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. 𝒱Collect P. 𝒱 = S}" for 𝒰
  proof -
    let ?𝒲 = "{V. 𝒱. 𝒱Collect 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. 𝒱Collect P  V  𝒱  (S  𝒰. 𝒱 = S)}" in exI)
      using that by (blast intro: *)
  qed
  have 2: "𝒰'{S. 𝒰Collect 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) (C𝒰. D𝒱. 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: "U𝒰. 𝒰. finite 𝒰  (𝒰  Collect P)  𝒰 = U"
      using S unfolding union_of_def by (auto simp: subset_eq)
    then obtain f where "U𝒰. 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) (C𝒰. D𝒱. 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" "T𝒰. 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" "T𝒰. 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" "T𝒰. 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  (X𝒰. 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" "T𝒰. 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  (X𝒰. 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" "T𝒰. 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  (X𝒰. 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" "T𝒰. 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. kn. 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  where : "U  𝒰. countable ( U)   U  Collect P  U = ( U)"
      by metis
    have "countable ( ( ` 𝒰))"
      using  countable 𝒰 by blast
    moreover have " ( ` 𝒰)  Collect P"
      by (simp add: Sup_le_iff )
    moreover have " ( ( ` 𝒰)) = S"
      by auto (metis Union_iff  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) (iI. 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) (iI. 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) (U𝒰. V𝒱. U  V)"
    by (meson countable 𝒰 countable 𝒱 countable_union_of_UN)
  moreover have "S  T = (U𝒰. V𝒱. 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