Theory Set_Idioms

(*  Title:      HOL/Library/Set_Idioms.thy
    Author:     Lawrence Paulson (but borrowed from HOL Light)
*)

section ‹Set Idioms›

theory Set_Idioms
imports Countable_Set

begin


subsection‹Idioms for being a suitable union/intersection of something›

definition union_of :: "('a set set ⇒ bool) ⇒ ('a set ⇒ bool) ⇒ 'a set ⇒ bool"
  (infixr ‹union'_of› 60)
  where "P union_of Q ≡ λS. ∃𝒰. P 𝒰 ∧ 𝒰 ⊆ Collect Q ∧ ⋃𝒰 = S"

definition intersection_of :: "('a set set ⇒ bool) ⇒ ('a set ⇒ bool) ⇒ 'a set ⇒ bool"
  (infixr ‹intersection'_of› 60)
  where "P intersection_of Q ≡ λS. ∃𝒰. P 𝒰 ∧ 𝒰 ⊆ Collect Q ∧ ⋂𝒰 = S"

definition arbitrary:: "'a set set ⇒ bool" where "arbitrary 𝒰 ≡ True"

lemma union_of_inc: "⟦P {S}; Q S⟧ ⟹ (P union_of Q) S"
  by (auto simp: union_of_def)

lemma intersection_of_inc:
   "⟦P {S}; Q S⟧ ⟹ (P intersection_of Q) S"
  by (auto simp: intersection_of_def)

lemma union_of_mono:
   "⟦(P union_of Q) S; ⋀x. Q x ⟹ Q' x⟧ ⟹ (P union_of Q') S"
  by (auto simp: union_of_def)

lemma intersection_of_mono:
   "⟦(P intersection_of Q) S; ⋀x. Q x ⟹ Q' x⟧ ⟹ (P intersection_of Q') S"
  by (auto simp: intersection_of_def)

lemma all_union_of:
     "(∀S. (P union_of Q) S ⟶ R S) ⟷ (∀T. P T ∧ T ⊆ Collect Q ⟶ R(⋃T))"
  by (auto simp: union_of_def)

lemma all_intersection_of:
     "(∀S. (P intersection_of Q) S ⟶ R S) ⟷ (∀T. P T ∧ T ⊆ Collect Q ⟶ R(⋂T))"
  by (auto simp: intersection_of_def)
             
lemma intersection_ofE:
     "⟦(P intersection_of Q) S; ⋀T. ⟦P T; T ⊆ Collect Q⟧ ⟹ R(⋂T)⟧ ⟹ R S"
  by (auto simp: intersection_of_def)

lemma union_of_empty:
     "P {} ⟹ (P union_of Q) {}"
  by (auto simp: union_of_def)

lemma intersection_of_empty:
     "P {} ⟹ (P intersection_of Q) UNIV"
  by (auto simp: intersection_of_def)

text‹ The arbitrary and finite cases›

lemma arbitrary_union_of_alt:
   "(arbitrary union_of Q) S ⟷ (∀x ∈ S. ∃U. Q U ∧ x ∈ U ∧ U ⊆ S)"
 (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
    by (force simp: union_of_def arbitrary_def)
next
  assume ?rhs
  then have "{U. Q U ∧ U ⊆ S} ⊆ Collect Q" "⋃{U. Q U ∧ U ⊆ S} = S"
    by auto
  then show ?lhs
    unfolding union_of_def arbitrary_def by blast
qed

lemma arbitrary_union_of_empty [simp]: "(arbitrary union_of P) {}"
  by (force simp: union_of_def arbitrary_def)

lemma arbitrary_intersection_of_empty [simp]:
  "(arbitrary intersection_of P) UNIV"
  by (force simp: intersection_of_def arbitrary_def)

lemma arbitrary_union_of_inc:
  "P S ⟹ (arbitrary union_of P) S"
  by (force simp: union_of_inc arbitrary_def)

lemma arbitrary_intersection_of_inc:
  "P S ⟹ (arbitrary intersection_of P) S"
  by (force simp: intersection_of_inc arbitrary_def)

lemma arbitrary_union_of_complement:
   "(arbitrary union_of P) S ⟷ (arbitrary intersection_of (λS. P(- S))) (- S)"  (is "?lhs = ?rhs")
proof
  assume ?lhs
  then obtain 𝒰 where "𝒰 ⊆ Collect P" "S = ⋃𝒰"
    by (auto simp: union_of_def arbitrary_def)
  then show ?rhs
    unfolding intersection_of_def arbitrary_def
    by (rule_tac x="uminus ` 𝒰" in exI) auto
next
  assume ?rhs
  then obtain 𝒰 where "𝒰 ⊆ {S. P (- S)}" "⋂𝒰 = - S"
    by (auto simp: union_of_def intersection_of_def arbitrary_def)
  then show ?lhs
    unfolding union_of_def arbitrary_def
    by (rule_tac x="uminus ` 𝒰" in exI) auto
qed

lemma arbitrary_intersection_of_complement:
   "(arbitrary intersection_of P) S ⟷ (arbitrary union_of (λS. P(- S))) (- S)"
  by (simp add: arbitrary_union_of_complement)

lemma arbitrary_union_of_idempot [simp]:
  "arbitrary union_of arbitrary union_of P = arbitrary union_of P"
proof -
  have 1: "∃𝒰'⊆Collect P. ⋃𝒰' = ⋃𝒰" if "𝒰 ⊆ {S. ∃𝒱⊆Collect P. ⋃𝒱 = S}" for 𝒰
  proof -
    let ?𝒲 = "{V. ∃𝒱. 𝒱⊆Collect P ∧ V ∈ 𝒱 ∧ (∃S ∈ 𝒰. ⋃𝒱 = S)}"
    have *: "⋀x U. ⟦x ∈ U; U ∈ 𝒰⟧ ⟹ x ∈ ⋃?𝒲"
      using that
      apply simp
      apply (drule subsetD, assumption, auto)
      done
    show ?thesis
    apply (rule_tac x="{V. ∃𝒱. 𝒱⊆Collect P ∧ V ∈ 𝒱 ∧ (∃S ∈ 𝒰. ⋃𝒱 = S)}" in exI)
      using that by (blast intro: *)
  qed
  have 2: "∃𝒰'⊆{S. ∃𝒰⊆Collect P. ⋃𝒰 = S}. ⋃𝒰' = ⋃𝒰" if "𝒰 ⊆ Collect P" for 𝒰
    by (metis (mono_tags, lifting) union_of_def arbitrary_union_of_inc that)
  show ?thesis
    unfolding union_of_def arbitrary_def by (force simp: 1 2)
qed

lemma arbitrary_intersection_of_idempot:
   "arbitrary intersection_of arbitrary intersection_of P = arbitrary intersection_of P" (is "?lhs = ?rhs")
proof -
  have "- ?lhs = - ?rhs"
    unfolding arbitrary_intersection_of_complement by simp
  then show ?thesis
    by simp
qed

lemma arbitrary_union_of_Union:
   "(⋀S. S ∈ 𝒰 ⟹ (arbitrary union_of P) S) ⟹ (arbitrary union_of P) (⋃𝒰)"
  by (metis union_of_def arbitrary_def arbitrary_union_of_idempot mem_Collect_eq subsetI)

lemma arbitrary_union_of_Un:
   "⟦(arbitrary union_of P) S; (arbitrary union_of P) T⟧
           ⟹ (arbitrary union_of P) (S ∪ T)"
  using arbitrary_union_of_Union [of "{S,T}"] by auto

lemma arbitrary_intersection_of_Inter:
   "(⋀S. S ∈ 𝒰 ⟹ (arbitrary intersection_of P) S) ⟹ (arbitrary intersection_of P) (⋂𝒰)"
  by (metis intersection_of_def arbitrary_def arbitrary_intersection_of_idempot mem_Collect_eq subsetI)

lemma arbitrary_intersection_of_Int:
   "⟦(arbitrary intersection_of P) S; (arbitrary intersection_of P) T⟧
           ⟹ (arbitrary intersection_of P) (S ∩ T)"
  using arbitrary_intersection_of_Inter [of "{S,T}"] by auto

lemma arbitrary_union_of_Int_eq:
  "(∀S T. (arbitrary union_of P) S ∧ (arbitrary union_of P) T
               ⟶ (arbitrary union_of P) (S ∩ T))
   ⟷ (∀S T. P S ∧ P T ⟶ (arbitrary union_of P) (S ∩ T))"  (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
    by (simp add: arbitrary_union_of_inc)
next
  assume R: ?rhs
  show ?lhs
  proof clarify
    fix S :: "'a set" and T :: "'a set"
    assume "(arbitrary union_of P) S" and "(arbitrary union_of P) T"
    then obtain 𝒰 𝒱 where *: "𝒰 ⊆ Collect P" "⋃𝒰 = S" "𝒱 ⊆ Collect P" "⋃𝒱 = T"
      by (auto simp: union_of_def)
    then have "(arbitrary union_of P) (⋃C∈𝒰. ⋃D∈𝒱. C ∩ D)"
     using R by (blast intro: arbitrary_union_of_Union)
   then show "(arbitrary union_of P) (S ∩ T)"
     by (simp add: Int_UN_distrib2 *)
 qed
qed

lemma arbitrary_intersection_of_Un_eq:
   "(∀S T. (arbitrary intersection_of P) S ∧ (arbitrary intersection_of P) T
               ⟶ (arbitrary intersection_of P) (S ∪ T)) ⟷
        (∀S T. P S ∧ P T ⟶ (arbitrary intersection_of P) (S ∪ T))"
  apply (simp add: arbitrary_intersection_of_complement)
  using arbitrary_union_of_Int_eq [of "λS. P (- S)"]
  by (metis (no_types, lifting) arbitrary_def double_compl union_of_inc)

lemma finite_union_of_empty [simp]: "(finite union_of P) {}"
  by (simp add: union_of_empty)

lemma finite_intersection_of_empty [simp]: "(finite intersection_of P) UNIV"
  by (simp add: intersection_of_empty)

lemma finite_union_of_inc:
   "P S ⟹ (finite union_of P) S"
  by (simp add: union_of_inc)

lemma finite_intersection_of_inc:
   "P S ⟹ (finite intersection_of P) S"
  by (simp add: intersection_of_inc)

lemma finite_union_of_complement:
  "(finite union_of P) S ⟷ (finite intersection_of (λS. P(- S))) (- S)"
  unfolding union_of_def intersection_of_def
  apply safe
   apply (rule_tac x="uminus ` 𝒰" in exI, fastforce)+
  done

lemma finite_intersection_of_complement:
   "(finite intersection_of P) S ⟷ (finite union_of (λS. P(- S))) (- S)"
  by (simp add: finite_union_of_complement)

lemma finite_union_of_idempot [simp]:
  "finite union_of finite union_of P = finite union_of P"
proof -
  have "(finite union_of P) S" if S: "(finite union_of finite union_of P) S" for S
  proof -
    obtain 𝒰 where "finite 𝒰" "S = ⋃𝒰" and 𝒰: "∀U∈𝒰. ∃𝒰. finite 𝒰 ∧ (𝒰 ⊆ Collect P) ∧ ⋃𝒰 = U"
      using S unfolding union_of_def by (auto simp: subset_eq)
    then obtain f where "∀U∈𝒰. finite (f U) ∧ (f U ⊆ Collect P) ∧ ⋃(f U) = U"
      by metis
    then show ?thesis
      unfolding union_of_def ‹S = ⋃𝒰›
      by (rule_tac x = "snd ` Sigma 𝒰 f" in exI) (fastforce simp: ‹finite 𝒰›)
  qed
  moreover
  have "(finite union_of finite union_of P) S" if "(finite union_of P) S" for S
    by (simp add: finite_union_of_inc that)
  ultimately show ?thesis
    by force
qed

lemma finite_intersection_of_idempot [simp]:
   "finite intersection_of finite intersection_of P = finite intersection_of P"
  by (force simp: finite_intersection_of_complement)

lemma finite_union_of_Union:
   "⟦finite 𝒰; ⋀S. S ∈ 𝒰 ⟹ (finite union_of P) S⟧ ⟹ (finite union_of P) (⋃𝒰)"
  using finite_union_of_idempot [of P]
  by (metis mem_Collect_eq subsetI union_of_def)

lemma finite_union_of_Un:
   "⟦(finite union_of P) S; (finite union_of P) T⟧ ⟹ (finite union_of P) (S ∪ T)"
  by (auto simp: union_of_def)

lemma finite_intersection_of_Inter:
   "⟦finite 𝒰; ⋀S. S ∈ 𝒰 ⟹ (finite intersection_of P) S⟧ ⟹ (finite intersection_of P) (⋂𝒰)"
  using finite_intersection_of_idempot [of P]
  by (metis intersection_of_def mem_Collect_eq subsetI)

lemma finite_intersection_of_Int:
   "⟦(finite intersection_of P) S; (finite intersection_of P) T⟧
           ⟹ (finite intersection_of P) (S ∩ T)"
  by (auto simp: intersection_of_def)

lemma finite_union_of_Int_eq:
   "(∀S T. (finite union_of P) S ∧ (finite union_of P) T ⟶ (finite union_of P) (S ∩ T))
    ⟷ (∀S T. P S ∧ P T ⟶ (finite union_of P) (S ∩ T))"
(is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
    by (simp add: finite_union_of_inc)
next
  assume R: ?rhs
  show ?lhs
  proof clarify
    fix S :: "'a set" and T :: "'a set"
    assume "(finite union_of P) S" and "(finite union_of P) T"
    then obtain 𝒰 𝒱 where *: "𝒰 ⊆ Collect P" "⋃𝒰 = S" "finite 𝒰" "𝒱 ⊆ Collect P" "⋃𝒱 = T" "finite 𝒱"
      by (auto simp: union_of_def)
    then have "(finite union_of P) (⋃C∈𝒰. ⋃D∈𝒱. C ∩ D)"
      using R
      by (blast intro: finite_union_of_Union)
   then show "(finite union_of P) (S ∩ T)"
     by (simp add: Int_UN_distrib2 *)
 qed
qed

lemma finite_intersection_of_Un_eq:
   "(∀S T. (finite intersection_of P) S ∧
               (finite intersection_of P) T
               ⟶ (finite intersection_of P) (S ∪ T)) ⟷
        (∀S T. P S ∧ P T ⟶ (finite intersection_of P) (S ∪ T))"
  apply (simp add: finite_intersection_of_complement)
  using finite_union_of_Int_eq [of "λS. P (- S)"]
  by (metis (no_types, lifting) double_compl)


abbreviation finite' :: "'a set ⇒ bool"
  where "finite' A ≡ finite A ∧ A ≠ {}"

lemma finite'_intersection_of_Int:
   "⟦(finite' intersection_of P) S; (finite' intersection_of P) T⟧
           ⟹ (finite' intersection_of P) (S ∩ T)"
  by (auto simp: intersection_of_def)

lemma finite'_intersection_of_inc:
   "P S ⟹ (finite' intersection_of P) S"
  by (simp add: intersection_of_inc)


subsection ‹The ``Relative to'' operator›

text‹A somewhat cheap but handy way of getting localized forms of various topological concepts
(open, closed, borel, fsigma, gdelta etc.)›

definition relative_to :: "['a set ⇒ bool, 'a set, 'a set] ⇒ bool" (infixl ‹relative'_to› 55)
  where "P relative_to S ≡ λT. ∃U. P U ∧ S ∩ U = T"

lemma relative_to_UNIV [simp]: "(P relative_to UNIV) S ⟷ P S"
  by (simp add: relative_to_def)

lemma relative_to_imp_subset:
   "(P relative_to S) T ⟹ T ⊆ S"
  by (auto simp: relative_to_def)

lemma all_relative_to: "(∀S. (P relative_to U) S ⟶ Q S) ⟷ (∀S. P S ⟶ Q(U ∩ S))"
  by (auto simp: relative_to_def)

lemma relative_toE: "⟦(P relative_to U) S; ⋀S. P S ⟹ Q(U ∩ S)⟧ ⟹ Q S"
  by (auto simp: relative_to_def)

lemma relative_to_inc:
   "P S ⟹ (P relative_to U) (U ∩ S)"
  by (auto simp: relative_to_def)

lemma relative_to_relative_to [simp]:
   "P relative_to S relative_to T = P relative_to (S ∩ T)"
  unfolding relative_to_def
  by auto

lemma relative_to_compl:
   "S ⊆ U ⟹ ((P relative_to U) (U - S) ⟷ ((λc. P(- c)) relative_to U) S)"
  unfolding relative_to_def
  by (metis Diff_Diff_Int Diff_eq double_compl inf.absorb_iff2)

lemma relative_to_subset_trans:
   "⟦(P relative_to U) S; S ⊆ T; T ⊆ U⟧ ⟹ (P relative_to T) S"
  unfolding relative_to_def by auto

lemma relative_to_mono:
   "⟦(P relative_to U) S; ⋀S. P S ⟹ Q S⟧ ⟹ (Q relative_to U) S"
  unfolding relative_to_def by auto

lemma relative_to_subset_inc: "⟦S ⊆ U; P S⟧ ⟹ (P relative_to U) S"
  unfolding relative_to_def by auto

lemma relative_to_Int:
   "⟦(P relative_to S) C; (P relative_to S) D; ⋀X Y. ⟦P X; P Y⟧ ⟹ P(X ∩ Y)⟧
         ⟹  (P relative_to S) (C ∩ D)"
  unfolding relative_to_def by auto

lemma relative_to_Un:
   "⟦(P relative_to S) C; (P relative_to S) D; ⋀X Y. ⟦P X; P Y⟧ ⟹ P(X ∪ Y)⟧
         ⟹  (P relative_to S) (C ∪ D)"
  unfolding relative_to_def by auto

lemma arbitrary_union_of_relative_to:
  "((arbitrary union_of P) relative_to U) = (arbitrary union_of (P relative_to U))" (is "?lhs = ?rhs")
proof -
  have "?rhs S" if L: "?lhs S" for S
  proof -
    obtain 𝒰 where "S = U ∩ ⋃𝒰" "𝒰 ⊆ Collect P"
      using L unfolding relative_to_def union_of_def by auto
    then show ?thesis
      unfolding relative_to_def union_of_def arbitrary_def
      by (rule_tac x="(λX. U ∩ X) ` 𝒰" in exI) auto
  qed
  moreover have "?lhs S" if R: "?rhs S" for S
  proof -
    obtain 𝒰 where "S = ⋃𝒰" "∀T∈𝒰. ∃V. P V ∧ U ∩ V = T"
      using R unfolding relative_to_def union_of_def by auto
    then obtain f where f: "⋀T. T ∈ 𝒰 ⟹ P (f T)" "⋀T. T ∈ 𝒰 ⟹ U ∩ (f T) = T"
      by metis
    then have "∃𝒰'⊆Collect P. ⋃𝒰' = ⋃ (f ` 𝒰)"
      by (metis image_subset_iff mem_Collect_eq)
    moreover have eq: "U ∩ ⋃ (f ` 𝒰) = ⋃𝒰"
      using f by auto
    ultimately show ?thesis
      unfolding relative_to_def union_of_def arbitrary_def ‹S = ⋃𝒰›
      by metis
  qed
  ultimately show ?thesis
    by blast
qed

lemma finite_union_of_relative_to:
  "((finite union_of P) relative_to U) = (finite union_of (P relative_to U))" (is "?lhs = ?rhs")
proof -
  have "?rhs S" if L: "?lhs S" for S
  proof -
    obtain 𝒰 where "S = U ∩ ⋃𝒰" "𝒰 ⊆ Collect P" "finite 𝒰"
      using L unfolding relative_to_def union_of_def by auto
    then show ?thesis
      unfolding relative_to_def union_of_def
      by (rule_tac x="(λX. U ∩ X) ` 𝒰" in exI) auto
  qed
  moreover have "?lhs S" if R: "?rhs S" for S
  proof -
    obtain 𝒰 where "S = ⋃𝒰" "∀T∈𝒰. ∃V. P V ∧ U ∩ V = T" "finite 𝒰"
      using R unfolding relative_to_def union_of_def by auto
    then obtain f where f: "⋀T. T ∈ 𝒰 ⟹ P (f T)" "⋀T. T ∈ 𝒰 ⟹ U ∩ (f T) = T"
      by metis
    then have "∃𝒰'⊆Collect P. ⋃𝒰' = ⋃ (f ` 𝒰)"
      by (metis image_subset_iff mem_Collect_eq)
    moreover have eq: "U ∩ ⋃ (f ` 𝒰) = ⋃𝒰"
      using f by auto
    ultimately show ?thesis
      using ‹finite 𝒰› f
      unfolding relative_to_def union_of_def ‹S = ⋃𝒰›
      by (rule_tac x="⋃ (f ` 𝒰)" in exI) (metis finite_imageI image_subsetI mem_Collect_eq)
  qed
  ultimately show ?thesis
    by blast
qed

lemma countable_union_of_relative_to:
  "((countable union_of P) relative_to U) = (countable union_of (P relative_to U))" (is "?lhs = ?rhs")
proof -
  have "?rhs S" if L: "?lhs S" for S
  proof -
    obtain 𝒰 where "S = U ∩ ⋃𝒰" "𝒰 ⊆ Collect P" "countable 𝒰"
      using L unfolding relative_to_def union_of_def by auto
    then show ?thesis
      unfolding relative_to_def union_of_def
      by (rule_tac x="(λX. U ∩ X) ` 𝒰" in exI) auto
  qed
  moreover have "?lhs S" if R: "?rhs S" for S
  proof -
    obtain 𝒰 where "S = ⋃𝒰" "∀T∈𝒰. ∃V. P V ∧ U ∩ V = T" "countable 𝒰"
      using R unfolding relative_to_def union_of_def by auto
    then obtain f where f: "⋀T. T ∈ 𝒰 ⟹ P (f T)" "⋀T. T ∈ 𝒰 ⟹ U ∩ (f T) = T"
      by metis
    then have "∃𝒰'⊆Collect P. ⋃𝒰' = ⋃ (f ` 𝒰)"
      by (metis image_subset_iff mem_Collect_eq)
    moreover have eq: "U ∩ ⋃ (f ` 𝒰) = ⋃𝒰"
      using f by auto
    ultimately show ?thesis
      using ‹countable 𝒰› f
      unfolding relative_to_def union_of_def ‹S = ⋃𝒰›
      by (rule_tac x="⋃ (f ` 𝒰)" in exI) (metis countable_image image_subsetI mem_Collect_eq)
  qed
  ultimately show ?thesis
    by blast
qed


lemma arbitrary_intersection_of_relative_to:
  "((arbitrary intersection_of P) relative_to U) = ((arbitrary intersection_of (P relative_to U)) relative_to U)" (is "?lhs = ?rhs")
proof -
  have "?rhs S" if L: "?lhs S" for S
  proof -
    obtain 𝒰 where 𝒰: "S = U ∩ ⋂𝒰" "𝒰 ⊆ Collect P"
      using L unfolding relative_to_def intersection_of_def by auto
    show ?thesis
      unfolding relative_to_def intersection_of_def arbitrary_def
    proof (intro exI conjI)
      show "U ∩ (⋂X∈𝒰. U ∩ X) = S" "(∩) U ` 𝒰 ⊆ {T. ∃Ua. P Ua ∧ U ∩ Ua = T}"
        using 𝒰 by blast+
    qed auto
  qed
  moreover have "?lhs S" if R: "?rhs S" for S
  proof -
    obtain 𝒰 where "S = U ∩ ⋂𝒰" "∀T∈𝒰. ∃V. P V ∧ U ∩ V = T"
      using R unfolding relative_to_def intersection_of_def  by auto
    then obtain f where f: "⋀T. T ∈ 𝒰 ⟹ P (f T)" "⋀T. T ∈ 𝒰 ⟹ U ∩ (f T) = T"
      by metis
    then have "f `  𝒰 ⊆ Collect P"
      by auto
    moreover have eq: "U ∩ ⋂(f ` 𝒰) = U ∩ ⋂𝒰"
      using f by auto
    ultimately show ?thesis
      unfolding relative_to_def intersection_of_def arbitrary_def ‹S = U ∩ ⋂𝒰›
      by auto
  qed
  ultimately show ?thesis
    by blast
qed

lemma finite_intersection_of_relative_to:
  "((finite intersection_of P) relative_to U) = ((finite intersection_of (P relative_to U)) relative_to U)" (is "?lhs = ?rhs")
proof -
  have "?rhs S" if L: "?lhs S" for S
  proof -
    obtain 𝒰 where 𝒰: "S = U ∩ ⋂𝒰" "𝒰 ⊆ Collect P" "finite 𝒰"
      using L unfolding relative_to_def intersection_of_def by auto
    show ?thesis
      unfolding relative_to_def intersection_of_def
    proof (intro exI conjI)
      show "U ∩ (⋂X∈𝒰. U ∩ X) = S" "(∩) U ` 𝒰 ⊆ {T. ∃Ua. P Ua ∧ U ∩ Ua = T}"
        using 𝒰 by blast+
      show "finite ((∩) U ` 𝒰)"
        by (simp add: ‹finite 𝒰›)
    qed auto
  qed
  moreover have "?lhs S" if R: "?rhs S" for S
  proof -
    obtain 𝒰 where "S = U ∩ ⋂𝒰" "∀T∈𝒰. ∃V. P V ∧ U ∩ V = T" "finite 𝒰"
      using R unfolding relative_to_def intersection_of_def  by auto
    then obtain f where f: "⋀T. T ∈ 𝒰 ⟹ P (f T)" "⋀T. T ∈ 𝒰 ⟹ U ∩ (f T) = T"
      by metis
    then have "f `  𝒰 ⊆ Collect P"
      by auto
    moreover have eq: "U ∩ ⋂ (f ` 𝒰) = U ∩ ⋂ 𝒰"
      using f by auto
    ultimately show ?thesis
      unfolding relative_to_def intersection_of_def ‹S = U ∩ ⋂𝒰›
      using ‹finite 𝒰›
      by auto
  qed
  ultimately show ?thesis
    by blast
qed

lemma countable_intersection_of_relative_to:
  "((countable intersection_of P) relative_to U) = ((countable intersection_of (P relative_to U)) relative_to U)" (is "?lhs = ?rhs")
proof -
  have "?rhs S" if L: "?lhs S" for S
  proof -
    obtain 𝒰 where 𝒰: "S = U ∩ ⋂𝒰" "𝒰 ⊆ Collect P" "countable 𝒰"
      using L unfolding relative_to_def intersection_of_def by auto
    show ?thesis
      unfolding relative_to_def intersection_of_def
    proof (intro exI conjI)
      show "U ∩ (⋂X∈𝒰. U ∩ X) = S" "(∩) U ` 𝒰 ⊆ {T. ∃Ua. P Ua ∧ U ∩ Ua = T}"
        using 𝒰 by blast+
      show "countable ((∩) U ` 𝒰)"
        by (simp add: ‹countable 𝒰›)
    qed auto
  qed
  moreover have "?lhs S" if R: "?rhs S" for S
  proof -
    obtain 𝒰 where "S = U ∩ ⋂𝒰" "∀T∈𝒰. ∃V. P V ∧ U ∩ V = T" "countable 𝒰"
      using R unfolding relative_to_def intersection_of_def  by auto
    then obtain f where f: "⋀T. T ∈ 𝒰 ⟹ P (f T)" "⋀T. T ∈ 𝒰 ⟹ U ∩ (f T) = T"
      by metis
    then have "f `  𝒰 ⊆ Collect P"
      by auto
    moreover have eq: "U ∩ ⋂ (f ` 𝒰) = U ∩ ⋂ 𝒰"
      using f by auto
    ultimately show ?thesis
      unfolding relative_to_def intersection_of_def ‹S = U ∩ ⋂𝒰›
      using ‹countable 𝒰› countable_image
      by auto
  qed
  ultimately show ?thesis
    by blast
qed

lemma countable_union_of_empty [simp]: "(countable union_of P) {}"
  by (simp add: union_of_empty)

lemma countable_intersection_of_empty [simp]: "(countable intersection_of P) UNIV"
  by (simp add: intersection_of_empty)

lemma countable_union_of_inc: "P S ⟹ (countable union_of P) S"
  by (simp add: union_of_inc)

lemma countable_intersection_of_inc: "P S ⟹ (countable intersection_of P) S"
  by (simp add: intersection_of_inc)

lemma countable_union_of_complement:
  "(countable union_of P) S ⟷ (countable intersection_of (λS. P(-S))) (-S)" 
  (is "?lhs=?rhs")
proof
  assume ?lhs
  then obtain 𝒰 where "countable 𝒰" and 𝒰: "𝒰 ⊆ Collect P" "⋃𝒰 = S"
    by (metis union_of_def)
  define 𝒰' where "𝒰' ≡ (λC. -C) ` 𝒰"
  have "𝒰' ⊆ {S. P (- S)}" "⋂𝒰' = -S"
    using 𝒰'_def 𝒰 by auto
  then show ?rhs
    unfolding intersection_of_def by (metis 𝒰'_def ‹countable 𝒰› countable_image)
next
  assume ?rhs
  then obtain 𝒰 where "countable 𝒰" and 𝒰: "𝒰 ⊆ {S. P (- S)}" "⋂𝒰 = -S"
    by (metis intersection_of_def)
  define 𝒰' where "𝒰' ≡ (λC. -C) ` 𝒰"
  have "𝒰' ⊆ Collect P" "⋃ 𝒰' = S"
    using 𝒰'_def 𝒰 by auto
  then show ?lhs
    unfolding union_of_def
    by (metis 𝒰'_def ‹countable 𝒰› countable_image)
qed

lemma countable_intersection_of_complement:
   "(countable intersection_of P) S ⟷ (countable union_of (λS. P(- S))) (- S)"
  by (simp add: countable_union_of_complement)

lemma countable_union_of_explicit:
  assumes "P {}"
  shows "(countable union_of P) S ⟷
         (∃T. (∀n::nat. P(T n)) ∧ ⋃(range T) = S)" (is "?lhs=?rhs")
proof
  assume ?lhs
  then obtain 𝒰 where "countable 𝒰" and 𝒰: "𝒰 ⊆ Collect P" "⋃𝒰 = S"
    by (metis union_of_def)
  then show ?rhs
    by (metis SUP_bot Sup_empty assms from_nat_into mem_Collect_eq range_from_nat_into subsetD)
next
  assume ?rhs
  then show ?lhs
    by (metis countableI_type countable_image image_subset_iff mem_Collect_eq union_of_def)
qed

lemma countable_union_of_ascending:
  assumes empty: "P {}" and Un: "⋀T U. ⟦P T; P U⟧ ⟹ P(T ∪ U)"
  shows "(countable union_of P) S ⟷
         (∃T. (∀n. P(T n)) ∧ (∀n. T n ⊆ T(Suc n)) ∧ ⋃(range T) = S)" (is "?lhs=?rhs")
proof
  assume ?lhs
  then obtain T where T: "⋀n::nat. P(T n)" "⋃(range T) = S"
    by (meson empty countable_union_of_explicit)
  have "P (⋃ (T ` {..n}))" for n
    by (induction n) (auto simp: atMost_Suc Un T)
  with T show ?rhs
    by (rule_tac x="λn. ⋃k≤n. T k" in exI) force
next
  assume ?rhs
  then show ?lhs
    using empty countable_union_of_explicit by auto
qed

lemma countable_union_of_idem [simp]:
  "countable union_of countable union_of P = countable union_of P"  (is "?lhs=?rhs")
proof
  fix S
  show "(countable union_of countable union_of P) S = (countable union_of P) S"
  proof
    assume L: "?lhs S"
    then obtain 𝒰 where "countable 𝒰" and 𝒰: "𝒰 ⊆ Collect (countable union_of P)" "⋃𝒰 = S"
      by (metis union_of_def)
    then have "∀U ∈ 𝒰. ∃𝒱. countable 𝒱 ∧ 𝒱 ⊆ Collect P ∧ U = ⋃𝒱"
      by (metis Ball_Collect union_of_def)
    then obtain ℱ where ℱ: "∀U ∈ 𝒰. countable (ℱ U) ∧ ℱ U ⊆ Collect P ∧ U = ⋃(ℱ U)"
      by metis
    have "countable (⋃ (ℱ ` 𝒰))"
      using ℱ ‹countable 𝒰› by blast
    moreover have "⋃ (ℱ ` 𝒰) ⊆ Collect P"
      by (simp add: Sup_le_iff ℱ)
    moreover have "⋃ (⋃ (ℱ ` 𝒰)) = S"
      by auto (metis Union_iff ℱ 𝒰(2))+
    ultimately show "?rhs S"
      by (meson union_of_def)
  qed (simp add: countable_union_of_inc)
qed

lemma countable_intersection_of_idem [simp]:
   "countable intersection_of countable intersection_of P =
        countable intersection_of P"
  by (force simp: countable_intersection_of_complement)

lemma countable_union_of_Union:
   "⟦countable 𝒰; ⋀S. S ∈ 𝒰 ⟹ (countable union_of P) S⟧
        ⟹ (countable union_of P) (⋃ 𝒰)"
  by (metis Ball_Collect countable_union_of_idem union_of_def)

lemma countable_union_of_UN:
   "⟦countable I; ⋀i. i ∈ I ⟹ (countable union_of P) (U i)⟧
        ⟹ (countable union_of P) (⋃i∈I. U i)"
  by (metis (mono_tags, lifting) countable_image countable_union_of_Union imageE)

lemma countable_union_of_Un:
  "⟦(countable union_of P) S; (countable union_of P) T⟧
           ⟹ (countable union_of P) (S ∪ T)"
  by (smt (verit) Union_Un_distrib countable_Un le_sup_iff union_of_def)

lemma countable_intersection_of_Inter:
   "⟦countable 𝒰; ⋀S. S ∈ 𝒰 ⟹ (countable intersection_of P) S⟧
        ⟹ (countable intersection_of P) (⋂ 𝒰)"
  by (metis countable_intersection_of_idem intersection_of_def mem_Collect_eq subsetI)

lemma countable_intersection_of_INT:
   "⟦countable I; ⋀i. i ∈ I ⟹ (countable intersection_of P) (U i)⟧
        ⟹ (countable intersection_of P) (⋂i∈I. U i)"
  by (metis (mono_tags, lifting) countable_image countable_intersection_of_Inter imageE)

lemma countable_intersection_of_inter:
   "⟦(countable intersection_of P) S; (countable intersection_of P) T⟧
           ⟹ (countable intersection_of P) (S ∩ T)"
  by (simp add: countable_intersection_of_complement countable_union_of_Un)

lemma countable_union_of_Int:
  assumes S: "(countable union_of P) S" and T: "(countable union_of P) T"
    and Int: "⋀S T. P S ∧ P T ⟹ P(S ∩ T)"
  shows "(countable union_of P) (S ∩ T)"
proof -
  obtain 𝒰 where "countable 𝒰" and 𝒰: "𝒰 ⊆ Collect P" "⋃𝒰 = S"
    using S by (metis union_of_def)
  obtain 𝒱 where "countable 𝒱" and 𝒱: "𝒱 ⊆ Collect P" "⋃𝒱 = T"
    using T by (metis union_of_def)
  have "⋀U V. ⟦U ∈ 𝒰; V ∈ 𝒱⟧ ⟹ (countable union_of P) (U ∩ V)"
    using 𝒰 𝒱 by (metis Ball_Collect countable_union_of_inc local.Int)
  then have "(countable union_of P) (⋃U∈𝒰. ⋃V∈𝒱. U ∩ V)"
    by (meson ‹countable 𝒰› ‹countable 𝒱› countable_union_of_UN)
  moreover have "S ∩ T = (⋃U∈𝒰. ⋃V∈𝒱. U ∩ V)"
    by (simp add: 𝒰 𝒱)
  ultimately show ?thesis
    by presburger
qed

lemma countable_intersection_of_union:
  assumes S: "(countable intersection_of P) S" and T: "(countable intersection_of P) T"
    and Un: "⋀S T. P S ∧ P T ⟹ P(S ∪ T)"
  shows "(countable intersection_of P) (S ∪ T)"
  by (metis (mono_tags, lifting) Compl_Int S T Un compl_sup countable_intersection_of_complement countable_union_of_Int)

end