# Theory Filter

theory Filter
imports Set_Interval Lifting_Set
```(*  Title:      HOL/Filter.thy
Author:     Brian Huffman
Author:     Johannes Hölzl
*)

section ‹Filters on predicates›

theory Filter
imports Set_Interval Lifting_Set
begin

subsection ‹Filters›

text ‹
This definition also allows non-proper filters.
›

locale is_filter =
fixes F :: "('a ⇒ bool) ⇒ bool"
assumes True: "F (λx. True)"
assumes conj: "F (λx. P x) ⟹ F (λx. Q x) ⟹ F (λx. P x ∧ Q x)"
assumes mono: "∀x. P x ⟶ Q x ⟹ F (λx. P x) ⟹ F (λx. Q x)"

typedef 'a filter = "{F :: ('a ⇒ bool) ⇒ bool. is_filter F}"
proof
show "(λx. True) ∈ ?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 ‹Eventually›

definition eventually :: "('a ⇒ bool) ⇒ 'a filter ⇒ bool"
where "eventually P F ⟷ Rep_filter F P"

syntax
"_eventually" :: "pttrn => 'a filter => bool => bool"  ("(3∀⇩F _ in _./ _)" [0, 0, 10] 10)
translations
"∀⇩Fx in F. P" == "CONST eventually (λx. 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' ⟷ (∀P. eventually P F = eventually P F')"
unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..

lemma eventually_True [simp]: "eventually (λx. True) F"
unfolding eventually_def
by (rule is_filter.True [OF is_filter_Rep_filter])

lemma always_eventually: "∀x. P x ⟹ eventually P F"
proof -
assume "∀x. P x" hence "P = (λx. True)" by (simp add: ext)
thus "eventually P F" by simp
qed

lemma eventuallyI: "(⋀x. P x) ⟹ eventually P F"
by (auto intro: always_eventually)

lemma eventually_mono:
"⟦eventually P F; ⋀x. P x ⟹ Q x⟧ ⟹ eventually Q F"
unfolding eventually_def
by (blast intro: is_filter.mono [OF is_filter_Rep_filter])

lemma eventually_conj:
assumes P: "eventually (λx. P x) F"
assumes Q: "eventually (λx. Q x) F"
shows "eventually (λx. P x ∧ Q x) F"
using assms unfolding eventually_def
by (rule is_filter.conj [OF is_filter_Rep_filter])

lemma eventually_mp:
assumes "eventually (λx. P x ⟶ Q x) F"
assumes "eventually (λx. P x) F"
shows "eventually (λx. Q x) F"
proof -
have "eventually (λx. (P x ⟶ Q x) ∧ P x) F"
using assms by (rule eventually_conj)
then show ?thesis
by (blast intro: eventually_mono)
qed

lemma eventually_rev_mp:
assumes "eventually (λx. P x) F"
assumes "eventually (λx. P x ⟶ Q x) F"
shows "eventually (λx. Q x) F"
using assms(2) assms(1) by (rule eventually_mp)

lemma eventually_conj_iff:
"eventually (λx. P x ∧ Q x) F ⟷ eventually P F ∧ eventually Q F"
by (auto intro: eventually_conj elim: eventually_rev_mp)

lemma eventually_elim2:
assumes "eventually (λi. P i) F"
assumes "eventually (λi. Q i) F"
assumes "⋀i. P i ⟹ Q i ⟹ R i"
shows "eventually (λi. R i) F"
using assms by (auto elim!: eventually_rev_mp)

lemma eventually_ball_finite_distrib:
"finite A ⟹ (eventually (λx. ∀y∈A. P x y) net) ⟷ (∀y∈A. eventually (λx. P x y) net)"
by (induction A rule: finite_induct) (auto simp: eventually_conj_iff)

lemma eventually_ball_finite:
"finite A ⟹ ∀y∈A. eventually (λx. P x y) net ⟹ eventually (λx. ∀y∈A. P x y) net"
by (auto simp: eventually_ball_finite_distrib)

lemma eventually_all_finite:
fixes P :: "'a ⇒ 'b::finite ⇒ bool"
assumes "⋀y. eventually (λx. P x y) net"
shows "eventually (λx. ∀y. P x y) net"
using eventually_ball_finite [of UNIV P] assms by simp

lemma eventually_ex: "(∀⇩Fx in F. ∃y. P x y) ⟷ (∃Y. ∀⇩Fx in F. P x (Y x))"
proof
assume "∀⇩Fx in F. ∃y. P x y"
then have "∀⇩Fx in F. P x (SOME y. P x y)"
by (auto intro: someI_ex eventually_mono)
then show "∃Y. ∀⇩Fx in F. P x (Y x)"
by auto
qed (auto intro: eventually_mono)

lemma not_eventually_impI: "eventually P F ⟹ ¬ eventually Q F ⟹ ¬ eventually (λx. P x ⟶ Q x) F"
by (auto intro: eventually_mp)

lemma not_eventuallyD: "¬ eventually P F ⟹ ∃x. ¬ P x"
by (metis always_eventually)

lemma eventually_subst:
assumes "eventually (λn. P n = Q n) F"
shows "eventually P F = eventually Q F" (is "?L = ?R")
proof -
from assms have "eventually (λx. P x ⟶ Q x) F"
and "eventually (λx. Q x ⟶ P x) F"
by (auto elim: eventually_mono)
then show ?thesis by (auto elim: eventually_elim2)
qed

subsection ‹ Frequently as dual to eventually ›

definition frequently :: "('a ⇒ bool) ⇒ 'a filter ⇒ bool"
where "frequently P F ⟷ ¬ eventually (λx. ¬ P x) F"

syntax
"_frequently" :: "pttrn ⇒ 'a filter ⇒ bool ⇒ bool"  ("(3∃⇩F _ in _./ _)" [0, 0, 10] 10)
translations
"∃⇩Fx in F. P" == "CONST frequently (λx. P) F"

lemma not_frequently_False [simp]: "¬ (∃⇩Fx in F. False)"

lemma frequently_ex: "∃⇩Fx in F. P x ⟹ ∃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: "∀⇩Fx in F. P x ⟶ Q x" and P: "∃⇩Fx in F. P x" shows "∃⇩Fx in F. Q x"
proof -
from ev have "eventually (λx. ¬ Q x ⟶ ¬ 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 "∃⇩Fx in F. P x"
assumes "∀⇩Fx in F. P x ⟶ Q x"
shows "∃⇩Fx in F. Q x"
using assms(2) assms(1) by (rule frequently_mp)

lemma frequently_mono: "(∀x. P x ⟶ Q x) ⟹ frequently P F ⟹ frequently Q F"
using frequently_mp[of P Q] by (simp add: always_eventually)

lemma frequently_elim1: "∃⇩Fx in F. P x ⟹ (⋀i. P i ⟹ Q i) ⟹ ∃⇩Fx in F. Q x"
by (metis frequently_mono)

lemma frequently_disj_iff: "(∃⇩Fx in F. P x ∨ Q x) ⟷ (∃⇩Fx in F. P x) ∨ (∃⇩Fx in F. Q x)"

lemma frequently_disj: "∃⇩Fx in F. P x ⟹ ∃⇩Fx in F. Q x ⟹ ∃⇩Fx in F. P x ∨ Q x"

lemma frequently_bex_finite_distrib:
assumes "finite A" shows "(∃⇩Fx in F. ∃y∈A. P x y) ⟷ (∃y∈A. ∃⇩Fx in F. P x y)"
using assms by induction (auto simp: frequently_disj_iff)

lemma frequently_bex_finite: "finite A ⟹ ∃⇩Fx in F. ∃y∈A. P x y ⟹ ∃y∈A. ∃⇩Fx in F. P x y"

lemma frequently_all: "(∃⇩Fx in F. ∀y. P x y) ⟷ (∀Y. ∃⇩Fx in F. P x (Y x))"
using eventually_ex[of "λx y. ¬ P x y" F] by (simp add: frequently_def)

lemma
shows not_eventually: "¬ eventually P F ⟷ (∃⇩Fx in F. ¬ P x)"
and not_frequently: "¬ frequently P F ⟷ (∀⇩Fx in F. ¬ P x)"
by (auto simp: frequently_def)

lemma frequently_imp_iff:
"(∃⇩Fx in F. P x ⟶ Q x) ⟷ (eventually P F ⟶ frequently Q F)"
unfolding imp_conv_disj frequently_disj_iff not_eventually[symmetric] ..

lemma eventually_frequently_const_simps:
"(∃⇩Fx in F. P x ∧ C) ⟷ (∃⇩Fx in F. P x) ∧ C"
"(∃⇩Fx in F. C ∧ P x) ⟷ C ∧ (∃⇩Fx in F. P x)"
"(∀⇩Fx in F. P x ∨ C) ⟷ (∀⇩Fx in F. P x) ∨ C"
"(∀⇩Fx in F. C ∨ P x) ⟷ C ∨ (∀⇩Fx in F. P x)"
"(∀⇩Fx in F. P x ⟶ C) ⟷ ((∃⇩Fx in F. P x) ⟶ C)"
"(∀⇩Fx in F. C ⟶ P x) ⟷ (C ⟶ (∀⇩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 ‹
fun eventually_elim_tac facts =
CONTEXT_SUBGOAL (fn (goal, i) => fn (ctxt, st) =>
let
val mp_facts = facts RL @{thms eventually_rev_mp}
val rule =
@{thm eventuallyI}
|> fold (fn mp_fact => fn th => th RS mp_fact) mp_facts
|> funpow (length facts) (fn th => @{thm impI} RS th)
val cases_prop =
Thm.prop_of (Rule_Cases.internalize_params (rule 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 [rule] i) (ctxt, st) end)
›

method_setup eventually_elim = ‹
Scan.succeed (fn _ => CONTEXT_METHOD (fn facts => eventually_elim_tac facts 1))
› "elimination of eventually quantifiers"

subsubsection ‹Finer-than relation›

text ‹@{term "F ≤ F'"} means that filter @{term F} is finer than
filter @{term F'}.›

instantiation filter :: (type) complete_lattice
begin

definition le_filter_def:
"F ≤ F' ⟷ (∀P. eventually P F' ⟶ eventually P F)"

definition
"(F :: 'a filter) < F' ⟷ F ≤ F' ∧ ¬ F' ≤ F"

definition
"top = Abs_filter (λP. ∀x. P x)"

definition
"bot = Abs_filter (λP. True)"

definition
"sup F F' = Abs_filter (λP. eventually P F ∧ eventually P F')"

definition
"inf F F' = Abs_filter
(λP. ∃Q R. eventually Q F ∧ eventually R F' ∧ (∀x. Q x ∧ R x ⟶ P x))"

definition
"Sup S = Abs_filter (λP. ∀F∈S. eventually P F)"

definition
"Inf S = Sup {F::'a filter. ∀F'∈S. F ≤ F'}"

lemma eventually_top [simp]: "eventually P top ⟷ (∀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') ⟷ eventually P F ∧ 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') ⟷
(∃Q R. eventually Q F ∧ eventually R F' ∧ (∀x. Q x ∧ R x ⟶ 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) ⟷ (∀F∈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' ⟷ F ≤ F' ∧ ¬ F' ≤ F"
by (rule less_filter_def) }
{ show "F ≤ F"
unfolding le_filter_def by simp }
{ assume "F ≤ F'" and "F' ≤ F''" thus "F ≤ F''"
unfolding le_filter_def by simp }
{ assume "F ≤ F'" and "F' ≤ F" thus "F = F'"
unfolding le_filter_def filter_eq_iff by fast }
{ show "inf F F' ≤ F" and "inf F F' ≤ F'"
unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
{ assume "F ≤ F'" and "F ≤ F''" thus "F ≤ inf F' F''"
unfolding le_filter_def eventually_inf
by (auto intro: eventually_mono [OF eventually_conj]) }
{ show "F ≤ sup F F'" and "F' ≤ sup F F'"
unfolding le_filter_def eventually_sup by simp_all }
{ assume "F ≤ F''" and "F' ≤ F''" thus "sup F F' ≤ F''"
unfolding le_filter_def eventually_sup by simp }
{ assume "F'' ∈ S" thus "Inf S ≤ F''"
unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
{ assume "⋀F'. F' ∈ S ⟹ F ≤ F'" thus "F ≤ Inf S"
unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
{ assume "F ∈ S" thus "F ≤ Sup S"
unfolding le_filter_def eventually_Sup by simp }
{ assume "⋀F. F ∈ S ⟹ F ≤ F'" thus "Sup S ≤ 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

instance filter :: (type) distrib_lattice
proof
fix F G H :: "'a filter"
show "sup F (inf G H) = inf (sup F G) (sup F H)"
proof (rule order.antisym)
show "inf (sup F G) (sup F H) ≤ sup F (inf G H)"
unfolding le_filter_def eventually_sup
proof safe
fix P assume 1: "eventually P F" and 2: "eventually P (inf G H)"
from 2 obtain Q R
where QR: "eventually Q G" "eventually R H" "⋀x. Q x ⟹ R x ⟹ P x"
by (auto simp: eventually_inf)
define Q' where "Q' = (λx. Q x ∨ P x)"
define R' where "R' = (λx. R x ∨ P x)"
from 1 have "eventually Q' F"
by (elim eventually_mono) (auto simp: Q'_def)
moreover from 1 have "eventually R' F"
by (elim eventually_mono) (auto simp: R'_def)
moreover from QR(1) have "eventually Q' G"
by (elim eventually_mono) (auto simp: Q'_def)
moreover from QR(2) have "eventually R' H"
by (elim eventually_mono)(auto simp: R'_def)
moreover from QR have "P x" if "Q' x" "R' x" for x
using that by (auto simp: Q'_def R'_def)
ultimately show "eventually P (inf (sup F G) (sup F H))"
by (auto simp: eventually_inf eventually_sup)
qed
qed (auto intro: inf.coboundedI1 inf.coboundedI2)
qed

lemma filter_leD:
"F ≤ F' ⟹ eventually P F' ⟹ eventually P F"
unfolding le_filter_def by simp

lemma filter_leI:
"(⋀P. eventually P F' ⟹ eventually P F) ⟹ F ≤ F'"
unfolding le_filter_def by simp

lemma eventually_False:
"eventually (λx. False) F ⟷ F = bot"
unfolding filter_eq_iff by (auto elim: eventually_rev_mp)

lemma eventually_frequently: "F ≠ bot ⟹ eventually P F ⟹ frequently P F"
using eventually_conj[of P F "λx. ¬ P x"]
by (auto simp add: frequently_def eventually_False)

lemma eventually_frequentlyE:
assumes "eventually P F"
assumes "eventually (λx. ¬ P x ∨ Q x) F" "F≠bot"
shows "frequently Q F"
proof -
have "eventually Q F"
using eventually_conj[OF assms(1,2),simplified] by (auto elim:eventually_mono)
then show ?thesis using eventually_frequently[OF ‹F≠bot›] by auto
qed

lemma eventually_const_iff: "eventually (λx. P) F ⟷ P ∨ F = bot"
by (cases P) (auto simp: eventually_False)

lemma eventually_const[simp]: "F ≠ bot ⟹ eventually (λx. P) F ⟷ P"

lemma frequently_const_iff: "frequently (λx. P) F ⟷ P ∧ F ≠ bot"

lemma frequently_const[simp]: "F ≠ bot ⟹ frequently (λx. P) F ⟷ P"

lemma eventually_happens: "eventually P net ⟹ net = bot ∨ (∃x. P x)"
by (metis frequentlyE eventually_frequently)

lemma eventually_happens':
assumes "F ≠ bot" "eventually P F"
shows   "∃x. P x"
using assms eventually_frequently frequentlyE by blast

abbreviation (input) trivial_limit :: "'a filter ⇒ bool"
where "trivial_limit F ≡ F = bot"

lemma trivial_limit_def: "trivial_limit F ⟷ eventually (λx. False) F"
by (rule eventually_False [symmetric])

lemma False_imp_not_eventually: "(∀x. ¬ P x ) ⟹ ¬ trivial_limit net ⟹ ¬ eventually (λx. P x) net"

lemma eventually_Inf: "eventually P (Inf B) ⟷ (∃X⊆B. finite X ∧ eventually P (Inf X))"
proof -
let ?F = "λP. ∃X⊆B. finite X ∧ eventually P (Inf X)"

{ fix P have "eventually P (Abs_filter ?F) ⟷ ?F P"
proof (rule eventually_Abs_filter is_filter.intro)+
show "?F (λx. 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 (λx. P x ∧ Q x)"
by (intro exI[of _ "X ∪ Y"])
(auto simp: Inf_union_distrib eventually_inf)
next
fix P Q
assume "?F P" then guess X ..
moreover assume "∀x. P x ⟶ 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 ≤ Abs_filter ?F"
by (auto simp: le_filter_def eventually_F dest: Inf_superset_mono)
next
fix F assume "F ∈ B" then show "Abs_filter ?F ≤ F"
by (auto simp add: le_filter_def eventually_F intro!: exI[of _ "{F}"])
qed
then show ?thesis
qed

lemma eventually_INF: "eventually P (⨅b∈B. F b) ⟷ (∃X⊆B. finite X ∧ eventually P (⨅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 "(⋀X. X ⊆ B ⟹ finite X ⟹ Inf X ≠ bot) ⟹ Inf B ≠ 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 ⇒ 'a filter"
shows "(⋀X. X ⊆ B ⟹ finite X ⟹ (⨅b∈X. F b) ≠ bot) ⟹ (⨅b∈B. F b) ≠ 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 ≠ {}" and base: "⋀F G. F ∈ B ⟹ G ∈ B ⟹ ∃x∈B. x ≤ inf F G"
shows "eventually P (Inf B) ⟷ (∃b∈B. eventually P b)"
proof (subst eventually_Inf, safe)
fix X assume "finite X" "X ⊆ B"
then have "∃b∈B. ∀x∈X. b ≤ x"
proof induct
case empty then show ?case
using ‹B ≠ {}› by auto
next
case (insert x X)
then obtain b where "b ∈ B" "⋀x. x ∈ X ⟹ b ≤ x"
by auto
with ‹insert x X ⊆ B› base[of b x] show ?case
by (auto intro: order_trans)
qed
then obtain b where "b ∈ B" "b ≤ Inf X"
by (auto simp: le_Inf_iff)
then show "eventually P (Inf X) ⟹ 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 ≠ {} ⟹ (⋀a b. a ∈ B ⟹ b ∈ B ⟹ ∃x∈B. F x ≤ inf (F a) (F b)) ⟹
eventually P (⨅b∈B. F b) ⟷ (∃b∈B. eventually P (F b))"
by (subst eventually_Inf_base) auto

lemma eventually_INF1: "i ∈ I ⟹ eventually P (F i) ⟹ eventually P (⨅i∈I. F i)"
using filter_leD[OF INF_lower] .

subsubsection ‹Map function for filters›

definition filtermap :: "('a ⇒ 'b) ⇒ 'a filter ⇒ 'b filter"
where "filtermap f F = Abs_filter (λP. eventually (λx. P (f x)) F)"

lemma eventually_filtermap:
"eventually P (filtermap f F) = eventually (λx. 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 (λx. x) F = F"

lemma filtermap_filtermap:
"filtermap f (filtermap g F) = filtermap (λx. f (g x)) F"

lemma filtermap_mono: "F ≤ F' ⟹ filtermap f F ≤ filtermap f F'"
unfolding le_filter_def eventually_filtermap by simp

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

lemma filtermap_bot_iff: "filtermap f F = bot ⟷ F = bot"

lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)"
by (simp add: filter_eq_iff eventually_filtermap eventually_sup)

lemma filtermap_SUP: "filtermap f (⨆b∈B. F b) = (⨆b∈B. filtermap f (F b))"
by (simp add: filter_eq_iff eventually_Sup eventually_filtermap)

lemma filtermap_inf: "filtermap f (inf F1 F2) ≤ inf (filtermap f F1) (filtermap f F2)"
by (intro inf_greatest filtermap_mono inf_sup_ord)

lemma filtermap_INF: "filtermap f (⨅b∈B. F b) ≤ (⨅b∈B. filtermap f (F b))"
by (rule INF_greatest, rule filtermap_mono, erule INF_lower)

subsubsection ‹Contravariant map function for filters›

definition filtercomap :: "('a ⇒ 'b) ⇒ 'b filter ⇒ 'a filter" where
"filtercomap f F = Abs_filter (λP. ∃Q. eventually Q F ∧ (∀x. Q (f x) ⟶ P x))"

lemma eventually_filtercomap:
"eventually P (filtercomap f F) ⟷ (∃Q. eventually Q F ∧ (∀x. Q (f x) ⟶ P x))"
unfolding filtercomap_def
proof (intro eventually_Abs_filter, unfold_locales, goal_cases)
case 1
show ?case by (auto intro!: exI[of _ "λ_. True"])
next
case (2 P Q)
from 2(1) guess P' by (elim exE conjE) note P' = this
from 2(2) guess Q' by (elim exE conjE) note Q' = this
show ?case
by (rule exI[of _ "λx. P' x ∧ Q' x"])
(insert P' Q', auto intro!: eventually_conj)
next
case (3 P Q)
thus ?case by blast
qed

lemma filtercomap_ident: "filtercomap (λx. x) F = F"
by (auto simp: filter_eq_iff eventually_filtercomap elim!: eventually_mono)

lemma filtercomap_filtercomap: "filtercomap f (filtercomap g F) = filtercomap (λx. g (f x)) F"
unfolding filter_eq_iff by (auto simp: eventually_filtercomap)

lemma filtercomap_mono: "F ≤ F' ⟹ filtercomap f F ≤ filtercomap f F'"
by (auto simp: eventually_filtercomap le_filter_def)

lemma filtercomap_bot [simp]: "filtercomap f bot = bot"
by (auto simp: filter_eq_iff eventually_filtercomap)

lemma filtercomap_top [simp]: "filtercomap f top = top"
by (auto simp: filter_eq_iff eventually_filtercomap)

lemma filtercomap_inf: "filtercomap f (inf F1 F2) = inf (filtercomap f F1) (filtercomap f F2)"
unfolding filter_eq_iff
proof safe
fix P
assume "eventually P (filtercomap f (F1 ⊓ F2))"
then obtain Q R S where *:
"eventually Q F1" "eventually R F2" "⋀x. Q x ⟹ R x ⟹ S x" "⋀x. S (f x) ⟹ P x"
unfolding eventually_filtercomap eventually_inf by blast
from * have "eventually (λx. Q (f x)) (filtercomap f F1)"
"eventually (λx. R (f x)) (filtercomap f F2)"
by (auto simp: eventually_filtercomap)
with * show "eventually P (filtercomap f F1 ⊓ filtercomap f F2)"
unfolding eventually_inf by blast
next
fix P
assume "eventually P (inf (filtercomap f F1) (filtercomap f F2))"
then obtain Q Q' R R' where *:
"eventually Q F1" "eventually R F2" "⋀x. Q (f x) ⟹ Q' x" "⋀x. R (f x) ⟹ R' x"
"⋀x. Q' x ⟹ R' x ⟹ P x"
unfolding eventually_filtercomap eventually_inf by blast
from * have "eventually (λx. Q x ∧ R x) (F1 ⊓ F2)" by (auto simp: eventually_inf)
with * show "eventually P (filtercomap f (F1 ⊓ F2))"
by (auto simp: eventually_filtercomap)
qed

lemma filtercomap_sup: "filtercomap f (sup F1 F2) ≥ sup (filtercomap f F1) (filtercomap f F2)"
by (intro sup_least filtercomap_mono inf_sup_ord)

lemma filtercomap_INF: "filtercomap f (⨅b∈B. F b) = (⨅b∈B. filtercomap f (F b))"
proof -
have *: "filtercomap f (⨅b∈B. F b) = (⨅b∈B. filtercomap f (F b))" if "finite B" for B
using that by induction (simp_all add: filtercomap_inf)
show ?thesis unfolding filter_eq_iff
proof
fix P
have "eventually P (⨅b∈B. filtercomap f (F b)) ⟷
(∃X. (X ⊆ B ∧ finite X) ∧ eventually P (⨅b∈X. filtercomap f (F b)))"
by (subst eventually_INF) blast
also have "… ⟷ (∃X. (X ⊆ B ∧ finite X) ∧ eventually P (filtercomap f (⨅b∈X. F b)))"
by (rule ex_cong) (simp add: *)
also have "… ⟷ eventually P (filtercomap f (INFIMUM B F))"
unfolding eventually_filtercomap by (subst eventually_INF) blast
finally show "eventually P (filtercomap f (INFIMUM B F)) =
eventually P (⨅b∈B. filtercomap f (F b))" ..
qed
qed

lemma filtercomap_SUP:
"filtercomap f (⨆b∈B. F b) ≥ (⨆b∈B. filtercomap f (F b))"
by (intro SUP_least filtercomap_mono SUP_upper)

lemma eventually_filtercomapI [intro]:
assumes "eventually P F"
shows   "eventually (λx. P (f x)) (filtercomap f F)"
using assms by (auto simp: eventually_filtercomap)

lemma filtermap_filtercomap: "filtermap f (filtercomap f F) ≤ F"
by (auto simp: le_filter_def eventually_filtermap eventually_filtercomap)

lemma filtercomap_filtermap: "filtercomap f (filtermap f F) ≥ F"
unfolding le_filter_def eventually_filtermap eventually_filtercomap
by (auto elim!: eventually_mono)

subsubsection ‹Standard filters›

definition principal :: "'a set ⇒ 'a filter" where
"principal S = Abs_filter (λP. ∀x∈S. P x)"

lemma eventually_principal: "eventually P (principal S) ⟷ (∀x∈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)) ⟷ eventually (λx. x ∈ s ⟶ 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 ⟷ X = {}"
by (auto simp add: filter_eq_iff eventually_principal)

lemma principal_le_iff[iff]: "principal A ≤ principal B ⟷ A ⊆ B"
by (auto simp: le_filter_def eventually_principal)

lemma le_principal: "F ≤ principal A ⟷ eventually (λx. x ∈ A) F"
unfolding le_filter_def eventually_principal
apply safe
apply (erule_tac x="λx. x ∈ A" in allE)
apply (auto elim: eventually_mono)
done

lemma principal_inject[iff]: "principal A = principal B ⟷ A = B"
unfolding eq_iff by simp

lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A ∪ B)"
unfolding filter_eq_iff eventually_sup eventually_principal by auto

lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A ∩ B)"
unfolding filter_eq_iff eventually_inf eventually_principal
by (auto intro: exI[of _ "λx. x ∈ A"] exI[of _ "λx. x ∈ B"])

lemma SUP_principal[simp]: "(⨆i∈I. principal (A i)) = principal (⋃i∈I. A i)"
unfolding filter_eq_iff eventually_Sup by (auto simp: eventually_principal)

lemma INF_principal_finite: "finite X ⟹ (⨅x∈X. principal (f x)) = principal (⋂x∈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

lemma filtercomap_principal[simp]: "filtercomap f (principal A) = principal (f -` A)"
unfolding filter_eq_iff eventually_filtercomap eventually_principal by fast

subsubsection ‹Order filters›

definition at_top :: "('a::order) filter"
where "at_top = (⨅k. principal {k ..})"

lemma at_top_sub: "at_top = (⨅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 ⟷ (∃N::'a::linorder. ∀n≥N. P n)"
unfolding at_top_def
by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)

lemma eventually_filtercomap_at_top_linorder:
"eventually P (filtercomap f at_top) ⟷ (∃N::'a::linorder. ∀x. f x ≥ N ⟶ P x)"
by (auto simp: eventually_filtercomap eventually_at_top_linorder)

lemma eventually_at_top_linorderI:
fixes c::"'a::linorder"
assumes "⋀x. c ≤ x ⟹ P x"
shows "eventually P at_top"
using assms by (auto simp: eventually_at_top_linorder)

lemma eventually_ge_at_top [simp]:
"eventually (λx. (c::_::linorder) ≤ x) at_top"
unfolding eventually_at_top_linorder by auto

lemma eventually_at_top_dense: "eventually P at_top ⟷ (∃N::'a::{no_top, linorder}. ∀n>N. P n)"
proof -
have "eventually P (⨅k. principal {k <..}) ⟷ (∃N::'a. ∀n>N. P n)"
by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
also have "(⨅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_filtercomap_at_top_dense:
"eventually P (filtercomap f at_top) ⟷ (∃N::'a::{no_top, linorder}. ∀x. f x > N ⟶ P x)"
by (auto simp: eventually_filtercomap eventually_at_top_dense)

lemma eventually_at_top_not_equal [simp]: "eventually (λx::'a::{no_top, linorder}. x ≠ c) at_top"
unfolding eventually_at_top_dense by auto

lemma eventually_gt_at_top [simp]: "eventually (λx. (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 (λx. ∀y≥x. P y) at_top"
proof -
from assms obtain x where "⋀y. y ≥ x ⟹ P y" by (auto simp: eventually_at_top_linorder)
hence "∀z≥y. P z" if "y ≥ 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 = (⨅k. principal {.. k})"

lemma at_bot_sub: "at_bot = (⨅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 ⇒ bool" shows "eventually P at_bot ⟷ (∃N. ∀n≤N. P n)"
unfolding at_bot_def
by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)

lemma eventually_filtercomap_at_bot_linorder:
"eventually P (filtercomap f at_bot) ⟷ (∃N::'a::linorder. ∀x. f x ≤ N ⟶ P x)"
by (auto simp: eventually_filtercomap eventually_at_bot_linorder)

lemma eventually_le_at_bot [simp]:
"eventually (λx. x ≤ (c::_::linorder)) at_bot"
unfolding eventually_at_bot_linorder by auto

lemma eventually_at_bot_dense: "eventually P at_bot ⟷ (∃N::'a::{no_bot, linorder}. ∀n<N. P n)"
proof -
have "eventually P (⨅k. principal {..< k}) ⟷ (∃N::'a. ∀n<N. P n)"
by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
also have "(⨅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_filtercomap_at_bot_dense:
"eventually P (filtercomap f at_bot) ⟷ (∃N::'a::{no_bot, linorder}. ∀x. f x < N ⟶ P x)"
by (auto simp: eventually_filtercomap eventually_at_bot_dense)

lemma eventually_at_bot_not_equal [simp]: "eventually (λx::'a::{no_bot, linorder}. x ≠ c) at_bot"
unfolding eventually_at_bot_dense by auto

lemma eventually_gt_at_bot [simp]:
"eventually (λx. x < (c::_::unbounded_dense_linorder)) at_bot"
unfolding eventually_at_bot_dense by auto

lemma trivial_limit_at_bot_linorder [simp]: "¬ 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]: "¬ trivial_limit (at_top ::('a::linorder) filter)"
unfolding trivial_limit_def
by (metis eventually_at_top_linorder order_refl)

subsection ‹Sequentially›

abbreviation sequentially :: "nat filter"
where "sequentially ≡ at_top"

lemma eventually_sequentially:
"eventually P sequentially ⟷ (∃N. ∀n≥N. P n)"
by (rule eventually_at_top_linorder)

lemma sequentially_bot [simp, intro]: "sequentially ≠ bot"
unfolding filter_eq_iff eventually_sequentially by auto

lemmas trivial_limit_sequentially = sequentially_bot

lemma eventually_False_sequentially [simp]:
"¬ eventually (λn. False) sequentially"

lemma le_sequentially:
"F ≤ sequentially ⟷ (∀N. eventually (λn. N ≤ n) F)"
by (simp add: at_top_def le_INF_iff le_principal)

lemma eventually_sequentiallyI [intro?]:
assumes "⋀x. c ≤ x ⟹ P x"
shows "eventually P sequentially"
using assms by (auto simp: eventually_sequentially)

lemma eventually_sequentially_Suc [simp]: "eventually (λi. P (Suc i)) sequentially ⟷ eventually P sequentially"
unfolding eventually_sequentially by (metis Suc_le_D Suc_le_mono le_Suc_eq)

lemma eventually_sequentially_seg [simp]: "eventually (λn. P (n + k)) sequentially ⟷ eventually P sequentially"
using eventually_sequentially_Suc[of "λn. P (n + k)" for k] by (induction k) auto

lemma filtermap_sequentually_ne_bot: "filtermap f sequentially ≠ bot"

subsection ‹The cofinite filter›

definition "cofinite = Abs_filter (λP. finite {x. ¬ P x})"

abbreviation Inf_many :: "('a ⇒ bool) ⇒ bool"  (binder "∃⇩∞" 10)
where "Inf_many P ≡ frequently P cofinite"

abbreviation Alm_all :: "('a ⇒ bool) ⇒ bool"  (binder "∀⇩∞" 10)
where "Alm_all P ≡ eventually P cofinite"

notation (ASCII)
Inf_many  (binder "INFM " 10) and
Alm_all  (binder "MOST " 10)

lemma eventually_cofinite: "eventually P cofinite ⟷ finite {x. ¬ P x}"
unfolding cofinite_def
proof (rule eventually_Abs_filter, rule is_filter.intro)
fix P Q :: "'a ⇒ bool" assume "finite {x. ¬ P x}" "finite {x. ¬ Q x}"
from finite_UnI[OF this] show "finite {x. ¬ (P x ∧ Q x)}"
by (rule rev_finite_subset) auto
next
fix P Q :: "'a ⇒ bool" assume P: "finite {x. ¬ P x}" and *: "∀x. P x ⟶ Q x"
from * show "finite {x. ¬ Q x}"
by (intro finite_subset[OF _ P]) auto
qed simp

lemma frequently_cofinite: "frequently P cofinite ⟷ ¬ finite {x. P x}"

lemma cofinite_bot[simp]: "cofinite = (bot::'a filter) ⟷ 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 ⇒ bool" assume [simp]: "finite {x. ¬ P x}"
show "∃N. ∀n≥N. P n"
proof cases
assume "{x. ¬ P x} ≠ {}" then show ?thesis
by (intro exI[of _ "Suc (Max {x. ¬ P x})"]) (auto simp: Suc_le_eq)
qed auto
next
fix P :: "nat ⇒ bool" and N :: nat assume "∀n≥N. P n"
then have "{x. ¬ P x} ⊆ {..< N}"
by (auto simp: not_le)
then show "finite {x. ¬ P x}"
by (blast intro: finite_subset)
qed

subsubsection ‹Product of filters›

definition prod_filter :: "'a filter ⇒ 'b filter ⇒ ('a × 'b) filter" (infixr "×⇩F" 80) where
"prod_filter F G =
(⨅(P, Q)∈{(P, Q). eventually P F ∧ eventually Q G}. principal {(x, y). P x ∧ Q y})"

lemma eventually_prod_filter: "eventually P (F ×⇩F G) ⟷
(∃Pf Pg. eventually Pf F ∧ eventually Pg G ∧ (∀x y. Pf x ⟶ Pg y ⟶ P (x, y)))"
unfolding prod_filter_def
proof (subst eventually_INF_base, goal_cases)
case 2
moreover have "eventually Pf F ⟹ eventually Qf F ⟹ eventually Pg G ⟹ eventually Qg G ⟹
∃P Q. eventually P F ∧ eventually Q G ∧
Collect P × Collect Q ⊆ Collect Pf × Collect Pg ∩ Collect Qf × 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 ≠ bot"
shows "(∀⇩F (x, y) in A ×⇩F B. P x) ⟷ (∀⇩F x in A. P x)"
unfolding eventually_prod_filter
proof safe
fix R Q
assume *: "∀⇩F x in A. R x" "∀⇩F x in B. Q x" "∀x y. R x ⟶ Q y ⟶ P x"
with ‹B ≠ bot› 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 "∃Pf Pg. eventually Pf A ∧ eventually Pg B ∧ (∀x y. Pf x ⟶ Pg y ⟶ P x)"
by (intro exI[of _ P] exI[of _ "λx. True"]) auto
qed

lemma eventually_prod2:
assumes "A ≠ bot"
shows "(∀⇩F (x, y) in A ×⇩F B. P y) ⟷ (∀⇩F y in B. P y)"
unfolding eventually_prod_filter
proof safe
fix R Q
assume *: "∀⇩F x in A. R x" "∀⇩F x in B. Q x" "∀x y. R x ⟶ Q y ⟶ P y"
with ‹A ≠ bot› 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 "∃Pf Pg. eventually Pf A ∧ eventually Pg B ∧ (∀x y. Pf x ⟶ Pg y ⟶ P y)"
by (intro exI[of _ P] exI[of _ "λx. True"]) auto
qed

lemma INF_filter_bot_base:
fixes F :: "'a ⇒ 'b filter"
assumes *: "⋀i j. i ∈ I ⟹ j ∈ I ⟹ ∃k∈I. F k ≤ F i ⊓ F j"
shows "(⨅i∈I. F i) = bot ⟷ (∃i∈I. F i = bot)"
proof (cases "∃i∈I. F i = bot")
case True
then have "(⨅i∈I. F i) ≤ bot"
by (auto intro: INF_lower2)
with True show ?thesis
by (auto simp: bot_unique)
next
case False
moreover have "(⨅i∈I. F i) ≠ bot"
proof (cases "I = {}")
case True
then show ?thesis
next
case False': False
show ?thesis
proof (rule INF_filter_not_bot)
fix J
assume "finite J" "J ⊆ I"
then have "∃k∈I. F k ≤ (⨅i∈J. F i)"
proof (induct J)
case empty
then show ?case
using ‹I ≠ {}› by auto
next
case (insert i J)
then obtain k where "k ∈ I" "F k ≤ (⨅i∈J. F i)" by auto
with insert *[of i k] show ?case
by auto
qed
with False show "(⨅i∈J. F i) ≠ ⊥"
by (auto simp: bot_unique)
qed
qed
ultimately show ?thesis
by auto
qed

lemma Collect_empty_eq_bot: "Collect P = {} ⟷ P = ⊥"
by auto

lemma prod_filter_eq_bot: "A ×⇩F B = bot ⟷ A = bot ∨ B = bot"
unfolding trivial_limit_def
proof
assume "∀⇩F x in A ×⇩F B. False"
then obtain Pf Pg
where Pf: "eventually (λx. Pf x) A" and Pg: "eventually (λy. Pg y) B"
and *: "∀x y. Pf x ⟶ Pg y ⟶ False"
unfolding eventually_prod_filter by fast
from * have "(∀x. ¬ Pf x) ∨ (∀y. ¬ Pg y)" by fast
with Pf Pg show "(∀⇩F x in A. False) ∨ (∀⇩F x in B. False)" by auto
next
assume "(∀⇩F x in A. False) ∨ (∀⇩F x in B. False)"
then show "∀⇩F x in A ×⇩F B. False"
unfolding eventually_prod_filter by (force intro: eventually_True)
qed

lemma prod_filter_mono: "F ≤ F' ⟹ G ≤ G' ⟹ F ×⇩F G ≤ F' ×⇩F G'"
by (auto simp: le_filter_def eventually_prod_filter)

lemma prod_filter_mono_iff:
assumes nAB: "A ≠ bot" "B ≠ bot"
shows "A ×⇩F B ≤ C ×⇩F D ⟷ A ≤ C ∧ B ≤ D"
proof safe
assume *: "A ×⇩F B ≤ C ×⇩F D"
with assms have "A ×⇩F B ≠ bot"
by (auto simp: bot_unique prod_filter_eq_bot)
with * have "C ×⇩F D ≠ bot"
by (auto simp: bot_unique)
then have nCD: "C ≠ bot" "D ≠ bot"
by (auto simp: prod_filter_eq_bot)

show "A ≤ C"
proof (rule filter_leI)
fix P assume "eventually P C" with *[THEN filter_leD, of "λ(x, y). P x"] show "eventually P A"
using nAB nCD by (simp add: eventually_prod1 eventually_prod2)
qed

show "B ≤ D"
proof (rule filter_leI)
fix P assume "eventually P D" with *[THEN filter_leD, of "λ(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 ×⇩F F) ⟷
(∃Q. eventually Q F ∧ (∀x y. Q x ⟶ Q y ⟶ 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 ×⇩F sequentially) ⟷ (∃N. ∀m ≥ N. ∀n ≥ N. P (n, m))"
unfolding eventually_prod_same eventually_sequentially by auto

lemma principal_prod_principal: "principal A ×⇩F principal B = principal (A × B)"
unfolding filter_eq_iff eventually_prod_filter eventually_principal
by (fast intro: exI[of _ "λx. x ∈ A"] exI[of _ "λx. x ∈ B"])

lemma le_prod_filterI:
"filtermap fst F ≤ A ⟹ filtermap snd F ≤ B ⟹ F ≤ A ×⇩F B"
unfolding le_filter_def eventually_filtermap eventually_prod_filter
by (force elim: eventually_elim2)

lemma filtermap_fst_prod_filter: "filtermap fst (A ×⇩F B) ≤ A"
unfolding le_filter_def eventually_filtermap eventually_prod_filter
by (force intro: eventually_True)

lemma filtermap_snd_prod_filter: "filtermap snd (A ×⇩F B) ≤ B"
unfolding le_filter_def eventually_filtermap eventually_prod_filter
by (force intro: eventually_True)

lemma prod_filter_INF:
assumes "I ≠ {}" and "J ≠ {}"
shows "(⨅i∈I. A i) ×⇩F (⨅j∈J. B j) = (⨅i∈I. ⨅j∈J. A i ×⇩F B j)"
proof (rule antisym)
from ‹I ≠ {}› obtain i where "i ∈ I" by auto
from ‹J ≠ {}› obtain j where "j ∈ J" by auto

show "(⨅i∈I. ⨅j∈J. A i ×⇩F B j) ≤ (⨅i∈I. A i) ×⇩F (⨅j∈J. B j)"
by (fast intro: le_prod_filterI INF_greatest INF_lower2
order_trans[OF filtermap_INF] `i ∈ I` `j ∈ J`
filtermap_fst_prod_filter filtermap_snd_prod_filter)
show "(⨅i∈I. A i) ×⇩F (⨅j∈J. B j) ≤ (⨅i∈I. ⨅j∈J. A i ×⇩F B j)"
by (intro INF_greatest prod_filter_mono INF_lower)
qed

lemma filtermap_Pair: "filtermap (λx. (f x, g x)) F ≤ filtermap f F ×⇩F filtermap g F"
by (rule le_prod_filterI, simp_all add: filtermap_filtermap)

lemma eventually_prodI: "eventually P F ⟹ eventually Q G ⟹ eventually (λx. P (fst x) ∧ Q (snd x)) (F ×⇩F G)"
unfolding eventually_prod_filter by auto

lemma prod_filter_INF1: "I ≠ {} ⟹ (⨅i∈I. A i) ×⇩F B = (⨅i∈I. A i ×⇩F B)"
using prod_filter_INF[of I "{B}" A "λx. x"] by simp

lemma prod_filter_INF2: "J ≠ {} ⟹ A ×⇩F (⨅i∈J. B i) = (⨅i∈J. A ×⇩F B i)"
using prod_filter_INF[of "{A}" J "λx. x" B] by simp

lemma prod_filtermap1: "prod_filter (filtermap f F) G = filtermap (apfst f) (prod_filter F G)"
apply(clarsimp simp add: filter_eq_iff eventually_filtermap eventually_prod_filter; safe)
subgoal by auto
subgoal for P Q R by(rule exI[where x="λy. ∃x. y = f x ∧ Q x"])(auto intro: eventually_mono)
done

lemma prod_filtermap2: "prod_filter F (filtermap g G) = filtermap (apsnd g) (prod_filter F G)"
apply(clarsimp simp add: filter_eq_iff eventually_filtermap eventually_prod_filter; safe)
subgoal by auto
subgoal for P Q R  by(auto intro: exI[where x="λy. ∃x. y = g x ∧ R x"] eventually_mono)
done

lemma prod_filter_assoc:
"prod_filter (prod_filter F G) H = filtermap (λ(x, y, z). ((x, y), z)) (prod_filter F (prod_filter G H))"
apply(clarsimp simp add: filter_eq_iff eventually_filtermap eventually_prod_filter; safe)
subgoal for P Q R S T by(auto 4 4 intro: exI[where x="λ(a, b). T a ∧ S b"])
subgoal for P Q R S T by(auto 4 3 intro: exI[where x="λ(a, b). Q a ∧ S b"])
done

lemma prod_filter_principal_singleton: "prod_filter (principal {x}) F = filtermap (Pair x) F"
by(fastforce simp add: filter_eq_iff eventually_prod_filter eventually_principal eventually_filtermap elim: eventually_mono intro: exI[where x="λa. a = x"])

lemma prod_filter_principal_singleton2: "prod_filter F (principal {x}) = filtermap (λa. (a, x)) F"
by(fastforce simp add: filter_eq_iff eventually_prod_filter eventually_principal eventually_filtermap elim: eventually_mono intro: exI[where x="λa. a = x"])

lemma prod_filter_commute: "prod_filter F G = filtermap prod.swap (prod_filter G F)"
by(auto simp add: filter_eq_iff eventually_prod_filter eventually_filtermap)

subsection ‹Limits›

definition filterlim :: "('a ⇒ 'b) ⇒ 'b filter ⇒ 'a filter ⇒ bool" where
"filterlim f F2 F1 ⟷ filtermap f F1 ≤ F2"

syntax
"_LIM" :: "pttrns ⇒ 'a ⇒ 'b ⇒ 'a ⇒ bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)

translations
"LIM x F1. f :> F2" == "CONST filterlim (λx. f) F2 F1"

lemma filterlim_top [simp]: "filterlim f top F"

lemma filterlim_iff:
"(LIM x F1. f x :> F2) ⟷ (∀P. eventually P F2 ⟶ eventually (λx. P (f x)) F1)"
unfolding filterlim_def le_filter_def eventually_filtermap ..

lemma filterlim_compose:
"filterlim g F3 F2 ⟹ filterlim f F2 F1 ⟹ filterlim (λx. g (f x)) F3 F1"
unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)

lemma filterlim_mono:
"filterlim f F2 F1 ⟹ F2 ≤ F2' ⟹ F1' ≤ F1 ⟹ filterlim f F2' F1'"
unfolding filterlim_def by (metis filtermap_mono order_trans)

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

lemma filterlim_cong:
"F1 = F1' ⟹ F2 = F2' ⟹ eventually (λx. f x = g x) F2 ⟹ 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 ≤ F'" "G' ≤ G"
assumes eq: "eventually (λx. 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 ⟹ filtermap f F ≤ filtermap f G ⟷ F ≤ G"
apply (safe intro!: filtermap_mono)
apply (auto simp: le_filter_def eventually_filtermap)
apply (erule_tac x="λx. P (inv f x)" in allE)
apply auto
done

lemma eventually_compose_filterlim:
assumes "eventually P F" "filterlim f F G"
shows "eventually (λx. P (f x)) G"
using assms by (simp add: filterlim_iff)

lemma filtermap_eq_strong: "inj f ⟹ filtermap f F = filtermap f G ⟷ F = G"

lemma filtermap_fun_inverse:
assumes g: "filterlim g F G"
assumes f: "filterlim f G F"
assumes ev: "eventually (λx. f (g x) = x) G"
shows "filtermap f F = G"
proof (rule antisym)
show "filtermap f F ≤ 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 "… ≤ filtermap f F"
using g by (intro filtermap_mono) (simp add: filterlim_def)
finally show "G ≤ filtermap f F" .
qed

lemma filterlim_principal:
"(LIM x F. f x :> principal S) ⟷ (eventually (λx. f x ∈ S) F)"
unfolding filterlim_def eventually_filtermap le_principal ..

lemma filterlim_inf:
"(LIM x F1. f x :> inf F2 F3) ⟷ ((LIM x F1. f x :> F2) ∧ (LIM x F1. f x :> F3))"
unfolding filterlim_def by simp

lemma filterlim_INF:
"(LIM x F. f x :> (⨅b∈B. G b)) ⟷ (∀b∈B. LIM x F. f x :> G b)"
unfolding filterlim_def le_INF_iff ..

lemma filterlim_INF_INF:
"(⋀m. m ∈ J ⟹ ∃i∈I. filtermap f (F i) ≤ G m) ⟹ LIM x (⨅i∈I. F i). f x :> (⨅j∈J. G j)"
unfolding filterlim_def by (rule order_trans[OF filtermap_INF INF_mono])

lemma filterlim_base:
"(⋀m x. m ∈ J ⟹ i m ∈ I) ⟹ (⋀m x. m ∈ J ⟹ x ∈ F (i m) ⟹ f x ∈ G m) ⟹
LIM x (⨅i∈I. principal (F i)). f x :> (⨅j∈J. principal (G j))"
by (force intro!: filterlim_INF_INF simp: image_subset_iff)

lemma filterlim_base_iff:
assumes "I ≠ {}" and chain: "⋀i j. i ∈ I ⟹ j ∈ I ⟹ F i ⊆ F j ∨ F j ⊆ F i"
shows "(LIM x (⨅i∈I. principal (F i)). f x :> ⨅j∈J. principal (G j)) ⟷
(∀j∈J. ∃i∈I. ∀x∈F i. f x ∈ G j)"
unfolding filterlim_INF filterlim_principal
proof (subst eventually_INF_base)
fix i j assume "i ∈ I" "j ∈ I"
with chain[OF this] show "∃x∈I. principal (F x) ≤ inf (principal (F i)) (principal (F j))"
by auto
qed (auto simp: eventually_principal ‹I ≠ {}›)

lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (λx. f (g x)) F1 F2"
unfolding filterlim_def filtermap_filtermap ..

lemma filterlim_sup:
"filterlim f F F1 ⟹ filterlim f F F2 ⟹ filterlim f F (sup F1 F2)"
unfolding filterlim_def filtermap_sup by auto

lemma filterlim_sequentially_Suc:
"(LIM x sequentially. f (Suc x) :> F) ⟷ (LIM x sequentially. f x :> F)"
unfolding filterlim_iff by (subst eventually_sequentially_Suc) simp

lemma filterlim_Suc: "filterlim Suc sequentially sequentially"

lemma filterlim_If:
"LIM x inf F (principal {x. P x}). f x :> G ⟹
LIM x inf F (principal {x. ¬ P x}). g x :> G ⟹
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 ⟹ LIM x F. g x :> H ⟹ LIM x F. (f x, g x) :> G ×⇩F H"
unfolding filterlim_def
by (rule order_trans[OF filtermap_Pair prod_filter_mono])

subsection ‹Limits to @{const at_top} and @{const at_bot}›

lemma filterlim_at_top:
fixes f :: "'a ⇒ ('b::linorder)"
shows "(LIM x F. f x :> at_top) ⟷ (∀Z. eventually (λx. Z ≤ 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 ⟹ eventually (λx. f x ≤ (g x::'a::linorder)) F ⟹
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 ⇒ ('b::unbounded_dense_linorder)"
shows "(LIM x F. f x :> at_top) ⟷ (∀Z. eventually (λx. 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 ⇒ ('b::linorder)" and c :: "'b"
shows "(LIM x F. f x :> at_top) ⟷ (∀Z≥c. eventually (λx. Z ≤ 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 ⇒ 'b::linorder"
assumes mono: "⋀x y. Q x ⟹ Q y ⟹ x ≤ y ⟹ f x ≤ f y"
assumes bij: "⋀x. P x ⟹ f (g x) = x" "⋀x. P x ⟹ 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: "⋀y. x ≤ y ⟹ 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 ≤ z"
with x have "P z" by auto
have "eventually (λx. g z ≤ x) at_top"
by (rule eventually_ge_at_top)
with Q show "eventually (λx. z ≤ f x) at_top"
by eventually_elim (metis mono bij ‹P z›)
qed
qed

lemma filterlim_at_top_gt:
fixes f :: "'a ⇒ ('b::unbounded_dense_linorder)" and c :: "'b"
shows "(LIM x F. f x :> at_top) ⟷ (∀Z>c. eventually (λx. Z ≤ 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 ⇒ ('b::linorder)"
shows "(LIM x F. f x :> at_bot) ⟷ (∀Z. eventually (λx. f x ≤ Z) F)"
by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_mono)

lemma filterlim_at_bot_dense:
fixes f :: "'a ⇒ ('b::{dense_linorder, no_bot})"
shows "(LIM x F. f x :> at_bot) ⟷ (∀Z. eventually (λx. 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 "∀Z. eventually (λx. f x ≤ Z) F"
hence "eventually (λx. f x ≤ Z') F" by auto
thus "eventually (λx. f x < Z) F"
apply (rule eventually_mono)
using 1 by auto
next
fix Z :: 'b
show "∀Z. eventually (λx. f x < Z) F ⟹ eventually (λx. f x ≤ 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 ⇒ ('b::linorder)" and c :: "'b"
shows "(LIM x F. f x :> at_bot) ⟷ (∀Z≤c. eventually (λx. Z ≥ f x) F)"
unfolding filterlim_at_bot
proof safe
fix Z assume *: "∀Z≤c. eventually (λx. Z ≥ f x) F"
with *[THEN spec, of "min Z c"] show "eventually (λx. Z ≥ f x) F"
by (auto elim!: eventually_mono)
qed simp

lemma filterlim_at_bot_lt:
fixes f :: "'a ⇒ ('b::unbounded_dense_linorder)" and c :: "'b"
shows "(LIM x F. f x :> at_bot) ⟷ (∀Z<c. eventually (λx. Z ≥ f x) F)"
by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans)

lemma filterlim_filtercomap [intro]: "filterlim f F (filtercomap f F)"
unfolding filterlim_def by (rule filtermap_filtercomap)

subsection ‹Setup @{typ "'a filter"} for lifting and transfer›

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

lemma filtermap_id' [simp]: "filtermap (λx. x) = (λF. F)"
using filtermap_id unfolding id_def .

context includes lifting_syntax
begin

definition map_filter_on :: "'a set ⇒ ('a ⇒ 'b) ⇒ 'a filter ⇒ 'b filter" where
"map_filter_on X f F = Abs_filter (λP. eventually (λx. P (f x) ∧ x ∈ X) F)"

lemma is_filter_map_filter_on:
"is_filter (λP. ∀⇩F x in F. P (f x) ∧ x ∈ X) ⟷ eventually (λx. x ∈ X) F"
proof(rule iffI; unfold_locales)
show "∀⇩F x in F. True ∧ x ∈ X" if "eventually (λx. x ∈ X) F" using that by simp
show "∀⇩F x in F. (P (f x) ∧ Q (f x)) ∧ x ∈ X" if "∀⇩F x in F. P (f x) ∧ x ∈ X" "∀⇩F x in F. Q (f x) ∧ x ∈ X" for P Q
using eventually_conj[OF that] by(auto simp add: conj_ac cong: conj_cong)
show "∀⇩F x in F. Q (f x) ∧ x ∈ X" if "∀x. P x ⟶ Q x" "∀⇩F x in F. P (f x) ∧ x ∈ X" for P Q
using that(2) by(rule eventually_mono)(use that(1) in auto)
show "eventually (λx. x ∈ X) F" if "is_filter (λP. ∀⇩F x in F. P (f x) ∧ x ∈ X)"
using is_filter.True[OF that] by simp
qed

lemma eventually_map_filter_on: "eventually P (map_filter_on X f F) = (∀⇩F x in F. P (f x) ∧ x ∈ X)"
if "eventually (λx. x ∈ X) F"
by(simp add: is_filter_map_filter_on map_filter_on_def eventually_Abs_filter that)

lemma map_filter_on_UNIV: "map_filter_on UNIV = filtermap"

lemma map_filter_on_comp: "map_filter_on X f (map_filter_on Y g F) = map_filter_on Y (f ∘ g) F"
if "g ` Y ⊆ X" and "eventually (λx. x ∈ Y) F"
unfolding map_filter_on_def using that(1)
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])

inductive rel_filter :: "('a ⇒ 'b ⇒ bool) ⇒ 'a filter ⇒ 'b filter ⇒ bool" for R F G where
"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"

lemma rel_filter_eq [relator_eq]: "rel_filter (=) = (=)"
proof(intro ext iffI)+
show "F = G" if "rel_filter (=) F G" for F G using that
by cases(clarsimp simp add: filter_eq_iff eventually_map_filter_on split_def cong: rev_conj_cong)
show "rel_filter (=) F G" if "F = G" for F G unfolding ‹F = G›
proof
let ?Z = "map_filter_on UNIV (λx. (x, x)) G"
have [simp]: "range (λx. (x, x)) ⊆ {(x, y). x = y}" by auto
show "map_filter_on {(x, y). x = y} fst ?Z = G" and "map_filter_on {(x, y). x = y} snd ?Z = G"
show "∀⇩F (x, y) in ?Z. x = y" by(simp add: eventually_map_filter_on)
qed
qed

lemma rel_filter_mono [relator_mono]: "rel_filter A ≤ rel_filter B" if le: "A ≤ B"
proof(clarify elim!: rel_filter.cases)
show "rel_filter B (map_filter_on {(x, y). A x y} fst Z) (map_filter_on {(x, y). A x y} snd Z)"
(is "rel_filter _ ?X ?Y") if "∀⇩F (x, y) in Z. A x y" for Z
proof
let ?Z = "map_filter_on {(x, y). A x y} id Z"
show "∀⇩F (x, y) in ?Z. B x y" using le that
by(simp add: eventually_map_filter_on le_fun_def split_def conj_commute cong: conj_cong)
have [simp]: "{(x, y). A x y} ⊆ {(x, y). B x y}" using le by auto
show "map_filter_on {(x, y). B x y} fst ?Z = ?X" "map_filter_on {(x, y). B x y} snd ?Z = ?Y"
using le that by(simp_all add: le_fun_def map_filter_on_comp)
qed
qed

lemma rel_filter_conversep: "rel_filter A¯¯ = (rel_filter A)¯¯"
proof(safe intro!: ext elim!: rel_filter.cases)
show *: "rel_filter A (map_filter_on {(x, y). A¯¯ x y} snd Z) (map_filter_on {(x, y). A¯¯ x y} fst Z)"
(is "rel_filter _ ?X ?Y") if "∀⇩F (x, y) in Z. A¯¯ x y" for A Z
proof
let ?Z = "map_filter_on {(x, y). A y x} prod.swap Z"
show "∀⇩F (x, y) in ?Z. A x y" using that by(simp add: eventually_map_filter_on)
have [simp]: "prod.swap ` {(x, y). A y x} ⊆ {(x, y). A x y}" by auto
show "map_filter_on {(x, y). A x y} fst ?Z = ?X" "map_filter_on {(x, y). A x y} snd ?Z = ?Y"
using that by(simp_all add: map_filter_on_comp o_def)
qed
show "rel_filter A¯¯ (map_filter_on {(x, y). A x y} snd Z) (map_filter_on {(x, y). A x y} fst Z)"
if "∀⇩F (x, y) in Z. A x y" for Z using *[of "A¯¯" Z] that by simp
qed

lemma rel_filter_distr [relator_distr]:
"rel_filter A OO rel_filter B = rel_filter (A OO B)"
proof(safe intro!: ext elim!: rel_filter.cases)
let ?AB = "{(x, y). (A OO B) x y}"
show "(rel_filter A OO rel_filter B)
(map_filter_on {(x, y). (A OO B) x y} fst Z) (map_filter_on {(x, y). (A OO B) x y} snd Z)"
(is "(_ OO _) ?F ?H") if "∀⇩F (x, y) in Z. (A OO B) x y" for Z
proof
let ?G = "map_filter_on ?AB (λ(x, y). SOME z. A x z ∧ B z y) Z"
show "rel_filter A ?F ?G"
proof
let ?Z = "map_filter_on ?AB (λ(x, y). (x, SOME z. A x z ∧ B z y)) Z"
show "∀⇩F (x, y) in ?Z. A x y" using that
by(auto simp add: eventually_map_filter_on split_def elim!: eventually_mono intro: someI2)
have [simp]: "(λp. (fst p, SOME z. A (fst p) z ∧ B z (snd p))) ` {p. (A OO B) (fst p) (snd p)} ⊆ {p. A (fst p) (snd p)}" by(auto intro: someI2)
show "map_filter_on {(x, y). A x y} fst ?Z = ?F" "map_filter_on {(x, y). A x y} snd ?Z = ?G"
using that by(simp_all add: map_filter_on_comp split_def o_def)
qed
show "rel_filter B ?G ?H"
proof
let ?Z = "map_filter_on ?AB (λ(x, y). (SOME z. A x z ∧ B z y, y)) Z"
show "∀⇩F (x, y) in ?Z. B x y" using that
by(auto simp add: eventually_map_filter_on split_def elim!: eventually_mono intro: someI2)
have [simp]: "(λp. (SOME z. A (fst p) z ∧ B z (snd p), snd p)) ` {p. (A OO B) (fst p) (snd p)} ⊆ {p. B (fst p) (snd p)}" by(auto intro: someI2)
show "map_filter_on {(x, y). B x y} fst ?Z = ?G" "map_filter_on {(x, y). B x y} snd ?Z = ?H"
using that by(simp_all add: map_filter_on_comp split_def o_def)
qed
qed

fix F G
assume F: "∀⇩F (x, y) in F. A x y" and G: "∀⇩F (x, y) in G. B x y"
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")
let ?X = "map_filter_on {(x, y). A x y} fst F"
and ?Z = "(map_filter_on {(x, y). B x y} snd G)"
have step: "∃P'≤P. ∃Q' ≤ Q. eventually P' F ∧ eventually Q' G ∧ {y. ∃x. P' (x, y)} = {y. ∃z. Q' (y, z)}"
if P: "eventually P F" and Q: "eventually Q G" for P Q
proof -
let ?P = "λ(x, y). P (x, y) ∧ A x y" and ?Q = "λ(y, z). Q (y, z) ∧ B y z"
define P' where "P' ≡ λ(x, y). ?P (x, y) ∧ (∃z. ?Q (y, z))"
define Q' where "Q' ≡ λ(y, z). ?Q (y, z) ∧ (∃x. ?P (x, y))"
have "P' ≤ P" "Q' ≤ Q" "{y. ∃x. P' (x, y)} = {y. ∃z. Q' (y, z)}"
moreover
from P Q F G have P': "eventually ?P F" and Q': "eventually ?Q G"
from P' F have "∀⇩F y in ?Y1. ∃x. P (x, y) ∧ A x y"
by(auto simp add: eventually_map_filter_on elim!: eventually_mono)
from this[folded eq] obtain Q'' where Q'': "eventually Q'' G"
and Q''P: "{y. ∃z. Q'' (y, z)} ⊆ {y. ∃x. ?P (x, y)}"
using G by(fastforce simp add: eventually_map_filter_on)
have "eventually (inf Q'' ?Q) G" using Q'' Q' by(auto intro: eventually_conj simp add: inf_fun_def)
then have "eventually Q' G" using Q''P  by(auto elim!: eventually_mono simp add: Q'_def)
moreover
from Q' G have "∀⇩F y in ?Y2. ∃z. Q (y, z) ∧ B y z"
by(auto simp add: eventually_map_filter_on elim!: eventually_mono)
from this[unfolded eq] obtain P'' where P'': "eventually P'' F"
and P''Q: "{y. ∃x. P'' (x, y)} ⊆ {y. ∃z. ?Q (y, z)}"
using F by(fastforce simp add: eventually_map_filter_on)
have "eventually (inf P'' ?P) F" using P'' P' by(auto intro: eventually_conj simp add: inf_fun_def)
then have "eventually P' F" using P''Q  by(auto elim!: eventually_mono simp add: P'_def)
ultimately show ?thesis by blast
qed

show "rel_filter (A OO B) ?X ?Z"
proof
let ?Y = "λY. ∃X Z. eventually X ?X ∧ eventually Z ?Z ∧ (λ(x, z). X x ∧ Z z ∧ (A OO B) x z) ≤ Y"
have Y: "is_filter ?Y"
proof
show "?Y (λ_. True)" by(auto simp add: le_fun_def intro: eventually_True)
show "?Y (λx. P x ∧ Q x)" if "?Y P" "?Y Q" for P Q using that
apply clarify
apply(intro exI conjI; (elim eventually_rev_mp; fold imp_conjL; intro always_eventually allI; rule imp_refl)?)
apply auto
done
show "?Y Q" if "?Y P" "∀x. P x ⟶ Q x" for P Q using that by blast
qed
define Y where "Y = Abs_filter ?Y"
have eventually_Y: "eventually P Y ⟷ ?Y P" for P
using eventually_Abs_filter[OF Y, of P] by(simp add: Y_def)
show YY: "∀⇩F (x, y) in Y. (A OO B) x y" using F G
by(auto simp add: eventually_Y eventually_map_filter_on eventually_conj_iff intro!: eventually_True)
have "?Y (λ(x, z). P x ∧ (A OO B) x z) ⟷ (∀⇩F (x, y) in F. P x ∧ A x y)" (is "?lhs = ?rhs") for P
proof
show ?lhs if ?rhs using G F that
by(auto 4 3 intro: exI[where x="λ_. True"] simp add: eventually_map_filter_on split_def)
assume ?lhs
then obtain X Z where "∀⇩F (x, y) in F. X x ∧ A x y"
and "∀⇩F (x, y) in G. Z y ∧ B x y"
and "(λ(x, z). X x ∧ Z z ∧ (A OO B) x z) ≤ (λ(x, z). P x ∧ (A OO B) x z)"
using F G by(auto simp add: eventually_map_filter_on split_def)
from step[OF this(1, 2)] this(3)
show ?rhs by(clarsimp elim!: eventually_rev_mp simp add: le_fun_def)(fastforce intro: always_eventually)
qed
then show "map_filter_on ?AB fst Y = ?X"

have "?Y (λ(x, z). P z ∧ (A OO B) x z) ⟷ (∀⇩F (x, y) in G. P y ∧ B x y)" (is "?lhs = ?rhs") for P
proof
show ?lhs if ?rhs using G F that
by(auto 4 3 intro: exI[where x="λ_. True"] simp add: eventually_map_filter_on split_def)
assume ?lhs
then obtain X Z where "∀⇩F (x, y) in F. X x ∧ A x y"
and "∀⇩F (x, y) in G. Z y ∧ B x y"
and "(λ(x, z). X x ∧ Z z ∧ (A OO B) x z) ≤ (λ(x, z). P z ∧ (A OO B) x z)"
using F G by(auto simp add: eventually_map_filter_on split_def)
from step[OF this(1, 2)] this(3)
show ?rhs by(clarsimp elim!: eventually_rev_mp simp add: le_fun_def)(fastforce intro: always_eventually)
qed
then show "map_filter_on ?AB snd Y = ?Z"
qed
qed

lemma filtermap_parametric: "((A ===> B) ===> rel_filter A ===> rel_filter B) filtermap filtermap"
proof(intro rel_funI; erule rel_filter.cases; hypsubst)
fix f g Z
assume fg: "(A ===> B) f g" and Z: "∀⇩F (x, y) in Z. A x y"
have "rel_filter B (map_filter_on {(x, y). A x y} (f ∘ fst) Z) (map_filter_on {(x, y). A x y} (g ∘ snd) Z)"
(is "rel_filter _ ?F ?G")
proof
let ?Z = "map_filter_on {(x, y). A x y} (map_prod f g) Z"
show "∀⇩F (x, y) in ?Z. B x y" using fg Z
by(auto simp add: eventually_map_filter_on split_def elim!: eventually_mono rel_funD)
have [simp]: "map_prod f g ` {p. A (fst p) (snd p)} ⊆ {p. B (fst p) (snd p)}"
using fg by(auto dest: rel_funD)
show "map_filter_on {(x, y). B x y} fst ?Z = ?F" "map_filter_on {(x, y). B x y} snd ?Z = ?G"
using Z by(auto simp add: map_filter_on_comp split_def)
qed
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))"
using Z by(simp add: map_filter_on_UNIV[symmetric] map_filter_on_comp)
qed

lemma rel_filter_Grp: "rel_filter (Grp UNIV f) = Grp UNIV (filtermap f)"
proof((intro antisym predicate2I; (elim GrpE; hypsubst)?), rule GrpI[OF _ UNIV_I])
fix F G
assume "rel_filter (Grp UNIV f) F G"
hence "rel_filter (=) (filtermap f F) (filtermap id G)"
by(rule filtermap_parametric[THEN rel_funD, THEN rel_funD, rotated])(simp add: Grp_def rel_fun_def)
thus "filtermap f F = G" by(simp add: rel_filter_eq)
next
fix F :: "'a filter"
have "rel_filter (=) F F" by(simp add: rel_filter_eq)
hence "rel_filter (Grp UNIV f) (filtermap id F) (filtermap f F)"
by(rule filtermap_parametric[THEN rel_funD, THEN rel_funD, rotated])(simp add: Grp_def rel_fun_def)
thus "rel_filter (Grp UNIV f) F (filtermap f F)" by simp
qed

lemma Quotient_filter [quot_map]:
"Quotient R Abs Rep T ⟹ Quotient (rel_filter R) (filtermap Abs) (filtermap Rep) (rel_filter T)"
unfolding Quotient_alt_def5 rel_filter_eq[symmetric] rel_filter_Grp[symmetric]

lemma left_total_rel_filter [transfer_rule]: "left_total A ⟹ left_total (rel_filter A)"
unfolding left_total_alt_def rel_filter_eq[symmetric] rel_filter_conversep[symmetric] rel_filter_distr
by(rule rel_filter_mono)

lemma right_total_rel_filter [transfer_rule]: "right_total A ⟹ right_total (rel_filter A)"
using left_total_rel_filter[of "A¯¯"] by(simp add: rel_filter_conversep)

lemma bi_total_rel_filter [transfer_rule]: "bi_total A ⟹ bi_total (rel_filter A)"
unfolding bi_total_alt_def by(simp add: left_total_rel_filter right_total_rel_filter)

lemma left_unique_rel_filter [transfer_rule]: "left_unique A ⟹ left_unique (rel_filter A)"
unfolding left_unique_alt_def rel_filter_eq[symmetric] rel_filter_conversep[symmetric] rel_filter_distr
by(rule rel_filter_mono)

lemma right_unique_rel_filter [transfer_rule]:
"right_unique A ⟹ right_unique (rel_filter A)"
using left_unique_rel_filter[of "A¯¯"] by(simp add: rel_filter_conversep)

lemma bi_unique_rel_filter [transfer_rule]: "bi_unique A ⟹ bi_unique (rel_filter A)"

lemma eventually_parametric [transfer_rule]:
"((A ===> (=)) ===> rel_filter A ===> (=)) eventually eventually"
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)

lemma frequently_parametric [transfer_rule]: "((A ===> (=)) ===> rel_filter A ===> (=)) frequently frequently"
unfolding frequently_def[abs_def] by transfer_prover

lemma is_filter_parametric [transfer_rule]:
assumes [transfer_rule]: "bi_total A"
assumes [transfer_rule]: "bi_unique A"
shows "(((A ===> (=)) ===> (=)) ===> (=)) is_filter is_filter"
unfolding is_filter_def by transfer_prover

lemma top_filter_parametric [transfer_rule]: "rel_filter A top top" if "bi_total A"
proof
let ?Z = "principal {(x, y). A x y}"
show "∀⇩F (x, y) in ?Z. A x y" by(simp add: eventually_principal)
show "map_filter_on {(x, y). A x y} fst ?Z = top" "map_filter_on {(x, y). A x y} snd ?Z = top"
using that by(auto simp add: filter_eq_iff eventually_map_filter_on eventually_principal bi_total_def)
qed

lemma bot_filter_parametric [transfer_rule]: "rel_filter A bot bot"
proof
show "∀⇩F (x, y) in bot. A x y" by simp
show "map_filter_on {(x, y). A x y} fst bot = bot" "map_filter_on {(x, y). A x y} snd bot = bot"
qed

lemma principal_parametric [transfer_rule]: "(rel_set A ===> rel_filter A) principal principal"
proof(rule rel_funI rel_filter.intros)+
fix S S'
assume *: "rel_set A S S'"
define SS' where "SS' = S × S' ∩ {(x, y). A x y}"
have SS': "SS' ⊆ {(x, y). A x y}" and [simp]: "S = fst ` SS'" "S' = snd ` SS'"
using * by(auto 4 3 dest: rel_setD1 rel_setD2 intro: rev_image_eqI simp add: SS'_def)
let ?Z = "principal SS'"
show "∀⇩F (x, y) in ?Z. A x y" using SS' by(auto simp add: eventually_principal)
then show "map_filter_on {(x, y). A x y} fst ?Z = principal S"
and "map_filter_on {(x, y). A x y} snd ?Z = principal S'"
by(auto simp add: filter_eq_iff eventually_map_filter_on eventually_principal)
qed

lemma sup_filter_parametric [transfer_rule]:
"(rel_filter A ===> rel_filter A ===> rel_filter A) sup sup"
proof(intro rel_funI; elim rel_filter.cases; hypsubst)
show "rel_filter A
(map_filter_on {(x, y). A x y} fst FG ⊔ map_filter_on {(x, y). A x y} fst FG')
(map_filter_on {(x, y). A x y} snd FG ⊔ map_filter_on {(x, y). A x y} snd FG')"
(is "rel_filter _ (sup ?F ?G) (sup ?F' ?G')")
if "∀⇩F (x, y) in FG. A x y" "∀⇩F (x, y) in FG'. A x y" for FG FG'
proof
let ?Z = "sup FG FG'"
show "∀⇩F (x, y) in ?Z. A x y" by(simp add: eventually_sup that)
then show "map_filter_on {(x, y). A x y} fst ?Z = sup ?F ?G"
and "map_filter_on {(x, y). A x y} snd ?Z = sup ?F' ?G'"
qed
qed

lemma Sup_filter_parametric [transfer_rule]: "(rel_set (rel_filter A) ===> rel_filter A) Sup Sup"
proof(rule rel_funI)
fix S S'
define SS' where "SS' = S × S' ∩ {(F, G). rel_filter A F G}"
assume "rel_set (rel_filter A) S S'"
then have SS': "SS' ⊆ {(F, G). rel_filter A F G}" and [simp]: "S = fst ` SS'" "S' = snd ` SS'"
by(auto 4 3 dest: rel_setD1 rel_setD2 intro: rev_image_eqI simp add: SS'_def)
from SS' obtain Z where Z: "⋀F G. (F, G) ∈ SS' ⟹
(∀⇩F (x, y) in Z F G. A x y) ∧
id F = map_filter_on {(x, y). A x y} fst (Z F G) ∧
id G = map_filter_on {(x, y). A x y} snd (Z F G)"
unfolding rel_filter.simps by atomize_elim((rule choice allI)+; auto)
have id: "eventually P F = eventually P (id F)" "eventually Q G = eventually Q (id G)"
if "(F, G) ∈ SS'" for P Q F G by simp_all
show "rel_filter A (Sup S) (Sup S')"
proof
let ?Z = "SUP (F, G):SS'. Z F G"
show *: "∀⇩F (x, y) in ?Z. A x y" using Z by(auto simp add: eventually_Sup)
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'"
unfolding filter_eq_iff
by(auto 4 4 simp add: id eventually_Sup eventually_map_filter_on *[simplified eventually_Sup] simp del: id_apply dest: Z)
qed
qed

context
fixes A :: "'a ⇒ 'b ⇒ bool"
assumes [transfer_rule]: "bi_unique A"
begin

lemma le_filter_parametric [transfer_rule]:
"(rel_filter A ===> rel_filter A ===> (=)) (≤) (≤)"
unfolding le_filter_def[abs_def] by transfer_prover

lemma less_filter_parametric [transfer_rule]:
"(rel_filter A ===> rel_filter A ===> (=)) (<) (<)"
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

lemma prod_filter_parametric [transfer_rule]: includes lifting_syntax shows
"(rel_filter R ===> rel_filter S ===> rel_filter (rel_prod R S)) prod_filter prod_filter"
proof(intro rel_funI; elim rel_filter.cases; hypsubst)
fix F G
assume F: "∀⇩F (x, y) in F. R x y" and G: "∀⇩F (x, y) in G. S x y"
show "rel_filter (rel_prod R S)
(map_filter_on {(x, y). R x y} fst F ×⇩F map_filter_on {(x, y). S x y} fst G)
(map_filter_on {(x, y). R x y} snd F ×⇩F map_filter_on {(x, y). S x y} snd G)"
(is "rel_filter ?RS ?F ?G")
proof
let ?Z = "filtermap (λ((a, b), (a', b')). ((a, a'), (b, b'))) (prod_filter F G)"
show *: "∀⇩F (x, y) in ?Z. rel_prod R S x y" using F G
by(auto simp add: eventually_filtermap split_beta eventually_prod_filter)
show "map_filter_on {(x, y). ?RS x y} fst ?Z = ?F"
using F G
apply(clarsimp simp add: filter_eq_iff eventually_map_filter_on *)
apply(subst eventually_map_filter_on; simp)+
apply(rule iffI; clarsimp)
subgoal for P P' P''
apply(rule exI[where x="λa. ∃b. P' (a, b) ∧ R a b"]; rule conjI)
subgoal by(fastforce elim: eventually_rev_mp eventually_mono)
subgoal
by(rule exI[where x="λa. ∃b. P'' (a, b) ∧ S a b"])(fastforce elim: eventually_rev_mp eventually_mono)
done
subgoal by fastforce
done
show "map_filter_on {(x, y). ?RS x y} snd ?Z = ?G"
using F G
apply(clarsimp simp add: filter_eq_iff eventually_map_filter_on *)
apply(subst eventually_map_filter_on; simp)+
apply(rule iffI; clarsimp)
subgoal for P P' P''
apply(rule exI[where x="λb. ∃a. P' (a, b) ∧ R a b"]; rule conjI)
subgoal by(fastforce elim: eventually_rev_mp eventually_mono)
subgoal
by(rule exI[where x="λb. ∃a. P'' (a, b) ∧ S a b"])(fastforce elim: eventually_rev_mp eventually_mono)
done
subgoal by fastforce
done
qed
qed

text ‹Code generation for filters›

definition abstract_filter :: "(unit ⇒ 'a filter) ⇒ '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 ⇒ _" "sup :: _ filter ⇒ _" "less_eq :: _ filter ⇒ _"
Abs_filter]]

declare filterlim_principal [code]
declare principal_prod_principal [code]
declare filtermap_principal [code]
declare filtercomap_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 ⟷ eventually P F"

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 ⇒ 'a filter ⇒ bool"
where "equal_filter F F' ⟷ F = F'"

lemma equal_filter [code]:
"HOL.equal (principal A) (principal B) ⟷ A = B"