src/HOL/Probability/Independent_Family.thy

more canonical names

(*  Title:      HOL/Probability/Independent_Family.thy
    Author:     Johannes Hölzl, TU München
    Author:     Sudeep Kanav, TU München
*)

section \<open>Independent families of events, event sets, and random variables\<close>

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 (case_bool 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

lemma (in prob_space) indep_eventsI:
  "(\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M) \<Longrightarrow> (\<And>J. J \<subseteq> I \<Longrightarrow> finite J \<Longrightarrow> J \<noteq> {} \<Longrightarrow> prob (\<Inter>i\<in>J. F i) = (\<Prod>i\<in>J. prob (F i))) \<Longrightarrow> indep_events F I"
  by (auto simp: indep_events_def)

definition (in prob_space)
  "indep_event A B \<longleftrightarrow> indep_events (case_bool 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 \<open>J \<noteq> {}\<close> 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 "case_bool 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 \<open>j \<in> J\<close> \<open>A j = X\<close> by (auto intro!: arg_cong[where f=prob] split: if_split_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 \<open>J \<noteq> {j}\<close> \<open>j \<in> J\<close> 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 \<open>A j = X\<close> by simp
              also have "\<dots> = (\<Prod>i\<in>J. prob (A i))"
                unfolding setprod.insert_remove[OF \<open>finite J\<close>, symmetric, of "\<lambda>i. prob  (A i)"]
                using \<open>j \<in> J\<close> 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: if_split_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.sets_into_space[of _ M] X \<open>J \<noteq> {}\<close>
            by (auto intro!: arg_cong[where f=prob] split: if_split_asm)
          also have "\<dots> = prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)"
            using J \<open>J \<noteq> {}\<close> \<open>j \<notin> J\<close> A_sets X sets.sets_into_space
            by (auto intro!: finite_measure_Diff sets.finite_INT split: if_split_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 \<open>finite J\<close> 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 \<open>j \<in> K\<close> 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 \<open>finite J\<close> \<open>j \<notin> J\<close> 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 \<open>J \<noteq> {}\<close> \<open>j \<notin> J\<close> \<open>j \<in> K\<close> by (auto intro!: arg_cong[where f=prob] split: if_split_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 \<open>finite J\<close> \<open>J \<noteq> {}\<close> \<open>j \<notin> J\<close> by (auto intro!: sets.Int)
          qed
          moreover { fix k
            from J A \<open>j \<in> K\<close> 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: if_split_asm)
            also have "\<dots> = prob (F k) * prob (\<Inter>i\<in>J. A i)"
              using J A \<open>j \<in> K\<close> 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 \<open>j \<in> K\<close> 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.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 \<open>j \<in> K\<close> by auto
        from \<open>indep_sets G K\<close>
        show "indep_sets (G(j := {X})) K"
          by (rule indep_sets_mono_sets) (insert \<open>X \<in> G j\<close>, 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: if_split_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 \<open>j \<in> K\<close> \<open>j \<notin> J\<close>, auto simp: G_def)
    qed (insert \<open>indep_sets F K\<close>, simp) }
  from this[OF \<open>indep_sets F J\<close> \<open>finite J\<close> 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.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 (case_bool Ma Mb) (case_bool 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 (case_bool A B) UNIV"
  { fix a b assume "a \<in> A" "b \<in> B"
    with indep_setsD[OF indep, of UNIV "case_bool 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 (case_bool 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 (case_bool A B) UNIV"
      by (rule \<open>indep_set A B\<close>[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_eventsI_indep_vars:
  assumes indep: "indep_vars N X I"
  assumes P: "\<And>i. i \<in> I \<Longrightarrow> {x\<in>space (N i). P i x} \<in> sets (N i)"
  shows "indep_events (\<lambda>i. {x\<in>space M. P i (X i x)}) I"
proof -
  have "indep_sets (\<lambda>i. {X i -` A \<inter> space M |A. A \<in> sets (N i)}) I"
    using indep unfolding indep_vars_def2 by auto
  then show ?thesis
    unfolding indep_events_def_alt
  proof (rule indep_sets_mono_sets)
    fix i assume "i \<in> I"
    then have "{{x \<in> space M. P i (X i x)}} = {X i -` {x\<in>space (N i). P i x} \<inter> space M}"
      using indep by (auto simp: indep_vars_def dest: measurable_space)
    also have "\<dots> \<subseteq> {X i -` A \<inter> space M |A. A \<in> sets (N i)}"
      using P[OF \<open>i \<in> I\<close>] by blast
    finally show "{{x \<in> space M. P i (X i x)}} \<subseteq> {X i -` A \<inter> space M |A. A \<in> sets (N i)}" .
  qed
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.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!: sets.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 \<open>K \<subseteq> J\<close> \<open>k \<in> K\<close> \<open>j \<in> K\<close> have "I k \<inter> I j = {}"
            unfolding disjoint_family_on_def by auto
          with L(2,3)[OF \<open>j \<in> K\<close>] L(2,3)[OF \<open>k \<in> K\<close>]
          show False using \<open>l \<in> L k\<close> \<open>l \<in> L j\<close> 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.UNION_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 ?f = "\<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 "Ka \<union> Kb = (Ka \<inter> Kb) \<union> (Kb - Ka) \<union> (Ka - Kb)"
        by blast
      moreover have "(\<Inter>x\<in>Ka \<inter> Kb. Ea x \<inter> Eb x) \<inter>
        (\<Inter>x\<in>Kb - Ka. Eb x) \<inter> (\<Inter>x\<in>Ka - Kb. Ea x) = (\<Inter>k\<in>Ka. Ea k) \<inter> (\<Inter>k\<in>Kb. Eb k)"
        by auto
      ultimately have "(\<Inter>k\<in>Ka \<union> Kb. ?f k) = (\<Inter>k\<in>Ka. Ea k) \<inter> (\<Inter>k\<in>Kb. Eb k)" (is "?lhs = ?rhs")
        by (simp only: image_Un Inter_Un_distrib) simp
      then have "a \<inter> b = (\<Inter>k\<in>Ka \<union> Kb. ?f k)"
        by (simp only: a(1) b(1))
      with a b \<open>j \<in> J\<close> Int_stableD[OF Int_stable] show "a \<inter> b \<in> ?E j"
        by (intro CollectI exI[of _ "Ka \<union> Kb"] exI[of _ ?f]) auto
    qed
  qed
  ultimately show ?thesis
    by (simp cong: indep_sets_cong)
qed

lemma (in prob_space) indep_vars_restrict:
  assumes ind: "indep_vars M' X I" and K: "\<And>j. j \<in> L \<Longrightarrow> K j \<subseteq> I" and J: "disjoint_family_on K L"
  shows "indep_vars (\<lambda>j. PiM (K j) M') (\<lambda>j \<omega>. restrict (\<lambda>i. X i \<omega>) (K j)) L"
  unfolding indep_vars_def
proof safe
  fix j assume "j \<in> L" then show "random_variable (Pi\<^sub>M (K j) M') (\<lambda>\<omega>. \<lambda>i\<in>K j. X i \<omega>)"
    using K ind by (auto simp: indep_vars_def intro!: measurable_restrict)
next
  have X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable M (M' i)"
    using ind by (auto simp: indep_vars_def)
  let ?proj = "\<lambda>j S. {(\<lambda>\<omega>. \<lambda>i\<in>K j. X i \<omega>) -` A \<inter> space M |A. A \<in> S}"
  let ?UN = "\<lambda>j. sigma_sets (space M) (\<Union>i\<in>K j. { X i -` A \<inter> space M| A. A \<in> sets (M' i) })"
  show "indep_sets (\<lambda>i. sigma_sets (space M) (?proj i (sets (Pi\<^sub>M (K i) M')))) L"
  proof (rule indep_sets_mono_sets)
    fix j assume j: "j \<in> L"
    have "sigma_sets (space M) (?proj j (sets (Pi\<^sub>M (K j) M'))) = 
      sigma_sets (space M) (sigma_sets (space M) (?proj j (prod_algebra (K j) M')))"
      using j K X[THEN measurable_space] unfolding sets_PiM
      by (subst sigma_sets_vimage_commute) (auto simp add: Pi_iff)
    also have "\<dots> = sigma_sets (space M) (?proj j (prod_algebra (K j) M'))"
      by (rule sigma_sets_sigma_sets_eq) auto
    also have "\<dots> \<subseteq> ?UN j"
    proof (rule sigma_sets_mono, safe del: disjE elim!: prod_algebraE)
      fix J E assume J: "finite J" "J \<noteq> {} \<or> K j = {}"  "J \<subseteq> K j" and E: "\<forall>i. i \<in> J \<longrightarrow> E i \<in> sets (M' i)"
      show "(\<lambda>\<omega>. \<lambda>i\<in>K j. X i \<omega>) -` prod_emb (K j) M' J (Pi\<^sub>E J E) \<inter> space M \<in> ?UN j"
      proof cases
        assume "K j = {}" with J show ?thesis
          by (auto simp add: sigma_sets_empty_eq prod_emb_def)
      next
        assume "K j \<noteq> {}" with J have "J \<noteq> {}"
          by auto
        { interpret sigma_algebra "space M" "?UN j"
            by (rule sigma_algebra_sigma_sets) auto 
          have "\<And>A. (\<And>i. i \<in> J \<Longrightarrow> A i \<in> ?UN j) \<Longrightarrow> INTER J A \<in> ?UN j"
            using \<open>finite J\<close> \<open>J \<noteq> {}\<close> by (rule finite_INT) blast }
        note INT = this

        from \<open>J \<noteq> {}\<close> J K E[rule_format, THEN sets.sets_into_space] j
        have "(\<lambda>\<omega>. \<lambda>i\<in>K j. X i \<omega>) -` prod_emb (K j) M' J (Pi\<^sub>E J E) \<inter> space M
          = (\<Inter>i\<in>J. X i -` E i \<inter> space M)"
          apply (subst prod_emb_PiE[OF _ ])
          apply auto []
          apply auto []
          apply (auto simp add: Pi_iff intro!: X[THEN measurable_space])
          apply (erule_tac x=i in ballE)
          apply auto
          done
        also have "\<dots> \<in> ?UN j"
          apply (rule INT)
          apply (rule sigma_sets.Basic)
          using \<open>J \<subseteq> K j\<close> E
          apply auto
          done
        finally show ?thesis .
      qed
    qed
    finally show "sigma_sets (space M) (?proj j (sets (Pi\<^sub>M (K j) M'))) \<subseteq> ?UN j" .
  next
    show "indep_sets ?UN L"
    proof (rule indep_sets_collect_sigma)
      show "indep_sets (\<lambda>i. {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) (\<Union>j\<in>L. K j)"
      proof (rule indep_sets_mono_index)
        show "indep_sets (\<lambda>i. {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"
          using ind unfolding indep_vars_def2 by auto
        show "(\<Union>l\<in>L. K l) \<subseteq> I"
          using K by auto
      qed
    next
      fix l i assume "l \<in> L" "i \<in> K l"
      show "Int_stable {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
        apply (auto simp: Int_stable_def)
        apply (rule_tac x="A \<inter> Aa" in exI)
        apply auto
        done
    qed fact
  qed
qed

lemma (in prob_space) indep_var_restrict:
  assumes ind: "indep_vars M' X I" and AB: "A \<inter> B = {}" "A \<subseteq> I" "B \<subseteq> I"
  shows "indep_var (PiM A M') (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) A) (PiM B M') (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) B)"
proof -
  have *:
    "case_bool (Pi\<^sub>M A M') (Pi\<^sub>M B M') = (\<lambda>b. PiM (case_bool A B b) M')"
    "case_bool (\<lambda>\<omega>. \<lambda>i\<in>A. X i \<omega>) (\<lambda>\<omega>. \<lambda>i\<in>B. X i \<omega>) = (\<lambda>b \<omega>. \<lambda>i\<in>case_bool A B b. X i \<omega>)"
    by (simp_all add: fun_eq_iff split: bool.split)
  show ?thesis
    unfolding indep_var_def * using AB
    by (intro indep_vars_restrict[OF ind]) (auto simp: disjoint_family_on_def split: bool.split)
qed