author | wenzelm |
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parent 71743 | 0239bee6bffd |
child 74325 | 8d0c2d74ad63 |
permissions | -rw-r--r-- |
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(* Title: HOL/Filter.thy |
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Author: Brian Huffman |
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Author: Johannes Hölzl |
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*) |
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section \<open>Filters on predicates\<close> |
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theory Filter |
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imports Set_Interval Lifting_Set |
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begin |
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subsection \<open>Filters\<close> |
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text \<open> |
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This definition also allows non-proper filters. |
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\<close> |
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locale is_filter = |
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fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool" |
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assumes True: "F (\<lambda>x. True)" |
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assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)" |
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assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)" |
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typedef 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}" |
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proof |
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show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro) |
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qed |
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lemma is_filter_Rep_filter: "is_filter (Rep_filter F)" |
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using Rep_filter [of F] by simp |
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lemma Abs_filter_inverse': |
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assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F" |
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using assms by (simp add: Abs_filter_inverse) |
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subsubsection \<open>Eventually\<close> |
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definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool" |
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where "eventually P F \<longleftrightarrow> Rep_filter F P" |
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syntax |
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"_eventually" :: "pttrn => 'a filter => bool => bool" ("(3\<forall>\<^sub>F _ in _./ _)" [0, 0, 10] 10) |
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translations |
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"\<forall>\<^sub>Fx in F. P" == "CONST eventually (\<lambda>x. P) F" |
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lemma eventually_Abs_filter: |
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assumes "is_filter F" shows "eventually P (Abs_filter F) = F P" |
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unfolding eventually_def using assms by (simp add: Abs_filter_inverse) |
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lemma filter_eq_iff: |
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shows "F = F' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')" |
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unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def .. |
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lemma eventually_True [simp]: "eventually (\<lambda>x. True) F" |
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unfolding eventually_def |
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by (rule is_filter.True [OF is_filter_Rep_filter]) |
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lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P F" |
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proof - |
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assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext) |
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thus "eventually P F" by simp |
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qed |
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lemma eventuallyI: "(\<And>x. P x) \<Longrightarrow> eventually P F" |
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by (auto intro: always_eventually) |
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lemma eventually_mono: |
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"\<lbrakk>eventually P F; \<And>x. P x \<Longrightarrow> Q x\<rbrakk> \<Longrightarrow> eventually Q F" |
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unfolding eventually_def |
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by (blast intro: is_filter.mono [OF is_filter_Rep_filter]) |
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lemma eventually_conj: |
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assumes P: "eventually (\<lambda>x. P x) F" |
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assumes Q: "eventually (\<lambda>x. Q x) F" |
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shows "eventually (\<lambda>x. P x \<and> Q x) F" |
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using assms unfolding eventually_def |
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by (rule is_filter.conj [OF is_filter_Rep_filter]) |
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lemma eventually_mp: |
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assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F" |
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assumes "eventually (\<lambda>x. P x) F" |
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shows "eventually (\<lambda>x. Q x) F" |
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proof - |
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have "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F" |
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using assms by (rule eventually_conj) |
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then show ?thesis |
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by (blast intro: eventually_mono) |
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qed |
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lemma eventually_rev_mp: |
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assumes "eventually (\<lambda>x. P x) F" |
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assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F" |
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shows "eventually (\<lambda>x. Q x) F" |
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using assms(2) assms(1) by (rule eventually_mp) |
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lemma eventually_conj_iff: |
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"eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F" |
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by (auto intro: eventually_conj elim: eventually_rev_mp) |
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lemma eventually_elim2: |
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assumes "eventually (\<lambda>i. P i) F" |
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assumes "eventually (\<lambda>i. Q i) F" |
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assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i" |
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shows "eventually (\<lambda>i. R i) F" |
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using assms by (auto elim!: eventually_rev_mp) |
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lemma eventually_ball_finite_distrib: |
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"finite A \<Longrightarrow> (eventually (\<lambda>x. \<forall>y\<in>A. P x y) net) \<longleftrightarrow> (\<forall>y\<in>A. eventually (\<lambda>x. P x y) net)" |
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by (induction A rule: finite_induct) (auto simp: eventually_conj_iff) |
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lemma eventually_ball_finite: |
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"finite A \<Longrightarrow> \<forall>y\<in>A. eventually (\<lambda>x. P x y) net \<Longrightarrow> eventually (\<lambda>x. \<forall>y\<in>A. P x y) net" |
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by (auto simp: eventually_ball_finite_distrib) |
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lemma eventually_all_finite: |
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fixes P :: "'a \<Rightarrow> 'b::finite \<Rightarrow> bool" |
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assumes "\<And>y. eventually (\<lambda>x. P x y) net" |
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shows "eventually (\<lambda>x. \<forall>y. P x y) net" |
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using eventually_ball_finite [of UNIV P] assms by simp |
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lemma eventually_ex: "(\<forall>\<^sub>Fx in F. \<exists>y. P x y) \<longleftrightarrow> (\<exists>Y. \<forall>\<^sub>Fx in F. P x (Y x))" |
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proof |
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assume "\<forall>\<^sub>Fx in F. \<exists>y. P x y" |
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then have "\<forall>\<^sub>Fx in F. P x (SOME y. P x y)" |
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by (auto intro: someI_ex eventually_mono) |
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then show "\<exists>Y. \<forall>\<^sub>Fx in F. P x (Y x)" |
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by auto |
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qed (auto intro: eventually_mono) |
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lemma not_eventually_impI: "eventually P F \<Longrightarrow> \<not> eventually Q F \<Longrightarrow> \<not> eventually (\<lambda>x. P x \<longrightarrow> Q x) F" |
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by (auto intro: eventually_mp) |
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lemma not_eventuallyD: "\<not> eventually P F \<Longrightarrow> \<exists>x. \<not> P x" |
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by (metis always_eventually) |
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lemma eventually_subst: |
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assumes "eventually (\<lambda>n. P n = Q n) F" |
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shows "eventually P F = eventually Q F" (is "?L = ?R") |
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proof - |
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from assms have "eventually (\<lambda>x. P x \<longrightarrow> Q x) F" |
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and "eventually (\<lambda>x. Q x \<longrightarrow> P x) F" |
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by (auto elim: eventually_mono) |
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then show ?thesis by (auto elim: eventually_elim2) |
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qed |
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subsection \<open> Frequently as dual to eventually \<close> |
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definition frequently :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool" |
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where "frequently P F \<longleftrightarrow> \<not> eventually (\<lambda>x. \<not> P x) F" |
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syntax |
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"_frequently" :: "pttrn \<Rightarrow> 'a filter \<Rightarrow> bool \<Rightarrow> bool" ("(3\<exists>\<^sub>F _ in _./ _)" [0, 0, 10] 10) |
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translations |
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"\<exists>\<^sub>Fx in F. P" == "CONST frequently (\<lambda>x. P) F" |
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lemma not_frequently_False [simp]: "\<not> (\<exists>\<^sub>Fx in F. False)" |
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by (simp add: frequently_def) |
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lemma frequently_ex: "\<exists>\<^sub>Fx in F. P x \<Longrightarrow> \<exists>x. P x" |
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by (auto simp: frequently_def dest: not_eventuallyD) |
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lemma frequentlyE: assumes "frequently P F" obtains x where "P x" |
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using frequently_ex[OF assms] by auto |
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lemma frequently_mp: |
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assumes ev: "\<forall>\<^sub>Fx in F. P x \<longrightarrow> Q x" and P: "\<exists>\<^sub>Fx in F. P x" shows "\<exists>\<^sub>Fx in F. Q x" |
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proof - |
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from ev have "eventually (\<lambda>x. \<not> Q x \<longrightarrow> \<not> P x) F" |
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by (rule eventually_rev_mp) (auto intro!: always_eventually) |
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from eventually_mp[OF this] P show ?thesis |
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by (auto simp: frequently_def) |
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qed |
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lemma frequently_rev_mp: |
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assumes "\<exists>\<^sub>Fx in F. P x" |
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assumes "\<forall>\<^sub>Fx in F. P x \<longrightarrow> Q x" |
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shows "\<exists>\<^sub>Fx in F. Q x" |
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using assms(2) assms(1) by (rule frequently_mp) |
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lemma frequently_mono: "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> frequently P F \<Longrightarrow> frequently Q F" |
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using frequently_mp[of P Q] by (simp add: always_eventually) |
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lemma frequently_elim1: "\<exists>\<^sub>Fx in F. P x \<Longrightarrow> (\<And>i. P i \<Longrightarrow> Q i) \<Longrightarrow> \<exists>\<^sub>Fx in F. Q x" |
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by (metis frequently_mono) |
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186 |
|
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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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189 |
|
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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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192 |
|
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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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196 |
|
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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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199 |
|
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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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202 |
|
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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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207 |
|
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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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211 |
|
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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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220 |
|
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Cauchy's integral formula for circles. Starting to fix eventually_mono.
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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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232 |
|
60758 | 233 |
ML \<open> |
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more general types Proof.method / context_tactic;
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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 |
67855 | 237 |
val mp_facts = facts RL @{thms eventually_rev_mp} |
238 |
val rule = |
|
239 |
@{thm eventuallyI} |
|
240 |
|> fold (fn mp_fact => fn th => th RS mp_fact) mp_facts |
|
241 |
|> funpow (length facts) (fn th => @{thm impI} RS th) |
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val cases_prop = |
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Thm.prop_of (Rule_Cases.internalize_params (rule RS Goal.init (Thm.cterm_of ctxt goal))) |
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val cases = Rule_Cases.make_common ctxt cases_prop [(("elim", []), [])] |
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in CONTEXT_CASES cases (resolve_tac ctxt [rule] i) (ctxt, st) end) |
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\<close> |
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|
60758 | 248 |
method_setup eventually_elim = \<open> |
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Scan.succeed (fn _ => CONTEXT_METHOD (fn facts => eventually_elim_tac facts 1)) |
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\<close> "elimination of eventually quantifiers" |
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|
60758 | 252 |
subsubsection \<open>Finer-than relation\<close> |
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|
69593 | 254 |
text \<open>\<^term>\<open>F \<le> F'\<close> means that filter \<^term>\<open>F\<close> is finer than |
255 |
filter \<^term>\<open>F'\<close>.\<close> |
|
60036 | 256 |
|
257 |
instantiation filter :: (type) complete_lattice |
|
258 |
begin |
|
259 |
||
260 |
definition le_filter_def: |
|
261 |
"F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)" |
|
262 |
||
263 |
definition |
|
264 |
"(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F" |
|
265 |
||
266 |
definition |
|
267 |
"top = Abs_filter (\<lambda>P. \<forall>x. P x)" |
|
268 |
||
269 |
definition |
|
270 |
"bot = Abs_filter (\<lambda>P. True)" |
|
271 |
||
272 |
definition |
|
273 |
"sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')" |
|
274 |
||
275 |
definition |
|
276 |
"inf F F' = Abs_filter |
|
277 |
(\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))" |
|
278 |
||
279 |
definition |
|
280 |
"Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)" |
|
281 |
||
282 |
definition |
|
283 |
"Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}" |
|
284 |
||
285 |
lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)" |
|
286 |
unfolding top_filter_def |
|
287 |
by (rule eventually_Abs_filter, rule is_filter.intro, auto) |
|
288 |
||
289 |
lemma eventually_bot [simp]: "eventually P bot" |
|
290 |
unfolding bot_filter_def |
|
291 |
by (subst eventually_Abs_filter, rule is_filter.intro, auto) |
|
292 |
||
293 |
lemma eventually_sup: |
|
294 |
"eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'" |
|
295 |
unfolding sup_filter_def |
|
296 |
by (rule eventually_Abs_filter, rule is_filter.intro) |
|
297 |
(auto elim!: eventually_rev_mp) |
|
298 |
||
299 |
lemma eventually_inf: |
|
300 |
"eventually P (inf F F') \<longleftrightarrow> |
|
301 |
(\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))" |
|
302 |
unfolding inf_filter_def |
|
71743 | 303 |
apply (rule eventually_Abs_filter [OF is_filter.intro]) |
304 |
apply (blast intro: eventually_True) |
|
305 |
apply (force elim!: eventually_conj)+ |
|
60036 | 306 |
done |
307 |
||
308 |
lemma eventually_Sup: |
|
309 |
"eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)" |
|
310 |
unfolding Sup_filter_def |
|
71743 | 311 |
apply (rule eventually_Abs_filter [OF is_filter.intro]) |
60036 | 312 |
apply (auto intro: eventually_conj elim!: eventually_rev_mp) |
313 |
done |
|
314 |
||
315 |
instance proof |
|
316 |
fix F F' F'' :: "'a filter" and S :: "'a filter set" |
|
317 |
{ show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F" |
|
318 |
by (rule less_filter_def) } |
|
319 |
{ show "F \<le> F" |
|
320 |
unfolding le_filter_def by simp } |
|
321 |
{ assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''" |
|
322 |
unfolding le_filter_def by simp } |
|
323 |
{ assume "F \<le> F'" and "F' \<le> F" thus "F = F'" |
|
324 |
unfolding le_filter_def filter_eq_iff by fast } |
|
325 |
{ show "inf F F' \<le> F" and "inf F F' \<le> F'" |
|
326 |
unfolding le_filter_def eventually_inf by (auto intro: eventually_True) } |
|
327 |
{ assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''" |
|
328 |
unfolding le_filter_def eventually_inf |
|
61810 | 329 |
by (auto intro: eventually_mono [OF eventually_conj]) } |
60036 | 330 |
{ show "F \<le> sup F F'" and "F' \<le> sup F F'" |
331 |
unfolding le_filter_def eventually_sup by simp_all } |
|
332 |
{ assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''" |
|
333 |
unfolding le_filter_def eventually_sup by simp } |
|
334 |
{ assume "F'' \<in> S" thus "Inf S \<le> F''" |
|
335 |
unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp } |
|
336 |
{ assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S" |
|
337 |
unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp } |
|
338 |
{ assume "F \<in> S" thus "F \<le> Sup S" |
|
339 |
unfolding le_filter_def eventually_Sup by simp } |
|
340 |
{ assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'" |
|
341 |
unfolding le_filter_def eventually_Sup by simp } |
|
342 |
{ show "Inf {} = (top::'a filter)" |
|
343 |
by (auto simp: top_filter_def Inf_filter_def Sup_filter_def) |
|
344 |
(metis (full_types) top_filter_def always_eventually eventually_top) } |
|
345 |
{ show "Sup {} = (bot::'a filter)" |
|
346 |
by (auto simp: bot_filter_def Sup_filter_def) } |
|
347 |
qed |
|
348 |
||
349 |
end |
|
350 |
||
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distrib_lattice instance for filters
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351 |
instance filter :: (type) distrib_lattice |
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352 |
proof |
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353 |
fix F G H :: "'a filter" |
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354 |
show "sup F (inf G H) = inf (sup F G) (sup F H)" |
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355 |
proof (rule order.antisym) |
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356 |
show "inf (sup F G) (sup F H) \<le> sup F (inf G H)" |
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357 |
unfolding le_filter_def eventually_sup |
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358 |
proof safe |
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359 |
fix P assume 1: "eventually P F" and 2: "eventually P (inf G H)" |
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360 |
from 2 obtain Q R |
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361 |
where QR: "eventually Q G" "eventually R H" "\<And>x. Q x \<Longrightarrow> R x \<Longrightarrow> P x" |
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362 |
by (auto simp: eventually_inf) |
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363 |
define Q' where "Q' = (\<lambda>x. Q x \<or> P x)" |
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364 |
define R' where "R' = (\<lambda>x. R x \<or> P x)" |
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|
365 |
from 1 have "eventually Q' F" |
454abfe923fe
distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents:
66162
diff
changeset
|
366 |
by (elim eventually_mono) (auto simp: Q'_def) |
454abfe923fe
distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents:
66162
diff
changeset
|
367 |
moreover from 1 have "eventually R' F" |
454abfe923fe
distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents:
66162
diff
changeset
|
368 |
by (elim eventually_mono) (auto simp: R'_def) |
454abfe923fe
distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents:
66162
diff
changeset
|
369 |
moreover from QR(1) have "eventually Q' G" |
454abfe923fe
distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents:
66162
diff
changeset
|
370 |
by (elim eventually_mono) (auto simp: Q'_def) |
454abfe923fe
distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents:
66162
diff
changeset
|
371 |
moreover from QR(2) have "eventually R' H" |
454abfe923fe
distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents:
66162
diff
changeset
|
372 |
by (elim eventually_mono)(auto simp: R'_def) |
454abfe923fe
distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents:
66162
diff
changeset
|
373 |
moreover from QR have "P x" if "Q' x" "R' x" for x |
454abfe923fe
distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents:
66162
diff
changeset
|
374 |
using that by (auto simp: Q'_def R'_def) |
454abfe923fe
distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents:
66162
diff
changeset
|
375 |
ultimately show "eventually P (inf (sup F G) (sup F H))" |
454abfe923fe
distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents:
66162
diff
changeset
|
376 |
by (auto simp: eventually_inf eventually_sup) |
454abfe923fe
distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents:
66162
diff
changeset
|
377 |
qed |
454abfe923fe
distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents:
66162
diff
changeset
|
378 |
qed (auto intro: inf.coboundedI1 inf.coboundedI2) |
454abfe923fe
distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents:
66162
diff
changeset
|
379 |
qed |
454abfe923fe
distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents:
66162
diff
changeset
|
380 |
|
454abfe923fe
distrib_lattice instance for filters
eberlm <eberlm@in.tum.de>
parents:
66162
diff
changeset
|
381 |
|
60036 | 382 |
lemma filter_leD: |
383 |
"F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F" |
|
384 |
unfolding le_filter_def by simp |
|
385 |
||
386 |
lemma filter_leI: |
|
387 |
"(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'" |
|
388 |
unfolding le_filter_def by simp |
|
389 |
||
390 |
lemma eventually_False: |
|
391 |
"eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot" |
|
392 |
unfolding filter_eq_iff by (auto elim: eventually_rev_mp) |
|
393 |
||
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
394 |
lemma eventually_frequently: "F \<noteq> bot \<Longrightarrow> eventually P F \<Longrightarrow> frequently P F" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
395 |
using eventually_conj[of P F "\<lambda>x. \<not> P x"] |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
396 |
by (auto simp add: frequently_def eventually_False) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
397 |
|
67706
4ddc49205f5d
Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents:
67616
diff
changeset
|
398 |
lemma eventually_frequentlyE: |
4ddc49205f5d
Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents:
67616
diff
changeset
|
399 |
assumes "eventually P F" |
4ddc49205f5d
Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents:
67616
diff
changeset
|
400 |
assumes "eventually (\<lambda>x. \<not> P x \<or> Q x) F" "F\<noteq>bot" |
4ddc49205f5d
Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents:
67616
diff
changeset
|
401 |
shows "frequently Q F" |
4ddc49205f5d
Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents:
67616
diff
changeset
|
402 |
proof - |
4ddc49205f5d
Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents:
67616
diff
changeset
|
403 |
have "eventually Q F" |
4ddc49205f5d
Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents:
67616
diff
changeset
|
404 |
using eventually_conj[OF assms(1,2),simplified] by (auto elim:eventually_mono) |
4ddc49205f5d
Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents:
67616
diff
changeset
|
405 |
then show ?thesis using eventually_frequently[OF \<open>F\<noteq>bot\<close>] by auto |
4ddc49205f5d
Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents:
67616
diff
changeset
|
406 |
qed |
4ddc49205f5d
Unified the order of zeros and poles; improved reasoning around non-essential singularites
Wenda Li <wl302@cam.ac.uk>
parents:
67616
diff
changeset
|
407 |
|
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
408 |
lemma eventually_const_iff: "eventually (\<lambda>x. P) F \<longleftrightarrow> P \<or> F = bot" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
409 |
by (cases P) (auto simp: eventually_False) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
410 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
411 |
lemma eventually_const[simp]: "F \<noteq> bot \<Longrightarrow> eventually (\<lambda>x. P) F \<longleftrightarrow> P" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
412 |
by (simp add: eventually_const_iff) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
413 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
414 |
lemma frequently_const_iff: "frequently (\<lambda>x. P) F \<longleftrightarrow> P \<and> F \<noteq> bot" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
415 |
by (simp add: frequently_def eventually_const_iff) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
416 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
417 |
lemma frequently_const[simp]: "F \<noteq> bot \<Longrightarrow> frequently (\<lambda>x. P) F \<longleftrightarrow> P" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
418 |
by (simp add: frequently_const_iff) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
419 |
|
61245 | 420 |
lemma eventually_happens: "eventually P net \<Longrightarrow> net = bot \<or> (\<exists>x. P x)" |
421 |
by (metis frequentlyE eventually_frequently) |
|
422 |
||
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
423 |
lemma eventually_happens': |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
424 |
assumes "F \<noteq> bot" "eventually P F" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
425 |
shows "\<exists>x. P x" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
426 |
using assms eventually_frequently frequentlyE by blast |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
427 |
|
60036 | 428 |
abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool" |
429 |
where "trivial_limit F \<equiv> F = bot" |
|
430 |
||
431 |
lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F" |
|
432 |
by (rule eventually_False [symmetric]) |
|
433 |
||
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
434 |
lemma False_imp_not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> \<not> trivial_limit net \<Longrightarrow> \<not> eventually (\<lambda>x. P x) net" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
435 |
by (simp add: eventually_False) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
436 |
|
60036 | 437 |
lemma eventually_Inf: "eventually P (Inf B) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X))" |
438 |
proof - |
|
439 |
let ?F = "\<lambda>P. \<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X)" |
|
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
440 |
|
60036 | 441 |
{ fix P have "eventually P (Abs_filter ?F) \<longleftrightarrow> ?F P" |
442 |
proof (rule eventually_Abs_filter is_filter.intro)+ |
|
443 |
show "?F (\<lambda>x. True)" |
|
444 |
by (rule exI[of _ "{}"]) (simp add: le_fun_def) |
|
445 |
next |
|
446 |
fix P Q |
|
447 |
assume "?F P" then guess X .. |
|
448 |
moreover |
|
449 |
assume "?F Q" then guess Y .. |
|
450 |
ultimately show "?F (\<lambda>x. P x \<and> Q x)" |
|
451 |
by (intro exI[of _ "X \<union> Y"]) |
|
452 |
(auto simp: Inf_union_distrib eventually_inf) |
|
453 |
next |
|
454 |
fix P Q |
|
455 |
assume "?F P" then guess X .. |
|
456 |
moreover assume "\<forall>x. P x \<longrightarrow> Q x" |
|
457 |
ultimately show "?F Q" |
|
61810 | 458 |
by (intro exI[of _ X]) (auto elim: eventually_mono) |
60036 | 459 |
qed } |
460 |
note eventually_F = this |
|
461 |
||
462 |
have "Inf B = Abs_filter ?F" |
|
463 |
proof (intro antisym Inf_greatest) |
|
464 |
show "Inf B \<le> Abs_filter ?F" |
|
465 |
by (auto simp: le_filter_def eventually_F dest: Inf_superset_mono) |
|
466 |
next |
|
467 |
fix F assume "F \<in> B" then show "Abs_filter ?F \<le> F" |
|
468 |
by (auto simp add: le_filter_def eventually_F intro!: exI[of _ "{F}"]) |
|
469 |
qed |
|
470 |
then show ?thesis |
|
471 |
by (simp add: eventually_F) |
|
472 |
qed |
|
473 |
||
67613 | 474 |
lemma eventually_INF: "eventually P (\<Sqinter>b\<in>B. F b) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (\<Sqinter>b\<in>X. F b))" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62123
diff
changeset
|
475 |
unfolding eventually_Inf [of P "F`B"] |
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62123
diff
changeset
|
476 |
by (metis finite_imageI image_mono finite_subset_image) |
60036 | 477 |
|
478 |
lemma Inf_filter_not_bot: |
|
479 |
fixes B :: "'a filter set" |
|
480 |
shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> Inf X \<noteq> bot) \<Longrightarrow> Inf B \<noteq> bot" |
|
481 |
unfolding trivial_limit_def eventually_Inf[of _ B] |
|
482 |
bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp |
|
483 |
||
484 |
lemma INF_filter_not_bot: |
|
485 |
fixes F :: "'i \<Rightarrow> 'a filter" |
|
67613 | 486 |
shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> (\<Sqinter>b\<in>X. F b) \<noteq> bot) \<Longrightarrow> (\<Sqinter>b\<in>B. F b) \<noteq> bot" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62123
diff
changeset
|
487 |
unfolding trivial_limit_def eventually_INF [of _ _ B] |
60036 | 488 |
bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp |
489 |
||
490 |
lemma eventually_Inf_base: |
|
491 |
assumes "B \<noteq> {}" and base: "\<And>F G. F \<in> B \<Longrightarrow> G \<in> B \<Longrightarrow> \<exists>x\<in>B. x \<le> inf F G" |
|
492 |
shows "eventually P (Inf B) \<longleftrightarrow> (\<exists>b\<in>B. eventually P b)" |
|
493 |
proof (subst eventually_Inf, safe) |
|
494 |
fix X assume "finite X" "X \<subseteq> B" |
|
495 |
then have "\<exists>b\<in>B. \<forall>x\<in>X. b \<le> x" |
|
496 |
proof induct |
|
497 |
case empty then show ?case |
|
60758 | 498 |
using \<open>B \<noteq> {}\<close> by auto |
60036 | 499 |
next |
500 |
case (insert x X) |
|
501 |
then obtain b where "b \<in> B" "\<And>x. x \<in> X \<Longrightarrow> b \<le> x" |
|
502 |
by auto |
|
60758 | 503 |
with \<open>insert x X \<subseteq> B\<close> base[of b x] show ?case |
60036 | 504 |
by (auto intro: order_trans) |
505 |
qed |
|
506 |
then obtain b where "b \<in> B" "b \<le> Inf X" |
|
507 |
by (auto simp: le_Inf_iff) |
|
508 |
then show "eventually P (Inf X) \<Longrightarrow> Bex B (eventually P)" |
|
509 |
by (intro bexI[of _ b]) (auto simp: le_filter_def) |
|
510 |
qed (auto intro!: exI[of _ "{x}" for x]) |
|
511 |
||
512 |
lemma eventually_INF_base: |
|
513 |
"B \<noteq> {} \<Longrightarrow> (\<And>a b. a \<in> B \<Longrightarrow> b \<in> B \<Longrightarrow> \<exists>x\<in>B. F x \<le> inf (F a) (F b)) \<Longrightarrow> |
|
67613 | 514 |
eventually P (\<Sqinter>b\<in>B. F b) \<longleftrightarrow> (\<exists>b\<in>B. eventually P (F b))" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62123
diff
changeset
|
515 |
by (subst eventually_Inf_base) auto |
60036 | 516 |
|
67613 | 517 |
lemma eventually_INF1: "i \<in> I \<Longrightarrow> eventually P (F i) \<Longrightarrow> eventually P (\<Sqinter>i\<in>I. F i)" |
62369 | 518 |
using filter_leD[OF INF_lower] . |
519 |
||
68860
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
520 |
lemma eventually_INF_finite: |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
521 |
assumes "finite A" |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
68860
diff
changeset
|
522 |
shows "eventually P (\<Sqinter> x\<in>A. F x) \<longleftrightarrow> |
68860
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
523 |
(\<exists>Q. (\<forall>x\<in>A. eventually (Q x) (F x)) \<and> (\<forall>y. (\<forall>x\<in>A. Q x y) \<longrightarrow> P y))" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
524 |
using assms |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
525 |
proof (induction arbitrary: P rule: finite_induct) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
526 |
case (insert a A P) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
527 |
from insert.hyps have [simp]: "x \<noteq> a" if "x \<in> A" for x |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
528 |
using that by auto |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
68860
diff
changeset
|
529 |
have "eventually P (\<Sqinter> x\<in>insert a A. F x) \<longleftrightarrow> |
68860
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
530 |
(\<exists>Q R S. eventually Q (F a) \<and> (( (\<forall>x\<in>A. eventually (S x) (F x)) \<and> |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
531 |
(\<forall>y. (\<forall>x\<in>A. S x y) \<longrightarrow> R y)) \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x)))" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
532 |
unfolding ex_simps by (simp add: eventually_inf insert.IH) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
533 |
also have "\<dots> \<longleftrightarrow> (\<exists>Q. (\<forall>x\<in>insert a A. eventually (Q x) (F x)) \<and> |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
534 |
(\<forall>y. (\<forall>x\<in>insert a A. Q x y) \<longrightarrow> P y))" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
535 |
proof (safe, goal_cases) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
536 |
case (1 Q R S) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
537 |
thus ?case using 1 by (intro exI[of _ "S(a := Q)"]) auto |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
538 |
next |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
539 |
case (2 Q) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
540 |
show ?case |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
541 |
by (rule exI[of _ "Q a"], rule exI[of _ "\<lambda>y. \<forall>x\<in>A. Q x y"], |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
542 |
rule exI[of _ "Q(a := (\<lambda>_. True))"]) (use 2 in auto) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
543 |
qed |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
544 |
finally show ?case . |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
545 |
qed auto |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
546 |
|
60758 | 547 |
subsubsection \<open>Map function for filters\<close> |
60036 | 548 |
|
549 |
definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter" |
|
550 |
where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)" |
|
551 |
||
552 |
lemma eventually_filtermap: |
|
553 |
"eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F" |
|
554 |
unfolding filtermap_def |
|
71743 | 555 |
apply (rule eventually_Abs_filter [OF is_filter.intro]) |
60036 | 556 |
apply (auto elim!: eventually_rev_mp) |
557 |
done |
|
558 |
||
559 |
lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F" |
|
560 |
by (simp add: filter_eq_iff eventually_filtermap) |
|
561 |
||
562 |
lemma filtermap_filtermap: |
|
563 |
"filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F" |
|
564 |
by (simp add: filter_eq_iff eventually_filtermap) |
|
565 |
||
566 |
lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'" |
|
567 |
unfolding le_filter_def eventually_filtermap by simp |
|
568 |
||
569 |
lemma filtermap_bot [simp]: "filtermap f bot = bot" |
|
570 |
by (simp add: filter_eq_iff eventually_filtermap) |
|
571 |
||
67956 | 572 |
lemma filtermap_bot_iff: "filtermap f F = bot \<longleftrightarrow> F = bot" |
573 |
by (simp add: trivial_limit_def eventually_filtermap) |
|
574 |
||
60036 | 575 |
lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)" |
67956 | 576 |
by (simp add: filter_eq_iff eventually_filtermap eventually_sup) |
577 |
||
578 |
lemma filtermap_SUP: "filtermap f (\<Squnion>b\<in>B. F b) = (\<Squnion>b\<in>B. filtermap f (F b))" |
|
579 |
by (simp add: filter_eq_iff eventually_Sup eventually_filtermap) |
|
60036 | 580 |
|
581 |
lemma filtermap_inf: "filtermap f (inf F1 F2) \<le> inf (filtermap f F1) (filtermap f F2)" |
|
67956 | 582 |
by (intro inf_greatest filtermap_mono inf_sup_ord) |
60036 | 583 |
|
67613 | 584 |
lemma filtermap_INF: "filtermap f (\<Sqinter>b\<in>B. F b) \<le> (\<Sqinter>b\<in>B. filtermap f (F b))" |
67956 | 585 |
by (rule INF_greatest, rule filtermap_mono, erule INF_lower) |
62101 | 586 |
|
66162 | 587 |
|
588 |
subsubsection \<open>Contravariant map function for filters\<close> |
|
589 |
||
590 |
definition filtercomap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter" where |
|
591 |
"filtercomap f F = Abs_filter (\<lambda>P. \<exists>Q. eventually Q F \<and> (\<forall>x. Q (f x) \<longrightarrow> P x))" |
|
592 |
||
593 |
lemma eventually_filtercomap: |
|
594 |
"eventually P (filtercomap f F) \<longleftrightarrow> (\<exists>Q. eventually Q F \<and> (\<forall>x. Q (f x) \<longrightarrow> P x))" |
|
595 |
unfolding filtercomap_def |
|
596 |
proof (intro eventually_Abs_filter, unfold_locales, goal_cases) |
|
597 |
case 1 |
|
598 |
show ?case by (auto intro!: exI[of _ "\<lambda>_. True"]) |
|
599 |
next |
|
600 |
case (2 P Q) |
|
601 |
from 2(1) guess P' by (elim exE conjE) note P' = this |
|
602 |
from 2(2) guess Q' by (elim exE conjE) note Q' = this |
|
603 |
show ?case |
|
604 |
by (rule exI[of _ "\<lambda>x. P' x \<and> Q' x"]) |
|
605 |
(insert P' Q', auto intro!: eventually_conj) |
|
606 |
next |
|
607 |
case (3 P Q) |
|
608 |
thus ?case by blast |
|
609 |
qed |
|
610 |
||
611 |
lemma filtercomap_ident: "filtercomap (\<lambda>x. x) F = F" |
|
612 |
by (auto simp: filter_eq_iff eventually_filtercomap elim!: eventually_mono) |
|
613 |
||
614 |
lemma filtercomap_filtercomap: "filtercomap f (filtercomap g F) = filtercomap (\<lambda>x. g (f x)) F" |
|
615 |
unfolding filter_eq_iff by (auto simp: eventually_filtercomap) |
|
616 |
||
617 |
lemma filtercomap_mono: "F \<le> F' \<Longrightarrow> filtercomap f F \<le> filtercomap f F'" |
|
618 |
by (auto simp: eventually_filtercomap le_filter_def) |
|
619 |
||
620 |
lemma filtercomap_bot [simp]: "filtercomap f bot = bot" |
|
621 |
by (auto simp: filter_eq_iff eventually_filtercomap) |
|
622 |
||
623 |
lemma filtercomap_top [simp]: "filtercomap f top = top" |
|
624 |
by (auto simp: filter_eq_iff eventually_filtercomap) |
|
625 |
||
626 |
lemma filtercomap_inf: "filtercomap f (inf F1 F2) = inf (filtercomap f F1) (filtercomap f F2)" |
|
627 |
unfolding filter_eq_iff |
|
628 |
proof safe |
|
629 |
fix P |
|
630 |
assume "eventually P (filtercomap f (F1 \<sqinter> F2))" |
|
631 |
then obtain Q R S where *: |
|
632 |
"eventually Q F1" "eventually R F2" "\<And>x. Q x \<Longrightarrow> R x \<Longrightarrow> S x" "\<And>x. S (f x) \<Longrightarrow> P x" |
|
633 |
unfolding eventually_filtercomap eventually_inf by blast |
|
634 |
from * have "eventually (\<lambda>x. Q (f x)) (filtercomap f F1)" |
|
635 |
"eventually (\<lambda>x. R (f x)) (filtercomap f F2)" |
|
636 |
by (auto simp: eventually_filtercomap) |
|
637 |
with * show "eventually P (filtercomap f F1 \<sqinter> filtercomap f F2)" |
|
638 |
unfolding eventually_inf by blast |
|
639 |
next |
|
640 |
fix P |
|
641 |
assume "eventually P (inf (filtercomap f F1) (filtercomap f F2))" |
|
642 |
then obtain Q Q' R R' where *: |
|
643 |
"eventually Q F1" "eventually R F2" "\<And>x. Q (f x) \<Longrightarrow> Q' x" "\<And>x. R (f x) \<Longrightarrow> R' x" |
|
644 |
"\<And>x. Q' x \<Longrightarrow> R' x \<Longrightarrow> P x" |
|
645 |
unfolding eventually_filtercomap eventually_inf by blast |
|
646 |
from * have "eventually (\<lambda>x. Q x \<and> R x) (F1 \<sqinter> F2)" by (auto simp: eventually_inf) |
|
647 |
with * show "eventually P (filtercomap f (F1 \<sqinter> F2))" |
|
648 |
by (auto simp: eventually_filtercomap) |
|
649 |
qed |
|
650 |
||
651 |
lemma filtercomap_sup: "filtercomap f (sup F1 F2) \<ge> sup (filtercomap f F1) (filtercomap f F2)" |
|
67956 | 652 |
by (intro sup_least filtercomap_mono inf_sup_ord) |
66162 | 653 |
|
67613 | 654 |
lemma filtercomap_INF: "filtercomap f (\<Sqinter>b\<in>B. F b) = (\<Sqinter>b\<in>B. filtercomap f (F b))" |
66162 | 655 |
proof - |
67613 | 656 |
have *: "filtercomap f (\<Sqinter>b\<in>B. F b) = (\<Sqinter>b\<in>B. filtercomap f (F b))" if "finite B" for B |
66162 | 657 |
using that by induction (simp_all add: filtercomap_inf) |
658 |
show ?thesis unfolding filter_eq_iff |
|
659 |
proof |
|
660 |
fix P |
|
67613 | 661 |
have "eventually P (\<Sqinter>b\<in>B. filtercomap f (F b)) \<longleftrightarrow> |
66162 | 662 |
(\<exists>X. (X \<subseteq> B \<and> finite X) \<and> eventually P (\<Sqinter>b\<in>X. filtercomap f (F b)))" |
663 |
by (subst eventually_INF) blast |
|
67613 | 664 |
also have "\<dots> \<longleftrightarrow> (\<exists>X. (X \<subseteq> B \<and> finite X) \<and> eventually P (filtercomap f (\<Sqinter>b\<in>X. F b)))" |
66162 | 665 |
by (rule ex_cong) (simp add: *) |
69275 | 666 |
also have "\<dots> \<longleftrightarrow> eventually P (filtercomap f (\<Sqinter>(F ` B)))" |
66162 | 667 |
unfolding eventually_filtercomap by (subst eventually_INF) blast |
69275 | 668 |
finally show "eventually P (filtercomap f (\<Sqinter>(F ` B))) = |
66162 | 669 |
eventually P (\<Sqinter>b\<in>B. filtercomap f (F b))" .. |
670 |
qed |
|
671 |
qed |
|
672 |
||
67956 | 673 |
lemma filtercomap_SUP: |
674 |
"filtercomap f (\<Squnion>b\<in>B. F b) \<ge> (\<Squnion>b\<in>B. filtercomap f (F b))" |
|
675 |
by (intro SUP_least filtercomap_mono SUP_upper) |
|
676 |
||
68860
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
677 |
lemma filtermap_le_iff_le_filtercomap: "filtermap f F \<le> G \<longleftrightarrow> F \<le> filtercomap f G" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
678 |
unfolding le_filter_def eventually_filtermap eventually_filtercomap |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
679 |
using eventually_mono by auto |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
680 |
|
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
681 |
lemma filtercomap_neq_bot: |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
682 |
assumes "\<And>P. eventually P F \<Longrightarrow> \<exists>x. P (f x)" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
683 |
shows "filtercomap f F \<noteq> bot" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
684 |
using assms by (auto simp: trivial_limit_def eventually_filtercomap) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
685 |
|
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
686 |
lemma filtercomap_neq_bot_surj: |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
687 |
assumes "F \<noteq> bot" and "surj f" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
688 |
shows "filtercomap f F \<noteq> bot" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
689 |
proof (rule filtercomap_neq_bot) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
690 |
fix P assume *: "eventually P F" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
691 |
show "\<exists>x. P (f x)" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
692 |
proof (rule ccontr) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
693 |
assume **: "\<not>(\<exists>x. P (f x))" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
694 |
from * have "eventually (\<lambda>_. False) F" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
695 |
proof eventually_elim |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
696 |
case (elim x) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
697 |
from \<open>surj f\<close> obtain y where "x = f y" by auto |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
698 |
with elim and ** show False by auto |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
699 |
qed |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
700 |
with assms show False by (simp add: trivial_limit_def) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
701 |
qed |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
702 |
qed |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
703 |
|
66162 | 704 |
lemma eventually_filtercomapI [intro]: |
705 |
assumes "eventually P F" |
|
706 |
shows "eventually (\<lambda>x. P (f x)) (filtercomap f F)" |
|
707 |
using assms by (auto simp: eventually_filtercomap) |
|
708 |
||
709 |
lemma filtermap_filtercomap: "filtermap f (filtercomap f F) \<le> F" |
|
710 |
by (auto simp: le_filter_def eventually_filtermap eventually_filtercomap) |
|
67956 | 711 |
|
66162 | 712 |
lemma filtercomap_filtermap: "filtercomap f (filtermap f F) \<ge> F" |
713 |
unfolding le_filter_def eventually_filtermap eventually_filtercomap |
|
714 |
by (auto elim!: eventually_mono) |
|
715 |
||
716 |
||
60758 | 717 |
subsubsection \<open>Standard filters\<close> |
60036 | 718 |
|
719 |
definition principal :: "'a set \<Rightarrow> 'a filter" where |
|
720 |
"principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)" |
|
721 |
||
722 |
lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)" |
|
723 |
unfolding principal_def |
|
724 |
by (rule eventually_Abs_filter, rule is_filter.intro) auto |
|
725 |
||
726 |
lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F" |
|
61810 | 727 |
unfolding eventually_inf eventually_principal by (auto elim: eventually_mono) |
60036 | 728 |
|
729 |
lemma principal_UNIV[simp]: "principal UNIV = top" |
|
730 |
by (auto simp: filter_eq_iff eventually_principal) |
|
731 |
||
732 |
lemma principal_empty[simp]: "principal {} = bot" |
|
733 |
by (auto simp: filter_eq_iff eventually_principal) |
|
734 |
||
735 |
lemma principal_eq_bot_iff: "principal X = bot \<longleftrightarrow> X = {}" |
|
736 |
by (auto simp add: filter_eq_iff eventually_principal) |
|
737 |
||
738 |
lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B" |
|
739 |
by (auto simp: le_filter_def eventually_principal) |
|
740 |
||
741 |
lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F" |
|
742 |
unfolding le_filter_def eventually_principal |
|
71743 | 743 |
by (force elim: eventually_mono) |
60036 | 744 |
|
745 |
lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B" |
|
746 |
unfolding eq_iff by simp |
|
747 |
||
748 |
lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)" |
|
749 |
unfolding filter_eq_iff eventually_sup eventually_principal by auto |
|
750 |
||
751 |
lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)" |
|
752 |
unfolding filter_eq_iff eventually_inf eventually_principal |
|
753 |
by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"]) |
|
754 |
||
67613 | 755 |
lemma SUP_principal[simp]: "(\<Squnion>i\<in>I. principal (A i)) = principal (\<Union>i\<in>I. A i)" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62123
diff
changeset
|
756 |
unfolding filter_eq_iff eventually_Sup by (auto simp: eventually_principal) |
60036 | 757 |
|
67613 | 758 |
lemma INF_principal_finite: "finite X \<Longrightarrow> (\<Sqinter>x\<in>X. principal (f x)) = principal (\<Inter>x\<in>X. f x)" |
60036 | 759 |
by (induct X rule: finite_induct) auto |
760 |
||
761 |
lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)" |
|
762 |
unfolding filter_eq_iff eventually_filtermap eventually_principal by simp |
|
66162 | 763 |
|
764 |
lemma filtercomap_principal[simp]: "filtercomap f (principal A) = principal (f -` A)" |
|
765 |
unfolding filter_eq_iff eventually_filtercomap eventually_principal by fast |
|
60036 | 766 |
|
60758 | 767 |
subsubsection \<open>Order filters\<close> |
60036 | 768 |
|
769 |
definition at_top :: "('a::order) filter" |
|
67613 | 770 |
where "at_top = (\<Sqinter>k. principal {k ..})" |
60036 | 771 |
|
67613 | 772 |
lemma at_top_sub: "at_top = (\<Sqinter>k\<in>{c::'a::linorder..}. principal {k ..})" |
60036 | 773 |
by (auto intro!: INF_eq max.cobounded1 max.cobounded2 simp: at_top_def) |
774 |
||
775 |
lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)" |
|
776 |
unfolding at_top_def |
|
777 |
by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2) |
|
778 |
||
66162 | 779 |
lemma eventually_filtercomap_at_top_linorder: |
780 |
"eventually P (filtercomap f at_top) \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>x. f x \<ge> N \<longrightarrow> P x)" |
|
781 |
by (auto simp: eventually_filtercomap eventually_at_top_linorder) |
|
782 |
||
63556 | 783 |
lemma eventually_at_top_linorderI: |
784 |
fixes c::"'a::linorder" |
|
785 |
assumes "\<And>x. c \<le> x \<Longrightarrow> P x" |
|
786 |
shows "eventually P at_top" |
|
787 |
using assms by (auto simp: eventually_at_top_linorder) |
|
788 |
||
65578
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
63967
diff
changeset
|
789 |
lemma eventually_ge_at_top [simp]: |
60036 | 790 |
"eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top" |
791 |
unfolding eventually_at_top_linorder by auto |
|
792 |
||
793 |
lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::{no_top, linorder}. \<forall>n>N. P n)" |
|
794 |
proof - |
|
67613 | 795 |
have "eventually P (\<Sqinter>k. principal {k <..}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n>N. P n)" |
60036 | 796 |
by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2) |
67613 | 797 |
also have "(\<Sqinter>k. principal {k::'a <..}) = at_top" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
798 |
unfolding at_top_def |
60036 | 799 |
by (intro INF_eq) (auto intro: less_imp_le simp: Ici_subset_Ioi_iff gt_ex) |
800 |
finally show ?thesis . |
|
801 |
qed |
|
66162 | 802 |
|
803 |
lemma eventually_filtercomap_at_top_dense: |
|
804 |
"eventually P (filtercomap f at_top) \<longleftrightarrow> (\<exists>N::'a::{no_top, linorder}. \<forall>x. f x > N \<longrightarrow> P x)" |
|
805 |
by (auto simp: eventually_filtercomap eventually_at_top_dense) |
|
60036 | 806 |
|
65578
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
63967
diff
changeset
|
807 |
lemma eventually_at_top_not_equal [simp]: "eventually (\<lambda>x::'a::{no_top, linorder}. x \<noteq> c) at_top" |
60721
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
808 |
unfolding eventually_at_top_dense by auto |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
809 |
|
65578
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
63967
diff
changeset
|
810 |
lemma eventually_gt_at_top [simp]: "eventually (\<lambda>x. (c::_::{no_top, linorder}) < x) at_top" |
60036 | 811 |
unfolding eventually_at_top_dense by auto |
812 |
||
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
813 |
lemma eventually_all_ge_at_top: |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
814 |
assumes "eventually P (at_top :: ('a :: linorder) filter)" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
815 |
shows "eventually (\<lambda>x. \<forall>y\<ge>x. P y) at_top" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
816 |
proof - |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
817 |
from assms obtain x where "\<And>y. y \<ge> x \<Longrightarrow> P y" by (auto simp: eventually_at_top_linorder) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
818 |
hence "\<forall>z\<ge>y. P z" if "y \<ge> x" for y using that by simp |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
819 |
thus ?thesis by (auto simp: eventually_at_top_linorder) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
820 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
821 |
|
60036 | 822 |
definition at_bot :: "('a::order) filter" |
67613 | 823 |
where "at_bot = (\<Sqinter>k. principal {.. k})" |
60036 | 824 |
|
67613 | 825 |
lemma at_bot_sub: "at_bot = (\<Sqinter>k\<in>{.. c::'a::linorder}. principal {.. k})" |
60036 | 826 |
by (auto intro!: INF_eq min.cobounded1 min.cobounded2 simp: at_bot_def) |
827 |
||
828 |
lemma eventually_at_bot_linorder: |
|
829 |
fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)" |
|
830 |
unfolding at_bot_def |
|
831 |
by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2) |
|
832 |
||
66162 | 833 |
lemma eventually_filtercomap_at_bot_linorder: |
834 |
"eventually P (filtercomap f at_bot) \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>x. f x \<le> N \<longrightarrow> P x)" |
|
835 |
by (auto simp: eventually_filtercomap eventually_at_bot_linorder) |
|
836 |
||
65578
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
63967
diff
changeset
|
837 |
lemma eventually_le_at_bot [simp]: |
60036 | 838 |
"eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot" |
839 |
unfolding eventually_at_bot_linorder by auto |
|
840 |
||
841 |
lemma eventually_at_bot_dense: "eventually P at_bot \<longleftrightarrow> (\<exists>N::'a::{no_bot, linorder}. \<forall>n<N. P n)" |
|
842 |
proof - |
|
67613 | 843 |
have "eventually P (\<Sqinter>k. principal {..< k}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n<N. P n)" |
60036 | 844 |
by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2) |
67613 | 845 |
also have "(\<Sqinter>k. principal {..< k::'a}) = at_bot" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
846 |
unfolding at_bot_def |
60036 | 847 |
by (intro INF_eq) (auto intro: less_imp_le simp: Iic_subset_Iio_iff lt_ex) |
848 |
finally show ?thesis . |
|
849 |
qed |
|
850 |
||
66162 | 851 |
lemma eventually_filtercomap_at_bot_dense: |
852 |
"eventually P (filtercomap f at_bot) \<longleftrightarrow> (\<exists>N::'a::{no_bot, linorder}. \<forall>x. f x < N \<longrightarrow> P x)" |
|
853 |
by (auto simp: eventually_filtercomap eventually_at_bot_dense) |
|
854 |
||
65578
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
63967
diff
changeset
|
855 |
lemma eventually_at_bot_not_equal [simp]: "eventually (\<lambda>x::'a::{no_bot, linorder}. x \<noteq> c) at_bot" |
60721
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
856 |
unfolding eventually_at_bot_dense by auto |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
857 |
|
65578
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
63967
diff
changeset
|
858 |
lemma eventually_gt_at_bot [simp]: |
60036 | 859 |
"eventually (\<lambda>x. x < (c::_::unbounded_dense_linorder)) at_bot" |
860 |
unfolding eventually_at_bot_dense by auto |
|
861 |
||
63967
2aa42596edc3
new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents:
63556
diff
changeset
|
862 |
lemma trivial_limit_at_bot_linorder [simp]: "\<not> trivial_limit (at_bot ::('a::linorder) filter)" |
60036 | 863 |
unfolding trivial_limit_def |
864 |
by (metis eventually_at_bot_linorder order_refl) |
|
865 |
||
63967
2aa42596edc3
new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents:
63556
diff
changeset
|
866 |
lemma trivial_limit_at_top_linorder [simp]: "\<not> trivial_limit (at_top ::('a::linorder) filter)" |
60036 | 867 |
unfolding trivial_limit_def |
868 |
by (metis eventually_at_top_linorder order_refl) |
|
869 |
||
60758 | 870 |
subsection \<open>Sequentially\<close> |
60036 | 871 |
|
872 |
abbreviation sequentially :: "nat filter" |
|
873 |
where "sequentially \<equiv> at_top" |
|
874 |
||
875 |
lemma eventually_sequentially: |
|
876 |
"eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)" |
|
877 |
by (rule eventually_at_top_linorder) |
|
878 |
||
879 |
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot" |
|
880 |
unfolding filter_eq_iff eventually_sequentially by auto |
|
881 |
||
882 |
lemmas trivial_limit_sequentially = sequentially_bot |
|
883 |
||
884 |
lemma eventually_False_sequentially [simp]: |
|
885 |
"\<not> eventually (\<lambda>n. False) sequentially" |
|
886 |
by (simp add: eventually_False) |
|
887 |
||
888 |
lemma le_sequentially: |
|
889 |
"F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)" |
|
890 |
by (simp add: at_top_def le_INF_iff le_principal) |
|
891 |
||
60974
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
892 |
lemma eventually_sequentiallyI [intro?]: |
60036 | 893 |
assumes "\<And>x. c \<le> x \<Longrightarrow> P x" |
894 |
shows "eventually P sequentially" |
|
895 |
using assms by (auto simp: eventually_sequentially) |
|
896 |
||
63967
2aa42596edc3
new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents:
63556
diff
changeset
|
897 |
lemma eventually_sequentially_Suc [simp]: "eventually (\<lambda>i. P (Suc i)) sequentially \<longleftrightarrow> eventually P sequentially" |
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
898 |
unfolding eventually_sequentially by (metis Suc_le_D Suc_le_mono le_Suc_eq) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
899 |
|
63967
2aa42596edc3
new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents:
63556
diff
changeset
|
900 |
lemma eventually_sequentially_seg [simp]: "eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially" |
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
901 |
using eventually_sequentially_Suc[of "\<lambda>n. P (n + k)" for k] by (induction k) auto |
60036 | 902 |
|
67956 | 903 |
lemma filtermap_sequentually_ne_bot: "filtermap f sequentially \<noteq> bot" |
904 |
by (simp add: filtermap_bot_iff) |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset
|
905 |
|
68860
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
906 |
subsection \<open>Increasing finite subsets\<close> |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
907 |
|
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
908 |
definition finite_subsets_at_top where |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
68860
diff
changeset
|
909 |
"finite_subsets_at_top A = (\<Sqinter> X\<in>{X. finite X \<and> X \<subseteq> A}. principal {Y. finite Y \<and> X \<subseteq> Y \<and> Y \<subseteq> A})" |
68860
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
910 |
|
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
911 |
lemma eventually_finite_subsets_at_top: |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
912 |
"eventually P (finite_subsets_at_top A) \<longleftrightarrow> |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
913 |
(\<exists>X. finite X \<and> X \<subseteq> A \<and> (\<forall>Y. finite Y \<and> X \<subseteq> Y \<and> Y \<subseteq> A \<longrightarrow> P Y))" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
914 |
unfolding finite_subsets_at_top_def |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
915 |
proof (subst eventually_INF_base, goal_cases) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
916 |
show "{X. finite X \<and> X \<subseteq> A} \<noteq> {}" by auto |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
917 |
next |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
918 |
case (2 B C) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
919 |
thus ?case by (intro bexI[of _ "B \<union> C"]) auto |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
920 |
qed (simp_all add: eventually_principal) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
921 |
|
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
922 |
lemma eventually_finite_subsets_at_top_weakI [intro]: |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
923 |
assumes "\<And>X. finite X \<Longrightarrow> X \<subseteq> A \<Longrightarrow> P X" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
924 |
shows "eventually P (finite_subsets_at_top A)" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
925 |
proof - |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
926 |
have "eventually (\<lambda>X. finite X \<and> X \<subseteq> A) (finite_subsets_at_top A)" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
927 |
by (auto simp: eventually_finite_subsets_at_top) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
928 |
thus ?thesis by eventually_elim (use assms in auto) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
929 |
qed |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
930 |
|
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
931 |
lemma finite_subsets_at_top_neq_bot [simp]: "finite_subsets_at_top A \<noteq> bot" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
932 |
proof - |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
933 |
have "\<not>eventually (\<lambda>x. False) (finite_subsets_at_top A)" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
934 |
by (auto simp: eventually_finite_subsets_at_top) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
935 |
thus ?thesis by auto |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
936 |
qed |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
937 |
|
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
938 |
lemma filtermap_image_finite_subsets_at_top: |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
939 |
assumes "inj_on f A" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
940 |
shows "filtermap ((`) f) (finite_subsets_at_top A) = finite_subsets_at_top (f ` A)" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
941 |
unfolding filter_eq_iff eventually_filtermap |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
942 |
proof (safe, goal_cases) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
943 |
case (1 P) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
944 |
then obtain X where X: "finite X" "X \<subseteq> A" "\<And>Y. finite Y \<Longrightarrow> X \<subseteq> Y \<Longrightarrow> Y \<subseteq> A \<Longrightarrow> P (f ` Y)" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
945 |
unfolding eventually_finite_subsets_at_top by force |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
946 |
show ?case unfolding eventually_finite_subsets_at_top eventually_filtermap |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
947 |
proof (rule exI[of _ "f ` X"], intro conjI allI impI, goal_cases) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
948 |
case (3 Y) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
949 |
with assms and X(1,2) have "P (f ` (f -` Y \<inter> A))" using X(1,2) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
950 |
by (intro X(3) finite_vimage_IntI) auto |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
951 |
also have "f ` (f -` Y \<inter> A) = Y" using assms 3 by blast |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
952 |
finally show ?case . |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
953 |
qed (insert assms X(1,2), auto intro!: finite_vimage_IntI) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
954 |
next |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
955 |
case (2 P) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
956 |
then obtain X where X: "finite X" "X \<subseteq> f ` A" "\<And>Y. finite Y \<Longrightarrow> X \<subseteq> Y \<Longrightarrow> Y \<subseteq> f ` A \<Longrightarrow> P Y" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
957 |
unfolding eventually_finite_subsets_at_top by force |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
958 |
show ?case unfolding eventually_finite_subsets_at_top eventually_filtermap |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
959 |
proof (rule exI[of _ "f -` X \<inter> A"], intro conjI allI impI, goal_cases) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
960 |
case (3 Y) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
961 |
with X(1,2) and assms show ?case by (intro X(3)) force+ |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
962 |
qed (insert assms X(1), auto intro!: finite_vimage_IntI) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
963 |
qed |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
964 |
|
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
965 |
lemma eventually_finite_subsets_at_top_finite: |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
966 |
assumes "finite A" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
967 |
shows "eventually P (finite_subsets_at_top A) \<longleftrightarrow> P A" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
968 |
unfolding eventually_finite_subsets_at_top using assms by force |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
969 |
|
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
970 |
lemma finite_subsets_at_top_finite: "finite A \<Longrightarrow> finite_subsets_at_top A = principal {A}" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
971 |
by (auto simp: filter_eq_iff eventually_finite_subsets_at_top_finite eventually_principal) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
972 |
|
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
973 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset
|
974 |
subsection \<open>The cofinite filter\<close> |
60039 | 975 |
|
976 |
definition "cofinite = Abs_filter (\<lambda>P. finite {x. \<not> P x})" |
|
977 |
||
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset
|
978 |
abbreviation Inf_many :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "\<exists>\<^sub>\<infinity>" 10) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset
|
979 |
where "Inf_many P \<equiv> frequently P cofinite" |
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
980 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset
|
981 |
abbreviation Alm_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "\<forall>\<^sub>\<infinity>" 10) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset
|
982 |
where "Alm_all P \<equiv> eventually P cofinite" |
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
983 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset
|
984 |
notation (ASCII) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset
|
985 |
Inf_many (binder "INFM " 10) and |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset
|
986 |
Alm_all (binder "MOST " 10) |
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
987 |
|
60039 | 988 |
lemma eventually_cofinite: "eventually P cofinite \<longleftrightarrow> finite {x. \<not> P x}" |
989 |
unfolding cofinite_def |
|
990 |
proof (rule eventually_Abs_filter, rule is_filter.intro) |
|
991 |
fix P Q :: "'a \<Rightarrow> bool" assume "finite {x. \<not> P x}" "finite {x. \<not> Q x}" |
|
992 |
from finite_UnI[OF this] show "finite {x. \<not> (P x \<and> Q x)}" |
|
993 |
by (rule rev_finite_subset) auto |
|
994 |
next |
|
995 |
fix P Q :: "'a \<Rightarrow> bool" assume P: "finite {x. \<not> P x}" and *: "\<forall>x. P x \<longrightarrow> Q x" |
|
996 |
from * show "finite {x. \<not> Q x}" |
|
997 |
by (intro finite_subset[OF _ P]) auto |
|
998 |
qed simp |
|
999 |
||
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
1000 |
lemma frequently_cofinite: "frequently P cofinite \<longleftrightarrow> \<not> finite {x. P x}" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
1001 |
by (simp add: frequently_def eventually_cofinite) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
1002 |
|
60039 | 1003 |
lemma cofinite_bot[simp]: "cofinite = (bot::'a filter) \<longleftrightarrow> finite (UNIV :: 'a set)" |
1004 |
unfolding trivial_limit_def eventually_cofinite by simp |
|
1005 |
||
1006 |
lemma cofinite_eq_sequentially: "cofinite = sequentially" |
|
1007 |
unfolding filter_eq_iff eventually_sequentially eventually_cofinite |
|
1008 |
proof safe |
|
1009 |
fix P :: "nat \<Rightarrow> bool" assume [simp]: "finite {x. \<not> P x}" |
|
1010 |
show "\<exists>N. \<forall>n\<ge>N. P n" |
|
1011 |
proof cases |
|
1012 |
assume "{x. \<not> P x} \<noteq> {}" then show ?thesis |
|
1013 |
by (intro exI[of _ "Suc (Max {x. \<not> P x})"]) (auto simp: Suc_le_eq) |
|
1014 |
qed auto |
|
1015 |
next |
|
1016 |
fix P :: "nat \<Rightarrow> bool" and N :: nat assume "\<forall>n\<ge>N. P n" |
|
1017 |
then have "{x. \<not> P x} \<subseteq> {..< N}" |
|
1018 |
by (auto simp: not_le) |
|
1019 |
then show "finite {x. \<not> P x}" |
|
1020 |
by (blast intro: finite_subset) |
|
1021 |
qed |
|
60036 | 1022 |
|
62101 | 1023 |
subsubsection \<open>Product of filters\<close> |
1024 |
||
1025 |
definition prod_filter :: "'a filter \<Rightarrow> 'b filter \<Rightarrow> ('a \<times> 'b) filter" (infixr "\<times>\<^sub>F" 80) where |
|
1026 |
"prod_filter F G = |
|
67613 | 1027 |
(\<Sqinter>(P, Q)\<in>{(P, Q). eventually P F \<and> eventually Q G}. principal {(x, y). P x \<and> Q y})" |
62101 | 1028 |
|
1029 |
lemma eventually_prod_filter: "eventually P (F \<times>\<^sub>F G) \<longleftrightarrow> |
|
1030 |
(\<exists>Pf Pg. eventually Pf F \<and> eventually Pg G \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P (x, y)))" |
|
1031 |
unfolding prod_filter_def |
|
1032 |
proof (subst eventually_INF_base, goal_cases) |
|
1033 |
case 2 |
|
1034 |
moreover have "eventually Pf F \<Longrightarrow> eventually Qf F \<Longrightarrow> eventually Pg G \<Longrightarrow> eventually Qg G \<Longrightarrow> |
|
1035 |
\<exists>P Q. eventually P F \<and> eventually Q G \<and> |
|
1036 |
Collect P \<times> Collect Q \<subseteq> Collect Pf \<times> Collect Pg \<inter> Collect Qf \<times> Collect Qg" for Pf Pg Qf Qg |
|
1037 |
by (intro conjI exI[of _ "inf Pf Qf"] exI[of _ "inf Pg Qg"]) |
|
1038 |
(auto simp: inf_fun_def eventually_conj) |
|
1039 |
ultimately show ?case |
|
1040 |
by auto |
|
1041 |
qed (auto simp: eventually_principal intro: eventually_True) |
|
1042 |
||
62367 | 1043 |
lemma eventually_prod1: |
1044 |
assumes "B \<noteq> bot" |
|
1045 |
shows "(\<forall>\<^sub>F (x, y) in A \<times>\<^sub>F B. P x) \<longleftrightarrow> (\<forall>\<^sub>F x in A. P x)" |
|
1046 |
unfolding eventually_prod_filter |
|
1047 |
proof safe |
|
63540 | 1048 |
fix R Q |
1049 |
assume *: "\<forall>\<^sub>F x in A. R x" "\<forall>\<^sub>F x in B. Q x" "\<forall>x y. R x \<longrightarrow> Q y \<longrightarrow> P x" |
|
1050 |
with \<open>B \<noteq> bot\<close> obtain y where "Q y" by (auto dest: eventually_happens) |
|
1051 |
with * show "eventually P A" |
|
62367 | 1052 |
by (force elim: eventually_mono) |
1053 |
next |
|
1054 |
assume "eventually P A" |
|
1055 |
then show "\<exists>Pf Pg. eventually Pf A \<and> eventually Pg B \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P x)" |
|
1056 |
by (intro exI[of _ P] exI[of _ "\<lambda>x. True"]) auto |
|
1057 |
qed |
|
1058 |
||
1059 |
lemma eventually_prod2: |
|
1060 |
assumes "A \<noteq> bot" |
|
1061 |
shows "(\<forall>\<^sub>F (x, y) in A \<times>\<^sub>F B. P y) \<longleftrightarrow> (\<forall>\<^sub>F y in B. P y)" |
|
1062 |
unfolding eventually_prod_filter |
|
1063 |
proof safe |
|
63540 | 1064 |
fix R Q |
1065 |
assume *: "\<forall>\<^sub>F x in A. R x" "\<forall>\<^sub>F x in B. Q x" "\<forall>x y. R x \<longrightarrow> Q y \<longrightarrow> P y" |
|
1066 |
with \<open>A \<noteq> bot\<close> obtain x where "R x" by (auto dest: eventually_happens) |
|
1067 |
with * show "eventually P B" |
|
62367 | 1068 |
by (force elim: eventually_mono) |
1069 |
next |
|
1070 |
assume "eventually P B" |
|
1071 |
then show "\<exists>Pf Pg. eventually Pf A \<and> eventually Pg B \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P y)" |
|
1072 |
by (intro exI[of _ P] exI[of _ "\<lambda>x. True"]) auto |
|
1073 |
qed |
|
1074 |
||
1075 |
lemma INF_filter_bot_base: |
|
1076 |
fixes F :: "'a \<Rightarrow> 'b filter" |
|
1077 |
assumes *: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> \<exists>k\<in>I. F k \<le> F i \<sqinter> F j" |
|
67613 | 1078 |
shows "(\<Sqinter>i\<in>I. F i) = bot \<longleftrightarrow> (\<exists>i\<in>I. F i = bot)" |
63540 | 1079 |
proof (cases "\<exists>i\<in>I. F i = bot") |
1080 |
case True |
|
67613 | 1081 |
then have "(\<Sqinter>i\<in>I. F i) \<le> bot" |
62367 | 1082 |
by (auto intro: INF_lower2) |
63540 | 1083 |
with True show ?thesis |
62367 | 1084 |
by (auto simp: bot_unique) |
1085 |
next |
|
63540 | 1086 |
case False |
67613 | 1087 |
moreover have "(\<Sqinter>i\<in>I. F i) \<noteq> bot" |
63540 | 1088 |
proof (cases "I = {}") |
1089 |
case True |
|
1090 |
then show ?thesis |
|
1091 |
by (auto simp add: filter_eq_iff) |
|
1092 |
next |
|
1093 |
case False': False |
|
62367 | 1094 |
show ?thesis |
1095 |
proof (rule INF_filter_not_bot) |
|
63540 | 1096 |
fix J |
1097 |
assume "finite J" "J \<subseteq> I" |
|
62367 | 1098 |
then have "\<exists>k\<in>I. F k \<le> (\<Sqinter>i\<in>J. F i)" |
63540 | 1099 |
proof (induct J) |
1100 |
case empty |
|
1101 |
then show ?case |
|
62367 | 1102 |
using \<open>I \<noteq> {}\<close> by auto |
1103 |
next |
|
1104 |
case (insert i J) |
|
63540 | 1105 |
then obtain k where "k \<in> I" "F k \<le> (\<Sqinter>i\<in>J. F i)" by auto |
1106 |
with insert *[of i k] show ?case |
|
62367 | 1107 |
by auto |
1108 |
qed |
|
63540 | 1109 |
with False show "(\<Sqinter>i\<in>J. F i) \<noteq> \<bottom>" |
62367 | 1110 |
by (auto simp: bot_unique) |
1111 |
qed |
|
63540 | 1112 |
qed |
62367 | 1113 |
ultimately show ?thesis |
1114 |
by auto |
|
1115 |
qed |
|
1116 |
||
1117 |
lemma Collect_empty_eq_bot: "Collect P = {} \<longleftrightarrow> P = \<bottom>" |
|
1118 |
by auto |
|
1119 |
||
1120 |
lemma prod_filter_eq_bot: "A \<times>\<^sub>F B = bot \<longleftrightarrow> A = bot \<or> B = bot" |
|
67956 | 1121 |
unfolding trivial_limit_def |
1122 |
proof |
|
1123 |
assume "\<forall>\<^sub>F x in A \<times>\<^sub>F B. False" |
|
1124 |
then obtain Pf Pg |
|
1125 |
where Pf: "eventually (\<lambda>x. Pf x) A" and Pg: "eventually (\<lambda>y. Pg y) B" |
|
1126 |
and *: "\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> False" |
|
1127 |
unfolding eventually_prod_filter by fast |
|
1128 |
from * have "(\<forall>x. \<not> Pf x) \<or> (\<forall>y. \<not> Pg y)" by fast |
|
1129 |
with Pf Pg show "(\<forall>\<^sub>F x in A. False) \<or> (\<forall>\<^sub>F x in B. False)" by auto |
|
62367 | 1130 |
next |
67956 | 1131 |
assume "(\<forall>\<^sub>F x in A. False) \<or> (\<forall>\<^sub>F x in B. False)" |
1132 |
then show "\<forall>\<^sub>F x in A \<times>\<^sub>F B. False" |
|
1133 |
unfolding eventually_prod_filter by (force intro: eventually_True) |
|
62367 | 1134 |
qed |
1135 |
||
62101 | 1136 |
lemma prod_filter_mono: "F \<le> F' \<Longrightarrow> G \<le> G' \<Longrightarrow> F \<times>\<^sub>F G \<le> F' \<times>\<^sub>F G'" |
1137 |
by (auto simp: le_filter_def eventually_prod_filter) |
|
1138 |
||
62367 | 1139 |
lemma prod_filter_mono_iff: |
1140 |
assumes nAB: "A \<noteq> bot" "B \<noteq> bot" |
|
1141 |
shows "A \<times>\<^sub>F B \<le> C \<times>\<^sub>F D \<longleftrightarrow> A \<le> C \<and> B \<le> D" |
|
1142 |
proof safe |
|
1143 |
assume *: "A \<times>\<^sub>F B \<le> C \<times>\<^sub>F D" |
|
63540 | 1144 |
with assms have "A \<times>\<^sub>F B \<noteq> bot" |
62367 | 1145 |
by (auto simp: bot_unique prod_filter_eq_bot) |
63540 | 1146 |
with * have "C \<times>\<^sub>F D \<noteq> bot" |
62367 | 1147 |
by (auto simp: bot_unique) |
1148 |
then have nCD: "C \<noteq> bot" "D \<noteq> bot" |
|
1149 |
by (auto simp: prod_filter_eq_bot) |
|
1150 |
||
1151 |
show "A \<le> C" |
|
1152 |
proof (rule filter_leI) |
|
1153 |
fix P assume "eventually P C" with *[THEN filter_leD, of "\<lambda>(x, y). P x"] show "eventually P A" |
|
1154 |
using nAB nCD by (simp add: eventually_prod1 eventually_prod2) |
|
1155 |
qed |
|
1156 |
||
1157 |
show "B \<le> D" |
|
1158 |
proof (rule filter_leI) |
|
1159 |
fix P assume "eventually P D" with *[THEN filter_leD, of "\<lambda>(x, y). P y"] show "eventually P B" |
|
1160 |
using nAB nCD by (simp add: eventually_prod1 eventually_prod2) |
|
1161 |
qed |
|
1162 |
qed (intro prod_filter_mono) |
|
1163 |
||
62101 | 1164 |
lemma eventually_prod_same: "eventually P (F \<times>\<^sub>F F) \<longleftrightarrow> |
1165 |
(\<exists>Q. eventually Q F \<and> (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y)))" |
|
71743 | 1166 |
unfolding eventually_prod_filter by (blast intro!: eventually_conj) |
62101 | 1167 |
|
1168 |
lemma eventually_prod_sequentially: |
|
1169 |
"eventually P (sequentially \<times>\<^sub>F sequentially) \<longleftrightarrow> (\<exists>N. \<forall>m \<ge> N. \<forall>n \<ge> N. P (n, m))" |
|
1170 |
unfolding eventually_prod_same eventually_sequentially by auto |
|
1171 |
||
1172 |
lemma principal_prod_principal: "principal A \<times>\<^sub>F principal B = principal (A \<times> B)" |
|
67956 | 1173 |
unfolding filter_eq_iff eventually_prod_filter eventually_principal |
1174 |
by (fast intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"]) |
|
1175 |
||
1176 |
lemma le_prod_filterI: |
|
1177 |
"filtermap fst F \<le> A \<Longrightarrow> filtermap snd F \<le> B \<Longrightarrow> F \<le> A \<times>\<^sub>F B" |
|
1178 |
unfolding le_filter_def eventually_filtermap eventually_prod_filter |
|
1179 |
by (force elim: eventually_elim2) |
|
1180 |
||
1181 |
lemma filtermap_fst_prod_filter: "filtermap fst (A \<times>\<^sub>F B) \<le> A" |
|
1182 |
unfolding le_filter_def eventually_filtermap eventually_prod_filter |
|
1183 |
by (force intro: eventually_True) |
|
1184 |
||
1185 |
lemma filtermap_snd_prod_filter: "filtermap snd (A \<times>\<^sub>F B) \<le> B" |
|
1186 |
unfolding le_filter_def eventually_filtermap eventually_prod_filter |
|
1187 |
by (force intro: eventually_True) |
|
62101 | 1188 |
|
62367 | 1189 |
lemma prod_filter_INF: |
67956 | 1190 |
assumes "I \<noteq> {}" and "J \<noteq> {}" |
67613 | 1191 |
shows "(\<Sqinter>i\<in>I. A i) \<times>\<^sub>F (\<Sqinter>j\<in>J. B j) = (\<Sqinter>i\<in>I. \<Sqinter>j\<in>J. A i \<times>\<^sub>F B j)" |
67956 | 1192 |
proof (rule antisym) |
62367 | 1193 |
from \<open>I \<noteq> {}\<close> obtain i where "i \<in> I" by auto |
1194 |
from \<open>J \<noteq> {}\<close> obtain j where "j \<in> J" by auto |
|
1195 |
||
1196 |
show "(\<Sqinter>i\<in>I. \<Sqinter>j\<in>J. A i \<times>\<^sub>F B j) \<le> (\<Sqinter>i\<in>I. A i) \<times>\<^sub>F (\<Sqinter>j\<in>J. B j)" |
|
67956 | 1197 |
by (fast intro: le_prod_filterI INF_greatest INF_lower2 |
69272 | 1198 |
order_trans[OF filtermap_INF] \<open>i \<in> I\<close> \<open>j \<in> J\<close> |
67956 | 1199 |
filtermap_fst_prod_filter filtermap_snd_prod_filter) |
1200 |
show "(\<Sqinter>i\<in>I. A i) \<times>\<^sub>F (\<Sqinter>j\<in>J. B j) \<le> (\<Sqinter>i\<in>I. \<Sqinter>j\<in>J. A i \<times>\<^sub>F B j)" |
|
1201 |
by (intro INF_greatest prod_filter_mono INF_lower) |
|
1202 |
qed |
|
62367 | 1203 |
|
1204 |
lemma filtermap_Pair: "filtermap (\<lambda>x. (f x, g x)) F \<le> filtermap f F \<times>\<^sub>F filtermap g F" |
|
67956 | 1205 |
by (rule le_prod_filterI, simp_all add: filtermap_filtermap) |
62367 | 1206 |
|
62369 | 1207 |
lemma eventually_prodI: "eventually P F \<Longrightarrow> eventually Q G \<Longrightarrow> eventually (\<lambda>x. P (fst x) \<and> Q (snd x)) (F \<times>\<^sub>F G)" |
67956 | 1208 |
unfolding eventually_prod_filter by auto |
62369 | 1209 |
|
67613 | 1210 |
lemma prod_filter_INF1: "I \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>I. A i) \<times>\<^sub>F B = (\<Sqinter>i\<in>I. A i \<times>\<^sub>F B)" |
62369 | 1211 |
using prod_filter_INF[of I "{B}" A "\<lambda>x. x"] by simp |
1212 |
||
67613 | 1213 |
lemma prod_filter_INF2: "J \<noteq> {} \<Longrightarrow> A \<times>\<^sub>F (\<Sqinter>i\<in>J. B i) = (\<Sqinter>i\<in>J. A \<times>\<^sub>F B i)" |
62369 | 1214 |
using prod_filter_INF[of "{A}" J "\<lambda>x. x" B] by simp |
1215 |
||
68667 | 1216 |
lemma prod_filtermap1: "prod_filter (filtermap f F) G = filtermap (apfst f) (prod_filter F G)" |
71743 | 1217 |
unfolding filter_eq_iff eventually_filtermap eventually_prod_filter |
1218 |
apply safe |
|
68667 | 1219 |
subgoal by auto |
1220 |
subgoal for P Q R by(rule exI[where x="\<lambda>y. \<exists>x. y = f x \<and> Q x"])(auto intro: eventually_mono) |
|
1221 |
done |
|
1222 |
||
1223 |
lemma prod_filtermap2: "prod_filter F (filtermap g G) = filtermap (apsnd g) (prod_filter F G)" |
|
71743 | 1224 |
unfolding filter_eq_iff eventually_filtermap eventually_prod_filter |
1225 |
apply safe |
|
68667 | 1226 |
subgoal by auto |
1227 |
subgoal for P Q R by(auto intro: exI[where x="\<lambda>y. \<exists>x. y = g x \<and> R x"] eventually_mono) |
|
1228 |
done |
|
1229 |
||
1230 |
lemma prod_filter_assoc: |
|
1231 |
"prod_filter (prod_filter F G) H = filtermap (\<lambda>(x, y, z). ((x, y), z)) (prod_filter F (prod_filter G H))" |
|
1232 |
apply(clarsimp simp add: filter_eq_iff eventually_filtermap eventually_prod_filter; safe) |
|
1233 |
subgoal for P Q R S T by(auto 4 4 intro: exI[where x="\<lambda>(a, b). T a \<and> S b"]) |
|
1234 |
subgoal for P Q R S T by(auto 4 3 intro: exI[where x="\<lambda>(a, b). Q a \<and> S b"]) |
|
1235 |
done |
|
1236 |
||
1237 |
lemma prod_filter_principal_singleton: "prod_filter (principal {x}) F = filtermap (Pair x) F" |
|
1238 |
by(fastforce simp add: filter_eq_iff eventually_prod_filter eventually_principal eventually_filtermap elim: eventually_mono intro: exI[where x="\<lambda>a. a = x"]) |
|
1239 |
||
1240 |
lemma prod_filter_principal_singleton2: "prod_filter F (principal {x}) = filtermap (\<lambda>a. (a, x)) F" |
|
1241 |
by(fastforce simp add: filter_eq_iff eventually_prod_filter eventually_principal eventually_filtermap elim: eventually_mono intro: exI[where x="\<lambda>a. a = x"]) |
|
1242 |
||
1243 |
lemma prod_filter_commute: "prod_filter F G = filtermap prod.swap (prod_filter G F)" |
|
1244 |
by(auto simp add: filter_eq_iff eventually_prod_filter eventually_filtermap) |
|
1245 |
||
60758 | 1246 |
subsection \<open>Limits\<close> |
60036 | 1247 |
|
1248 |
definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where |
|
1249 |
"filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2" |
|
1250 |
||
1251 |
syntax |
|
1252 |
"_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10) |
|
1253 |
||
1254 |
translations |
|
62367 | 1255 |
"LIM x F1. f :> F2" == "CONST filterlim (\<lambda>x. f) F2 F1" |
60036 | 1256 |
|
62379
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62378
diff
changeset
|
1257 |
lemma filterlim_top [simp]: "filterlim f top F" |
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62378
diff
changeset
|
1258 |
by (simp add: filterlim_def) |
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62378
diff
changeset
|
1259 |
|
60036 | 1260 |
lemma filterlim_iff: |
1261 |
"(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)" |
|
1262 |
unfolding filterlim_def le_filter_def eventually_filtermap .. |
|
1263 |
||
1264 |
lemma filterlim_compose: |
|
1265 |
"filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1" |
|
1266 |
unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans) |
|
1267 |
||
1268 |
lemma filterlim_mono: |
|
1269 |
"filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'" |
|
1270 |
unfolding filterlim_def by (metis filtermap_mono order_trans) |
|
1271 |
||
1272 |
lemma filterlim_ident: "LIM x F. x :> F" |
|
1273 |
by (simp add: filterlim_def filtermap_ident) |
|
1274 |
||
1275 |
lemma filterlim_cong: |
|
1276 |
"F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'" |
|
1277 |
by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2) |
|
1278 |
||
1279 |
lemma filterlim_mono_eventually: |
|
1280 |
assumes "filterlim f F G" and ord: "F \<le> F'" "G' \<le> G" |
|
1281 |
assumes eq: "eventually (\<lambda>x. f x = f' x) G'" |
|
1282 |
shows "filterlim f' F' G'" |
|
71743 | 1283 |
proof - |
1284 |
have "filterlim f F' G'" |
|
1285 |
by (simp add: filterlim_mono[OF _ ord] assms) |
|
1286 |
then show ?thesis |
|
1287 |
by (rule filterlim_cong[OF refl refl eq, THEN iffD1]) |
|
1288 |
qed |
|
60036 | 1289 |
|
1290 |
lemma filtermap_mono_strong: "inj f \<Longrightarrow> filtermap f F \<le> filtermap f G \<longleftrightarrow> F \<le> G" |
|
67956 | 1291 |
apply (safe intro!: filtermap_mono) |
60036 | 1292 |
apply (auto simp: le_filter_def eventually_filtermap) |
1293 |
apply (erule_tac x="\<lambda>x. P (inv f x)" in allE) |
|
1294 |
apply auto |
|
1295 |
done |
|
1296 |
||
67950
99eaa5cedbb7
Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents:
67855
diff
changeset
|
1297 |
lemma eventually_compose_filterlim: |
99eaa5cedbb7
Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents:
67855
diff
changeset
|
1298 |
assumes "eventually P F" "filterlim f F G" |
99eaa5cedbb7
Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents:
67855
diff
changeset
|
1299 |
shows "eventually (\<lambda>x. P (f x)) G" |
99eaa5cedbb7
Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents:
67855
diff
changeset
|
1300 |
using assms by (simp add: filterlim_iff) |
99eaa5cedbb7
Added some simple facts about limits
Manuel Eberl <eberlm@in.tum.de>
parents:
67855
diff
changeset
|
1301 |
|
60036 | 1302 |
lemma filtermap_eq_strong: "inj f \<Longrightarrow> filtermap f F = filtermap f G \<longleftrightarrow> F = G" |
1303 |
by (simp add: filtermap_mono_strong eq_iff) |
|
1304 |
||
60721
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
1305 |
lemma filtermap_fun_inverse: |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
1306 |
assumes g: "filterlim g F G" |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
1307 |
assumes f: "filterlim f G F" |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
1308 |
assumes ev: "eventually (\<lambda>x. f (g x) = x) G" |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
1309 |
shows "filtermap f F = G" |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
1310 |
proof (rule antisym) |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
1311 |
show "filtermap f F \<le> G" |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
1312 |
using f unfolding filterlim_def . |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
1313 |
have "G = filtermap f (filtermap g G)" |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
1314 |
using ev by (auto elim: eventually_elim2 simp: filter_eq_iff eventually_filtermap) |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
1315 |
also have "\<dots> \<le> filtermap f F" |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
1316 |
using g by (intro filtermap_mono) (simp add: filterlim_def) |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
1317 |
finally show "G \<le> filtermap f F" . |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
1318 |
qed |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
1319 |
|
60036 | 1320 |
lemma filterlim_principal: |
1321 |
"(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)" |
|
1322 |
unfolding filterlim_def eventually_filtermap le_principal .. |
|
1323 |
||
68860
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1324 |
lemma filterlim_filtercomap [intro]: "filterlim f F (filtercomap f F)" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1325 |
unfolding filterlim_def by (rule filtermap_filtercomap) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1326 |
|
60036 | 1327 |
lemma filterlim_inf: |
1328 |
"(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))" |
|
1329 |
unfolding filterlim_def by simp |
|
1330 |
||
1331 |
lemma filterlim_INF: |
|
67613 | 1332 |
"(LIM x F. f x :> (\<Sqinter>b\<in>B. G b)) \<longleftrightarrow> (\<forall>b\<in>B. LIM x F. f x :> G b)" |
60036 | 1333 |
unfolding filterlim_def le_INF_iff .. |
1334 |
||
1335 |
lemma filterlim_INF_INF: |
|
67613 | 1336 |
"(\<And>m. m \<in> J \<Longrightarrow> \<exists>i\<in>I. filtermap f (F i) \<le> G m) \<Longrightarrow> LIM x (\<Sqinter>i\<in>I. F i). f x :> (\<Sqinter>j\<in>J. G j)" |
60036 | 1337 |
unfolding filterlim_def by (rule order_trans[OF filtermap_INF INF_mono]) |
1338 |
||
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
68860
diff
changeset
|
1339 |
lemma filterlim_INF': "x \<in> A \<Longrightarrow> filterlim f F (G x) \<Longrightarrow> filterlim f F (\<Sqinter> x\<in>A. G x)" |
68860
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1340 |
unfolding filterlim_def by (rule order.trans[OF filtermap_mono[OF INF_lower]]) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1341 |
|
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1342 |
lemma filterlim_filtercomap_iff: "filterlim f (filtercomap g G) F \<longleftrightarrow> filterlim (g \<circ> f) G F" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1343 |
by (simp add: filterlim_def filtermap_le_iff_le_filtercomap filtercomap_filtercomap o_def) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1344 |
|
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1345 |
lemma filterlim_iff_le_filtercomap: "filterlim f F G \<longleftrightarrow> G \<le> filtercomap f F" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1346 |
by (simp add: filterlim_def filtermap_le_iff_le_filtercomap) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1347 |
|
60036 | 1348 |
lemma filterlim_base: |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
1349 |
"(\<And>m x. m \<in> J \<Longrightarrow> i m \<in> I) \<Longrightarrow> (\<And>m x. m \<in> J \<Longrightarrow> x \<in> F (i m) \<Longrightarrow> f x \<in> G m) \<Longrightarrow> |
67613 | 1350 |
LIM x (\<Sqinter>i\<in>I. principal (F i)). f x :> (\<Sqinter>j\<in>J. principal (G j))" |
60036 | 1351 |
by (force intro!: filterlim_INF_INF simp: image_subset_iff) |
1352 |
||
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
1353 |
lemma filterlim_base_iff: |
60036 | 1354 |
assumes "I \<noteq> {}" and chain: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> F i \<subseteq> F j \<or> F j \<subseteq> F i" |
67613 | 1355 |
shows "(LIM x (\<Sqinter>i\<in>I. principal (F i)). f x :> \<Sqinter>j\<in>J. principal (G j)) \<longleftrightarrow> |
60036 | 1356 |
(\<forall>j\<in>J. \<exists>i\<in>I. \<forall>x\<in>F i. f x \<in> G j)" |
1357 |
unfolding filterlim_INF filterlim_principal |
|
1358 |
proof (subst eventually_INF_base) |
|
1359 |
fix i j assume "i \<in> I" "j \<in> I" |
|
1360 |
with chain[OF this] show "\<exists>x\<in>I. principal (F x) \<le> inf (principal (F i)) (principal (F j))" |
|
1361 |
by auto |
|
60758 | 1362 |
qed (auto simp: eventually_principal \<open>I \<noteq> {}\<close>) |
60036 | 1363 |
|
1364 |
lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2" |
|
1365 |
unfolding filterlim_def filtermap_filtermap .. |
|
1366 |
||
1367 |
lemma filterlim_sup: |
|
1368 |
"filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)" |
|
1369 |
unfolding filterlim_def filtermap_sup by auto |
|
1370 |
||
1371 |
lemma filterlim_sequentially_Suc: |
|
1372 |
"(LIM x sequentially. f (Suc x) :> F) \<longleftrightarrow> (LIM x sequentially. f x :> F)" |
|
1373 |
unfolding filterlim_iff by (subst eventually_sequentially_Suc) simp |
|
1374 |
||
1375 |
lemma filterlim_Suc: "filterlim Suc sequentially sequentially" |
|
63967
2aa42596edc3
new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents:
63556
diff
changeset
|
1376 |
by (simp add: filterlim_iff eventually_sequentially) |
60036 | 1377 |
|
60182
e1ea5a6379c9
generalized tends over powr; added DERIV rule for powr
hoelzl
parents:
60040
diff
changeset
|
1378 |
lemma filterlim_If: |
e1ea5a6379c9
generalized tends over powr; added DERIV rule for powr
hoelzl
parents:
60040
diff
changeset
|
1379 |
"LIM x inf F (principal {x. P x}). f x :> G \<Longrightarrow> |
e1ea5a6379c9
generalized tends over powr; added DERIV rule for powr
hoelzl
parents:
60040
diff
changeset
|
1380 |
LIM x inf F (principal {x. \<not> P x}). g x :> G \<Longrightarrow> |
e1ea5a6379c9
generalized tends over powr; added DERIV rule for powr
hoelzl
parents:
60040
diff
changeset
|
1381 |
LIM x F. if P x then f x else g x :> G" |
e1ea5a6379c9
generalized tends over powr; added DERIV rule for powr
hoelzl
parents:
60040
diff
changeset
|
1382 |
unfolding filterlim_iff eventually_inf_principal by (auto simp: eventually_conj_iff) |
60036 | 1383 |
|
62367 | 1384 |
lemma filterlim_Pair: |
1385 |
"LIM x F. f x :> G \<Longrightarrow> LIM x F. g x :> H \<Longrightarrow> LIM x F. (f x, g x) :> G \<times>\<^sub>F H" |
|
1386 |
unfolding filterlim_def |
|
1387 |
by (rule order_trans[OF filtermap_Pair prod_filter_mono]) |
|
1388 |
||
69593 | 1389 |
subsection \<open>Limits to \<^const>\<open>at_top\<close> and \<^const>\<open>at_bot\<close>\<close> |
60036 | 1390 |
|
1391 |
lemma filterlim_at_top: |
|
1392 |
fixes f :: "'a \<Rightarrow> ('b::linorder)" |
|
1393 |
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)" |
|
61810 | 1394 |
by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_mono) |
60036 | 1395 |
|
1396 |
lemma filterlim_at_top_mono: |
|
1397 |
"LIM x F. f x :> at_top \<Longrightarrow> eventually (\<lambda>x. f x \<le> (g x::'a::linorder)) F \<Longrightarrow> |
|
1398 |
LIM x F. g x :> at_top" |
|
1399 |
by (auto simp: filterlim_at_top elim: eventually_elim2 intro: order_trans) |
|
1400 |
||
1401 |
lemma filterlim_at_top_dense: |
|
1402 |
fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" |
|
1403 |
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)" |
|
61810 | 1404 |
by (metis eventually_mono[of _ F] eventually_gt_at_top order_less_imp_le |
60036 | 1405 |
filterlim_at_top[of f F] filterlim_iff[of f at_top F]) |
1406 |
||
1407 |
lemma filterlim_at_top_ge: |
|
1408 |
fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b" |
|
1409 |
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)" |
|
1410 |
unfolding at_top_sub[of c] filterlim_INF by (auto simp add: filterlim_principal) |
|
1411 |
||
1412 |
lemma filterlim_at_top_at_top: |
|
1413 |
fixes f :: "'a::linorder \<Rightarrow> 'b::linorder" |
|
1414 |
assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y" |
|
1415 |
assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)" |
|
1416 |
assumes Q: "eventually Q at_top" |
|
1417 |
assumes P: "eventually P at_top" |
|
1418 |
shows "filterlim f at_top at_top" |
|
1419 |
proof - |
|
1420 |
from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y" |
|
1421 |
unfolding eventually_at_top_linorder by auto |
|
1422 |
show ?thesis |
|
1423 |
proof (intro filterlim_at_top_ge[THEN iffD2] allI impI) |
|
1424 |
fix z assume "x \<le> z" |
|
1425 |
with x have "P z" by auto |
|
1426 |
have "eventually (\<lambda>x. g z \<le> x) at_top" |
|
1427 |
by (rule eventually_ge_at_top) |
|
1428 |
with Q show "eventually (\<lambda>x. z \<le> f x) at_top" |
|
60758 | 1429 |
by eventually_elim (metis mono bij \<open>P z\<close>) |
60036 | 1430 |
qed |
1431 |
qed |
|
1432 |
||
1433 |
lemma filterlim_at_top_gt: |
|
1434 |
fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b" |
|
1435 |
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)" |
|
1436 |
by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge) |
|
1437 |
||
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
1438 |
lemma filterlim_at_bot: |
60036 | 1439 |
fixes f :: "'a \<Rightarrow> ('b::linorder)" |
1440 |
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)" |
|
61810 | 1441 |
by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_mono) |
60036 | 1442 |
|
1443 |
lemma filterlim_at_bot_dense: |
|
1444 |
fixes f :: "'a \<Rightarrow> ('b::{dense_linorder, no_bot})" |
|
1445 |
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x < Z) F)" |
|
1446 |
proof (auto simp add: filterlim_at_bot[of f F]) |
|
1447 |
fix Z :: 'b |
|
1448 |
from lt_ex [of Z] obtain Z' where 1: "Z' < Z" .. |
|
1449 |
assume "\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F" |
|
1450 |
hence "eventually (\<lambda>x. f x \<le> Z') F" by auto |
|
1451 |
thus "eventually (\<lambda>x. f x < Z) F" |
|
71743 | 1452 |
by (rule eventually_mono) (use 1 in auto) |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
1453 |
next |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
1454 |
fix Z :: 'b |
60036 | 1455 |
show "\<forall>Z. eventually (\<lambda>x. f x < Z) F \<Longrightarrow> eventually (\<lambda>x. f x \<le> Z) F" |
61810 | 1456 |
by (drule spec [of _ Z], erule eventually_mono, auto simp add: less_imp_le) |
60036 | 1457 |
qed |
1458 |
||
1459 |
lemma filterlim_at_bot_le: |
|
1460 |
fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b" |
|
1461 |
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)" |
|
1462 |
unfolding filterlim_at_bot |
|
1463 |
proof safe |
|
1464 |
fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F" |
|
1465 |
with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F" |
|
61810 | 1466 |
by (auto elim!: eventually_mono) |
60036 | 1467 |
qed simp |
1468 |
||
1469 |
lemma filterlim_at_bot_lt: |
|
1470 |
fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b" |
|
1471 |
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)" |
|
1472 |
by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans) |
|
66162 | 1473 |
|
68860
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1474 |
lemma filterlim_finite_subsets_at_top: |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1475 |
"filterlim f (finite_subsets_at_top A) F \<longleftrightarrow> |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1476 |
(\<forall>X. finite X \<and> X \<subseteq> A \<longrightarrow> eventually (\<lambda>y. finite (f y) \<and> X \<subseteq> f y \<and> f y \<subseteq> A) F)" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1477 |
(is "?lhs = ?rhs") |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1478 |
proof |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1479 |
assume ?lhs |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1480 |
thus ?rhs |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1481 |
proof (safe, goal_cases) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1482 |
case (1 X) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1483 |
hence *: "(\<forall>\<^sub>F x in F. P (f x))" if "eventually P (finite_subsets_at_top A)" for P |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1484 |
using that by (auto simp: filterlim_def le_filter_def eventually_filtermap) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1485 |
have "\<forall>\<^sub>F Y in finite_subsets_at_top A. finite Y \<and> X \<subseteq> Y \<and> Y \<subseteq> A" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1486 |
using 1 unfolding eventually_finite_subsets_at_top by force |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1487 |
thus ?case by (intro *) auto |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1488 |
qed |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1489 |
next |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1490 |
assume rhs: ?rhs |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1491 |
show ?lhs unfolding filterlim_def le_filter_def eventually_finite_subsets_at_top |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1492 |
proof (safe, goal_cases) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1493 |
case (1 P X) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1494 |
with rhs have "\<forall>\<^sub>F y in F. finite (f y) \<and> X \<subseteq> f y \<and> f y \<subseteq> A" by auto |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1495 |
thus "eventually P (filtermap f F)" unfolding eventually_filtermap |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1496 |
by eventually_elim (insert 1, auto) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1497 |
qed |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1498 |
qed |
60036 | 1499 |
|
68860
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1500 |
lemma filterlim_atMost_at_top: |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1501 |
"filterlim (\<lambda>n. {..n}) (finite_subsets_at_top (UNIV :: nat set)) at_top" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1502 |
unfolding filterlim_finite_subsets_at_top |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1503 |
proof (safe, goal_cases) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1504 |
case (1 X) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1505 |
then obtain n where n: "X \<subseteq> {..n}" by (auto simp: finite_nat_set_iff_bounded_le) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1506 |
show ?case using eventually_ge_at_top[of n] |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1507 |
by eventually_elim (insert n, auto) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1508 |
qed |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1509 |
|
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1510 |
lemma filterlim_lessThan_at_top: |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1511 |
"filterlim (\<lambda>n. {..<n}) (finite_subsets_at_top (UNIV :: nat set)) at_top" |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1512 |
unfolding filterlim_finite_subsets_at_top |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1513 |
proof (safe, goal_cases) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1514 |
case (1 X) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1515 |
then obtain n where n: "X \<subseteq> {..<n}" by (auto simp: finite_nat_set_iff_bounded) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1516 |
show ?case using eventually_ge_at_top[of n] |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1517 |
by eventually_elim (insert n, auto) |
f443ec10447d
Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68667
diff
changeset
|
1518 |
qed |
60036 | 1519 |
|
69593 | 1520 |
subsection \<open>Setup \<^typ>\<open>'a filter\<close> for lifting and transfer\<close> |
60036 | 1521 |
|
1522 |
lemma filtermap_id [simp, id_simps]: "filtermap id = id" |
|
1523 |
by(simp add: fun_eq_iff id_def filtermap_ident) |
|
1524 |
||
1525 |
lemma filtermap_id' [simp]: "filtermap (\<lambda>x. x) = (\<lambda>F. F)" |
|
1526 |
using filtermap_id unfolding id_def . |
|
1527 |
||
67616
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1528 |
context includes lifting_syntax |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1529 |
begin |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1530 |
|
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1531 |
definition map_filter_on :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter" where |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1532 |
"map_filter_on X f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x) \<and> x \<in> X) F)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1533 |
|
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1534 |
lemma is_filter_map_filter_on: |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1535 |
"is_filter (\<lambda>P. \<forall>\<^sub>F x in F. P (f x) \<and> x \<in> X) \<longleftrightarrow> eventually (\<lambda>x. x \<in> X) F" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1536 |
proof(rule iffI; unfold_locales) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1537 |
show "\<forall>\<^sub>F x in F. True \<and> x \<in> X" if "eventually (\<lambda>x. x \<in> X) F" using that by simp |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1538 |
show "\<forall>\<^sub>F x in F. (P (f x) \<and> Q (f x)) \<and> x \<in> X" if "\<forall>\<^sub>F x in F. P (f x) \<and> x \<in> X" "\<forall>\<^sub>F x in F. Q (f x) \<and> x \<in> X" for P Q |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1539 |
using eventually_conj[OF that] by(auto simp add: conj_ac cong: conj_cong) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1540 |
show "\<forall>\<^sub>F x in F. Q (f x) \<and> x \<in> X" if "\<forall>x. P x \<longrightarrow> Q x" "\<forall>\<^sub>F x in F. P (f x) \<and> x \<in> X" for P Q |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1541 |
using that(2) by(rule eventually_mono)(use that(1) in auto) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1542 |
show "eventually (\<lambda>x. x \<in> X) F" if "is_filter (\<lambda>P. \<forall>\<^sub>F x in F. P (f x) \<and> x \<in> X)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1543 |
using is_filter.True[OF that] by simp |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1544 |
qed |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1545 |
|
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1546 |
lemma eventually_map_filter_on: "eventually P (map_filter_on X f F) = (\<forall>\<^sub>F x in F. P (f x) \<and> x \<in> X)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1547 |
if "eventually (\<lambda>x. x \<in> X) F" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1548 |
by(simp add: is_filter_map_filter_on map_filter_on_def eventually_Abs_filter that) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1549 |
|
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1550 |
lemma map_filter_on_UNIV: "map_filter_on UNIV = filtermap" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1551 |
by(simp add: map_filter_on_def filtermap_def fun_eq_iff) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1552 |
|
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1553 |
lemma map_filter_on_comp: "map_filter_on X f (map_filter_on Y g F) = map_filter_on Y (f \<circ> g) F" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1554 |
if "g ` Y \<subseteq> X" and "eventually (\<lambda>x. x \<in> Y) F" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1555 |
unfolding map_filter_on_def using that(1) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1556 |
by(auto simp add: eventually_Abs_filter that(2) is_filter_map_filter_on intro!: arg_cong[where f=Abs_filter] arg_cong2[where f=eventually]) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1557 |
|
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1558 |
inductive rel_filter :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> 'b filter \<Rightarrow> bool" for R F G where |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1559 |
"rel_filter R F G" if "eventually (case_prod R) Z" "map_filter_on {(x, y). R x y} fst Z = F" "map_filter_on {(x, y). R x y} snd Z = G" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1560 |
|
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1561 |
lemma rel_filter_eq [relator_eq]: "rel_filter (=) = (=)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1562 |
proof(intro ext iffI)+ |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1563 |
show "F = G" if "rel_filter (=) F G" for F G using that |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1564 |
by cases(clarsimp simp add: filter_eq_iff eventually_map_filter_on split_def cong: rev_conj_cong) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1565 |
show "rel_filter (=) F G" if "F = G" for F G unfolding \<open>F = G\<close> |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1566 |
proof |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1567 |
let ?Z = "map_filter_on UNIV (\<lambda>x. (x, x)) G" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1568 |
have [simp]: "range (\<lambda>x. (x, x)) \<subseteq> {(x, y). x = y}" by auto |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1569 |
show "map_filter_on {(x, y). x = y} fst ?Z = G" and "map_filter_on {(x, y). x = y} snd ?Z = G" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1570 |
by(simp_all add: map_filter_on_comp)(simp_all add: map_filter_on_UNIV o_def) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1571 |
show "\<forall>\<^sub>F (x, y) in ?Z. x = y" by(simp add: eventually_map_filter_on) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1572 |
qed |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1573 |
qed |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1574 |
|
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1575 |
lemma rel_filter_mono [relator_mono]: "rel_filter A \<le> rel_filter B" if le: "A \<le> B" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1576 |
proof(clarify elim!: rel_filter.cases) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1577 |
show "rel_filter B (map_filter_on {(x, y). A x y} fst Z) (map_filter_on {(x, y). A x y} snd Z)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1578 |
(is "rel_filter _ ?X ?Y") if "\<forall>\<^sub>F (x, y) in Z. A x y" for Z |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1579 |
proof |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1580 |
let ?Z = "map_filter_on {(x, y). A x y} id Z" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1581 |
show "\<forall>\<^sub>F (x, y) in ?Z. B x y" using le that |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1582 |
by(simp add: eventually_map_filter_on le_fun_def split_def conj_commute cong: conj_cong) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1583 |
have [simp]: "{(x, y). A x y} \<subseteq> {(x, y). B x y}" using le by auto |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1584 |
show "map_filter_on {(x, y). B x y} fst ?Z = ?X" "map_filter_on {(x, y). B x y} snd ?Z = ?Y" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1585 |
using le that by(simp_all add: le_fun_def map_filter_on_comp) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1586 |
qed |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1587 |
qed |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1588 |
|
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1589 |
lemma rel_filter_conversep: "rel_filter A\<inverse>\<inverse> = (rel_filter A)\<inverse>\<inverse>" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1590 |
proof(safe intro!: ext elim!: rel_filter.cases) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1591 |
show *: "rel_filter A (map_filter_on {(x, y). A\<inverse>\<inverse> x y} snd Z) (map_filter_on {(x, y). A\<inverse>\<inverse> x y} fst Z)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1592 |
(is "rel_filter _ ?X ?Y") if "\<forall>\<^sub>F (x, y) in Z. A\<inverse>\<inverse> x y" for A Z |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1593 |
proof |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1594 |
let ?Z = "map_filter_on {(x, y). A y x} prod.swap Z" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1595 |
show "\<forall>\<^sub>F (x, y) in ?Z. A x y" using that by(simp add: eventually_map_filter_on) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1596 |
have [simp]: "prod.swap ` {(x, y). A y x} \<subseteq> {(x, y). A x y}" by auto |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1597 |
show "map_filter_on {(x, y). A x y} fst ?Z = ?X" "map_filter_on {(x, y). A x y} snd ?Z = ?Y" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1598 |
using that by(simp_all add: map_filter_on_comp o_def) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1599 |
qed |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1600 |
show "rel_filter A\<inverse>\<inverse> (map_filter_on {(x, y). A x y} snd Z) (map_filter_on {(x, y). A x y} fst Z)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1601 |
if "\<forall>\<^sub>F (x, y) in Z. A x y" for Z using *[of "A\<inverse>\<inverse>" Z] that by simp |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1602 |
qed |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1603 |
|
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1604 |
lemma rel_filter_distr [relator_distr]: |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1605 |
"rel_filter A OO rel_filter B = rel_filter (A OO B)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1606 |
proof(safe intro!: ext elim!: rel_filter.cases) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1607 |
let ?AB = "{(x, y). (A OO B) x y}" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1608 |
show "(rel_filter A OO rel_filter B) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1609 |
(map_filter_on {(x, y). (A OO B) x y} fst Z) (map_filter_on {(x, y). (A OO B) x y} snd Z)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1610 |
(is "(_ OO _) ?F ?H") if "\<forall>\<^sub>F (x, y) in Z. (A OO B) x y" for Z |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1611 |
proof |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1612 |
let ?G = "map_filter_on ?AB (\<lambda>(x, y). SOME z. A x z \<and> B z y) Z" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1613 |
show "rel_filter A ?F ?G" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1614 |
proof |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1615 |
let ?Z = "map_filter_on ?AB (\<lambda>(x, y). (x, SOME z. A x z \<and> B z y)) Z" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1616 |
show "\<forall>\<^sub>F (x, y) in ?Z. A x y" using that |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1617 |
by(auto simp add: eventually_map_filter_on split_def elim!: eventually_mono intro: someI2) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1618 |
have [simp]: "(\<lambda>p. (fst p, SOME z. A (fst p) z \<and> B z (snd p))) ` {p. (A OO B) (fst p) (snd p)} \<subseteq> {p. A (fst p) (snd p)}" by(auto intro: someI2) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1619 |
show "map_filter_on {(x, y). A x y} fst ?Z = ?F" "map_filter_on {(x, y). A x y} snd ?Z = ?G" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1620 |
using that by(simp_all add: map_filter_on_comp split_def o_def) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1621 |
qed |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1622 |
show "rel_filter B ?G ?H" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1623 |
proof |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1624 |
let ?Z = "map_filter_on ?AB (\<lambda>(x, y). (SOME z. A x z \<and> B z y, y)) Z" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1625 |
show "\<forall>\<^sub>F (x, y) in ?Z. B x y" using that |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1626 |
by(auto simp add: eventually_map_filter_on split_def elim!: eventually_mono intro: someI2) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1627 |
have [simp]: "(\<lambda>p. (SOME z. A (fst p) z \<and> B z (snd p), snd p)) ` {p. (A OO B) (fst p) (snd p)} \<subseteq> {p. B (fst p) (snd p)}" by(auto intro: someI2) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1628 |
show "map_filter_on {(x, y). B x y} fst ?Z = ?G" "map_filter_on {(x, y). B x y} snd ?Z = ?H" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1629 |
using that by(simp_all add: map_filter_on_comp split_def o_def) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1630 |
qed |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1631 |
qed |
60036 | 1632 |
|
1633 |
fix F G |
|
67616
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1634 |
assume F: "\<forall>\<^sub>F (x, y) in F. A x y" and G: "\<forall>\<^sub>F (x, y) in G. B x y" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1635 |
and eq: "map_filter_on {(x, y). B x y} fst G = map_filter_on {(x, y). A x y} snd F" (is "?Y2 = ?Y1") |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1636 |
let ?X = "map_filter_on {(x, y). A x y} fst F" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1637 |
and ?Z = "(map_filter_on {(x, y). B x y} snd G)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1638 |
have step: "\<exists>P'\<le>P. \<exists>Q' \<le> Q. eventually P' F \<and> eventually Q' G \<and> {y. \<exists>x. P' (x, y)} = {y. \<exists>z. Q' (y, z)}" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1639 |
if P: "eventually P F" and Q: "eventually Q G" for P Q |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1640 |
proof - |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1641 |
let ?P = "\<lambda>(x, y). P (x, y) \<and> A x y" and ?Q = "\<lambda>(y, z). Q (y, z) \<and> B y z" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1642 |
define P' where "P' \<equiv> \<lambda>(x, y). ?P (x, y) \<and> (\<exists>z. ?Q (y, z))" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1643 |
define Q' where "Q' \<equiv> \<lambda>(y, z). ?Q (y, z) \<and> (\<exists>x. ?P (x, y))" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1644 |
have "P' \<le> P" "Q' \<le> Q" "{y. \<exists>x. P' (x, y)} = {y. \<exists>z. Q' (y, z)}" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1645 |
by(auto simp add: P'_def Q'_def) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1646 |
moreover |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1647 |
from P Q F G have P': "eventually ?P F" and Q': "eventually ?Q G" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1648 |
by(simp_all add: eventually_conj_iff split_def) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1649 |
from P' F have "\<forall>\<^sub>F y in ?Y1. \<exists>x. P (x, y) \<and> A x y" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1650 |
by(auto simp add: eventually_map_filter_on elim!: eventually_mono) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1651 |
from this[folded eq] obtain Q'' where Q'': "eventually Q'' G" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1652 |
and Q''P: "{y. \<exists>z. Q'' (y, z)} \<subseteq> {y. \<exists>x. ?P (x, y)}" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1653 |
using G by(fastforce simp add: eventually_map_filter_on) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1654 |
have "eventually (inf Q'' ?Q) G" using Q'' Q' by(auto intro: eventually_conj simp add: inf_fun_def) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1655 |
then have "eventually Q' G" using Q''P by(auto elim!: eventually_mono simp add: Q'_def) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1656 |
moreover |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1657 |
from Q' G have "\<forall>\<^sub>F y in ?Y2. \<exists>z. Q (y, z) \<and> B y z" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1658 |
by(auto simp add: eventually_map_filter_on elim!: eventually_mono) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1659 |
from this[unfolded eq] obtain P'' where P'': "eventually P'' F" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1660 |
and P''Q: "{y. \<exists>x. P'' (x, y)} \<subseteq> {y. \<exists>z. ?Q (y, z)}" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1661 |
using F by(fastforce simp add: eventually_map_filter_on) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1662 |
have "eventually (inf P'' ?P) F" using P'' P' by(auto intro: eventually_conj simp add: inf_fun_def) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1663 |
then have "eventually P' F" using P''Q by(auto elim!: eventually_mono simp add: P'_def) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1664 |
ultimately show ?thesis by blast |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1665 |
qed |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1666 |
|
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1667 |
show "rel_filter (A OO B) ?X ?Z" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1668 |
proof |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1669 |
let ?Y = "\<lambda>Y. \<exists>X Z. eventually X ?X \<and> eventually Z ?Z \<and> (\<lambda>(x, z). X x \<and> Z z \<and> (A OO B) x z) \<le> Y" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1670 |
have Y: "is_filter ?Y" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1671 |
proof |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1672 |
show "?Y (\<lambda>_. True)" by(auto simp add: le_fun_def intro: eventually_True) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1673 |
show "?Y (\<lambda>x. P x \<and> Q x)" if "?Y P" "?Y Q" for P Q using that |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1674 |
apply clarify |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1675 |
apply(intro exI conjI; (elim eventually_rev_mp; fold imp_conjL; intro always_eventually allI; rule imp_refl)?) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1676 |
apply auto |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1677 |
done |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1678 |
show "?Y Q" if "?Y P" "\<forall>x. P x \<longrightarrow> Q x" for P Q using that by blast |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1679 |
qed |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1680 |
define Y where "Y = Abs_filter ?Y" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1681 |
have eventually_Y: "eventually P Y \<longleftrightarrow> ?Y P" for P |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1682 |
using eventually_Abs_filter[OF Y, of P] by(simp add: Y_def) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1683 |
show YY: "\<forall>\<^sub>F (x, y) in Y. (A OO B) x y" using F G |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1684 |
by(auto simp add: eventually_Y eventually_map_filter_on eventually_conj_iff intro!: eventually_True) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1685 |
have "?Y (\<lambda>(x, z). P x \<and> (A OO B) x z) \<longleftrightarrow> (\<forall>\<^sub>F (x, y) in F. P x \<and> A x y)" (is "?lhs = ?rhs") for P |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1686 |
proof |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1687 |
show ?lhs if ?rhs using G F that |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1688 |
by(auto 4 3 intro: exI[where x="\<lambda>_. True"] simp add: eventually_map_filter_on split_def) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1689 |
assume ?lhs |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1690 |
then obtain X Z where "\<forall>\<^sub>F (x, y) in F. X x \<and> A x y" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1691 |
and "\<forall>\<^sub>F (x, y) in G. Z y \<and> B x y" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1692 |
and "(\<lambda>(x, z). X x \<and> Z z \<and> (A OO B) x z) \<le> (\<lambda>(x, z). P x \<and> (A OO B) x z)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1693 |
using F G by(auto simp add: eventually_map_filter_on split_def) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1694 |
from step[OF this(1, 2)] this(3) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1695 |
show ?rhs by(clarsimp elim!: eventually_rev_mp simp add: le_fun_def)(fastforce intro: always_eventually) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1696 |
qed |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1697 |
then show "map_filter_on ?AB fst Y = ?X" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1698 |
by(simp add: filter_eq_iff YY eventually_map_filter_on)(simp add: eventually_Y eventually_map_filter_on F G; simp add: split_def) |
60036 | 1699 |
|
67616
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1700 |
have "?Y (\<lambda>(x, z). P z \<and> (A OO B) x z) \<longleftrightarrow> (\<forall>\<^sub>F (x, y) in G. P y \<and> B x y)" (is "?lhs = ?rhs") for P |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1701 |
proof |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1702 |
show ?lhs if ?rhs using G F that |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1703 |
by(auto 4 3 intro: exI[where x="\<lambda>_. True"] simp add: eventually_map_filter_on split_def) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1704 |
assume ?lhs |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1705 |
then obtain X Z where "\<forall>\<^sub>F (x, y) in F. X x \<and> A x y" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1706 |
and "\<forall>\<^sub>F (x, y) in G. Z y \<and> B x y" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1707 |
and "(\<lambda>(x, z). X x \<and> Z z \<and> (A OO B) x z) \<le> (\<lambda>(x, z). P z \<and> (A OO B) x z)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1708 |
using F G by(auto simp add: eventually_map_filter_on split_def) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1709 |
from step[OF this(1, 2)] this(3) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1710 |
show ?rhs by(clarsimp elim!: eventually_rev_mp simp add: le_fun_def)(fastforce intro: always_eventually) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1711 |
qed |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1712 |
then show "map_filter_on ?AB snd Y = ?Z" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1713 |
by(simp add: filter_eq_iff YY eventually_map_filter_on)(simp add: eventually_Y eventually_map_filter_on F G; simp add: split_def) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1714 |
qed |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1715 |
qed |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1716 |
|
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1717 |
lemma filtermap_parametric: "((A ===> B) ===> rel_filter A ===> rel_filter B) filtermap filtermap" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1718 |
proof(intro rel_funI; erule rel_filter.cases; hypsubst) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1719 |
fix f g Z |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1720 |
assume fg: "(A ===> B) f g" and Z: "\<forall>\<^sub>F (x, y) in Z. A x y" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1721 |
have "rel_filter B (map_filter_on {(x, y). A x y} (f \<circ> fst) Z) (map_filter_on {(x, y). A x y} (g \<circ> snd) Z)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1722 |
(is "rel_filter _ ?F ?G") |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1723 |
proof |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1724 |
let ?Z = "map_filter_on {(x, y). A x y} (map_prod f g) Z" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1725 |
show "\<forall>\<^sub>F (x, y) in ?Z. B x y" using fg Z |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1726 |
by(auto simp add: eventually_map_filter_on split_def elim!: eventually_mono rel_funD) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1727 |
have [simp]: "map_prod f g ` {p. A (fst p) (snd p)} \<subseteq> {p. B (fst p) (snd p)}" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1728 |
using fg by(auto dest: rel_funD) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1729 |
show "map_filter_on {(x, y). B x y} fst ?Z = ?F" "map_filter_on {(x, y). B x y} snd ?Z = ?G" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1730 |
using Z by(auto simp add: map_filter_on_comp split_def) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1731 |
qed |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1732 |
thus "rel_filter B (filtermap f (map_filter_on {(x, y). A x y} fst Z)) (filtermap g (map_filter_on {(x, y). A x y} snd Z))" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1733 |
using Z by(simp add: map_filter_on_UNIV[symmetric] map_filter_on_comp) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1734 |
qed |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1735 |
|
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1736 |
lemma rel_filter_Grp: "rel_filter (Grp UNIV f) = Grp UNIV (filtermap f)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1737 |
proof((intro antisym predicate2I; (elim GrpE; hypsubst)?), rule GrpI[OF _ UNIV_I]) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1738 |
fix F G |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1739 |
assume "rel_filter (Grp UNIV f) F G" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1740 |
hence "rel_filter (=) (filtermap f F) (filtermap id G)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1741 |
by(rule filtermap_parametric[THEN rel_funD, THEN rel_funD, rotated])(simp add: Grp_def rel_fun_def) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1742 |
thus "filtermap f F = G" by(simp add: rel_filter_eq) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1743 |
next |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1744 |
fix F :: "'a filter" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1745 |
have "rel_filter (=) F F" by(simp add: rel_filter_eq) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1746 |
hence "rel_filter (Grp UNIV f) (filtermap id F) (filtermap f F)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1747 |
by(rule filtermap_parametric[THEN rel_funD, THEN rel_funD, rotated])(simp add: Grp_def rel_fun_def) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1748 |
thus "rel_filter (Grp UNIV f) F (filtermap f F)" by simp |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1749 |
qed |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1750 |
|
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1751 |
lemma Quotient_filter [quot_map]: |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1752 |
"Quotient R Abs Rep T \<Longrightarrow> Quotient (rel_filter R) (filtermap Abs) (filtermap Rep) (rel_filter T)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1753 |
unfolding Quotient_alt_def5 rel_filter_eq[symmetric] rel_filter_Grp[symmetric] |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1754 |
by(simp add: rel_filter_distr[symmetric] rel_filter_conversep[symmetric] rel_filter_mono) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1755 |
|
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1756 |
lemma left_total_rel_filter [transfer_rule]: "left_total A \<Longrightarrow> left_total (rel_filter A)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1757 |
unfolding left_total_alt_def rel_filter_eq[symmetric] rel_filter_conversep[symmetric] rel_filter_distr |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1758 |
by(rule rel_filter_mono) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1759 |
|
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1760 |
lemma right_total_rel_filter [transfer_rule]: "right_total A \<Longrightarrow> right_total (rel_filter A)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1761 |
using left_total_rel_filter[of "A\<inverse>\<inverse>"] by(simp add: rel_filter_conversep) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1762 |
|
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1763 |
lemma bi_total_rel_filter [transfer_rule]: "bi_total A \<Longrightarrow> bi_total (rel_filter A)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1764 |
unfolding bi_total_alt_def by(simp add: left_total_rel_filter right_total_rel_filter) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1765 |
|
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1766 |
lemma left_unique_rel_filter [transfer_rule]: "left_unique A \<Longrightarrow> left_unique (rel_filter A)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1767 |
unfolding left_unique_alt_def rel_filter_eq[symmetric] rel_filter_conversep[symmetric] rel_filter_distr |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1768 |
by(rule rel_filter_mono) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1769 |
|
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1770 |
lemma right_unique_rel_filter [transfer_rule]: |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1771 |
"right_unique A \<Longrightarrow> right_unique (rel_filter A)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1772 |
using left_unique_rel_filter[of "A\<inverse>\<inverse>"] by(simp add: rel_filter_conversep) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1773 |
|
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1774 |
lemma bi_unique_rel_filter [transfer_rule]: "bi_unique A \<Longrightarrow> bi_unique (rel_filter A)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1775 |
by(simp add: bi_unique_alt_def left_unique_rel_filter right_unique_rel_filter) |
60036 | 1776 |
|
1777 |
lemma eventually_parametric [transfer_rule]: |
|
67399 | 1778 |
"((A ===> (=)) ===> rel_filter A ===> (=)) eventually eventually" |
67616
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1779 |
by(auto 4 4 intro!: rel_funI elim!: rel_filter.cases simp add: eventually_map_filter_on dest: rel_funD intro: always_eventually elim!: eventually_rev_mp) |
60036 | 1780 |
|
67616
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1781 |
lemma frequently_parametric [transfer_rule]: "((A ===> (=)) ===> rel_filter A ===> (=)) frequently frequently" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1782 |
unfolding frequently_def[abs_def] by transfer_prover |
60036 | 1783 |
|
1784 |
lemma is_filter_parametric [transfer_rule]: |
|
67956 | 1785 |
assumes [transfer_rule]: "bi_total A" |
1786 |
assumes [transfer_rule]: "bi_unique A" |
|
1787 |
shows "(((A ===> (=)) ===> (=)) ===> (=)) is_filter is_filter" |
|
1788 |
unfolding is_filter_def by transfer_prover |
|
60036 | 1789 |
|
67616
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1790 |
lemma top_filter_parametric [transfer_rule]: "rel_filter A top top" if "bi_total A" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1791 |
proof |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1792 |
let ?Z = "principal {(x, y). A x y}" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1793 |
show "\<forall>\<^sub>F (x, y) in ?Z. A x y" by(simp add: eventually_principal) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1794 |
show "map_filter_on {(x, y). A x y} fst ?Z = top" "map_filter_on {(x, y). A x y} snd ?Z = top" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1795 |
using that by(auto simp add: filter_eq_iff eventually_map_filter_on eventually_principal bi_total_def) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1796 |
qed |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1797 |
|
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1798 |
lemma bot_filter_parametric [transfer_rule]: "rel_filter A bot bot" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1799 |
proof |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1800 |
show "\<forall>\<^sub>F (x, y) in bot. A x y" by simp |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1801 |
show "map_filter_on {(x, y). A x y} fst bot = bot" "map_filter_on {(x, y). A x y} snd bot = bot" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1802 |
by(simp_all add: filter_eq_iff eventually_map_filter_on) |
60036 | 1803 |
qed |
1804 |
||
67616
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1805 |
lemma principal_parametric [transfer_rule]: "(rel_set A ===> rel_filter A) principal principal" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1806 |
proof(rule rel_funI rel_filter.intros)+ |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1807 |
fix S S' |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1808 |
assume *: "rel_set A S S'" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1809 |
define SS' where "SS' = S \<times> S' \<inter> {(x, y). A x y}" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1810 |
have SS': "SS' \<subseteq> {(x, y). A x y}" and [simp]: "S = fst ` SS'" "S' = snd ` SS'" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1811 |
using * by(auto 4 3 dest: rel_setD1 rel_setD2 intro: rev_image_eqI simp add: SS'_def) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1812 |
let ?Z = "principal SS'" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1813 |
show "\<forall>\<^sub>F (x, y) in ?Z. A x y" using SS' by(auto simp add: eventually_principal) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1814 |
then show "map_filter_on {(x, y). A x y} fst ?Z = principal S" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1815 |
and "map_filter_on {(x, y). A x y} snd ?Z = principal S'" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1816 |
by(auto simp add: filter_eq_iff eventually_map_filter_on eventually_principal) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1817 |
qed |
60036 | 1818 |
|
67616
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1819 |
lemma sup_filter_parametric [transfer_rule]: |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1820 |
"(rel_filter A ===> rel_filter A ===> rel_filter A) sup sup" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1821 |
proof(intro rel_funI; elim rel_filter.cases; hypsubst) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1822 |
show "rel_filter A |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1823 |
(map_filter_on {(x, y). A x y} fst FG \<squnion> map_filter_on {(x, y). A x y} fst FG') |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1824 |
(map_filter_on {(x, y). A x y} snd FG \<squnion> map_filter_on {(x, y). A x y} snd FG')" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1825 |
(is "rel_filter _ (sup ?F ?G) (sup ?F' ?G')") |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1826 |
if "\<forall>\<^sub>F (x, y) in FG. A x y" "\<forall>\<^sub>F (x, y) in FG'. A x y" for FG FG' |
60036 | 1827 |
proof |
67616
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1828 |
let ?Z = "sup FG FG'" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1829 |
show "\<forall>\<^sub>F (x, y) in ?Z. A x y" by(simp add: eventually_sup that) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1830 |
then show "map_filter_on {(x, y). A x y} fst ?Z = sup ?F ?G" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1831 |
and "map_filter_on {(x, y). A x y} snd ?Z = sup ?F' ?G'" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1832 |
by(simp_all add: filter_eq_iff eventually_map_filter_on eventually_sup) |
60036 | 1833 |
qed |
1834 |
qed |
|
1835 |
||
67616
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1836 |
lemma Sup_filter_parametric [transfer_rule]: "(rel_set (rel_filter A) ===> rel_filter A) Sup Sup" |
60036 | 1837 |
proof(rule rel_funI) |
1838 |
fix S S' |
|
67616
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1839 |
define SS' where "SS' = S \<times> S' \<inter> {(F, G). rel_filter A F G}" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1840 |
assume "rel_set (rel_filter A) S S'" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1841 |
then have SS': "SS' \<subseteq> {(F, G). rel_filter A F G}" and [simp]: "S = fst ` SS'" "S' = snd ` SS'" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1842 |
by(auto 4 3 dest: rel_setD1 rel_setD2 intro: rev_image_eqI simp add: SS'_def) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1843 |
from SS' obtain Z where Z: "\<And>F G. (F, G) \<in> SS' \<Longrightarrow> |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1844 |
(\<forall>\<^sub>F (x, y) in Z F G. A x y) \<and> |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1845 |
id F = map_filter_on {(x, y). A x y} fst (Z F G) \<and> |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1846 |
id G = map_filter_on {(x, y). A x y} snd (Z F G)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1847 |
unfolding rel_filter.simps by atomize_elim((rule choice allI)+; auto) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1848 |
have id: "eventually P F = eventually P (id F)" "eventually Q G = eventually Q (id G)" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1849 |
if "(F, G) \<in> SS'" for P Q F G by simp_all |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1850 |
show "rel_filter A (Sup S) (Sup S')" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1851 |
proof |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
68860
diff
changeset
|
1852 |
let ?Z = "\<Squnion>(F, G)\<in>SS'. Z F G" |
67616
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1853 |
show *: "\<forall>\<^sub>F (x, y) in ?Z. A x y" using Z by(auto simp add: eventually_Sup) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1854 |
show "map_filter_on {(x, y). A x y} fst ?Z = Sup S" "map_filter_on {(x, y). A x y} snd ?Z = Sup S'" |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1855 |
unfolding filter_eq_iff |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1856 |
by(auto 4 4 simp add: id eventually_Sup eventually_map_filter_on *[simplified eventually_Sup] simp del: id_apply dest: Z) |
1d005f514417
strengthen filter relator to canonical categorical definition with better properties
Andreas Lochbihler
parents:
67613
diff
changeset
|
1857 |
qed |
66162 | 1858 |
qed |
1859 |
||
60036 | 1860 |
context |
1861 |
fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" |
|
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
1862 |
assumes [transfer_rule]: "bi_unique A" |
60036 | 1863 |
begin |
1864 |
||
1865 |
lemma le_filter_parametric [transfer_rule]: |
|
67399 | 1866 |
"(rel_filter A ===> rel_filter A ===> (=)) (\<le>) (\<le>)" |
60036 | 1867 |
unfolding le_filter_def[abs_def] by transfer_prover |
1868 |
||
1869 |
lemma less_filter_parametric [transfer_rule]: |
|
67399 | 1870 |
"(rel_filter A ===> rel_filter A ===> (=)) (<) (<)" |
60036 | 1871 |
unfolding less_filter_def[abs_def] by transfer_prover |
1872 |
||
1873 |
context |
|
1874 |
assumes [transfer_rule]: "bi_total A" |
|
1875 |
begin |
|
1876 |
||
1877 |
lemma Inf_filter_parametric [transfer_rule]: |
|
1878 |
"(rel_set (rel_filter A) ===> rel_filter A) Inf Inf" |
|
1879 |
unfolding Inf_filter_def[abs_def] by transfer_prover |
|
1880 |
||
1881 |
lemma inf_filter_parametric [transfer_rule]: |
|
1882 |
"(rel_filter A ===> rel_filter A ===> rel_filter A) inf inf" |
|
1883 |
proof(intro rel_funI)+ |
|
1884 |
fix F F' G G' |
|
1885 |
assume [transfer_rule]: "rel_filter A F F'" "rel_filter A G G'" |
|
1886 |
have "rel_filter A (Inf {F, G}) (Inf {F', G'})" by transfer_prover |
|
1887 |
thus "rel_filter A (inf F G) (inf F' G')" by simp |
|
1888 |
qed |
|
1889 |
||
1890 |
end |
|
1891 |
||
1892 |
end |
|
1893 |
||
1894 |
end |
|
1895 |
||
70927 | 1896 |
context |
1897 |
includes lifting_syntax |
|
1898 |
begin |
|
1899 |
||
1900 |
lemma prod_filter_parametric [transfer_rule]: |
|
68667 | 1901 |
"(rel_filter R ===> rel_filter S ===> rel_filter (rel_prod R S)) prod_filter prod_filter" |
1902 |
proof(intro rel_funI; elim rel_filter.cases; hypsubst) |
|
1903 |
fix F G |
|
1904 |
assume F: "\<forall>\<^sub>F (x, y) in F. R x y" and G: "\<forall>\<^sub>F (x, y) in G. S x y" |
|
1905 |
show "rel_filter (rel_prod R S) |
|
1906 |
(map_filter_on {(x, y). R x y} fst F \<times>\<^sub>F map_filter_on {(x, y). S x y} fst G) |
|
1907 |
(map_filter_on {(x, y). R x y} snd F \<times>\<^sub>F map_filter_on {(x, y). S x y} snd G)" |
|
1908 |
(is "rel_filter ?RS ?F ?G") |
|
1909 |
proof |
|
1910 |
let ?Z = "filtermap (\<lambda>((a, b), (a', b')). ((a, a'), (b, b'))) (prod_filter F G)" |
|
1911 |
show *: "\<forall>\<^sub>F (x, y) in ?Z. rel_prod R S x y" using F G |
|
1912 |
by(auto simp add: eventually_filtermap split_beta eventually_prod_filter) |
|
1913 |
show "map_filter_on {(x, y). ?RS x y} fst ?Z = ?F" |
|
1914 |
using F G |
|
1915 |
apply(clarsimp simp add: filter_eq_iff eventually_map_filter_on *) |
|
1916 |
apply(simp add: eventually_filtermap split_beta eventually_prod_filter) |
|
1917 |
apply(subst eventually_map_filter_on; simp)+ |
|
1918 |
apply(rule iffI; clarsimp) |
|
1919 |
subgoal for P P' P'' |
|
1920 |
apply(rule exI[where x="\<lambda>a. \<exists>b. P' (a, b) \<and> R a b"]; rule conjI) |
|
1921 |
subgoal by(fastforce elim: eventually_rev_mp eventually_mono) |
|
1922 |
subgoal |
|
1923 |
by(rule exI[where x="\<lambda>a. \<exists>b. P'' (a, b) \<and> S a b"])(fastforce elim: eventually_rev_mp eventually_mono) |
|
1924 |
done |
|
1925 |
subgoal by fastforce |
|
1926 |
done |
|
1927 |
show "map_filter_on {(x, y). ?RS x y} snd ?Z = ?G" |
|
1928 |
using F G |
|
1929 |
apply(clarsimp simp add: filter_eq_iff eventually_map_filter_on *) |
|
1930 |
apply(simp add: eventually_filtermap split_beta eventually_prod_filter) |
|
1931 |
apply(subst eventually_map_filter_on; simp)+ |
|
1932 |
apply(rule iffI; clarsimp) |
|
1933 |
subgoal for P P' P'' |
|
1934 |
apply(rule exI[where x="\<lambda>b. \<exists>a. P' (a, b) \<and> R a b"]; rule conjI) |
|
1935 |
subgoal by(fastforce elim: eventually_rev_mp eventually_mono) |
|
1936 |
subgoal |
|
1937 |
by(rule exI[where x="\<lambda>b. \<exists>a. P'' (a, b) \<and> S a b"])(fastforce elim: eventually_rev_mp eventually_mono) |
|
1938 |
done |
|
1939 |
subgoal by fastforce |
|
1940 |
done |
|
1941 |
qed |
|
1942 |
qed |
|
1943 |
||
70927 | 1944 |
end |
1945 |
||
1946 |
||
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1947 |
text \<open>Code generation for filters\<close> |
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1948 |
|
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1949 |
definition abstract_filter :: "(unit \<Rightarrow> 'a filter) \<Rightarrow> 'a filter" |
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1950 |
where [simp]: "abstract_filter f = f ()" |
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1951 |
|
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1952 |
code_datatype principal abstract_filter |
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1953 |
|
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1954 |
hide_const (open) abstract_filter |
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1955 |
|
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1956 |
declare [[code drop: filterlim prod_filter filtermap eventually |
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1957 |
"inf :: _ filter \<Rightarrow> _" "sup :: _ filter \<Rightarrow> _" "less_eq :: _ filter \<Rightarrow> _" |
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1958 |
Abs_filter]] |
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1959 |
|
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1960 |
declare filterlim_principal [code] |
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1961 |
declare principal_prod_principal [code] |
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1962 |
declare filtermap_principal [code] |
66162 | 1963 |
declare filtercomap_principal [code] |
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1964 |
declare eventually_principal [code] |
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1965 |
declare inf_principal [code] |
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1966 |
declare sup_principal [code] |
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1967 |
declare principal_le_iff [code] |
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1968 |
|
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1969 |
lemma Rep_filter_iff_eventually [simp, code]: |
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1970 |
"Rep_filter F P \<longleftrightarrow> eventually P F" |
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1971 |
by (simp add: eventually_def) |
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1972 |
|
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1973 |
lemma bot_eq_principal_empty [code]: |
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1974 |
"bot = principal {}" |
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1975 |
by simp |
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1976 |
|
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1977 |
lemma top_eq_principal_UNIV [code]: |
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1978 |
"top = principal UNIV" |
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1979 |
by simp |
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1980 |
|
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1981 |
instantiation filter :: (equal) equal |
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1982 |
begin |
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1983 |
|
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1984 |
definition equal_filter :: "'a filter \<Rightarrow> 'a filter \<Rightarrow> bool" |
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1985 |
where "equal_filter F F' \<longleftrightarrow> F = F'" |
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1986 |
|
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1987 |
lemma equal_filter [code]: |
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1988 |
"HOL.equal (principal A) (principal B) \<longleftrightarrow> A = B" |
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|
1989 |
by (simp add: equal_filter_def) |
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1990 |
|
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1991 |
instance |
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1992 |
by standard (simp add: equal_filter_def) |
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|
1993 |
|
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|
1994 |
end |
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1995 |
|
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|
1996 |
end |