Theory HOL-Analysis.Measurable

(*  Title:      HOL/Analysis/Measurable.thy
    Author:     Johannes Hölzl <hoelzl@in.tum.de>
*)
section ‹Measurability Prover›
theory Measurable
  imports
    Sigma_Algebra
    "HOL-Library.Order_Continuity"
begin


lemma (in algebra) sets_Collect_finite_All:
  assumes "i. i  S  {xΩ. P i x}  M" "finite S"
  shows "{xΩ. iS. P i x}  M"
proof -
  have "{xΩ. iS. P i x} = (if S = {} then Ω else iS. {xΩ. 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  {xspace 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 = {xspace M. P x}" by auto
  finally show "{xspace M. P x}  sets M" .
next
  assume P: "{xspace M. P x}  sets M"
  moreover
  { fix X
    have "X  Pow (UNIV :: bool set)" by simp
    then have "P -` X  space M = {xspace 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: "{xspace 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_setuptag important› measurable = Scan.lift (Scan.succeed (METHOD o Measurable.measurable_tac))
  "measurability prover"

simproc_setuptag important› measurable ("A  sets M" | "f  measurable M N") =
  K Measurable.proc

setup Global_Theory.add_thms_dynamic (bindingmeasurable, 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  {xspace 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  (iX. N x i))"
    "(i. i  X  pred M (λx. x  N x i))  pred M (λx. x  (iX. N x i))"
    "(i. i  X  pred M (λx. P x i))  pred M (λx. iX. P x i)"
    "(i. i  X  pred M (λx. P x i))  pred M (λx. iX. 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  (iI. N x i))"
  "finite I  (i. i  I  pred M (λx. x  N x i))  pred M (λx. x  (iI. N x i))"
  "finite I  (i. i  I  pred M (λx. P x i))  pred M (λx. iI. P x i)"
  "finite I  (i. i  I  pred M (λx. P x i))  pred M (λx. iI. 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)  (iI. N i)  sets M"
  "I  {}  (i :: 'i :: countable. i  I  N i  sets M)  (iI. 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 "{xspace M. x  A  B}  sets M" by auto
  also have "{xspace 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]:
  "iI. A i  sets (M i)  iI  A i  sets (M i)"
  by auto

lemma pred_sets_Ball[measurable_dest]:
  "iI. A i  sets (N i)  iI  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. {xspace 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))"
  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. ni. 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" "nj. ¬ 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 = {xspace M. Max {i. P i x} = n}"
    by auto
  also have " =
    {xspace M. if (i. ni. 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. ni. 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" "nj. ¬ 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 = {xspace M. Min {i. P i x} = n}"
    by auto
  also have " =
    {xspace M. if (i. ni. 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. {xspace 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 = {xspace 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}. {xspace 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. iX. P x i)"
    "(i. i  X  Measurable.pred M (λx. P x i))  Measurable.pred M (λx. iX. P x i)"
  unfolding pred_def
  by (auto intro!: sets.sets_Collect_countable_All' sets.sets_Collect_countable_Ex' assms)

subsectiontag unimportant› ‹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 iI. F i x)  measurable M (count_space UNIV)"
  unfolding measurable_count_space_eq2_countable
proof (safe intro!: UNIV_I)
  fix a
  have "(λx. SUP iI. F i x) -` {a}  space M =
    {xspace M. (iI. F i x  a)  (b. (iI. F i x  b)  a  b)}"
    unfolding SUP_le_iff[symmetric] by auto
  also have "  sets M"
    by measurable
  finally show "(λx. SUP iI. 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 iI. F i x)  measurable M (count_space UNIV)"
  unfolding measurable_count_space_eq2_countable
proof (safe intro!: UNIV_I)
  fix a
  have "(λx. INF iI. F i x) -` {a}  space M =
    {xspace M. (iI. a  F i x)  (b. (iI. b  F i x)  b  a)}"
    unfolding le_INF_iff[symmetric] by auto
  also have "  sets M"
    by measurable
  finally show "(λx. INF iI. 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 simp: image_comp)
  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 simp: image_comp)
  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 simp: image_comp)
  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 simp: image_comp)
  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 = {xspace 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
  define f where "f x = (THE i. P i x)" for x
  define undef where "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" "iI. ¬ 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 = (iI  X. {xspace M. P i x}) 
     (if undef  X then space M - (iI. {xspace 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. ∃!iI. 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 = 0
    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 =
    {xspace M. if finite (F x) then if F x = {} then a = 0
      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)

lemma measurable_compose_countable_restrict:
  assumes P: "countable {i. P i}"
    and f: "f  M M count_space UNIV"
    and Q: "i. P i  pred M (Q i)"
  shows "pred M (λx. P (f x)  Q (f x) x)"
proof -
  have P_f: "{x  space M. P (f x)}  sets M"
    unfolding pred_def[symmetric] by (rule measurable_compose[OF f]) simp
  have "pred (restrict_space M {xspace M. P (f x)}) (λx. Q (f x) x)"
  proof (rule measurable_compose_countable'[where g=f, OF _ _ P])
    show "f  restrict_space M {xspace M. P (f x)} M count_space {i. P i}"
      by (rule measurable_count_space_extend[OF subset_UNIV])
         (auto simp: space_restrict_space intro!: measurable_restrict_space1 f)
  qed (auto intro!: measurable_restrict_space1 Q)
  then show ?thesis
    unfolding pred_restrict_space[OF P_f] by (simp cong: measurable_cong)
qed

lemma measurable_limsup [measurable (raw)]:
  assumes [measurable]: "n. A n  sets M"
  shows "limsup A  sets M"
by (subst limsup_INF_SUP, auto)

lemma measurable_liminf [measurable (raw)]:
  assumes [measurable]: "n. A n  sets M"
  shows "liminf A  sets M"
by (subst liminf_SUP_INF, auto)

lemma measurable_case_enat[measurable (raw)]:
  assumes f: "f  M M count_space UNIV" and g: "i. g i  M M N" and h: "h  M M N"
  shows "(λx. case f x of enat i  g i x |   h x)  M M N"
  apply (rule measurable_compose_countable[OF _ f])
  subgoal for i
    by (cases i) (auto intro: g h)
  done

hide_const (open) pred

end