# 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)"
by (simp add: frequently_def)

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)"
by (simp add: frequently_def eventually_conj_iff)

lemma frequently_disj: "∃⇩Fx in F. P x ⟹ ∃⇩Fx in F. Q x ⟹ ∃⇩Fx in F. P x ∨ Q x"
by (simp add: frequently_disj_iff)

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"
by (simp add: frequently_bex_finite_distrib)

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_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)
›

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_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"
by (simp add: eventually_const_iff)

lemma frequently_const_iff: "frequently (λx. P) F ⟷ P ∧ F ≠ bot"
by (simp add: frequently_def eventually_const_iff)

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

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"
by (simp add: eventually_False)

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
by (simp add: eventually_F)
qed

lemma eventually_INF: "eventually P (INF b:B. F b) ⟷ (∃X⊆B. finite X ∧ 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 "(⋀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 ⟹ (INF b:X. F b) ≠ bot) ⟹ (INF 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 (INF 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 (INF i:I. F i)"
using filter_leD[OF INF_lower] .

lemma eventually_INF_mono:
assumes *: "∀⇩F x in ⨅i∈I. F i. P x"
assumes T1: "⋀Q R P. (⋀x. Q x ∧ R x ⟶ P x) ⟹ (⋀x. T Q x ⟹ T R x ⟹ T P x)"
assumes T2: "⋀P. (⋀x. P x) ⟹ (⋀x. T P x)"
assumes **: "⋀i P. i ∈ I ⟹ ∀⇩F x in F i. P x ⟹ ∀⇩F x in F' i. T P x"
shows "∀⇩F x in ⨅i∈I. F' i. T P x"
proof -
from * obtain X where X: "finite X" "X ⊆ I" "∀⇩F x in ⨅i∈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 "∀⇩F x in ⨅i∈I. F' i. T P x"
by (subst eventually_INF) auto
qed

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"
by (simp add: filter_eq_iff eventually_filtermap)

lemma filtermap_filtermap:
"filtermap f (filtermap g F) = filtermap (λx. f (g x)) F"
by (simp add: filter_eq_iff eventually_filtermap)

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"
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) ≤ 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) ≤ (INF b:B. filtermap f (F b))"
proof -
{ fix X :: "'c set" assume "finite X"
then have "filtermap f (INFIMUM X F) ≤ (INF b:X. filtermap f (F b))"
proof induct
case (insert x X)
have "filtermap f (INF a:insert x X. F a) ≤ inf (filtermap f (F x)) (filtermap f (INF a:X. F a))"
by (rule order_trans[OF _ filtermap_inf]) simp
also have "… ≤ 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 ‹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)"
unfolding le_filter_def
proof safe
fix P
assume "eventually P (filtercomap f (sup F1 F2))"
thus "eventually P (sup (filtercomap f F1) (filtercomap f F2))"
by (auto simp: filter_eq_iff eventually_filtercomap eventually_sup)
qed

lemma filtercomap_INF: "filtercomap f (INF b:B. F b) = (INF b:B. filtercomap f (F b))"
proof -
have *: "filtercomap f (INF b:B. F b) = (INF 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 (INF 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 (INF 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_finite:
"finite B ⟹ filtercomap f (SUP b:B. F b) ≥ (SUP b:B. filtercomap f (F b))"
by (induction B rule: finite_induct)
(auto intro: order_trans[OF _ order_trans[OF _ filtercomap_sup]] filtercomap_mono)

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]: "(SUP 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 ⟹ (INF 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 = (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 ⟷ (∃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 (INF 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 "(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_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 = (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 ⇒ 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 (INF 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 "(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_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"
by (simp add: eventually_False)

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

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}"
by (simp add: frequently_def eventually_cofinite)

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›

lemma filtermap_sequentually_ne_bot: "filtermap f sequentially ≠ bot"
by (auto simp add: filter_eq_iff eventually_filtermap eventually_sequentially)

definition prod_filter :: "'a filter ⇒ 'b filter ⇒ ('a × 'b) filter" (infixr "×⇩F" 80) where
"prod_filter F G =
(INF (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 "(INF i:I. F i) = bot ⟷ (∃i∈I. F i = bot)"
proof (cases "∃i∈I. F i = bot")
case True
then have "(INF i:I. F i) ≤ 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) ≠ 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 ⊆ 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 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 "∀⇩F x in A. A1 x" "∀⇩F x in A. A2 x" "∀⇩F x in B. B1 x" "∀⇩F x in B. B2 x"
then show "∃x. eventually x A ∧ (∃y. eventually y B ∧ Collect x × Collect y ⊆ Collect A1 × Collect B1 ∧ Collect x × Collect y ⊆ Collect A2 × Collect B2)"
by (intro exI[of _ "λx. A1 x ∧ A2 x"] exI[of _ "λx. B1 x ∧ B2 x"] conjI)
(auto simp: eventually_conj_iff)
next
show "(∃x. eventually x A ∧ (∃y. eventually y B ∧ (x = (λx. False) ∨ y = (λx. False)))) = (A = ⊥ ∨ B = ⊥)"
by (auto simp: trivial_limit_def 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)"
apply (simp add: filter_eq_iff eventually_prod_filter eventually_principal)
apply safe
apply blast
apply (intro conjI exI[of _ "λx. x ∈ A"] exI[of _ "λx. x ∈ B"])
apply auto
done

lemma prod_filter_INF:
assumes "I ≠ {}" "J ≠ {}"
shows "(INF i:I. A i) ×⇩F (INF j:J. B j) = (INF i:I. INF j:J. A i ×⇩F B j)"
proof (safe intro!: antisym INF_greatest)
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)"
unfolding prod_filter_def
proof (safe intro!: INF_greatest)
fix P Q assume P: "∀⇩F x in ⨅i∈I. A i. P x" and Q: "∀⇩F x in ⨅j∈J. B j. Q x"
let ?X = "(⨅i∈I. ⨅j∈J. ⨅(P, Q)∈{(P, Q). (∀⇩F x in A i. P x) ∧ (∀⇩F x in B j. Q x)}. principal {(x, y). P x ∧ Q y})"
have "?X ≤ principal {x. P (fst x)} ⊓ principal {x. Q (snd x)}"
proof (intro inf_greatest)
have "?X ≤ (⨅i∈I. ⨅P∈{P. eventually P (A i)}. principal {x. P (fst x)})"
by (auto intro!: INF_greatest INF_lower2[of j] INF_lower2 ‹j∈J› INF_lower2[of "(_, λx. True)"])
also have "… ≤ principal {x. P (fst x)}"
unfolding le_principal
proof (rule eventually_INF_mono[OF P])
fix i P assume "i ∈ I" "eventually P (A i)"
then show "∀⇩F x in ⨅P∈{P. eventually P (A i)}. principal {x. P (fst x)}. x ∈ {x. P (fst x)}"
unfolding le_principal[symmetric] by (auto intro!: INF_lower)
qed auto
finally show "?X ≤ principal {x. P (fst x)}" .

have "?X ≤ (⨅i∈J. ⨅P∈{P. eventually P (B i)}. principal {x. P (snd x)})"
by (auto intro!: INF_greatest INF_lower2[of i] INF_lower2 ‹i∈I› INF_lower2[of "(λx. True, _)"])
also have "… ≤ principal {x. Q (snd x)}"
unfolding le_principal
proof (rule eventually_INF_mono[OF Q])
fix j Q assume "j ∈ J" "eventually Q (B j)"
then show "∀⇩F x in ⨅P∈{P. eventually P (B j)}. principal {x. P (snd x)}. x ∈ {x. Q (snd x)}"
unfolding le_principal[symmetric] by (auto intro!: INF_lower)
qed auto
finally show "?X ≤ principal {x. Q (snd x)}" .
qed
also have "… = principal {(x, y). P x ∧ Q y}"
by auto
finally show "?X ≤ principal {(x, y). P x ∧ Q y}" .
qed
qed (intro prod_filter_mono INF_lower)

lemma filtermap_Pair: "filtermap (λx. (f x, g x)) F ≤ filtermap f F ×⇩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 ⟹ eventually Q G ⟹ eventually (λx. P (fst x) ∧ Q (snd x)) (F ×⇩F G)"
unfolding prod_filter_def
by (intro eventually_INF1[of "(P, Q)"]) (auto simp: eventually_principal)

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

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"
by (simp add: filterlim_def)

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"
by (simp add: filterlim_def filtermap_ident)

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 (auto 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 filtermap_eq_strong: "inj f ⟹ filtermap f F = filtermap f G ⟷ 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 (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 :> (INF 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 (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:
"(⋀m x. m ∈ J ⟹ i m ∈ I) ⟹ (⋀m x. m ∈ J ⟹ x ∈ F (i m) ⟹ f x ∈ G m) ⟹
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 ≠ {}" and chain: "⋀i j. i ∈ I ⟹ j ∈ I ⟹ F i ⊆ F j ∨ F j ⊆ F i"
shows "(LIM x (INF i:I. principal (F i)). f x :> INF 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"
by (simp add: filterlim_iff eventually_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›

context includes lifting_syntax
begin

definition rel_filter :: "('a ⇒ 'b ⇒ bool) ⇒ 'a filter ⇒ 'b filter ⇒ 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 ⟷
((R ===> op =) ===> op =) (λP. eventually P F) (λP. 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 (λx. x) = (λF. 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 *: "⋀x y. T x y ⟹ 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 *: "⋀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="λP. 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 ≤ B ⟹ rel_filter A ≤ 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¯¯ = (rel_filter A)¯¯"
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 (λ_. True) = G (λx. True)" by transfer_prover
thus "G (λx. True)" by(simp add: True)
next
fix P' Q'
assume "G P'" "G Q'"
moreover
from bi_total_fun[OF ‹bi_unique A› 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 (λx. P x ∧ Q x)" by(simp add: conj)
moreover have "F (λx. P x ∧ Q x) = G (λx. P' x ∧ Q' x)" by transfer_prover
ultimately show "G (λx. P' x ∧ Q' x)" by simp
next
fix P' Q'
assume "∀x. P' x ⟶ Q' x" "G P'"
moreover
from bi_total_fun[OF ‹bi_unique A› 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 "(∀x. P x ⟶ Q x) ⟷ (∀x. P' x ⟶ 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]:
"⟦ bi_total A; bi_unique A ⟧
⟹ (((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 ‹bi_total A› bi_unique_eq] bi_total_eq]
obtain G where [transfer_rule]: "((A ===> op =) ===> op =) (λP. eventually P F) G"
unfolding  bi_total_def by blast
moreover have "is_filter (λP. eventually P F) ⟷ 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 "∃G. rel_filter A F G" ..
qed

lemma right_total_rel_filter [transfer_rule]:
"⟦ bi_total A; bi_unique A ⟧ ⟹ right_total (rel_filter A)"
using left_total_rel_filter[of "A¯¯"] 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 ⇒ 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 ⟹ right_unique (rel_filter A)"
using left_unique_rel_filter[of "A¯¯"] by simp

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

lemma filtermap_parametric [transfer_rule]:
"((A ===> B) ===> rel_filter A ===> rel_filter B) filtermap filtermap"
proof (intro rel_funI)
fix f g F G assume [transfer_rule]: "(A ===> B) f g" "rel_filter A F G"
show "rel_filter B (filtermap f F) (filtermap g G)"
unfolding rel_filter_eventually eventually_filtermap by transfer_prover
qed

(* TODO: Are those assumptions needed? *)
lemma filtercomap_parametric [transfer_rule]:
assumes [transfer_rule]: "bi_unique B" "bi_total A"
shows   "((A ===> B) ===> rel_filter B ===> rel_filter A) filtercomap filtercomap"
proof (intro rel_funI)
fix f g F G assume [transfer_rule]: "(A ===> B) f g" "rel_filter B F G"
show "rel_filter A (filtercomap f F) (filtercomap g G)"
unfolding rel_filter_eventually eventually_filtercomap by transfer_prover
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 ===> op =) op ≤ op ≤"
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 ‹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"
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 ⇒ 'a filter ⇒ bool"
where "equal_filter F F' ⟷ F = F'"

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

instance
by standard (simp add: equal_filter_def)

end

end
```