Theory Measurable

(*  Title:      HOL/Probability/Measurable.thy
    Author:     Johannes Hölzl <hoelzl@in.tum.de>
*)
theory Measurable
  imports
    Sigma_Algebra
    "~~/src/HOL/Library/Order_Continuity"
begin

subsection ‹Measurability prover›

lemma (in algebra) sets_Collect_finite_All:
  assumes "⋀i. i ∈ S ⟹ {x∈Ω. P i x} ∈ M" "finite S"
  shows "{x∈Ω. ∀i∈S. P i x} ∈ M"
proof -
  have "{x∈Ω. ∀i∈S. P i x} = (if S = {} then Ω else ⋂i∈S. {x∈Ω. P i x})"
    by auto
  with assms show ?thesis by (auto intro!: sets_Collect_finite_All')
qed

abbreviation "pred M P ≡ P ∈ measurable M (count_space (UNIV::bool set))"

lemma pred_def: "pred M P ⟷ {x∈space M. P x} ∈ sets M"
proof
  assume "pred M P"
  then have "P -` {True} ∩ space M ∈ sets M"
    by (auto simp: measurable_count_space_eq2)
  also have "P -` {True} ∩ space M = {x∈space M. P x}" by auto
  finally show "{x∈space M. P x} ∈ sets M" .
next
  assume P: "{x∈space M. P x} ∈ sets M"
  moreover
  { fix X
    have "X ∈ Pow (UNIV :: bool set)" by simp
    then have "P -` X ∩ space M = {x∈space M. ((X = {True} ⟶ P x) ∧ (X = {False} ⟶ ¬ P x) ∧ X ≠ {})}"
      unfolding UNIV_bool Pow_insert Pow_empty by auto
    then have "P -` X ∩ space M ∈ sets M"
      by (auto intro!: sets.sets_Collect_neg sets.sets_Collect_imp sets.sets_Collect_conj sets.sets_Collect_const P) }
  then show "pred M P"
    by (auto simp: measurable_def)
qed

lemma pred_sets1: "{x∈space M. P x} ∈ sets M ⟹ f ∈ measurable N M ⟹ pred N (λx. P (f x))"
  by (rule measurable_compose[where f=f and N=M]) (auto simp: pred_def)

lemma pred_sets2: "A ∈ sets N ⟹ f ∈ measurable M N ⟹ pred M (λx. f x ∈ A)"
  by (rule measurable_compose[where f=f and N=N]) (auto simp: pred_def Int_def[symmetric])

ML_file "measurable.ML"

attribute_setup measurable = ‹
  Scan.lift (
    (Args.add >> K true || Args.del >> K false || Scan.succeed true) --
    Scan.optional (Args.parens (
      Scan.optional (Args.$$$ "raw" >> K true) false --
      Scan.optional (Args.$$$ "generic" >> K Measurable.Generic) Measurable.Concrete))
    (false, Measurable.Concrete) >>
    Measurable.measurable_thm_attr)
› "declaration of measurability theorems"

attribute_setup measurable_dest = Measurable.dest_thm_attr
  "add dest rule to measurability prover"

attribute_setup measurable_cong = Measurable.cong_thm_attr
  "add congurence rules to measurability prover"

method_setup measurable = ‹ Scan.lift (Scan.succeed (METHOD o Measurable.measurable_tac)) ›
  "measurability prover"

simproc_setup measurable ("A ∈ sets M" | "f ∈ measurable M N") = ‹K Measurable.simproc›

setup ‹
  Global_Theory.add_thms_dynamic (@{binding measurable}, Measurable.get_all)
›

declare
  pred_sets1[measurable_dest]
  pred_sets2[measurable_dest]
  sets.sets_into_space[measurable_dest]

declare
  sets.top[measurable]
  sets.empty_sets[measurable (raw)]
  sets.Un[measurable (raw)]
  sets.Diff[measurable (raw)]

declare
  measurable_count_space[measurable (raw)]
  measurable_ident[measurable (raw)]
  measurable_id[measurable (raw)]
  measurable_const[measurable (raw)]
  measurable_If[measurable (raw)]
  measurable_comp[measurable (raw)]
  measurable_sets[measurable (raw)]

declare measurable_cong_sets[measurable_cong]
declare sets_restrict_space_cong[measurable_cong]
declare sets_restrict_UNIV[measurable_cong]

lemma predE[measurable (raw)]:
  "pred M P ⟹ {x∈space M. P x} ∈ sets M"
  unfolding pred_def .

lemma pred_intros_imp'[measurable (raw)]:
  "(K ⟹ pred M (λx. P x)) ⟹ pred M (λx. K ⟶ P x)"
  by (cases K) auto

lemma pred_intros_conj1'[measurable (raw)]:
  "(K ⟹ pred M (λx. P x)) ⟹ pred M (λx. K ∧ P x)"
  by (cases K) auto

lemma pred_intros_conj2'[measurable (raw)]:
  "(K ⟹ pred M (λx. P x)) ⟹ pred M (λx. P x ∧ K)"
  by (cases K) auto

lemma pred_intros_disj1'[measurable (raw)]:
  "(¬ K ⟹ pred M (λx. P x)) ⟹ pred M (λx. K ∨ P x)"
  by (cases K) auto

lemma pred_intros_disj2'[measurable (raw)]:
  "(¬ K ⟹ pred M (λx. P x)) ⟹ pred M (λx. P x ∨ K)"
  by (cases K) auto

lemma pred_intros_logic[measurable (raw)]:
  "pred M (λx. x ∈ space M)"
  "pred M (λx. P x) ⟹ pred M (λx. ¬ P x)"
  "pred M (λx. Q x) ⟹ pred M (λx. P x) ⟹ pred M (λx. Q x ∧ P x)"
  "pred M (λx. Q x) ⟹ pred M (λx. P x) ⟹ pred M (λx. Q x ⟶ P x)"
  "pred M (λx. Q x) ⟹ pred M (λx. P x) ⟹ pred M (λx. Q x ∨ P x)"
  "pred M (λx. Q x) ⟹ pred M (λx. P x) ⟹ pred M (λx. Q x = P x)"
  "pred M (λx. f x ∈ UNIV)"
  "pred M (λx. f x ∈ {})"
  "pred M (λx. P' (f x) x) ⟹ pred M (λx. f x ∈ {y. P' y x})"
  "pred M (λx. f x ∈ (B x)) ⟹ pred M (λx. f x ∈ - (B x))"
  "pred M (λx. f x ∈ (A x)) ⟹ pred M (λx. f x ∈ (B x)) ⟹ pred M (λx. f x ∈ (A x) - (B x))"
  "pred M (λx. f x ∈ (A x)) ⟹ pred M (λx. f x ∈ (B x)) ⟹ pred M (λx. f x ∈ (A x) ∩ (B x))"
  "pred M (λx. f x ∈ (A x)) ⟹ pred M (λx. f x ∈ (B x)) ⟹ pred M (λx. f x ∈ (A x) ∪ (B x))"
  "pred M (λx. g x (f x) ∈ (X x)) ⟹ pred M (λx. f x ∈ (g x) -` (X x))"
  by (auto simp: iff_conv_conj_imp pred_def)

lemma pred_intros_countable[measurable (raw)]:
  fixes P :: "'a ⇒ 'i :: countable ⇒ bool"
  shows
    "(⋀i. pred M (λx. P x i)) ⟹ pred M (λx. ∀i. P x i)"
    "(⋀i. pred M (λx. P x i)) ⟹ pred M (λx. ∃i. P x i)"
  by (auto intro!: sets.sets_Collect_countable_All sets.sets_Collect_countable_Ex simp: pred_def)

lemma pred_intros_countable_bounded[measurable (raw)]:
  fixes X :: "'i :: countable set"
  shows
    "(⋀i. i ∈ X ⟹ pred M (λx. x ∈ N x i)) ⟹ pred M (λx. x ∈ (⋂i∈X. N x i))"
    "(⋀i. i ∈ X ⟹ pred M (λx. x ∈ N x i)) ⟹ pred M (λx. x ∈ (⋃i∈X. N x i))"
    "(⋀i. i ∈ X ⟹ pred M (λx. P x i)) ⟹ pred M (λx. ∀i∈X. P x i)"
    "(⋀i. i ∈ X ⟹ pred M (λx. P x i)) ⟹ pred M (λx. ∃i∈X. P x i)"
  by simp_all (auto simp: Bex_def Ball_def)

lemma pred_intros_finite[measurable (raw)]:
  "finite I ⟹ (⋀i. i ∈ I ⟹ pred M (λx. x ∈ N x i)) ⟹ pred M (λx. x ∈ (⋂i∈I. N x i))"
  "finite I ⟹ (⋀i. i ∈ I ⟹ pred M (λx. x ∈ N x i)) ⟹ pred M (λx. x ∈ (⋃i∈I. N x i))"
  "finite I ⟹ (⋀i. i ∈ I ⟹ pred M (λx. P x i)) ⟹ pred M (λx. ∀i∈I. P x i)"
  "finite I ⟹ (⋀i. i ∈ I ⟹ pred M (λx. P x i)) ⟹ pred M (λx. ∃i∈I. P x i)"
  by (auto intro!: sets.sets_Collect_finite_Ex sets.sets_Collect_finite_All simp: iff_conv_conj_imp pred_def)

lemma countable_Un_Int[measurable (raw)]:
  "(⋀i :: 'i :: countable. i ∈ I ⟹ N i ∈ sets M) ⟹ (⋃i∈I. N i) ∈ sets M"
  "I ≠ {} ⟹ (⋀i :: 'i :: countable. i ∈ I ⟹ N i ∈ sets M) ⟹ (⋂i∈I. N i) ∈ sets M"
  by auto

declare
  finite_UN[measurable (raw)]
  finite_INT[measurable (raw)]

lemma sets_Int_pred[measurable (raw)]:
  assumes space: "A ∩ B ⊆ space M" and [measurable]: "pred M (λx. x ∈ A)" "pred M (λx. x ∈ B)"
  shows "A ∩ B ∈ sets M"
proof -
  have "{x∈space M. x ∈ A ∩ B} ∈ sets M" by auto
  also have "{x∈space M. x ∈ A ∩ B} = A ∩ B"
    using space by auto
  finally show ?thesis .
qed

lemma [measurable (raw generic)]:
  assumes f: "f ∈ measurable M N" and c: "c ∈ space N ⟹ {c} ∈ sets N"
  shows pred_eq_const1: "pred M (λx. f x = c)"
    and pred_eq_const2: "pred M (λx. c = f x)"
proof -
  show "pred M (λx. f x = c)"
  proof cases
    assume "c ∈ space N"
    with measurable_sets[OF f c] show ?thesis
      by (auto simp: Int_def conj_commute pred_def)
  next
    assume "c ∉ space N"
    with f[THEN measurable_space] have "{x ∈ space M. f x = c} = {}" by auto
    then show ?thesis by (auto simp: pred_def cong: conj_cong)
  qed
  then show "pred M (λx. c = f x)"
    by (simp add: eq_commute)
qed

lemma pred_count_space_const1[measurable (raw)]:
  "f ∈ measurable M (count_space UNIV) ⟹ Measurable.pred M (λx. f x = c)"
  by (intro pred_eq_const1[where N="count_space UNIV"]) (auto )

lemma pred_count_space_const2[measurable (raw)]:
  "f ∈ measurable M (count_space UNIV) ⟹ Measurable.pred M (λx. c = f x)"
  by (intro pred_eq_const2[where N="count_space UNIV"]) (auto )

lemma pred_le_const[measurable (raw generic)]:
  assumes f: "f ∈ measurable M N" and c: "{.. c} ∈ sets N" shows "pred M (λx. f x ≤ c)"
  using measurable_sets[OF f c]
  by (auto simp: Int_def conj_commute eq_commute pred_def)

lemma pred_const_le[measurable (raw generic)]:
  assumes f: "f ∈ measurable M N" and c: "{c ..} ∈ sets N" shows "pred M (λx. c ≤ f x)"
  using measurable_sets[OF f c]
  by (auto simp: Int_def conj_commute eq_commute pred_def)

lemma pred_less_const[measurable (raw generic)]:
  assumes f: "f ∈ measurable M N" and c: "{..< c} ∈ sets N" shows "pred M (λx. f x < c)"
  using measurable_sets[OF f c]
  by (auto simp: Int_def conj_commute eq_commute pred_def)

lemma pred_const_less[measurable (raw generic)]:
  assumes f: "f ∈ measurable M N" and c: "{c <..} ∈ sets N" shows "pred M (λx. c < f x)"
  using measurable_sets[OF f c]
  by (auto simp: Int_def conj_commute eq_commute pred_def)

declare
  sets.Int[measurable (raw)]

lemma pred_in_If[measurable (raw)]:
  "(P ⟹ pred M (λx. x ∈ A x)) ⟹ (¬ P ⟹ pred M (λx. x ∈ B x)) ⟹
    pred M (λx. x ∈ (if P then A x else B x))"
  by auto

lemma sets_range[measurable_dest]:
  "A ` I ⊆ sets M ⟹ i ∈ I ⟹ A i ∈ sets M"
  by auto

lemma pred_sets_range[measurable_dest]:
  "A ` I ⊆ sets N ⟹ i ∈ I ⟹ f ∈ measurable M N ⟹ pred M (λx. f x ∈ A i)"
  using pred_sets2[OF sets_range] by auto

lemma sets_All[measurable_dest]:
  "∀i. A i ∈ sets (M i) ⟹ A i ∈ sets (M i)"
  by auto

lemma pred_sets_All[measurable_dest]:
  "∀i. A i ∈ sets (N i) ⟹ f ∈ measurable M (N i) ⟹ pred M (λx. f x ∈ A i)"
  using pred_sets2[OF sets_All, of A N f] by auto

lemma sets_Ball[measurable_dest]:
  "∀i∈I. A i ∈ sets (M i) ⟹ i∈I ⟹ A i ∈ sets (M i)"
  by auto

lemma pred_sets_Ball[measurable_dest]:
  "∀i∈I. A i ∈ sets (N i) ⟹ i∈I ⟹ f ∈ measurable M (N i) ⟹ pred M (λx. f x ∈ A i)"
  using pred_sets2[OF sets_Ball, of _ _ _ f] by auto

lemma measurable_finite[measurable (raw)]:
  fixes S :: "'a ⇒ nat set"
  assumes [measurable]: "⋀i. {x∈space M. i ∈ S x} ∈ sets M"
  shows "pred M (λx. finite (S x))"
  unfolding finite_nat_set_iff_bounded by (simp add: Ball_def)

lemma measurable_Least[measurable]:
  assumes [measurable]: "(⋀i::nat. (λx. P i x) ∈ measurable M (count_space UNIV))"q
  shows "(λx. LEAST i. P i x) ∈ measurable M (count_space UNIV)"
  unfolding measurable_def by (safe intro!: sets_Least) simp_all

lemma measurable_Max_nat[measurable (raw)]:
  fixes P :: "nat ⇒ 'a ⇒ bool"
  assumes [measurable]: "⋀i. Measurable.pred M (P i)"
  shows "(λx. Max {i. P i x}) ∈ measurable M (count_space UNIV)"
  unfolding measurable_count_space_eq2_countable
proof safe
  fix n

  { fix x assume "∀i. ∃n≥i. P n x"
    then have "infinite {i. P i x}"
      unfolding infinite_nat_iff_unbounded_le by auto
    then have "Max {i. P i x} = the None"
      by (rule Max.infinite) }
  note 1 = this

  { fix x i j assume "P i x" "∀n≥j. ¬ P n x"
    then have "finite {i. P i x}"
      by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
    with ‹P i x› have "P (Max {i. P i x}) x" "i ≤ Max {i. P i x}" "finite {i. P i x}"
      using Max_in[of "{i. P i x}"] by auto }
  note 2 = this

  have "(λx. Max {i. P i x}) -` {n} ∩ space M = {x∈space M. Max {i. P i x} = n}"
    by auto
  also have "… =
    {x∈space M. if (∀i. ∃n≥i. P n x) then the None = n else
      if (∃i. P i x) then P n x ∧ (∀i>n. ¬ P i x)
      else Max {} = n}"
    by (intro arg_cong[where f=Collect] ext conj_cong)
       (auto simp add: 1 2 not_le[symmetric] intro!: Max_eqI)
  also have "… ∈ sets M"
    by measurable
  finally show "(λx. Max {i. P i x}) -` {n} ∩ space M ∈ sets M" .
qed simp

lemma measurable_Min_nat[measurable (raw)]:
  fixes P :: "nat ⇒ 'a ⇒ bool"
  assumes [measurable]: "⋀i. Measurable.pred M (P i)"
  shows "(λx. Min {i. P i x}) ∈ measurable M (count_space UNIV)"
  unfolding measurable_count_space_eq2_countable
proof safe
  fix n

  { fix x assume "∀i. ∃n≥i. P n x"
    then have "infinite {i. P i x}"
      unfolding infinite_nat_iff_unbounded_le by auto
    then have "Min {i. P i x} = the None"
      by (rule Min.infinite) }
  note 1 = this

  { fix x i j assume "P i x" "∀n≥j. ¬ P n x"
    then have "finite {i. P i x}"
      by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
    with ‹P i x› have "P (Min {i. P i x}) x" "Min {i. P i x} ≤ i" "finite {i. P i x}"
      using Min_in[of "{i. P i x}"] by auto }
  note 2 = this

  have "(λx. Min {i. P i x}) -` {n} ∩ space M = {x∈space M. Min {i. P i x} = n}"
    by auto
  also have "… =
    {x∈space M. if (∀i. ∃n≥i. P n x) then the None = n else
      if (∃i. P i x) then P n x ∧ (∀i<n. ¬ P i x)
      else Min {} = n}"
    by (intro arg_cong[where f=Collect] ext conj_cong)
       (auto simp add: 1 2 not_le[symmetric] intro!: Min_eqI)
  also have "… ∈ sets M"
    by measurable
  finally show "(λx. Min {i. P i x}) -` {n} ∩ space M ∈ sets M" .
qed simp

lemma measurable_count_space_insert[measurable (raw)]:
  "s ∈ S ⟹ A ∈ sets (count_space S) ⟹ insert s A ∈ sets (count_space S)"
  by simp

lemma sets_UNIV [measurable (raw)]: "A ∈ sets (count_space UNIV)"
  by simp

lemma measurable_card[measurable]:
  fixes S :: "'a ⇒ nat set"
  assumes [measurable]: "⋀i. {x∈space M. i ∈ S x} ∈ sets M"
  shows "(λx. card (S x)) ∈ measurable M (count_space UNIV)"
  unfolding measurable_count_space_eq2_countable
proof safe
  fix n show "(λx. card (S x)) -` {n} ∩ space M ∈ sets M"
  proof (cases n)
    case 0
    then have "(λx. card (S x)) -` {n} ∩ space M = {x∈space M. infinite (S x) ∨ (∀i. i ∉ S x)}"
      by auto
    also have "… ∈ sets M"
      by measurable
    finally show ?thesis .
  next
    case (Suc i)
    then have "(λx. card (S x)) -` {n} ∩ space M =
      (⋃F∈{A∈{A. finite A}. card A = n}. {x∈space M. (∀i. i ∈ S x ⟷ i ∈ F)})"
      unfolding set_eq_iff[symmetric] Collect_bex_eq[symmetric] by (auto intro: card_ge_0_finite)
    also have "… ∈ sets M"
      by (intro sets.countable_UN' countable_Collect countable_Collect_finite) auto
    finally show ?thesis .
  qed
qed rule

lemma measurable_pred_countable[measurable (raw)]:
  assumes "countable X"
  shows
    "(⋀i. i ∈ X ⟹ Measurable.pred M (λx. P x i)) ⟹ Measurable.pred M (λx. ∀i∈X. P x i)"
    "(⋀i. i ∈ X ⟹ Measurable.pred M (λx. P x i)) ⟹ Measurable.pred M (λx. ∃i∈X. P x i)"
  unfolding pred_def
  by (auto intro!: sets.sets_Collect_countable_All' sets.sets_Collect_countable_Ex' assms)

subsection ‹Measurability for (co)inductive predicates›

lemma measurable_bot[measurable]: "bot ∈ measurable M (count_space UNIV)"
  by (simp add: bot_fun_def)

lemma measurable_top[measurable]: "top ∈ measurable M (count_space UNIV)"
  by (simp add: top_fun_def)

lemma measurable_SUP[measurable]:
  fixes F :: "'i ⇒ 'a ⇒ 'b::{complete_lattice, countable}"
  assumes [simp]: "countable I"
  assumes [measurable]: "⋀i. i ∈ I ⟹ F i ∈ measurable M (count_space UNIV)"
  shows "(λx. SUP i:I. F i x) ∈ measurable M (count_space UNIV)"
  unfolding measurable_count_space_eq2_countable
proof (safe intro!: UNIV_I)
  fix a
  have "(λx. SUP i:I. F i x) -` {a} ∩ space M =
    {x∈space M. (∀i∈I. F i x ≤ a) ∧ (∀b. (∀i∈I. F i x ≤ b) ⟶ a ≤ b)}"
    unfolding SUP_le_iff[symmetric] by auto
  also have "… ∈ sets M"
    by measurable
  finally show "(λx. SUP i:I. F i x) -` {a} ∩ space M ∈ sets M" .
qed

lemma measurable_INF[measurable]:
  fixes F :: "'i ⇒ 'a ⇒ 'b::{complete_lattice, countable}"
  assumes [simp]: "countable I"
  assumes [measurable]: "⋀i. i ∈ I ⟹ F i ∈ measurable M (count_space UNIV)"
  shows "(λx. INF i:I. F i x) ∈ measurable M (count_space UNIV)"
  unfolding measurable_count_space_eq2_countable
proof (safe intro!: UNIV_I)
  fix a
  have "(λx. INF i:I. F i x) -` {a} ∩ space M =
    {x∈space M. (∀i∈I. a ≤ F i x) ∧ (∀b. (∀i∈I. b ≤ F i x) ⟶ b ≤ a)}"
    unfolding le_INF_iff[symmetric] by auto
  also have "… ∈ sets M"
    by measurable
  finally show "(λx. INF i:I. F i x) -` {a} ∩ space M ∈ sets M" .
qed

lemma measurable_lfp_coinduct[consumes 1, case_names continuity step]:
  fixes F :: "('a ⇒ 'b) ⇒ ('a ⇒ 'b::{complete_lattice, countable})"
  assumes "P M"
  assumes F: "sup_continuous F"
  assumes *: "⋀M A. P M ⟹ (⋀N. P N ⟹ A ∈ measurable N (count_space UNIV)) ⟹ F A ∈ measurable M (count_space UNIV)"
  shows "lfp F ∈ measurable M (count_space UNIV)"
proof -
  { fix i from ‹P M› have "((F ^^ i) bot) ∈ measurable M (count_space UNIV)"
      by (induct i arbitrary: M) (auto intro!: *) }
  then have "(λx. SUP i. (F ^^ i) bot x) ∈ measurable M (count_space UNIV)"
    by measurable
  also have "(λx. SUP i. (F ^^ i) bot x) = lfp F"
    by (subst sup_continuous_lfp) (auto intro: F)
  finally show ?thesis .
qed

lemma measurable_lfp:
  fixes F :: "('a ⇒ 'b) ⇒ ('a ⇒ 'b::{complete_lattice, countable})"
  assumes F: "sup_continuous F"
  assumes *: "⋀A. A ∈ measurable M (count_space UNIV) ⟹ F A ∈ measurable M (count_space UNIV)"
  shows "lfp F ∈ measurable M (count_space UNIV)"
  by (coinduction rule: measurable_lfp_coinduct[OF _ F]) (blast intro: *)

lemma measurable_gfp_coinduct[consumes 1, case_names continuity step]:
  fixes F :: "('a ⇒ 'b) ⇒ ('a ⇒ 'b::{complete_lattice, countable})"
  assumes "P M"
  assumes F: "inf_continuous F"
  assumes *: "⋀M A. P M ⟹ (⋀N. P N ⟹ A ∈ measurable N (count_space UNIV)) ⟹ F A ∈ measurable M (count_space UNIV)"
  shows "gfp F ∈ measurable M (count_space UNIV)"
proof -
  { fix i from ‹P M› have "((F ^^ i) top) ∈ measurable M (count_space UNIV)"
      by (induct i arbitrary: M) (auto intro!: *) }
  then have "(λx. INF i. (F ^^ i) top x) ∈ measurable M (count_space UNIV)"
    by measurable
  also have "(λx. INF i. (F ^^ i) top x) = gfp F"
    by (subst inf_continuous_gfp) (auto intro: F)
  finally show ?thesis .
qed

lemma measurable_gfp:
  fixes F :: "('a ⇒ 'b) ⇒ ('a ⇒ 'b::{complete_lattice, countable})"
  assumes F: "inf_continuous F"
  assumes *: "⋀A. A ∈ measurable M (count_space UNIV) ⟹ F A ∈ measurable M (count_space UNIV)"
  shows "gfp F ∈ measurable M (count_space UNIV)"
  by (coinduction rule: measurable_gfp_coinduct[OF _ F]) (blast intro: *)

lemma measurable_lfp2_coinduct[consumes 1, case_names continuity step]:
  fixes F :: "('a ⇒ 'c ⇒ 'b) ⇒ ('a ⇒ 'c ⇒ 'b::{complete_lattice, countable})"
  assumes "P M s"
  assumes F: "sup_continuous F"
  assumes *: "⋀M A s. P M s ⟹ (⋀N t. P N t ⟹ A t ∈ measurable N (count_space UNIV)) ⟹ F A s ∈ measurable M (count_space UNIV)"
  shows "lfp F s ∈ measurable M (count_space UNIV)"
proof -
  { fix i from ‹P M s› have "(λx. (F ^^ i) bot s x) ∈ measurable M (count_space UNIV)"
      by (induct i arbitrary: M s) (auto intro!: *) }
  then have "(λx. SUP i. (F ^^ i) bot s x) ∈ measurable M (count_space UNIV)"
    by measurable
  also have "(λx. SUP i. (F ^^ i) bot s x) = lfp F s"
    by (subst sup_continuous_lfp) (auto simp: F)
  finally show ?thesis .
qed

lemma measurable_gfp2_coinduct[consumes 1, case_names continuity step]:
  fixes F :: "('a ⇒ 'c ⇒ 'b) ⇒ ('a ⇒ 'c ⇒ 'b::{complete_lattice, countable})"
  assumes "P M s"
  assumes F: "inf_continuous F"
  assumes *: "⋀M A s. P M s ⟹ (⋀N t. P N t ⟹ A t ∈ measurable N (count_space UNIV)) ⟹ F A s ∈ measurable M (count_space UNIV)"
  shows "gfp F s ∈ measurable M (count_space UNIV)"
proof -
  { fix i from ‹P M s› have "(λx. (F ^^ i) top s x) ∈ measurable M (count_space UNIV)"
      by (induct i arbitrary: M s) (auto intro!: *) }
  then have "(λx. INF i. (F ^^ i) top s x) ∈ measurable M (count_space UNIV)"
    by measurable
  also have "(λx. INF i. (F ^^ i) top s x) = gfp F s"
    by (subst inf_continuous_gfp) (auto simp: F)
  finally show ?thesis .
qed

lemma measurable_enat_coinduct:
  fixes f :: "'a ⇒ enat"
  assumes "R f"
  assumes *: "⋀f. R f ⟹ ∃g h i P. R g ∧ f = (λx. if P x then h x else eSuc (g (i x))) ∧
    Measurable.pred M P ∧
    i ∈ measurable M M ∧
    h ∈ measurable M (count_space UNIV)"
  shows "f ∈ measurable M (count_space UNIV)"
proof (simp add: measurable_count_space_eq2_countable, rule )
  fix a :: enat
  have "f -` {a} ∩ space M = {x∈space M. f x = a}"
    by auto
  { fix i :: nat
    from ‹R f› have "Measurable.pred M (λx. f x = enat i)"
    proof (induction i arbitrary: f)
      case 0
      from *[OF this] obtain g h i P
        where f: "f = (λx. if P x then h x else eSuc (g (i x)))" and
          [measurable]: "Measurable.pred M P" "i ∈ measurable M M" "h ∈ measurable M (count_space UNIV)"
        by auto
      have "Measurable.pred M (λx. P x ∧ h x = 0)"
        by measurable
      also have "(λx. P x ∧ h x = 0) = (λx. f x = enat 0)"
        by (auto simp: f zero_enat_def[symmetric])
      finally show ?case .
    next
      case (Suc n)
      from *[OF Suc.prems] obtain g h i P
        where f: "f = (λx. if P x then h x else eSuc (g (i x)))" and "R g" and
          M[measurable]: "Measurable.pred M P" "i ∈ measurable M M" "h ∈ measurable M (count_space UNIV)"
        by auto
      have "(λx. f x = enat (Suc n)) =
        (λx. (P x ⟶ h x = enat (Suc n)) ∧ (¬ P x ⟶ g (i x) = enat n))"
        by (auto simp: f zero_enat_def[symmetric] eSuc_enat[symmetric])
      also have "Measurable.pred M …"
        by (intro pred_intros_logic measurable_compose[OF M(2)] Suc ‹R g›) measurable
      finally show ?case .
    qed
    then have "f -` {enat i} ∩ space M ∈ sets M"
      by (simp add: pred_def Int_def conj_commute) }
  note fin = this
  show "f -` {a} ∩ space M ∈ sets M"
  proof (cases a)
    case infinity
    then have "f -` {a} ∩ space M = space M - (⋃n. f -` {enat n} ∩ space M)"
      by auto
    also have "… ∈ sets M"
      by (intro sets.Diff sets.top sets.Un sets.countable_UN) (auto intro!: fin)
    finally show ?thesis .
  qed (simp add: fin)
qed

lemma measurable_THE:
  fixes P :: "'a ⇒ 'b ⇒ bool"
  assumes [measurable]: "⋀i. Measurable.pred M (P i)"
  assumes I[simp]: "countable I" "⋀i x. x ∈ space M ⟹ P i x ⟹ i ∈ I"
  assumes unique: "⋀x i j. x ∈ space M ⟹ P i x ⟹ P j x ⟹ i = j"
  shows "(λx. THE i. P i x) ∈ measurable M (count_space UNIV)"
  unfolding measurable_def
proof safe
  fix X
  def f ≡ "λx. THE i. P i x" def undef ≡ "THE i::'a. False"
  { fix i x assume "x ∈ space M" "P i x" then have "f x = i"
      unfolding f_def using unique by auto }
  note f_eq = this
  { fix x assume "x ∈ space M" "∀i∈I. ¬ P i x"
    then have "⋀i. ¬ P i x"
      using I(2)[of x] by auto
    then have "f x = undef"
      by (auto simp: undef_def f_def) }
  then have "f -` X ∩ space M = (⋃i∈I ∩ X. {x∈space M. P i x}) ∪
     (if undef ∈ X then space M - (⋃i∈I. {x∈space M. P i x}) else {})"
    by (auto dest: f_eq)
  also have "… ∈ sets M"
    by (auto intro!: sets.Diff sets.countable_UN')
  finally show "f -` X ∩ space M ∈ sets M" .
qed simp

lemma measurable_Ex1[measurable (raw)]:
  assumes [simp]: "countable I" and [measurable]: "⋀i. i ∈ I ⟹ Measurable.pred M (P i)"
  shows "Measurable.pred M (λx. ∃!i∈I. P i x)"
  unfolding bex1_def by measurable

lemma measurable_Sup_nat[measurable (raw)]:
  fixes F :: "'a ⇒ nat set"
  assumes [measurable]: "⋀i. Measurable.pred M (λx. i ∈ F x)"
  shows "(λx. Sup (F x)) ∈ M →⇩M count_space UNIV"
proof (clarsimp simp add: measurable_count_space_eq2_countable)
  fix a
  have F_empty_iff: "F x = {} ⟷ (∀i. i ∉ F x)" for x
    by auto
  have "Measurable.pred M (λx. if finite (F x) then if F x = {} then a = Max {}
    else a ∈ F x ∧ (∀j. j ∈ F x ⟶ j ≤ a) else a = the None)"
    unfolding finite_nat_set_iff_bounded Ball_def F_empty_iff by measurable
  moreover have "(λx. Sup (F x)) -` {a} ∩ space M =
    {x∈space M. if finite (F x) then if F x = {} then a = Max {}
      else a ∈ F x ∧ (∀j. j ∈ F x ⟶ j ≤ a) else a = the None}"
    by (intro set_eqI)
       (auto simp: Sup_nat_def Max.infinite intro!: Max_in Max_eqI)
  ultimately show "(λx. Sup (F x)) -` {a} ∩ space M ∈ sets M"
    by auto
qed

lemma measurable_if_split[measurable (raw)]:
  "(c ⟹ Measurable.pred M f) ⟹ (¬ c ⟹ Measurable.pred M g) ⟹
   Measurable.pred M (if c then f else g)"
  by simp

lemma pred_restrict_space:
  assumes "S ∈ sets M"
  shows "Measurable.pred (restrict_space M S) P ⟷ Measurable.pred M (λx. x ∈ S ∧ P x)"
  unfolding pred_def sets_Collect_restrict_space_iff[OF assms] ..

lemma measurable_predpow[measurable]:
  assumes "Measurable.pred M T"
  assumes "⋀Q. Measurable.pred M Q ⟹ Measurable.pred M (R Q)"
  shows "Measurable.pred M ((R ^^ n) T)"
  by (induct n) (auto intro: assms)

hide_const (open) pred

end