src/HOL/Probability/Independent_Family.thy
author hoelzl
Thu May 26 14:11:57 2011 +0200 (2011-05-26)
changeset 42981 fe7f5a26e4c6
parent 42861 16375b493b64
child 42982 fa0ac7bee9ac
permissions -rw-r--r--
add lemma indep_sets_collect_sigma
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(*  Title:      HOL/Probability/Independent_Family.thy
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    Author:     Johannes Hölzl, TU München
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*)
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header {* Independent families of events, event sets, and random variables *}
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theory Independent_Family
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  imports Probability_Measure
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begin
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definition (in prob_space)
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  "indep_events A I \<longleftrightarrow> (A`I \<subseteq> sets M) \<and>
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    (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j)))"
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definition (in prob_space)
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  "indep_event A B \<longleftrightarrow> indep_events (bool_case A B) UNIV"
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definition (in prob_space)
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  "indep_sets F I \<longleftrightarrow> (\<forall>i\<in>I. F i \<subseteq> sets M) \<and>
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    (\<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))))"
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definition (in prob_space)
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  "indep_set A B \<longleftrightarrow> indep_sets (bool_case A B) UNIV"
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definition (in prob_space)
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  "indep_rv M' X I \<longleftrightarrow>
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    (\<forall>i\<in>I. random_variable (M' i) (X i)) \<and>
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    indep_sets (\<lambda>i. sigma_sets (space M) { X i -` A \<inter> space M | A. A \<in> sets (M' i)}) I"
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lemma (in prob_space) indep_sets_cong:
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  "I = J \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> F i = G i) \<Longrightarrow> indep_sets F I \<longleftrightarrow> indep_sets G J"
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  by (simp add: indep_sets_def, intro conj_cong all_cong imp_cong ball_cong) blast+
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lemma (in prob_space) indep_events_finite_index_events:
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  "indep_events F I \<longleftrightarrow> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_events F J)"
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  by (auto simp: indep_events_def)
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lemma (in prob_space) indep_sets_finite_index_sets:
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  "indep_sets F I \<longleftrightarrow> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J)"
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proof (intro iffI allI impI)
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  assume *: "\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J"
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  show "indep_sets F I" unfolding indep_sets_def
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  proof (intro conjI ballI allI impI)
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    fix i assume "i \<in> I"
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    with *[THEN spec, of "{i}"] show "F i \<subseteq> events"
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      by (auto simp: indep_sets_def)
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  qed (insert *, auto simp: indep_sets_def)
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qed (auto simp: indep_sets_def)
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lemma (in prob_space) indep_sets_mono_index:
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  "J \<subseteq> I \<Longrightarrow> indep_sets F I \<Longrightarrow> indep_sets F J"
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  unfolding indep_sets_def by auto
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lemma (in prob_space) indep_sets_mono_sets:
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  assumes indep: "indep_sets F I"
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  assumes mono: "\<And>i. i\<in>I \<Longrightarrow> G i \<subseteq> F i"
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  shows "indep_sets G I"
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proof -
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  have "(\<forall>i\<in>I. F i \<subseteq> events) \<Longrightarrow> (\<forall>i\<in>I. G i \<subseteq> events)"
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    using mono by auto
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  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)"
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    using mono by (auto simp: Pi_iff)
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  ultimately show ?thesis
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    using indep by (auto simp: indep_sets_def)
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qed
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lemma (in prob_space) indep_setsI:
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  assumes "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events"
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    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))"
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  shows "indep_sets F I"
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  using assms unfolding indep_sets_def by (auto simp: Pi_iff)
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lemma (in prob_space) indep_setsD:
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  assumes "indep_sets F I" and "J \<subseteq> I" "J \<noteq> {}" "finite J" "\<forall>j\<in>J. A j \<in> F j"
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  shows "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
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  using assms unfolding indep_sets_def by auto
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lemma dynkin_systemI':
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  assumes 1: "\<And> A. A \<in> sets M \<Longrightarrow> A \<subseteq> space M"
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  assumes empty: "{} \<in> sets M"
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  assumes Diff: "\<And> A. A \<in> sets M \<Longrightarrow> space M - A \<in> sets M"
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  assumes 2: "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> sets M
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          \<Longrightarrow> (\<Union>i::nat. A i) \<in> sets M"
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  shows "dynkin_system M"
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proof -
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  from Diff[OF empty] have "space M \<in> sets M" by auto
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  from 1 this Diff 2 show ?thesis
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    by (intro dynkin_systemI) auto
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qed
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lemma (in prob_space) indep_sets_dynkin:
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  assumes indep: "indep_sets F I"
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  shows "indep_sets (\<lambda>i. sets (dynkin \<lparr> space = space M, sets = F i \<rparr>)) I"
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    (is "indep_sets ?F I")
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proof (subst indep_sets_finite_index_sets, intro allI impI ballI)
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  fix J assume "finite J" "J \<subseteq> I" "J \<noteq> {}"
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  with indep have "indep_sets F J"
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    by (subst (asm) indep_sets_finite_index_sets) auto
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  { fix J K assume "indep_sets F K"
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    let "?G S i" = "if i \<in> S then ?F i else F i"
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    assume "finite J" "J \<subseteq> K"
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    then have "indep_sets (?G J) K"
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    proof induct
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      case (insert j J)
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      moreover def G \<equiv> "?G J"
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      ultimately have G: "indep_sets G K" "\<And>i. i \<in> K \<Longrightarrow> G i \<subseteq> events" and "j \<in> K"
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        by (auto simp: indep_sets_def)
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      let ?D = "{E\<in>events. indep_sets (G(j := {E})) K }"
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      { fix X assume X: "X \<in> events"
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        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)
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          \<Longrightarrow> prob ((\<Inter>i\<in>J. A i) \<inter> X) = prob X * (\<Prod>i\<in>J. prob (A i))"
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        have "indep_sets (G(j := {X})) K"
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        proof (rule indep_setsI)
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          fix i assume "i \<in> K" then show "(G(j:={X})) i \<subseteq> events"
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            using G X by auto
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        next
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          fix A J assume J: "J \<noteq> {}" "J \<subseteq> K" "finite J" "\<forall>i\<in>J. A i \<in> (G(j := {X})) i"
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          show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
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          proof cases
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            assume "j \<in> J"
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            with J have "A j = X" by auto
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            show ?thesis
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            proof cases
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              assume "J = {j}" then show ?thesis by simp
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            next
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              assume "J \<noteq> {j}"
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              have "prob (\<Inter>i\<in>J. A i) = prob ((\<Inter>i\<in>J-{j}. A i) \<inter> X)"
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                using `j \<in> J` `A j = X` by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
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              also have "\<dots> = prob X * (\<Prod>i\<in>J-{j}. prob (A i))"
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              proof (rule indep)
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                show "J - {j} \<noteq> {}" "J - {j} \<subseteq> K" "finite (J - {j})" "j \<notin> J - {j}"
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                  using J `J \<noteq> {j}` `j \<in> J` by auto
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                show "\<forall>i\<in>J - {j}. A i \<in> G i"
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                  using J by auto
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              qed
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              also have "\<dots> = prob (A j) * (\<Prod>i\<in>J-{j}. prob (A i))"
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                using `A j = X` by simp
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              also have "\<dots> = (\<Prod>i\<in>J. prob (A i))"
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                unfolding setprod.insert_remove[OF `finite J`, symmetric, of "\<lambda>i. prob  (A i)"]
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                using `j \<in> J` by (simp add: insert_absorb)
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              finally show ?thesis .
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            qed
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          next
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            assume "j \<notin> J"
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            with J have "\<forall>i\<in>J. A i \<in> G i" by (auto split: split_if_asm)
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            with J show ?thesis
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              by (intro indep_setsD[OF G(1)]) auto
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          qed
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        qed }
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      note indep_sets_insert = this
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      have "dynkin_system \<lparr> space = space M, sets = ?D \<rparr>"
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      proof (rule dynkin_systemI', simp_all, safe)
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        show "indep_sets (G(j := {{}})) K"
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          by (rule indep_sets_insert) auto
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      next
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        fix X assume X: "X \<in> events" and G': "indep_sets (G(j := {X})) K"
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        show "indep_sets (G(j := {space M - X})) K"
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        proof (rule indep_sets_insert)
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          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"
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          then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events"
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            using G by auto
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          have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) =
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              prob ((\<Inter>j\<in>J. A j) - (\<Inter>i\<in>insert j J. (A(j := X)) i))"
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            using A_sets sets_into_space X `J \<noteq> {}`
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            by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
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          also have "\<dots> = prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)"
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            using J `J \<noteq> {}` `j \<notin> J` A_sets X sets_into_space
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            by (auto intro!: finite_measure_Diff finite_INT split: split_if_asm)
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          finally have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) =
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              prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)" .
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          moreover {
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            have "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
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              using J A `finite J` by (intro indep_setsD[OF G(1)]) auto
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            then have "prob (\<Inter>j\<in>J. A j) = prob (space M) * (\<Prod>i\<in>J. prob (A i))"
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              using prob_space by simp }
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          moreover {
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            have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = (\<Prod>i\<in>insert j J. prob ((A(j := X)) i))"
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              using J A `j \<in> K` by (intro indep_setsD[OF G']) auto
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            then have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = prob X * (\<Prod>i\<in>J. prob (A i))"
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              using `finite J` `j \<notin> J` by (auto intro!: setprod_cong) }
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          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))"
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            by (simp add: field_simps)
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          also have "\<dots> = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))"
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            using X A by (simp add: finite_measure_compl)
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          finally show "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))" .
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        qed (insert X, auto)
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      next
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        fix F :: "nat \<Rightarrow> 'a set" assume disj: "disjoint_family F" and "range F \<subseteq> ?D"
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        then have F: "\<And>i. F i \<in> events" "\<And>i. indep_sets (G(j:={F i})) K" by auto
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        show "indep_sets (G(j := {\<Union>k. F k})) K"
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        proof (rule indep_sets_insert)
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          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"
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          then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events"
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            using G by auto
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          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))"
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            using `J \<noteq> {}` `j \<notin> J` `j \<in> K` by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
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          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))"
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          proof (rule finite_measure_UNION)
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            show "disjoint_family (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i)"
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              using disj by (rule disjoint_family_on_bisimulation) auto
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            show "range (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i) \<subseteq> events"
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              using A_sets F `finite J` `J \<noteq> {}` `j \<notin> J` by (auto intro!: Int)
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          qed
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          moreover { fix k
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            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))"
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              by (subst indep_setsD[OF F(2)]) (auto intro!: setprod_cong split: split_if_asm)
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            also have "\<dots> = prob (F k) * prob (\<Inter>i\<in>J. A i)"
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              using J A `j \<in> K` by (subst indep_setsD[OF G(1)]) auto
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            finally have "prob (\<Inter>i\<in>insert j J. (A(j := F k)) i) = prob (F k) * prob (\<Inter>i\<in>J. A i)" . }
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          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)))"
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            by simp
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          moreover
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          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))"
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            using disj F(1) by (intro finite_measure_UNION sums_mult2) auto
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          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)))"
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            using J A `j \<in> K` by (subst indep_setsD[OF G(1), symmetric]) auto
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          ultimately
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          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))"
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            by (auto dest!: sums_unique)
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        qed (insert F, auto)
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      qed (insert sets_into_space, auto)
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      then have mono: "sets (dynkin \<lparr>space = space M, sets = G j\<rparr>) \<subseteq>
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        sets \<lparr>space = space M, sets = {E \<in> events. indep_sets (G(j := {E})) K}\<rparr>"
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      proof (rule dynkin_system.dynkin_subset, simp_all, safe)
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        fix X assume "X \<in> G j"
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        then show "X \<in> events" using G `j \<in> K` by auto
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        from `indep_sets G K`
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        show "indep_sets (G(j := {X})) K"
hoelzl@42861
   229
          by (rule indep_sets_mono_sets) (insert `X \<in> G j`, auto)
hoelzl@42861
   230
      qed
hoelzl@42861
   231
      have "indep_sets (G(j:=?D)) K"
hoelzl@42861
   232
      proof (rule indep_setsI)
hoelzl@42861
   233
        fix i assume "i \<in> K" then show "(G(j := ?D)) i \<subseteq> events"
hoelzl@42861
   234
          using G(2) by auto
hoelzl@42861
   235
      next
hoelzl@42861
   236
        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"
hoelzl@42861
   237
        show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
hoelzl@42861
   238
        proof cases
hoelzl@42861
   239
          assume "j \<in> J"
hoelzl@42861
   240
          with A have indep: "indep_sets (G(j := {A j})) K" by auto
hoelzl@42861
   241
          from J A show ?thesis
hoelzl@42861
   242
            by (intro indep_setsD[OF indep]) auto
hoelzl@42861
   243
        next
hoelzl@42861
   244
          assume "j \<notin> J"
hoelzl@42861
   245
          with J A have "\<forall>i\<in>J. A i \<in> G i" by (auto split: split_if_asm)
hoelzl@42861
   246
          with J show ?thesis
hoelzl@42861
   247
            by (intro indep_setsD[OF G(1)]) auto
hoelzl@42861
   248
        qed
hoelzl@42861
   249
      qed
hoelzl@42861
   250
      then have "indep_sets (G(j:=sets (dynkin \<lparr>space = space M, sets = G j\<rparr>))) K"
hoelzl@42861
   251
        by (rule indep_sets_mono_sets) (insert mono, auto)
hoelzl@42861
   252
      then show ?case
hoelzl@42861
   253
        by (rule indep_sets_mono_sets) (insert `j \<in> K` `j \<notin> J`, auto simp: G_def)
hoelzl@42861
   254
    qed (insert `indep_sets F K`, simp) }
hoelzl@42861
   255
  from this[OF `indep_sets F J` `finite J` subset_refl]
hoelzl@42861
   256
  show "indep_sets (\<lambda>i. sets (dynkin \<lparr> space = space M, sets = F i \<rparr>)) J"
hoelzl@42861
   257
    by (rule indep_sets_mono_sets) auto
hoelzl@42861
   258
qed
hoelzl@42861
   259
hoelzl@42861
   260
lemma (in prob_space) indep_sets_sigma:
hoelzl@42861
   261
  assumes indep: "indep_sets F I"
hoelzl@42861
   262
  assumes stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable \<lparr> space = space M, sets = F i \<rparr>"
hoelzl@42861
   263
  shows "indep_sets (\<lambda>i. sets (sigma \<lparr> space = space M, sets = F i \<rparr>)) I"
hoelzl@42861
   264
proof -
hoelzl@42861
   265
  from indep_sets_dynkin[OF indep]
hoelzl@42861
   266
  show ?thesis
hoelzl@42861
   267
  proof (rule indep_sets_mono_sets, subst sigma_eq_dynkin, simp_all add: stable)
hoelzl@42861
   268
    fix i assume "i \<in> I"
hoelzl@42861
   269
    with indep have "F i \<subseteq> events" by (auto simp: indep_sets_def)
hoelzl@42861
   270
    with sets_into_space show "F i \<subseteq> Pow (space M)" by auto
hoelzl@42861
   271
  qed
hoelzl@42861
   272
qed
hoelzl@42861
   273
hoelzl@42861
   274
lemma (in prob_space) indep_sets_sigma_sets:
hoelzl@42861
   275
  assumes "indep_sets F I"
hoelzl@42861
   276
  assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable \<lparr> space = space M, sets = F i \<rparr>"
hoelzl@42861
   277
  shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I"
hoelzl@42861
   278
  using indep_sets_sigma[OF assms] by (simp add: sets_sigma)
hoelzl@42861
   279
hoelzl@42861
   280
lemma (in prob_space) indep_sets2_eq:
hoelzl@42981
   281
  "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)"
hoelzl@42981
   282
  unfolding indep_set_def
hoelzl@42861
   283
proof (intro iffI ballI conjI)
hoelzl@42861
   284
  assume indep: "indep_sets (bool_case A B) UNIV"
hoelzl@42861
   285
  { fix a b assume "a \<in> A" "b \<in> B"
hoelzl@42861
   286
    with indep_setsD[OF indep, of UNIV "bool_case a b"]
hoelzl@42861
   287
    show "prob (a \<inter> b) = prob a * prob b"
hoelzl@42861
   288
      unfolding UNIV_bool by (simp add: ac_simps) }
hoelzl@42861
   289
  from indep show "A \<subseteq> events" "B \<subseteq> events"
hoelzl@42861
   290
    unfolding indep_sets_def UNIV_bool by auto
hoelzl@42861
   291
next
hoelzl@42861
   292
  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)"
hoelzl@42861
   293
  show "indep_sets (bool_case A B) UNIV"
hoelzl@42861
   294
  proof (rule indep_setsI)
hoelzl@42861
   295
    fix i show "(case i of True \<Rightarrow> A | False \<Rightarrow> B) \<subseteq> events"
hoelzl@42861
   296
      using * by (auto split: bool.split)
hoelzl@42861
   297
  next
hoelzl@42861
   298
    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)"
hoelzl@42861
   299
    then have "J = {True} \<or> J = {False} \<or> J = {True,False}"
hoelzl@42861
   300
      by (auto simp: UNIV_bool)
hoelzl@42861
   301
    then show "prob (\<Inter>j\<in>J. X j) = (\<Prod>j\<in>J. prob (X j))"
hoelzl@42861
   302
      using X * by auto
hoelzl@42861
   303
  qed
hoelzl@42861
   304
qed
hoelzl@42861
   305
hoelzl@42981
   306
lemma (in prob_space) indep_set_sigma_sets:
hoelzl@42981
   307
  assumes "indep_set A B"
hoelzl@42861
   308
  assumes A: "Int_stable \<lparr> space = space M, sets = A \<rparr>"
hoelzl@42861
   309
  assumes B: "Int_stable \<lparr> space = space M, sets = B \<rparr>"
hoelzl@42981
   310
  shows "indep_set (sigma_sets (space M) A) (sigma_sets (space M) B)"
hoelzl@42861
   311
proof -
hoelzl@42861
   312
  have "indep_sets (\<lambda>i. sigma_sets (space M) (case i of True \<Rightarrow> A | False \<Rightarrow> B)) UNIV"
hoelzl@42861
   313
  proof (rule indep_sets_sigma_sets)
hoelzl@42861
   314
    show "indep_sets (bool_case A B) UNIV"
hoelzl@42981
   315
      by (rule `indep_set A B`[unfolded indep_set_def])
hoelzl@42861
   316
    fix i show "Int_stable \<lparr>space = space M, sets = case i of True \<Rightarrow> A | False \<Rightarrow> B\<rparr>"
hoelzl@42861
   317
      using A B by (cases i) auto
hoelzl@42861
   318
  qed
hoelzl@42861
   319
  then show ?thesis
hoelzl@42981
   320
    unfolding indep_set_def
hoelzl@42861
   321
    by (rule indep_sets_mono_sets) (auto split: bool.split)
hoelzl@42861
   322
qed
hoelzl@42861
   323
hoelzl@42981
   324
lemma (in prob_space) indep_sets_collect_sigma:
hoelzl@42981
   325
  fixes I :: "'j \<Rightarrow> 'i set" and J :: "'j set" and E :: "'i \<Rightarrow> 'a set set"
hoelzl@42981
   326
  assumes indep: "indep_sets E (\<Union>j\<in>J. I j)"
hoelzl@42981
   327
  assumes Int_stable: "\<And>i j. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> Int_stable \<lparr>space = space M, sets = E i\<rparr>"
hoelzl@42981
   328
  assumes disjoint: "disjoint_family_on I J"
hoelzl@42981
   329
  shows "indep_sets (\<lambda>j. sigma_sets (space M) (\<Union>i\<in>I j. E i)) J"
hoelzl@42981
   330
proof -
hoelzl@42981
   331
  let "?E j" = "{\<Inter>k\<in>K. E' k| E' K. finite K \<and> K \<noteq> {} \<and> K \<subseteq> I j \<and> (\<forall>k\<in>K. E' k \<in> E k) }"
hoelzl@42981
   332
hoelzl@42981
   333
  from indep have E: "\<And>j i. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> E i \<subseteq> sets M"
hoelzl@42981
   334
    unfolding indep_sets_def by auto
hoelzl@42981
   335
  { fix j
hoelzl@42981
   336
    let ?S = "sigma \<lparr> space = space M, sets = (\<Union>i\<in>I j. E i) \<rparr>"
hoelzl@42981
   337
    assume "j \<in> J"
hoelzl@42981
   338
    from E[OF this] interpret S: sigma_algebra ?S
hoelzl@42981
   339
      using sets_into_space by (intro sigma_algebra_sigma) auto
hoelzl@42981
   340
hoelzl@42981
   341
    have "sigma_sets (space M) (\<Union>i\<in>I j. E i) = sigma_sets (space M) (?E j)"
hoelzl@42981
   342
    proof (rule sigma_sets_eqI)
hoelzl@42981
   343
      fix A assume "A \<in> (\<Union>i\<in>I j. E i)"
hoelzl@42981
   344
      then guess i ..
hoelzl@42981
   345
      then show "A \<in> sigma_sets (space M) (?E j)"
hoelzl@42981
   346
        by (auto intro!: sigma_sets.intros exI[of _ "{i}"] exI[of _ "\<lambda>i. A"])
hoelzl@42981
   347
    next
hoelzl@42981
   348
      fix A assume "A \<in> ?E j"
hoelzl@42981
   349
      then obtain E' K where "finite K" "K \<noteq> {}" "K \<subseteq> I j" "\<And>k. k \<in> K \<Longrightarrow> E' k \<in> E k"
hoelzl@42981
   350
        and A: "A = (\<Inter>k\<in>K. E' k)"
hoelzl@42981
   351
        by auto
hoelzl@42981
   352
      then have "A \<in> sets ?S" unfolding A
hoelzl@42981
   353
        by (safe intro!: S.finite_INT)
hoelzl@42981
   354
           (auto simp: sets_sigma intro!: sigma_sets.Basic)
hoelzl@42981
   355
      then show "A \<in> sigma_sets (space M) (\<Union>i\<in>I j. E i)"
hoelzl@42981
   356
        by (simp add: sets_sigma)
hoelzl@42981
   357
    qed }
hoelzl@42981
   358
  moreover have "indep_sets (\<lambda>j. sigma_sets (space M) (?E j)) J"
hoelzl@42981
   359
  proof (rule indep_sets_sigma_sets)
hoelzl@42981
   360
    show "indep_sets ?E J"
hoelzl@42981
   361
    proof (intro indep_setsI)
hoelzl@42981
   362
      fix j assume "j \<in> J" with E show "?E j \<subseteq> events" by (force  intro!: finite_INT)
hoelzl@42981
   363
    next
hoelzl@42981
   364
      fix K A assume K: "K \<noteq> {}" "K \<subseteq> J" "finite K"
hoelzl@42981
   365
        and "\<forall>j\<in>K. A j \<in> ?E j"
hoelzl@42981
   366
      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)"
hoelzl@42981
   367
        by simp
hoelzl@42981
   368
      from bchoice[OF this] guess E' ..
hoelzl@42981
   369
      from bchoice[OF this] obtain L
hoelzl@42981
   370
        where A: "\<And>j. j\<in>K \<Longrightarrow> A j = (\<Inter>l\<in>L j. E' j l)"
hoelzl@42981
   371
        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"
hoelzl@42981
   372
        and E': "\<And>j l. j\<in>K \<Longrightarrow> l \<in> L j \<Longrightarrow> E' j l \<in> E l"
hoelzl@42981
   373
        by auto
hoelzl@42981
   374
hoelzl@42981
   375
      { fix k l j assume "k \<in> K" "j \<in> K" "l \<in> L j" "l \<in> L k"
hoelzl@42981
   376
        have "k = j"
hoelzl@42981
   377
        proof (rule ccontr)
hoelzl@42981
   378
          assume "k \<noteq> j"
hoelzl@42981
   379
          with disjoint `K \<subseteq> J` `k \<in> K` `j \<in> K` have "I k \<inter> I j = {}"
hoelzl@42981
   380
            unfolding disjoint_family_on_def by auto
hoelzl@42981
   381
          with L(2,3)[OF `j \<in> K`] L(2,3)[OF `k \<in> K`]
hoelzl@42981
   382
          show False using `l \<in> L k` `l \<in> L j` by auto
hoelzl@42981
   383
        qed }
hoelzl@42981
   384
      note L_inj = this
hoelzl@42981
   385
hoelzl@42981
   386
      def k \<equiv> "\<lambda>l. (SOME k. k \<in> K \<and> l \<in> L k)"
hoelzl@42981
   387
      { fix x j l assume *: "j \<in> K" "l \<in> L j"
hoelzl@42981
   388
        have "k l = j" unfolding k_def
hoelzl@42981
   389
        proof (rule some_equality)
hoelzl@42981
   390
          fix k assume "k \<in> K \<and> l \<in> L k"
hoelzl@42981
   391
          with * L_inj show "k = j" by auto
hoelzl@42981
   392
        qed (insert *, simp) }
hoelzl@42981
   393
      note k_simp[simp] = this
hoelzl@42981
   394
      let "?E' l" = "E' (k l) l"
hoelzl@42981
   395
      have "prob (\<Inter>j\<in>K. A j) = prob (\<Inter>l\<in>(\<Union>k\<in>K. L k). ?E' l)"
hoelzl@42981
   396
        by (auto simp: A intro!: arg_cong[where f=prob])
hoelzl@42981
   397
      also have "\<dots> = (\<Prod>l\<in>(\<Union>k\<in>K. L k). prob (?E' l))"
hoelzl@42981
   398
        using L K E' by (intro indep_setsD[OF indep]) (simp_all add: UN_mono)
hoelzl@42981
   399
      also have "\<dots> = (\<Prod>j\<in>K. \<Prod>l\<in>L j. prob (E' j l))"
hoelzl@42981
   400
        using K L L_inj by (subst setprod_UN_disjoint) auto
hoelzl@42981
   401
      also have "\<dots> = (\<Prod>j\<in>K. prob (A j))"
hoelzl@42981
   402
        using K L E' by (auto simp add: A intro!: setprod_cong indep_setsD[OF indep, symmetric]) blast
hoelzl@42981
   403
      finally show "prob (\<Inter>j\<in>K. A j) = (\<Prod>j\<in>K. prob (A j))" .
hoelzl@42981
   404
    qed
hoelzl@42981
   405
  next
hoelzl@42981
   406
    fix j assume "j \<in> J"
hoelzl@42981
   407
    show "Int_stable \<lparr> space = space M, sets = ?E j \<rparr>"
hoelzl@42981
   408
    proof (rule Int_stableI)
hoelzl@42981
   409
      fix a assume "a \<in> ?E j" then obtain Ka Ea
hoelzl@42981
   410
        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
hoelzl@42981
   411
      fix b assume "b \<in> ?E j" then obtain Kb Eb
hoelzl@42981
   412
        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
hoelzl@42981
   413
      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 {})"
hoelzl@42981
   414
      have "a \<inter> b = INTER (Ka \<union> Kb) ?A"
hoelzl@42981
   415
        by (simp add: a b set_eq_iff) auto
hoelzl@42981
   416
      with a b `j \<in> J` Int_stableD[OF Int_stable] show "a \<inter> b \<in> ?E j"
hoelzl@42981
   417
        by (intro CollectI exI[of _ "Ka \<union> Kb"] exI[of _ ?A]) auto
hoelzl@42981
   418
    qed
hoelzl@42981
   419
  qed
hoelzl@42981
   420
  ultimately show ?thesis
hoelzl@42981
   421
    by (simp cong: indep_sets_cong)
hoelzl@42981
   422
qed
hoelzl@42981
   423
hoelzl@42861
   424
end