src/HOL/Probability/Independent_Family.thy
author hoelzl
Thu May 26 14:12:06 2011 +0200 (2011-05-26)
changeset 42987 73e2d802ea41
parent 42985 1fb670792708
child 42988 d8f3fc934ff6
permissions -rw-r--r--
add lemma indep_rv_finite
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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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lemma INT_decseq_offset:
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  assumes "decseq F"
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  shows "(\<Inter>i. F i) = (\<Inter>i\<in>{n..}. F i)"
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proof safe
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  fix x i assume x: "x \<in> (\<Inter>i\<in>{n..}. F i)"
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  show "x \<in> F i"
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  proof cases
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    from x have "x \<in> F n" by auto
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    also assume "i \<le> n" with `decseq F` have "F n \<subseteq> F i"
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      unfolding decseq_def by simp
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    finally show ?thesis .
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  qed (insert x, simp)
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qed auto
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definition (in prob_space)
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  "indep_events A I \<longleftrightarrow> (A`I \<subseteq> events) \<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> events) \<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[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_sets_singleton_iff_indep_events:
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  "indep_sets (\<lambda>i. {F i}) I \<longleftrightarrow> indep_events F I"
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  unfolding indep_sets_def indep_events_def
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  by (simp, intro conj_cong ball_cong all_cong imp_cong) (auto simp: Pi_iff)
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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 (in prob_space) indep_setI:
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  assumes ev: "A \<subseteq> events" "B \<subseteq> events"
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    and indep: "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> prob (a \<inter> b) = prob a * prob b"
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  shows "indep_set A B"
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  unfolding indep_set_def
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proof (rule indep_setsI)
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  fix F J assume "J \<noteq> {}" "J \<subseteq> UNIV"
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    and F: "\<forall>j\<in>J. F j \<in> (case j of True \<Rightarrow> A | False \<Rightarrow> B)"
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  have "J \<in> Pow UNIV" by auto
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  with F `J \<noteq> {}` indep[of "F True" "F False"]
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  show "prob (\<Inter>j\<in>J. F j) = (\<Prod>j\<in>J. prob (F j))"
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    unfolding UNIV_bool Pow_insert by (auto simp: ac_simps)
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qed (auto split: bool.split simp: ev)
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lemma (in prob_space) indep_setD:
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  assumes indep: "indep_set A B" and ev: "a \<in> A" "b \<in> B"
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  shows "prob (a \<inter> b) = prob a * prob b"
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  using indep[unfolded indep_set_def, THEN indep_setsD, of UNIV "bool_case a b"] ev
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  by (simp add: ac_simps UNIV_bool)
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lemma (in prob_space)
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  assumes indep: "indep_set A B"
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  shows indep_setD_ev1: "A \<subseteq> events"
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    and indep_setD_ev2: "B \<subseteq> events"
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  using indep unfolding indep_set_def indep_sets_def UNIV_bool 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 cong del: indep_sets_cong, 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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   241
            using `J \<noteq> {}` `j \<notin> J` `j \<in> K` by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
hoelzl@42861
   242
          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))"
hoelzl@42861
   243
          proof (rule finite_measure_UNION)
hoelzl@42861
   244
            show "disjoint_family (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i)"
hoelzl@42861
   245
              using disj by (rule disjoint_family_on_bisimulation) auto
hoelzl@42861
   246
            show "range (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i) \<subseteq> events"
hoelzl@42861
   247
              using A_sets F `finite J` `J \<noteq> {}` `j \<notin> J` by (auto intro!: Int)
hoelzl@42861
   248
          qed
hoelzl@42861
   249
          moreover { fix k
hoelzl@42861
   250
            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))"
hoelzl@42861
   251
              by (subst indep_setsD[OF F(2)]) (auto intro!: setprod_cong split: split_if_asm)
hoelzl@42861
   252
            also have "\<dots> = prob (F k) * prob (\<Inter>i\<in>J. A i)"
hoelzl@42861
   253
              using J A `j \<in> K` by (subst indep_setsD[OF G(1)]) auto
hoelzl@42861
   254
            finally have "prob (\<Inter>i\<in>insert j J. (A(j := F k)) i) = prob (F k) * prob (\<Inter>i\<in>J. A i)" . }
hoelzl@42861
   255
          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)))"
hoelzl@42861
   256
            by simp
hoelzl@42861
   257
          moreover
hoelzl@42861
   258
          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))"
hoelzl@42861
   259
            using disj F(1) by (intro finite_measure_UNION sums_mult2) auto
hoelzl@42861
   260
          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)))"
hoelzl@42861
   261
            using J A `j \<in> K` by (subst indep_setsD[OF G(1), symmetric]) auto
hoelzl@42861
   262
          ultimately
hoelzl@42861
   263
          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))"
hoelzl@42861
   264
            by (auto dest!: sums_unique)
hoelzl@42861
   265
        qed (insert F, auto)
hoelzl@42861
   266
      qed (insert sets_into_space, auto)
hoelzl@42861
   267
      then have mono: "sets (dynkin \<lparr>space = space M, sets = G j\<rparr>) \<subseteq>
hoelzl@42861
   268
        sets \<lparr>space = space M, sets = {E \<in> events. indep_sets (G(j := {E})) K}\<rparr>"
hoelzl@42987
   269
      proof (rule dynkin_system.dynkin_subset, simp_all cong del: indep_sets_cong, safe)
hoelzl@42861
   270
        fix X assume "X \<in> G j"
hoelzl@42861
   271
        then show "X \<in> events" using G `j \<in> K` by auto
hoelzl@42861
   272
        from `indep_sets G K`
hoelzl@42861
   273
        show "indep_sets (G(j := {X})) K"
hoelzl@42861
   274
          by (rule indep_sets_mono_sets) (insert `X \<in> G j`, auto)
hoelzl@42861
   275
      qed
hoelzl@42861
   276
      have "indep_sets (G(j:=?D)) K"
hoelzl@42861
   277
      proof (rule indep_setsI)
hoelzl@42861
   278
        fix i assume "i \<in> K" then show "(G(j := ?D)) i \<subseteq> events"
hoelzl@42861
   279
          using G(2) by auto
hoelzl@42861
   280
      next
hoelzl@42861
   281
        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
   282
        show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
hoelzl@42861
   283
        proof cases
hoelzl@42861
   284
          assume "j \<in> J"
hoelzl@42861
   285
          with A have indep: "indep_sets (G(j := {A j})) K" by auto
hoelzl@42861
   286
          from J A show ?thesis
hoelzl@42861
   287
            by (intro indep_setsD[OF indep]) auto
hoelzl@42861
   288
        next
hoelzl@42861
   289
          assume "j \<notin> J"
hoelzl@42861
   290
          with J A have "\<forall>i\<in>J. A i \<in> G i" by (auto split: split_if_asm)
hoelzl@42861
   291
          with J show ?thesis
hoelzl@42861
   292
            by (intro indep_setsD[OF G(1)]) auto
hoelzl@42861
   293
        qed
hoelzl@42861
   294
      qed
hoelzl@42861
   295
      then have "indep_sets (G(j:=sets (dynkin \<lparr>space = space M, sets = G j\<rparr>))) K"
hoelzl@42861
   296
        by (rule indep_sets_mono_sets) (insert mono, auto)
hoelzl@42861
   297
      then show ?case
hoelzl@42861
   298
        by (rule indep_sets_mono_sets) (insert `j \<in> K` `j \<notin> J`, auto simp: G_def)
hoelzl@42861
   299
    qed (insert `indep_sets F K`, simp) }
hoelzl@42861
   300
  from this[OF `indep_sets F J` `finite J` subset_refl]
hoelzl@42861
   301
  show "indep_sets (\<lambda>i. sets (dynkin \<lparr> space = space M, sets = F i \<rparr>)) J"
hoelzl@42861
   302
    by (rule indep_sets_mono_sets) auto
hoelzl@42861
   303
qed
hoelzl@42861
   304
hoelzl@42861
   305
lemma (in prob_space) indep_sets_sigma:
hoelzl@42861
   306
  assumes indep: "indep_sets F I"
hoelzl@42861
   307
  assumes stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable \<lparr> space = space M, sets = F i \<rparr>"
hoelzl@42861
   308
  shows "indep_sets (\<lambda>i. sets (sigma \<lparr> space = space M, sets = F i \<rparr>)) I"
hoelzl@42861
   309
proof -
hoelzl@42861
   310
  from indep_sets_dynkin[OF indep]
hoelzl@42861
   311
  show ?thesis
hoelzl@42861
   312
  proof (rule indep_sets_mono_sets, subst sigma_eq_dynkin, simp_all add: stable)
hoelzl@42861
   313
    fix i assume "i \<in> I"
hoelzl@42861
   314
    with indep have "F i \<subseteq> events" by (auto simp: indep_sets_def)
hoelzl@42861
   315
    with sets_into_space show "F i \<subseteq> Pow (space M)" by auto
hoelzl@42861
   316
  qed
hoelzl@42861
   317
qed
hoelzl@42861
   318
hoelzl@42861
   319
lemma (in prob_space) indep_sets_sigma_sets:
hoelzl@42861
   320
  assumes "indep_sets F I"
hoelzl@42861
   321
  assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable \<lparr> space = space M, sets = F i \<rparr>"
hoelzl@42861
   322
  shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I"
hoelzl@42861
   323
  using indep_sets_sigma[OF assms] by (simp add: sets_sigma)
hoelzl@42861
   324
hoelzl@42987
   325
lemma (in prob_space) indep_sets_sigma_sets_iff:
hoelzl@42987
   326
  assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable \<lparr> space = space M, sets = F i \<rparr>"
hoelzl@42987
   327
  shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I \<longleftrightarrow> indep_sets F I"
hoelzl@42987
   328
proof
hoelzl@42987
   329
  assume "indep_sets F I" then show "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I"
hoelzl@42987
   330
    by (rule indep_sets_sigma_sets) fact
hoelzl@42987
   331
next
hoelzl@42987
   332
  assume "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I" then show "indep_sets F I"
hoelzl@42987
   333
    by (rule indep_sets_mono_sets) (intro subsetI sigma_sets.Basic)
hoelzl@42987
   334
qed
hoelzl@42987
   335
hoelzl@42861
   336
lemma (in prob_space) indep_sets2_eq:
hoelzl@42981
   337
  "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
   338
  unfolding indep_set_def
hoelzl@42861
   339
proof (intro iffI ballI conjI)
hoelzl@42861
   340
  assume indep: "indep_sets (bool_case A B) UNIV"
hoelzl@42861
   341
  { fix a b assume "a \<in> A" "b \<in> B"
hoelzl@42861
   342
    with indep_setsD[OF indep, of UNIV "bool_case a b"]
hoelzl@42861
   343
    show "prob (a \<inter> b) = prob a * prob b"
hoelzl@42861
   344
      unfolding UNIV_bool by (simp add: ac_simps) }
hoelzl@42861
   345
  from indep show "A \<subseteq> events" "B \<subseteq> events"
hoelzl@42861
   346
    unfolding indep_sets_def UNIV_bool by auto
hoelzl@42861
   347
next
hoelzl@42861
   348
  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
   349
  show "indep_sets (bool_case A B) UNIV"
hoelzl@42861
   350
  proof (rule indep_setsI)
hoelzl@42861
   351
    fix i show "(case i of True \<Rightarrow> A | False \<Rightarrow> B) \<subseteq> events"
hoelzl@42861
   352
      using * by (auto split: bool.split)
hoelzl@42861
   353
  next
hoelzl@42861
   354
    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
   355
    then have "J = {True} \<or> J = {False} \<or> J = {True,False}"
hoelzl@42861
   356
      by (auto simp: UNIV_bool)
hoelzl@42861
   357
    then show "prob (\<Inter>j\<in>J. X j) = (\<Prod>j\<in>J. prob (X j))"
hoelzl@42861
   358
      using X * by auto
hoelzl@42861
   359
  qed
hoelzl@42861
   360
qed
hoelzl@42861
   361
hoelzl@42981
   362
lemma (in prob_space) indep_set_sigma_sets:
hoelzl@42981
   363
  assumes "indep_set A B"
hoelzl@42861
   364
  assumes A: "Int_stable \<lparr> space = space M, sets = A \<rparr>"
hoelzl@42861
   365
  assumes B: "Int_stable \<lparr> space = space M, sets = B \<rparr>"
hoelzl@42981
   366
  shows "indep_set (sigma_sets (space M) A) (sigma_sets (space M) B)"
hoelzl@42861
   367
proof -
hoelzl@42861
   368
  have "indep_sets (\<lambda>i. sigma_sets (space M) (case i of True \<Rightarrow> A | False \<Rightarrow> B)) UNIV"
hoelzl@42861
   369
  proof (rule indep_sets_sigma_sets)
hoelzl@42861
   370
    show "indep_sets (bool_case A B) UNIV"
hoelzl@42981
   371
      by (rule `indep_set A B`[unfolded indep_set_def])
hoelzl@42861
   372
    fix i show "Int_stable \<lparr>space = space M, sets = case i of True \<Rightarrow> A | False \<Rightarrow> B\<rparr>"
hoelzl@42861
   373
      using A B by (cases i) auto
hoelzl@42861
   374
  qed
hoelzl@42861
   375
  then show ?thesis
hoelzl@42981
   376
    unfolding indep_set_def
hoelzl@42861
   377
    by (rule indep_sets_mono_sets) (auto split: bool.split)
hoelzl@42861
   378
qed
hoelzl@42861
   379
hoelzl@42981
   380
lemma (in prob_space) indep_sets_collect_sigma:
hoelzl@42981
   381
  fixes I :: "'j \<Rightarrow> 'i set" and J :: "'j set" and E :: "'i \<Rightarrow> 'a set set"
hoelzl@42981
   382
  assumes indep: "indep_sets E (\<Union>j\<in>J. I j)"
hoelzl@42981
   383
  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
   384
  assumes disjoint: "disjoint_family_on I J"
hoelzl@42981
   385
  shows "indep_sets (\<lambda>j. sigma_sets (space M) (\<Union>i\<in>I j. E i)) J"
hoelzl@42981
   386
proof -
hoelzl@42981
   387
  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
   388
hoelzl@42983
   389
  from indep have E: "\<And>j i. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> E i \<subseteq> events"
hoelzl@42981
   390
    unfolding indep_sets_def by auto
hoelzl@42981
   391
  { fix j
hoelzl@42981
   392
    let ?S = "sigma \<lparr> space = space M, sets = (\<Union>i\<in>I j. E i) \<rparr>"
hoelzl@42981
   393
    assume "j \<in> J"
hoelzl@42981
   394
    from E[OF this] interpret S: sigma_algebra ?S
hoelzl@42981
   395
      using sets_into_space by (intro sigma_algebra_sigma) auto
hoelzl@42981
   396
hoelzl@42981
   397
    have "sigma_sets (space M) (\<Union>i\<in>I j. E i) = sigma_sets (space M) (?E j)"
hoelzl@42981
   398
    proof (rule sigma_sets_eqI)
hoelzl@42981
   399
      fix A assume "A \<in> (\<Union>i\<in>I j. E i)"
hoelzl@42981
   400
      then guess i ..
hoelzl@42981
   401
      then show "A \<in> sigma_sets (space M) (?E j)"
hoelzl@42981
   402
        by (auto intro!: sigma_sets.intros exI[of _ "{i}"] exI[of _ "\<lambda>i. A"])
hoelzl@42981
   403
    next
hoelzl@42981
   404
      fix A assume "A \<in> ?E j"
hoelzl@42981
   405
      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
   406
        and A: "A = (\<Inter>k\<in>K. E' k)"
hoelzl@42981
   407
        by auto
hoelzl@42981
   408
      then have "A \<in> sets ?S" unfolding A
hoelzl@42981
   409
        by (safe intro!: S.finite_INT)
hoelzl@42981
   410
           (auto simp: sets_sigma intro!: sigma_sets.Basic)
hoelzl@42981
   411
      then show "A \<in> sigma_sets (space M) (\<Union>i\<in>I j. E i)"
hoelzl@42981
   412
        by (simp add: sets_sigma)
hoelzl@42981
   413
    qed }
hoelzl@42981
   414
  moreover have "indep_sets (\<lambda>j. sigma_sets (space M) (?E j)) J"
hoelzl@42981
   415
  proof (rule indep_sets_sigma_sets)
hoelzl@42981
   416
    show "indep_sets ?E J"
hoelzl@42981
   417
    proof (intro indep_setsI)
hoelzl@42981
   418
      fix j assume "j \<in> J" with E show "?E j \<subseteq> events" by (force  intro!: finite_INT)
hoelzl@42981
   419
    next
hoelzl@42981
   420
      fix K A assume K: "K \<noteq> {}" "K \<subseteq> J" "finite K"
hoelzl@42981
   421
        and "\<forall>j\<in>K. A j \<in> ?E j"
hoelzl@42981
   422
      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
   423
        by simp
hoelzl@42981
   424
      from bchoice[OF this] guess E' ..
hoelzl@42981
   425
      from bchoice[OF this] obtain L
hoelzl@42981
   426
        where A: "\<And>j. j\<in>K \<Longrightarrow> A j = (\<Inter>l\<in>L j. E' j l)"
hoelzl@42981
   427
        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
   428
        and E': "\<And>j l. j\<in>K \<Longrightarrow> l \<in> L j \<Longrightarrow> E' j l \<in> E l"
hoelzl@42981
   429
        by auto
hoelzl@42981
   430
hoelzl@42981
   431
      { fix k l j assume "k \<in> K" "j \<in> K" "l \<in> L j" "l \<in> L k"
hoelzl@42981
   432
        have "k = j"
hoelzl@42981
   433
        proof (rule ccontr)
hoelzl@42981
   434
          assume "k \<noteq> j"
hoelzl@42981
   435
          with disjoint `K \<subseteq> J` `k \<in> K` `j \<in> K` have "I k \<inter> I j = {}"
hoelzl@42981
   436
            unfolding disjoint_family_on_def by auto
hoelzl@42981
   437
          with L(2,3)[OF `j \<in> K`] L(2,3)[OF `k \<in> K`]
hoelzl@42981
   438
          show False using `l \<in> L k` `l \<in> L j` by auto
hoelzl@42981
   439
        qed }
hoelzl@42981
   440
      note L_inj = this
hoelzl@42981
   441
hoelzl@42981
   442
      def k \<equiv> "\<lambda>l. (SOME k. k \<in> K \<and> l \<in> L k)"
hoelzl@42981
   443
      { fix x j l assume *: "j \<in> K" "l \<in> L j"
hoelzl@42981
   444
        have "k l = j" unfolding k_def
hoelzl@42981
   445
        proof (rule some_equality)
hoelzl@42981
   446
          fix k assume "k \<in> K \<and> l \<in> L k"
hoelzl@42981
   447
          with * L_inj show "k = j" by auto
hoelzl@42981
   448
        qed (insert *, simp) }
hoelzl@42981
   449
      note k_simp[simp] = this
hoelzl@42981
   450
      let "?E' l" = "E' (k l) l"
hoelzl@42981
   451
      have "prob (\<Inter>j\<in>K. A j) = prob (\<Inter>l\<in>(\<Union>k\<in>K. L k). ?E' l)"
hoelzl@42981
   452
        by (auto simp: A intro!: arg_cong[where f=prob])
hoelzl@42981
   453
      also have "\<dots> = (\<Prod>l\<in>(\<Union>k\<in>K. L k). prob (?E' l))"
hoelzl@42981
   454
        using L K E' by (intro indep_setsD[OF indep]) (simp_all add: UN_mono)
hoelzl@42981
   455
      also have "\<dots> = (\<Prod>j\<in>K. \<Prod>l\<in>L j. prob (E' j l))"
hoelzl@42981
   456
        using K L L_inj by (subst setprod_UN_disjoint) auto
hoelzl@42981
   457
      also have "\<dots> = (\<Prod>j\<in>K. prob (A j))"
hoelzl@42981
   458
        using K L E' by (auto simp add: A intro!: setprod_cong indep_setsD[OF indep, symmetric]) blast
hoelzl@42981
   459
      finally show "prob (\<Inter>j\<in>K. A j) = (\<Prod>j\<in>K. prob (A j))" .
hoelzl@42981
   460
    qed
hoelzl@42981
   461
  next
hoelzl@42981
   462
    fix j assume "j \<in> J"
hoelzl@42981
   463
    show "Int_stable \<lparr> space = space M, sets = ?E j \<rparr>"
hoelzl@42981
   464
    proof (rule Int_stableI)
hoelzl@42981
   465
      fix a assume "a \<in> ?E j" then obtain Ka Ea
hoelzl@42981
   466
        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
   467
      fix b assume "b \<in> ?E j" then obtain Kb Eb
hoelzl@42981
   468
        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
   469
      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
   470
      have "a \<inter> b = INTER (Ka \<union> Kb) ?A"
hoelzl@42981
   471
        by (simp add: a b set_eq_iff) auto
hoelzl@42981
   472
      with a b `j \<in> J` Int_stableD[OF Int_stable] show "a \<inter> b \<in> ?E j"
hoelzl@42981
   473
        by (intro CollectI exI[of _ "Ka \<union> Kb"] exI[of _ ?A]) auto
hoelzl@42981
   474
    qed
hoelzl@42981
   475
  qed
hoelzl@42981
   476
  ultimately show ?thesis
hoelzl@42981
   477
    by (simp cong: indep_sets_cong)
hoelzl@42981
   478
qed
hoelzl@42981
   479
hoelzl@42982
   480
definition (in prob_space) terminal_events where
hoelzl@42982
   481
  "terminal_events A = (\<Inter>n. sigma_sets (space M) (UNION {n..} A))"
hoelzl@42982
   482
hoelzl@42982
   483
lemma (in prob_space) terminal_events_sets:
hoelzl@42983
   484
  assumes A: "\<And>i. A i \<subseteq> events"
hoelzl@42982
   485
  assumes "\<And>i::nat. sigma_algebra \<lparr>space = space M, sets = A i\<rparr>"
hoelzl@42982
   486
  assumes X: "X \<in> terminal_events A"
hoelzl@42983
   487
  shows "X \<in> events"
hoelzl@42982
   488
proof -
hoelzl@42982
   489
  let ?A = "(\<Inter>n. sigma_sets (space M) (UNION {n..} A))"
hoelzl@42982
   490
  interpret A: sigma_algebra "\<lparr>space = space M, sets = A i\<rparr>" for i by fact
hoelzl@42982
   491
  from X have "\<And>n. X \<in> sigma_sets (space M) (UNION {n..} A)" by (auto simp: terminal_events_def)
hoelzl@42982
   492
  from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp
hoelzl@42983
   493
  then show "X \<in> events"
hoelzl@42982
   494
    by induct (insert A, auto)
hoelzl@42982
   495
qed
hoelzl@42982
   496
hoelzl@42982
   497
lemma (in prob_space) sigma_algebra_terminal_events:
hoelzl@42982
   498
  assumes "\<And>i::nat. sigma_algebra \<lparr>space = space M, sets = A i\<rparr>"
hoelzl@42982
   499
  shows "sigma_algebra \<lparr> space = space M, sets = terminal_events A \<rparr>"
hoelzl@42982
   500
  unfolding terminal_events_def
hoelzl@42982
   501
proof (simp add: sigma_algebra_iff2, safe)
hoelzl@42982
   502
  let ?A = "(\<Inter>n. sigma_sets (space M) (UNION {n..} A))"
hoelzl@42982
   503
  interpret A: sigma_algebra "\<lparr>space = space M, sets = A i\<rparr>" for i by fact
hoelzl@42982
   504
  { fix X x assume "X \<in> ?A" "x \<in> X" 
hoelzl@42982
   505
    then have "\<And>n. X \<in> sigma_sets (space M) (UNION {n..} A)" by auto
hoelzl@42982
   506
    from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp
hoelzl@42982
   507
    then have "X \<subseteq> space M"
hoelzl@42982
   508
      by induct (insert A.sets_into_space, auto)
hoelzl@42982
   509
    with `x \<in> X` show "x \<in> space M" by auto }
hoelzl@42982
   510
  { fix F :: "nat \<Rightarrow> 'a set" and n assume "range F \<subseteq> ?A"
hoelzl@42982
   511
    then show "(UNION UNIV F) \<in> sigma_sets (space M) (UNION {n..} A)"
hoelzl@42982
   512
      by (intro sigma_sets.Union) auto }
hoelzl@42982
   513
qed (auto intro!: sigma_sets.Compl sigma_sets.Empty)
hoelzl@42982
   514
hoelzl@42982
   515
lemma (in prob_space) kolmogorov_0_1_law:
hoelzl@42982
   516
  fixes A :: "nat \<Rightarrow> 'a set set"
hoelzl@42983
   517
  assumes A: "\<And>i. A i \<subseteq> events"
hoelzl@42982
   518
  assumes "\<And>i::nat. sigma_algebra \<lparr>space = space M, sets = A i\<rparr>"
hoelzl@42982
   519
  assumes indep: "indep_sets A UNIV"
hoelzl@42982
   520
  and X: "X \<in> terminal_events A"
hoelzl@42982
   521
  shows "prob X = 0 \<or> prob X = 1"
hoelzl@42982
   522
proof -
hoelzl@42983
   523
  let ?D = "\<lparr> space = space M, sets = {D \<in> events. prob (X \<inter> D) = prob X * prob D} \<rparr>"
hoelzl@42982
   524
  interpret A: sigma_algebra "\<lparr>space = space M, sets = A i\<rparr>" for i by fact
hoelzl@42982
   525
  interpret T: sigma_algebra "\<lparr> space = space M, sets = terminal_events A \<rparr>"
hoelzl@42982
   526
    by (rule sigma_algebra_terminal_events) fact
hoelzl@42982
   527
  have "X \<subseteq> space M" using T.space_closed X by auto
hoelzl@42982
   528
hoelzl@42983
   529
  have X_in: "X \<in> events"
hoelzl@42982
   530
    by (rule terminal_events_sets) fact+
hoelzl@42982
   531
hoelzl@42982
   532
  interpret D: dynkin_system ?D
hoelzl@42982
   533
  proof (rule dynkin_systemI)
hoelzl@42982
   534
    fix D assume "D \<in> sets ?D" then show "D \<subseteq> space ?D"
hoelzl@42982
   535
      using sets_into_space by auto
hoelzl@42982
   536
  next
hoelzl@42982
   537
    show "space ?D \<in> sets ?D"
hoelzl@42982
   538
      using prob_space `X \<subseteq> space M` by (simp add: Int_absorb2)
hoelzl@42982
   539
  next
hoelzl@42982
   540
    fix A assume A: "A \<in> sets ?D"
hoelzl@42982
   541
    have "prob (X \<inter> (space M - A)) = prob (X - (X \<inter> A))"
hoelzl@42982
   542
      using `X \<subseteq> space M` by (auto intro!: arg_cong[where f=prob])
hoelzl@42982
   543
    also have "\<dots> = prob X - prob (X \<inter> A)"
hoelzl@42982
   544
      using X_in A by (intro finite_measure_Diff) auto
hoelzl@42982
   545
    also have "\<dots> = prob X * prob (space M) - prob X * prob A"
hoelzl@42982
   546
      using A prob_space by auto
hoelzl@42982
   547
    also have "\<dots> = prob X * prob (space M - A)"
hoelzl@42982
   548
      using X_in A sets_into_space
hoelzl@42982
   549
      by (subst finite_measure_Diff) (auto simp: field_simps)
hoelzl@42982
   550
    finally show "space ?D - A \<in> sets ?D"
hoelzl@42982
   551
      using A `X \<subseteq> space M` by auto
hoelzl@42982
   552
  next
hoelzl@42982
   553
    fix F :: "nat \<Rightarrow> 'a set" assume dis: "disjoint_family F" and "range F \<subseteq> sets ?D"
hoelzl@42982
   554
    then have F: "range F \<subseteq> events" "\<And>i. prob (X \<inter> F i) = prob X * prob (F i)"
hoelzl@42982
   555
      by auto
hoelzl@42982
   556
    have "(\<lambda>i. prob (X \<inter> F i)) sums prob (\<Union>i. X \<inter> F i)"
hoelzl@42982
   557
    proof (rule finite_measure_UNION)
hoelzl@42982
   558
      show "range (\<lambda>i. X \<inter> F i) \<subseteq> events"
hoelzl@42982
   559
        using F X_in by auto
hoelzl@42982
   560
      show "disjoint_family (\<lambda>i. X \<inter> F i)"
hoelzl@42982
   561
        using dis by (rule disjoint_family_on_bisimulation) auto
hoelzl@42982
   562
    qed
hoelzl@42982
   563
    with F have "(\<lambda>i. prob X * prob (F i)) sums prob (X \<inter> (\<Union>i. F i))"
hoelzl@42982
   564
      by simp
hoelzl@42982
   565
    moreover have "(\<lambda>i. prob X * prob (F i)) sums (prob X * prob (\<Union>i. F i))"
hoelzl@42982
   566
      by (intro mult_right.sums finite_measure_UNION F dis)
hoelzl@42982
   567
    ultimately have "prob (X \<inter> (\<Union>i. F i)) = prob X * prob (\<Union>i. F i)"
hoelzl@42982
   568
      by (auto dest!: sums_unique)
hoelzl@42982
   569
    with F show "(\<Union>i. F i) \<in> sets ?D"
hoelzl@42982
   570
      by auto
hoelzl@42982
   571
  qed
hoelzl@42982
   572
hoelzl@42982
   573
  { fix n
hoelzl@42982
   574
    have "indep_sets (\<lambda>b. sigma_sets (space M) (\<Union>m\<in>bool_case {..n} {Suc n..} b. A m)) UNIV"
hoelzl@42982
   575
    proof (rule indep_sets_collect_sigma)
hoelzl@42982
   576
      have *: "(\<Union>b. case b of True \<Rightarrow> {..n} | False \<Rightarrow> {Suc n..}) = UNIV" (is "?U = _")
hoelzl@42982
   577
        by (simp split: bool.split add: set_eq_iff) (metis not_less_eq_eq)
hoelzl@42982
   578
      with indep show "indep_sets A ?U" by simp
hoelzl@42982
   579
      show "disjoint_family (bool_case {..n} {Suc n..})"
hoelzl@42982
   580
        unfolding disjoint_family_on_def by (auto split: bool.split)
hoelzl@42982
   581
      fix m
hoelzl@42982
   582
      show "Int_stable \<lparr>space = space M, sets = A m\<rparr>"
hoelzl@42982
   583
        unfolding Int_stable_def using A.Int by auto
hoelzl@42982
   584
    qed
hoelzl@42982
   585
    also have "(\<lambda>b. sigma_sets (space M) (\<Union>m\<in>bool_case {..n} {Suc n..} b. A m)) = 
hoelzl@42982
   586
      bool_case (sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) (sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m))"
hoelzl@42982
   587
      by (auto intro!: ext split: bool.split)
hoelzl@42982
   588
    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))"
hoelzl@42982
   589
      unfolding indep_set_def by simp
hoelzl@42982
   590
hoelzl@42982
   591
    have "sigma_sets (space M) (\<Union>m\<in>{..n}. A m) \<subseteq> sets ?D"
hoelzl@42982
   592
    proof (simp add: subset_eq, rule)
hoelzl@42982
   593
      fix D assume D: "D \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)"
hoelzl@42982
   594
      have "X \<in> sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m)"
hoelzl@42982
   595
        using X unfolding terminal_events_def by simp
hoelzl@42982
   596
      from indep_setD[OF indep D this] indep_setD_ev1[OF indep] D
hoelzl@42982
   597
      show "D \<in> events \<and> prob (X \<inter> D) = prob X * prob D"
hoelzl@42982
   598
        by (auto simp add: ac_simps)
hoelzl@42982
   599
    qed }
hoelzl@42982
   600
  then have "(\<Union>n. sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) \<subseteq> sets ?D" (is "?A \<subseteq> _")
hoelzl@42982
   601
    by auto
hoelzl@42982
   602
hoelzl@42982
   603
  have "sigma \<lparr> space = space M, sets = ?A \<rparr> =
hoelzl@42982
   604
    dynkin \<lparr> space = space M, sets = ?A \<rparr>" (is "sigma ?UA = dynkin ?UA")
hoelzl@42982
   605
  proof (rule sigma_eq_dynkin)
hoelzl@42982
   606
    { fix B n assume "B \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)"
hoelzl@42982
   607
      then have "B \<subseteq> space M"
hoelzl@42982
   608
        by induct (insert A sets_into_space, auto) }
hoelzl@42982
   609
    then show "sets ?UA \<subseteq> Pow (space ?UA)" by auto
hoelzl@42982
   610
    show "Int_stable ?UA"
hoelzl@42982
   611
    proof (rule Int_stableI)
hoelzl@42982
   612
      fix a assume "a \<in> ?A" then guess n .. note a = this
hoelzl@42982
   613
      fix b assume "b \<in> ?A" then guess m .. note b = this
hoelzl@42982
   614
      interpret Amn: sigma_algebra "sigma \<lparr>space = space M, sets = (\<Union>i\<in>{..max m n}. A i)\<rparr>"
hoelzl@42982
   615
        using A sets_into_space by (intro sigma_algebra_sigma) auto
hoelzl@42982
   616
      have "sigma_sets (space M) (\<Union>i\<in>{..n}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
hoelzl@42982
   617
        by (intro sigma_sets_subseteq UN_mono) auto
hoelzl@42982
   618
      with a have "a \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto
hoelzl@42982
   619
      moreover
hoelzl@42982
   620
      have "sigma_sets (space M) (\<Union>i\<in>{..m}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
hoelzl@42982
   621
        by (intro sigma_sets_subseteq UN_mono) auto
hoelzl@42982
   622
      with b have "b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto
hoelzl@42982
   623
      ultimately have "a \<inter> b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
hoelzl@42982
   624
        using Amn.Int[of a b] by (simp add: sets_sigma)
hoelzl@42982
   625
      then show "a \<inter> b \<in> (\<Union>n. sigma_sets (space M) (\<Union>i\<in>{..n}. A i))" by auto
hoelzl@42982
   626
    qed
hoelzl@42982
   627
  qed
hoelzl@42982
   628
  moreover have "sets (dynkin ?UA) \<subseteq> sets ?D"
hoelzl@42982
   629
  proof (rule D.dynkin_subset)
hoelzl@42982
   630
    show "sets ?UA \<subseteq> sets ?D" using `?A \<subseteq> sets ?D` by auto
hoelzl@42982
   631
  qed simp
hoelzl@42982
   632
  ultimately have "sets (sigma ?UA) \<subseteq> sets ?D" by simp
hoelzl@42982
   633
  moreover
hoelzl@42982
   634
  have "\<And>n. sigma_sets (space M) (\<Union>i\<in>{n..}. A i) \<subseteq> sigma_sets (space M) ?A"
hoelzl@42982
   635
    by (intro sigma_sets_subseteq UN_mono) (auto intro: sigma_sets.Basic)
hoelzl@42982
   636
  then have "terminal_events A \<subseteq> sets (sigma ?UA)"
hoelzl@42982
   637
    unfolding sets_sigma terminal_events_def by auto
hoelzl@42982
   638
  moreover note `X \<in> terminal_events A`
hoelzl@42982
   639
  ultimately have "X \<in> sets ?D" by auto
hoelzl@42982
   640
  then show ?thesis by auto
hoelzl@42982
   641
qed
hoelzl@42982
   642
hoelzl@42985
   643
lemma (in prob_space) borel_0_1_law:
hoelzl@42985
   644
  fixes F :: "nat \<Rightarrow> 'a set"
hoelzl@42985
   645
  assumes F: "range F \<subseteq> events" "indep_events F UNIV"
hoelzl@42985
   646
  shows "prob (\<Inter>n. \<Union>m\<in>{n..}. F m) = 0 \<or> prob (\<Inter>n. \<Union>m\<in>{n..}. F m) = 1"
hoelzl@42985
   647
proof (rule kolmogorov_0_1_law[of "\<lambda>i. sigma_sets (space M) { F i }"])
hoelzl@42985
   648
  show "\<And>i. sigma_sets (space M) {F i} \<subseteq> events"
hoelzl@42985
   649
    using F(1) sets_into_space
hoelzl@42985
   650
    by (subst sigma_sets_singleton) auto
hoelzl@42985
   651
  { fix i show "sigma_algebra \<lparr>space = space M, sets = sigma_sets (space M) {F i}\<rparr>"
hoelzl@42985
   652
      using sigma_algebra_sigma[of "\<lparr>space = space M, sets = {F i}\<rparr>"] F sets_into_space
hoelzl@42985
   653
      by (auto simp add: sigma_def) }
hoelzl@42985
   654
  show "indep_sets (\<lambda>i. sigma_sets (space M) {F i}) UNIV"
hoelzl@42985
   655
  proof (rule indep_sets_sigma_sets)
hoelzl@42985
   656
    show "indep_sets (\<lambda>i. {F i}) UNIV"
hoelzl@42985
   657
      unfolding indep_sets_singleton_iff_indep_events by fact
hoelzl@42985
   658
    fix i show "Int_stable \<lparr>space = space M, sets = {F i}\<rparr>"
hoelzl@42985
   659
      unfolding Int_stable_def by simp
hoelzl@42985
   660
  qed
hoelzl@42985
   661
  let "?Q n" = "\<Union>i\<in>{n..}. F i"
hoelzl@42985
   662
  show "(\<Inter>n. \<Union>m\<in>{n..}. F m) \<in> terminal_events (\<lambda>i. sigma_sets (space M) {F i})"
hoelzl@42985
   663
    unfolding terminal_events_def
hoelzl@42985
   664
  proof
hoelzl@42985
   665
    fix j
hoelzl@42985
   666
    interpret S: sigma_algebra "sigma \<lparr> space = space M, sets = (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})\<rparr>"
hoelzl@42985
   667
      using order_trans[OF F(1) space_closed]
hoelzl@42985
   668
      by (intro sigma_algebra_sigma) (simp add: sigma_sets_singleton subset_eq)
hoelzl@42985
   669
    have "(\<Inter>n. ?Q n) = (\<Inter>n\<in>{j..}. ?Q n)"
hoelzl@42985
   670
      by (intro decseq_SucI INT_decseq_offset UN_mono) auto
hoelzl@42985
   671
    also have "\<dots> \<in> sets (sigma \<lparr> space = space M, sets = (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})\<rparr>)"
hoelzl@42985
   672
      using order_trans[OF F(1) space_closed]
hoelzl@42985
   673
      by (safe intro!: S.countable_INT S.countable_UN)
hoelzl@42985
   674
         (auto simp: sets_sigma sigma_sets_singleton intro!: sigma_sets.Basic bexI)
hoelzl@42985
   675
    finally show "(\<Inter>n. ?Q n) \<in> sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})"
hoelzl@42985
   676
      by (simp add: sets_sigma)
hoelzl@42985
   677
  qed
hoelzl@42985
   678
qed
hoelzl@42985
   679
hoelzl@42987
   680
lemma (in prob_space) indep_sets_finite:
hoelzl@42987
   681
  assumes I: "I \<noteq> {}" "finite I"
hoelzl@42987
   682
    and F: "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events" "\<And>i. i \<in> I \<Longrightarrow> space M \<in> F i"
hoelzl@42987
   683
  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)))"
hoelzl@42987
   684
proof
hoelzl@42987
   685
  assume *: "indep_sets F I"
hoelzl@42987
   686
  from I show "\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j))"
hoelzl@42987
   687
    by (intro indep_setsD[OF *] ballI) auto
hoelzl@42987
   688
next
hoelzl@42987
   689
  assume indep: "\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j))"
hoelzl@42987
   690
  show "indep_sets F I"
hoelzl@42987
   691
  proof (rule indep_setsI[OF F(1)])
hoelzl@42987
   692
    fix A J assume J: "J \<noteq> {}" "J \<subseteq> I" "finite J"
hoelzl@42987
   693
    assume A: "\<forall>j\<in>J. A j \<in> F j"
hoelzl@42987
   694
    let "?A j" = "if j \<in> J then A j else space M"
hoelzl@42987
   695
    have "prob (\<Inter>j\<in>I. ?A j) = prob (\<Inter>j\<in>J. A j)"
hoelzl@42987
   696
      using subset_trans[OF F(1) space_closed] J A
hoelzl@42987
   697
      by (auto intro!: arg_cong[where f=prob] split: split_if_asm) blast
hoelzl@42987
   698
    also
hoelzl@42987
   699
    from A F have "(\<lambda>j. if j \<in> J then A j else space M) \<in> Pi I F" (is "?A \<in> _")
hoelzl@42987
   700
      by (auto split: split_if_asm)
hoelzl@42987
   701
    with indep have "prob (\<Inter>j\<in>I. ?A j) = (\<Prod>j\<in>I. prob (?A j))"
hoelzl@42987
   702
      by auto
hoelzl@42987
   703
    also have "\<dots> = (\<Prod>j\<in>J. prob (A j))"
hoelzl@42987
   704
      unfolding if_distrib setprod.If_cases[OF `finite I`]
hoelzl@42987
   705
      using prob_space `J \<subseteq> I` by (simp add: Int_absorb1 setprod_1)
hoelzl@42987
   706
    finally show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" ..
hoelzl@42987
   707
  qed
hoelzl@42987
   708
qed
hoelzl@42987
   709
hoelzl@42987
   710
lemma (in prob_space) indep_rv_finite:
hoelzl@42987
   711
  fixes I :: "'i set"
hoelzl@42987
   712
  assumes I: "I \<noteq> {}" "finite I"
hoelzl@42987
   713
    and rv: "\<And>i. i \<in> I \<Longrightarrow> random_variable (sigma (M' i)) (X i)"
hoelzl@42987
   714
    and Int_stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (M' i)"
hoelzl@42987
   715
    and space: "\<And>i. i \<in> I \<Longrightarrow> space (M' i) \<in> sets (M' i)"
hoelzl@42987
   716
  shows "indep_rv (\<lambda>i. sigma (M' i)) X I \<longleftrightarrow>
hoelzl@42987
   717
    (\<forall>A\<in>(\<Pi> i\<in>I. sets (M' i)). prob (\<Inter>j\<in>I. X j -` A j \<inter> space M) = (\<Prod>j\<in>I. distribution (X j) (A j)))"
hoelzl@42987
   718
proof -
hoelzl@42987
   719
  from rv have X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> space M \<rightarrow> space (M' i)"
hoelzl@42987
   720
    unfolding measurable_def by simp
hoelzl@42987
   721
hoelzl@42987
   722
  { fix i assume "i\<in>I"
hoelzl@42987
   723
    have "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (sigma (M' i))}
hoelzl@42987
   724
      = sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
hoelzl@42987
   725
      unfolding sigma_sets_vimage_commute[OF X, OF `i \<in> I`]
hoelzl@42987
   726
      by (subst sigma_sets_sigma_sets_eq) auto }
hoelzl@42987
   727
  note this[simp]
hoelzl@42987
   728
hoelzl@42987
   729
  { fix i assume "i\<in>I"
hoelzl@42987
   730
    have "Int_stable \<lparr>space = space M, sets = {X i -` A \<inter> space M |A. A \<in> sets (M' i)}\<rparr>"
hoelzl@42987
   731
    proof (rule Int_stableI)
hoelzl@42987
   732
      fix a assume "a \<in> {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
hoelzl@42987
   733
      then obtain A where "a = X i -` A \<inter> space M" "A \<in> sets (M' i)" by auto
hoelzl@42987
   734
      moreover
hoelzl@42987
   735
      fix b assume "b \<in> {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
hoelzl@42987
   736
      then obtain B where "b = X i -` B \<inter> space M" "B \<in> sets (M' i)" by auto
hoelzl@42987
   737
      moreover
hoelzl@42987
   738
      have "(X i -` A \<inter> space M) \<inter> (X i -` B \<inter> space M) = X i -` (A \<inter> B) \<inter> space M" by auto
hoelzl@42987
   739
      moreover note Int_stable[OF `i \<in> I`]
hoelzl@42987
   740
      ultimately
hoelzl@42987
   741
      show "a \<inter> b \<in> {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
hoelzl@42987
   742
        by (auto simp del: vimage_Int intro!: exI[of _ "A \<inter> B"] dest: Int_stableD)
hoelzl@42987
   743
    qed }
hoelzl@42987
   744
  note indep_sets_sigma_sets_iff[OF this, simp]
hoelzl@42987
   745
 
hoelzl@42987
   746
  { fix i assume "i \<in> I"
hoelzl@42987
   747
    { fix A assume "A \<in> sets (M' i)"
hoelzl@42987
   748
      then have "A \<in> sets (sigma (M' i))" by (auto simp: sets_sigma intro: sigma_sets.Basic)
hoelzl@42987
   749
      moreover
hoelzl@42987
   750
      from rv[OF `i\<in>I`] have "X i \<in> measurable M (sigma (M' i))" by auto
hoelzl@42987
   751
      ultimately
hoelzl@42987
   752
      have "X i -` A \<inter> space M \<in> sets M" by (auto intro: measurable_sets) }
hoelzl@42987
   753
    with X[OF `i\<in>I`] space[OF `i\<in>I`]
hoelzl@42987
   754
    have "{X i -` A \<inter> space M |A. A \<in> sets (M' i)} \<subseteq> events"
hoelzl@42987
   755
      "space M \<in> {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
hoelzl@42987
   756
      by (auto intro!: exI[of _ "space (M' i)"]) }
hoelzl@42987
   757
  note indep_sets_finite[OF I this, simp]
hoelzl@42987
   758
  
hoelzl@42987
   759
  have "(\<forall>A\<in>\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> sets (M' i)}. prob (INTER I A) = (\<Prod>j\<in>I. prob (A j))) =
hoelzl@42987
   760
    (\<forall>A\<in>\<Pi> i\<in>I. sets (M' i). prob ((\<Inter>j\<in>I. X j -` A j) \<inter> space M) = (\<Prod>x\<in>I. prob (X x -` A x \<inter> space M)))"
hoelzl@42987
   761
    (is "?L = ?R")
hoelzl@42987
   762
  proof safe
hoelzl@42987
   763
    fix A assume ?L and A: "A \<in> (\<Pi> i\<in>I. sets (M' i))"
hoelzl@42987
   764
    from `?L`[THEN bspec, of "\<lambda>i. X i -` A i \<inter> space M"] A `I \<noteq> {}`
hoelzl@42987
   765
    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))"
hoelzl@42987
   766
      by (auto simp add: Pi_iff)
hoelzl@42987
   767
  next
hoelzl@42987
   768
    fix A assume ?R and A: "A \<in> (\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> sets (M' i)})"
hoelzl@42987
   769
    from A have "\<forall>i\<in>I. \<exists>B. A i = X i -` B \<inter> space M \<and> B \<in> sets (M' i)" by auto
hoelzl@42987
   770
    from bchoice[OF this] obtain B where B: "\<forall>i\<in>I. A i = X i -` B i \<inter> space M"
hoelzl@42987
   771
      "B \<in> (\<Pi> i\<in>I. sets (M' i))" by auto
hoelzl@42987
   772
    from `?R`[THEN bspec, OF B(2)] B(1) `I \<noteq> {}`
hoelzl@42987
   773
    show "prob (INTER I A) = (\<Prod>j\<in>I. prob (A j))"
hoelzl@42987
   774
      by simp
hoelzl@42987
   775
  qed
hoelzl@42987
   776
  then show ?thesis using `I \<noteq> {}`
hoelzl@42987
   777
    by (simp add: rv distribution_def indep_rv_def)
hoelzl@42987
   778
qed
hoelzl@42987
   779
hoelzl@42861
   780
end