src/HOL/Probability/Independent_Family.thy
author haftmann
Sat Jun 28 09:16:42 2014 +0200 (2014-06-28)
changeset 57418 6ab1c7cb0b8d
parent 57235 b0b9a10e4bf4
child 57447 87429bdecad5
permissions -rw-r--r--
fact consolidation
     1 (*  Title:      HOL/Probability/Independent_Family.thy
     2     Author:     Johannes Hölzl, TU München
     3     Author:     Sudeep Kanav, TU München
     4 *)
     5 
     6 header {* Independent families of events, event sets, and random variables *}
     7 
     8 theory Independent_Family
     9   imports Probability_Measure Infinite_Product_Measure
    10 begin
    11 
    12 definition (in prob_space)
    13   "indep_sets F I \<longleftrightarrow> (\<forall>i\<in>I. F i \<subseteq> events) \<and>
    14     (\<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))))"
    15 
    16 definition (in prob_space)
    17   "indep_set A B \<longleftrightarrow> indep_sets (case_bool A B) UNIV"
    18 
    19 definition (in prob_space)
    20   indep_events_def_alt: "indep_events A I \<longleftrightarrow> indep_sets (\<lambda>i. {A i}) I"
    21 
    22 lemma (in prob_space) indep_events_def:
    23   "indep_events A I \<longleftrightarrow> (A`I \<subseteq> events) \<and>
    24     (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j)))"
    25   unfolding indep_events_def_alt indep_sets_def
    26   apply (simp add: Ball_def Pi_iff image_subset_iff_funcset)
    27   apply (intro conj_cong refl arg_cong[where f=All] ext imp_cong)
    28   apply auto
    29   done
    30 
    31 definition (in prob_space)
    32   "indep_event A B \<longleftrightarrow> indep_events (case_bool A B) UNIV"
    33 
    34 lemma (in prob_space) indep_sets_cong:
    35   "I = J \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> F i = G i) \<Longrightarrow> indep_sets F I \<longleftrightarrow> indep_sets G J"
    36   by (simp add: indep_sets_def, intro conj_cong all_cong imp_cong ball_cong) blast+
    37 
    38 lemma (in prob_space) indep_events_finite_index_events:
    39   "indep_events F I \<longleftrightarrow> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_events F J)"
    40   by (auto simp: indep_events_def)
    41 
    42 lemma (in prob_space) indep_sets_finite_index_sets:
    43   "indep_sets F I \<longleftrightarrow> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J)"
    44 proof (intro iffI allI impI)
    45   assume *: "\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J"
    46   show "indep_sets F I" unfolding indep_sets_def
    47   proof (intro conjI ballI allI impI)
    48     fix i assume "i \<in> I"
    49     with *[THEN spec, of "{i}"] show "F i \<subseteq> events"
    50       by (auto simp: indep_sets_def)
    51   qed (insert *, auto simp: indep_sets_def)
    52 qed (auto simp: indep_sets_def)
    53 
    54 lemma (in prob_space) indep_sets_mono_index:
    55   "J \<subseteq> I \<Longrightarrow> indep_sets F I \<Longrightarrow> indep_sets F J"
    56   unfolding indep_sets_def by auto
    57 
    58 lemma (in prob_space) indep_sets_mono_sets:
    59   assumes indep: "indep_sets F I"
    60   assumes mono: "\<And>i. i\<in>I \<Longrightarrow> G i \<subseteq> F i"
    61   shows "indep_sets G I"
    62 proof -
    63   have "(\<forall>i\<in>I. F i \<subseteq> events) \<Longrightarrow> (\<forall>i\<in>I. G i \<subseteq> events)"
    64     using mono by auto
    65   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)"
    66     using mono by (auto simp: Pi_iff)
    67   ultimately show ?thesis
    68     using indep by (auto simp: indep_sets_def)
    69 qed
    70 
    71 lemma (in prob_space) indep_sets_mono:
    72   assumes indep: "indep_sets F I"
    73   assumes mono: "J \<subseteq> I" "\<And>i. i\<in>J \<Longrightarrow> G i \<subseteq> F i"
    74   shows "indep_sets G J"
    75   apply (rule indep_sets_mono_sets)
    76   apply (rule indep_sets_mono_index)
    77   apply (fact +)
    78   done
    79 
    80 lemma (in prob_space) indep_setsI:
    81   assumes "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events"
    82     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))"
    83   shows "indep_sets F I"
    84   using assms unfolding indep_sets_def by (auto simp: Pi_iff)
    85 
    86 lemma (in prob_space) indep_setsD:
    87   assumes "indep_sets F I" and "J \<subseteq> I" "J \<noteq> {}" "finite J" "\<forall>j\<in>J. A j \<in> F j"
    88   shows "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
    89   using assms unfolding indep_sets_def by auto
    90 
    91 lemma (in prob_space) indep_setI:
    92   assumes ev: "A \<subseteq> events" "B \<subseteq> events"
    93     and indep: "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> prob (a \<inter> b) = prob a * prob b"
    94   shows "indep_set A B"
    95   unfolding indep_set_def
    96 proof (rule indep_setsI)
    97   fix F J assume "J \<noteq> {}" "J \<subseteq> UNIV"
    98     and F: "\<forall>j\<in>J. F j \<in> (case j of True \<Rightarrow> A | False \<Rightarrow> B)"
    99   have "J \<in> Pow UNIV" by auto
   100   with F `J \<noteq> {}` indep[of "F True" "F False"]
   101   show "prob (\<Inter>j\<in>J. F j) = (\<Prod>j\<in>J. prob (F j))"
   102     unfolding UNIV_bool Pow_insert by (auto simp: ac_simps)
   103 qed (auto split: bool.split simp: ev)
   104 
   105 lemma (in prob_space) indep_setD:
   106   assumes indep: "indep_set A B" and ev: "a \<in> A" "b \<in> B"
   107   shows "prob (a \<inter> b) = prob a * prob b"
   108   using indep[unfolded indep_set_def, THEN indep_setsD, of UNIV "case_bool a b"] ev
   109   by (simp add: ac_simps UNIV_bool)
   110 
   111 lemma (in prob_space)
   112   assumes indep: "indep_set A B"
   113   shows indep_setD_ev1: "A \<subseteq> events"
   114     and indep_setD_ev2: "B \<subseteq> events"
   115   using indep unfolding indep_set_def indep_sets_def UNIV_bool by auto
   116 
   117 lemma (in prob_space) indep_sets_dynkin:
   118   assumes indep: "indep_sets F I"
   119   shows "indep_sets (\<lambda>i. dynkin (space M) (F i)) I"
   120     (is "indep_sets ?F I")
   121 proof (subst indep_sets_finite_index_sets, intro allI impI ballI)
   122   fix J assume "finite J" "J \<subseteq> I" "J \<noteq> {}"
   123   with indep have "indep_sets F J"
   124     by (subst (asm) indep_sets_finite_index_sets) auto
   125   { fix J K assume "indep_sets F K"
   126     let ?G = "\<lambda>S i. if i \<in> S then ?F i else F i"
   127     assume "finite J" "J \<subseteq> K"
   128     then have "indep_sets (?G J) K"
   129     proof induct
   130       case (insert j J)
   131       moreover def G \<equiv> "?G J"
   132       ultimately have G: "indep_sets G K" "\<And>i. i \<in> K \<Longrightarrow> G i \<subseteq> events" and "j \<in> K"
   133         by (auto simp: indep_sets_def)
   134       let ?D = "{E\<in>events. indep_sets (G(j := {E})) K }"
   135       { fix X assume X: "X \<in> events"
   136         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)
   137           \<Longrightarrow> prob ((\<Inter>i\<in>J. A i) \<inter> X) = prob X * (\<Prod>i\<in>J. prob (A i))"
   138         have "indep_sets (G(j := {X})) K"
   139         proof (rule indep_setsI)
   140           fix i assume "i \<in> K" then show "(G(j:={X})) i \<subseteq> events"
   141             using G X by auto
   142         next
   143           fix A J assume J: "J \<noteq> {}" "J \<subseteq> K" "finite J" "\<forall>i\<in>J. A i \<in> (G(j := {X})) i"
   144           show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
   145           proof cases
   146             assume "j \<in> J"
   147             with J have "A j = X" by auto
   148             show ?thesis
   149             proof cases
   150               assume "J = {j}" then show ?thesis by simp
   151             next
   152               assume "J \<noteq> {j}"
   153               have "prob (\<Inter>i\<in>J. A i) = prob ((\<Inter>i\<in>J-{j}. A i) \<inter> X)"
   154                 using `j \<in> J` `A j = X` by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
   155               also have "\<dots> = prob X * (\<Prod>i\<in>J-{j}. prob (A i))"
   156               proof (rule indep)
   157                 show "J - {j} \<noteq> {}" "J - {j} \<subseteq> K" "finite (J - {j})" "j \<notin> J - {j}"
   158                   using J `J \<noteq> {j}` `j \<in> J` by auto
   159                 show "\<forall>i\<in>J - {j}. A i \<in> G i"
   160                   using J by auto
   161               qed
   162               also have "\<dots> = prob (A j) * (\<Prod>i\<in>J-{j}. prob (A i))"
   163                 using `A j = X` by simp
   164               also have "\<dots> = (\<Prod>i\<in>J. prob (A i))"
   165                 unfolding setprod.insert_remove[OF `finite J`, symmetric, of "\<lambda>i. prob  (A i)"]
   166                 using `j \<in> J` by (simp add: insert_absorb)
   167               finally show ?thesis .
   168             qed
   169           next
   170             assume "j \<notin> J"
   171             with J have "\<forall>i\<in>J. A i \<in> G i" by (auto split: split_if_asm)
   172             with J show ?thesis
   173               by (intro indep_setsD[OF G(1)]) auto
   174           qed
   175         qed }
   176       note indep_sets_insert = this
   177       have "dynkin_system (space M) ?D"
   178       proof (rule dynkin_systemI', simp_all cong del: indep_sets_cong, safe)
   179         show "indep_sets (G(j := {{}})) K"
   180           by (rule indep_sets_insert) auto
   181       next
   182         fix X assume X: "X \<in> events" and G': "indep_sets (G(j := {X})) K"
   183         show "indep_sets (G(j := {space M - X})) K"
   184         proof (rule indep_sets_insert)
   185           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"
   186           then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events"
   187             using G by auto
   188           have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) =
   189               prob ((\<Inter>j\<in>J. A j) - (\<Inter>i\<in>insert j J. (A(j := X)) i))"
   190             using A_sets sets.sets_into_space[of _ M] X `J \<noteq> {}`
   191             by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
   192           also have "\<dots> = prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)"
   193             using J `J \<noteq> {}` `j \<notin> J` A_sets X sets.sets_into_space
   194             by (auto intro!: finite_measure_Diff sets.finite_INT split: split_if_asm)
   195           finally have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) =
   196               prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)" .
   197           moreover {
   198             have "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
   199               using J A `finite J` by (intro indep_setsD[OF G(1)]) auto
   200             then have "prob (\<Inter>j\<in>J. A j) = prob (space M) * (\<Prod>i\<in>J. prob (A i))"
   201               using prob_space by simp }
   202           moreover {
   203             have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = (\<Prod>i\<in>insert j J. prob ((A(j := X)) i))"
   204               using J A `j \<in> K` by (intro indep_setsD[OF G']) auto
   205             then have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = prob X * (\<Prod>i\<in>J. prob (A i))"
   206               using `finite J` `j \<notin> J` by (auto intro!: setprod.cong) }
   207           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))"
   208             by (simp add: field_simps)
   209           also have "\<dots> = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))"
   210             using X A by (simp add: finite_measure_compl)
   211           finally show "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))" .
   212         qed (insert X, auto)
   213       next
   214         fix F :: "nat \<Rightarrow> 'a set" assume disj: "disjoint_family F" and "range F \<subseteq> ?D"
   215         then have F: "\<And>i. F i \<in> events" "\<And>i. indep_sets (G(j:={F i})) K" by auto
   216         show "indep_sets (G(j := {\<Union>k. F k})) K"
   217         proof (rule indep_sets_insert)
   218           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"
   219           then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events"
   220             using G by auto
   221           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))"
   222             using `J \<noteq> {}` `j \<notin> J` `j \<in> K` by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
   223           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))"
   224           proof (rule finite_measure_UNION)
   225             show "disjoint_family (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i)"
   226               using disj by (rule disjoint_family_on_bisimulation) auto
   227             show "range (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i) \<subseteq> events"
   228               using A_sets F `finite J` `J \<noteq> {}` `j \<notin> J` by (auto intro!: sets.Int)
   229           qed
   230           moreover { fix k
   231             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))"
   232               by (subst indep_setsD[OF F(2)]) (auto intro!: setprod.cong split: split_if_asm)
   233             also have "\<dots> = prob (F k) * prob (\<Inter>i\<in>J. A i)"
   234               using J A `j \<in> K` by (subst indep_setsD[OF G(1)]) auto
   235             finally have "prob (\<Inter>i\<in>insert j J. (A(j := F k)) i) = prob (F k) * prob (\<Inter>i\<in>J. A i)" . }
   236           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)))"
   237             by simp
   238           moreover
   239           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))"
   240             using disj F(1) by (intro finite_measure_UNION sums_mult2) auto
   241           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)))"
   242             using J A `j \<in> K` by (subst indep_setsD[OF G(1), symmetric]) auto
   243           ultimately
   244           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))"
   245             by (auto dest!: sums_unique)
   246         qed (insert F, auto)
   247       qed (insert sets.sets_into_space, auto)
   248       then have mono: "dynkin (space M) (G j) \<subseteq> {E \<in> events. indep_sets (G(j := {E})) K}"
   249       proof (rule dynkin_system.dynkin_subset, safe)
   250         fix X assume "X \<in> G j"
   251         then show "X \<in> events" using G `j \<in> K` by auto
   252         from `indep_sets G K`
   253         show "indep_sets (G(j := {X})) K"
   254           by (rule indep_sets_mono_sets) (insert `X \<in> G j`, auto)
   255       qed
   256       have "indep_sets (G(j:=?D)) K"
   257       proof (rule indep_setsI)
   258         fix i assume "i \<in> K" then show "(G(j := ?D)) i \<subseteq> events"
   259           using G(2) by auto
   260       next
   261         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"
   262         show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
   263         proof cases
   264           assume "j \<in> J"
   265           with A have indep: "indep_sets (G(j := {A j})) K" by auto
   266           from J A show ?thesis
   267             by (intro indep_setsD[OF indep]) auto
   268         next
   269           assume "j \<notin> J"
   270           with J A have "\<forall>i\<in>J. A i \<in> G i" by (auto split: split_if_asm)
   271           with J show ?thesis
   272             by (intro indep_setsD[OF G(1)]) auto
   273         qed
   274       qed
   275       then have "indep_sets (G(j := dynkin (space M) (G j))) K"
   276         by (rule indep_sets_mono_sets) (insert mono, auto)
   277       then show ?case
   278         by (rule indep_sets_mono_sets) (insert `j \<in> K` `j \<notin> J`, auto simp: G_def)
   279     qed (insert `indep_sets F K`, simp) }
   280   from this[OF `indep_sets F J` `finite J` subset_refl]
   281   show "indep_sets ?F J"
   282     by (rule indep_sets_mono_sets) auto
   283 qed
   284 
   285 lemma (in prob_space) indep_sets_sigma:
   286   assumes indep: "indep_sets F I"
   287   assumes stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)"
   288   shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I"
   289 proof -
   290   from indep_sets_dynkin[OF indep]
   291   show ?thesis
   292   proof (rule indep_sets_mono_sets, subst sigma_eq_dynkin, simp_all add: stable)
   293     fix i assume "i \<in> I"
   294     with indep have "F i \<subseteq> events" by (auto simp: indep_sets_def)
   295     with sets.sets_into_space show "F i \<subseteq> Pow (space M)" by auto
   296   qed
   297 qed
   298 
   299 lemma (in prob_space) indep_sets_sigma_sets_iff:
   300   assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)"
   301   shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I \<longleftrightarrow> indep_sets F I"
   302 proof
   303   assume "indep_sets F I" then show "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I"
   304     by (rule indep_sets_sigma) fact
   305 next
   306   assume "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I" then show "indep_sets F I"
   307     by (rule indep_sets_mono_sets) (intro subsetI sigma_sets.Basic)
   308 qed
   309 
   310 definition (in prob_space)
   311   indep_vars_def2: "indep_vars M' X I \<longleftrightarrow>
   312     (\<forall>i\<in>I. random_variable (M' i) (X i)) \<and>
   313     indep_sets (\<lambda>i. { X i -` A \<inter> space M | A. A \<in> sets (M' i)}) I"
   314 
   315 definition (in prob_space)
   316   "indep_var Ma A Mb B \<longleftrightarrow> indep_vars (case_bool Ma Mb) (case_bool A B) UNIV"
   317 
   318 lemma (in prob_space) indep_vars_def:
   319   "indep_vars M' X I \<longleftrightarrow>
   320     (\<forall>i\<in>I. random_variable (M' i) (X i)) \<and>
   321     indep_sets (\<lambda>i. sigma_sets (space M) { X i -` A \<inter> space M | A. A \<in> sets (M' i)}) I"
   322   unfolding indep_vars_def2
   323   apply (rule conj_cong[OF refl])
   324   apply (rule indep_sets_sigma_sets_iff[symmetric])
   325   apply (auto simp: Int_stable_def)
   326   apply (rule_tac x="A \<inter> Aa" in exI)
   327   apply auto
   328   done
   329 
   330 lemma (in prob_space) indep_var_eq:
   331   "indep_var S X T Y \<longleftrightarrow>
   332     (random_variable S X \<and> random_variable T Y) \<and>
   333     indep_set
   334       (sigma_sets (space M) { X -` A \<inter> space M | A. A \<in> sets S})
   335       (sigma_sets (space M) { Y -` A \<inter> space M | A. A \<in> sets T})"
   336   unfolding indep_var_def indep_vars_def indep_set_def UNIV_bool
   337   by (intro arg_cong2[where f="op \<and>"] arg_cong2[where f=indep_sets] ext)
   338      (auto split: bool.split)
   339 
   340 lemma (in prob_space) indep_sets2_eq:
   341   "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)"
   342   unfolding indep_set_def
   343 proof (intro iffI ballI conjI)
   344   assume indep: "indep_sets (case_bool A B) UNIV"
   345   { fix a b assume "a \<in> A" "b \<in> B"
   346     with indep_setsD[OF indep, of UNIV "case_bool a b"]
   347     show "prob (a \<inter> b) = prob a * prob b"
   348       unfolding UNIV_bool by (simp add: ac_simps) }
   349   from indep show "A \<subseteq> events" "B \<subseteq> events"
   350     unfolding indep_sets_def UNIV_bool by auto
   351 next
   352   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)"
   353   show "indep_sets (case_bool A B) UNIV"
   354   proof (rule indep_setsI)
   355     fix i show "(case i of True \<Rightarrow> A | False \<Rightarrow> B) \<subseteq> events"
   356       using * by (auto split: bool.split)
   357   next
   358     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)"
   359     then have "J = {True} \<or> J = {False} \<or> J = {True,False}"
   360       by (auto simp: UNIV_bool)
   361     then show "prob (\<Inter>j\<in>J. X j) = (\<Prod>j\<in>J. prob (X j))"
   362       using X * by auto
   363   qed
   364 qed
   365 
   366 lemma (in prob_space) indep_set_sigma_sets:
   367   assumes "indep_set A B"
   368   assumes A: "Int_stable A" and B: "Int_stable B"
   369   shows "indep_set (sigma_sets (space M) A) (sigma_sets (space M) B)"
   370 proof -
   371   have "indep_sets (\<lambda>i. sigma_sets (space M) (case i of True \<Rightarrow> A | False \<Rightarrow> B)) UNIV"
   372   proof (rule indep_sets_sigma)
   373     show "indep_sets (case_bool A B) UNIV"
   374       by (rule `indep_set A B`[unfolded indep_set_def])
   375     fix i show "Int_stable (case i of True \<Rightarrow> A | False \<Rightarrow> B)"
   376       using A B by (cases i) auto
   377   qed
   378   then show ?thesis
   379     unfolding indep_set_def
   380     by (rule indep_sets_mono_sets) (auto split: bool.split)
   381 qed
   382 
   383 lemma (in prob_space) indep_sets_collect_sigma:
   384   fixes I :: "'j \<Rightarrow> 'i set" and J :: "'j set" and E :: "'i \<Rightarrow> 'a set set"
   385   assumes indep: "indep_sets E (\<Union>j\<in>J. I j)"
   386   assumes Int_stable: "\<And>i j. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> Int_stable (E i)"
   387   assumes disjoint: "disjoint_family_on I J"
   388   shows "indep_sets (\<lambda>j. sigma_sets (space M) (\<Union>i\<in>I j. E i)) J"
   389 proof -
   390   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) }"
   391 
   392   from indep have E: "\<And>j i. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> E i \<subseteq> events"
   393     unfolding indep_sets_def by auto
   394   { fix j
   395     let ?S = "sigma_sets (space M) (\<Union>i\<in>I j. E i)"
   396     assume "j \<in> J"
   397     from E[OF this] interpret S: sigma_algebra "space M" ?S
   398       using sets.sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto
   399 
   400     have "sigma_sets (space M) (\<Union>i\<in>I j. E i) = sigma_sets (space M) (?E j)"
   401     proof (rule sigma_sets_eqI)
   402       fix A assume "A \<in> (\<Union>i\<in>I j. E i)"
   403       then guess i ..
   404       then show "A \<in> sigma_sets (space M) (?E j)"
   405         by (auto intro!: sigma_sets.intros(2-) exI[of _ "{i}"] exI[of _ "\<lambda>i. A"])
   406     next
   407       fix A assume "A \<in> ?E j"
   408       then obtain E' K where "finite K" "K \<noteq> {}" "K \<subseteq> I j" "\<And>k. k \<in> K \<Longrightarrow> E' k \<in> E k"
   409         and A: "A = (\<Inter>k\<in>K. E' k)"
   410         by auto
   411       then have "A \<in> ?S" unfolding A
   412         by (safe intro!: S.finite_INT) auto
   413       then show "A \<in> sigma_sets (space M) (\<Union>i\<in>I j. E i)"
   414         by simp
   415     qed }
   416   moreover have "indep_sets (\<lambda>j. sigma_sets (space M) (?E j)) J"
   417   proof (rule indep_sets_sigma)
   418     show "indep_sets ?E J"
   419     proof (intro indep_setsI)
   420       fix j assume "j \<in> J" with E show "?E j \<subseteq> events" by (force  intro!: sets.finite_INT)
   421     next
   422       fix K A assume K: "K \<noteq> {}" "K \<subseteq> J" "finite K"
   423         and "\<forall>j\<in>K. A j \<in> ?E j"
   424       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)"
   425         by simp
   426       from bchoice[OF this] guess E' ..
   427       from bchoice[OF this] obtain L
   428         where A: "\<And>j. j\<in>K \<Longrightarrow> A j = (\<Inter>l\<in>L j. E' j l)"
   429         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"
   430         and E': "\<And>j l. j\<in>K \<Longrightarrow> l \<in> L j \<Longrightarrow> E' j l \<in> E l"
   431         by auto
   432 
   433       { fix k l j assume "k \<in> K" "j \<in> K" "l \<in> L j" "l \<in> L k"
   434         have "k = j"
   435         proof (rule ccontr)
   436           assume "k \<noteq> j"
   437           with disjoint `K \<subseteq> J` `k \<in> K` `j \<in> K` have "I k \<inter> I j = {}"
   438             unfolding disjoint_family_on_def by auto
   439           with L(2,3)[OF `j \<in> K`] L(2,3)[OF `k \<in> K`]
   440           show False using `l \<in> L k` `l \<in> L j` by auto
   441         qed }
   442       note L_inj = this
   443 
   444       def k \<equiv> "\<lambda>l. (SOME k. k \<in> K \<and> l \<in> L k)"
   445       { fix x j l assume *: "j \<in> K" "l \<in> L j"
   446         have "k l = j" unfolding k_def
   447         proof (rule some_equality)
   448           fix k assume "k \<in> K \<and> l \<in> L k"
   449           with * L_inj show "k = j" by auto
   450         qed (insert *, simp) }
   451       note k_simp[simp] = this
   452       let ?E' = "\<lambda>l. E' (k l) l"
   453       have "prob (\<Inter>j\<in>K. A j) = prob (\<Inter>l\<in>(\<Union>k\<in>K. L k). ?E' l)"
   454         by (auto simp: A intro!: arg_cong[where f=prob])
   455       also have "\<dots> = (\<Prod>l\<in>(\<Union>k\<in>K. L k). prob (?E' l))"
   456         using L K E' by (intro indep_setsD[OF indep]) (simp_all add: UN_mono)
   457       also have "\<dots> = (\<Prod>j\<in>K. \<Prod>l\<in>L j. prob (E' j l))"
   458         using K L L_inj by (subst setprod.UNION_disjoint) auto
   459       also have "\<dots> = (\<Prod>j\<in>K. prob (A j))"
   460         using K L E' by (auto simp add: A intro!: setprod.cong indep_setsD[OF indep, symmetric]) blast
   461       finally show "prob (\<Inter>j\<in>K. A j) = (\<Prod>j\<in>K. prob (A j))" .
   462     qed
   463   next
   464     fix j assume "j \<in> J"
   465     show "Int_stable (?E j)"
   466     proof (rule Int_stableI)
   467       fix a assume "a \<in> ?E j" then obtain Ka Ea
   468         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
   469       fix b assume "b \<in> ?E j" then obtain Kb Eb
   470         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
   471       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 {})"
   472       have "a \<inter> b = INTER (Ka \<union> Kb) ?A"
   473         by (simp add: a b set_eq_iff) auto
   474       with a b `j \<in> J` Int_stableD[OF Int_stable] show "a \<inter> b \<in> ?E j"
   475         by (intro CollectI exI[of _ "Ka \<union> Kb"] exI[of _ ?A]) auto
   476     qed
   477   qed
   478   ultimately show ?thesis
   479     by (simp cong: indep_sets_cong)
   480 qed
   481 
   482 lemma (in prob_space) indep_vars_restrict:
   483   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"
   484   shows "indep_vars (\<lambda>j. PiM (K j) M') (\<lambda>j \<omega>. restrict (\<lambda>i. X i \<omega>) (K j)) L"
   485   unfolding indep_vars_def
   486 proof safe
   487   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>)"
   488     using K ind by (auto simp: indep_vars_def intro!: measurable_restrict)
   489 next
   490   have X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable M (M' i)"
   491     using ind by (auto simp: indep_vars_def)
   492   let ?proj = "\<lambda>j S. {(\<lambda>\<omega>. \<lambda>i\<in>K j. X i \<omega>) -` A \<inter> space M |A. A \<in> S}"
   493   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) })"
   494   show "indep_sets (\<lambda>i. sigma_sets (space M) (?proj i (sets (Pi\<^sub>M (K i) M')))) L"
   495   proof (rule indep_sets_mono_sets)
   496     fix j assume j: "j \<in> L"
   497     have "sigma_sets (space M) (?proj j (sets (Pi\<^sub>M (K j) M'))) = 
   498       sigma_sets (space M) (sigma_sets (space M) (?proj j (prod_algebra (K j) M')))"
   499       using j K X[THEN measurable_space] unfolding sets_PiM
   500       by (subst sigma_sets_vimage_commute) (auto simp add: Pi_iff)
   501     also have "\<dots> = sigma_sets (space M) (?proj j (prod_algebra (K j) M'))"
   502       by (rule sigma_sets_sigma_sets_eq) auto
   503     also have "\<dots> \<subseteq> ?UN j"
   504     proof (rule sigma_sets_mono, safe del: disjE elim!: prod_algebraE)
   505       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)"
   506       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"
   507       proof cases
   508         assume "K j = {}" with J show ?thesis
   509           by (auto simp add: sigma_sets_empty_eq prod_emb_def)
   510       next
   511         assume "K j \<noteq> {}" with J have "J \<noteq> {}"
   512           by auto
   513         { interpret sigma_algebra "space M" "?UN j"
   514             by (rule sigma_algebra_sigma_sets) auto 
   515           have "\<And>A. (\<And>i. i \<in> J \<Longrightarrow> A i \<in> ?UN j) \<Longrightarrow> INTER J A \<in> ?UN j"
   516             using `finite J` `J \<noteq> {}` by (rule finite_INT) blast }
   517         note INT = this
   518 
   519         from `J \<noteq> {}` J K E[rule_format, THEN sets.sets_into_space] j
   520         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
   521           = (\<Inter>i\<in>J. X i -` E i \<inter> space M)"
   522           apply (subst prod_emb_PiE[OF _ ])
   523           apply auto []
   524           apply auto []
   525           apply (auto simp add: Pi_iff intro!: X[THEN measurable_space])
   526           apply (erule_tac x=i in ballE)
   527           apply auto
   528           done
   529         also have "\<dots> \<in> ?UN j"
   530           apply (rule INT)
   531           apply (rule sigma_sets.Basic)
   532           using `J \<subseteq> K j` E
   533           apply auto
   534           done
   535         finally show ?thesis .
   536       qed
   537     qed
   538     finally show "sigma_sets (space M) (?proj j (sets (Pi\<^sub>M (K j) M'))) \<subseteq> ?UN j" .
   539   next
   540     show "indep_sets ?UN L"
   541     proof (rule indep_sets_collect_sigma)
   542       show "indep_sets (\<lambda>i. {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) (\<Union>j\<in>L. K j)"
   543       proof (rule indep_sets_mono_index)
   544         show "indep_sets (\<lambda>i. {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"
   545           using ind unfolding indep_vars_def2 by auto
   546         show "(\<Union>l\<in>L. K l) \<subseteq> I"
   547           using K by auto
   548       qed
   549     next
   550       fix l i assume "l \<in> L" "i \<in> K l"
   551       show "Int_stable {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
   552         apply (auto simp: Int_stable_def)
   553         apply (rule_tac x="A \<inter> Aa" in exI)
   554         apply auto
   555         done
   556     qed fact
   557   qed
   558 qed
   559 
   560 lemma (in prob_space) indep_var_restrict:
   561   assumes ind: "indep_vars M' X I" and AB: "A \<inter> B = {}" "A \<subseteq> I" "B \<subseteq> I"
   562   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)"
   563 proof -
   564   have *:
   565     "case_bool (Pi\<^sub>M A M') (Pi\<^sub>M B M') = (\<lambda>b. PiM (case_bool A B b) M')"
   566     "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>)"
   567     by (simp_all add: fun_eq_iff split: bool.split)
   568   show ?thesis
   569     unfolding indep_var_def * using AB
   570     by (intro indep_vars_restrict[OF ind]) (auto simp: disjoint_family_on_def split: bool.split)
   571 qed
   572 
   573 lemma (in prob_space) indep_vars_subset:
   574   assumes "indep_vars M' X I" "J \<subseteq> I"
   575   shows "indep_vars M' X J"
   576   using assms unfolding indep_vars_def indep_sets_def
   577   by auto
   578 
   579 lemma (in prob_space) indep_vars_cong:
   580   "I = J \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> X i = Y i) \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> M' i = N' i) \<Longrightarrow> indep_vars M' X I \<longleftrightarrow> indep_vars N' Y J"
   581   unfolding indep_vars_def2 by (intro conj_cong indep_sets_cong) auto
   582 
   583 definition (in prob_space) tail_events where
   584   "tail_events A = (\<Inter>n. sigma_sets (space M) (UNION {n..} A))"
   585 
   586 lemma (in prob_space) tail_events_sets:
   587   assumes A: "\<And>i::nat. A i \<subseteq> events"
   588   shows "tail_events A \<subseteq> events"
   589 proof
   590   fix X assume X: "X \<in> tail_events A"
   591   let ?A = "(\<Inter>n. sigma_sets (space M) (UNION {n..} A))"
   592   from X have "\<And>n::nat. X \<in> sigma_sets (space M) (UNION {n..} A)" by (auto simp: tail_events_def)
   593   from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp
   594   then show "X \<in> events"
   595     by induct (insert A, auto)
   596 qed
   597 
   598 lemma (in prob_space) sigma_algebra_tail_events:
   599   assumes "\<And>i::nat. sigma_algebra (space M) (A i)"
   600   shows "sigma_algebra (space M) (tail_events A)"
   601   unfolding tail_events_def
   602 proof (simp add: sigma_algebra_iff2, safe)
   603   let ?A = "(\<Inter>n. sigma_sets (space M) (UNION {n..} A))"
   604   interpret A: sigma_algebra "space M" "A i" for i by fact
   605   { fix X x assume "X \<in> ?A" "x \<in> X"
   606     then have "\<And>n. X \<in> sigma_sets (space M) (UNION {n..} A)" by auto
   607     from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp
   608     then have "X \<subseteq> space M"
   609       by induct (insert A.sets_into_space, auto)
   610     with `x \<in> X` show "x \<in> space M" by auto }
   611   { fix F :: "nat \<Rightarrow> 'a set" and n assume "range F \<subseteq> ?A"
   612     then show "(UNION UNIV F) \<in> sigma_sets (space M) (UNION {n..} A)"
   613       by (intro sigma_sets.Union) auto }
   614 qed (auto intro!: sigma_sets.Compl sigma_sets.Empty)
   615 
   616 lemma (in prob_space) kolmogorov_0_1_law:
   617   fixes A :: "nat \<Rightarrow> 'a set set"
   618   assumes "\<And>i::nat. sigma_algebra (space M) (A i)"
   619   assumes indep: "indep_sets A UNIV"
   620   and X: "X \<in> tail_events A"
   621   shows "prob X = 0 \<or> prob X = 1"
   622 proof -
   623   have A: "\<And>i. A i \<subseteq> events"
   624     using indep unfolding indep_sets_def by simp
   625 
   626   let ?D = "{D \<in> events. prob (X \<inter> D) = prob X * prob D}"
   627   interpret A: sigma_algebra "space M" "A i" for i by fact
   628   interpret T: sigma_algebra "space M" "tail_events A"
   629     by (rule sigma_algebra_tail_events) fact
   630   have "X \<subseteq> space M" using T.space_closed X by auto
   631 
   632   have X_in: "X \<in> events"
   633     using tail_events_sets A X by auto
   634 
   635   interpret D: dynkin_system "space M" ?D
   636   proof (rule dynkin_systemI)
   637     fix D assume "D \<in> ?D" then show "D \<subseteq> space M"
   638       using sets.sets_into_space by auto
   639   next
   640     show "space M \<in> ?D"
   641       using prob_space `X \<subseteq> space M` by (simp add: Int_absorb2)
   642   next
   643     fix A assume A: "A \<in> ?D"
   644     have "prob (X \<inter> (space M - A)) = prob (X - (X \<inter> A))"
   645       using `X \<subseteq> space M` by (auto intro!: arg_cong[where f=prob])
   646     also have "\<dots> = prob X - prob (X \<inter> A)"
   647       using X_in A by (intro finite_measure_Diff) auto
   648     also have "\<dots> = prob X * prob (space M) - prob X * prob A"
   649       using A prob_space by auto
   650     also have "\<dots> = prob X * prob (space M - A)"
   651       using X_in A sets.sets_into_space
   652       by (subst finite_measure_Diff) (auto simp: field_simps)
   653     finally show "space M - A \<in> ?D"
   654       using A `X \<subseteq> space M` by auto
   655   next
   656     fix F :: "nat \<Rightarrow> 'a set" assume dis: "disjoint_family F" and "range F \<subseteq> ?D"
   657     then have F: "range F \<subseteq> events" "\<And>i. prob (X \<inter> F i) = prob X * prob (F i)"
   658       by auto
   659     have "(\<lambda>i. prob (X \<inter> F i)) sums prob (\<Union>i. X \<inter> F i)"
   660     proof (rule finite_measure_UNION)
   661       show "range (\<lambda>i. X \<inter> F i) \<subseteq> events"
   662         using F X_in by auto
   663       show "disjoint_family (\<lambda>i. X \<inter> F i)"
   664         using dis by (rule disjoint_family_on_bisimulation) auto
   665     qed
   666     with F have "(\<lambda>i. prob X * prob (F i)) sums prob (X \<inter> (\<Union>i. F i))"
   667       by simp
   668     moreover have "(\<lambda>i. prob X * prob (F i)) sums (prob X * prob (\<Union>i. F i))"
   669       by (intro sums_mult finite_measure_UNION F dis)
   670     ultimately have "prob (X \<inter> (\<Union>i. F i)) = prob X * prob (\<Union>i. F i)"
   671       by (auto dest!: sums_unique)
   672     with F show "(\<Union>i. F i) \<in> ?D"
   673       by auto
   674   qed
   675 
   676   { fix n
   677     have "indep_sets (\<lambda>b. sigma_sets (space M) (\<Union>m\<in>case_bool {..n} {Suc n..} b. A m)) UNIV"
   678     proof (rule indep_sets_collect_sigma)
   679       have *: "(\<Union>b. case b of True \<Rightarrow> {..n} | False \<Rightarrow> {Suc n..}) = UNIV" (is "?U = _")
   680         by (simp split: bool.split add: set_eq_iff) (metis not_less_eq_eq)
   681       with indep show "indep_sets A ?U" by simp
   682       show "disjoint_family (case_bool {..n} {Suc n..})"
   683         unfolding disjoint_family_on_def by (auto split: bool.split)
   684       fix m
   685       show "Int_stable (A m)"
   686         unfolding Int_stable_def using A.Int by auto
   687     qed
   688     also have "(\<lambda>b. sigma_sets (space M) (\<Union>m\<in>case_bool {..n} {Suc n..} b. A m)) =
   689       case_bool (sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) (sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m))"
   690       by (auto intro!: ext split: bool.split)
   691     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))"
   692       unfolding indep_set_def by simp
   693 
   694     have "sigma_sets (space M) (\<Union>m\<in>{..n}. A m) \<subseteq> ?D"
   695     proof (simp add: subset_eq, rule)
   696       fix D assume D: "D \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)"
   697       have "X \<in> sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m)"
   698         using X unfolding tail_events_def by simp
   699       from indep_setD[OF indep D this] indep_setD_ev1[OF indep] D
   700       show "D \<in> events \<and> prob (X \<inter> D) = prob X * prob D"
   701         by (auto simp add: ac_simps)
   702     qed }
   703   then have "(\<Union>n. sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) \<subseteq> ?D" (is "?A \<subseteq> _")
   704     by auto
   705 
   706   note `X \<in> tail_events A`
   707   also {
   708     have "\<And>n. sigma_sets (space M) (\<Union>i\<in>{n..}. A i) \<subseteq> sigma_sets (space M) ?A"
   709       by (intro sigma_sets_subseteq UN_mono) auto
   710    then have "tail_events A \<subseteq> sigma_sets (space M) ?A"
   711       unfolding tail_events_def by auto }
   712   also have "sigma_sets (space M) ?A = dynkin (space M) ?A"
   713   proof (rule sigma_eq_dynkin)
   714     { fix B n assume "B \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)"
   715       then have "B \<subseteq> space M"
   716         by induct (insert A sets.sets_into_space[of _ M], auto) }
   717     then show "?A \<subseteq> Pow (space M)" by auto
   718     show "Int_stable ?A"
   719     proof (rule Int_stableI)
   720       fix a assume "a \<in> ?A" then guess n .. note a = this
   721       fix b assume "b \<in> ?A" then guess m .. note b = this
   722       interpret Amn: sigma_algebra "space M" "sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
   723         using A sets.sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto
   724       have "sigma_sets (space M) (\<Union>i\<in>{..n}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
   725         by (intro sigma_sets_subseteq UN_mono) auto
   726       with a have "a \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto
   727       moreover
   728       have "sigma_sets (space M) (\<Union>i\<in>{..m}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
   729         by (intro sigma_sets_subseteq UN_mono) auto
   730       with b have "b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto
   731       ultimately have "a \<inter> b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
   732         using Amn.Int[of a b] by simp
   733       then show "a \<inter> b \<in> (\<Union>n. sigma_sets (space M) (\<Union>i\<in>{..n}. A i))" by auto
   734     qed
   735   qed
   736   also have "dynkin (space M) ?A \<subseteq> ?D"
   737     using `?A \<subseteq> ?D` by (auto intro!: D.dynkin_subset)
   738   finally show ?thesis by auto
   739 qed
   740 
   741 lemma (in prob_space) borel_0_1_law:
   742   fixes F :: "nat \<Rightarrow> 'a set"
   743   assumes F2: "indep_events F UNIV"
   744   shows "prob (\<Inter>n. \<Union>m\<in>{n..}. F m) = 0 \<or> prob (\<Inter>n. \<Union>m\<in>{n..}. F m) = 1"
   745 proof (rule kolmogorov_0_1_law[of "\<lambda>i. sigma_sets (space M) { F i }"])
   746   have F1: "range F \<subseteq> events"
   747     using F2 by (simp add: indep_events_def subset_eq)
   748   { fix i show "sigma_algebra (space M) (sigma_sets (space M) {F i})"
   749       using sigma_algebra_sigma_sets[of "{F i}" "space M"] F1 sets.sets_into_space
   750       by auto }
   751   show "indep_sets (\<lambda>i. sigma_sets (space M) {F i}) UNIV"
   752   proof (rule indep_sets_sigma)
   753     show "indep_sets (\<lambda>i. {F i}) UNIV"
   754       unfolding indep_events_def_alt[symmetric] by fact
   755     fix i show "Int_stable {F i}"
   756       unfolding Int_stable_def by simp
   757   qed
   758   let ?Q = "\<lambda>n. \<Union>i\<in>{n..}. F i"
   759   show "(\<Inter>n. \<Union>m\<in>{n..}. F m) \<in> tail_events (\<lambda>i. sigma_sets (space M) {F i})"
   760     unfolding tail_events_def
   761   proof
   762     fix j
   763     interpret S: sigma_algebra "space M" "sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})"
   764       using order_trans[OF F1 sets.space_closed]
   765       by (intro sigma_algebra_sigma_sets) (simp add: sigma_sets_singleton subset_eq)
   766     have "(\<Inter>n. ?Q n) = (\<Inter>n\<in>{j..}. ?Q n)"
   767       by (intro decseq_SucI INT_decseq_offset UN_mono) auto
   768     also have "\<dots> \<in> sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})"
   769       using order_trans[OF F1 sets.space_closed]
   770       by (safe intro!: S.countable_INT S.countable_UN)
   771          (auto simp: sigma_sets_singleton intro!: sigma_sets.Basic bexI)
   772     finally show "(\<Inter>n. ?Q n) \<in> sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})"
   773       by simp
   774   qed
   775 qed
   776 
   777 lemma (in prob_space) indep_sets_finite:
   778   assumes I: "I \<noteq> {}" "finite I"
   779     and F: "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events" "\<And>i. i \<in> I \<Longrightarrow> space M \<in> F i"
   780   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)))"
   781 proof
   782   assume *: "indep_sets F I"
   783   from I show "\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j))"
   784     by (intro indep_setsD[OF *] ballI) auto
   785 next
   786   assume indep: "\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j))"
   787   show "indep_sets F I"
   788   proof (rule indep_setsI[OF F(1)])
   789     fix A J assume J: "J \<noteq> {}" "J \<subseteq> I" "finite J"
   790     assume A: "\<forall>j\<in>J. A j \<in> F j"
   791     let ?A = "\<lambda>j. if j \<in> J then A j else space M"
   792     have "prob (\<Inter>j\<in>I. ?A j) = prob (\<Inter>j\<in>J. A j)"
   793       using subset_trans[OF F(1) sets.space_closed] J A
   794       by (auto intro!: arg_cong[where f=prob] split: split_if_asm) blast
   795     also
   796     from A F have "(\<lambda>j. if j \<in> J then A j else space M) \<in> Pi I F" (is "?A \<in> _")
   797       by (auto split: split_if_asm)
   798     with indep have "prob (\<Inter>j\<in>I. ?A j) = (\<Prod>j\<in>I. prob (?A j))"
   799       by auto
   800     also have "\<dots> = (\<Prod>j\<in>J. prob (A j))"
   801       unfolding if_distrib setprod.If_cases[OF `finite I`]
   802       using prob_space `J \<subseteq> I` by (simp add: Int_absorb1 setprod.neutral_const)
   803     finally show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" ..
   804   qed
   805 qed
   806 
   807 lemma (in prob_space) indep_vars_finite:
   808   fixes I :: "'i set"
   809   assumes I: "I \<noteq> {}" "finite I"
   810     and M': "\<And>i. i \<in> I \<Longrightarrow> sets (M' i) = sigma_sets (space (M' i)) (E i)"
   811     and rv: "\<And>i. i \<in> I \<Longrightarrow> random_variable (M' i) (X i)"
   812     and Int_stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (E i)"
   813     and space: "\<And>i. i \<in> I \<Longrightarrow> space (M' i) \<in> E i" and closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M' i))"
   814   shows "indep_vars M' X I \<longleftrightarrow>
   815     (\<forall>A\<in>(\<Pi> i\<in>I. E i). prob (\<Inter>j\<in>I. X j -` A j \<inter> space M) = (\<Prod>j\<in>I. prob (X j -` A j \<inter> space M)))"
   816 proof -
   817   from rv have X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> space M \<rightarrow> space (M' i)"
   818     unfolding measurable_def by simp
   819 
   820   { fix i assume "i\<in>I"
   821     from closed[OF `i \<in> I`]
   822     have "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}
   823       = sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> E i}"
   824       unfolding sigma_sets_vimage_commute[OF X, OF `i \<in> I`, symmetric] M'[OF `i \<in> I`]
   825       by (subst sigma_sets_sigma_sets_eq) auto }
   826   note sigma_sets_X = this
   827 
   828   { fix i assume "i\<in>I"
   829     have "Int_stable {X i -` A \<inter> space M |A. A \<in> E i}"
   830     proof (rule Int_stableI)
   831       fix a assume "a \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
   832       then obtain A where "a = X i -` A \<inter> space M" "A \<in> E i" by auto
   833       moreover
   834       fix b assume "b \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
   835       then obtain B where "b = X i -` B \<inter> space M" "B \<in> E i" by auto
   836       moreover
   837       have "(X i -` A \<inter> space M) \<inter> (X i -` B \<inter> space M) = X i -` (A \<inter> B) \<inter> space M" by auto
   838       moreover note Int_stable[OF `i \<in> I`]
   839       ultimately
   840       show "a \<inter> b \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
   841         by (auto simp del: vimage_Int intro!: exI[of _ "A \<inter> B"] dest: Int_stableD)
   842     qed }
   843   note indep_sets_X = indep_sets_sigma_sets_iff[OF this]
   844 
   845   { fix i assume "i \<in> I"
   846     { fix A assume "A \<in> E i"
   847       with M'[OF `i \<in> I`] have "A \<in> sets (M' i)" by auto
   848       moreover
   849       from rv[OF `i\<in>I`] have "X i \<in> measurable M (M' i)" by auto
   850       ultimately
   851       have "X i -` A \<inter> space M \<in> sets M" by (auto intro: measurable_sets) }
   852     with X[OF `i\<in>I`] space[OF `i\<in>I`]
   853     have "{X i -` A \<inter> space M |A. A \<in> E i} \<subseteq> events"
   854       "space M \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
   855       by (auto intro!: exI[of _ "space (M' i)"]) }
   856   note indep_sets_finite_X = indep_sets_finite[OF I this]
   857 
   858   have "(\<forall>A\<in>\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> E i}. prob (INTER I A) = (\<Prod>j\<in>I. prob (A j))) =
   859     (\<forall>A\<in>\<Pi> i\<in>I. E i. prob ((\<Inter>j\<in>I. X j -` A j) \<inter> space M) = (\<Prod>x\<in>I. prob (X x -` A x \<inter> space M)))"
   860     (is "?L = ?R")
   861   proof safe
   862     fix A assume ?L and A: "A \<in> (\<Pi> i\<in>I. E i)"
   863     from `?L`[THEN bspec, of "\<lambda>i. X i -` A i \<inter> space M"] A `I \<noteq> {}`
   864     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))"
   865       by (auto simp add: Pi_iff)
   866   next
   867     fix A assume ?R and A: "A \<in> (\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> E i})"
   868     from A have "\<forall>i\<in>I. \<exists>B. A i = X i -` B \<inter> space M \<and> B \<in> E i" by auto
   869     from bchoice[OF this] obtain B where B: "\<forall>i\<in>I. A i = X i -` B i \<inter> space M"
   870       "B \<in> (\<Pi> i\<in>I. E i)" by auto
   871     from `?R`[THEN bspec, OF B(2)] B(1) `I \<noteq> {}`
   872     show "prob (INTER I A) = (\<Prod>j\<in>I. prob (A j))"
   873       by simp
   874   qed
   875   then show ?thesis using `I \<noteq> {}`
   876     by (simp add: rv indep_vars_def indep_sets_X sigma_sets_X indep_sets_finite_X cong: indep_sets_cong)
   877 qed
   878 
   879 lemma (in prob_space) indep_vars_compose:
   880   assumes "indep_vars M' X I"
   881   assumes rv: "\<And>i. i \<in> I \<Longrightarrow> Y i \<in> measurable (M' i) (N i)"
   882   shows "indep_vars N (\<lambda>i. Y i \<circ> X i) I"
   883   unfolding indep_vars_def
   884 proof
   885   from rv `indep_vars M' X I`
   886   show "\<forall>i\<in>I. random_variable (N i) (Y i \<circ> X i)"
   887     by (auto simp: indep_vars_def)
   888 
   889   have "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"
   890     using `indep_vars M' X I` by (simp add: indep_vars_def)
   891   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"
   892   proof (rule indep_sets_mono_sets)
   893     fix i assume "i \<in> I"
   894     with `indep_vars M' X I` have X: "X i \<in> space M \<rightarrow> space (M' i)"
   895       unfolding indep_vars_def measurable_def by auto
   896     { fix A assume "A \<in> sets (N i)"
   897       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)"
   898         by (intro exI[of _ "Y i -` A \<inter> space (M' i)"])
   899            (auto simp: vimage_comp intro!: measurable_sets rv `i \<in> I` funcset_mem[OF X]) }
   900     then show "sigma_sets (space M) {(Y i \<circ> X i) -` A \<inter> space M |A. A \<in> sets (N i)} \<subseteq>
   901       sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
   902       by (intro sigma_sets_subseteq) (auto simp: vimage_comp)
   903   qed
   904 qed
   905 
   906 lemma (in prob_space) indep_var_compose:
   907   assumes "indep_var M1 X1 M2 X2" "Y1 \<in> measurable M1 N1" "Y2 \<in> measurable M2 N2"
   908   shows "indep_var N1 (Y1 \<circ> X1) N2 (Y2 \<circ> X2)"
   909 proof -
   910   have "indep_vars (case_bool N1 N2) (\<lambda>b. case_bool Y1 Y2 b \<circ> case_bool X1 X2 b) UNIV"
   911     using assms
   912     by (intro indep_vars_compose[where M'="case_bool M1 M2"])
   913        (auto simp: indep_var_def split: bool.split)
   914   also have "(\<lambda>b. case_bool Y1 Y2 b \<circ> case_bool X1 X2 b) = case_bool (Y1 \<circ> X1) (Y2 \<circ> X2)"
   915     by (simp add: fun_eq_iff split: bool.split)
   916   finally show ?thesis
   917     unfolding indep_var_def .
   918 qed
   919 
   920 lemma (in prob_space) indep_vars_Min:
   921   fixes X :: "'i \<Rightarrow> 'a \<Rightarrow> real"
   922   assumes I: "finite I" "i \<notin> I" and indep: "indep_vars (\<lambda>_. borel) X (insert i I)"
   923   shows "indep_var borel (X i) borel (\<lambda>\<omega>. Min ((\<lambda>i. X i \<omega>)`I))"
   924 proof -
   925   have "indep_var
   926     borel ((\<lambda>f. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) {i}))
   927     borel ((\<lambda>f. Min (f`I)) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) I))"
   928     using I by (intro indep_var_compose[OF indep_var_restrict[OF indep]] borel_measurable_Min) auto
   929   also have "((\<lambda>f. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) {i})) = X i"
   930     by auto
   931   also have "((\<lambda>f. Min (f`I)) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) I)) = (\<lambda>\<omega>. Min ((\<lambda>i. X i \<omega>)`I))"
   932     by (auto cong: rev_conj_cong)
   933   finally show ?thesis
   934     unfolding indep_var_def .
   935 qed
   936 
   937 lemma (in prob_space) indep_vars_setsum:
   938   fixes X :: "'i \<Rightarrow> 'a \<Rightarrow> real"
   939   assumes I: "finite I" "i \<notin> I" and indep: "indep_vars (\<lambda>_. borel) X (insert i I)"
   940   shows "indep_var borel (X i) borel (\<lambda>\<omega>. \<Sum>i\<in>I. X i \<omega>)"
   941 proof -
   942   have "indep_var 
   943     borel ((\<lambda>f. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) {i}))
   944     borel ((\<lambda>f. \<Sum>i\<in>I. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) I))"
   945     using I by (intro indep_var_compose[OF indep_var_restrict[OF indep]] ) auto
   946   also have "((\<lambda>f. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) {i})) = X i"
   947     by auto
   948   also have "((\<lambda>f. \<Sum>i\<in>I. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) I)) = (\<lambda>\<omega>. \<Sum>i\<in>I. X i \<omega>)"
   949     by (auto cong: rev_conj_cong)
   950   finally show ?thesis .
   951 qed
   952 
   953 lemma (in prob_space) indep_vars_setprod:
   954   fixes X :: "'i \<Rightarrow> 'a \<Rightarrow> real"
   955   assumes I: "finite I" "i \<notin> I" and indep: "indep_vars (\<lambda>_. borel) X (insert i I)"
   956   shows "indep_var borel (X i) borel (\<lambda>\<omega>. \<Prod>i\<in>I. X i \<omega>)"
   957 proof -
   958   have "indep_var 
   959     borel ((\<lambda>f. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) {i}))
   960     borel ((\<lambda>f. \<Prod>i\<in>I. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) I))"
   961     using I by (intro indep_var_compose[OF indep_var_restrict[OF indep]] ) auto
   962   also have "((\<lambda>f. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) {i})) = X i"
   963     by auto
   964   also have "((\<lambda>f. \<Prod>i\<in>I. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) I)) = (\<lambda>\<omega>. \<Prod>i\<in>I. X i \<omega>)"
   965     by (auto cong: rev_conj_cong)
   966   finally show ?thesis .
   967 qed
   968 
   969 lemma (in prob_space) indep_varsD_finite:
   970   assumes X: "indep_vars M' X I"
   971   assumes I: "I \<noteq> {}" "finite I" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M' i)"
   972   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))"
   973 proof (rule indep_setsD)
   974   show "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"
   975     using X by (auto simp: indep_vars_def)
   976   show "I \<subseteq> I" "I \<noteq> {}" "finite I" using I by auto
   977   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)}"
   978     using I by auto
   979 qed
   980 
   981 lemma (in prob_space) indep_varsD:
   982   assumes X: "indep_vars M' X I"
   983   assumes I: "J \<noteq> {}" "finite J" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M' i)"
   984   shows "prob (\<Inter>i\<in>J. X i -` A i \<inter> space M) = (\<Prod>i\<in>J. prob (X i -` A i \<inter> space M))"
   985 proof (rule indep_setsD)
   986   show "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"
   987     using X by (auto simp: indep_vars_def)
   988   show "\<forall>i\<in>J. X i -` A i \<inter> space M \<in> sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
   989     using I by auto
   990 qed fact+
   991 
   992 lemma (in prob_space) indep_vars_iff_distr_eq_PiM:
   993   fixes I :: "'i set" and X :: "'i \<Rightarrow> 'a \<Rightarrow> 'b"
   994   assumes "I \<noteq> {}"
   995   assumes rv: "\<And>i. random_variable (M' i) (X i)"
   996   shows "indep_vars M' X I \<longleftrightarrow>
   997     distr M (\<Pi>\<^sub>M i\<in>I. M' i) (\<lambda>x. \<lambda>i\<in>I. X i x) = (\<Pi>\<^sub>M i\<in>I. distr M (M' i) (X i))"
   998 proof -
   999   let ?P = "\<Pi>\<^sub>M i\<in>I. M' i"
  1000   let ?X = "\<lambda>x. \<lambda>i\<in>I. X i x"
  1001   let ?D = "distr M ?P ?X"
  1002   have X: "random_variable ?P ?X" by (intro measurable_restrict rv)
  1003   interpret D: prob_space ?D by (intro prob_space_distr X)
  1004 
  1005   let ?D' = "\<lambda>i. distr M (M' i) (X i)"
  1006   let ?P' = "\<Pi>\<^sub>M i\<in>I. distr M (M' i) (X i)"
  1007   interpret D': prob_space "?D' i" for i by (intro prob_space_distr rv)
  1008   interpret P: product_prob_space ?D' I ..
  1009     
  1010   show ?thesis
  1011   proof
  1012     assume "indep_vars M' X I"
  1013     show "?D = ?P'"
  1014     proof (rule measure_eqI_generator_eq)
  1015       show "Int_stable (prod_algebra I M')"
  1016         by (rule Int_stable_prod_algebra)
  1017       show "prod_algebra I M' \<subseteq> Pow (space ?P)"
  1018         using prod_algebra_sets_into_space by (simp add: space_PiM)
  1019       show "sets ?D = sigma_sets (space ?P) (prod_algebra I M')"
  1020         by (simp add: sets_PiM space_PiM)
  1021       show "sets ?P' = sigma_sets (space ?P) (prod_algebra I M')"
  1022         by (simp add: sets_PiM space_PiM cong: prod_algebra_cong)
  1023       let ?A = "\<lambda>i. \<Pi>\<^sub>E i\<in>I. space (M' i)"
  1024       show "range ?A \<subseteq> prod_algebra I M'" "(\<Union>i. ?A i) = space (Pi\<^sub>M I M')"
  1025         by (auto simp: space_PiM intro!: space_in_prod_algebra cong: prod_algebra_cong)
  1026       { fix i show "emeasure ?D (\<Pi>\<^sub>E i\<in>I. space (M' i)) \<noteq> \<infinity>" by auto }
  1027     next
  1028       fix E assume E: "E \<in> prod_algebra I M'"
  1029       from prod_algebraE[OF E] guess J Y . note J = this
  1030 
  1031       from E have "E \<in> sets ?P" by (auto simp: sets_PiM)
  1032       then have "emeasure ?D E = emeasure M (?X -` E \<inter> space M)"
  1033         by (simp add: emeasure_distr X)
  1034       also have "?X -` E \<inter> space M = (\<Inter>i\<in>J. X i -` Y i \<inter> space M)"
  1035         using J `I \<noteq> {}` measurable_space[OF rv] by (auto simp: prod_emb_def PiE_iff split: split_if_asm)
  1036       also have "emeasure M (\<Inter>i\<in>J. X i -` Y i \<inter> space M) = (\<Prod> i\<in>J. emeasure M (X i -` Y i \<inter> space M))"
  1037         using `indep_vars M' X I` J `I \<noteq> {}` using indep_varsD[of M' X I J]
  1038         by (auto simp: emeasure_eq_measure setprod_ereal)
  1039       also have "\<dots> = (\<Prod> i\<in>J. emeasure (?D' i) (Y i))"
  1040         using rv J by (simp add: emeasure_distr)
  1041       also have "\<dots> = emeasure ?P' E"
  1042         using P.emeasure_PiM_emb[of J Y] J by (simp add: prod_emb_def)
  1043       finally show "emeasure ?D E = emeasure ?P' E" .
  1044     qed
  1045   next
  1046     assume "?D = ?P'"
  1047     show "indep_vars M' X I" unfolding indep_vars_def
  1048     proof (intro conjI indep_setsI ballI rv)
  1049       fix i show "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)} \<subseteq> events"
  1050         by (auto intro!: sets.sigma_sets_subset measurable_sets rv)
  1051     next
  1052       fix J Y' assume J: "J \<noteq> {}" "J \<subseteq> I" "finite J"
  1053       assume Y': "\<forall>j\<in>J. Y' j \<in> sigma_sets (space M) {X j -` A \<inter> space M |A. A \<in> sets (M' j)}"
  1054       have "\<forall>j\<in>J. \<exists>Y. Y' j = X j -` Y \<inter> space M \<and> Y \<in> sets (M' j)"
  1055       proof
  1056         fix j assume "j \<in> J"
  1057         from Y'[rule_format, OF this] rv[of j]
  1058         show "\<exists>Y. Y' j = X j -` Y \<inter> space M \<and> Y \<in> sets (M' j)"
  1059           by (subst (asm) sigma_sets_vimage_commute[symmetric, of _ _ "space (M' j)"])
  1060              (auto dest: measurable_space simp: sets.sigma_sets_eq)
  1061       qed
  1062       from bchoice[OF this] obtain Y where
  1063         Y: "\<And>j. j \<in> J \<Longrightarrow> Y' j = X j -` Y j \<inter> space M" "\<And>j. j \<in> J \<Longrightarrow> Y j \<in> sets (M' j)" by auto
  1064       let ?E = "prod_emb I M' J (Pi\<^sub>E J Y)"
  1065       from Y have "(\<Inter>j\<in>J. Y' j) = ?X -` ?E \<inter> space M"
  1066         using J `I \<noteq> {}` measurable_space[OF rv] by (auto simp: prod_emb_def PiE_iff split: split_if_asm)
  1067       then have "emeasure M (\<Inter>j\<in>J. Y' j) = emeasure M (?X -` ?E \<inter> space M)"
  1068         by simp
  1069       also have "\<dots> = emeasure ?D ?E"
  1070         using Y  J by (intro emeasure_distr[symmetric] X sets_PiM_I) auto
  1071       also have "\<dots> = emeasure ?P' ?E"
  1072         using `?D = ?P'` by simp
  1073       also have "\<dots> = (\<Prod> i\<in>J. emeasure (?D' i) (Y i))"
  1074         using P.emeasure_PiM_emb[of J Y] J Y by (simp add: prod_emb_def)
  1075       also have "\<dots> = (\<Prod> i\<in>J. emeasure M (Y' i))"
  1076         using rv J Y by (simp add: emeasure_distr)
  1077       finally have "emeasure M (\<Inter>j\<in>J. Y' j) = (\<Prod> i\<in>J. emeasure M (Y' i))" .
  1078       then show "prob (\<Inter>j\<in>J. Y' j) = (\<Prod> i\<in>J. prob (Y' i))"
  1079         by (auto simp: emeasure_eq_measure setprod_ereal)
  1080     qed
  1081   qed
  1082 qed
  1083 
  1084 lemma (in prob_space) indep_varD:
  1085   assumes indep: "indep_var Ma A Mb B"
  1086   assumes sets: "Xa \<in> sets Ma" "Xb \<in> sets Mb"
  1087   shows "prob ((\<lambda>x. (A x, B x)) -` (Xa \<times> Xb) \<inter> space M) =
  1088     prob (A -` Xa \<inter> space M) * prob (B -` Xb \<inter> space M)"
  1089 proof -
  1090   have "prob ((\<lambda>x. (A x, B x)) -` (Xa \<times> Xb) \<inter> space M) =
  1091     prob (\<Inter>i\<in>UNIV. (case_bool A B i -` case_bool Xa Xb i \<inter> space M))"
  1092     by (auto intro!: arg_cong[where f=prob] simp: UNIV_bool)
  1093   also have "\<dots> = (\<Prod>i\<in>UNIV. prob (case_bool A B i -` case_bool Xa Xb i \<inter> space M))"
  1094     using indep unfolding indep_var_def
  1095     by (rule indep_varsD) (auto split: bool.split intro: sets)
  1096   also have "\<dots> = prob (A -` Xa \<inter> space M) * prob (B -` Xb \<inter> space M)"
  1097     unfolding UNIV_bool by simp
  1098   finally show ?thesis .
  1099 qed
  1100 
  1101 lemma (in prob_space) prob_indep_random_variable:
  1102   assumes ind[simp]: "indep_var N X N Y"
  1103   assumes [simp]: "A \<in> sets N" "B \<in> sets N"
  1104   shows "\<P>(x in M. X x \<in> A \<and> Y x \<in> B) = \<P>(x in M. X x \<in> A) * \<P>(x in M. Y x \<in> B)"
  1105 proof-
  1106   have  " \<P>(x in M. (X x)\<in>A \<and>  (Y x)\<in> B ) = prob ((\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M)" 
  1107     by (auto intro!: arg_cong[where f= prob])
  1108   also have "...=  prob (X -` A \<inter> space M) * prob (Y -` B \<inter> space M)"  
  1109     by (auto intro!: indep_varD[where Ma=N and Mb=N])     
  1110   also have "... = \<P>(x in M. X x \<in> A) * \<P>(x in M. Y x \<in> B)"
  1111     by (auto intro!: arg_cong2[where f= "op *"] arg_cong[where f= prob])
  1112   finally show ?thesis .
  1113 qed
  1114 
  1115 lemma (in prob_space)
  1116   assumes "indep_var S X T Y"
  1117   shows indep_var_rv1: "random_variable S X"
  1118     and indep_var_rv2: "random_variable T Y"
  1119 proof -
  1120   have "\<forall>i\<in>UNIV. random_variable (case_bool S T i) (case_bool X Y i)"
  1121     using assms unfolding indep_var_def indep_vars_def by auto
  1122   then show "random_variable S X" "random_variable T Y"
  1123     unfolding UNIV_bool by auto
  1124 qed
  1125 
  1126 lemma (in prob_space) indep_var_distribution_eq:
  1127   "indep_var S X T Y \<longleftrightarrow> random_variable S X \<and> random_variable T Y \<and>
  1128     distr M S X \<Otimes>\<^sub>M distr M T Y = distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))" (is "_ \<longleftrightarrow> _ \<and> _ \<and> ?S \<Otimes>\<^sub>M ?T = ?J")
  1129 proof safe
  1130   assume "indep_var S X T Y"
  1131   then show rvs: "random_variable S X" "random_variable T Y"
  1132     by (blast dest: indep_var_rv1 indep_var_rv2)+
  1133   then have XY: "random_variable (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))"
  1134     by (rule measurable_Pair)
  1135 
  1136   interpret X: prob_space ?S by (rule prob_space_distr) fact
  1137   interpret Y: prob_space ?T by (rule prob_space_distr) fact
  1138   interpret XY: pair_prob_space ?S ?T ..
  1139   show "?S \<Otimes>\<^sub>M ?T = ?J"
  1140   proof (rule pair_measure_eqI)
  1141     show "sigma_finite_measure ?S" ..
  1142     show "sigma_finite_measure ?T" ..
  1143 
  1144     fix A B assume A: "A \<in> sets ?S" and B: "B \<in> sets ?T"
  1145     have "emeasure ?J (A \<times> B) = emeasure M ((\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M)"
  1146       using A B by (intro emeasure_distr[OF XY]) auto
  1147     also have "\<dots> = emeasure M (X -` A \<inter> space M) * emeasure M (Y -` B \<inter> space M)"
  1148       using indep_varD[OF `indep_var S X T Y`, of A B] A B by (simp add: emeasure_eq_measure)
  1149     also have "\<dots> = emeasure ?S A * emeasure ?T B"
  1150       using rvs A B by (simp add: emeasure_distr)
  1151     finally show "emeasure ?S A * emeasure ?T B = emeasure ?J (A \<times> B)" by simp
  1152   qed simp
  1153 next
  1154   assume rvs: "random_variable S X" "random_variable T Y"
  1155   then have XY: "random_variable (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))"
  1156     by (rule measurable_Pair)
  1157 
  1158   let ?S = "distr M S X" and ?T = "distr M T Y"
  1159   interpret X: prob_space ?S by (rule prob_space_distr) fact
  1160   interpret Y: prob_space ?T by (rule prob_space_distr) fact
  1161   interpret XY: pair_prob_space ?S ?T ..
  1162 
  1163   assume "?S \<Otimes>\<^sub>M ?T = ?J"
  1164 
  1165   { fix S and X
  1166     have "Int_stable {X -` A \<inter> space M |A. A \<in> sets S}"
  1167     proof (safe intro!: Int_stableI)
  1168       fix A B assume "A \<in> sets S" "B \<in> sets S"
  1169       then show "\<exists>C. (X -` A \<inter> space M) \<inter> (X -` B \<inter> space M) = (X -` C \<inter> space M) \<and> C \<in> sets S"
  1170         by (intro exI[of _ "A \<inter> B"]) auto
  1171     qed }
  1172   note Int_stable = this
  1173 
  1174   show "indep_var S X T Y" unfolding indep_var_eq
  1175   proof (intro conjI indep_set_sigma_sets Int_stable rvs)
  1176     show "indep_set {X -` A \<inter> space M |A. A \<in> sets S} {Y -` A \<inter> space M |A. A \<in> sets T}"
  1177     proof (safe intro!: indep_setI)
  1178       { fix A assume "A \<in> sets S" then show "X -` A \<inter> space M \<in> sets M"
  1179         using `X \<in> measurable M S` by (auto intro: measurable_sets) }
  1180       { fix A assume "A \<in> sets T" then show "Y -` A \<inter> space M \<in> sets M"
  1181         using `Y \<in> measurable M T` by (auto intro: measurable_sets) }
  1182     next
  1183       fix A B assume ab: "A \<in> sets S" "B \<in> sets T"
  1184       then have "ereal (prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M))) = emeasure ?J (A \<times> B)"
  1185         using XY by (auto simp add: emeasure_distr emeasure_eq_measure intro!: arg_cong[where f="prob"])
  1186       also have "\<dots> = emeasure (?S \<Otimes>\<^sub>M ?T) (A \<times> B)"
  1187         unfolding `?S \<Otimes>\<^sub>M ?T = ?J` ..
  1188       also have "\<dots> = emeasure ?S A * emeasure ?T B"
  1189         using ab by (simp add: Y.emeasure_pair_measure_Times)
  1190       finally show "prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)) =
  1191         prob (X -` A \<inter> space M) * prob (Y -` B \<inter> space M)"
  1192         using rvs ab by (simp add: emeasure_eq_measure emeasure_distr)
  1193     qed
  1194   qed
  1195 qed
  1196 
  1197 lemma (in prob_space) distributed_joint_indep:
  1198   assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
  1199   assumes X: "distributed M S X Px" and Y: "distributed M T Y Py"
  1200   assumes indep: "indep_var S X T Y"
  1201   shows "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) (\<lambda>(x, y). Px x * Py y)"
  1202   using indep_var_distribution_eq[of S X T Y] indep
  1203   by (intro distributed_joint_indep'[OF S T X Y]) auto
  1204 
  1205 lemma (in prob_space) indep_vars_nn_integral:
  1206   assumes I: "finite I" "indep_vars (\<lambda>_. borel) X I" "\<And>i \<omega>. i \<in> I \<Longrightarrow> 0 \<le> X i \<omega>"
  1207   shows "(\<integral>\<^sup>+\<omega>. (\<Prod>i\<in>I. X i \<omega>) \<partial>M) = (\<Prod>i\<in>I. \<integral>\<^sup>+\<omega>. X i \<omega> \<partial>M)"
  1208 proof cases
  1209   assume "I \<noteq> {}"
  1210   def Y \<equiv> "\<lambda>i \<omega>. if i \<in> I then X i \<omega> else 0"
  1211   { fix i have "i \<in> I \<Longrightarrow> random_variable borel (X i)"
  1212     using I(2) by (cases "i\<in>I") (auto simp: indep_vars_def) }
  1213   note rv_X = this
  1214 
  1215   { fix i have "random_variable borel (Y i)"
  1216     using I(2) by (cases "i\<in>I") (auto simp: Y_def rv_X) }
  1217   note rv_Y = this[measurable]
  1218     
  1219   interpret Y: prob_space "distr M borel (Y i)" for i
  1220     using I(2) by (cases "i \<in> I") (auto intro!: prob_space_distr simp: Y_def indep_vars_def)
  1221   interpret product_sigma_finite "\<lambda>i. distr M borel (Y i)"
  1222     ..
  1223   
  1224   have indep_Y: "indep_vars (\<lambda>i. borel) Y I"
  1225     by (rule indep_vars_cong[THEN iffD1, OF _ _ _ I(2)]) (auto simp: Y_def)
  1226 
  1227   have "(\<integral>\<^sup>+\<omega>. (\<Prod>i\<in>I. X i \<omega>) \<partial>M) = (\<integral>\<^sup>+\<omega>. (\<Prod>i\<in>I. max 0 (Y i \<omega>)) \<partial>M)"
  1228     using I(3) by (auto intro!: nn_integral_cong setprod.cong simp add: Y_def max_def)
  1229   also have "\<dots> = (\<integral>\<^sup>+\<omega>. (\<Prod>i\<in>I. max 0 (\<omega> i)) \<partial>distr M (Pi\<^sub>M I (\<lambda>i. borel)) (\<lambda>x. \<lambda>i\<in>I. Y i x))"
  1230     by (subst nn_integral_distr) auto
  1231   also have "\<dots> = (\<integral>\<^sup>+\<omega>. (\<Prod>i\<in>I. max 0 (\<omega> i)) \<partial>Pi\<^sub>M I (\<lambda>i. distr M borel (Y i)))"
  1232     unfolding indep_vars_iff_distr_eq_PiM[THEN iffD1, OF `I \<noteq> {}` rv_Y indep_Y] ..
  1233   also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^sup>+\<omega>. max 0 \<omega> \<partial>distr M borel (Y i)))"
  1234     by (rule product_nn_integral_setprod) (auto intro: `finite I`)
  1235   also have "\<dots> = (\<Prod>i\<in>I. \<integral>\<^sup>+\<omega>. X i \<omega> \<partial>M)"
  1236     by (intro setprod.cong nn_integral_cong)
  1237        (auto simp: nn_integral_distr nn_integral_max_0 Y_def rv_X)
  1238   finally show ?thesis .
  1239 qed (simp add: emeasure_space_1)
  1240 
  1241 lemma (in prob_space)
  1242   fixes X :: "'i \<Rightarrow> 'a \<Rightarrow> 'b::{real_normed_field, banach, second_countable_topology}"
  1243   assumes I: "finite I" "indep_vars (\<lambda>_. borel) X I" "\<And>i. i \<in> I \<Longrightarrow> integrable M (X i)"
  1244   shows indep_vars_lebesgue_integral: "(\<integral>\<omega>. (\<Prod>i\<in>I. X i \<omega>) \<partial>M) = (\<Prod>i\<in>I. \<integral>\<omega>. X i \<omega> \<partial>M)" (is ?eq)
  1245     and indep_vars_integrable: "integrable M (\<lambda>\<omega>. (\<Prod>i\<in>I. X i \<omega>))" (is ?int)
  1246 proof (induct rule: case_split)
  1247   assume "I \<noteq> {}"
  1248   def Y \<equiv> "\<lambda>i \<omega>. if i \<in> I then X i \<omega> else 0"
  1249   { fix i have "i \<in> I \<Longrightarrow> random_variable borel (X i)"
  1250     using I(2) by (cases "i\<in>I") (auto simp: indep_vars_def) }
  1251   note rv_X = this[measurable]
  1252 
  1253   { fix i have "random_variable borel (Y i)"
  1254     using I(2) by (cases "i\<in>I") (auto simp: Y_def rv_X) }
  1255   note rv_Y = this[measurable]
  1256 
  1257   { fix i have "integrable M (Y i)"
  1258     using I(3) by (cases "i\<in>I") (auto simp: Y_def) }
  1259   note int_Y = this
  1260     
  1261   interpret Y: prob_space "distr M borel (Y i)" for i
  1262     using I(2) by (cases "i \<in> I") (auto intro!: prob_space_distr simp: Y_def indep_vars_def)
  1263   interpret product_sigma_finite "\<lambda>i. distr M borel (Y i)"
  1264     ..
  1265   
  1266   have indep_Y: "indep_vars (\<lambda>i. borel) Y I"
  1267     by (rule indep_vars_cong[THEN iffD1, OF _ _ _ I(2)]) (auto simp: Y_def)
  1268 
  1269   have "(\<integral>\<omega>. (\<Prod>i\<in>I. X i \<omega>) \<partial>M) = (\<integral>\<omega>. (\<Prod>i\<in>I. Y i \<omega>) \<partial>M)"
  1270     using I(3) by (simp add: Y_def)
  1271   also have "\<dots> = (\<integral>\<omega>. (\<Prod>i\<in>I. \<omega> i) \<partial>distr M (Pi\<^sub>M I (\<lambda>i. borel)) (\<lambda>x. \<lambda>i\<in>I. Y i x))"
  1272     by (subst integral_distr) auto
  1273   also have "\<dots> = (\<integral>\<omega>. (\<Prod>i\<in>I. \<omega> i) \<partial>Pi\<^sub>M I (\<lambda>i. distr M borel (Y i)))"
  1274     unfolding indep_vars_iff_distr_eq_PiM[THEN iffD1, OF `I \<noteq> {}` rv_Y indep_Y] ..
  1275   also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<omega>. \<omega> \<partial>distr M borel (Y i)))"
  1276     by (rule product_integral_setprod) (auto intro: `finite I` simp: integrable_distr_eq int_Y)
  1277   also have "\<dots> = (\<Prod>i\<in>I. \<integral>\<omega>. X i \<omega> \<partial>M)"
  1278     by (intro setprod.cong integral_cong)
  1279        (auto simp: integral_distr Y_def rv_X)
  1280   finally show ?eq .
  1281 
  1282   have "integrable (distr M (Pi\<^sub>M I (\<lambda>i. borel)) (\<lambda>x. \<lambda>i\<in>I. Y i x)) (\<lambda>\<omega>. (\<Prod>i\<in>I. \<omega> i))"
  1283     unfolding indep_vars_iff_distr_eq_PiM[THEN iffD1, OF `I \<noteq> {}` rv_Y indep_Y]
  1284     by (intro product_integrable_setprod[OF `finite I`])
  1285        (simp add: integrable_distr_eq int_Y)
  1286   then show ?int
  1287     by (simp add: integrable_distr_eq Y_def)
  1288 qed (simp_all add: prob_space)
  1289 
  1290 lemma (in prob_space)
  1291   fixes X1 X2 :: "'a \<Rightarrow> 'b::{real_normed_field, banach, second_countable_topology}"
  1292   assumes "indep_var borel X1 borel X2" "integrable M X1" "integrable M X2"
  1293   shows indep_var_lebesgue_integral: "(\<integral>\<omega>. X1 \<omega> * X2 \<omega> \<partial>M) = (\<integral>\<omega>. X1 \<omega> \<partial>M) * (\<integral>\<omega>. X2 \<omega> \<partial>M)" (is ?eq)
  1294     and indep_var_integrable: "integrable M (\<lambda>\<omega>. X1 \<omega> * X2 \<omega>)" (is ?int)
  1295 unfolding indep_var_def
  1296 proof -
  1297   have *: "(\<lambda>\<omega>. X1 \<omega> * X2 \<omega>) = (\<lambda>\<omega>. \<Prod>i\<in>UNIV. (case_bool X1 X2 i \<omega>))"
  1298     by (simp add: UNIV_bool mult_commute)
  1299   have **: "(\<lambda> _. borel) = case_bool borel borel"
  1300     by (rule ext, metis (full_types) bool.simps(3) bool.simps(4))
  1301   show ?eq
  1302     apply (subst *)
  1303     apply (subst indep_vars_lebesgue_integral)
  1304     apply (auto)
  1305     apply (subst **, subst indep_var_def [symmetric], rule assms)
  1306     apply (simp split: bool.split add: assms)
  1307     by (simp add: UNIV_bool mult_commute)
  1308   show ?int
  1309     apply (subst *)
  1310     apply (rule indep_vars_integrable)
  1311     apply auto
  1312     apply (subst **, subst indep_var_def [symmetric], rule assms)
  1313     by (simp split: bool.split add: assms)
  1314 qed
  1315 
  1316 end