter A F G" ..


955 
qed


956 


957 
lemma right_total_rel_filter [transfer_rule]:


958 
"\<lbrakk> bi_total A; bi_unique A \<rbrakk> \<Longrightarrow> right_total (rel_filter A)"


959 
using left_total_rel_filter[of "A\<inverse>\<inverse>"] by simp


960 


961 
lemma bi_total_rel_filter [transfer_rule]:


962 
assumes "bi_total A" "bi_unique A"


963 
shows "bi_total (rel_filter A)"


964 
unfolding bi_total_alt_def using assms


965 
by(simp add: left_total_rel_filter right_total_rel_filter)


966 


967 
lemma left_unique_rel_filter [transfer_rule]:


968 
assumes "left_unique A"


969 
shows "left_unique (rel_filter A)"


970 
proof(rule left_uniqueI)


971 
fix F F' G


972 
assume [transfer_rule]: "rel_filter A F G" "rel_filter A F' G"


973 
show "F = F'"


974 
unfolding filter_eq_iff


975 
proof


976 
fix P :: "'a \<Rightarrow> bool"


977 
obtain P' where [transfer_rule]: "(A ===> op =) P P'"


978 
using left_total_fun[OF assms left_total_eq] unfolding left_total_def by blast


979 
have "eventually P F = eventually P' G"


980 
and "eventually P F' = eventually P' G" by transfer_prover+


981 
thus "eventually P F = eventually P F'" by simp


982 
qed


983 
qed


984 


985 
lemma right_unique_rel_filter [transfer_rule]:


986 
"right_unique A \<Longrightarrow> right_unique (rel_filter A)"


987 
using left_unique_rel_filter[of "A\<inverse>\<inverse>"] by simp


988 


989 
lemma bi_unique_rel_filter [transfer_rule]:


990 
"bi_unique A \<Longrightarrow> bi_unique (rel_filter A)"


991 
by(simp add: bi_unique_alt_def left_unique_rel_filter right_unique_rel_filter)


992 


993 
lemma top_filter_parametric [transfer_rule]:


994 
"bi_total A \<Longrightarrow> (rel_filter A) top top"


995 
by(simp add: rel_filter_eventually All_transfer)


996 


997 
lemma bot_filter_parametric [transfer_rule]: "(rel_filter A) bot bot"


998 
by(simp add: rel_filter_eventually rel_fun_def)


999 


1000 
lemma sup_filter_parametric [transfer_rule]:


1001 
"(rel_filter A ===> rel_filter A ===> rel_filter A) sup sup"


1002 
by(fastforce simp add: rel_filter_eventually[abs_def] eventually_sup dest: rel_funD)


1003 


1004 
lemma Sup_filter_parametric [transfer_rule]:


1005 
"(rel_set (rel_filter A) ===> rel_filter A) Sup Sup"


1006 
proof(rule rel_funI)


1007 
fix S T


1008 
assume [transfer_rule]: "rel_set (rel_filter A) S T"


1009 
show "rel_filter A (Sup S) (Sup T)"


1010 
by(simp add: rel_filter_eventually eventually_Sup) transfer_prover


1011 
qed


1012 


1013 
lemma principal_parametric [transfer_rule]:


1014 
"(rel_set A ===> rel_filter A) principal principal"


1015 
proof(rule rel_funI)


1016 
fix S S'


1017 
assume [transfer_rule]: "rel_set A S S'"


1018 
show "rel_filter A (principal S) (principal S')"


1019 
by(simp add: rel_filter_eventually eventually_principal) transfer_prover


1020 
qed


1021 


1022 
context


1023 
fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"


1024 
assumes [transfer_rule]: "bi_unique A"


1025 
begin


1026 


1027 
lemma le_filter_parametric [transfer_rule]:


1028 
"(rel_filter A ===> rel_filter A ===> op =) op \<le> op \<le>"


1029 
unfolding le_filter_def[abs_def] by transfer_prover


1030 


1031 
lemma less_filter_parametric [transfer_rule]:


1032 
"(rel_filter A ===> rel_filter A ===> op =) op < op <"


1033 
unfolding less_filter_def[abs_def] by transfer_prover


1034 


1035 
context


1036 
assumes [transfer_rule]: "bi_total A"


1037 
begin


1038 


1039 
lemma Inf_filter_parametric [transfer_rule]:


1040 
"(rel_set (rel_filter A) ===> rel_filter A) Inf Inf"


1041 
unfolding Inf_filter_def[abs_def] by transfer_prover


1042 


1043 
lemma inf_filter_parametric [transfer_rule]:


1044 
"(rel_filter A ===> rel_filter A ===> rel_filter A) inf inf"


1045 
proof(intro rel_funI)+


1046 
fix F F' G G'


1047 
assume [transfer_rule]: "rel_filter A F F'" "rel_filter A G G'"


1048 
have "rel_filter A (Inf {F, G}) (Inf {F', G'})" by transfer_prover


1049 
thus "rel_filter A (inf F G) (inf F' G')" by simp


1050 
qed


1051 


1052 
end


1053 


1054 
end


1055 


1056 
end


1057 


1058 
end 