1(*  Title:      HOL/Probability/Independent_Family.thy
2    Author:     Johannes H��lzl, TU M��nchen
3    Author:     Sudeep Kanav, TU M��nchen
4*)
5
6section \<open>Independent families of events, event sets, and random variables\<close>
7
8theory Independent_Family
9  imports Infinite_Product_Measure
10begin
11
12definition (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
16definition (in prob_space)
17  "indep_set A B \<longleftrightarrow> indep_sets (case_bool A B) UNIV"
18
19definition (in prob_space)
20  indep_events_def_alt: "indep_events A I \<longleftrightarrow> indep_sets (\<lambda>i. {A i}) I"
21
22lemma (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
31lemma (in prob_space) indep_eventsI:
32  "(\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M) \<Longrightarrow> (\<And>J. J \<subseteq> I \<Longrightarrow> finite J \<Longrightarrow> J \<noteq> {} \<Longrightarrow> prob (\<Inter>i\<in>J. F i) = (\<Prod>i\<in>J. prob (F i))) \<Longrightarrow> indep_events F I"
33  by (auto simp: indep_events_def)
34
35definition (in prob_space)
36  "indep_event A B \<longleftrightarrow> indep_events (case_bool A B) UNIV"
37
38lemma (in prob_space) indep_sets_cong:
39  "I = J \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> F i = G i) \<Longrightarrow> indep_sets F I \<longleftrightarrow> indep_sets G J"
40  by (simp add: indep_sets_def, intro conj_cong all_cong imp_cong ball_cong) blast+
41
42lemma (in prob_space) indep_events_finite_index_events:
43  "indep_events F I \<longleftrightarrow> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_events F J)"
44  by (auto simp: indep_events_def)
45
46lemma (in prob_space) indep_sets_finite_index_sets:
47  "indep_sets F I \<longleftrightarrow> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J)"
48proof (intro iffI allI impI)
49  assume *: "\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J"
50  show "indep_sets F I" unfolding indep_sets_def
51  proof (intro conjI ballI allI impI)
52    fix i assume "i \<in> I"
53    with *[THEN spec, of "{i}"] show "F i \<subseteq> events"
54      by (auto simp: indep_sets_def)
55  qed (insert *, auto simp: indep_sets_def)
56qed (auto simp: indep_sets_def)
57
58lemma (in prob_space) indep_sets_mono_index:
59  "J \<subseteq> I \<Longrightarrow> indep_sets F I \<Longrightarrow> indep_sets F J"
60  unfolding indep_sets_def by auto
61
62lemma (in prob_space) indep_sets_mono_sets:
63  assumes indep: "indep_sets F I"
64  assumes mono: "\<And>i. i\<in>I \<Longrightarrow> G i \<subseteq> F i"
65  shows "indep_sets G I"
66proof -
67  have "(\<forall>i\<in>I. F i \<subseteq> events) \<Longrightarrow> (\<forall>i\<in>I. G i \<subseteq> events)"
68    using mono by auto
69  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)"
70    using mono by (auto simp: Pi_iff)
71  ultimately show ?thesis
72    using indep by (auto simp: indep_sets_def)
73qed
74
75lemma (in prob_space) indep_sets_mono:
76  assumes indep: "indep_sets F I"
77  assumes mono: "J \<subseteq> I" "\<And>i. i\<in>J \<Longrightarrow> G i \<subseteq> F i"
78  shows "indep_sets G J"
79  apply (rule indep_sets_mono_sets)
80  apply (rule indep_sets_mono_index)
81  apply (fact +)
82  done
83
84lemma (in prob_space) indep_setsI:
85  assumes "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events"
86    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))"
87  shows "indep_sets F I"
88  using assms unfolding indep_sets_def by (auto simp: Pi_iff)
89
90lemma (in prob_space) indep_setsD:
91  assumes "indep_sets F I" and "J \<subseteq> I" "J \<noteq> {}" "finite J" "\<forall>j\<in>J. A j \<in> F j"
92  shows "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
93  using assms unfolding indep_sets_def by auto
94
95lemma (in prob_space) indep_setI:
96  assumes ev: "A \<subseteq> events" "B \<subseteq> events"
97    and indep: "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> prob (a \<inter> b) = prob a * prob b"
98  shows "indep_set A B"
99  unfolding indep_set_def
100proof (rule indep_setsI)
101  fix F J assume "J \<noteq> {}" "J \<subseteq> UNIV"
102    and F: "\<forall>j\<in>J. F j \<in> (case j of True \<Rightarrow> A | False \<Rightarrow> B)"
103  have "J \<in> Pow UNIV" by auto
104  with F \<open>J \<noteq> {}\<close> indep[of "F True" "F False"]
105  show "prob (\<Inter>j\<in>J. F j) = (\<Prod>j\<in>J. prob (F j))"
106    unfolding UNIV_bool Pow_insert by (auto simp: ac_simps)
107qed (auto split: bool.split simp: ev)
108
109lemma (in prob_space) indep_setD:
110  assumes indep: "indep_set A B" and ev: "a \<in> A" "b \<in> B"
111  shows "prob (a \<inter> b) = prob a * prob b"
112  using indep[unfolded indep_set_def, THEN indep_setsD, of UNIV "case_bool a b"] ev
113  by (simp add: ac_simps UNIV_bool)
114
115lemma (in prob_space)
116  assumes indep: "indep_set A B"
117  shows indep_setD_ev1: "A \<subseteq> events"
118    and indep_setD_ev2: "B \<subseteq> events"
119  using indep unfolding indep_set_def indep_sets_def UNIV_bool by auto
120
121lemma (in prob_space) indep_sets_Dynkin:
122  assumes indep: "indep_sets F I"
123  shows "indep_sets (\<lambda>i. Dynkin (space M) (F i)) I"
124    (is "indep_sets ?F I")
125proof (subst indep_sets_finite_index_sets, intro allI impI ballI)
126  fix J assume "finite J" "J \<subseteq> I" "J \<noteq> {}"
127  with indep have "indep_sets F J"
128    by (subst (asm) indep_sets_finite_index_sets) auto
129  { fix J K assume "indep_sets F K"
130    let ?G = "\<lambda>S i. if i \<in> S then ?F i else F i"
131    assume "finite J" "J \<subseteq> K"
132    then have "indep_sets (?G J) K"
133    proof induct
134      case (insert j J)
135      moreover define G where "G = ?G J"
136      ultimately have G: "indep_sets G K" "\<And>i. i \<in> K \<Longrightarrow> G i \<subseteq> events" and "j \<in> K"
137        by (auto simp: indep_sets_def)
138      let ?D = "{E\<in>events. indep_sets (G(j := {E})) K }"
139      { fix X assume X: "X \<in> events"
140        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)
141          \<Longrightarrow> prob ((\<Inter>i\<in>J. A i) \<inter> X) = prob X * (\<Prod>i\<in>J. prob (A i))"
142        have "indep_sets (G(j := {X})) K"
143        proof (rule indep_setsI)
144          fix i assume "i \<in> K" then show "(G(j:={X})) i \<subseteq> events"
145            using G X by auto
146        next
147          fix A J assume J: "J \<noteq> {}" "J \<subseteq> K" "finite J" "\<forall>i\<in>J. A i \<in> (G(j := {X})) i"
148          show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
149          proof cases
150            assume "j \<in> J"
151            with J have "A j = X" by auto
152            show ?thesis
153            proof cases
154              assume "J = {j}" then show ?thesis by simp
155            next
156              assume "J \<noteq> {j}"
157              have "prob (\<Inter>i\<in>J. A i) = prob ((\<Inter>i\<in>J-{j}. A i) \<inter> X)"
158                using \<open>j \<in> J\<close> \<open>A j = X\<close> by (auto intro!: arg_cong[where f=prob] split: if_split_asm)
159              also have "\<dots> = prob X * (\<Prod>i\<in>J-{j}. prob (A i))"
160              proof (rule indep)
161                show "J - {j} \<noteq> {}" "J - {j} \<subseteq> K" "finite (J - {j})" "j \<notin> J - {j}"
162                  using J \<open>J \<noteq> {j}\<close> \<open>j \<in> J\<close> by auto
163                show "\<forall>i\<in>J - {j}. A i \<in> G i"
164                  using J by auto
165              qed
166              also have "\<dots> = prob (A j) * (\<Prod>i\<in>J-{j}. prob (A i))"
167                using \<open>A j = X\<close> by simp
168              also have "\<dots> = (\<Prod>i\<in>J. prob (A i))"
169                unfolding prod.insert_remove[OF \<open>finite J\<close>, symmetric, of "\<lambda>i. prob  (A i)"]
170                using \<open>j \<in> J\<close> by (simp add: insert_absorb)
171              finally show ?thesis .
172            qed
173          next
174            assume "j \<notin> J"
175            with J have "\<forall>i\<in>J. A i \<in> G i" by (auto split: if_split_asm)
176            with J show ?thesis
177              by (intro indep_setsD[OF G(1)]) auto
178          qed
179        qed }
180      note indep_sets_insert = this
181      have "Dynkin_system (space M) ?D"
182      proof (rule Dynkin_systemI', simp_all cong del: indep_sets_cong, safe)
183        show "indep_sets (G(j := {{}})) K"
184          by (rule indep_sets_insert) auto
185      next
186        fix X assume X: "X \<in> events" and G': "indep_sets (G(j := {X})) K"
187        show "indep_sets (G(j := {space M - X})) K"
188        proof (rule indep_sets_insert)
189          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"
190          then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events"
191            using G by auto
192          have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) =
193              prob ((\<Inter>j\<in>J. A j) - (\<Inter>i\<in>insert j J. (A(j := X)) i))"
194            using A_sets sets.sets_into_space[of _ M] X \<open>J \<noteq> {}\<close>
195            by (auto intro!: arg_cong[where f=prob] split: if_split_asm)
196          also have "\<dots> = prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)"
197            using J \<open>J \<noteq> {}\<close> \<open>j \<notin> J\<close> A_sets X sets.sets_into_space
198            by (auto intro!: finite_measure_Diff sets.finite_INT split: if_split_asm)
199          finally have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) =
200              prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)" .
201          moreover {
202            have "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
203              using J A \<open>finite J\<close> by (intro indep_setsD[OF G(1)]) auto
204            then have "prob (\<Inter>j\<in>J. A j) = prob (space M) * (\<Prod>i\<in>J. prob (A i))"
205              using prob_space by simp }
206          moreover {
207            have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = (\<Prod>i\<in>insert j J. prob ((A(j := X)) i))"
208              using J A \<open>j \<in> K\<close> by (intro indep_setsD[OF G']) auto
209            then have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = prob X * (\<Prod>i\<in>J. prob (A i))"
210              using \<open>finite J\<close> \<open>j \<notin> J\<close> by (auto intro!: prod.cong) }
211          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))"
212            by (simp add: field_simps)
213          also have "\<dots> = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))"
214            using X A by (simp add: finite_measure_compl)
215          finally show "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))" .
216        qed (insert X, auto)
217      next
218        fix F :: "nat \<Rightarrow> 'a set" assume disj: "disjoint_family F" and "range F \<subseteq> ?D"
219        then have F: "\<And>i. F i \<in> events" "\<And>i. indep_sets (G(j:={F i})) K" by auto
220        show "indep_sets (G(j := {\<Union>k. F k})) K"
221        proof (rule indep_sets_insert)
222          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"
223          then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events"
224            using G by auto
225          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))"
226            using \<open>J \<noteq> {}\<close> \<open>j \<notin> J\<close> \<open>j \<in> K\<close> by (auto intro!: arg_cong[where f=prob] split: if_split_asm)
227          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))"
228          proof (rule finite_measure_UNION)
229            show "disjoint_family (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i)"
230              using disj by (rule disjoint_family_on_bisimulation) auto
231            show "range (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i) \<subseteq> events"
232              using A_sets F \<open>finite J\<close> \<open>J \<noteq> {}\<close> \<open>j \<notin> J\<close> by (auto intro!: sets.Int)
233          qed
234          moreover { fix k
235            from J A \<open>j \<in> K\<close> have "prob (\<Inter>i\<in>insert j J. (A(j := F k)) i) = prob (F k) * (\<Prod>i\<in>J. prob (A i))"
236              by (subst indep_setsD[OF F(2)]) (auto intro!: prod.cong split: if_split_asm)
237            also have "\<dots> = prob (F k) * prob (\<Inter>i\<in>J. A i)"
238              using J A \<open>j \<in> K\<close> by (subst indep_setsD[OF G(1)]) auto
239            finally have "prob (\<Inter>i\<in>insert j J. (A(j := F k)) i) = prob (F k) * prob (\<Inter>i\<in>J. A i)" . }
240          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)))"
241            by simp
242          moreover
243          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))"
244            using disj F(1) by (intro finite_measure_UNION sums_mult2) auto
245          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)))"
246            using J A \<open>j \<in> K\<close> by (subst indep_setsD[OF G(1), symmetric]) auto
247          ultimately
248          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))"
249            by (auto dest!: sums_unique)
250        qed (insert F, auto)
251      qed (insert sets.sets_into_space, auto)
252      then have mono: "Dynkin (space M) (G j) \<subseteq> {E \<in> events. indep_sets (G(j := {E})) K}"
253      proof (rule Dynkin_system.Dynkin_subset, safe)
254        fix X assume "X \<in> G j"
255        then show "X \<in> events" using G \<open>j \<in> K\<close> by auto
256        from \<open>indep_sets G K\<close>
257        show "indep_sets (G(j := {X})) K"
258          by (rule indep_sets_mono_sets) (insert \<open>X \<in> G j\<close>, auto)
259      qed
260      have "indep_sets (G(j:=?D)) K"
261      proof (rule indep_setsI)
262        fix i assume "i \<in> K" then show "(G(j := ?D)) i \<subseteq> events"
263          using G(2) by auto
264      next
265        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"
266        show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
267        proof cases
268          assume "j \<in> J"
269          with A have indep: "indep_sets (G(j := {A j})) K" by auto
270          from J A show ?thesis
271            by (intro indep_setsD[OF indep]) auto
272        next
273          assume "j \<notin> J"
274          with J A have "\<forall>i\<in>J. A i \<in> G i" by (auto split: if_split_asm)
275          with J show ?thesis
276            by (intro indep_setsD[OF G(1)]) auto
277        qed
278      qed
279      then have "indep_sets (G(j := Dynkin (space M) (G j))) K"
280        by (rule indep_sets_mono_sets) (insert mono, auto)
281      then show ?case
282        by (rule indep_sets_mono_sets) (insert \<open>j \<in> K\<close> \<open>j \<notin> J\<close>, auto simp: G_def)
283    qed (insert \<open>indep_sets F K\<close>, simp) }
284  from this[OF \<open>indep_sets F J\<close> \<open>finite J\<close> subset_refl]
285  show "indep_sets ?F J"
286    by (rule indep_sets_mono_sets) auto
287qed
288
289lemma (in prob_space) indep_sets_sigma:
290  assumes indep: "indep_sets F I"
291  assumes stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)"
292  shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I"
293proof -
294  from indep_sets_Dynkin[OF indep]
295  show ?thesis
296  proof (rule indep_sets_mono_sets, subst sigma_eq_Dynkin, simp_all add: stable)
297    fix i assume "i \<in> I"
298    with indep have "F i \<subseteq> events" by (auto simp: indep_sets_def)
299    with sets.sets_into_space show "F i \<subseteq> Pow (space M)" by auto
300  qed
301qed
302
303lemma (in prob_space) indep_sets_sigma_sets_iff:
304  assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)"
305  shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I \<longleftrightarrow> indep_sets F I"
306proof
307  assume "indep_sets F I" then show "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I"
308    by (rule indep_sets_sigma) fact
309next
310  assume "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I" then show "indep_sets F I"
311    by (rule indep_sets_mono_sets) (intro subsetI sigma_sets.Basic)
312qed
313
314definition (in prob_space)
315  indep_vars_def2: "indep_vars M' X I \<longleftrightarrow>
316    (\<forall>i\<in>I. random_variable (M' i) (X i)) \<and>
317    indep_sets (\<lambda>i. { X i -` A \<inter> space M | A. A \<in> sets (M' i)}) I"
318
319definition (in prob_space)
320  "indep_var Ma A Mb B \<longleftrightarrow> indep_vars (case_bool Ma Mb) (case_bool A B) UNIV"
321
322lemma (in prob_space) indep_vars_def:
323  "indep_vars M' X I \<longleftrightarrow>
324    (\<forall>i\<in>I. random_variable (M' i) (X i)) \<and>
325    indep_sets (\<lambda>i. sigma_sets (space M) { X i -` A \<inter> space M | A. A \<in> sets (M' i)}) I"
326  unfolding indep_vars_def2
327  apply (rule conj_cong[OF refl])
328  apply (rule indep_sets_sigma_sets_iff[symmetric])
329  apply (auto simp: Int_stable_def)
330  apply (rule_tac x="A \<inter> Aa" in exI)
331  apply auto
332  done
333
334lemma (in prob_space) indep_var_eq:
335  "indep_var S X T Y \<longleftrightarrow>
336    (random_variable S X \<and> random_variable T Y) \<and>
337    indep_set
338      (sigma_sets (space M) { X -` A \<inter> space M | A. A \<in> sets S})
339      (sigma_sets (space M) { Y -` A \<inter> space M | A. A \<in> sets T})"
340  unfolding indep_var_def indep_vars_def indep_set_def UNIV_bool
341  by (intro arg_cong2[where f="(\<and>)"] arg_cong2[where f=indep_sets] ext)
342     (auto split: bool.split)
343
344lemma (in prob_space) indep_sets2_eq:
345  "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)"
346  unfolding indep_set_def
347proof (intro iffI ballI conjI)
348  assume indep: "indep_sets (case_bool A B) UNIV"
349  { fix a b assume "a \<in> A" "b \<in> B"
350    with indep_setsD[OF indep, of UNIV "case_bool a b"]
351    show "prob (a \<inter> b) = prob a * prob b"
352      unfolding UNIV_bool by (simp add: ac_simps) }
353  from indep show "A \<subseteq> events" "B \<subseteq> events"
354    unfolding indep_sets_def UNIV_bool by auto
355next
356  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)"
357  show "indep_sets (case_bool A B) UNIV"
358  proof (rule indep_setsI)
359    fix i show "(case i of True \<Rightarrow> A | False \<Rightarrow> B) \<subseteq> events"
360      using * by (auto split: bool.split)
361  next
362    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)"
363    then have "J = {True} \<or> J = {False} \<or> J = {True,False}"
364      by (auto simp: UNIV_bool)
365    then show "prob (\<Inter>j\<in>J. X j) = (\<Prod>j\<in>J. prob (X j))"
366      using X * by auto
367  qed
368qed
369
370lemma (in prob_space) indep_set_sigma_sets:
371  assumes "indep_set A B"
372  assumes A: "Int_stable A" and B: "Int_stable B"
373  shows "indep_set (sigma_sets (space M) A) (sigma_sets (space M) B)"
374proof -
375  have "indep_sets (\<lambda>i. sigma_sets (space M) (case i of True \<Rightarrow> A | False \<Rightarrow> B)) UNIV"
376  proof (rule indep_sets_sigma)
377    show "indep_sets (case_bool A B) UNIV"
378      by (rule \<open>indep_set A B\<close>[unfolded indep_set_def])
379    fix i show "Int_stable (case i of True \<Rightarrow> A | False \<Rightarrow> B)"
380      using A B by (cases i) auto
381  qed
382  then show ?thesis
383    unfolding indep_set_def
384    by (rule indep_sets_mono_sets) (auto split: bool.split)
385qed
386
387lemma (in prob_space) indep_eventsI_indep_vars:
388  assumes indep: "indep_vars N X I"
389  assumes P: "\<And>i. i \<in> I \<Longrightarrow> {x\<in>space (N i). P i x} \<in> sets (N i)"
390  shows "indep_events (\<lambda>i. {x\<in>space M. P i (X i x)}) I"
391proof -
392  have "indep_sets (\<lambda>i. {X i -` A \<inter> space M |A. A \<in> sets (N i)}) I"
393    using indep unfolding indep_vars_def2 by auto
394  then show ?thesis
395    unfolding indep_events_def_alt
396  proof (rule indep_sets_mono_sets)
397    fix i assume "i \<in> I"
398    then have "{{x \<in> space M. P i (X i x)}} = {X i -` {x\<in>space (N i). P i x} \<inter> space M}"
399      using indep by (auto simp: indep_vars_def dest: measurable_space)
400    also have "\<dots> \<subseteq> {X i -` A \<inter> space M |A. A \<in> sets (N i)}"
401      using P[OF \<open>i \<in> I\<close>] by blast
402    finally show "{{x \<in> space M. P i (X i x)}} \<subseteq> {X i -` A \<inter> space M |A. A \<in> sets (N i)}" .
403  qed
404qed
405
406lemma (in prob_space) indep_sets_collect_sigma:
407  fixes I :: "'j \<Rightarrow> 'i set" and J :: "'j set" and E :: "'i \<Rightarrow> 'a set set"
408  assumes indep: "indep_sets E (\<Union>j\<in>J. I j)"
409  assumes Int_stable: "\<And>i j. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> Int_stable (E i)"
410  assumes disjoint: "disjoint_family_on I J"
411  shows "indep_sets (\<lambda>j. sigma_sets (space M) (\<Union>i\<in>I j. E i)) J"
412proof -
413  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) }"
414
415  from indep have E: "\<And>j i. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> E i \<subseteq> events"
416    unfolding indep_sets_def by auto
417  { fix j
418    let ?S = "sigma_sets (space M) (\<Union>i\<in>I j. E i)"
419    assume "j \<in> J"
420    from E[OF this] interpret S: sigma_algebra "space M" ?S
421      using sets.sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto
422
423    have "sigma_sets (space M) (\<Union>i\<in>I j. E i) = sigma_sets (space M) (?E j)"
424    proof (rule sigma_sets_eqI)
425      fix A assume "A \<in> (\<Union>i\<in>I j. E i)"
426      then guess i ..
427      then show "A \<in> sigma_sets (space M) (?E j)"
428        by (auto intro!: sigma_sets.intros(2-) exI[of _ "{i}"] exI[of _ "\<lambda>i. A"])
429    next
430      fix A assume "A \<in> ?E j"
431      then obtain E' K where "finite K" "K \<noteq> {}" "K \<subseteq> I j" "\<And>k. k \<in> K \<Longrightarrow> E' k \<in> E k"
432        and A: "A = (\<Inter>k\<in>K. E' k)"
433        by auto
434      then have "A \<in> ?S" unfolding A
435        by (safe intro!: S.finite_INT) auto
436      then show "A \<in> sigma_sets (space M) (\<Union>i\<in>I j. E i)"
437        by simp
438    qed }
439  moreover have "indep_sets (\<lambda>j. sigma_sets (space M) (?E j)) J"
440  proof (rule indep_sets_sigma)
441    show "indep_sets ?E J"
442    proof (intro indep_setsI)
443      fix j assume "j \<in> J" with E show "?E j \<subseteq> events" by (force  intro!: sets.finite_INT)
444    next
445      fix K A assume K: "K \<noteq> {}" "K \<subseteq> J" "finite K"
446        and "\<forall>j\<in>K. A j \<in> ?E j"
447      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)"
448        by simp
449      from bchoice[OF this] guess E' ..
450      from bchoice[OF this] obtain L
451        where A: "\<And>j. j\<in>K \<Longrightarrow> A j = (\<Inter>l\<in>L j. E' j l)"
452        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"
453        and E': "\<And>j l. j\<in>K \<Longrightarrow> l \<in> L j \<Longrightarrow> E' j l \<in> E l"
454        by auto
455
456      { fix k l j assume "k \<in> K" "j \<in> K" "l \<in> L j" "l \<in> L k"
457        have "k = j"
458        proof (rule ccontr)
459          assume "k \<noteq> j"
460          with disjoint \<open>K \<subseteq> J\<close> \<open>k \<in> K\<close> \<open>j \<in> K\<close> have "I k \<inter> I j = {}"
461            unfolding disjoint_family_on_def by auto
462          with L(2,3)[OF \<open>j \<in> K\<close>] L(2,3)[OF \<open>k \<in> K\<close>]
463          show False using \<open>l \<in> L k\<close> \<open>l \<in> L j\<close> by auto
464        qed }
465      note L_inj = this
466
467      define k where "k l = (SOME k. k \<in> K \<and> l \<in> L k)" for l
468      { fix x j l assume *: "j \<in> K" "l \<in> L j"
469        have "k l = j" unfolding k_def
470        proof (rule some_equality)
471          fix k assume "k \<in> K \<and> l \<in> L k"
472          with * L_inj show "k = j" by auto
473        qed (insert *, simp) }
474      note k_simp[simp] = this
475      let ?E' = "\<lambda>l. E' (k l) l"
476      have "prob (\<Inter>j\<in>K. A j) = prob (\<Inter>l\<in>(\<Union>k\<in>K. L k). ?E' l)"
477        by (auto simp: A intro!: arg_cong[where f=prob])
478      also have "\<dots> = (\<Prod>l\<in>(\<Union>k\<in>K. L k). prob (?E' l))"
479        using L K E' by (intro indep_setsD[OF indep]) (simp_all add: UN_mono)
480      also have "\<dots> = (\<Prod>j\<in>K. \<Prod>l\<in>L j. prob (E' j l))"
481        using K L L_inj by (subst prod.UNION_disjoint) auto
482      also have "\<dots> = (\<Prod>j\<in>K. prob (A j))"
483        using K L E' by (auto simp add: A intro!: prod.cong indep_setsD[OF indep, symmetric]) blast
484      finally show "prob (\<Inter>j\<in>K. A j) = (\<Prod>j\<in>K. prob (A j))" .
485    qed
486  next
487    fix j assume "j \<in> J"
488    show "Int_stable (?E j)"
489    proof (rule Int_stableI)
490      fix a assume "a \<in> ?E j" then obtain Ka Ea
491        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
492      fix b assume "b \<in> ?E j" then obtain Kb Eb
493        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
494      let ?f = "\<lambda>k. (if k \<in> Ka \<inter> Kb then Ea k \<inter> Eb k else if k \<in> Kb then Eb k else if k \<in> Ka then Ea k else {})"
495      have "Ka \<union> Kb = (Ka \<inter> Kb) \<union> (Kb - Ka) \<union> (Ka - Kb)"
496        by blast
497      moreover have "(\<Inter>x\<in>Ka \<inter> Kb. Ea x \<inter> Eb x) \<inter>
498        (\<Inter>x\<in>Kb - Ka. Eb x) \<inter> (\<Inter>x\<in>Ka - Kb. Ea x) = (\<Inter>k\<in>Ka. Ea k) \<inter> (\<Inter>k\<in>Kb. Eb k)"
499        by auto
500      ultimately have "(\<Inter>k\<in>Ka \<union> Kb. ?f k) = (\<Inter>k\<in>Ka. Ea k) \<inter> (\<Inter>k\<in>Kb. Eb k)" (is "?lhs = ?rhs")
501        by (simp only: image_Un Inter_Un_distrib) simp
502      then have "a \<inter> b = (\<Inter>k\<in>Ka \<union> Kb. ?f k)"
503        by (simp only: a(1) b(1))
504      with a b \<open>j \<in> J\<close> Int_stableD[OF Int_stable] show "a \<inter> b \<in> ?E j"
505        by (intro CollectI exI[of _ "Ka \<union> Kb"] exI[of _ ?f]) auto
506    qed
507  qed
508  ultimately show ?thesis
509    by (simp cong: indep_sets_cong)
510qed
511
512lemma (in prob_space) indep_vars_restrict:
513  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"
514  shows "indep_vars (\<lambda>j. PiM (K j) M') (\<lambda>j \<omega>. restrict (\<lambda>i. X i \<omega>) (K j)) L"
515  unfolding indep_vars_def
516proof safe
517  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>)"
518    using K ind by (auto simp: indep_vars_def intro!: measurable_restrict)
519next
520  have X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable M (M' i)"
521    using ind by (auto simp: indep_vars_def)
522  let ?proj = "\<lambda>j S. {(\<lambda>\<omega>. \<lambda>i\<in>K j. X i \<omega>) -` A \<inter> space M |A. A \<in> S}"
523  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) })"
524  show "indep_sets (\<lambda>i. sigma_sets (space M) (?proj i (sets (Pi\<^sub>M (K i) M')))) L"
525  proof (rule indep_sets_mono_sets)
526    fix j assume j: "j \<in> L"
527    have "sigma_sets (space M) (?proj j (sets (Pi\<^sub>M (K j) M'))) =
528      sigma_sets (space M) (sigma_sets (space M) (?proj j (prod_algebra (K j) M')))"
529      using j K X[THEN measurable_space] unfolding sets_PiM
530      by (subst sigma_sets_vimage_commute) (auto simp add: Pi_iff)
531    also have "\<dots> = sigma_sets (space M) (?proj j (prod_algebra (K j) M'))"
532      by (rule sigma_sets_sigma_sets_eq) auto
533    also have "\<dots> \<subseteq> ?UN j"
534    proof (rule sigma_sets_mono, safe del: disjE elim!: prod_algebraE)
535      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)"
536      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"
537      proof cases
538        assume "K j = {}" with J show ?thesis
539          by (auto simp add: sigma_sets_empty_eq prod_emb_def)
540      next
541        assume "K j \<noteq> {}" with J have "J \<noteq> {}"
542          by auto
543        { interpret sigma_algebra "space M" "?UN j"
544            by (rule sigma_algebra_sigma_sets) auto
545          have "\<And>A. (\<And>i. i \<in> J \<Longrightarrow> A i \<in> ?UN j) \<Longrightarrow> \<Inter>(A ` J) \<in> ?UN j"
546            using \<open>finite J\<close> \<open>J \<noteq> {}\<close> by (rule finite_INT) blast }
547        note INT = this
548
549        from \<open>J \<noteq> {}\<close> J K E[rule_format, THEN sets.sets_into_space] j
550        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
551          = (\<Inter>i\<in>J. X i -` E i \<inter> space M)"
552          apply (subst prod_emb_PiE[OF _ ])
553          apply auto []
554          apply auto []
555          apply (auto simp add: Pi_iff intro!: X[THEN measurable_space])
556          apply (erule_tac x=i in ballE)
557          apply auto
558          done
559        also have "\<dots> \<in> ?UN j"
560          apply (rule INT)
561          apply (rule sigma_sets.Basic)
562          using \<open>J \<subseteq> K j\<close> E
563          apply auto
564          done
565        finally show ?thesis .
566      qed
567    qed
568    finally show "sigma_sets (space M) (?proj j (sets (Pi\<^sub>M (K j) M'))) \<subseteq> ?UN j" .
569  next
570    show "indep_sets ?UN L"
571    proof (rule indep_sets_collect_sigma)
572      show "indep_sets (\<lambda>i. {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) (\<Union>j\<in>L. K j)"
573      proof (rule indep_sets_mono_index)
574        show "indep_sets (\<lambda>i. {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"
575          using ind unfolding indep_vars_def2 by auto
576        show "(\<Union>l\<in>L. K l) \<subseteq> I"
577          using K by auto
578      qed
579    next
580      fix l i assume "l \<in> L" "i \<in> K l"
581      show "Int_stable {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
582        apply (auto simp: Int_stable_def)
583        apply (rule_tac x="A \<inter> Aa" in exI)
584        apply auto
585        done
586    qed fact
587  qed
588qed
589
590lemma (in prob_space) indep_var_restrict:
591  assumes ind: "indep_vars M' X I" and AB: "A \<inter> B = {}" "A \<subseteq> I" "B \<subseteq> I"
592  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)"
593proof -
594  have *:
595    "case_bool (Pi\<^sub>M A M') (Pi\<^sub>M B M') = (\<lambda>b. PiM (case_bool A B b) M')"
596    "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>)"
597    by (simp_all add: fun_eq_iff split: bool.split)
598  show ?thesis
599    unfolding indep_var_def * using AB
600    by (intro indep_vars_restrict[OF ind]) (auto simp: disjoint_family_on_def split: bool.split)
601qed
602
603lemma (in prob_space) indep_vars_subset:
604  assumes "indep_vars M' X I" "J \<subseteq> I"
605  shows "indep_vars M' X J"
606  using assms unfolding indep_vars_def indep_sets_def
607  by auto
608
609lemma (in prob_space) indep_vars_cong:
610  "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"
611  unfolding indep_vars_def2 by (intro conj_cong indep_sets_cong) auto
612
613definition (in prob_space) tail_events where
614  "tail_events A = (\<Inter>n. sigma_sets (space M) (\<Union> (A ` {n..})))"
615
616lemma (in prob_space) tail_events_sets:
617  assumes A: "\<And>i::nat. A i \<subseteq> events"
618  shows "tail_events A \<subseteq> events"
619proof
620  fix X assume X: "X \<in> tail_events A"
621  let ?A = "(\<Inter>n. sigma_sets (space M) (\<Union> (A ` {n..})))"
622  from X have "\<And>n::nat. X \<in> sigma_sets (space M) (\<Union> (A ` {n..}))" by (auto simp: tail_events_def)
623  from this[of 0] have "X \<in> sigma_sets (space M) (\<Union>(A ` UNIV))" by simp
624  then show "X \<in> events"
625    by induct (insert A, auto)
626qed
627
628lemma (in prob_space) sigma_algebra_tail_events:
629  assumes "\<And>i::nat. sigma_algebra (space M) (A i)"
630  shows "sigma_algebra (space M) (tail_events A)"
631  unfolding tail_events_def
632proof (simp add: sigma_algebra_iff2, safe)
633  let ?A = "(\<Inter>n. sigma_sets (space M) (\<Union> (A ` {n..})))"
634  interpret A: sigma_algebra "space M" "A i" for i by fact
635  { fix X x assume "X \<in> ?A" "x \<in> X"
636    then have "\<And>n. X \<in> sigma_sets (space M) (\<Union> (A ` {n..}))" by auto
637    from this[of 0] have "X \<in> sigma_sets (space M) (\<Union>(A ` UNIV))" by simp
638    then have "X \<subseteq> space M"
639      by induct (insert A.sets_into_space, auto)
640    with \<open>x \<in> X\<close> show "x \<in> space M" by auto }
641  { fix F :: "nat \<Rightarrow> 'a set" and n assume "range F \<subseteq> ?A"
642    then show "(\<Union>(F ` UNIV)) \<in> sigma_sets (space M) (\<Union> (A ` {n..}))"
643      by (intro sigma_sets.Union) auto }
644qed (auto intro!: sigma_sets.Compl sigma_sets.Empty)
645
646lemma (in prob_space) kolmogorov_0_1_law:
647  fixes A :: "nat \<Rightarrow> 'a set set"
648  assumes "\<And>i::nat. sigma_algebra (space M) (A i)"
649  assumes indep: "indep_sets A UNIV"
650  and X: "X \<in> tail_events A"
651  shows "prob X = 0 \<or> prob X = 1"
652proof -
653  have A: "\<And>i. A i \<subseteq> events"
654    using indep unfolding indep_sets_def by simp
655
656  let ?D = "{D \<in> events. prob (X \<inter> D) = prob X * prob D}"
657  interpret A: sigma_algebra "space M" "A i" for i by fact
658  interpret T: sigma_algebra "space M" "tail_events A"
659    by (rule sigma_algebra_tail_events) fact
660  have "X \<subseteq> space M" using T.space_closed X by auto
661
662  have X_in: "X \<in> events"
663    using tail_events_sets A X by auto
664
665  interpret D: Dynkin_system "space M" ?D
666  proof (rule Dynkin_systemI)
667    fix D assume "D \<in> ?D" then show "D \<subseteq> space M"
668      using sets.sets_into_space by auto
669  next
670    show "space M \<in> ?D"
671      using prob_space \<open>X \<subseteq> space M\<close> by (simp add: Int_absorb2)
672  next
673    fix A assume A: "A \<in> ?D"
674    have "prob (X \<inter> (space M - A)) = prob (X - (X \<inter> A))"
675      using \<open>X \<subseteq> space M\<close> by (auto intro!: arg_cong[where f=prob])
676    also have "\<dots> = prob X - prob (X \<inter> A)"
677      using X_in A by (intro finite_measure_Diff) auto
678    also have "\<dots> = prob X * prob (space M) - prob X * prob A"
679      using A prob_space by auto
680    also have "\<dots> = prob X * prob (space M - A)"
681      using X_in A sets.sets_into_space
682      by (subst finite_measure_Diff) (auto simp: field_simps)
683    finally show "space M - A \<in> ?D"
684      using A \<open>X \<subseteq> space M\<close> by auto
685  next
686    fix F :: "nat \<Rightarrow> 'a set" assume dis: "disjoint_family F" and "range F \<subseteq> ?D"
687    then have F: "range F \<subseteq> events" "\<And>i. prob (X \<inter> F i) = prob X * prob (F i)"
688      by auto
689    have "(\<lambda>i. prob (X \<inter> F i)) sums prob (\<Union>i. X \<inter> F i)"
690    proof (rule finite_measure_UNION)
691      show "range (\<lambda>i. X \<inter> F i) \<subseteq> events"
692        using F X_in by auto
693      show "disjoint_family (\<lambda>i. X \<inter> F i)"
694        using dis by (rule disjoint_family_on_bisimulation) auto
695    qed
696    with F have "(\<lambda>i. prob X * prob (F i)) sums prob (X \<inter> (\<Union>i. F i))"
697      by simp
698    moreover have "(\<lambda>i. prob X * prob (F i)) sums (prob X * prob (\<Union>i. F i))"
699      by (intro sums_mult finite_measure_UNION F dis)
700    ultimately have "prob (X \<inter> (\<Union>i. F i)) = prob X * prob (\<Union>i. F i)"
701      by (auto dest!: sums_unique)
702    with F show "(\<Union>i. F i) \<in> ?D"
703      by auto
704  qed
705
706  { fix n
707    have "indep_sets (\<lambda>b. sigma_sets (space M) (\<Union>m\<in>case_bool {..n} {Suc n..} b. A m)) UNIV"
708    proof (rule indep_sets_collect_sigma)
709      have *: "(\<Union>b. case b of True \<Rightarrow> {..n} | False \<Rightarrow> {Suc n..}) = UNIV" (is "?U = _")
710        by (simp split: bool.split add: set_eq_iff) (metis not_less_eq_eq)
711      with indep show "indep_sets A ?U" by simp
712      show "disjoint_family (case_bool {..n} {Suc n..})"
713        unfolding disjoint_family_on_def by (auto split: bool.split)
714      fix m
715      show "Int_stable (A m)"
716        unfolding Int_stable_def using A.Int by auto
717    qed
718    also have "(\<lambda>b. sigma_sets (space M) (\<Union>m\<in>case_bool {..n} {Suc n..} b. A m)) =
719      case_bool (sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) (sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m))"
720      by (auto intro!: ext split: bool.split)
721    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))"
722      unfolding indep_set_def by simp
723
724    have "sigma_sets (space M) (\<Union>m\<in>{..n}. A m) \<subseteq> ?D"
725    proof (simp add: subset_eq, rule)
726      fix D assume D: "D \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)"
727      have "X \<in> sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m)"
728        using X unfolding tail_events_def by simp
729      from indep_setD[OF indep D this] indep_setD_ev1[OF indep] D
730      show "D \<in> events \<and> prob (X \<inter> D) = prob X * prob D"
731        by (auto simp add: ac_simps)
732    qed }
733  then have "(\<Union>n. sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) \<subseteq> ?D" (is "?A \<subseteq> _")
734    by auto
735
736  note \<open>X \<in> tail_events A\<close>
737  also {
738    have "\<And>n. sigma_sets (space M) (\<Union>i\<in>{n..}. A i) \<subseteq> sigma_sets (space M) ?A"
739      by (intro sigma_sets_subseteq UN_mono) auto
740   then have "tail_events A \<subseteq> sigma_sets (space M) ?A"
741      unfolding tail_events_def by auto }
742  also have "sigma_sets (space M) ?A = Dynkin (space M) ?A"
743  proof (rule sigma_eq_Dynkin)
744    { fix B n assume "B \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)"
745      then have "B \<subseteq> space M"
746        by induct (insert A sets.sets_into_space[of _ M], auto) }
747    then show "?A \<subseteq> Pow (space M)" by auto
748    show "Int_stable ?A"
749    proof (rule Int_stableI)
750      fix a assume "a \<in> ?A" then guess n .. note a = this
751      fix b assume "b \<in> ?A" then guess m .. note b = this
752      interpret Amn: sigma_algebra "space M" "sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
753        using A sets.sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto
754      have "sigma_sets (space M) (\<Union>i\<in>{..n}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
755        by (intro sigma_sets_subseteq UN_mono) auto
756      with a have "a \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto
757      moreover
758      have "sigma_sets (space M) (\<Union>i\<in>{..m}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
759        by (intro sigma_sets_subseteq UN_mono) auto
760      with b have "b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto
761      ultimately have "a \<inter> b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
762        using Amn.Int[of a b] by simp
763      then show "a \<inter> b \<in> (\<Union>n. sigma_sets (space M) (\<Union>i\<in>{..n}. A i))" by auto
764    qed
765  qed
766  also have "Dynkin (space M) ?A \<subseteq> ?D"
767    using \<open>?A \<subseteq> ?D\<close> by (auto intro!: D.Dynkin_subset)
768  finally show ?thesis by auto
769qed
770
771lemma (in prob_space) borel_0_1_law:
772  fixes F :: "nat \<Rightarrow> 'a set"
773  assumes F2: "indep_events F UNIV"
774  shows "prob (\<Inter>n. \<Union>m\<in>{n..}. F m) = 0 \<or> prob (\<Inter>n. \<Union>m\<in>{n..}. F m) = 1"
775proof (rule kolmogorov_0_1_law[of "\<lambda>i. sigma_sets (space M) { F i }"])
776  have F1: "range F \<subseteq> events"
777    using F2 by (simp add: indep_events_def subset_eq)
778  { fix i show "sigma_algebra (space M) (sigma_sets (space M) {F i})"
779      using sigma_algebra_sigma_sets[of "{F i}" "space M"] F1 sets.sets_into_space
780      by auto }
781  show "indep_sets (\<lambda>i. sigma_sets (space M) {F i}) UNIV"
782  proof (rule indep_sets_sigma)
783    show "indep_sets (\<lambda>i. {F i}) UNIV"
784      unfolding indep_events_def_alt[symmetric] by fact
785    fix i show "Int_stable {F i}"
786      unfolding Int_stable_def by simp
787  qed
788  let ?Q = "\<lambda>n. \<Union>i\<in>{n..}. F i"
789  show "(\<Inter>n. \<Union>m\<in>{n..}. F m) \<in> tail_events (\<lambda>i. sigma_sets (space M) {F i})"
790    unfolding tail_events_def
791  proof
792    fix j
793    interpret S: sigma_algebra "space M" "sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})"
794      using order_trans[OF F1 sets.space_closed]
795      by (intro sigma_algebra_sigma_sets) (simp add: sigma_sets_singleton subset_eq)
796    have "(\<Inter>n. ?Q n) = (\<Inter>n\<in>{j..}. ?Q n)"
797      by (intro decseq_SucI INT_decseq_offset UN_mono) auto
798    also have "\<dots> \<in> sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})"
799      using order_trans[OF F1 sets.space_closed]
800      by (safe intro!: S.countable_INT S.countable_UN)
801         (auto simp: sigma_sets_singleton intro!: sigma_sets.Basic bexI)
802    finally show "(\<Inter>n. ?Q n) \<in> sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})"
803      by simp
804  qed
805qed
806
807lemma (in prob_space) borel_0_1_law_AE:
808  fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
809  assumes "indep_events (\<lambda>m. {x\<in>space M. P m x}) UNIV" (is "indep_events ?P _")
810  shows "(AE x in M. infinite {m. P m x}) \<or> (AE x in M. finite {m. P m x})"
811proof -
812  have [measurable]: "\<And>m. {x\<in>space M. P m x} \<in> sets M"
813    using assms by (auto simp: indep_events_def)
814  have *: "(\<Inter>n. \<Union>m\<in>{n..}. {x \<in> space M. P m x}) \<in> events"
815    by simp
816  from assms have "prob (\<Inter>n. \<Union>m\<in>{n..}. ?P m) = 0 \<or> prob (\<Inter>n. \<Union>m\<in>{n..}. ?P m) = 1"
817    by (rule borel_0_1_law)
818  also have "prob (\<Inter>n. \<Union>m\<in>{n..}. ?P m) = 1 \<longleftrightarrow> (AE x in M. infinite {m. P m x})"
819    using * by (simp add: prob_eq_1)
820      (simp add: Bex_def infinite_nat_iff_unbounded_le)
821  also have "prob (\<Inter>n. \<Union>m\<in>{n..}. ?P m) = 0 \<longleftrightarrow> (AE x in M. finite {m. P m x})"
822    using * by (simp add: prob_eq_0)
823      (auto simp add: Ball_def finite_nat_iff_bounded not_less [symmetric])
824  finally show ?thesis
825    by blast
826qed
827
828lemma (in prob_space) indep_sets_finite:
829  assumes I: "I \<noteq> {}" "finite I"
830    and F: "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events" "\<And>i. i \<in> I \<Longrightarrow> space M \<in> F i"
831  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)))"
832proof
833  assume *: "indep_sets F I"
834  from I show "\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j))"
835    by (intro indep_setsD[OF *] ballI) auto
836next
837  assume indep: "\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j))"
838  show "indep_sets F I"
839  proof (rule indep_setsI[OF F(1)])
840    fix A J assume J: "J \<noteq> {}" "J \<subseteq> I" "finite J"
841    assume A: "\<forall>j\<in>J. A j \<in> F j"
842    let ?A = "\<lambda>j. if j \<in> J then A j else space M"
843    have "prob (\<Inter>j\<in>I. ?A j) = prob (\<Inter>j\<in>J. A j)"
844      using subset_trans[OF F(1) sets.space_closed] J A
845      by (auto intro!: arg_cong[where f=prob] split: if_split_asm) blast
846    also
847    from A F have "(\<lambda>j. if j \<in> J then A j else space M) \<in> Pi I F" (is "?A \<in> _")
848      by (auto split: if_split_asm)
849    with indep have "prob (\<Inter>j\<in>I. ?A j) = (\<Prod>j\<in>I. prob (?A j))"
850      by auto
851    also have "\<dots> = (\<Prod>j\<in>J. prob (A j))"
852      unfolding if_distrib prod.If_cases[OF \<open>finite I\<close>]
853      using prob_space \<open>J \<subseteq> I\<close> by (simp add: Int_absorb1 prod.neutral_const)
854    finally show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" ..
855  qed
856qed
857
858lemma (in prob_space) indep_vars_finite:
859  fixes I :: "'i set"
860  assumes I: "I \<noteq> {}" "finite I"
861    and M': "\<And>i. i \<in> I \<Longrightarrow> sets (M' i) = sigma_sets (space (M' i)) (E i)"
862    and rv: "\<And>i. i \<in> I \<Longrightarrow> random_variable (M' i) (X i)"
863    and Int_stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (E i)"
864    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))"
865  shows "indep_vars M' X I \<longleftrightarrow>
866    (\<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)))"
867proof -
868  from rv have X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> space M \<rightarrow> space (M' i)"
869    unfolding measurable_def by simp
870
871  { fix i assume "i\<in>I"
872    from closed[OF \<open>i \<in> I\<close>]
873    have "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}
874      = sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> E i}"
875      unfolding sigma_sets_vimage_commute[OF X, OF \<open>i \<in> I\<close>, symmetric] M'[OF \<open>i \<in> I\<close>]
876      by (subst sigma_sets_sigma_sets_eq) auto }
877  note sigma_sets_X = this
878
879  { fix i assume "i\<in>I"
880    have "Int_stable {X i -` A \<inter> space M |A. A \<in> E i}"
881    proof (rule Int_stableI)
882      fix a assume "a \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
883      then obtain A where "a = X i -` A \<inter> space M" "A \<in> E i" by auto
884      moreover
885      fix b assume "b \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
886      then obtain B where "b = X i -` B \<inter> space M" "B \<in> E i" by auto
887      moreover
888      have "(X i -` A \<inter> space M) \<inter> (X i -` B \<inter> space M) = X i -` (A \<inter> B) \<inter> space M" by auto
889      moreover note Int_stable[OF \<open>i \<in> I\<close>]
890      ultimately
891      show "a \<inter> b \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
892        by (auto simp del: vimage_Int intro!: exI[of _ "A \<inter> B"] dest: Int_stableD)
893    qed }
894  note indep_sets_X = indep_sets_sigma_sets_iff[OF this]
895
896  { fix i assume "i \<in> I"
897    { fix A assume "A \<in> E i"
898      with M'[OF \<open>i \<in> I\<close>] have "A \<in> sets (M' i)" by auto
899      moreover
900      from rv[OF \<open>i\<in>I\<close>] have "X i \<in> measurable M (M' i)" by auto
901      ultimately
902      have "X i -` A \<inter> space M \<in> sets M" by (auto intro: measurable_sets) }
903    with X[OF \<open>i\<in>I\<close>] space[OF \<open>i\<in>I\<close>]
904    have "{X i -` A \<inter> space M |A. A \<in> E i} \<subseteq> events"
905      "space M \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
906      by (auto intro!: exI[of _ "space (M' i)"]) }
907  note indep_sets_finite_X = indep_sets_finite[OF I this]
908
909  have "(\<forall>A\<in>\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> E i}. prob (\<Inter>(A ` I)) = (\<Prod>j\<in>I. prob (A j))) =
910    (\<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)))"
911    (is "?L = ?R")
912  proof safe
913    fix A assume ?L and A: "A \<in> (\<Pi> i\<in>I. E i)"
914    from \<open>?L\<close>[THEN bspec, of "\<lambda>i. X i -` A i \<inter> space M"] A \<open>I \<noteq> {}\<close>
915    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))"
916      by (auto simp add: Pi_iff)
917  next
918    fix A assume ?R and A: "A \<in> (\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> E i})"
919    from A have "\<forall>i\<in>I. \<exists>B. A i = X i -` B \<inter> space M \<and> B \<in> E i" by auto
920    from bchoice[OF this] obtain B where B: "\<forall>i\<in>I. A i = X i -` B i \<inter> space M"
921      "B \<in> (\<Pi> i\<in>I. E i)" by auto
922    from \<open>?R\<close>[THEN bspec, OF B(2)] B(1) \<open>I \<noteq> {}\<close>
923    show "prob (\<Inter>(A ` I)) = (\<Prod>j\<in>I. prob (A j))"
924      by simp
925  qed
926  then show ?thesis using \<open>I \<noteq> {}\<close>
927    by (simp add: rv indep_vars_def indep_sets_X sigma_sets_X indep_sets_finite_X cong: indep_sets_cong)
928qed
929
930lemma (in prob_space) indep_vars_compose:
931  assumes "indep_vars M' X I"
932  assumes rv: "\<And>i. i \<in> I \<Longrightarrow> Y i \<in> measurable (M' i) (N i)"
933  shows "indep_vars N (\<lambda>i. Y i \<circ> X i) I"
934  unfolding indep_vars_def
935proof
936  from rv \<open>indep_vars M' X I\<close>
937  show "\<forall>i\<in>I. random_variable (N i) (Y i \<circ> X i)"
938    by (auto simp: indep_vars_def)
939
940  have "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"
941    using \<open>indep_vars M' X I\<close> by (simp add: indep_vars_def)
942  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"
943  proof (rule indep_sets_mono_sets)
944    fix i assume "i \<in> I"
945    with \<open>indep_vars M' X I\<close> have X: "X i \<in> space M \<rightarrow> space (M' i)"
946      unfolding indep_vars_def measurable_def by auto
947    { fix A assume "A \<in> sets (N i)"
948      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)"
949        by (intro exI[of _ "Y i -` A \<inter> space (M' i)"])
950           (auto simp: vimage_comp intro!: measurable_sets rv \<open>i \<in> I\<close> funcset_mem[OF X]) }
951    then show "sigma_sets (space M) {(Y i \<circ> X i) -` A \<inter> space M |A. A \<in> sets (N i)} \<subseteq>
952      sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
953      by (intro sigma_sets_subseteq) (auto simp: vimage_comp)
954  qed
955qed
956
957lemma (in prob_space) indep_vars_compose2:
958  assumes "indep_vars M' X I"
959  assumes rv: "\<And>i. i \<in> I \<Longrightarrow> Y i \<in> measurable (M' i) (N i)"
960  shows "indep_vars N (\<lambda>i x. Y i (X i x)) I"
961  using indep_vars_compose [OF assms] by (simp add: comp_def)
962
963lemma (in prob_space) indep_var_compose:
964  assumes "indep_var M1 X1 M2 X2" "Y1 \<in> measurable M1 N1" "Y2 \<in> measurable M2 N2"
965  shows "indep_var N1 (Y1 \<circ> X1) N2 (Y2 \<circ> X2)"
966proof -
967  have "indep_vars (case_bool N1 N2) (\<lambda>b. case_bool Y1 Y2 b \<circ> case_bool X1 X2 b) UNIV"
968    using assms
969    by (intro indep_vars_compose[where M'="case_bool M1 M2"])
970       (auto simp: indep_var_def split: bool.split)
971  also have "(\<lambda>b. case_bool Y1 Y2 b \<circ> case_bool X1 X2 b) = case_bool (Y1 \<circ> X1) (Y2 \<circ> X2)"
972    by (simp add: fun_eq_iff split: bool.split)
973  finally show ?thesis
974    unfolding indep_var_def .
975qed
976
977lemma (in prob_space) indep_vars_Min:
978  fixes X :: "'i \<Rightarrow> 'a \<Rightarrow> real"
979  assumes I: "finite I" "i \<notin> I" and indep: "indep_vars (\<lambda>_. borel) X (insert i I)"
980  shows "indep_var borel (X i) borel (\<lambda>\<omega>. Min ((\<lambda>i. X i \<omega>)`I))"
981proof -
982  have "indep_var
983    borel ((\<lambda>f. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) {i}))
984    borel ((\<lambda>f. Min (f`I)) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) I))"
985    using I by (intro indep_var_compose[OF indep_var_restrict[OF indep]] borel_measurable_Min) auto
986  also have "((\<lambda>f. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) {i})) = X i"
987    by auto
988  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))"
989    by (auto cong: rev_conj_cong)
990  finally show ?thesis
991    unfolding indep_var_def .
992qed
993
994lemma (in prob_space) indep_vars_sum:
995  fixes X :: "'i \<Rightarrow> 'a \<Rightarrow> real"
996  assumes I: "finite I" "i \<notin> I" and indep: "indep_vars (\<lambda>_. borel) X (insert i I)"
997  shows "indep_var borel (X i) borel (\<lambda>\<omega>. \<Sum>i\<in>I. X i \<omega>)"
998proof -
999  have "indep_var
1000    borel ((\<lambda>f. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) {i}))
1001    borel ((\<lambda>f. \<Sum>i\<in>I. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) I))"
1002    using I by (intro indep_var_compose[OF indep_var_restrict[OF indep]] ) auto
1003  also have "((\<lambda>f. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) {i})) = X i"
1004    by auto
1005  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>)"
1006    by (auto cong: rev_conj_cong)
1007  finally show ?thesis .
1008qed
1009
1010lemma (in prob_space) indep_vars_prod:
1011  fixes X :: "'i \<Rightarrow> 'a \<Rightarrow> real"
1012  assumes I: "finite I" "i \<notin> I" and indep: "indep_vars (\<lambda>_. borel) X (insert i I)"
1013  shows "indep_var borel (X i) borel (\<lambda>\<omega>. \<Prod>i\<in>I. X i \<omega>)"
1014proof -
1015  have "indep_var
1016    borel ((\<lambda>f. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) {i}))
1017    borel ((\<lambda>f. \<Prod>i\<in>I. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) I))"
1018    using I by (intro indep_var_compose[OF indep_var_restrict[OF indep]] ) auto
1019  also have "((\<lambda>f. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) {i})) = X i"
1020    by auto
1021  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>)"
1022    by (auto cong: rev_conj_cong)
1023  finally show ?thesis .
1024qed
1025
1026lemma (in prob_space) indep_varsD_finite:
1027  assumes X: "indep_vars M' X I"
1028  assumes I: "I \<noteq> {}" "finite I" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M' i)"
1029  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))"
1030proof (rule indep_setsD)
1031  show "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"
1032    using X by (auto simp: indep_vars_def)
1033  show "I \<subseteq> I" "I \<noteq> {}" "finite I" using I by auto
1034  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)}"
1035    using I by auto
1036qed
1037
1038lemma (in prob_space) indep_varsD:
1039  assumes X: "indep_vars M' X I"
1040  assumes I: "J \<noteq> {}" "finite J" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M' i)"
1041  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))"
1042proof (rule indep_setsD)
1043  show "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"
1044    using X by (auto simp: indep_vars_def)
1045  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)}"
1046    using I by auto
1047qed fact+
1048
1049lemma (in prob_space) indep_vars_iff_distr_eq_PiM:
1050  fixes I :: "'i set" and X :: "'i \<Rightarrow> 'a \<Rightarrow> 'b"
1051  assumes "I \<noteq> {}"
1052  assumes rv: "\<And>i. random_variable (M' i) (X i)"
1053  shows "indep_vars M' X I \<longleftrightarrow>
1054    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))"
1055proof -
1056  let ?P = "\<Pi>\<^sub>M i\<in>I. M' i"
1057  let ?X = "\<lambda>x. \<lambda>i\<in>I. X i x"
1058  let ?D = "distr M ?P ?X"
1059  have X: "random_variable ?P ?X" by (intro measurable_restrict rv)
1060  interpret D: prob_space ?D by (intro prob_space_distr X)
1061
1062  let ?D' = "\<lambda>i. distr M (M' i) (X i)"
1063  let ?P' = "\<Pi>\<^sub>M i\<in>I. distr M (M' i) (X i)"
1064  interpret D': prob_space "?D' i" for i by (intro prob_space_distr rv)
1065  interpret P: product_prob_space ?D' I ..
1066
1067  show ?thesis
1068  proof
1069    assume "indep_vars M' X I"
1070    show "?D = ?P'"
1071    proof (rule measure_eqI_generator_eq)
1072      show "Int_stable (prod_algebra I M')"
1073        by (rule Int_stable_prod_algebra)
1074      show "prod_algebra I M' \<subseteq> Pow (space ?P)"
1075        using prod_algebra_sets_into_space by (simp add: space_PiM)
1076      show "sets ?D = sigma_sets (space ?P) (prod_algebra I M')"
1077        by (simp add: sets_PiM space_PiM)
1078      show "sets ?P' = sigma_sets (space ?P) (prod_algebra I M')"
1079        by (simp add: sets_PiM space_PiM cong: prod_algebra_cong)
1080      let ?A = "\<lambda>i. \<Pi>\<^sub>E i\<in>I. space (M' i)"
1081      show "range ?A \<subseteq> prod_algebra I M'" "(\<Union>i. ?A i) = space (Pi\<^sub>M I M')"
1082        by (auto simp: space_PiM intro!: space_in_prod_algebra cong: prod_algebra_cong)
1083      { fix i show "emeasure ?D (\<Pi>\<^sub>E i\<in>I. space (M' i)) \<noteq> \<infinity>" by auto }
1084    next
1085      fix E assume E: "E \<in> prod_algebra I M'"
1086      from prod_algebraE[OF E] guess J Y . note J = this
1087
1088      from E have "E \<in> sets ?P" by (auto simp: sets_PiM)
1089      then have "emeasure ?D E = emeasure M (?X -` E \<inter> space M)"
1090        by (simp add: emeasure_distr X)
1091      also have "?X -` E \<inter> space M = (\<Inter>i\<in>J. X i -` Y i \<inter> space M)"
1092        using J \<open>I \<noteq> {}\<close> measurable_space[OF rv] by (auto simp: prod_emb_def PiE_iff split: if_split_asm)
1093      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))"
1094        using \<open>indep_vars M' X I\<close> J \<open>I \<noteq> {}\<close> using indep_varsD[of M' X I J]
1095        by (auto simp: emeasure_eq_measure prod_ennreal measure_nonneg prod_nonneg)
1096      also have "\<dots> = (\<Prod> i\<in>J. emeasure (?D' i) (Y i))"
1097        using rv J by (simp add: emeasure_distr)
1098      also have "\<dots> = emeasure ?P' E"
1099        using P.emeasure_PiM_emb[of J Y] J by (simp add: prod_emb_def)
1100      finally show "emeasure ?D E = emeasure ?P' E" .
1101    qed
1102  next
1103    assume "?D = ?P'"
1104    show "indep_vars M' X I" unfolding indep_vars_def
1105    proof (intro conjI indep_setsI ballI rv)
1106      fix i show "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)} \<subseteq> events"
1107        by (auto intro!: sets.sigma_sets_subset measurable_sets rv)
1108    next
1109      fix J Y' assume J: "J \<noteq> {}" "J \<subseteq> I" "finite J"
1110      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)}"
1111      have "\<forall>j\<in>J. \<exists>Y. Y' j = X j -` Y \<inter> space M \<and> Y \<in> sets (M' j)"
1112      proof
1113        fix j assume "j \<in> J"
1114        from Y'[rule_format, OF this] rv[of j]
1115        show "\<exists>Y. Y' j = X j -` Y \<inter> space M \<and> Y \<in> sets (M' j)"
1116          by (subst (asm) sigma_sets_vimage_commute[symmetric, of _ _ "space (M' j)"])
1117             (auto dest: measurable_space simp: sets.sigma_sets_eq)
1118      qed
1119      from bchoice[OF this] obtain Y where
1120        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
1121      let ?E = "prod_emb I M' J (Pi\<^sub>E J Y)"
1122      from Y have "(\<Inter>j\<in>J. Y' j) = ?X -` ?E \<inter> space M"
1123        using J \<open>I \<noteq> {}\<close> measurable_space[OF rv] by (auto simp: prod_emb_def PiE_iff split: if_split_asm)
1124      then have "emeasure M (\<Inter>j\<in>J. Y' j) = emeasure M (?X -` ?E \<inter> space M)"
1125        by simp
1126      also have "\<dots> = emeasure ?D ?E"
1127        using Y  J by (intro emeasure_distr[symmetric] X sets_PiM_I) auto
1128      also have "\<dots> = emeasure ?P' ?E"
1129        using \<open>?D = ?P'\<close> by simp
1130      also have "\<dots> = (\<Prod> i\<in>J. emeasure (?D' i) (Y i))"
1131        using P.emeasure_PiM_emb[of J Y] J Y by (simp add: prod_emb_def)
1132      also have "\<dots> = (\<Prod> i\<in>J. emeasure M (Y' i))"
1133        using rv J Y by (simp add: emeasure_distr)
1134      finally have "emeasure M (\<Inter>j\<in>J. Y' j) = (\<Prod> i\<in>J. emeasure M (Y' i))" .
1135      then show "prob (\<Inter>j\<in>J. Y' j) = (\<Prod> i\<in>J. prob (Y' i))"
1136        by (auto simp: emeasure_eq_measure prod_ennreal measure_nonneg prod_nonneg)
1137    qed
1138  qed
1139qed
1140
1141lemma (in prob_space) indep_varD:
1142  assumes indep: "indep_var Ma A Mb B"
1143  assumes sets: "Xa \<in> sets Ma" "Xb \<in> sets Mb"
1144  shows "prob ((\<lambda>x. (A x, B x)) -` (Xa \<times> Xb) \<inter> space M) =
1145    prob (A -` Xa \<inter> space M) * prob (B -` Xb \<inter> space M)"
1146proof -
1147  have "prob ((\<lambda>x. (A x, B x)) -` (Xa \<times> Xb) \<inter> space M) =
1148    prob (\<Inter>i\<in>UNIV. (case_bool A B i -` case_bool Xa Xb i \<inter> space M))"
1149    by (auto intro!: arg_cong[where f=prob] simp: UNIV_bool)
1150  also have "\<dots> = (\<Prod>i\<in>UNIV. prob (case_bool A B i -` case_bool Xa Xb i \<inter> space M))"
1151    using indep unfolding indep_var_def
1152    by (rule indep_varsD) (auto split: bool.split intro: sets)
1153  also have "\<dots> = prob (A -` Xa \<inter> space M) * prob (B -` Xb \<inter> space M)"
1154    unfolding UNIV_bool by simp
1155  finally show ?thesis .
1156qed
1157
1158lemma (in prob_space) prob_indep_random_variable:
1159  assumes ind[simp]: "indep_var N X N Y"
1160  assumes [simp]: "A \<in> sets N" "B \<in> sets N"
1161  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)"
1162proof-
1163  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)"
1164    by (auto intro!: arg_cong[where f= prob])
1165  also have "...=  prob (X -` A \<inter> space M) * prob (Y -` B \<inter> space M)"
1166    by (auto intro!: indep_varD[where Ma=N and Mb=N])
1167  also have "... = \<P>(x in M. X x \<in> A) * \<P>(x in M. Y x \<in> B)"
1168    by (auto intro!: arg_cong2[where f= "(*)"] arg_cong[where f= prob])
1169  finally show ?thesis .
1170qed
1171
1172lemma (in prob_space)
1173  assumes "indep_var S X T Y"
1174  shows indep_var_rv1: "random_variable S X"
1175    and indep_var_rv2: "random_variable T Y"
1176proof -
1177  have "\<forall>i\<in>UNIV. random_variable (case_bool S T i) (case_bool X Y i)"
1178    using assms unfolding indep_var_def indep_vars_def by auto
1179  then show "random_variable S X" "random_variable T Y"
1180    unfolding UNIV_bool by auto
1181qed
1182
1183lemma (in prob_space) indep_var_distribution_eq:
1184  "indep_var S X T Y \<longleftrightarrow> random_variable S X \<and> random_variable T Y \<and>
1185    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")
1186proof safe
1187  assume "indep_var S X T Y"
1188  then show rvs: "random_variable S X" "random_variable T Y"
1189    by (blast dest: indep_var_rv1 indep_var_rv2)+
1190  then have XY: "random_variable (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))"
1191    by (rule measurable_Pair)
1192
1193  interpret X: prob_space ?S by (rule prob_space_distr) fact
1194  interpret Y: prob_space ?T by (rule prob_space_distr) fact
1195  interpret XY: pair_prob_space ?S ?T ..
1196  show "?S \<Otimes>\<^sub>M ?T = ?J"
1197  proof (rule pair_measure_eqI)
1198    show "sigma_finite_measure ?S" ..
1199    show "sigma_finite_measure ?T" ..
1200
1201    fix A B assume A: "A \<in> sets ?S" and B: "B \<in> sets ?T"
1202    have "emeasure ?J (A \<times> B) = emeasure M ((\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M)"
1203      using A B by (intro emeasure_distr[OF XY]) auto
1204    also have "\<dots> = emeasure M (X -` A \<inter> space M) * emeasure M (Y -` B \<inter> space M)"
1205      using indep_varD[OF \<open>indep_var S X T Y\<close>, of A B] A B
1206      by (simp add: emeasure_eq_measure measure_nonneg ennreal_mult)
1207    also have "\<dots> = emeasure ?S A * emeasure ?T B"
1208      using rvs A B by (simp add: emeasure_distr)
1209    finally show "emeasure ?S A * emeasure ?T B = emeasure ?J (A \<times> B)" by simp
1210  qed simp
1211next
1212  assume rvs: "random_variable S X" "random_variable T Y"
1213  then have XY: "random_variable (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))"
1214    by (rule measurable_Pair)
1215
1216  let ?S = "distr M S X" and ?T = "distr M T Y"
1217  interpret X: prob_space ?S by (rule prob_space_distr) fact
1218  interpret Y: prob_space ?T by (rule prob_space_distr) fact
1219  interpret XY: pair_prob_space ?S ?T ..
1220
1221  assume "?S \<Otimes>\<^sub>M ?T = ?J"
1222
1223  { fix S and X
1224    have "Int_stable {X -` A \<inter> space M |A. A \<in> sets S}"
1225    proof (safe intro!: Int_stableI)
1226      fix A B assume "A \<in> sets S" "B \<in> sets S"
1227      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"
1228        by (intro exI[of _ "A \<inter> B"]) auto
1229    qed }
1230  note Int_stable = this
1231
1232  show "indep_var S X T Y" unfolding indep_var_eq
1233  proof (intro conjI indep_set_sigma_sets Int_stable rvs)
1234    show "indep_set {X -` A \<inter> space M |A. A \<in> sets S} {Y -` A \<inter> space M |A. A \<in> sets T}"
1235    proof (safe intro!: indep_setI)
1236      { fix A assume "A \<in> sets S" then show "X -` A \<inter> space M \<in> sets M"
1237        using \<open>X \<in> measurable M S\<close> by (auto intro: measurable_sets) }
1238      { fix A assume "A \<in> sets T" then show "Y -` A \<inter> space M \<in> sets M"
1239        using \<open>Y \<in> measurable M T\<close> by (auto intro: measurable_sets) }
1240    next
1241      fix A B assume ab: "A \<in> sets S" "B \<in> sets T"
1242      then have "prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)) = emeasure ?J (A \<times> B)"
1243        using XY by (auto simp add: emeasure_distr emeasure_eq_measure measure_nonneg intro!: arg_cong[where f="prob"])
1244      also have "\<dots> = emeasure (?S \<Otimes>\<^sub>M ?T) (A \<times> B)"
1245        unfolding \<open>?S \<Otimes>\<^sub>M ?T = ?J\<close> ..
1246      also have "\<dots> = emeasure ?S A * emeasure ?T B"
1247        using ab by (simp add: Y.emeasure_pair_measure_Times)
1248      finally show "prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)) =
1249        prob (X -` A \<inter> space M) * prob (Y -` B \<inter> space M)"
1250        using rvs ab by (simp add: emeasure_eq_measure emeasure_distr measure_nonneg ennreal_mult[symmetric])
1251    qed
1252  qed
1253qed
1254
1255lemma (in prob_space) distributed_joint_indep:
1256  assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
1257  assumes X: "distributed M S X Px" and Y: "distributed M T Y Py"
1258  assumes indep: "indep_var S X T Y"
1259  shows "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) (\<lambda>(x, y). Px x * Py y)"
1260  using indep_var_distribution_eq[of S X T Y] indep
1261  by (intro distributed_joint_indep'[OF S T X Y]) auto
1262
1263lemma (in prob_space) indep_vars_nn_integral:
1264  assumes I: "finite I" "indep_vars (\<lambda>_. borel) X I" "\<And>i \<omega>. i \<in> I \<Longrightarrow> 0 \<le> X i \<omega>"
1265  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)"
1266proof cases
1267  assume "I \<noteq> {}"
1268  define Y where [abs_def]: "Y i \<omega> = (if i \<in> I then X i \<omega> else 0)" for i \<omega>
1269  { fix i have "i \<in> I \<Longrightarrow> random_variable borel (X i)"
1270    using I(2) by (cases "i\<in>I") (auto simp: indep_vars_def) }
1271  note rv_X = this
1272
1273  { fix i have "random_variable borel (Y i)"
1274    using I(2) by (cases "i\<in>I") (auto simp: Y_def rv_X) }
1275  note rv_Y = this[measurable]
1276
1277  interpret Y: prob_space "distr M borel (Y i)" for i
1278    using I(2) by (cases "i \<in> I") (auto intro!: prob_space_distr simp: indep_vars_def prob_space_return)
1279  interpret product_sigma_finite "\<lambda>i. distr M borel (Y i)"
1280    ..
1281
1282  have indep_Y: "indep_vars (\<lambda>i. borel) Y I"
1283    by (rule indep_vars_cong[THEN iffD1, OF _ _ _ I(2)]) (auto simp: Y_def)
1284
1285  have "(\<integral>\<^sup>+\<omega>. (\<Prod>i\<in>I. X i \<omega>) \<partial>M) = (\<integral>\<^sup>+\<omega>. (\<Prod>i\<in>I. Y i \<omega>) \<partial>M)"
1286    using I(3) by (auto intro!: nn_integral_cong prod.cong simp add: Y_def max_def)
1287  also have "\<dots> = (\<integral>\<^sup>+\<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))"
1288    by (subst nn_integral_distr) auto
1289  also have "\<dots> = (\<integral>\<^sup>+\<omega>. (\<Prod>i\<in>I. \<omega> i) \<partial>Pi\<^sub>M I (\<lambda>i. distr M borel (Y i)))"
1290    unfolding indep_vars_iff_distr_eq_PiM[THEN iffD1, OF \<open>I \<noteq> {}\<close> rv_Y indep_Y] ..
1291  also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^sup>+\<omega>. \<omega> \<partial>distr M borel (Y i)))"
1292    by (rule product_nn_integral_prod) (auto intro: \<open>finite I\<close>)
1293  also have "\<dots> = (\<Prod>i\<in>I. \<integral>\<^sup>+\<omega>. X i \<omega> \<partial>M)"
1294    by (intro prod.cong nn_integral_cong) (auto simp: nn_integral_distr Y_def rv_X)
1295  finally show ?thesis .
1296qed (simp add: emeasure_space_1)
1297
1298lemma (in prob_space)
1299  fixes X :: "'i \<Rightarrow> 'a \<Rightarrow> 'b::{real_normed_field, banach, second_countable_topology}"
1300  assumes I: "finite I" "indep_vars (\<lambda>_. borel) X I" "\<And>i. i \<in> I \<Longrightarrow> integrable M (X i)"
1301  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)
1302    and indep_vars_integrable: "integrable M (\<lambda>\<omega>. (\<Prod>i\<in>I. X i \<omega>))" (is ?int)
1303proof (induct rule: case_split)
1304  assume "I \<noteq> {}"
1305  define Y where [abs_def]: "Y i \<omega> = (if i \<in> I then X i \<omega> else 0)" for i \<omega>
1306  { fix i have "i \<in> I \<Longrightarrow> random_variable borel (X i)"
1307    using I(2) by (cases "i\<in>I") (auto simp: indep_vars_def) }
1308  note rv_X = this[measurable]
1309
1310  { fix i have "random_variable borel (Y i)"
1311    using I(2) by (cases "i\<in>I") (auto simp: Y_def rv_X) }
1312  note rv_Y = this[measurable]
1313
1314  { fix i have "integrable M (Y i)"
1315    using I(3) by (cases "i\<in>I") (auto simp: Y_def) }
1316  note int_Y = this
1317
1318  interpret Y: prob_space "distr M borel (Y i)" for i
1319    using I(2) by (cases "i \<in> I") (auto intro!: prob_space_distr simp: indep_vars_def prob_space_return)
1320  interpret product_sigma_finite "\<lambda>i. distr M borel (Y i)"
1321    ..
1322
1323  have indep_Y: "indep_vars (\<lambda>i. borel) Y I"
1324    by (rule indep_vars_cong[THEN iffD1, OF _ _ _ I(2)]) (auto simp: Y_def)
1325
1326  have "(\<integral>\<omega>. (\<Prod>i\<in>I. X i \<omega>) \<partial>M) = (\<integral>\<omega>. (\<Prod>i\<in>I. Y i \<omega>) \<partial>M)"
1327    using I(3) by (simp add: Y_def)
1328  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))"
1329    by (subst integral_distr) auto
1330  also have "\<dots> = (\<integral>\<omega>. (\<Prod>i\<in>I. \<omega> i) \<partial>Pi\<^sub>M I (\<lambda>i. distr M borel (Y i)))"
1331    unfolding indep_vars_iff_distr_eq_PiM[THEN iffD1, OF \<open>I \<noteq> {}\<close> rv_Y indep_Y] ..
1332  also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<omega>. \<omega> \<partial>distr M borel (Y i)))"
1333    by (rule product_integral_prod) (auto intro: \<open>finite I\<close> simp: integrable_distr_eq int_Y)
1334  also have "\<dots> = (\<Prod>i\<in>I. \<integral>\<omega>. X i \<omega> \<partial>M)"
1335    by (intro prod.cong integral_cong)
1336       (auto simp: integral_distr Y_def rv_X)
1337  finally show ?eq .
1338
1339  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))"
1340    unfolding indep_vars_iff_distr_eq_PiM[THEN iffD1, OF \<open>I \<noteq> {}\<close> rv_Y indep_Y]
1341    by (intro product_integrable_prod[OF \<open>finite I\<close>])
1342       (simp add: integrable_distr_eq int_Y)
1343  then show ?int
1344    by (simp add: integrable_distr_eq Y_def)
1345qed (simp_all add: prob_space)
1346
1347lemma (in prob_space)
1348  fixes X1 X2 :: "'a \<Rightarrow> 'b::{real_normed_field, banach, second_countable_topology}"
1349  assumes "indep_var borel X1 borel X2" "integrable M X1" "integrable M X2"
1350  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)
1351    and indep_var_integrable: "integrable M (\<lambda>\<omega>. X1 \<omega> * X2 \<omega>)" (is ?int)
1352unfolding indep_var_def
1353proof -
1354  have *: "(\<lambda>\<omega>. X1 \<omega> * X2 \<omega>) = (\<lambda>\<omega>. \<Prod>i\<in>UNIV. (case_bool X1 X2 i \<omega>))"
1355    by (simp add: UNIV_bool mult.commute)
1356  have **: "(\<lambda> _. borel) = case_bool borel borel"
1357    by (rule ext, metis (full_types) bool.simps(3) bool.simps(4))
1358  show ?eq
1359    apply (subst *)
1360    apply (subst indep_vars_lebesgue_integral)
1361    apply (auto)
1362    apply (subst **, subst indep_var_def [symmetric], rule assms)
1363    apply (simp split: bool.split add: assms)
1364    by (simp add: UNIV_bool mult.commute)
1365  show ?int
1366    apply (subst *)
1367    apply (rule indep_vars_integrable)
1368    apply auto
1369    apply (subst **, subst indep_var_def [symmetric], rule assms)
1370    by (simp split: bool.split add: assms)
1371qed
1372
1373end
1374