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

hide_const (open) Order_Continuity.continuous

subsection {* Measurability prover *}

lemma (in algebra) sets_Collect_finite_All:
  assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S"
  shows "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} \<in> M"
proof -
  have "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} = (if S = {} then \<Omega> else \<Inter>i\<in>S. {x\<in>\<Omega>. P i x})"
    by auto
  with assms show ?thesis by (auto intro!: sets_Collect_finite_All')
qed

abbreviation "pred M P \<equiv> P \<in> measurable M (count_space (UNIV::bool set))"

lemma pred_def: "pred M P \<longleftrightarrow> {x\<in>space M. P x} \<in> sets M"
proof
  assume "pred M P"
  then have "P -` {True} \<inter> space M \<in> sets M"
    by (auto simp: measurable_count_space_eq2)
  also have "P -` {True} \<inter> space M = {x\<in>space M. P x}" by auto
  finally show "{x\<in>space M. P x} \<in> sets M" .
next
  assume P: "{x\<in>space M. P x} \<in> sets M"
  moreover
  { fix X
    have "X \<in> Pow (UNIV :: bool set)" by simp
    then have "P -` X \<inter> space M = {x\<in>space M. ((X = {True} \<longrightarrow> P x) \<and> (X = {False} \<longrightarrow> \<not> P x) \<and> X \<noteq> {})}"
      unfolding UNIV_bool Pow_insert Pow_empty by auto
    then have "P -` X \<inter> space M \<in> 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\<in>space M. P x} \<in> sets M \<Longrightarrow> f \<in> measurable N M \<Longrightarrow> pred N (\<lambda>x. P (f x))"
  by (rule measurable_compose[where f=f and N=M]) (auto simp: pred_def)

lemma pred_sets2: "A \<in> sets N \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> pred M (\<lambda>x. f x \<in> 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_app = Measurable.app_thm_attr
  "add application rule to measurability prover"

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

method_setup measurable = \<open> Scan.lift (Scan.succeed (METHOD o Measurable.measurable_tac)) \<close>
  "measurability prover"

simproc_setup measurable ("A \<in> sets M" | "f \<in> measurable M N") = {* K Measurable.simproc *}

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

declare
  measurable_compose_rev[measurable_dest]
  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]

lemma predE[measurable (raw)]: 
  "pred M P \<Longrightarrow> {x\<in>space M. P x} \<in> sets M"
  unfolding pred_def .

lemma pred_intros_imp'[measurable (raw)]:
  "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<longrightarrow> P x)"
  by (cases K) auto

lemma pred_intros_conj1'[measurable (raw)]:
  "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<and> P x)"
  by (cases K) auto

lemma pred_intros_conj2'[measurable (raw)]:
  "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<and> K)"
  by (cases K) auto

lemma pred_intros_disj1'[measurable (raw)]:
  "(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<or> P x)"
  by (cases K) auto

lemma pred_intros_disj2'[measurable (raw)]:
  "(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<or> K)"
  by (cases K) auto

lemma pred_intros_logic[measurable (raw)]:
  "pred M (\<lambda>x. x \<in> space M)"
  "pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. \<not> P x)"
  "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<and> P x)"
  "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<longrightarrow> P x)"
  "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<or> P x)"
  "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x = P x)"
  "pred M (\<lambda>x. f x \<in> UNIV)"
  "pred M (\<lambda>x. f x \<in> {})"
  "pred M (\<lambda>x. P' (f x) x) \<Longrightarrow> pred M (\<lambda>x. f x \<in> {y. P' y x})"
  "pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> - (B x))"
  "pred M (\<lambda>x. f x \<in> (A x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (A x) - (B x))"
  "pred M (\<lambda>x. f x \<in> (A x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (A x) \<inter> (B x))"
  "pred M (\<lambda>x. f x \<in> (A x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (A x) \<union> (B x))"
  "pred M (\<lambda>x. g x (f x) \<in> (X x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (g x) -` (X x))"
  by (auto simp: iff_conv_conj_imp pred_def)

lemma pred_intros_countable[measurable (raw)]:
  fixes P :: "'a \<Rightarrow> 'i :: countable \<Rightarrow> bool"
  shows 
    "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i. P x i)"
    "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>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 
    "(\<And>i. i \<in> X \<Longrightarrow> pred M (\<lambda>x. x \<in> N x i)) \<Longrightarrow> pred M (\<lambda>x. x \<in> (\<Inter>i\<in>X. N x i))"
    "(\<And>i. i \<in> X \<Longrightarrow> pred M (\<lambda>x. x \<in> N x i)) \<Longrightarrow> pred M (\<lambda>x. x \<in> (\<Union>i\<in>X. N x i))"
    "(\<And>i. i \<in> X \<Longrightarrow> pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i\<in>X. P x i)"
    "(\<And>i. i \<in> X \<Longrightarrow> pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i\<in>X. P x i)"
  by (auto simp: Bex_def Ball_def)

lemma pred_intros_finite[measurable (raw)]:
  "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> pred M (\<lambda>x. x \<in> N x i)) \<Longrightarrow> pred M (\<lambda>x. x \<in> (\<Inter>i\<in>I. N x i))"
  "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> pred M (\<lambda>x. x \<in> N x i)) \<Longrightarrow> pred M (\<lambda>x. x \<in> (\<Union>i\<in>I. N x i))"
  "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i\<in>I. P x i)"
  "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i\<in>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)]:
  "(\<And>i :: 'i :: countable. i \<in> I \<Longrightarrow> N i \<in> sets M) \<Longrightarrow> (\<Union>i\<in>I. N i) \<in> sets M"
  "I \<noteq> {} \<Longrightarrow> (\<And>i :: 'i :: countable. i \<in> I \<Longrightarrow> N i \<in> sets M) \<Longrightarrow> (\<Inter>i\<in>I. N i) \<in> sets M"
  by auto

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

lemma sets_Int_pred[measurable (raw)]:
  assumes space: "A \<inter> B \<subseteq> space M" and [measurable]: "pred M (\<lambda>x. x \<in> A)" "pred M (\<lambda>x. x \<in> B)"
  shows "A \<inter> B \<in> sets M"
proof -
  have "{x\<in>space M. x \<in> A \<inter> B} \<in> sets M" by auto
  also have "{x\<in>space M. x \<in> A \<inter> B} = A \<inter> B"
    using space by auto
  finally show ?thesis .
qed

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

lemma pred_count_space_const1[measurable (raw)]:
  "f \<in> measurable M (count_space UNIV) \<Longrightarrow> Measurable.pred M (\<lambda>x. f x = c)"
  by (intro pred_eq_const1[where N="count_space UNIV"]) (auto )

lemma pred_count_space_const2[measurable (raw)]:
  "f \<in> measurable M (count_space UNIV) \<Longrightarrow> Measurable.pred M (\<lambda>x. c = f x)"
  by (intro pred_eq_const2[where N="count_space UNIV"]) (auto )

lemma pred_le_const[measurable (raw generic)]:
  assumes f: "f \<in> measurable M N" and c: "{.. c} \<in> sets N" shows "pred M (\<lambda>x. f x \<le> 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 \<in> measurable M N" and c: "{c ..} \<in> sets N" shows "pred M (\<lambda>x. c \<le> 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 \<in> measurable M N" and c: "{..< c} \<in> sets N" shows "pred M (\<lambda>x. 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 \<in> measurable M N" and c: "{c <..} \<in> sets N" shows "pred M (\<lambda>x. 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 \<Longrightarrow> pred M (\<lambda>x. x \<in> A x)) \<Longrightarrow> (\<not> P \<Longrightarrow> pred M (\<lambda>x. x \<in> B x)) \<Longrightarrow>
    pred M (\<lambda>x. x \<in> (if P then A x else B x))"
  by auto

lemma sets_range[measurable_dest]:
  "A ` I \<subseteq> sets M \<Longrightarrow> i \<in> I \<Longrightarrow> A i \<in> sets M"
  by auto

lemma pred_sets_range[measurable_dest]:
  "A ` I \<subseteq> sets N \<Longrightarrow> i \<in> I \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> pred M (\<lambda>x. f x \<in> A i)"
  using pred_sets2[OF sets_range] by auto

lemma sets_All[measurable_dest]:
  "\<forall>i. A i \<in> sets (M i) \<Longrightarrow> A i \<in> sets (M i)"
  by auto

lemma pred_sets_All[measurable_dest]:
  "\<forall>i. A i \<in> sets (N i) \<Longrightarrow> f \<in> measurable M (N i) \<Longrightarrow> pred M (\<lambda>x. f x \<in> A i)"
  using pred_sets2[OF sets_All, of A N f] by auto

lemma sets_Ball[measurable_dest]:
  "\<forall>i\<in>I. A i \<in> sets (M i) \<Longrightarrow> i\<in>I \<Longrightarrow> A i \<in> sets (M i)"
  by auto

lemma pred_sets_Ball[measurable_dest]:
  "\<forall>i\<in>I. A i \<in> sets (N i) \<Longrightarrow> i\<in>I \<Longrightarrow> f \<in> measurable M (N i) \<Longrightarrow> pred M (\<lambda>x. f x \<in> A i)"
  using pred_sets2[OF sets_Ball, of _ _ _ f] by auto

lemma measurable_finite[measurable (raw)]:
  fixes S :: "'a \<Rightarrow> nat set"
  assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
  shows "pred M (\<lambda>x. finite (S x))"
  unfolding finite_nat_set_iff_bounded by (simp add: Ball_def)

lemma measurable_Least[measurable]:
  assumes [measurable]: "(\<And>i::nat. (\<lambda>x. P i x) \<in> measurable M (count_space UNIV))"q
  shows "(\<lambda>x. LEAST i. P i x) \<in> measurable M (count_space UNIV)"
  unfolding measurable_def by (safe intro!: sets_Least) simp_all

lemma measurable_Max_nat[measurable (raw)]: 
  fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
  assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
  shows "(\<lambda>x. Max {i. P i x}) \<in> measurable M (count_space UNIV)"
  unfolding measurable_count_space_eq2_countable
proof safe
  fix n

  { fix x assume "\<forall>i. \<exists>n\<ge>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" "\<forall>n\<ge>j. \<not> 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 \<le> Max {i. P i x}" "finite {i. P i x}"
      using Max_in[of "{i. P i x}"] by auto }
  note 2 = this

  have "(\<lambda>x. Max {i. P i x}) -` {n} \<inter> space M = {x\<in>space M. Max {i. P i x} = n}"
    by auto
  also have "\<dots> = 
    {x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else 
      if (\<exists>i. P i x) then P n x \<and> (\<forall>i>n. \<not> 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 "\<dots> \<in> sets M"
    by measurable
  finally show "(\<lambda>x. Max {i. P i x}) -` {n} \<inter> space M \<in> sets M" .
qed simp

lemma measurable_Min_nat[measurable (raw)]: 
  fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
  assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
  shows "(\<lambda>x. Min {i. P i x}) \<in> measurable M (count_space UNIV)"
  unfolding measurable_count_space_eq2_countable
proof safe
  fix n

  { fix x assume "\<forall>i. \<exists>n\<ge>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" "\<forall>n\<ge>j. \<not> 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} \<le> i" "finite {i. P i x}"
      using Min_in[of "{i. P i x}"] by auto }
  note 2 = this

  have "(\<lambda>x. Min {i. P i x}) -` {n} \<inter> space M = {x\<in>space M. Min {i. P i x} = n}"
    by auto
  also have "\<dots> = 
    {x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else 
      if (\<exists>i. P i x) then P n x \<and> (\<forall>i<n. \<not> 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 "\<dots> \<in> sets M"
    by measurable
  finally show "(\<lambda>x. Min {i. P i x}) -` {n} \<inter> space M \<in> sets M" .
qed simp

lemma measurable_count_space_insert[measurable (raw)]:
  "s \<in> S \<Longrightarrow> A \<in> sets (count_space S) \<Longrightarrow> insert s A \<in> sets (count_space S)"
  by simp

lemma sets_UNIV [measurable (raw)]: "A \<in> sets (count_space UNIV)"
  by simp

lemma measurable_card[measurable]:
  fixes S :: "'a \<Rightarrow> nat set"
  assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
  shows "(\<lambda>x. card (S x)) \<in> measurable M (count_space UNIV)"
  unfolding measurable_count_space_eq2_countable
proof safe
  fix n show "(\<lambda>x. card (S x)) -` {n} \<inter> space M \<in> sets M"
  proof (cases n)
    case 0
    then have "(\<lambda>x. card (S x)) -` {n} \<inter> space M = {x\<in>space M. infinite (S x) \<or> (\<forall>i. i \<notin> S x)}"
      by auto
    also have "\<dots> \<in> sets M"
      by measurable
    finally show ?thesis .
  next
    case (Suc i)
    then have "(\<lambda>x. card (S x)) -` {n} \<inter> space M =
      (\<Union>F\<in>{A\<in>{A. finite A}. card A = n}. {x\<in>space M. (\<forall>i. i \<in> S x \<longleftrightarrow> i \<in> F)})"
      unfolding set_eq_iff[symmetric] Collect_bex_eq[symmetric] by (auto intro: card_ge_0_finite)
    also have "\<dots> \<in> sets M"
      by (intro sets.countable_UN' countable_Collect countable_Collect_finite) auto
    finally show ?thesis .
  qed
qed rule

subsection {* Measurability for (co)inductive predicates *}

lemma measurable_lfp:
  assumes "Order_Continuity.continuous F"
  assumes *: "\<And>A. pred M A \<Longrightarrow> pred M (F A)"
  shows "pred M (lfp F)"
proof -
  { fix i have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. False) x)"
      by (induct i) (auto intro!: *) }
  then have "Measurable.pred M (\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x)"
    by measurable
  also have "(\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x) = (SUP i. (F ^^ i) bot)"
    by (auto simp add: bot_fun_def)
  also have "\<dots> = lfp F"
    by (rule continuous_lfp[symmetric]) fact
  finally show ?thesis .
qed

lemma measurable_gfp:
  assumes "Order_Continuity.down_continuous F"
  assumes *: "\<And>A. pred M A \<Longrightarrow> pred M (F A)"
  shows "pred M (gfp F)"
proof -
  { fix i have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. True) x)"
      by (induct i) (auto intro!: *) }
  then have "Measurable.pred M (\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x)"
    by measurable
  also have "(\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x) = (INF i. (F ^^ i) top)"
    by (auto simp add: top_fun_def)
  also have "\<dots> = gfp F"
    by (rule down_continuous_gfp[symmetric]) fact
  finally show ?thesis .
qed

lemma measurable_lfp_coinduct[consumes 1, case_names continuity step]:
  assumes "P M"
  assumes "Order_Continuity.continuous F"
  assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"
  shows "Measurable.pred M (lfp F)"
proof -
  { fix i from `P M` have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. False) x)"
      by (induct i arbitrary: M) (auto intro!: *) }
  then have "Measurable.pred M (\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x)"
    by measurable
  also have "(\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x) = (SUP i. (F ^^ i) bot)"
    by (auto simp add: bot_fun_def)
  also have "\<dots> = lfp F"
    by (rule continuous_lfp[symmetric]) fact
  finally show ?thesis .
qed

lemma measurable_gfp_coinduct[consumes 1, case_names continuity step]:
  assumes "P M"
  assumes "Order_Continuity.down_continuous F"
  assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"
  shows "Measurable.pred M (gfp F)"
proof -
  { fix i from `P M` have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. True) x)"
      by (induct i arbitrary: M) (auto intro!: *) }
  then have "Measurable.pred M (\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x)"
    by measurable
  also have "(\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x) = (INF i. (F ^^ i) top)"
    by (auto simp add: top_fun_def)
  also have "\<dots> = gfp F"
    by (rule down_continuous_gfp[symmetric]) fact
  finally show ?thesis .
qed

lemma measurable_lfp2_coinduct[consumes 1, case_names continuity step]:
  assumes "P M s"
  assumes "Order_Continuity.continuous F"
  assumes *: "\<And>M A s. P M s \<Longrightarrow> (\<And>N t. P N t \<Longrightarrow> Measurable.pred N (A t)) \<Longrightarrow> Measurable.pred M (F A s)"
  shows "Measurable.pred M (lfp F s)"
proof -
  { fix i from `P M s` have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>t x. False) s x)"
      by (induct i arbitrary: M s) (auto intro!: *) }
  then have "Measurable.pred M (\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>t x. False) s x)"
    by measurable
  also have "(\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>t x. False) s x) = (SUP i. (F ^^ i) bot) s"
    by (auto simp add: bot_fun_def)
  also have "(SUP i. (F ^^ i) bot) = lfp F"
    by (rule continuous_lfp[symmetric]) fact
  finally show ?thesis .
qed

lemma measurable_gfp2_coinduct[consumes 1, case_names continuity step]:
  assumes "P M s"
  assumes "Order_Continuity.down_continuous F"
  assumes *: "\<And>M A s. P M s \<Longrightarrow> (\<And>N t. P N t \<Longrightarrow> Measurable.pred N (A t)) \<Longrightarrow> Measurable.pred M (F A s)"
  shows "Measurable.pred M (gfp F s)"
proof -
  { fix i from `P M s` have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>t x. True) s x)"
      by (induct i arbitrary: M s) (auto intro!: *) }
  then have "Measurable.pred M (\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>t x. True) s x)"
    by measurable
  also have "(\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>t x. True) s x) = (INF i. (F ^^ i) top) s"
    by (auto simp add: top_fun_def)
  also have "(INF i. (F ^^ i) top) = gfp F"
    by (rule down_continuous_gfp[symmetric]) fact
  finally show ?thesis .
qed

lemma measurable_enat_coinduct:
  fixes f :: "'a \<Rightarrow> enat"
  assumes "R f"
  assumes *: "\<And>f. R f \<Longrightarrow> \<exists>g h i P. R g \<and> f = (\<lambda>x. if P x then h x else eSuc (g (i x))) \<and> 
    Measurable.pred M P \<and>
    i \<in> measurable M M \<and>
    h \<in> measurable M (count_space UNIV)"
  shows "f \<in> measurable M (count_space UNIV)"
proof (simp add: measurable_count_space_eq2_countable, rule )
  fix a :: enat
  have "f -` {a} \<inter> space M = {x\<in>space M. f x = a}"
    by auto
  { fix i :: nat
    from `R f` have "Measurable.pred M (\<lambda>x. f x = enat i)"
    proof (induction i arbitrary: f)
      case 0
      from *[OF this] obtain g h i P
        where f: "f = (\<lambda>x. if P x then h x else eSuc (g (i x)))" and
          [measurable]: "Measurable.pred M P" "i \<in> measurable M M" "h \<in> measurable M (count_space UNIV)"
        by auto
      have "Measurable.pred M (\<lambda>x. P x \<and> h x = 0)"
        by measurable
      also have "(\<lambda>x. P x \<and> h x = 0) = (\<lambda>x. 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 = (\<lambda>x. if P x then h x else eSuc (g (i x)))" and "R g" and
          M[measurable]: "Measurable.pred M P" "i \<in> measurable M M" "h \<in> measurable M (count_space UNIV)"
        by auto
      have "(\<lambda>x. f x = enat (Suc n)) =
        (\<lambda>x. (P x \<longrightarrow> h x = enat (Suc n)) \<and> (\<not> P x \<longrightarrow> g (i x) = enat n))"
        by (auto simp: f zero_enat_def[symmetric] eSuc_enat[symmetric])
      also have "Measurable.pred M \<dots>"
        by (intro pred_intros_logic measurable_compose[OF M(2)] Suc `R g`) measurable
      finally show ?case .
    qed
    then have "f -` {enat i} \<inter> space M \<in> sets M"
      by (simp add: pred_def Int_def conj_commute) }
  note fin = this
  show "f -` {a} \<inter> space M \<in> sets M"
  proof (cases a)
    case infinity
    then have "f -` {a} \<inter> space M = space M - (\<Union>n. f -` {enat n} \<inter> space M)"
      by auto
    also have "\<dots> \<in> 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_pred_countable[measurable (raw)]:
  assumes "countable X"
  shows 
    "(\<And>i. i \<in> X \<Longrightarrow> Measurable.pred M (\<lambda>x. P x i)) \<Longrightarrow> Measurable.pred M (\<lambda>x. \<forall>i\<in>X. P x i)"
    "(\<And>i. i \<in> X \<Longrightarrow> Measurable.pred M (\<lambda>x. P x i)) \<Longrightarrow> Measurable.pred M (\<lambda>x. \<exists>i\<in>X. P x i)"
  unfolding pred_def
  by (auto intro!: sets.sets_Collect_countable_All' sets.sets_Collect_countable_Ex' assms)

lemma measurable_THE:
  fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
  assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
  assumes I[simp]: "countable I" "\<And>i x. x \<in> space M \<Longrightarrow> P i x \<Longrightarrow> i \<in> I"
  assumes unique: "\<And>x i j. x \<in> space M \<Longrightarrow> P i x \<Longrightarrow> P j x \<Longrightarrow> i = j"
  shows "(\<lambda>x. THE i. P i x) \<in> measurable M (count_space UNIV)"
  unfolding measurable_def
proof safe
  fix X
  def f \<equiv> "\<lambda>x. THE i. P i x" def undef \<equiv> "THE i::'a. False"
  { fix i x assume "x \<in> 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 \<in> space M" "\<forall>i\<in>I. \<not> P i x"
    then have "\<And>i. \<not> 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 \<inter> space M = (\<Union>i\<in>I \<inter> X. {x\<in>space M. P i x}) \<union>
     (if undef \<in> X then space M - (\<Union>i\<in>I. {x\<in>space M. P i x}) else {})"
    by (auto dest: f_eq)
  also have "\<dots> \<in> sets M"
    by (auto intro!: sets.Diff sets.countable_UN')
  finally show "f -` X \<inter> space M \<in> sets M" .
qed simp

lemma measurable_bot[measurable]: "Measurable.pred M bot"
  by (simp add: bot_fun_def)

lemma measurable_top[measurable]: "Measurable.pred M top"
  by (simp add: top_fun_def)

lemma measurable_Ex1[measurable (raw)]:
  assumes [simp]: "countable I" and [measurable]: "\<And>i. i \<in> I \<Longrightarrow> Measurable.pred M (P i)"
  shows "Measurable.pred M (\<lambda>x. \<exists>!i\<in>I. P i x)"
  unfolding bex1_def by measurable

lemma measurable_split_if[measurable (raw)]:
  "(c \<Longrightarrow> Measurable.pred M f) \<Longrightarrow> (\<not> c \<Longrightarrow> Measurable.pred M g) \<Longrightarrow>
   Measurable.pred M (if c then f else g)"
  by simp

lemma pred_restrict_space:
  assumes "S \<in> sets M"
  shows "Measurable.pred (restrict_space M S) P \<longleftrightarrow> Measurable.pred M (\<lambda>x. x \<in> S \<and> P x)"
  unfolding pred_def sets_Collect_restrict_space_iff[OF assms] ..

lemma measurable_predpow[measurable]:
  assumes "Measurable.pred M T"
  assumes "\<And>Q. Measurable.pred M Q \<Longrightarrow> Measurable.pred M (R Q)"
  shows "Measurable.pred M ((R ^^ n) T)"
  by (induct n) (auto intro: assms)

hide_const (open) pred

end

