(*  Title:      HOL/Probability/Independent_Family.thy

    Author:     Johannes Hölzl, TU München

*)

header {* Independent families of events, event sets, and random variables *}

theory Independent_Family

  imports Probability_Measure Infinite_Product_Measure

begin

definition (in prob_space)

  "indep_sets F I \<longleftrightarrow> (\<forall>i\<in>I. F i \<subseteq> events) \<and>

    (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> (\<forall>A\<in>Pi J F. prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))))"

definition (in prob_space)

  "indep_set A B \<longleftrightarrow> indep_sets (bool_case A B) UNIV"

definition (in prob_space)

  indep_events_def_alt: "indep_events A I \<longleftrightarrow> indep_sets (\<lambda>i. {A i}) I"

lemma (in prob_space) indep_events_def:

  "indep_events A I \<longleftrightarrow> (A`I \<subseteq> events) \<and>

    (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j)))"

  unfolding indep_events_def_alt indep_sets_def

  apply (simp add: Ball_def Pi_iff image_subset_iff_funcset)

  apply (intro conj_cong refl arg_cong[where f=All] ext imp_cong)

  apply auto

  done

definition (in prob_space)

  "indep_event A B \<longleftrightarrow> indep_events (bool_case A B) UNIV"

lemma (in prob_space) indep_sets_cong:

  "I = J \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> F i = G i) \<Longrightarrow> indep_sets F I \<longleftrightarrow> indep_sets G J"

  by (simp add: indep_sets_def, intro conj_cong all_cong imp_cong ball_cong) blast+

lemma (in prob_space) indep_events_finite_index_events:

  "indep_events F I \<longleftrightarrow> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_events F J)"

  by (auto simp: indep_events_def)

lemma (in prob_space) indep_sets_finite_index_sets:

  "indep_sets F I \<longleftrightarrow> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J)"

proof (intro iffI allI impI)

  assume *: "\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J"

  show "indep_sets F I" unfolding indep_sets_def

  proof (intro conjI ballI allI impI)

    fix i assume "i \<in> I"

    with *[THEN spec, of "{i}"] show "F i \<subseteq> events"

      by (auto simp: indep_sets_def)

  qed (insert *, auto simp: indep_sets_def)

qed (auto simp: indep_sets_def)

lemma (in prob_space) indep_sets_mono_index:

  "J \<subseteq> I \<Longrightarrow> indep_sets F I \<Longrightarrow> indep_sets F J"

  unfolding indep_sets_def by auto

lemma (in prob_space) indep_sets_mono_sets:

  assumes indep: "indep_sets F I"

  assumes mono: "\<And>i. i\<in>I \<Longrightarrow> G i \<subseteq> F i"

  shows "indep_sets G I"

proof -

  have "(\<forall>i\<in>I. F i \<subseteq> events) \<Longrightarrow> (\<forall>i\<in>I. G i \<subseteq> events)"

    using mono by auto

  moreover have "\<And>A J. J \<subseteq> I \<Longrightarrow> A \<in> (\<Pi> j\<in>J. G j) \<Longrightarrow> A \<in> (\<Pi> j\<in>J. F j)"

    using mono by (auto simp: Pi_iff)

  ultimately show ?thesis

    using indep by (auto simp: indep_sets_def)

qed

lemma (in prob_space) indep_sets_mono:

  assumes indep: "indep_sets F I"

  assumes mono: "J \<subseteq> I" "\<And>i. i\<in>J \<Longrightarrow> G i \<subseteq> F i"

  shows "indep_sets G J"

  apply (rule indep_sets_mono_sets)

  apply (rule indep_sets_mono_index)

  apply (fact +)

  done

lemma (in prob_space) indep_setsI:

  assumes "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events"

    and "\<And>A J. J \<noteq> {} \<Longrightarrow> J \<subseteq> I \<Longrightarrow> finite J \<Longrightarrow> (\<forall>j\<in>J. A j \<in> F j) \<Longrightarrow> prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"

  shows "indep_sets F I"

  using assms unfolding indep_sets_def by (auto simp: Pi_iff)

lemma (in prob_space) indep_setsD:

  assumes "indep_sets F I" and "J \<subseteq> I" "J \<noteq> {}" "finite J" "\<forall>j\<in>J. A j \<in> F j"

  shows "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"

  using assms unfolding indep_sets_def by auto

lemma (in prob_space) indep_setI:

  assumes ev: "A \<subseteq> events" "B \<subseteq> events"

    and indep: "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> prob (a \<inter> b) = prob a * prob b"

  shows "indep_set A B"

  unfolding indep_set_def

proof (rule indep_setsI)

  fix F J assume "J \<noteq> {}" "J \<subseteq> UNIV"

    and F: "\<forall>j\<in>J. F j \<in> (case j of True \<Rightarrow> A | False \<Rightarrow> B)"

  have "J \<in> Pow UNIV" by auto

  with F `J \<noteq> {}` indep[of "F True" "F False"]

  show "prob (\<Inter>j\<in>J. F j) = (\<Prod>j\<in>J. prob (F j))"

    unfolding UNIV_bool Pow_insert by (auto simp: ac_simps)

qed (auto split: bool.split simp: ev)

lemma (in prob_space) indep_setD:

  assumes indep: "indep_set A B" and ev: "a \<in> A" "b \<in> B"

  shows "prob (a \<inter> b) = prob a * prob b"

  using indep[unfolded indep_set_def, THEN indep_setsD, of UNIV "bool_case a b"] ev

  by (simp add: ac_simps UNIV_bool)

lemma (in prob_space)

  assumes indep: "indep_set A B"

  shows indep_setD_ev1: "A \<subseteq> events"

    and indep_setD_ev2: "B \<subseteq> events"

  using indep unfolding indep_set_def indep_sets_def UNIV_bool by auto

lemma (in prob_space) indep_sets_dynkin:

  assumes indep: "indep_sets F I"

  shows "indep_sets (\<lambda>i. dynkin (space M) (F i)) I"

    (is "indep_sets ?F I")

proof (subst indep_sets_finite_index_sets, intro allI impI ballI)

  fix J assume "finite J" "J \<subseteq> I" "J \<noteq> {}"

  with indep have "indep_sets F J"

    by (subst (asm) indep_sets_finite_index_sets) auto

  { fix J K assume "indep_sets F K"

    let ?G = "\<lambda>S i. if i \<in> S then ?F i else F i"

    assume "finite J" "J \<subseteq> K"

    then have "indep_sets (?G J) K"

    proof induct

      case (insert j J)

      moreover def G \<equiv> "?G J"

      ultimately have G: "indep_sets G K" "\<And>i. i \<in> K \<Longrightarrow> G i \<subseteq> events" and "j \<in> K"

        by (auto simp: indep_sets_def)

      let ?D = "{E\<in>events. indep_sets (G(j := {E})) K }"

      { fix X assume X: "X \<in> events"

        assume indep: "\<And>J A. J \<noteq> {} \<Longrightarrow> J \<subseteq> K \<Longrightarrow> finite J \<Longrightarrow> j \<notin> J \<Longrightarrow> (\<forall>i\<in>J. A i \<in> G i)

          \<Longrightarrow> prob ((\<Inter>i\<in>J. A i) \<inter> X) = prob X * (\<Prod>i\<in>J. prob (A i))"

        have "indep_sets (G(j := {X})) K"

        proof (rule indep_setsI)

          fix i assume "i \<in> K" then show "(G(j:={X})) i \<subseteq> events"

            using G X by auto

        next

          fix A J assume J: "J \<noteq> {}" "J \<subseteq> K" "finite J" "\<forall>i\<in>J. A i \<in> (G(j := {X})) i"

          show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"

          proof cases

            assume "j \<in> J"

            with J have "A j = X" by auto

            show ?thesis

            proof cases

              assume "J = {j}" then show ?thesis by simp

            next

              assume "J \<noteq> {j}"

              have "prob (\<Inter>i\<in>J. A i) = prob ((\<Inter>i\<in>J-{j}. A i) \<inter> X)"

                using `j \<in> J` `A j = X` by (auto intro!: arg_cong[where f=prob] split: split_if_asm)

              also have "\<dots> = prob X * (\<Prod>i\<in>J-{j}. prob (A i))"

              proof (rule indep)

                show "J - {j} \<noteq> {}" "J - {j} \<subseteq> K" "finite (J - {j})" "j \<notin> J - {j}"

                  using J `J \<noteq> {j}` `j \<in> J` by auto

                show "\<forall>i\<in>J - {j}. A i \<in> G i"

                  using J by auto

              qed

              also have "\<dots> = prob (A j) * (\<Prod>i\<in>J-{j}. prob (A i))"

                using `A j = X` by simp

              also have "\<dots> = (\<Prod>i\<in>J. prob (A i))"

                unfolding setprod.insert_remove[OF `finite J`, symmetric, of "\<lambda>i. prob  (A i)"]

                using `j \<in> J` by (simp add: insert_absorb)

              finally show ?thesis .

            qed

          next

            assume "j \<notin> J"

            with J have "\<forall>i\<in>J. A i \<in> G i" by (auto split: split_if_asm)

            with J show ?thesis

              by (intro indep_setsD[OF G(1)]) auto

          qed

        qed }

      note indep_sets_insert = this

      have "dynkin_system (space M) ?D"

      proof (rule dynkin_systemI', simp_all cong del: indep_sets_cong, safe)

        show "indep_sets (G(j := {{}})) K"

          by (rule indep_sets_insert) auto

      next

        fix X assume X: "X \<in> events" and G': "indep_sets (G(j := {X})) K"

        show "indep_sets (G(j := {space M - X})) K"

        proof (rule indep_sets_insert)

          fix J A assume J: "J \<noteq> {}" "J \<subseteq> K" "finite J" "j \<notin> J" and A: "\<forall>i\<in>J. A i \<in> G i"

          then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events"

            using G by auto

          have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) =

              prob ((\<Inter>j\<in>J. A j) - (\<Inter>i\<in>insert j J. (A(j := X)) i))"

            using A_sets sets_into_space[of _ M] X `J \<noteq> {}`

            by (auto intro!: arg_cong[where f=prob] split: split_if_asm)

          also have "\<dots> = prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)"

            using J `J \<noteq> {}` `j \<notin> J` A_sets X sets_into_space

            by (auto intro!: finite_measure_Diff finite_INT split: split_if_asm)

          finally have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) =

              prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)" .

          moreover {

            have "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"

              using J A `finite J` by (intro indep_setsD[OF G(1)]) auto

            then have "prob (\<Inter>j\<in>J. A j) = prob (space M) * (\<Prod>i\<in>J. prob (A i))"

              using prob_space by simp }

          moreover {

            have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = (\<Prod>i\<in>insert j J. prob ((A(j := X)) i))"

              using J A `j \<in> K` by (intro indep_setsD[OF G']) auto

            then have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = prob X * (\<Prod>i\<in>J. prob (A i))"

              using `finite J` `j \<notin> J` by (auto intro!: setprod_cong) }

          ultimately have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) = (prob (space M) - prob X) * (\<Prod>i\<in>J. prob (A i))"

            by (simp add: field_simps)

          also have "\<dots> = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))"

            using X A by (simp add: finite_measure_compl)

          finally show "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))" .

        qed (insert X, auto)

      next

        fix F :: "nat \<Rightarrow> 'a set" assume disj: "disjoint_family F" and "range F \<subseteq> ?D"

        then have F: "\<And>i. F i \<in> events" "\<And>i. indep_sets (G(j:={F i})) K" by auto

        show "indep_sets (G(j := {\<Union>k. F k})) K"

        proof (rule indep_sets_insert)

          fix J A assume J: "j \<notin> J" "J \<noteq> {}" "J \<subseteq> K" "finite J" and A: "\<forall>i\<in>J. A i \<in> G i"

          then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events"

            using G by auto

          have "prob ((\<Inter>j\<in>J. A j) \<inter> (\<Union>k. F k)) = prob (\<Union>k. (\<Inter>i\<in>insert j J. (A(j := F k)) i))"

            using `J \<noteq> {}` `j \<notin> J` `j \<in> K` by (auto intro!: arg_cong[where f=prob] split: split_if_asm)

          moreover have "(\<lambda>k. prob (\<Inter>i\<in>insert j J. (A(j := F k)) i)) sums prob (\<Union>k. (\<Inter>i\<in>insert j J. (A(j := F k)) i))"

          proof (rule finite_measure_UNION)

            show "disjoint_family (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i)"

              using disj by (rule disjoint_family_on_bisimulation) auto

            show "range (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i) \<subseteq> events"

              using A_sets F `finite J` `J \<noteq> {}` `j \<notin> J` by (auto intro!: Int)

          qed

          moreover { fix k

            from J A `j \<in> K` have "prob (\<Inter>i\<in>insert j J. (A(j := F k)) i) = prob (F k) * (\<Prod>i\<in>J. prob (A i))"

              by (subst indep_setsD[OF F(2)]) (auto intro!: setprod_cong split: split_if_asm)

            also have "\<dots> = prob (F k) * prob (\<Inter>i\<in>J. A i)"

              using J A `j \<in> K` by (subst indep_setsD[OF G(1)]) auto

            finally have "prob (\<Inter>i\<in>insert j J. (A(j := F k)) i) = prob (F k) * prob (\<Inter>i\<in>J. A i)" . }

          ultimately have "(\<lambda>k. prob (F k) * prob (\<Inter>i\<in>J. A i)) sums (prob ((\<Inter>j\<in>J. A j) \<inter> (\<Union>k. F k)))"

            by simp

          moreover

          have "(\<lambda>k. prob (F k) * prob (\<Inter>i\<in>J. A i)) sums (prob (\<Union>k. F k) * prob (\<Inter>i\<in>J. A i))"

            using disj F(1) by (intro finite_measure_UNION sums_mult2) auto

          then have "(\<lambda>k. prob (F k) * prob (\<Inter>i\<in>J. A i)) sums (prob (\<Union>k. F k) * (\<Prod>i\<in>J. prob (A i)))"

            using J A `j \<in> K` by (subst indep_setsD[OF G(1), symmetric]) auto

          ultimately

          show "prob ((\<Inter>j\<in>J. A j) \<inter> (\<Union>k. F k)) = prob (\<Union>k. F k) * (\<Prod>j\<in>J. prob (A j))"

            by (auto dest!: sums_unique)

        qed (insert F, auto)

      qed (insert sets_into_space, auto)

      then have mono: "dynkin (space M) (G j) \<subseteq> {E \<in> events. indep_sets (G(j := {E})) K}"

      proof (rule dynkin_system.dynkin_subset, safe)

        fix X assume "X \<in> G j"

        then show "X \<in> events" using G `j \<in> K` by auto

        from `indep_sets G K`

        show "indep_sets (G(j := {X})) K"

          by (rule indep_sets_mono_sets) (insert `X \<in> G j`, auto)

      qed

      have "indep_sets (G(j:=?D)) K"

      proof (rule indep_setsI)

        fix i assume "i \<in> K" then show "(G(j := ?D)) i \<subseteq> events"

          using G(2) by auto

      next

        fix A J assume J: "J\<noteq>{}" "J \<subseteq> K" "finite J" and A: "\<forall>i\<in>J. A i \<in> (G(j := ?D)) i"

        show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"

        proof cases

          assume "j \<in> J"

          with A have indep: "indep_sets (G(j := {A j})) K" by auto

          from J A show ?thesis

            by (intro indep_setsD[OF indep]) auto

        next

          assume "j \<notin> J"

          with J A have "\<forall>i\<in>J. A i \<in> G i" by (auto split: split_if_asm)

          with J show ?thesis

            by (intro indep_setsD[OF G(1)]) auto

        qed

      qed

      then have "indep_sets (G(j := dynkin (space M) (G j))) K"

        by (rule indep_sets_mono_sets) (insert mono, auto)

      then show ?case

        by (rule indep_sets_mono_sets) (insert `j \<in> K` `j \<notin> J`, auto simp: G_def)

    qed (insert `indep_sets F K`, simp) }

  from this[OF `indep_sets F J` `finite J` subset_refl]

  show "indep_sets ?F J"

    by (rule indep_sets_mono_sets) auto

qed

lemma (in prob_space) indep_sets_sigma:

  assumes indep: "indep_sets F I"

  assumes stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)"

  shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I"

proof -

  from indep_sets_dynkin[OF indep]

  show ?thesis

  proof (rule indep_sets_mono_sets, subst sigma_eq_dynkin, simp_all add: stable)

    fix i assume "i \<in> I"

    with indep have "F i \<subseteq> events" by (auto simp: indep_sets_def)

    with sets_into_space show "F i \<subseteq> Pow (space M)" by auto

  qed

qed

lemma (in prob_space) indep_sets_sigma_sets_iff:

  assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)"

  shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I \<longleftrightarrow> indep_sets F I"

proof

  assume "indep_sets F I" then show "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I"

    by (rule indep_sets_sigma) fact

next

  assume "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I" then show "indep_sets F I"

    by (rule indep_sets_mono_sets) (intro subsetI sigma_sets.Basic)

qed

definition (in prob_space)

  indep_vars_def2: "indep_vars M' X I \<longleftrightarrow>

    (\<forall>i\<in>I. random_variable (M' i) (X i)) \<and>

    indep_sets (\<lambda>i. { X i -` A \<inter> space M | A. A \<in> sets (M' i)}) I"

definition (in prob_space)

  "indep_var Ma A Mb B \<longleftrightarrow> indep_vars (bool_case Ma Mb) (bool_case A B) UNIV"

lemma (in prob_space) indep_vars_def:

  "indep_vars M' X I \<longleftrightarrow>

    (\<forall>i\<in>I. random_variable (M' i) (X i)) \<and>

    indep_sets (\<lambda>i. sigma_sets (space M) { X i -` A \<inter> space M | A. A \<in> sets (M' i)}) I"

  unfolding indep_vars_def2

  apply (rule conj_cong[OF refl])

  apply (rule indep_sets_sigma_sets_iff[symmetric])

  apply (auto simp: Int_stable_def)

  apply (rule_tac x="A \<inter> Aa" in exI)

  apply auto

  done

lemma (in prob_space) indep_var_eq:

  "indep_var S X T Y \<longleftrightarrow>

    (random_variable S X \<and> random_variable T Y) \<and>

    indep_set

      (sigma_sets (space M) { X -` A \<inter> space M | A. A \<in> sets S})

      (sigma_sets (space M) { Y -` A \<inter> space M | A. A \<in> sets T})"

  unfolding indep_var_def indep_vars_def indep_set_def UNIV_bool

  by (intro arg_cong2[where f="op \<and>"] arg_cong2[where f=indep_sets] ext)

     (auto split: bool.split)

lemma (in prob_space) indep_sets2_eq:

  "indep_set A B \<longleftrightarrow> A \<subseteq> events \<and> B \<subseteq> events \<and> (\<forall>a\<in>A. \<forall>b\<in>B. prob (a \<inter> b) = prob a * prob b)"

  unfolding indep_set_def

proof (intro iffI ballI conjI)

  assume indep: "indep_sets (bool_case A B) UNIV"

  { fix a b assume "a \<in> A" "b \<in> B"

    with indep_setsD[OF indep, of UNIV "bool_case a b"]

    show "prob (a \<inter> b) = prob a * prob b"

      unfolding UNIV_bool by (simp add: ac_simps) }

  from indep show "A \<subseteq> events" "B \<subseteq> events"

    unfolding indep_sets_def UNIV_bool by auto

next

  assume *: "A \<subseteq> events \<and> B \<subseteq> events \<and> (\<forall>a\<in>A. \<forall>b\<in>B. prob (a \<inter> b) = prob a * prob b)"

  show "indep_sets (bool_case A B) UNIV"

  proof (rule indep_setsI)

    fix i show "(case i of True \<Rightarrow> A | False \<Rightarrow> B) \<subseteq> events"

      using * by (auto split: bool.split)

  next

    fix J X assume "J \<noteq> {}" "J \<subseteq> UNIV" and X: "\<forall>j\<in>J. X j \<in> (case j of True \<Rightarrow> A | False \<Rightarrow> B)"

    then have "J = {True} \<or> J = {False} \<or> J = {True,False}"

      by (auto simp: UNIV_bool)

    then show "prob (\<Inter>j\<in>J. X j) = (\<Prod>j\<in>J. prob (X j))"

      using X * by auto

  qed

qed

lemma (in prob_space) indep_set_sigma_sets:

  assumes "indep_set A B"

  assumes A: "Int_stable A" and B: "Int_stable B"

  shows "indep_set (sigma_sets (space M) A) (sigma_sets (space M) B)"

proof -

  have "indep_sets (\<lambda>i. sigma_sets (space M) (case i of True \<Rightarrow> A | False \<Rightarrow> B)) UNIV"

  proof (rule indep_sets_sigma)

    show "indep_sets (bool_case A B) UNIV"

      by (rule `indep_set A B`[unfolded indep_set_def])

    fix i show "Int_stable (case i of True \<Rightarrow> A | False \<Rightarrow> B)"

      using A B by (cases i) auto

  qed

  then show ?thesis

    unfolding indep_set_def

    by (rule indep_sets_mono_sets) (auto split: bool.split)

qed

lemma (in prob_space) indep_sets_collect_sigma:

  fixes I :: "'j \<Rightarrow> 'i set" and J :: "'j set" and E :: "'i \<Rightarrow> 'a set set"

  assumes indep: "indep_sets E (\<Union>j\<in>J. I j)"

  assumes Int_stable: "\<And>i j. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> Int_stable (E i)"

  assumes disjoint: "disjoint_family_on I J"

  shows "indep_sets (\<lambda>j. sigma_sets (space M) (\<Union>i\<in>I j. E i)) J"

proof -

  let ?E = "\<lambda>j. {\<Inter>k\<in>K. E' k| E' K. finite K \<and> K \<noteq> {} \<and> K \<subseteq> I j \<and> (\<forall>k\<in>K. E' k \<in> E k) }"

  from indep have E: "\<And>j i. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> E i \<subseteq> events"

    unfolding indep_sets_def by auto

  { fix j

    let ?S = "sigma_sets (space M) (\<Union>i\<in>I j. E i)"

    assume "j \<in> J"

    from E[OF this] interpret S: sigma_algebra "space M" ?S

      using sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto

    have "sigma_sets (space M) (\<Union>i\<in>I j. E i) = sigma_sets (space M) (?E j)"

    proof (rule sigma_sets_eqI)

      fix A assume "A \<in> (\<Union>i\<in>I j. E i)"

      then guess i ..

      then show "A \<in> sigma_sets (space M) (?E j)"

        by (auto intro!: sigma_sets.intros(2-) exI[of _ "{i}"] exI[of _ "\<lambda>i. A"])

    next

      fix A assume "A \<in> ?E j"

      then obtain E' K where "finite K" "K \<noteq> {}" "K \<subseteq> I j" "\<And>k. k \<in> K \<Longrightarrow> E' k \<in> E k"

        and A: "A = (\<Inter>k\<in>K. E' k)"

        by auto

      then have "A \<in> ?S" unfolding A

        by (safe intro!: S.finite_INT) auto

      then show "A \<in> sigma_sets (space M) (\<Union>i\<in>I j. E i)"

        by simp

    qed }

  moreover have "indep_sets (\<lambda>j. sigma_sets (space M) (?E j)) J"

  proof (rule indep_sets_sigma)

    show "indep_sets ?E J"

    proof (intro indep_setsI)

      fix j assume "j \<in> J" with E show "?E j \<subseteq> events" by (force  intro!: finite_INT)

    next

      fix K A assume K: "K \<noteq> {}" "K \<subseteq> J" "finite K"

        and "\<forall>j\<in>K. A j \<in> ?E j"

      then have "\<forall>j\<in>K. \<exists>E' L. A j = (\<Inter>l\<in>L. E' l) \<and> finite L \<and> L \<noteq> {} \<and> L \<subseteq> I j \<and> (\<forall>l\<in>L. E' l \<in> E l)"

        by simp

      from bchoice[OF this] guess E' ..

      from bchoice[OF this] obtain L

        where A: "\<And>j. j\<in>K \<Longrightarrow> A j = (\<Inter>l\<in>L j. E' j l)"

        and L: "\<And>j. j\<in>K \<Longrightarrow> finite (L j)" "\<And>j. j\<in>K \<Longrightarrow> L j \<noteq> {}" "\<And>j. j\<in>K \<Longrightarrow> L j \<subseteq> I j"

        and E': "\<And>j l. j\<in>K \<Longrightarrow> l \<in> L j \<Longrightarrow> E' j l \<in> E l"

        by auto

      { fix k l j assume "k \<in> K" "j \<in> K" "l \<in> L j" "l \<in> L k"

        have "k = j"

        proof (rule ccontr)

          assume "k \<noteq> j"

          with disjoint `K \<subseteq> J` `k \<in> K` `j \<in> K` have "I k \<inter> I j = {}"

            unfolding disjoint_family_on_def by auto

          with L(2,3)[OF `j \<in> K`] L(2,3)[OF `k \<in> K`]

          show False using `l \<in> L k` `l \<in> L j` by auto

        qed }

      note L_inj = this

      def k \<equiv> "\<lambda>l. (SOME k. k \<in> K \<and> l \<in> L k)"

      { fix x j l assume *: "j \<in> K" "l \<in> L j"

        have "k l = j" unfolding k_def

        proof (rule some_equality)

          fix k assume "k \<in> K \<and> l \<in> L k"

          with * L_inj show "k = j" by auto

        qed (insert *, simp) }

      note k_simp[simp] = this

      let ?E' = "\<lambda>l. E' (k l) l"

      have "prob (\<Inter>j\<in>K. A j) = prob (\<Inter>l\<in>(\<Union>k\<in>K. L k). ?E' l)"

        by (auto simp: A intro!: arg_cong[where f=prob])

      also have "\<dots> = (\<Prod>l\<in>(\<Union>k\<in>K. L k). prob (?E' l))"

        using L K E' by (intro indep_setsD[OF indep]) (simp_all add: UN_mono)

      also have "\<dots> = (\<Prod>j\<in>K. \<Prod>l\<in>L j. prob (E' j l))"

        using K L L_inj by (subst setprod_UN_disjoint) auto

      also have "\<dots> = (\<Prod>j\<in>K. prob (A j))"

        using K L E' by (auto simp add: A intro!: setprod_cong indep_setsD[OF indep, symmetric]) blast

      finally show "prob (\<Inter>j\<in>K. A j) = (\<Prod>j\<in>K. prob (A j))" .

    qed

  next

    fix j assume "j \<in> J"

    show "Int_stable (?E j)"

    proof (rule Int_stableI)

      fix a assume "a \<in> ?E j" then obtain Ka Ea

        where a: "a = (\<Inter>k\<in>Ka. Ea k)" "finite Ka" "Ka \<noteq> {}" "Ka \<subseteq> I j" "\<And>k. k\<in>Ka \<Longrightarrow> Ea k \<in> E k" by auto

      fix b assume "b \<in> ?E j" then obtain Kb Eb

        where b: "b = (\<Inter>k\<in>Kb. Eb k)" "finite Kb" "Kb \<noteq> {}" "Kb \<subseteq> I j" "\<And>k. k\<in>Kb \<Longrightarrow> Eb k \<in> E k" by auto

      let ?A = "\<lambda>k. (if k \<in> Ka \<inter> Kb then Ea k \<inter> Eb k else if k \<in> Kb then Eb k else if k \<in> Ka then Ea k else {})"

      have "a \<inter> b = INTER (Ka \<union> Kb) ?A"

        by (simp add: a b set_eq_iff) auto

      with a b `j \<in> J` Int_stableD[OF Int_stable] show "a \<inter> b \<in> ?E j"

        by (intro CollectI exI[of _ "Ka \<union> Kb"] exI[of _ ?A]) auto

    qed

  qed

  ultimately show ?thesis

    by (simp cong: indep_sets_cong)

qed

definition (in prob_space) tail_events where

  "tail_events A = (\<Inter>n. sigma_sets (space M) (UNION {n..} A))"

lemma (in prob_space) tail_events_sets:

  assumes A: "\<And>i::nat. A i \<subseteq> events"

  shows "tail_events A \<subseteq> events"

proof

  fix X assume X: "X \<in> tail_events A"

  let ?A = "(\<Inter>n. sigma_sets (space M) (UNION {n..} A))"

  from X have "\<And>n::nat. X \<in> sigma_sets (space M) (UNION {n..} A)" by (auto simp: tail_events_def)

  from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp

  then show "X \<in> events"

    by induct (insert A, auto)

qed

lemma (in prob_space) sigma_algebra_tail_events:

  assumes "\<And>i::nat. sigma_algebra (space M) (A i)"

  shows "sigma_algebra (space M) (tail_events A)"

  unfolding tail_events_def

proof (simp add: sigma_algebra_iff2, safe)

  let ?A = "(\<Inter>n. sigma_sets (space M) (UNION {n..} A))"

  interpret A: sigma_algebra "space M" "A i" for i by fact

  { fix X x assume "X \<in> ?A" "x \<in> X"

    then have "\<And>n. X \<in> sigma_sets (space M) (UNION {n..} A)" by auto

    from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp

    then have "X \<subseteq> space M"

      by induct (insert A.sets_into_space, auto)

    with `x \<in> X` show "x \<in> space M" by auto }

  { fix F :: "nat \<Rightarrow> 'a set" and n assume "range F \<subseteq> ?A"

    then show "(UNION UNIV F) \<in> sigma_sets (space M) (UNION {n..} A)"

      by (intro sigma_sets.Union) auto }

qed (auto intro!: sigma_sets.Compl sigma_sets.Empty)

lemma (in prob_space) kolmogorov_0_1_law:

  fixes A :: "nat \<Rightarrow> 'a set set"

  assumes "\<And>i::nat. sigma_algebra (space M) (A i)"

  assumes indep: "indep_sets A UNIV"

  and X: "X \<in> tail_events A"

  shows "prob X = 0 \<or> prob X = 1"

proof -

  have A: "\<And>i. A i \<subseteq> events"

    using indep unfolding indep_sets_def by simp

  let ?D = "{D \<in> events. prob (X \<inter> D) = prob X * prob D}"

  interpret A: sigma_algebra "space M" "A i" for i by fact

  interpret T: sigma_algebra "space M" "tail_events A"

    by (rule sigma_algebra_tail_events) fact

  have "X \<subseteq> space M" using T.space_closed X by auto

  have X_in: "X \<in> events"

    using tail_events_sets A X by auto

  interpret D: dynkin_system "space M" ?D

  proof (rule dynkin_systemI)

    fix D assume "D \<in> ?D" then show "D \<subseteq> space M"

      using sets_into_space by auto

  next

    show "space M \<in> ?D"

      using prob_space `X \<subseteq> space M` by (simp add: Int_absorb2)

  next

    fix A assume A: "A \<in> ?D"

    have "prob (X \<inter> (space M - A)) = prob (X - (X \<inter> A))"

      using `X \<subseteq> space M` by (auto intro!: arg_cong[where f=prob])

    also have "\<dots> = prob X - prob (X \<inter> A)"

      using X_in A by (intro finite_measure_Diff) auto

    also have "\<dots> = prob X * prob (space M) - prob X * prob A"

      using A prob_space by auto

    also have "\<dots> = prob X * prob (space M - A)"

      using X_in A sets_into_space

      by (subst finite_measure_Diff) (auto simp: field_simps)

    finally show "space M - A \<in> ?D"

      using A `X \<subseteq> space M` by auto

  next

    fix F :: "nat \<Rightarrow> 'a set" assume dis: "disjoint_family F" and "range F \<subseteq> ?D"

    then have F: "range F \<subseteq> events" "\<And>i. prob (X \<inter> F i) = prob X * prob (F i)"

      by auto

    have "(\<lambda>i. prob (X \<inter> F i)) sums prob (\<Union>i. X \<inter> F i)"

    proof (rule finite_measure_UNION)

      show "range (\<lambda>i. X \<inter> F i) \<subseteq> events"

        using F X_in by auto

      show "disjoint_family (\<lambda>i. X \<inter> F i)"

        using dis by (rule disjoint_family_on_bisimulation) auto

    qed

    with F have "(\<lambda>i. prob X * prob (F i)) sums prob (X \<inter> (\<Union>i. F i))"

      by simp

    moreover have "(\<lambda>i. prob X * prob (F i)) sums (prob X * prob (\<Union>i. F i))"

      by (intro sums_mult finite_measure_UNION F dis)

    ultimately have "prob (X \<inter> (\<Union>i. F i)) = prob X * prob (\<Union>i. F i)"

      by (auto dest!: sums_unique)

    with F show "(\<Union>i. F i) \<in> ?D"

      by auto

  qed

  { fix n

    have "indep_sets (\<lambda>b. sigma_sets (space M) (\<Union>m\<in>bool_case {..n} {Suc n..} b. A m)) UNIV"

    proof (rule indep_sets_collect_sigma)

      have *: "(\<Union>b. case b of True \<Rightarrow> {..n} | False \<Rightarrow> {Suc n..}) = UNIV" (is "?U = _")

        by (simp split: bool.split add: set_eq_iff) (metis not_less_eq_eq)

      with indep show "indep_sets A ?U" by simp

      show "disjoint_family (bool_case {..n} {Suc n..})"

        unfolding disjoint_family_on_def by (auto split: bool.split)

      fix m

      show "Int_stable (A m)"

        unfolding Int_stable_def using A.Int by auto

    qed

    also have "(\<lambda>b. sigma_sets (space M) (\<Union>m\<in>bool_case {..n} {Suc n..} b. A m)) =

      bool_case (sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) (sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m))"

      by (auto intro!: ext split: bool.split)

    finally have indep: "indep_set (sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) (sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m))"

      unfolding indep_set_def by simp

    have "sigma_sets (space M) (\<Union>m\<in>{..n}. A m) \<subseteq> ?D"

    proof (simp add: subset_eq, rule)

      fix D assume D: "D \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)"

      have "X \<in> sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m)"

        using X unfolding tail_events_def by simp

      from indep_setD[OF indep D this] indep_setD_ev1[OF indep] D

      show "D \<in> events \<and> prob (X \<inter> D) = prob X * prob D"

        by (auto simp add: ac_simps)

    qed }

  then have "(\<Union>n. sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) \<subseteq> ?D" (is "?A \<subseteq> _")

    by auto

  note `X \<in> tail_events A`

  also {

    have "\<And>n. sigma_sets (space M) (\<Union>i\<in>{n..}. A i) \<subseteq> sigma_sets (space M) ?A"

      by (intro sigma_sets_subseteq UN_mono) auto

   then have "tail_events A \<subseteq> sigma_sets (space M) ?A"

      unfolding tail_events_def by auto }

  also have "sigma_sets (space M) ?A = dynkin (space M) ?A"

  proof (rule sigma_eq_dynkin)

    { fix B n assume "B \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)"

      then have "B \<subseteq> space M"

        by induct (insert A sets_into_space[of _ M], auto) }

    then show "?A \<subseteq> Pow (space M)" by auto

    show "Int_stable ?A"

    proof (rule Int_stableI)

      fix a assume "a \<in> ?A" then guess n .. note a = this

      fix b assume "b \<in> ?A" then guess m .. note b = this

      interpret Amn: sigma_algebra "space M" "sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"

        using A sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto

      have "sigma_sets (space M) (\<Union>i\<in>{..n}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"

        by (intro sigma_sets_subseteq UN_mono) auto

      with a have "a \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto

      moreover

      have "sigma_sets (space M) (\<Union>i\<in>{..m}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"

        by (intro sigma_sets_subseteq UN_mono) auto

      with b have "b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto

      ultimately have "a \<inter> b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"

        using Amn.Int[of a b] by simp

      then show "a \<inter> b \<in> (\<Union>n. sigma_sets (space M) (\<Union>i\<in>{..n}. A i))" by auto

    qed

  qed

  also have "dynkin (space M) ?A \<subseteq> ?D"

    using `?A \<subseteq> ?D` by (auto intro!: D.dynkin_subset)

  finally show ?thesis by auto

qed

lemma (in prob_space) borel_0_1_law:

  fixes F :: "nat \<Rightarrow> 'a set"

  assumes F2: "indep_events F UNIV"

  shows "prob (\<Inter>n. \<Union>m\<in>{n..}. F m) = 0 \<or> prob (\<Inter>n. \<Union>m\<in>{n..}. F m) = 1"

proof (rule kolmogorov_0_1_law[of "\<lambda>i. sigma_sets (space M) { F i }"])

  have F1: "range F \<subseteq> events"

    using F2 by (simp add: indep_events_def subset_eq)

  { fix i show "sigma_algebra (space M) (sigma_sets (space M) {F i})"

      using sigma_algebra_sigma_sets[of "{F i}" "space M"] F1 sets_into_space

      by auto }

  show "indep_sets (\<lambda>i. sigma_sets (space M) {F i}) UNIV"

  proof (rule indep_sets_sigma)

    show "indep_sets (\<lambda>i. {F i}) UNIV"

      unfolding indep_events_def_alt[symmetric] by fact

    fix i show "Int_stable {F i}"

      unfolding Int_stable_def by simp

  qed

  let ?Q = "\<lambda>n. \<Union>i\<in>{n..}. F i"

  show "(\<Inter>n. \<Union>m\<in>{n..}. F m) \<in> tail_events (\<lambda>i. sigma_sets (space M) {F i})"

    unfolding tail_events_def

  proof

    fix j

    interpret S: sigma_algebra "space M" "sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})"

      using order_trans[OF F1 space_closed]

      by (intro sigma_algebra_sigma_sets) (simp add: sigma_sets_singleton subset_eq)

    have "(\<Inter>n. ?Q n) = (\<Inter>n\<in>{j..}. ?Q n)"

      by (intro decseq_SucI INT_decseq_offset UN_mono) auto

    also have "\<dots> \<in> sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})"

      using order_trans[OF F1 space_closed]

      by (safe intro!: S.countable_INT S.countable_UN)

         (auto simp: sigma_sets_singleton intro!: sigma_sets.Basic bexI)

    finally show "(\<Inter>n. ?Q n) \<in> sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})"

      by simp

  qed

qed

lemma (in prob_space) indep_sets_finite:

  assumes I: "I \<noteq> {}" "finite I"

    and F: "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events" "\<And>i. i \<in> I \<Longrightarrow> space M \<in> F i"

  shows "indep_sets F I \<longleftrightarrow> (\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j)))"

proof

  assume *: "indep_sets F I"

  from I show "\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j))"

    by (intro indep_setsD[OF *] ballI) auto

next

  assume indep: "\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j))"

  show "indep_sets F I"

  proof (rule indep_setsI[OF F(1)])

    fix A J assume J: "J \<noteq> {}" "J \<subseteq> I" "finite J"

    assume A: "\<forall>j\<in>J. A j \<in> F j"

    let ?A = "\<lambda>j. if j \<in> J then A j else space M"

    have "prob (\<Inter>j\<in>I. ?A j) = prob (\<Inter>j\<in>J. A j)"

      using subset_trans[OF F(1) space_closed] J A

      by (auto intro!: arg_cong[where f=prob] split: split_if_asm) blast

    also

    from A F have "(\<lambda>j. if j \<in> J then A j else space M) \<in> Pi I F" (is "?A \<in> _")

      by (auto split: split_if_asm)

    with indep have "prob (\<Inter>j\<in>I. ?A j) = (\<Prod>j\<in>I. prob (?A j))"

      by auto

    also have "\<dots> = (\<Prod>j\<in>J. prob (A j))"

      unfolding if_distrib setprod.If_cases[OF `finite I`]

      using prob_space `J \<subseteq> I` by (simp add: Int_absorb1 setprod_1)

    finally show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" ..

  qed

qed

lemma (in prob_space) indep_vars_finite:

  fixes I :: "'i set"

  assumes I: "I \<noteq> {}" "finite I"

    and M': "\<And>i. i \<in> I \<Longrightarrow> sets (M' i) = sigma_sets (space (M' i)) (E i)"

    and rv: "\<And>i. i \<in> I \<Longrightarrow> random_variable (M' i) (X i)"

    and Int_stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (E i)"

    and space: "\<And>i. i \<in> I \<Longrightarrow> space (M' i) \<in> E i" and closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M' i))"

  shows "indep_vars M' X I \<longleftrightarrow>

    (\<forall>A\<in>(\<Pi> i\<in>I. E i). prob (\<Inter>j\<in>I. X j -` A j \<inter> space M) = (\<Prod>j\<in>I. prob (X j -` A j \<inter> space M)))"

proof -

  from rv have X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> space M \<rightarrow> space (M' i)"

    unfolding measurable_def by simp

  { fix i assume "i\<in>I"

    from closed[OF `i \<in> I`]

    have "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}

      = sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> E i}"

      unfolding sigma_sets_vimage_commute[OF X, OF `i \<in> I`, symmetric] M'[OF `i \<in> I`]

      by (subst sigma_sets_sigma_sets_eq) auto }

  note sigma_sets_X = this

  { fix i assume "i\<in>I"

    have "Int_stable {X i -` A \<inter> space M |A. A \<in> E i}"

    proof (rule Int_stableI)

      fix a assume "a \<in> {X i -` A \<inter> space M |A. A \<in> E i}"

      then obtain A where "a = X i -` A \<inter> space M" "A \<in> E i" by auto

      moreover

      fix b assume "b \<in> {X i -` A \<inter> space M |A. A \<in> E i}"

      then obtain B where "b = X i -` B \<inter> space M" "B \<in> E i" by auto

      moreover

      have "(X i -` A \<inter> space M) \<inter> (X i -` B \<inter> space M) = X i -` (A \<inter> B) \<inter> space M" by auto

      moreover note Int_stable[OF `i \<in> I`]

      ultimately

      show "a \<inter> b \<in> {X i -` A \<inter> space M |A. A \<in> E i}"

        by (auto simp del: vimage_Int intro!: exI[of _ "A \<inter> B"] dest: Int_stableD)

    qed }

  note indep_sets_X = indep_sets_sigma_sets_iff[OF this]

  { fix i assume "i \<in> I"

    { fix A assume "A \<in> E i"

      with M'[OF `i \<in> I`] have "A \<in> sets (M' i)" by auto

      moreover

      from rv[OF `i\<in>I`] have "X i \<in> measurable M (M' i)" by auto

      ultimately

      have "X i -` A \<inter> space M \<in> sets M" by (auto intro: measurable_sets) }

    with X[OF `i\<in>I`] space[OF `i\<in>I`]

    have "{X i -` A \<inter> space M |A. A \<in> E i} \<subseteq> events"

      "space M \<in> {X i -` A \<inter> space M |A. A \<in> E i}"

      by (auto intro!: exI[of _ "space (M' i)"]) }

  note indep_sets_finite_X = indep_sets_finite[OF I this]

  have "(\<forall>A\<in>\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> E i}. prob (INTER I A) = (\<Prod>j\<in>I. prob (A j))) =

    (\<forall>A\<in>\<Pi> i\<in>I. E i. prob ((\<Inter>j\<in>I. X j -` A j) \<inter> space M) = (\<Prod>x\<in>I. prob (X x -` A x \<inter> space M)))"

    (is "?L = ?R")

  proof safe

    fix A assume ?L and A: "A \<in> (\<Pi> i\<in>I. E i)"

    from `?L`[THEN bspec, of "\<lambda>i. X i -` A i \<inter> space M"] A `I \<noteq> {}`

    show "prob ((\<Inter>j\<in>I. X j -` A j) \<inter> space M) = (\<Prod>x\<in>I. prob (X x -` A x \<inter> space M))"

      by (auto simp add: Pi_iff)

  next

    fix A assume ?R and A: "A \<in> (\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> E i})"

    from A have "\<forall>i\<in>I. \<exists>B. A i = X i -` B \<inter> space M \<and> B \<in> E i" by auto

    from bchoice[OF this] obtain B where B: "\<forall>i\<in>I. A i = X i -` B i \<inter> space M"

      "B \<in> (\<Pi> i\<in>I. E i)" by auto

    from `?R`[THEN bspec, OF B(2)] B(1) `I \<noteq> {}`

    show "prob (INTER I A) = (\<Prod>j\<in>I. prob (A j))"

      by simp

  qed

  then show ?thesis using `I \<noteq> {}`

    by (simp add: rv indep_vars_def indep_sets_X sigma_sets_X indep_sets_finite_X cong: indep_sets_cong)

qed

lemma (in prob_space) indep_vars_compose:

  assumes "indep_vars M' X I"

  assumes rv: "\<And>i. i \<in> I \<Longrightarrow> Y i \<in> measurable (M' i) (N i)"

  shows "indep_vars N (\<lambda>i. Y i \<circ> X i) I"

  unfolding indep_vars_def

proof

  from rv `indep_vars M' X I`

  show "\<forall>i\<in>I. random_variable (N i) (Y i \<circ> X i)"

    by (auto simp: indep_vars_def)

  have "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"

    using `indep_vars M' X I` by (simp add: indep_vars_def)

  then show "indep_sets (\<lambda>i. sigma_sets (space M) {(Y i \<circ> X i) -` A \<inter> space M |A. A \<in> sets (N i)}) I"

  proof (rule indep_sets_mono_sets)

    fix i assume "i \<in> I"

    with `indep_vars M' X I` have X: "X i \<in> space M \<rightarrow> space (M' i)"

      unfolding indep_vars_def measurable_def by auto

    { fix A assume "A \<in> sets (N i)"

      then have "\<exists>B. (Y i \<circ> X i) -` A \<inter> space M = X i -` B \<inter> space M \<and> B \<in> sets (M' i)"

        by (intro exI[of _ "Y i -` A \<inter> space (M' i)"])

           (auto simp: vimage_compose intro!: measurable_sets rv `i \<in> I` funcset_mem[OF X]) }

    then show "sigma_sets (space M) {(Y i \<circ> X i) -` A \<inter> space M |A. A \<in> sets (N i)} \<subseteq>

      sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"

      by (intro sigma_sets_subseteq) (auto simp: vimage_compose)

  qed

qed

lemma (in prob_space) indep_varsD_finite:

  assumes X: "indep_vars M' X I"

  assumes I: "I \<noteq> {}" "finite I" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M' i)"

  shows "prob (\<Inter>i\<in>I. X i -` A i \<inter> space M) = (\<Prod>i\<in>I. prob (X i -` A i \<inter> space M))"

proof (rule indep_setsD)

  show "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"

    using X by (auto simp: indep_vars_def)

  show "I \<subseteq> I" "I \<noteq> {}" "finite I" using I by auto

  show "\<forall>i\<in>I. X i -` A i \<inter> space M \<in> sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"

    using I by auto

qed

lemma (in prob_space) indep_varsD:

  assumes X: "indep_vars M' X I"

  assumes I: "J \<noteq> {}" "finite J" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M' i)"

  shows "prob (\<Inter>i\<in>J. X i -` A i \<inter> space M) = (\<Prod>i\<in>J. prob (X i -` A i \<inter> space M))"

proof (rule indep_setsD)

  show "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"

    using X by (auto simp: indep_vars_def)

  show "\<forall>i\<in>J. X i -` A i \<inter> space M \<in> sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"

    using I by auto

qed fact+

lemma (in prob_space) indep_vars_iff_distr_eq_PiM:

  fixes I :: "'i set" and X :: "'i \<Rightarrow> 'a \<Rightarrow> 'b"

  assumes "I \<noteq> {}"

  assumes rv: "\<And>i. random_variable (M' i) (X i)"

  shows "indep_vars M' X I \<longleftrightarrow>

    distr M (\<Pi>\<^isub>M i\<in>I. M' i) (\<lambda>x. \<lambda>i\<in>I. X i x) = (\<Pi>\<^isub>M i\<in>I. distr M (M' i) (X i))"

proof -

  let ?P = "\<Pi>\<^isub>M i\<in>I. M' i"

  let ?X = "\<lambda>x. \<lambda>i\<in>I. X i x"

  let ?D = "distr M ?P ?X"

  have X: "random_variable ?P ?X" by (intro measurable_restrict rv)

  interpret D: prob_space ?D by (intro prob_space_distr X)

  let ?D' = "\<lambda>i. distr M (M' i) (X i)"

  let ?P' = "\<Pi>\<^isub>M i\<in>I. distr M (M' i) (X i)"

  interpret D': prob_space "?D' i" for i by (intro prob_space_distr rv)

  interpret P: product_prob_space ?D' I ..

  show ?thesis

  proof

    assume "indep_vars M' X I"

    show "?D = ?P'"

    proof (rule measure_eqI_generator_eq)

      show "Int_stable (prod_algebra I M')"

        by (rule Int_stable_prod_algebra)

      show "prod_algebra I M' \<subseteq> Pow (space ?P)"

        using prod_algebra_sets_into_space by (simp add: space_PiM)

      show "sets ?D = sigma_sets (space ?P) (prod_algebra I M')"

        by (simp add: sets_PiM space_PiM)

      show "sets ?P' = sigma_sets (space ?P) (prod_algebra I M')"

        by (simp add: sets_PiM space_PiM cong: prod_algebra_cong)

      let ?A = "\<lambda>i. \<Pi>\<^isub>E i\<in>I. space (M' i)"

      show "range ?A \<subseteq> prod_algebra I M'" "(\<Union>i. ?A i) = space (Pi\<^isub>M I M')"

        by (auto simp: space_PiM intro!: space_in_prod_algebra cong: prod_algebra_cong)

      { fix i show "emeasure ?D (\<Pi>\<^isub>E i\<in>I. space (M' i)) \<noteq> \<infinity>" by auto }

    next

      fix E assume E: "E \<in> prod_algebra I M'"

      from prod_algebraE[OF E] guess J Y . note J = this

      from E have "E \<in> sets ?P" by (auto simp: sets_PiM)

      then have "emeasure ?D E = emeasure M (?X -` E \<inter> space M)"

        by (simp add: emeasure_distr X)

      also have "?X -` E \<inter> space M = (\<Inter>i\<in>J. X i -` Y i \<inter> space M)"

        using J `I \<noteq> {}` measurable_space[OF rv] by (auto simp: prod_emb_def PiE_iff split: split_if_asm)

      also have "emeasure M (\<Inter>i\<in>J. X i -` Y i \<inter> space M) = (\<Prod> i\<in>J. emeasure M (X i -` Y i \<inter> space M))"

        using `indep_vars M' X I` J `I \<noteq> {}` using indep_varsD[of M' X I J]

        by (auto simp: emeasure_eq_measure setprod_ereal)

      also have "\<dots> = (\<Prod> i\<in>J. emeasure (?D' i) (Y i))"

        using rv J by (simp add: emeasure_distr)

      also have "\<dots> = emeasure ?P' E"

        using P.emeasure_PiM_emb[of J Y] J by (simp add: prod_emb_def)

      finally show "emeasure ?D E = emeasure ?P' E" .

    qed

  next

    assume "?D = ?P'"

    show "indep_vars M' X I" unfolding indep_vars_def

    proof (intro conjI indep_setsI ballI rv)

      fix i show "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)} \<subseteq> events"

        by (auto intro!: sigma_sets_subset measurable_sets rv)

    next

      fix J Y' assume J: "J \<noteq> {}" "J \<subseteq> I" "finite J"

      assume Y': "\<forall>j\<in>J. Y' j \<in> sigma_sets (space M) {X j -` A \<inter> space M |A. A \<in> sets (M' j)}"

      have "\<forall>j\<in>J. \<exists>Y. Y' j = X j -` Y \<inter> space M \<and> Y \<in> sets (M' j)"

      proof

        fix j assume "j \<in> J"

        from Y'[rule_format, OF this] rv[of j]

        show "\<exists>Y. Y' j = X j -` Y \<inter> space M \<and> Y \<in> sets (M' j)"

          by (subst (asm) sigma_sets_vimage_commute[symmetric, of _ _ "space (M' j)"])

             (auto dest: measurable_space simp: sigma_sets_eq)

      qed

      from bchoice[OF this] obtain Y where

        Y: "\<And>j. j \<in> J \<Longrightarrow> Y' j = X j -` Y j \<inter> space M" "\<And>j. j \<in> J \<Longrightarrow> Y j \<in> sets (M' j)" by auto

      let ?E = "prod_emb I M' J (Pi\<^isub>E J Y)"

      from Y have "(\<Inter>j\<in>J. Y' j) = ?X -` ?E \<inter> space M"

        using J `I \<noteq> {}` measurable_space[OF rv] by (auto simp: prod_emb_def PiE_iff split: split_if_asm)

      then have "emeasure M (\<Inter>j\<in>J. Y' j) = emeasure M (?X -` ?E \<inter> space M)"

        by simp

      also have "\<dots> = emeasure ?D ?E"

        using Y  J by (intro emeasure_distr[symmetric] X sets_PiM_I) auto

      also have "\<dots> = emeasure ?P' ?E"

        using `?D = ?P'` by simp

      also have "\<dots> = (\<Prod> i\<in>J. emeasure (?D' i) (Y i))"

        using P.emeasure_PiM_emb[of J Y] J Y by (simp add: prod_emb_def)

      also have "\<dots> = (\<Prod> i\<in>J. emeasure M (Y' i))"

        using rv J Y by (simp add: emeasure_distr)

      finally have "emeasure M (\<Inter>j\<in>J. Y' j) = (\<Prod> i\<in>J. emeasure M (Y' i))" .

      then show "prob (\<Inter>j\<in>J. Y' j) = (\<Prod> i\<in>J. prob (Y' i))"

        by (auto simp: emeasure_eq_measure setprod_ereal)

    qed

  qed

qed

lemma (in prob_space) indep_varD:

  assumes indep: "indep_var Ma A Mb B"

  assumes sets: "Xa \<in> sets Ma" "Xb \<in> sets Mb"

  shows "prob ((\<lambda>x. (A x, B x)) -` (Xa \<times> Xb) \<inter> space M) =

    prob (A -` Xa \<inter> space M) * prob (B -` Xb \<inter> space M)"

proof -

  have "prob ((\<lambda>x. (A x, B x)) -` (Xa \<times> Xb) \<inter> space M) =

    prob (\<Inter>i\<in>UNIV. (bool_case A B i -` bool_case Xa Xb i \<inter> space M))"

    by (auto intro!: arg_cong[where f=prob] simp: UNIV_bool)

  also have "\<dots> = (\<Prod>i\<in>UNIV. prob (bool_case A B i -` bool_case Xa Xb i \<inter> space M))"

    using indep unfolding indep_var_def

    by (rule indep_varsD) (auto split: bool.split intro: sets)

  also have "\<dots> = prob (A -` Xa \<inter> space M) * prob (B -` Xb \<inter> space M)"

    unfolding UNIV_bool by simp

  finally show ?thesis .

qed

lemma (in prob_space)

  assumes "indep_var S X T Y"

  shows indep_var_rv1: "random_variable S X"

    and indep_var_rv2: "random_variable T Y"

proof -

  have "\<forall>i\<in>UNIV. random_variable (bool_case S T i) (bool_case X Y i)"

    using assms unfolding indep_var_def indep_vars_def by auto

  then show "random_variable S X" "random_variable T Y"

    unfolding UNIV_bool by auto

qed

lemma (in prob_space) indep_var_distribution_eq:

  "indep_var S X T Y \<longleftrightarrow> random_variable S X \<and> random_variable T Y \<and>

    distr M S X \<Otimes>\<^isub>M distr M T Y = distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))" (is "_ \<longleftrightarrow> _ \<and> _ \<and> ?S \<Otimes>\<^isub>M ?T = ?J")

proof safe

  assume "indep_var S X T Y"

  then show rvs: "random_variable S X" "random_variable T Y"

    by (blast dest: indep_var_rv1 indep_var_rv2)+

  then have XY: "random_variable (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"

    by (rule measurable_Pair)

  interpret X: prob_space ?S by (rule prob_space_distr) fact

  interpret Y: prob_space ?T by (rule prob_space_distr) fact

  interpret XY: pair_prob_space ?S ?T ..

  show "?S \<Otimes>\<^isub>M ?T = ?J"

  proof (rule pair_measure_eqI)

    show "sigma_finite_measure ?S" ..

    show "sigma_finite_measure ?T" ..

    fix A B assume A: "A \<in> sets ?S" and B: "B \<in> sets ?T"

    have "emeasure ?J (A \<times> B) = emeasure M ((\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M)"

      using A B by (intro emeasure_distr[OF XY]) auto

    also have "\<dots> = emeasure M (X -` A \<inter> space M) * emeasure M (Y -` B \<inter> space M)"

      using indep_varD[OF `indep_var S X T Y`, of A B] A B by (simp add: emeasure_eq_measure)

    also have "\<dots> = emeasure ?S A * emeasure ?T B"

      using rvs A B by (simp add: emeasure_distr)

    finally show "emeasure ?S A * emeasure ?T B = emeasure ?J (A \<times> B)" by simp

  qed simp

next

  assume rvs: "random_variable S X" "random_variable T Y"

  then have XY: "random_variable (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"

    by (rule measurable_Pair)

  let ?S = "distr M S X" and ?T = "distr M T Y"

  interpret X: prob_space ?S by (rule prob_space_distr) fact

  interpret Y: prob_space ?T by (rule prob_space_distr) fact

  interpret XY: pair_prob_space ?S ?T ..

  assume "?S \<Otimes>\<^isub>M ?T = ?J"

  { fix S and X

    have "Int_stable {X -` A \<inter> space M |A. A \<in> sets S}"

    proof (safe intro!: Int_stableI)

      fix A B assume "A \<in> sets S" "B \<in> sets S"

      then show "\<exists>C. (X -` A \<inter> space M) \<inter> (X -` B \<inter> space M) = (X -` C \<inter> space M) \<and> C \<in> sets S"

        by (intro exI[of _ "A \<inter> B"]) auto

    qed }

  note Int_stable = this