(*  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

 begin

 lemma INT_decseq_offset:

   assumes "decseq F"

   shows "(\<Inter>i. F i) = (\<Inter>i\<in>{n..}. F i)"

 proof safe

   fix x i assume x: "x \<in> (\<Inter>i\<in>{n..}. F i)"

   show "x \<in> F i"

   proof cases

     from x have "x \<in> F n" by auto

     also assume "i \<le> n" with `decseq F` have "F n \<subseteq> F i"

       unfolding decseq_def by simp

     finally show ?thesis .

   qed (insert x, simp)

 qed auto

 definition (in prob_space)

   "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)))"

 definition (in prob_space)

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

 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_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"

 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_sets_cong[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_sets_singleton_iff_indep_events:

   "indep_sets (\<lambda>i. {F i}) I \<longleftrightarrow> indep_events F I"

   unfolding indep_sets_def indep_events_def

   by (simp, intro conj_cong ball_cong all_cong imp_cong) (auto simp: Pi_iff)

 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_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) 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)

   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. sets (dynkin \<lparr> space = space M, sets = F i \<rparr>)) 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 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 \<lparr> space = space M, sets = ?D \<rparr>"

       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 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: "sets (dynkin \<lparr>space = space M, sets = G j\<rparr>) \<subseteq>

         sets \<lparr>space = space M, sets = {E \<in> events. indep_sets (G(j := {E})) K}\<rparr>"

       proof (rule dynkin_system.dynkin_subset, simp_all cong del: indep_sets_cong, 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:=sets (dynkin \<lparr>space = space M, sets = G j\<rparr>))) 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 (\<lambda>i. sets (dynkin \<lparr> space = space M, sets = F i \<rparr>)) 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 \<lparr> space = space M, sets = F i \<rparr>"

   shows "indep_sets (\<lambda>i. sets (sigma \<lparr> space = space M, sets = F i \<rparr>)) 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:

   assumes "indep_sets F I"

   assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable \<lparr> space = space M, sets = F i \<rparr>"

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

   using indep_sets_sigma[OF assms] by (simp add: sets_sigma)

 lemma (in prob_space) indep_sets_sigma_sets_iff:

   assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable \<lparr> space = space M, sets = F i \<rparr>"

   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_sets) 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

 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 \<lparr> space = space M, sets = A \<rparr>"

   assumes B: "Int_stable \<lparr> space = space M, sets = B \<rparr>"

   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_sets)

     show "indep_sets (bool_case A B) UNIV"

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

     fix i show "Int_stable \<lparr>space = space M, sets = case i of True \<Rightarrow> A | False \<Rightarrow> B\<rparr>"

       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 \<lparr>space = space M, sets = E i\<rparr>"

   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 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 \<lparr> space = space M, sets = (\<Union>i\<in>I j. E i) \<rparr>"

     assume "j \<in> J"

     from E[OF this] interpret S: sigma_algebra ?S

       using sets_into_space by (intro sigma_algebra_sigma) 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 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> sets ?S" unfolding A

         by (safe intro!: S.finite_INT)

            (auto simp: sets_sigma intro!: sigma_sets.Basic)

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

         by (simp add: sets_sigma)

     qed }

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

   proof (rule indep_sets_sigma_sets)

     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' 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 \<lparr> space = space M, sets = ?E j \<rparr>"

     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) terminal_events where

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

 lemma (in prob_space) terminal_events_sets:

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

   assumes "\<And>i::nat. sigma_algebra \<lparr>space = space M, sets = A i\<rparr>"

   assumes X: "X \<in> terminal_events A"

   shows "X \<in> events"

 proof -

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

   interpret A: sigma_algebra "\<lparr>space = space M, sets = A i\<rparr>" for i by fact

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

   assumes "\<And>i::nat. sigma_algebra \<lparr>space = space M, sets = A i\<rparr>"

   shows "sigma_algebra \<lparr> space = space M, sets = terminal_events A \<rparr>"

   unfolding terminal_events_def

 proof (simp add: sigma_algebra_iff2, safe)

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

   interpret A: sigma_algebra "\<lparr>space = space M, sets = A i\<rparr>" 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 A: "\<And>i. A i \<subseteq> events"

   assumes "\<And>i::nat. sigma_algebra \<lparr>space = space M, sets = A i\<rparr>"

   assumes indep: "indep_sets A UNIV"

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

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

 proof -

   let ?D = "\<lparr> space = space M, sets = {D \<in> events. prob (X \<inter> D) = prob X * prob D} \<rparr>"

   interpret A: sigma_algebra "\<lparr>space = space M, sets = A i\<rparr>" for i by fact

   interpret T: sigma_algebra "\<lparr> space = space M, sets = terminal_events A \<rparr>"

     by (rule sigma_algebra_terminal_events) fact

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

   have X_in: "X \<in> events"

     by (rule terminal_events_sets) fact+

   interpret D: dynkin_system ?D

   proof (rule dynkin_systemI)

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

       using sets_into_space by auto

   next

     show "space ?D \<in> sets ?D"

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

   next

     fix A assume A: "A \<in> sets ?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 ?D - A \<in> sets ?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> sets ?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> sets ?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 \<lparr>space = space M, sets = A m\<rparr>"

         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> sets ?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 terminal_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> sets ?D" (is "?A \<subseteq> _")

     by auto

   have "sigma \<lparr> space = space M, sets = ?A \<rparr> =

     dynkin \<lparr> space = space M, sets = ?A \<rparr>" (is "sigma ?UA = dynkin ?UA")

   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, auto) }

     then show "sets ?UA \<subseteq> Pow (space ?UA)" by auto

     show "Int_stable ?UA"

     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 "sigma \<lparr>space = space M, sets = (\<Union>i\<in>{..max m n}. A i)\<rparr>"

         using A sets_into_space by (intro sigma_algebra_sigma) 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 add: sets_sigma)

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

     qed

   qed

   moreover have "sets (dynkin ?UA) \<subseteq> sets ?D"

   proof (rule D.dynkin_subset)

     show "sets ?UA \<subseteq> sets ?D" using `?A \<subseteq> sets ?D` by auto

   qed simp

   ultimately have "sets (sigma ?UA) \<subseteq> sets ?D" by simp

   moreover

   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 intro: sigma_sets.Basic)

   then have "terminal_events A \<subseteq> sets (sigma ?UA)"

     unfolding sets_sigma terminal_events_def by auto

   moreover note `X \<in> terminal_events A`

   ultimately have "X \<in> sets ?D" by auto

   then show ?thesis by auto

 qed

 lemma (in prob_space) borel_0_1_law:

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

   assumes F: "range F \<subseteq> events" "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 }"])

   show "\<And>i. sigma_sets (space M) {F i} \<subseteq> events"

     using F(1) sets_into_space

     by (subst sigma_sets_singleton) auto

   { fix i show "sigma_algebra \<lparr>space = space M, sets = sigma_sets (space M) {F i}\<rparr>"

       using sigma_algebra_sigma[of "\<lparr>space = space M, sets = {F i}\<rparr>"] F sets_into_space

       by (auto simp add: sigma_def) }

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

   proof (rule indep_sets_sigma_sets)

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

       unfolding indep_sets_singleton_iff_indep_events by fact

     fix i show "Int_stable \<lparr>space = space M, sets = {F i}\<rparr>"

       unfolding Int_stable_def by simp

   qed

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

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

     unfolding terminal_events_def

   proof

     fix j

     interpret S: sigma_algebra "sigma \<lparr> space = space M, sets = (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})\<rparr>"

       using order_trans[OF F(1) space_closed]

       by (intro sigma_algebra_sigma) (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> sets (sigma \<lparr> space = space M, sets = (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})\<rparr>)"

       using order_trans[OF F(1) space_closed]

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

          (auto simp: sets_sigma 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 add: sets_sigma)

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

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

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

   shows "indep_vars (\<lambda>i. sigma (M' i)) X I \<longleftrightarrow>

     (\<forall>A\<in>(\<Pi> i\<in>I. sets (M' 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"

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

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

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

       by (subst sigma_sets_sigma_sets_eq) auto }

   note this[simp]

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

     have "Int_stable \<lparr>space = space M, sets = {X i -` A \<inter> space M |A. A \<in> sets (M' i)}\<rparr>"

     proof (rule Int_stableI)

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

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

       moreover

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

       then obtain B where "b = X i -` B \<inter> space M" "B \<in> sets (M' 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> sets (M' i)}"

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

     qed }

   note indep_sets_sigma_sets_iff[OF this, simp]

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

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

       then have "A \<in> sets (sigma (M' i))" by (auto simp: sets_sigma intro: sigma_sets.Basic)

       moreover

       from rv[OF `i\<in>I`] have "X i \<in> measurable M (sigma (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> sets (M' i)} \<subseteq> events"

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

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

   note indep_sets_finite[OF I this, simp]

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

     (\<forall>A\<in>\<Pi> i\<in>I. sets (M' 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. sets (M' 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> sets (M' i)})"

     from A have "\<forall>i\<in>I. \<exists>B. A i = X i -` B \<inter> space M \<and> B \<in> sets (M' 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. sets (M' 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)

 qed

 lemma (in prob_space) indep_vars_compose:

   assumes "indep_vars M' X I"

   assumes rv:

     "\<And>i. i \<in> I \<Longrightarrow> sigma_algebra (N i)"

     "\<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 intro!: measurable_comp 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:

   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 intro: sigma_sets.Basic)

 qed

 lemma (in prob_space) indep_distribution_eq_measure:

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

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

   shows "indep_vars M' X I \<longleftrightarrow>

     (\<forall>A\<in>sets (\<Pi>\<^isub>M i\<in>I. (M' i \<lparr> measure := ereal\<circ>distribution (X i) \<rparr>)).

       distribution (\<lambda>x. \<lambda>i\<in>I. X i x) A =

       finite_measure.\<mu>' (\<Pi>\<^isub>M i\<in>I. (M' i \<lparr> measure := ereal\<circ>distribution (X i) \<rparr>)) A)"

     (is "_ \<longleftrightarrow> (\<forall>X\<in>_. distribution ?D X = finite_measure.\<mu>' (Pi\<^isub>M I ?M) X)")

 proof -

   interpret M': prob_space "?M i" for i

     using rv by (rule distribution_prob_space)

   interpret P: finite_product_prob_space ?M I

     proof qed fact

   let ?D' = "(Pi\<^isub>M I ?M) \<lparr> measure := ereal \<circ> distribution ?D \<rparr>"

   have "random_variable P.P ?D"

     using `finite I` rv by (intro random_variable_restrict) auto

   then interpret D: prob_space ?D'

     by (rule distribution_prob_space)

   show ?thesis

   proof (intro iffI ballI)

     assume "indep_vars M' X I"

     fix A assume "A \<in> sets P.P"

     moreover

     have "D.prob A = P.prob A"

     proof (rule prob_space_unique_Int_stable)

       show "prob_space ?D'" by default

       show "prob_space (Pi\<^isub>M I ?M)" by default

       show "Int_stable P.G" using M'.Int

         by (intro Int_stable_product_algebra_generator) (simp add: Int_stable_def)

       show "space P.G \<in> sets P.G"

         using M'.top by (simp add: product_algebra_generator_def)

       show "space ?D' = space P.G"  "sets ?D' = sets (sigma P.G)"

         by (simp_all add: product_algebra_def product_algebra_generator_def sets_sigma)

       show "space P.P = space P.G" "sets P.P = sets (sigma P.G)"

         by (simp_all add: product_algebra_def)

       show "A \<in> sets (sigma P.G)"

         using `A \<in> sets P.P` by (simp add: product_algebra_def)

       fix E assume E: "E \<in> sets P.G"

       then have "E \<in> sets P.P"

         by (simp add: sets_sigma sigma_sets.Basic product_algebra_def)

       then have "D.prob E = distribution ?D E"

         unfolding D.\<mu>'_def by simp

       also

       from E obtain F where "E = Pi\<^isub>E I F" and F: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets (M' i)"

         by (auto simp: product_algebra_generator_def)

       with `I \<noteq> {}` have "distribution ?D E = prob (\<Inter>i\<in>I. X i -` F i \<inter> space M)"

         using `I \<noteq> {}` by (auto intro!: arg_cong[where f=prob] simp: Pi_iff distribution_def)

       also have "\<dots> = (\<Prod>i\<in>I. prob (X i -` F i \<inter> space M))"

         using `indep_vars M' X I` I F by (rule indep_varsD)

       also have "\<dots> = P.prob E"

         using F by (simp add: `E = Pi\<^isub>E I F` P.prob_times M'.\<mu>'_def distribution_def)

       finally show "D.prob E = P.prob E" .

     qed

     ultimately show "distribution ?D A = P.prob A"

       by (simp add: D.\<mu>'_def)

   next

     assume eq: "\<forall>A\<in>sets P.P. distribution ?D A = P.prob A"

     have [simp]: "\<And>i. sigma (M' i) = M' i"

       using rv by (intro sigma_algebra.sigma_eq) simp

     have "indep_vars (\<lambda>i. sigma (M' i)) X I"

     proof (subst indep_vars_finite[OF I])

       fix i assume [simp]: "i \<in> I"

       show "random_variable (sigma (M' i)) (X i)"

         using rv[of i] by simp

       show "Int_stable (M' i)" "space (M' i) \<in> sets (M' i)"

         using M'.Int[of _ i] M'.top by (auto simp: Int_stable_def)

     next

       show "\<forall>A\<in>\<Pi> i\<in>I. sets (M' 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

         fix A assume A: "A \<in> (\<Pi> i\<in>I. sets (M' i))"

         then have A_in_P: "(Pi\<^isub>E I A) \<in> sets P.P"

           by (auto intro!: product_algebraI)

         have "prob (\<Inter>j\<in>I. X j -` A j \<inter> space M) = distribution ?D (Pi\<^isub>E I A)"

           using `I \<noteq> {}`by (auto intro!: arg_cong[where f=prob] simp: Pi_iff distribution_def)

         also have "\<dots> = P.prob (Pi\<^isub>E I A)" using A_in_P eq by simp

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

           using A by (intro P.prob_times) auto

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

           using A by (auto intro!: setprod_cong simp: M'.\<mu>'_def Pi_iff distribution_def)

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

       qed

     qed

     then show "indep_vars M' X I"

       by simp

   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_distributionD:

   assumes indep: "indep_var S X T Y"

   defines "P \<equiv> S\<lparr>measure := ereal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := ereal\<circ>distribution Y\<rparr>"

   assumes "A \<in> sets P"

   shows "joint_distribution X Y A = finite_measure.\<mu>' P A"

 proof -

   from indep have rvs: "random_variable S X" "random_variable T Y"

     by (blast dest: indep_var_rv1 indep_var_rv2)+

   let ?S = "S\<lparr>measure := ereal\<circ>distribution X\<rparr>"

   let ?T = "T\<lparr>measure := ereal\<circ>distribution Y\<rparr>"

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

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

   interpret XY: pair_prob_space ?S ?T by default

   let ?J = "XY.P\<lparr> measure := ereal \<circ> joint_distribution X Y \<rparr>"

   interpret J: prob_space ?J

     by (rule joint_distribution_prob_space) (simp_all add: rvs)

   have "finite_measure.\<mu>' (XY.P\<lparr> measure := ereal \<circ> joint_distribution X Y \<rparr>) A = XY.\<mu>' A"

   proof (rule prob_space_unique_Int_stable)

     show "Int_stable (pair_measure_generator ?S ?T)" (is "Int_stable ?P")

       by fact

     show "space ?P \<in> sets ?P"

       unfolding space_pair_measure[simplified pair_measure_def space_sigma]

       using X.top Y.top by (auto intro!: pair_measure_generatorI)

     show "prob_space ?J" by default

     show "space ?J = space ?P"

       by (simp add: pair_measure_generator_def space_pair_measure)

     show "sets ?J = sets (sigma ?P)"

       by (simp add: pair_measure_def)

     show "prob_space XY.P" by default

     show "space XY.P = space ?P" "sets XY.P = sets (sigma ?P)"

       by (simp_all add: pair_measure_generator_def pair_measure_def)

     show "A \<in> sets (sigma ?P)"

       using `A \<in> sets P` unfolding P_def pair_measure_def by simp

     fix X assume "X \<in> sets ?P"

     then obtain A B where "A \<in> sets S" "B \<in> sets T" "X = A \<times> B"

       by (auto simp: sets_pair_measure_generator)

     then show "J.\<mu>' X = XY.\<mu>' X"

       unfolding J.\<mu>'_def XY.\<mu>'_def using indep

       by (simp add: XY.pair_measure_times)

          (simp add: distribution_def indep_varD)

   qed

   then show ?thesis

     using `A \<in> sets P` unfolding P_def J.\<mu>'_def XY.\<mu>'_def by simp

 qed

 end