(* Title: HOL/Filter.thy
Author: Brian Huffman
Author: Johannes Hölzl
*)
section \<open>Filters on predicates\<close>
theory Filter
imports Set_Interval Lifting_Set
begin
subsection \<open>Filters\<close>
text \<open>
This definition also allows non-proper filters.
\<close>
locale is_filter =
fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
assumes True: "F (\<lambda>x. True)"
assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)"
assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)"
typedef 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}"
proof
show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro)
qed
lemma is_filter_Rep_filter: "is_filter (Rep_filter F)"
using Rep_filter [of F] by simp
lemma Abs_filter_inverse':
assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F"
using assms by (simp add: Abs_filter_inverse)
subsubsection \<open>Eventually\<close>
definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
where "eventually P F \<longleftrightarrow> Rep_filter F P"
syntax
"_eventually" :: "pttrn => 'a filter => bool => bool" ("(3\<forall>\<^sub>F _ in _./ _)" [0, 0, 10] 10)
translations
"\<forall>\<^sub>Fx in F. P" == "CONST eventually (\<lambda>x. P) F"
lemma eventually_Abs_filter:
assumes "is_filter F" shows "eventually P (Abs_filter F) = F P"
unfolding eventually_def using assms by (simp add: Abs_filter_inverse)
lemma filter_eq_iff:
shows "F = F' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')"
unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..
lemma eventually_True [simp]: "eventually (\<lambda>x. True) F"
unfolding eventually_def
by (rule is_filter.True [OF is_filter_Rep_filter])
lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P F"
proof -
assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
thus "eventually P F" by simp
qed
lemma eventuallyI: "(\<And>x. P x) \<Longrightarrow> eventually P F"
by (auto intro: always_eventually)
lemma eventually_mono:
"\<lbrakk>eventually P F; \<And>x. P x \<Longrightarrow> Q x\<rbrakk> \<Longrightarrow> eventually Q F"
unfolding eventually_def
by (blast intro: is_filter.mono [OF is_filter_Rep_filter])
lemma eventually_conj:
assumes P: "eventually (\<lambda>x. P x) F"
assumes Q: "eventually (\<lambda>x. Q x) F"
shows "eventually (\<lambda>x. P x \<and> Q x) F"
using assms unfolding eventually_def
by (rule is_filter.conj [OF is_filter_Rep_filter])
lemma eventually_mp:
assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
assumes "eventually (\<lambda>x. P x) F"
shows "eventually (\<lambda>x. Q x) F"
proof -
have "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F"
using assms by (rule eventually_conj)
then show ?thesis
by (blast intro: eventually_mono)
qed
lemma eventually_rev_mp:
assumes "eventually (\<lambda>x. P x) F"
assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
shows "eventually (\<lambda>x. Q x) F"
using assms(2) assms(1) by (rule eventually_mp)
lemma eventually_conj_iff:
"eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F"
by (auto intro: eventually_conj elim: eventually_rev_mp)
lemma eventually_elim2:
assumes "eventually (\<lambda>i. P i) F"
assumes "eventually (\<lambda>i. Q i) F"
assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
shows "eventually (\<lambda>i. R i) F"
using assms by (auto elim!: eventually_rev_mp)
lemma eventually_ball_finite_distrib:
"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)"
by (induction A rule: finite_induct) (auto simp: eventually_conj_iff)
lemma eventually_ball_finite:
"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"
by (auto simp: eventually_ball_finite_distrib)
lemma eventually_all_finite:
fixes P :: "'a \<Rightarrow> 'b::finite \<Rightarrow> bool"
assumes "\<And>y. eventually (\<lambda>x. P x y) net"
shows "eventually (\<lambda>x. \<forall>y. P x y) net"
using eventually_ball_finite [of UNIV P] assms by simp
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))"
proof
assume "\<forall>\<^sub>Fx in F. \<exists>y. P x y"
then have "\<forall>\<^sub>Fx in F. P x (SOME y. P x y)"
by (auto intro: someI_ex eventually_mono)
then show "\<exists>Y. \<forall>\<^sub>Fx in F. P x (Y x)"
by auto
qed (auto intro: eventually_mono)
lemma not_eventually_impI: "eventually P F \<Longrightarrow> \<not> eventually Q F \<Longrightarrow> \<not> eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
by (auto intro: eventually_mp)
lemma not_eventuallyD: "\<not> eventually P F \<Longrightarrow> \<exists>x. \<not> P x"
by (metis always_eventually)
lemma eventually_subst:
assumes "eventually (\<lambda>n. P n = Q n) F"
shows "eventually P F = eventually Q F" (is "?L = ?R")
proof -
from assms have "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
and "eventually (\<lambda>x. Q x \<longrightarrow> P x) F"
by (auto elim: eventually_mono)
then show ?thesis by (auto elim: eventually_elim2)
qed
subsection \<open> Frequently as dual to eventually \<close>
definition frequently :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
where "frequently P F \<longleftrightarrow> \<not> eventually (\<lambda>x. \<not> P x) F"
syntax
"_frequently" :: "pttrn \<Rightarrow> 'a filter \<Rightarrow> bool \<Rightarrow> bool" ("(3\<exists>\<^sub>F _ in _./ _)" [0, 0, 10] 10)
translations
"\<exists>\<^sub>Fx in F. P" == "CONST frequently (\<lambda>x. P) F"
lemma not_frequently_False [simp]: "\<not> (\<exists>\<^sub>Fx in F. False)"
by (simp add: frequently_def)
lemma frequently_ex: "\<exists>\<^sub>Fx in F. P x \<Longrightarrow> \<exists>x. P x"
by (auto simp: frequently_def dest: not_eventuallyD)
lemma frequentlyE: assumes "frequently P F" obtains x where "P x"
using frequently_ex[OF assms] by auto
lemma frequently_mp:
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"
proof -
from ev have "eventually (\<lambda>x. \<not> Q x \<longrightarrow> \<not> P x) F"
by (rule eventually_rev_mp) (auto intro!: always_eventually)
from eventually_mp[OF this] P show ?thesis
by (auto simp: frequently_def)
qed
lemma frequently_rev_mp:
assumes "\<exists>\<^sub>Fx in F. P x"
assumes "\<forall>\<^sub>Fx in F. P x \<longrightarrow> Q x"
shows "\<exists>\<^sub>Fx in F. Q x"
using assms(2) assms(1) by (rule frequently_mp)
lemma frequently_mono: "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> frequently P F \<Longrightarrow> frequently Q F"
using frequently_mp[of P Q] by (simp add: always_eventually)
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"
by (metis frequently_mono)
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)"
by (simp add: frequently_def eventually_conj_iff)
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"
by (simp add: frequently_disj_iff)
lemma frequently_bex_finite_distrib:
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)"
using assms by induction (auto simp: frequently_disj_iff)
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"
by (simp add: frequently_bex_finite_distrib)
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))"
using eventually_ex[of "\<lambda>x y. \<not> P x y" F] by (simp add: frequently_def)
lemma
shows not_eventually: "\<not> eventually P F \<longleftrightarrow> (\<exists>\<^sub>Fx in F. \<not> P x)"
and not_frequently: "\<not> frequently P F \<longleftrightarrow> (\<forall>\<^sub>Fx in F. \<not> P x)"
by (auto simp: frequently_def)
lemma frequently_imp_iff:
"(\<exists>\<^sub>Fx in F. P x \<longrightarrow> Q x) \<longleftrightarrow> (eventually P F \<longrightarrow> frequently Q F)"
unfolding imp_conv_disj frequently_disj_iff not_eventually[symmetric] ..
lemma eventually_frequently_const_simps:
"(\<exists>\<^sub>Fx in F. P x \<and> C) \<longleftrightarrow> (\<exists>\<^sub>Fx in F. P x) \<and> C"
"(\<exists>\<^sub>Fx in F. C \<and> P x) \<longleftrightarrow> C \<and> (\<exists>\<^sub>Fx in F. P x)"
"(\<forall>\<^sub>Fx in F. P x \<or> C) \<longleftrightarrow> (\<forall>\<^sub>Fx in F. P x) \<or> C"
"(\<forall>\<^sub>Fx in F. C \<or> P x) \<longleftrightarrow> C \<or> (\<forall>\<^sub>Fx in F. P x)"
"(\<forall>\<^sub>Fx in F. P x \<longrightarrow> C) \<longleftrightarrow> ((\<exists>\<^sub>Fx in F. P x) \<longrightarrow> C)"
"(\<forall>\<^sub>Fx in F. C \<longrightarrow> P x) \<longleftrightarrow> (C \<longrightarrow> (\<forall>\<^sub>Fx in F. P x))"
by (cases C; simp add: not_frequently)+
lemmas eventually_frequently_simps =
eventually_frequently_const_simps
not_eventually
eventually_conj_iff
eventually_ball_finite_distrib
eventually_ex
not_frequently
frequently_disj_iff
frequently_bex_finite_distrib
frequently_all
frequently_imp_iff
ML \<open>
fun eventually_elim_tac facts =
CONTEXT_SUBGOAL (fn (goal, i) => fn (ctxt, st) =>
let
val mp_thms = facts RL @{thms eventually_rev_mp}
val raw_elim_thm =
(@{thm allI} RS @{thm always_eventually})
|> fold (fn thm1 => fn thm2 => thm2 RS thm1) mp_thms
|> fold (fn _ => fn thm => @{thm impI} RS thm) facts
val cases_prop =
Thm.prop_of
(Rule_Cases.internalize_params (raw_elim_thm RS Goal.init (Thm.cterm_of ctxt goal)))
val cases = Rule_Cases.make_common ctxt cases_prop [(("elim", []), [])]
in CONTEXT_CASES cases (resolve_tac ctxt [raw_elim_thm] i) (ctxt, st) end)
\<close>
method_setup eventually_elim = \<open>
Scan.succeed (fn _ => CONTEXT_METHOD (fn facts => eventually_elim_tac facts 1))
\<close> "elimination of eventually quantifiers"
subsubsection \<open>Finer-than relation\<close>
text \<open>@{term "F \<le> F'"} means that filter @{term F} is finer than
filter @{term F'}.\<close>
instantiation filter :: (type) complete_lattice
begin
definition le_filter_def:
"F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"
definition
"(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
definition
"top = Abs_filter (\<lambda>P. \<forall>x. P x)"
definition
"bot = Abs_filter (\<lambda>P. True)"
definition
"sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"
definition
"inf F F' = Abs_filter
(\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
definition
"Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)"
definition
"Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}"
lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
unfolding top_filter_def
by (rule eventually_Abs_filter, rule is_filter.intro, auto)
lemma eventually_bot [simp]: "eventually P bot"
unfolding bot_filter_def
by (subst eventually_Abs_filter, rule is_filter.intro, auto)
lemma eventually_sup:
"eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'"
unfolding sup_filter_def
by (rule eventually_Abs_filter, rule is_filter.intro)
(auto elim!: eventually_rev_mp)
lemma eventually_inf:
"eventually P (inf F F') \<longleftrightarrow>
(\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
unfolding inf_filter_def
apply (rule eventually_Abs_filter, rule is_filter.intro)
apply (fast intro: eventually_True)
apply clarify
apply (intro exI conjI)
apply (erule (1) eventually_conj)
apply (erule (1) eventually_conj)
apply simp
apply auto
done
lemma eventually_Sup:
"eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)"
unfolding Sup_filter_def
apply (rule eventually_Abs_filter, rule is_filter.intro)
apply (auto intro: eventually_conj elim!: eventually_rev_mp)
done
instance proof
fix F F' F'' :: "'a filter" and S :: "'a filter set"
{ show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
by (rule less_filter_def) }
{ show "F \<le> F"
unfolding le_filter_def by simp }
{ assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"
unfolding le_filter_def by simp }
{ assume "F \<le> F'" and "F' \<le> F" thus "F = F'"
unfolding le_filter_def filter_eq_iff by fast }
{ show "inf F F' \<le> F" and "inf F F' \<le> F'"
unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
{ assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"
unfolding le_filter_def eventually_inf
by (auto intro: eventually_mono [OF eventually_conj]) }
{ show "F \<le> sup F F'" and "F' \<le> sup F F'"
unfolding le_filter_def eventually_sup by simp_all }
{ assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"
unfolding le_filter_def eventually_sup by simp }
{ assume "F'' \<in> S" thus "Inf S \<le> F''"
unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
{ assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"
unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
{ assume "F \<in> S" thus "F \<le> Sup S"
unfolding le_filter_def eventually_Sup by simp }
{ assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'"
unfolding le_filter_def eventually_Sup by simp }
{ show "Inf {} = (top::'a filter)"
by (auto simp: top_filter_def Inf_filter_def Sup_filter_def)
(metis (full_types) top_filter_def always_eventually eventually_top) }
{ show "Sup {} = (bot::'a filter)"
by (auto simp: bot_filter_def Sup_filter_def) }
qed
end
lemma filter_leD:
"F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"
unfolding le_filter_def by simp
lemma filter_leI:
"(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"
unfolding le_filter_def by simp
lemma eventually_False:
"eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot"
unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
lemma eventually_frequently: "F \<noteq> bot \<Longrightarrow> eventually P F \<Longrightarrow> frequently P F"
using eventually_conj[of P F "\<lambda>x. \<not> P x"]
by (auto simp add: frequently_def eventually_False)
lemma eventually_const_iff: "eventually (\<lambda>x. P) F \<longleftrightarrow> P \<or> F = bot"
by (cases P) (auto simp: eventually_False)
lemma eventually_const[simp]: "F \<noteq> bot \<Longrightarrow> eventually (\<lambda>x. P) F \<longleftrightarrow> P"
by (simp add: eventually_const_iff)
lemma frequently_const_iff: "frequently (\<lambda>x. P) F \<longleftrightarrow> P \<and> F \<noteq> bot"
by (simp add: frequently_def eventually_const_iff)
lemma frequently_const[simp]: "F \<noteq> bot \<Longrightarrow> frequently (\<lambda>x. P) F \<longleftrightarrow> P"
by (simp add: frequently_const_iff)
lemma eventually_happens: "eventually P net \<Longrightarrow> net = bot \<or> (\<exists>x. P x)"
by (metis frequentlyE eventually_frequently)
lemma eventually_happens':
assumes "F \<noteq> bot" "eventually P F"
shows "\<exists>x. P x"
using assms eventually_frequently frequentlyE by blast
abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"
where "trivial_limit F \<equiv> F = bot"
lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"
by (rule eventually_False [symmetric])
lemma False_imp_not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> \<not> trivial_limit net \<Longrightarrow> \<not> eventually (\<lambda>x. P x) net"
by (simp add: eventually_False)
lemma eventually_Inf: "eventually P (Inf B) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X))"
proof -
let ?F = "\<lambda>P. \<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X)"
{ fix P have "eventually P (Abs_filter ?F) \<longleftrightarrow> ?F P"
proof (rule eventually_Abs_filter is_filter.intro)+
show "?F (\<lambda>x. True)"
by (rule exI[of _ "{}"]) (simp add: le_fun_def)
next
fix P Q
assume "?F P" then guess X ..
moreover
assume "?F Q" then guess Y ..
ultimately show "?F (\<lambda>x. P x \<and> Q x)"
by (intro exI[of _ "X \<union> Y"])
(auto simp: Inf_union_distrib eventually_inf)
next
fix P Q
assume "?F P" then guess X ..
moreover assume "\<forall>x. P x \<longrightarrow> Q x"
ultimately show "?F Q"
by (intro exI[of _ X]) (auto elim: eventually_mono)
qed }
note eventually_F = this
have "Inf B = Abs_filter ?F"
proof (intro antisym Inf_greatest)
show "Inf B \<le> Abs_filter ?F"
by (auto simp: le_filter_def eventually_F dest: Inf_superset_mono)
next
fix F assume "F \<in> B" then show "Abs_filter ?F \<le> F"
by (auto simp add: le_filter_def eventually_F intro!: exI[of _ "{F}"])
qed
then show ?thesis
by (simp add: eventually_F)
qed
lemma eventually_INF: "eventually P (INF b:B. F b) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (INF b:X. F b))"
unfolding eventually_Inf [of P "F`B"]
by (metis finite_imageI image_mono finite_subset_image)
lemma Inf_filter_not_bot:
fixes B :: "'a filter set"
shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> Inf X \<noteq> bot) \<Longrightarrow> Inf B \<noteq> bot"
unfolding trivial_limit_def eventually_Inf[of _ B]
bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
lemma INF_filter_not_bot:
fixes F :: "'i \<Rightarrow> 'a filter"
shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> (INF b:X. F b) \<noteq> bot) \<Longrightarrow> (INF b:B. F b) \<noteq> bot"
unfolding trivial_limit_def eventually_INF [of _ _ B]
bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
lemma eventually_Inf_base:
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"
shows "eventually P (Inf B) \<longleftrightarrow> (\<exists>b\<in>B. eventually P b)"
proof (subst eventually_Inf, safe)
fix X assume "finite X" "X \<subseteq> B"
then have "\<exists>b\<in>B. \<forall>x\<in>X. b \<le> x"
proof induct
case empty then show ?case
using \<open>B \<noteq> {}\<close> by auto
next
case (insert x X)
then obtain b where "b \<in> B" "\<And>x. x \<in> X \<Longrightarrow> b \<le> x"
by auto
with \<open>insert x X \<subseteq> B\<close> base[of b x] show ?case
by (auto intro: order_trans)
qed
then obtain b where "b \<in> B" "b \<le> Inf X"
by (auto simp: le_Inf_iff)
then show "eventually P (Inf X) \<Longrightarrow> Bex B (eventually P)"
by (intro bexI[of _ b]) (auto simp: le_filter_def)
qed (auto intro!: exI[of _ "{x}" for x])
lemma eventually_INF_base:
"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>
eventually P (INF b:B. F b) \<longleftrightarrow> (\<exists>b\<in>B. eventually P (F b))"
by (subst eventually_Inf_base) auto
lemma eventually_INF1: "i \<in> I \<Longrightarrow> eventually P (F i) \<Longrightarrow> eventually P (INF i:I. F i)"
using filter_leD[OF INF_lower] .
lemma eventually_INF_mono:
assumes *: "\<forall>\<^sub>F x in \<Sqinter>i\<in>I. F i. P x"
assumes T1: "\<And>Q R P. (\<And>x. Q x \<and> R x \<longrightarrow> P x) \<Longrightarrow> (\<And>x. T Q x \<Longrightarrow> T R x \<Longrightarrow> T P x)"
assumes T2: "\<And>P. (\<And>x. P x) \<Longrightarrow> (\<And>x. T P x)"
assumes **: "\<And>i P. i \<in> I \<Longrightarrow> \<forall>\<^sub>F x in F i. P x \<Longrightarrow> \<forall>\<^sub>F x in F' i. T P x"
shows "\<forall>\<^sub>F x in \<Sqinter>i\<in>I. F' i. T P x"
proof -
from * obtain X where X: "finite X" "X \<subseteq> I" "\<forall>\<^sub>F x in \<Sqinter>i\<in>X. F i. P x"
unfolding eventually_INF[of _ _ I] by auto
then have "eventually (T P) (INFIMUM X F')"
apply (induction X arbitrary: P)
apply (auto simp: eventually_inf T2)
subgoal for x S P Q R
apply (intro exI[of _ "T Q"])
apply (auto intro!: **) []
apply (intro exI[of _ "T R"])
apply (auto intro: T1) []
done
done
with X show "\<forall>\<^sub>F x in \<Sqinter>i\<in>I. F' i. T P x"
by (subst eventually_INF) auto
qed
subsubsection \<open>Map function for filters\<close>
definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
lemma eventually_filtermap:
"eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
unfolding filtermap_def
apply (rule eventually_Abs_filter)
apply (rule is_filter.intro)
apply (auto elim!: eventually_rev_mp)
done
lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
by (simp add: filter_eq_iff eventually_filtermap)
lemma filtermap_filtermap:
"filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
by (simp add: filter_eq_iff eventually_filtermap)
lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"
unfolding le_filter_def eventually_filtermap by simp
lemma filtermap_bot [simp]: "filtermap f bot = bot"
by (simp add: filter_eq_iff eventually_filtermap)
lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)"
by (auto simp: filter_eq_iff eventually_filtermap eventually_sup)
lemma filtermap_inf: "filtermap f (inf F1 F2) \<le> inf (filtermap f F1) (filtermap f F2)"
by (auto simp: le_filter_def eventually_filtermap eventually_inf)
lemma filtermap_INF: "filtermap f (INF b:B. F b) \<le> (INF b:B. filtermap f (F b))"
proof -
{ fix X :: "'c set" assume "finite X"
then have "filtermap f (INFIMUM X F) \<le> (INF b:X. filtermap f (F b))"
proof induct
case (insert x X)
have "filtermap f (INF a:insert x X. F a) \<le> inf (filtermap f (F x)) (filtermap f (INF a:X. F a))"
by (rule order_trans[OF _ filtermap_inf]) simp
also have "\<dots> \<le> inf (filtermap f (F x)) (INF a:X. filtermap f (F a))"
by (intro inf_mono insert order_refl)
finally show ?case
by simp
qed simp }
then show ?thesis
unfolding le_filter_def eventually_filtermap
by (subst (1 2) eventually_INF) auto
qed
subsubsection \<open>Standard filters\<close>
definition principal :: "'a set \<Rightarrow> 'a filter" where
"principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)"
lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)"
unfolding principal_def
by (rule eventually_Abs_filter, rule is_filter.intro) auto
lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F"
unfolding eventually_inf eventually_principal by (auto elim: eventually_mono)
lemma principal_UNIV[simp]: "principal UNIV = top"
by (auto simp: filter_eq_iff eventually_principal)
lemma principal_empty[simp]: "principal {} = bot"
by (auto simp: filter_eq_iff eventually_principal)
lemma principal_eq_bot_iff: "principal X = bot \<longleftrightarrow> X = {}"
by (auto simp add: filter_eq_iff eventually_principal)
lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B"
by (auto simp: le_filter_def eventually_principal)
lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F"
unfolding le_filter_def eventually_principal
apply safe
apply (erule_tac x="\<lambda>x. x \<in> A" in allE)
apply (auto elim: eventually_mono)
done
lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B"
unfolding eq_iff by simp
lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)"
unfolding filter_eq_iff eventually_sup eventually_principal by auto
lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)"
unfolding filter_eq_iff eventually_inf eventually_principal
by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"])
lemma SUP_principal[simp]: "(SUP i : I. principal (A i)) = principal (\<Union>i\<in>I. A i)"
unfolding filter_eq_iff eventually_Sup by (auto simp: eventually_principal)
lemma INF_principal_finite: "finite X \<Longrightarrow> (INF x:X. principal (f x)) = principal (\<Inter>x\<in>X. f x)"
by (induct X rule: finite_induct) auto
lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)"
unfolding filter_eq_iff eventually_filtermap eventually_principal by simp
subsubsection \<open>Order filters\<close>
definition at_top :: "('a::order) filter"
where "at_top = (INF k. principal {k ..})"
lemma at_top_sub: "at_top = (INF k:{c::'a::linorder..}. principal {k ..})"
by (auto intro!: INF_eq max.cobounded1 max.cobounded2 simp: at_top_def)
lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)"
unfolding at_top_def
by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
lemma eventually_at_top_linorderI:
fixes c::"'a::linorder"
assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
shows "eventually P at_top"
using assms by (auto simp: eventually_at_top_linorder)
lemma eventually_ge_at_top:
"eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top"
unfolding eventually_at_top_linorder by auto
lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::{no_top, linorder}. \<forall>n>N. P n)"
proof -
have "eventually P (INF k. principal {k <..}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n>N. P n)"
by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
also have "(INF k. principal {k::'a <..}) = at_top"
unfolding at_top_def
by (intro INF_eq) (auto intro: less_imp_le simp: Ici_subset_Ioi_iff gt_ex)
finally show ?thesis .
qed
lemma eventually_at_top_not_equal: "eventually (\<lambda>x::'a::{no_top, linorder}. x \<noteq> c) at_top"
unfolding eventually_at_top_dense by auto
lemma eventually_gt_at_top: "eventually (\<lambda>x. (c::_::{no_top, linorder}) < x) at_top"
unfolding eventually_at_top_dense by auto
lemma eventually_all_ge_at_top:
assumes "eventually P (at_top :: ('a :: linorder) filter)"
shows "eventually (\<lambda>x. \<forall>y\<ge>x. P y) at_top"
proof -
from assms obtain x where "\<And>y. y \<ge> x \<Longrightarrow> P y" by (auto simp: eventually_at_top_linorder)
hence "\<forall>z\<ge>y. P z" if "y \<ge> x" for y using that by simp
thus ?thesis by (auto simp: eventually_at_top_linorder)
qed
definition at_bot :: "('a::order) filter"
where "at_bot = (INF k. principal {.. k})"
lemma at_bot_sub: "at_bot = (INF k:{.. c::'a::linorder}. principal {.. k})"
by (auto intro!: INF_eq min.cobounded1 min.cobounded2 simp: at_bot_def)
lemma eventually_at_bot_linorder:
fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"
unfolding at_bot_def
by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
lemma eventually_le_at_bot:
"eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot"
unfolding eventually_at_bot_linorder by auto
lemma eventually_at_bot_dense: "eventually P at_bot \<longleftrightarrow> (\<exists>N::'a::{no_bot, linorder}. \<forall>n<N. P n)"
proof -
have "eventually P (INF k. principal {..< k}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n<N. P n)"
by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
also have "(INF k. principal {..< k::'a}) = at_bot"
unfolding at_bot_def
by (intro INF_eq) (auto intro: less_imp_le simp: Iic_subset_Iio_iff lt_ex)
finally show ?thesis .
qed
lemma eventually_at_bot_not_equal: "eventually (\<lambda>x::'a::{no_bot, linorder}. x \<noteq> c) at_bot"
unfolding eventually_at_bot_dense by auto
lemma eventually_gt_at_bot:
"eventually (\<lambda>x. x < (c::_::unbounded_dense_linorder)) at_bot"
unfolding eventually_at_bot_dense by auto
lemma trivial_limit_at_bot_linorder [simp]: "\<not> trivial_limit (at_bot ::('a::linorder) filter)"
unfolding trivial_limit_def
by (metis eventually_at_bot_linorder order_refl)
lemma trivial_limit_at_top_linorder [simp]: "\<not> trivial_limit (at_top ::('a::linorder) filter)"
unfolding trivial_limit_def
by (metis eventually_at_top_linorder order_refl)
subsection \<open>Sequentially\<close>
abbreviation sequentially :: "nat filter"
where "sequentially \<equiv> at_top"
lemma eventually_sequentially:
"eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
by (rule eventually_at_top_linorder)
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
unfolding filter_eq_iff eventually_sequentially by auto
lemmas trivial_limit_sequentially = sequentially_bot
lemma eventually_False_sequentially [simp]:
"\<not> eventually (\<lambda>n. False) sequentially"
by (simp add: eventually_False)
lemma le_sequentially:
"F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
by (simp add: at_top_def le_INF_iff le_principal)
lemma eventually_sequentiallyI [intro?]:
assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
shows "eventually P sequentially"
using assms by (auto simp: eventually_sequentially)
lemma eventually_sequentially_Suc [simp]: "eventually (\<lambda>i. P (Suc i)) sequentially \<longleftrightarrow> eventually P sequentially"
unfolding eventually_sequentially by (metis Suc_le_D Suc_le_mono le_Suc_eq)
lemma eventually_sequentially_seg [simp]: "eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially"
using eventually_sequentially_Suc[of "\<lambda>n. P (n + k)" for k] by (induction k) auto
subsection \<open>The cofinite filter\<close>
definition "cofinite = Abs_filter (\<lambda>P. finite {x. \<not> P x})"
abbreviation Inf_many :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "\<exists>\<^sub>\<infinity>" 10)
where "Inf_many P \<equiv> frequently P cofinite"
abbreviation Alm_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "\<forall>\<^sub>\<infinity>" 10)
where "Alm_all P \<equiv> eventually P cofinite"
notation (ASCII)
Inf_many (binder "INFM " 10) and
Alm_all (binder "MOST " 10)
lemma eventually_cofinite: "eventually P cofinite \<longleftrightarrow> finite {x. \<not> P x}"
unfolding cofinite_def
proof (rule eventually_Abs_filter, rule is_filter.intro)
fix P Q :: "'a \<Rightarrow> bool" assume "finite {x. \<not> P x}" "finite {x. \<not> Q x}"
from finite_UnI[OF this] show "finite {x. \<not> (P x \<and> Q x)}"
by (rule rev_finite_subset) auto
next
fix P Q :: "'a \<Rightarrow> bool" assume P: "finite {x. \<not> P x}" and *: "\<forall>x. P x \<longrightarrow> Q x"
from * show "finite {x. \<not> Q x}"
by (intro finite_subset[OF _ P]) auto
qed simp
lemma frequently_cofinite: "frequently P cofinite \<longleftrightarrow> \<not> finite {x. P x}"
by (simp add: frequently_def eventually_cofinite)
lemma cofinite_bot[simp]: "cofinite = (bot::'a filter) \<longleftrightarrow> finite (UNIV :: 'a set)"
unfolding trivial_limit_def eventually_cofinite by simp
lemma cofinite_eq_sequentially: "cofinite = sequentially"
unfolding filter_eq_iff eventually_sequentially eventually_cofinite
proof safe
fix P :: "nat \<Rightarrow> bool" assume [simp]: "finite {x. \<not> P x}"
show "\<exists>N. \<forall>n\<ge>N. P n"
proof cases
assume "{x. \<not> P x} \<noteq> {}" then show ?thesis
by (intro exI[of _ "Suc (Max {x. \<not> P x})"]) (auto simp: Suc_le_eq)
qed auto
next
fix P :: "nat \<Rightarrow> bool" and N :: nat assume "\<forall>n\<ge>N. P n"
then have "{x. \<not> P x} \<subseteq> {..< N}"
by (auto simp: not_le)
then show "finite {x. \<not> P x}"
by (blast intro: finite_subset)
qed
subsubsection \<open>Product of filters\<close>
lemma filtermap_sequentually_ne_bot: "filtermap f sequentially \<noteq> bot"
by (auto simp add: filter_eq_iff eventually_filtermap eventually_sequentially)
definition prod_filter :: "'a filter \<Rightarrow> 'b filter \<Rightarrow> ('a \<times> 'b) filter" (infixr "\<times>\<^sub>F" 80) where
"prod_filter F G =
(INF (P, Q):{(P, Q). eventually P F \<and> eventually Q G}. principal {(x, y). P x \<and> Q y})"
lemma eventually_prod_filter: "eventually P (F \<times>\<^sub>F G) \<longleftrightarrow>
(\<exists>Pf Pg. eventually Pf F \<and> eventually Pg G \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P (x, y)))"
unfolding prod_filter_def
proof (subst eventually_INF_base, goal_cases)
case 2
moreover have "eventually Pf F \<Longrightarrow> eventually Qf F \<Longrightarrow> eventually Pg G \<Longrightarrow> eventually Qg G \<Longrightarrow>
\<exists>P Q. eventually P F \<and> eventually Q G \<and>
Collect P \<times> Collect Q \<subseteq> Collect Pf \<times> Collect Pg \<inter> Collect Qf \<times> Collect Qg" for Pf Pg Qf Qg
by (intro conjI exI[of _ "inf Pf Qf"] exI[of _ "inf Pg Qg"])
(auto simp: inf_fun_def eventually_conj)
ultimately show ?case
by auto
qed (auto simp: eventually_principal intro: eventually_True)
lemma eventually_prod1:
assumes "B \<noteq> bot"
shows "(\<forall>\<^sub>F (x, y) in A \<times>\<^sub>F B. P x) \<longleftrightarrow> (\<forall>\<^sub>F x in A. P x)"
unfolding eventually_prod_filter
proof safe
fix R Q
assume *: "\<forall>\<^sub>F x in A. R x" "\<forall>\<^sub>F x in B. Q x" "\<forall>x y. R x \<longrightarrow> Q y \<longrightarrow> P x"
with \<open>B \<noteq> bot\<close> obtain y where "Q y" by (auto dest: eventually_happens)
with * show "eventually P A"
by (force elim: eventually_mono)
next
assume "eventually P A"
then show "\<exists>Pf Pg. eventually Pf A \<and> eventually Pg B \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P x)"
by (intro exI[of _ P] exI[of _ "\<lambda>x. True"]) auto
qed
lemma eventually_prod2:
assumes "A \<noteq> bot"
shows "(\<forall>\<^sub>F (x, y) in A \<times>\<^sub>F B. P y) \<longleftrightarrow> (\<forall>\<^sub>F y in B. P y)"
unfolding eventually_prod_filter
proof safe
fix R Q
assume *: "\<forall>\<^sub>F x in A. R x" "\<forall>\<^sub>F x in B. Q x" "\<forall>x y. R x \<longrightarrow> Q y \<longrightarrow> P y"
with \<open>A \<noteq> bot\<close> obtain x where "R x" by (auto dest: eventually_happens)
with * show "eventually P B"
by (force elim: eventually_mono)
next
assume "eventually P B"
then show "\<exists>Pf Pg. eventually Pf A \<and> eventually Pg B \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P y)"
by (intro exI[of _ P] exI[of _ "\<lambda>x. True"]) auto
qed
lemma INF_filter_bot_base:
fixes F :: "'a \<Rightarrow> 'b filter"
assumes *: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> \<exists>k\<in>I. F k \<le> F i \<sqinter> F j"
shows "(INF i:I. F i) = bot \<longleftrightarrow> (\<exists>i\<in>I. F i = bot)"
proof (cases "\<exists>i\<in>I. F i = bot")
case True
then have "(INF i:I. F i) \<le> bot"
by (auto intro: INF_lower2)
with True show ?thesis
by (auto simp: bot_unique)
next
case False
moreover have "(INF i:I. F i) \<noteq> bot"
proof (cases "I = {}")
case True
then show ?thesis
by (auto simp add: filter_eq_iff)
next
case False': False
show ?thesis
proof (rule INF_filter_not_bot)
fix J
assume "finite J" "J \<subseteq> I"
then have "\<exists>k\<in>I. F k \<le> (\<Sqinter>i\<in>J. F i)"
proof (induct J)
case empty
then show ?case
using \<open>I \<noteq> {}\<close> by auto
next
case (insert i J)
then obtain k where "k \<in> I" "F k \<le> (\<Sqinter>i\<in>J. F i)" by auto
with insert *[of i k] show ?case
by auto
qed
with False show "(\<Sqinter>i\<in>J. F i) \<noteq> \<bottom>"
by (auto simp: bot_unique)
qed
qed
ultimately show ?thesis
by auto
qed
lemma Collect_empty_eq_bot: "Collect P = {} \<longleftrightarrow> P = \<bottom>"
by auto
lemma prod_filter_eq_bot: "A \<times>\<^sub>F B = bot \<longleftrightarrow> A = bot \<or> B = bot"
unfolding prod_filter_def
proof (subst INF_filter_bot_base; clarsimp simp: principal_eq_bot_iff Collect_empty_eq_bot bot_fun_def simp del: Collect_empty_eq)
fix A1 A2 B1 B2 assume "\<forall>\<^sub>F x in A. A1 x" "\<forall>\<^sub>F x in A. A2 x" "\<forall>\<^sub>F x in B. B1 x" "\<forall>\<^sub>F x in B. B2 x"
then show "\<exists>x. eventually x A \<and> (\<exists>y. eventually y B \<and> Collect x \<times> Collect y \<subseteq> Collect A1 \<times> Collect B1 \<and> Collect x \<times> Collect y \<subseteq> Collect A2 \<times> Collect B2)"
by (intro exI[of _ "\<lambda>x. A1 x \<and> A2 x"] exI[of _ "\<lambda>x. B1 x \<and> B2 x"] conjI)
(auto simp: eventually_conj_iff)
next
show "(\<exists>x. eventually x A \<and> (\<exists>y. eventually y B \<and> (x = (\<lambda>x. False) \<or> y = (\<lambda>x. False)))) = (A = \<bottom> \<or> B = \<bottom>)"
by (auto simp: trivial_limit_def intro: eventually_True)
qed
lemma prod_filter_mono: "F \<le> F' \<Longrightarrow> G \<le> G' \<Longrightarrow> F \<times>\<^sub>F G \<le> F' \<times>\<^sub>F G'"
by (auto simp: le_filter_def eventually_prod_filter)
lemma prod_filter_mono_iff:
assumes nAB: "A \<noteq> bot" "B \<noteq> bot"
shows "A \<times>\<^sub>F B \<le> C \<times>\<^sub>F D \<longleftrightarrow> A \<le> C \<and> B \<le> D"
proof safe
assume *: "A \<times>\<^sub>F B \<le> C \<times>\<^sub>F D"
with assms have "A \<times>\<^sub>F B \<noteq> bot"
by (auto simp: bot_unique prod_filter_eq_bot)
with * have "C \<times>\<^sub>F D \<noteq> bot"
by (auto simp: bot_unique)
then have nCD: "C \<noteq> bot" "D \<noteq> bot"
by (auto simp: prod_filter_eq_bot)
show "A \<le> C"
proof (rule filter_leI)
fix P assume "eventually P C" with *[THEN filter_leD, of "\<lambda>(x, y). P x"] show "eventually P A"
using nAB nCD by (simp add: eventually_prod1 eventually_prod2)
qed
show "B \<le> D"
proof (rule filter_leI)
fix P assume "eventually P D" with *[THEN filter_leD, of "\<lambda>(x, y). P y"] show "eventually P B"
using nAB nCD by (simp add: eventually_prod1 eventually_prod2)
qed
qed (intro prod_filter_mono)
lemma eventually_prod_same: "eventually P (F \<times>\<^sub>F F) \<longleftrightarrow>
(\<exists>Q. eventually Q F \<and> (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y)))"
unfolding eventually_prod_filter
apply safe
apply (rule_tac x="inf Pf Pg" in exI)
apply (auto simp: inf_fun_def intro!: eventually_conj)
done
lemma eventually_prod_sequentially:
"eventually P (sequentially \<times>\<^sub>F sequentially) \<longleftrightarrow> (\<exists>N. \<forall>m \<ge> N. \<forall>n \<ge> N. P (n, m))"
unfolding eventually_prod_same eventually_sequentially by auto
lemma principal_prod_principal: "principal A \<times>\<^sub>F principal B = principal (A \<times> B)"
apply (simp add: filter_eq_iff eventually_prod_filter eventually_principal)
apply safe
apply blast
apply (intro conjI exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"])
apply auto
done
lemma prod_filter_INF:
assumes "I \<noteq> {}" "J \<noteq> {}"
shows "(INF i:I. A i) \<times>\<^sub>F (INF j:J. B j) = (INF i:I. INF j:J. A i \<times>\<^sub>F B j)"
proof (safe intro!: antisym INF_greatest)
from \<open>I \<noteq> {}\<close> obtain i where "i \<in> I" by auto
from \<open>J \<noteq> {}\<close> obtain j where "j \<in> J" by auto
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)"
unfolding prod_filter_def
proof (safe intro!: INF_greatest)
fix P Q assume P: "\<forall>\<^sub>F x in \<Sqinter>i\<in>I. A i. P x" and Q: "\<forall>\<^sub>F x in \<Sqinter>j\<in>J. B j. Q x"
let ?X = "(\<Sqinter>i\<in>I. \<Sqinter>j\<in>J. \<Sqinter>(P, Q)\<in>{(P, Q). (\<forall>\<^sub>F x in A i. P x) \<and> (\<forall>\<^sub>F x in B j. Q x)}. principal {(x, y). P x \<and> Q y})"
have "?X \<le> principal {x. P (fst x)} \<sqinter> principal {x. Q (snd x)}"
proof (intro inf_greatest)
have "?X \<le> (\<Sqinter>i\<in>I. \<Sqinter>P\<in>{P. eventually P (A i)}. principal {x. P (fst x)})"
by (auto intro!: INF_greatest INF_lower2[of j] INF_lower2 \<open>j\<in>J\<close> INF_lower2[of "(_, \<lambda>x. True)"])
also have "\<dots> \<le> principal {x. P (fst x)}"
unfolding le_principal
proof (rule eventually_INF_mono[OF P])
fix i P assume "i \<in> I" "eventually P (A i)"
then show "\<forall>\<^sub>F x in \<Sqinter>P\<in>{P. eventually P (A i)}. principal {x. P (fst x)}. x \<in> {x. P (fst x)}"
unfolding le_principal[symmetric] by (auto intro!: INF_lower)
qed auto
finally show "?X \<le> principal {x. P (fst x)}" .
have "?X \<le> (\<Sqinter>i\<in>J. \<Sqinter>P\<in>{P. eventually P (B i)}. principal {x. P (snd x)})"
by (auto intro!: INF_greatest INF_lower2[of i] INF_lower2 \<open>i\<in>I\<close> INF_lower2[of "(\<lambda>x. True, _)"])
also have "\<dots> \<le> principal {x. Q (snd x)}"
unfolding le_principal
proof (rule eventually_INF_mono[OF Q])
fix j Q assume "j \<in> J" "eventually Q (B j)"
then show "\<forall>\<^sub>F x in \<Sqinter>P\<in>{P. eventually P (B j)}. principal {x. P (snd x)}. x \<in> {x. Q (snd x)}"
unfolding le_principal[symmetric] by (auto intro!: INF_lower)
qed auto
finally show "?X \<le> principal {x. Q (snd x)}" .
qed
also have "\<dots> = principal {(x, y). P x \<and> Q y}"
by auto
finally show "?X \<le> principal {(x, y). P x \<and> Q y}" .
qed
qed (intro prod_filter_mono INF_lower)
lemma filtermap_Pair: "filtermap (\<lambda>x. (f x, g x)) F \<le> filtermap f F \<times>\<^sub>F filtermap g F"
by (simp add: le_filter_def eventually_filtermap eventually_prod_filter)
(auto elim: eventually_elim2)
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)"
unfolding prod_filter_def
by (intro eventually_INF1[of "(P, Q)"]) (auto simp: eventually_principal)
lemma prod_filter_INF1: "I \<noteq> {} \<Longrightarrow> (INF i:I. A i) \<times>\<^sub>F B = (INF i:I. A i \<times>\<^sub>F B)"
using prod_filter_INF[of I "{B}" A "\<lambda>x. x"] by simp
lemma prod_filter_INF2: "J \<noteq> {} \<Longrightarrow> A \<times>\<^sub>F (INF i:J. B i) = (INF i:J. A \<times>\<^sub>F B i)"
using prod_filter_INF[of "{A}" J "\<lambda>x. x" B] by simp
subsection \<open>Limits\<close>
definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where
"filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
syntax
"_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)
translations
"LIM x F1. f :> F2" == "CONST filterlim (\<lambda>x. f) F2 F1"
lemma filterlim_top [simp]: "filterlim f top F"
by (simp add: filterlim_def)
lemma filterlim_iff:
"(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)"
unfolding filterlim_def le_filter_def eventually_filtermap ..
lemma filterlim_compose:
"filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1"
unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)
lemma filterlim_mono:
"filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'"
unfolding filterlim_def by (metis filtermap_mono order_trans)
lemma filterlim_ident: "LIM x F. x :> F"
by (simp add: filterlim_def filtermap_ident)
lemma filterlim_cong:
"F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'"
by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2)
lemma filterlim_mono_eventually:
assumes "filterlim f F G" and ord: "F \<le> F'" "G' \<le> G"
assumes eq: "eventually (\<lambda>x. f x = f' x) G'"
shows "filterlim f' F' G'"
apply (rule filterlim_cong[OF refl refl eq, THEN iffD1])
apply (rule filterlim_mono[OF _ ord])
apply fact
done
lemma filtermap_mono_strong: "inj f \<Longrightarrow> filtermap f F \<le> filtermap f G \<longleftrightarrow> F \<le> G"
apply (auto intro!: filtermap_mono) []
apply (auto simp: le_filter_def eventually_filtermap)
apply (erule_tac x="\<lambda>x. P (inv f x)" in allE)
apply auto
done
lemma filtermap_eq_strong: "inj f \<Longrightarrow> filtermap f F = filtermap f G \<longleftrightarrow> F = G"
by (simp add: filtermap_mono_strong eq_iff)
lemma filtermap_fun_inverse:
assumes g: "filterlim g F G"
assumes f: "filterlim f G F"
assumes ev: "eventually (\<lambda>x. f (g x) = x) G"
shows "filtermap f F = G"
proof (rule antisym)
show "filtermap f F \<le> G"
using f unfolding filterlim_def .
have "G = filtermap f (filtermap g G)"
using ev by (auto elim: eventually_elim2 simp: filter_eq_iff eventually_filtermap)
also have "\<dots> \<le> filtermap f F"
using g by (intro filtermap_mono) (simp add: filterlim_def)
finally show "G \<le> filtermap f F" .
qed
lemma filterlim_principal:
"(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)"
unfolding filterlim_def eventually_filtermap le_principal ..
lemma filterlim_inf:
"(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))"
unfolding filterlim_def by simp
lemma filterlim_INF:
"(LIM x F. f x :> (INF b:B. G b)) \<longleftrightarrow> (\<forall>b\<in>B. LIM x F. f x :> G b)"
unfolding filterlim_def le_INF_iff ..
lemma filterlim_INF_INF:
"(\<And>m. m \<in> J \<Longrightarrow> \<exists>i\<in>I. filtermap f (F i) \<le> G m) \<Longrightarrow> LIM x (INF i:I. F i). f x :> (INF j:J. G j)"
unfolding filterlim_def by (rule order_trans[OF filtermap_INF INF_mono])
lemma filterlim_base:
"(\<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>
LIM x (INF i:I. principal (F i)). f x :> (INF j:J. principal (G j))"
by (force intro!: filterlim_INF_INF simp: image_subset_iff)
lemma filterlim_base_iff:
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"
shows "(LIM x (INF i:I. principal (F i)). f x :> INF j:J. principal (G j)) \<longleftrightarrow>
(\<forall>j\<in>J. \<exists>i\<in>I. \<forall>x\<in>F i. f x \<in> G j)"
unfolding filterlim_INF filterlim_principal
proof (subst eventually_INF_base)
fix i j assume "i \<in> I" "j \<in> I"
with chain[OF this] show "\<exists>x\<in>I. principal (F x) \<le> inf (principal (F i)) (principal (F j))"
by auto
qed (auto simp: eventually_principal \<open>I \<noteq> {}\<close>)
lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2"
unfolding filterlim_def filtermap_filtermap ..
lemma filterlim_sup:
"filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)"
unfolding filterlim_def filtermap_sup by auto
lemma filterlim_sequentially_Suc:
"(LIM x sequentially. f (Suc x) :> F) \<longleftrightarrow> (LIM x sequentially. f x :> F)"
unfolding filterlim_iff by (subst eventually_sequentially_Suc) simp
lemma filterlim_Suc: "filterlim Suc sequentially sequentially"
by (simp add: filterlim_iff eventually_sequentially)
lemma filterlim_If:
"LIM x inf F (principal {x. P x}). f x :> G \<Longrightarrow>
LIM x inf F (principal {x. \<not> P x}). g x :> G \<Longrightarrow>
LIM x F. if P x then f x else g x :> G"
unfolding filterlim_iff eventually_inf_principal by (auto simp: eventually_conj_iff)
lemma filterlim_Pair:
"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"
unfolding filterlim_def
by (rule order_trans[OF filtermap_Pair prod_filter_mono])
subsection \<open>Limits to @{const at_top} and @{const at_bot}\<close>
lemma filterlim_at_top:
fixes f :: "'a \<Rightarrow> ('b::linorder)"
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)"
by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_mono)
lemma filterlim_at_top_mono:
"LIM x F. f x :> at_top \<Longrightarrow> eventually (\<lambda>x. f x \<le> (g x::'a::linorder)) F \<Longrightarrow>
LIM x F. g x :> at_top"
by (auto simp: filterlim_at_top elim: eventually_elim2 intro: order_trans)
lemma filterlim_at_top_dense:
fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)"
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)"
by (metis eventually_mono[of _ F] eventually_gt_at_top order_less_imp_le
filterlim_at_top[of f F] filterlim_iff[of f at_top F])
lemma filterlim_at_top_ge:
fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)"
unfolding at_top_sub[of c] filterlim_INF by (auto simp add: filterlim_principal)
lemma filterlim_at_top_at_top:
fixes f :: "'a::linorder \<Rightarrow> 'b::linorder"
assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
assumes Q: "eventually Q at_top"
assumes P: "eventually P at_top"
shows "filterlim f at_top at_top"
proof -
from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
unfolding eventually_at_top_linorder by auto
show ?thesis
proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
fix z assume "x \<le> z"
with x have "P z" by auto
have "eventually (\<lambda>x. g z \<le> x) at_top"
by (rule eventually_ge_at_top)
with Q show "eventually (\<lambda>x. z \<le> f x) at_top"
by eventually_elim (metis mono bij \<open>P z\<close>)
qed
qed
lemma filterlim_at_top_gt:
fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)"
by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge)
lemma filterlim_at_bot:
fixes f :: "'a \<Rightarrow> ('b::linorder)"
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)"
by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_mono)
lemma filterlim_at_bot_dense:
fixes f :: "'a \<Rightarrow> ('b::{dense_linorder, no_bot})"
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x < Z) F)"
proof (auto simp add: filterlim_at_bot[of f F])
fix Z :: 'b
from lt_ex [of Z] obtain Z' where 1: "Z' < Z" ..
assume "\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F"
hence "eventually (\<lambda>x. f x \<le> Z') F" by auto
thus "eventually (\<lambda>x. f x < Z) F"
apply (rule eventually_mono)
using 1 by auto
next
fix Z :: 'b
show "\<forall>Z. eventually (\<lambda>x. f x < Z) F \<Longrightarrow> eventually (\<lambda>x. f x \<le> Z) F"
by (drule spec [of _ Z], erule eventually_mono, auto simp add: less_imp_le)
qed
lemma filterlim_at_bot_le:
fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)"
unfolding filterlim_at_bot
proof safe
fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F"
with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F"
by (auto elim!: eventually_mono)
qed simp
lemma filterlim_at_bot_lt:
fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)"
by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans)
subsection \<open>Setup @{typ "'a filter"} for lifting and transfer\<close>
context includes lifting_syntax
begin
definition rel_filter :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> 'b filter \<Rightarrow> bool"
where "rel_filter R F G = ((R ===> op =) ===> op =) (Rep_filter F) (Rep_filter G)"
lemma rel_filter_eventually:
"rel_filter R F G \<longleftrightarrow>
((R ===> op =) ===> op =) (\<lambda>P. eventually P F) (\<lambda>P. eventually P G)"
by(simp add: rel_filter_def eventually_def)
lemma filtermap_id [simp, id_simps]: "filtermap id = id"
by(simp add: fun_eq_iff id_def filtermap_ident)
lemma filtermap_id' [simp]: "filtermap (\<lambda>x. x) = (\<lambda>F. F)"
using filtermap_id unfolding id_def .
lemma Quotient_filter [quot_map]:
assumes Q: "Quotient R Abs Rep T"
shows "Quotient (rel_filter R) (filtermap Abs) (filtermap Rep) (rel_filter T)"
unfolding Quotient_alt_def
proof(intro conjI strip)
from Q have *: "\<And>x y. T x y \<Longrightarrow> Abs x = y"
unfolding Quotient_alt_def by blast
fix F G
assume "rel_filter T F G"
thus "filtermap Abs F = G" unfolding filter_eq_iff
by(auto simp add: eventually_filtermap rel_filter_eventually * rel_funI del: iffI elim!: rel_funD)
next
from Q have *: "\<And>x. T (Rep x) x" unfolding Quotient_alt_def by blast
fix F
show "rel_filter T (filtermap Rep F) F"
by(auto elim: rel_funD intro: * intro!: ext arg_cong[where f="\<lambda>P. eventually P F"] rel_funI
del: iffI simp add: eventually_filtermap rel_filter_eventually)
qed(auto simp add: map_fun_def o_def eventually_filtermap filter_eq_iff fun_eq_iff rel_filter_eventually
fun_quotient[OF fun_quotient[OF Q identity_quotient] identity_quotient, unfolded Quotient_alt_def])
lemma eventually_parametric [transfer_rule]:
"((A ===> op =) ===> rel_filter A ===> op =) eventually eventually"
by(simp add: rel_fun_def rel_filter_eventually)
lemma frequently_parametric [transfer_rule]:
"((A ===> op =) ===> rel_filter A ===> op =) frequently frequently"
unfolding frequently_def[abs_def] by transfer_prover
lemma rel_filter_eq [relator_eq]: "rel_filter op = = op ="
by(auto simp add: rel_filter_eventually rel_fun_eq fun_eq_iff filter_eq_iff)
lemma rel_filter_mono [relator_mono]:
"A \<le> B \<Longrightarrow> rel_filter A \<le> rel_filter B"
unfolding rel_filter_eventually[abs_def]
by(rule le_funI)+(intro fun_mono fun_mono[THEN le_funD, THEN le_funD] order.refl)
lemma rel_filter_conversep [simp]: "rel_filter A\<inverse>\<inverse> = (rel_filter A)\<inverse>\<inverse>"
apply (simp add: rel_filter_eventually fun_eq_iff rel_fun_def)
apply (safe; metis)
done
lemma is_filter_parametric_aux:
assumes "is_filter F"
assumes [transfer_rule]: "bi_total A" "bi_unique A"
and [transfer_rule]: "((A ===> op =) ===> op =) F G"
shows "is_filter G"
proof -
interpret is_filter F by fact
show ?thesis
proof
have "F (\<lambda>_. True) = G (\<lambda>x. True)" by transfer_prover
thus "G (\<lambda>x. True)" by(simp add: True)
next
fix P' Q'
assume "G P'" "G Q'"
moreover
from bi_total_fun[OF \<open>bi_unique A\<close> bi_total_eq, unfolded bi_total_def]
obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast
have "F P = G P'" "F Q = G Q'" by transfer_prover+
ultimately have "F (\<lambda>x. P x \<and> Q x)" by(simp add: conj)
moreover have "F (\<lambda>x. P x \<and> Q x) = G (\<lambda>x. P' x \<and> Q' x)" by transfer_prover
ultimately show "G (\<lambda>x. P' x \<and> Q' x)" by simp
next
fix P' Q'
assume "\<forall>x. P' x \<longrightarrow> Q' x" "G P'"
moreover
from bi_total_fun[OF \<open>bi_unique A\<close> bi_total_eq, unfolded bi_total_def]
obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast
have "F P = G P'" by transfer_prover
moreover have "(\<forall>x. P x \<longrightarrow> Q x) \<longleftrightarrow> (\<forall>x. P' x \<longrightarrow> Q' x)" by transfer_prover
ultimately have "F Q" by(simp add: mono)
moreover have "F Q = G Q'" by transfer_prover
ultimately show "G Q'" by simp
qed
qed
lemma is_filter_parametric [transfer_rule]:
"\<lbrakk> bi_total A; bi_unique A \<rbrakk>
\<Longrightarrow> (((A ===> op =) ===> op =) ===> op =) is_filter is_filter"
apply(rule rel_funI)
apply(rule iffI)
apply(erule (3) is_filter_parametric_aux)
apply(erule is_filter_parametric_aux[where A="conversep A"])
apply (simp_all add: rel_fun_def)
apply metis
done
lemma left_total_rel_filter [transfer_rule]:
assumes [transfer_rule]: "bi_total A" "bi_unique A"
shows "left_total (rel_filter A)"
proof(rule left_totalI)
fix F :: "'a filter"
from bi_total_fun[OF bi_unique_fun[OF \<open>bi_total A\<close> bi_unique_eq] bi_total_eq]
obtain G where [transfer_rule]: "((A ===> op =) ===> op =) (\<lambda>P. eventually P F) G"
unfolding bi_total_def by blast
moreover have "is_filter (\<lambda>P. eventually P F) \<longleftrightarrow> is_filter G" by transfer_prover
hence "is_filter G" by(simp add: eventually_def is_filter_Rep_filter)
ultimately have "rel_filter A F (Abs_filter G)"
by(simp add: rel_filter_eventually eventually_Abs_filter)
thus "\<exists>G. rel_filter A F G" ..
qed
lemma right_total_rel_filter [transfer_rule]:
"\<lbrakk> bi_total A; bi_unique A \<rbrakk> \<Longrightarrow> right_total (rel_filter A)"
using left_total_rel_filter[of "A\<inverse>\<inverse>"] by simp
lemma bi_total_rel_filter [transfer_rule]:
assumes "bi_total A" "bi_unique A"
shows "bi_total (rel_filter A)"
unfolding bi_total_alt_def using assms
by(simp add: left_total_rel_filter right_total_rel_filter)
lemma left_unique_rel_filter [transfer_rule]:
assumes "left_unique A"
shows "left_unique (rel_filter A)"
proof(rule left_uniqueI)
fix F F' G
assume [transfer_rule]: "rel_filter A F G" "rel_filter A F' G"
show "F = F'"
unfolding filter_eq_iff
proof
fix P :: "'a \<Rightarrow> bool"
obtain P' where [transfer_rule]: "(A ===> op =) P P'"
using left_total_fun[OF assms left_total_eq] unfolding left_total_def by blast
have "eventually P F = eventually P' G"
and "eventually P F' = eventually P' G" by transfer_prover+
thus "eventually P F = eventually P F'" by simp
qed
qed
lemma right_unique_rel_filter [transfer_rule]:
"right_unique A \<Longrightarrow> right_unique (rel_filter A)"
using left_unique_rel_filter[of "A\<inverse>\<inverse>"] by simp
lemma bi_unique_rel_filter [transfer_rule]:
"bi_unique A \<Longrightarrow> bi_unique (rel_filter A)"
by(simp add: bi_unique_alt_def left_unique_rel_filter right_unique_rel_filter)
lemma top_filter_parametric [transfer_rule]:
"bi_total A \<Longrightarrow> (rel_filter A) top top"
by(simp add: rel_filter_eventually All_transfer)
lemma bot_filter_parametric [transfer_rule]: "(rel_filter A) bot bot"
by(simp add: rel_filter_eventually rel_fun_def)
lemma sup_filter_parametric [transfer_rule]:
"(rel_filter A ===> rel_filter A ===> rel_filter A) sup sup"
by(fastforce simp add: rel_filter_eventually[abs_def] eventually_sup dest: rel_funD)
lemma Sup_filter_parametric [transfer_rule]:
"(rel_set (rel_filter A) ===> rel_filter A) Sup Sup"
proof(rule rel_funI)
fix S T
assume [transfer_rule]: "rel_set (rel_filter A) S T"
show "rel_filter A (Sup S) (Sup T)"
by(simp add: rel_filter_eventually eventually_Sup) transfer_prover
qed
lemma principal_parametric [transfer_rule]:
"(rel_set A ===> rel_filter A) principal principal"
proof(rule rel_funI)
fix S S'
assume [transfer_rule]: "rel_set A S S'"
show "rel_filter A (principal S) (principal S')"
by(simp add: rel_filter_eventually eventually_principal) transfer_prover
qed
context
fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
assumes [transfer_rule]: "bi_unique A"
begin
lemma le_filter_parametric [transfer_rule]:
"(rel_filter A ===> rel_filter A ===> op =) op \<le> op \<le>"
unfolding le_filter_def[abs_def] by transfer_prover
lemma less_filter_parametric [transfer_rule]:
"(rel_filter A ===> rel_filter A ===> op =) op < op <"
unfolding less_filter_def[abs_def] by transfer_prover
context
assumes [transfer_rule]: "bi_total A"
begin
lemma Inf_filter_parametric [transfer_rule]:
"(rel_set (rel_filter A) ===> rel_filter A) Inf Inf"
unfolding Inf_filter_def[abs_def] by transfer_prover
lemma inf_filter_parametric [transfer_rule]:
"(rel_filter A ===> rel_filter A ===> rel_filter A) inf inf"
proof(intro rel_funI)+
fix F F' G G'
assume [transfer_rule]: "rel_filter A F F'" "rel_filter A G G'"
have "rel_filter A (Inf {F, G}) (Inf {F', G'})" by transfer_prover
thus "rel_filter A (inf F G) (inf F' G')" by simp
qed
end
end
end
text \<open>Code generation for filters\<close>
definition abstract_filter :: "(unit \<Rightarrow> 'a filter) \<Rightarrow> 'a filter"
where [simp]: "abstract_filter f = f ()"
code_datatype principal abstract_filter
hide_const (open) abstract_filter
declare [[code drop: filterlim prod_filter filtermap eventually
"inf :: _ filter \<Rightarrow> _" "sup :: _ filter \<Rightarrow> _" "less_eq :: _ filter \<Rightarrow> _"
Abs_filter]]
declare filterlim_principal [code]
declare principal_prod_principal [code]
declare filtermap_principal [code]
declare eventually_principal [code]
declare inf_principal [code]
declare sup_principal [code]
declare principal_le_iff [code]
lemma Rep_filter_iff_eventually [simp, code]:
"Rep_filter F P \<longleftrightarrow> eventually P F"
by (simp add: eventually_def)
lemma bot_eq_principal_empty [code]:
"bot = principal {}"
by simp
lemma top_eq_principal_UNIV [code]:
"top = principal UNIV"
by simp
instantiation filter :: (equal) equal
begin
definition equal_filter :: "'a filter \<Rightarrow> 'a filter \<Rightarrow> bool"
where "equal_filter F F' \<longleftrightarrow> F = F'"
lemma equal_filter [code]:
"HOL.equal (principal A) (principal B) \<longleftrightarrow> A = B"
by (simp add: equal_filter_def)
instance
by standard (simp add: equal_filter_def)
end
end