(*  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

 definition (in prob_space)

   "indep_events A I \<longleftrightarrow> (A`I \<subseteq> sets M) \<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> sets M) \<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_rv 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"

 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_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> sets M"

     and indep_setD_ev2: "B \<subseteq> sets M"

   using indep unfolding indep_set_def indep_sets_def UNIV_bool by auto

 lemma dynkin_systemI':

   assumes 1: "\<And> A. A \<in> sets M \<Longrightarrow> A \<subseteq> space M"

   assumes empty: "{} \<in> sets M"

   assumes Diff: "\<And> A. A \<in> sets M \<Longrightarrow> space M - A \<in> sets M"

   assumes 2: "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> sets M

           \<Longrightarrow> (\<Union>i::nat. A i) \<in> sets M"

   shows "dynkin_system M"

 proof -

   from Diff[OF empty] have "space M \<in> sets M" by auto

   from 1 this Diff 2 show ?thesis

     by (intro dynkin_systemI) auto

 qed

 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, 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, 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_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> sets M"

     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> sets M"

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

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

   shows "X \<in> sets M"

 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> sets M"

     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> sets M"

   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> sets M. 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> sets M"

     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 mult_right.sums 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

 end