Theory Filter

(*  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"  ("(3F _ 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. yA. P x y) net)  (yA. eventually (λx. P x y) net)"
  by (induction A rule: finite_induct) (auto simp: eventually_conj_iff)

lemma eventually_ball_finite:
  "finite A  yA. eventually (λx. P x y) net  eventually (λx. yA. 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"  ("(3F _ 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. yA. P x y)  (yA. Fx in F. P x y)"
  using assms by induction (auto simp: frequently_disj_iff)

lemma frequently_bex_finite: "finite A  Fx in F. yA. P x y  yA. 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_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 termF  F' means that filter termF is finer than
filter termF'.

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. FS. 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 [OF is_filter.intro])
  apply (blast intro: eventually_True)
   apply (force elim!: eventually_conj)+
  done

lemma eventually_Sup:
  "eventually P (Sup S)  (FS. eventually P F)"
  unfolding Sup_filter_def
  apply (rule eventually_Abs_filter [OF 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" "Fbot"
  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 Fbot] 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"
  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)  (XB. finite X  eventually P (Inf X))"
proof -
  let ?F = "λP. XB. finite X  eventually P (Inf X)"

  have eventually_F: "eventually P (Abs_filter ?F)  ?F P" for 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" "?F Q"
    then obtain X Y where
      "X  B" "finite X" "eventually P ( X)"
      "Y  B" "finite Y" "eventually Q ( Y)" by blast
    then 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 obtain X where "X  B" "finite X" "eventually P ( X)"
      by blast
    moreover assume "x. P x  Q x"
    ultimately show "?F Q"
      by (intro exI[of _ X]) (auto elim: eventually_mono)
  qed

  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 (bB. F b)  (XB. finite X  eventually P (bX. 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  (bX. F b)  bot)  (bB. 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  xB. x  inf F G"
  shows "eventually P (Inf B)  (bB. eventually P b)"
proof (subst eventually_Inf, safe)
  fix X assume "finite X" "X  B"
  then have "bB. xX. 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  xB. F x  inf (F a) (F b)) 
    eventually P (bB. F b)  (bB. eventually P (F b))"
  by (subst eventually_Inf_base) auto

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

lemma eventually_INF_finite:
  assumes "finite A"
  shows   "eventually P ( xA. F x) 
             (Q. (xA. eventually (Q x) (F x))  (y. (xA. Q x y)  P y))" 
  using assms
proof (induction arbitrary: P rule: finite_induct)
  case (insert a A P)
  from insert.hyps have [simp]: "x  a" if "x  A" for x
    using that by auto
  have "eventually P ( xinsert a A. F x) 
          (Q R S. eventually Q (F a)  (( (xA. eventually (S x) (F x)) 
            (y. (xA. S x y)  R y))  (x. Q x  R x  P x)))"
    unfolding ex_simps by (simp add: eventually_inf insert.IH)
  also have "  (Q. (xinsert a A. eventually (Q x) (F x)) 
                           (y. (xinsert a A. Q x y)  P y))"
  proof (safe, goal_cases)
    case (1 Q R S)
    thus ?case using 1 by (intro exI[of _ "S(a := Q)"]) auto
  next
    case (2 Q)
    show ?case
      by (rule exI[of _ "Q a"], rule exI[of _ "λy. xA. Q x y"],
          rule exI[of _ "Q(a := (λ_. True))"]) (use 2 in auto)
  qed
  finally show ?case .
qed auto

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 [OF 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_bot_iff: "filtermap f F = bot  F = bot"
  by (simp add: trivial_limit_def eventually_filtermap)

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 (bB. F b) = (bB. 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 (bB. F b)  (bB. 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)
  then obtain P' Q' where P'Q':
    "eventually P' F" "x. P' (f x)  P x"
    "eventually Q' F" "x. Q' (f x)  Q x"
    by (elim exE conjE)
  show ?case
    by (rule exI[of _ "λx. P' x  Q' x"]) (use P'Q' in 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 (bB. F b) = (bB. filtercomap f (F b))"
proof -
  have *: "filtercomap f (bB. F b) = (bB. 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 (bB. filtercomap f (F b))  
            (X. (X  B  finite X)  eventually P (bX. filtercomap f (F b)))"
      by (subst eventually_INF) blast
    also have "  (X. (X  B  finite X)  eventually P (filtercomap f (bX. F b)))"
      by (rule ex_cong) (simp add: *)
    also have "  eventually P (filtercomap f ((F ` B)))"
      unfolding eventually_filtercomap by (subst eventually_INF) blast
    finally show "eventually P (filtercomap f ((F ` B))) = 
                    eventually P (bB. filtercomap f (F b))" ..
  qed
qed

lemma filtercomap_SUP:
  "filtercomap f (bB. F b)  (bB. filtercomap f (F b))"
  by (intro SUP_least filtercomap_mono SUP_upper)

lemma filtermap_le_iff_le_filtercomap: "filtermap f F  G  F  filtercomap f G"
  unfolding le_filter_def eventually_filtermap eventually_filtercomap
  using eventually_mono by auto

lemma filtercomap_neq_bot:
  assumes "P. eventually P F  x. P (f x)"
  shows   "filtercomap f F  bot"
  using assms by (auto simp: trivial_limit_def eventually_filtercomap)

lemma filtercomap_neq_bot_surj:
  assumes "F  bot" and "surj f"
  shows   "filtercomap f F  bot"
proof (rule filtercomap_neq_bot)
  fix P assume *: "eventually P F"
  show "x. P (f x)"
  proof (rule ccontr)
    assume **: "¬(x. P (f x))"
    from * have "eventually (λ_. False) F"
    proof eventually_elim
      case (elim x)
      from surj f obtain y where "x = f y" by auto
      with elim and ** show False by auto
    qed
    with assms show False by (simp add: trivial_limit_def)
  qed
qed

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. xS. P x)"

lemma eventually_principal: "eventually P (principal S)  (xS. 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
  by (force elim: eventually_mono)

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]: "(iI. principal (A i)) = principal (iI. A i)"
  unfolding filter_eq_iff eventually_Sup by (auto simp: eventually_principal)

lemma INF_principal_finite: "finite X  (xX. principal (f x)) = principal (xX. 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. nN. 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. yx. P y) at_top"
proof -
  from assms obtain x where "y. y  x  P y" by (auto simp: eventually_at_top_linorder)
  hence "zy. 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. nN. 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. nN. 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

lemma filtermap_sequentually_ne_bot: "filtermap f sequentially  bot"
  by (simp add: filtermap_bot_iff)

subsection Increasing finite subsets

definition finite_subsets_at_top where
  "finite_subsets_at_top A = ( X{X. finite X  X  A}. principal {Y. finite Y  X  Y  Y  A})"

lemma eventually_finite_subsets_at_top:
  "eventually P (finite_subsets_at_top A) 
     (X. finite X  X  A  (Y. finite Y  X  Y  Y  A  P Y))"
  unfolding finite_subsets_at_top_def
proof (subst eventually_INF_base, goal_cases)
  show "{X. finite X  X  A}  {}" by auto
next
  case (2 B C)
  thus ?case by (intro bexI[of _ "B  C"]) auto
qed (simp_all add: eventually_principal)

lemma eventually_finite_subsets_at_top_weakI [intro]:
  assumes "X. finite X  X  A  P X"
  shows   "eventually P (finite_subsets_at_top A)"
proof -
  have "eventually (λX. finite X  X  A) (finite_subsets_at_top A)"
    by (auto simp: eventually_finite_subsets_at_top)
  thus ?thesis by eventually_elim (use assms in auto)
qed

lemma finite_subsets_at_top_neq_bot [simp]: "finite_subsets_at_top A  bot"
proof -
  have "¬eventually (λx. False) (finite_subsets_at_top A)"
    by (auto simp: eventually_finite_subsets_at_top)
  thus ?thesis by auto
qed

lemma filtermap_image_finite_subsets_at_top:
  assumes "inj_on f A"
  shows   "filtermap ((`) f) (finite_subsets_at_top A) = finite_subsets_at_top (f ` A)"
  unfolding filter_eq_iff eventually_filtermap
proof (safe, goal_cases)
  case (1 P)
  then obtain X where X: "finite X" "X  A" "Y. finite Y  X  Y  Y  A  P (f ` Y)"
    unfolding eventually_finite_subsets_at_top by force
  show ?case unfolding eventually_finite_subsets_at_top eventually_filtermap
  proof (rule exI[of _ "f ` X"], intro conjI allI impI, goal_cases)
    case (3 Y)
    with assms and X(1,2) have "P (f ` (f -` Y  A))" using X(1,2)
      by (intro X(3) finite_vimage_IntI) auto
    also have "f ` (f -` Y  A) = Y" using assms 3 by blast
    finally show ?case .
  qed (insert assms X(1,2), auto intro!: finite_vimage_IntI)
next
  case (2 P)
  then obtain X where X: "finite X" "X  f ` A" "Y. finite Y  X  Y  Y  f ` A  P Y"
    unfolding eventually_finite_subsets_at_top by force
  show ?case unfolding eventually_finite_subsets_at_top eventually_filtermap
  proof (rule exI[of _ "f -` X  A"], intro conjI allI impI, goal_cases)
    case (3 Y)
    with X(1,2) and assms show ?case by (intro X(3)) force+
  qed (insert assms X(1), auto intro!: finite_vimage_IntI)
qed

lemma eventually_finite_subsets_at_top_finite:
  assumes "finite A"
  shows   "eventually P (finite_subsets_at_top A)  P A"
  unfolding eventually_finite_subsets_at_top using assms by force

lemma finite_subsets_at_top_finite: "finite A  finite_subsets_at_top A = principal {A}"
  by (auto simp: filter_eq_iff eventually_finite_subsets_at_top_finite eventually_principal)


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. nN. 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 "nN. 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  kI. F k  F i  F j"
  shows "(iI. F i) = bot  (iI. F i = bot)"
proof (cases "iI. F i = bot")
  case True
  then have "(iI. F i)  bot"
    by (auto intro: INF_lower2)
  with True show ?thesis
    by (auto simp: bot_unique)
next
  case False
  moreover have "(iI. 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 "kI. F k  (iJ. 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  (iJ. F i)" by auto
        with insert *[of i k] show ?case
          by auto
      qed
      with False show "(iJ. 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 by (blast intro!: eventually_conj)

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 "(iI. A i) ×F (jJ. B j) = (iI. jJ. 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 "(iI. jJ. A i ×F B j)  (iI. A i) ×F (jJ. 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 "(iI. A i) ×F (jJ. B j)  (iI. jJ. 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  {}  (iI. A i) ×F B = (iI. A i ×F B)"
  using prod_filter_INF[of I "{B}" A "λx. x"] by simp

lemma prod_filter_INF2: "J  {}  A ×F (iJ. B i) = (iJ. 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)"
  unfolding filter_eq_iff eventually_filtermap eventually_prod_filter
  apply 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)"
  unfolding filter_eq_iff eventually_filtermap eventually_prod_filter
  apply 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"
  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'"
proof -
  have "filterlim f F' G'"
    by (simp add: filterlim_mono[OF _ ord] assms)
  then show ?thesis
    by (rule filterlim_cong[OF refl refl eq, THEN iffD1])
qed

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

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 :> (bB. G b))  (bB. LIM x F. f x :> G b)"
  unfolding filterlim_def le_INF_iff ..

lemma filterlim_INF_INF:
  "(m. m  J  iI. filtermap f (F i)  G m)  LIM x (iI. F i). f x :> (jJ. G j)"
  unfolding filterlim_def by (rule order_trans[OF filtermap_INF INF_mono])

lemma filterlim_INF': "x  A  filterlim f F (G x)  filterlim f F ( xA. G x)"
  unfolding filterlim_def by (rule order.trans[OF filtermap_mono[OF INF_lower]])

lemma filterlim_filtercomap_iff: "filterlim f (filtercomap g G) F  filterlim (g  f) G F"
  by (simp add: filterlim_def filtermap_le_iff_le_filtercomap filtercomap_filtercomap o_def)

lemma filterlim_iff_le_filtercomap: "filterlim f F G  G  filtercomap f F"
  by (simp add: filterlim_def filtermap_le_iff_le_filtercomap)

lemma filterlim_base:
  "(m x. m  J  i m  I)  (m x. m  J  x  F (i m)  f x  G m) 
    LIM x (iI. principal (F i)). f x :> (jJ. 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 (iI. principal (F i)). f x :> jJ. principal (G j)) 
    (jJ. iI. xF 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 "xI. 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 constat_top and constat_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)  (Zc. 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"
    by (rule eventually_mono) (use 1 in 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)  (Zc. eventually (λx. Z  f x) F)"
  unfolding filterlim_at_bot
proof safe
  fix Z assume *: "Zc. 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_finite_subsets_at_top:
  "filterlim f (finite_subsets_at_top A) F 
     (X. finite X  X  A  eventually (λy. finite (f y)  X  f y  f y  A) F)"
  (is "?lhs = ?rhs")
proof 
  assume ?lhs
  thus ?rhs
  proof (safe, goal_cases)
    case (1 X)
    hence *: "(F x in F. P (f x))" if "eventually P (finite_subsets_at_top A)" for P
      using that by (auto simp: filterlim_def le_filter_def eventually_filtermap)
    have "F Y in finite_subsets_at_top A. finite Y  X  Y  Y  A"
      using 1 unfolding eventually_finite_subsets_at_top by force
    thus ?case by (intro *) auto
  qed
next
  assume rhs: ?rhs
  show ?lhs unfolding filterlim_def le_filter_def eventually_finite_subsets_at_top
  proof (safe, goal_cases)
    case (1 P X)
    with rhs have "F y in F. finite (f y)  X  f y  f y  A" by auto
    thus "eventually P (filtermap f F)" unfolding eventually_filtermap
      by eventually_elim (insert 1, auto)
  qed
qed

lemma filterlim_atMost_at_top:
  "filterlim (λn. {..n}) (finite_subsets_at_top (UNIV :: nat set)) at_top"
  unfolding filterlim_finite_subsets_at_top
proof (safe, goal_cases)
  case (1 X)
  then obtain n where n: "X  {..n}" by (auto simp: finite_nat_set_iff_bounded_le)
  show ?case using eventually_ge_at_top[of n]
    by eventually_elim (insert n, auto)
qed

lemma filterlim_lessThan_at_top:
  "filterlim (λn. {..<n}) (finite_subsets_at_top (UNIV :: nat set)) at_top"
  unfolding filterlim_finite_subsets_at_top
proof (safe, goal_cases)
  case (1 X)
  then obtain n where n: "X  {..<n}" by (auto simp: finite_nat_set_iff_bounded)
  show ?case using eventually_ge_at_top[of n]
    by eventually_elim (insert n, auto)
qed

subsection Setup typ'a filter for lifting and transfer

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 .

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))