Theory Free_Ultrafilter

theory Free_Ultrafilter
imports Infinite_Set
(*  Title:      HOL/Nonstandard_Analysis/Free_Ultrafilter.thy
    Author:     Jacques D. Fleuriot, University of Cambridge
    Author:     Lawrence C Paulson
    Author:     Brian Huffman
*)

section ‹Filters and Ultrafilters›

theory Free_Ultrafilter
  imports "HOL-Library.Infinite_Set"
begin


subsection ‹Definitions and basic properties›

subsubsection ‹Ultrafilters›

locale ultrafilter =
  fixes F :: "'a filter"
  assumes proper: "F ≠ bot"
  assumes ultra: "eventually P F ∨ eventually (λx. ¬ P x) F"
begin

lemma eventually_imp_frequently: "frequently P F ⟹ eventually P F"
  using ultra[of P] by (simp add: frequently_def)

lemma frequently_eq_eventually: "frequently P F = eventually P F"
  using eventually_imp_frequently eventually_frequently[OF proper] ..

lemma eventually_disj_iff: "eventually (λx. P x ∨ Q x) F ⟷ eventually P F ∨ eventually Q F"
  unfolding frequently_eq_eventually[symmetric] frequently_disj_iff ..

lemma eventually_all_iff: "eventually (λx. ∀y. P x y) F = (∀Y. eventually (λx. P x (Y x)) F)"
  using frequently_all[of P F] by (simp add: frequently_eq_eventually)

lemma eventually_imp_iff: "eventually (λx. P x ⟶ Q x) F ⟷ (eventually P F ⟶ eventually Q F)"
  using frequently_imp_iff[of P Q F] by (simp add: frequently_eq_eventually)

lemma eventually_iff_iff: "eventually (λx. P x ⟷ Q x) F ⟷ (eventually P F ⟷ eventually Q F)"
  unfolding iff_conv_conj_imp eventually_conj_iff eventually_imp_iff by simp

lemma eventually_not_iff: "eventually (λx. ¬ P x) F ⟷ ¬ eventually P F"
  unfolding not_eventually frequently_eq_eventually ..

end


subsection ‹Maximal filter = Ultrafilter›

text ‹
   A filter ‹F› is an ultrafilter iff it is a maximal filter,
   i.e. whenever ‹G› is a filter and @{prop "F ⊆ G"} then @{prop "F = G"}
›

text ‹
  Lemma that shows existence of an extension to what was assumed to
  be a maximal filter. Will be used to derive contradiction in proof of
  property of ultrafilter.
›

lemma extend_filter: "frequently P F ⟹ inf F (principal {x. P x}) ≠ bot"
  by (simp add: trivial_limit_def eventually_inf_principal not_eventually)

lemma max_filter_ultrafilter:
  assumes "F ≠ bot"
  assumes max: "⋀G. G ≠ bot ⟹ G ≤ F ⟹ F = G"
  shows "ultrafilter F"
proof
  show "eventually P F ∨ (∀Fx in F. ¬ P x)" for P
  proof (rule disjCI)
    assume "¬ (∀Fx in F. ¬ P x)"
    then have "inf F (principal {x. P x}) ≠ bot"
      by (simp add: not_eventually extend_filter)
    then have F: "F = inf F (principal {x. P x})"
      by (rule max) simp
    show "eventually P F"
      by (subst F) (simp add: eventually_inf_principal)
  qed
qed fact

lemma le_filter_frequently: "F ≤ G ⟷ (∀P. frequently P F ⟶ frequently P G)"
  unfolding frequently_def le_filter_def
  apply auto
  apply (erule_tac x="λx. ¬ P x" in allE)
  apply auto
  done

lemma (in ultrafilter) max_filter:
  assumes G: "G ≠ bot"
    and sub: "G ≤ F"
  shows "F = G"
proof (rule antisym)
  show "F ≤ G"
    using sub
    by (auto simp: le_filter_frequently[of F] frequently_eq_eventually le_filter_def[of G]
             intro!: eventually_frequently G proper)
qed fact


subsection ‹Ultrafilter Theorem›

lemma ex_max_ultrafilter:
  fixes F :: "'a filter"
  assumes F: "F ≠ bot"
  shows "∃U≤F. ultrafilter U"
proof -
  let ?X = "{G. G ≠ bot ∧ G ≤ F}"
  let ?R = "{(b, a). a ≠ bot ∧ a ≤ b ∧ b ≤ F}"

  have bot_notin_R: "c ∈ Chains ?R ⟹ bot ∉ c" for c
    by (auto simp: Chains_def)

  have [simp]: "Field ?R = ?X"
    by (auto simp: Field_def bot_unique)

  have "∃m∈Field ?R. ∀a∈Field ?R. (m, a) ∈ ?R ⟶ a = m" (is "∃m∈?A. ?B m")
  proof (rule Zorns_po_lemma)
    show "Partial_order ?R"
      by (auto simp: partial_order_on_def preorder_on_def
          antisym_def refl_on_def trans_def Field_def bot_unique)
    show "∀C∈Chains ?R. ∃u∈Field ?R. ∀a∈C. (a, u) ∈ ?R"
    proof (safe intro!: bexI del: notI)
      fix c x
      assume c: "c ∈ Chains ?R"

      have Inf_c: "Inf c ≠ bot" "Inf c ≤ F" if "c ≠ {}"
      proof -
        from c that have "Inf c = bot ⟷ (∃x∈c. x = bot)"
          unfolding trivial_limit_def by (intro eventually_Inf_base) (auto simp: Chains_def)
        with c show "Inf c ≠ bot"
          by (simp add: bot_notin_R)
        from that obtain x where "x ∈ c" by auto
        with c show "Inf c ≤ F"
          by (auto intro!: Inf_lower2[of x] simp: Chains_def)
      qed
      then have [simp]: "inf F (Inf c) = (if c = {} then F else Inf c)"
        using c by (auto simp add: inf_absorb2)

      from c show "inf F (Inf c) ≠ bot"
        by (simp add: F Inf_c)
      from c show "inf F (Inf c) ∈ Field ?R"
        by (simp add: Chains_def Inf_c F)

      assume "x ∈ c"
      with c show "inf F (Inf c) ≤ x" "x ≤ F"
        by (auto intro: Inf_lower simp: Chains_def)
    qed
  qed
  then obtain U where U: "U ∈ ?A" "?B U" ..
  show ?thesis
  proof
    from U show "U ≤ F ∧ ultrafilter U"
      by (auto intro!: max_filter_ultrafilter)
  qed
qed


subsubsection ‹Free Ultrafilters›

text ‹There exists a free ultrafilter on any infinite set.›

locale freeultrafilter = ultrafilter +
  assumes infinite: "eventually P F ⟹ infinite {x. P x}"
begin

lemma finite: "finite {x. P x} ⟹ ¬ eventually P F"
  by (erule contrapos_pn) (erule infinite)

lemma finite': "finite {x. ¬ P x} ⟹ eventually P F"
  by (drule finite) (simp add: not_eventually frequently_eq_eventually)

lemma le_cofinite: "F ≤ cofinite"
  by (intro filter_leI)
    (auto simp add: eventually_cofinite not_eventually frequently_eq_eventually dest!: finite)

lemma singleton: "¬ eventually (λx. x = a) F"
  by (rule finite) simp

lemma singleton': "¬ eventually (op = a) F"
  by (rule finite) simp

lemma ultrafilter: "ultrafilter F" ..

end

lemma freeultrafilter_Ex:
  assumes [simp]: "infinite (UNIV :: 'a set)"
  shows "∃U::'a filter. freeultrafilter U"
proof -
  from ex_max_ultrafilter[of "cofinite :: 'a filter"]
  obtain U :: "'a filter" where "U ≤ cofinite" "ultrafilter U"
    by auto
  interpret ultrafilter U by fact
  have "freeultrafilter U"
  proof
    fix P
    assume "eventually P U"
    with proper have "frequently P U"
      by (rule eventually_frequently)
    then have "frequently P cofinite"
      using ‹U ≤ cofinite› by (simp add: le_filter_frequently)
    then show "infinite {x. P x}"
      by (simp add: frequently_cofinite)
  qed
  then show ?thesis ..
qed

end