src/HOL/Probability/Regularity.thy
 changeset 50125 4319691be975 parent 50089 1badf63e5d97 child 50244 de72bbe42190
```     1.1 --- a/src/HOL/Probability/Regularity.thy	Mon Nov 19 16:09:11 2012 +0100
1.2 +++ b/src/HOL/Probability/Regularity.thy	Mon Nov 19 18:01:48 2012 +0100
1.3 @@ -210,11 +210,9 @@
1.4      ultimately
1.5      show "?thesis e " by (auto simp: sU)
1.6    qed
1.7 -  have closed_in_D: "\<And>A. closed A \<Longrightarrow> ?inner A \<and> ?outer A"
1.8 -  proof
1.9 -    fix A::"'a set" assume "closed A" hence "A \<in> sets borel" by (simp add: compact_imp_closed)
1.10 +  { fix A::"'a set" assume "closed A" hence "A \<in> sets borel" by (simp add: compact_imp_closed)
1.11      hence [simp]: "A \<in> sets M" by (simp add: sb)
1.12 -    show "?inner A"
1.13 +    have "?inner A"
1.14      proof (rule approx_inner)
1.15        fix e::real assume "e > 0"
1.16        from approx_space[OF this] obtain K where
1.17 @@ -237,7 +235,7 @@
1.18        ultimately show "\<exists>K \<subseteq> A. compact K \<and> emeasure M A \<le> emeasure M K + ereal e"
1.19          by blast
1.20      qed simp
1.21 -    show "?outer A"
1.22 +    have "?outer A"
1.23      proof cases
1.24        assume "A \<noteq> {}"
1.25        let ?G = "\<lambda>d. {x. infdist x A < d}"
1.26 @@ -288,25 +286,25 @@
1.27          by (rule INF_greatest) (auto intro!: emeasure_mono simp: sb)
1.28        ultimately show ?thesis by simp
1.29      qed (auto intro!: ereal_INFI)
1.30 -  qed
1.31 -  let ?D = "{B \<in> sets M. ?inner B \<and> ?outer B}"
1.32 -  interpret dynkin: dynkin_system "space M" ?D
1.33 -  proof (rule dynkin_systemI)
1.34 -    have "{U::'a set. space M \<subseteq> U \<and> open U} = {space M}" by (auto simp add: sU)
1.35 -    hence "?outer (space M)" by (simp add: min_def INF_def)
1.36 -    moreover
1.37 -    have "?inner (space M)"
1.38 -    proof (rule ereal_approx_SUP)
1.39 -      fix e::real assume "0 < e"
1.40 -      thus "\<exists>K\<in>{K. K \<subseteq> space M \<and> compact K}. emeasure M (space M) \<le> emeasure M K + ereal e"
1.41 -        by (rule approx_space)
1.42 -    qed (auto intro: emeasure_mono simp: sU sb intro!: exI[where x="{}"])
1.43 -    ultimately show "space M \<in> ?D" by (simp add: sU sb)
1.44 +    note `?inner A` `?outer A` }
1.45 +  note closed_in_D = this
1.46 +  from `B \<in> sets borel`
1.47 +  have "Int_stable (Collect closed)" "Collect closed \<subseteq> Pow UNIV" "B \<in> sigma_sets UNIV (Collect closed)"
1.48 +    by (auto simp: Int_stable_def borel_eq_closed)
1.49 +  then show "?inner B" "?outer B"
1.50 +  proof (induct B rule: sigma_sets_induct_disjoint)
1.51 +    case empty
1.52 +    { case 1 show ?case by (intro ereal_SUPI[symmetric]) auto }
1.53 +    { case 2 show ?case by (intro ereal_INFI[symmetric]) (auto elim!: meta_allE[of _ "{}"]) }
1.54    next
1.55 -    fix B assume "B \<in> ?D" thus "B \<subseteq> space M" by (simp add: sU)
1.56 -    from `B \<in> ?D` have [simp]: "B \<in> sets M" and "?inner B" "?outer B" by auto
1.57 -    hence inner: "emeasure M B = (SUP K:{K. K \<subseteq> B \<and> compact K}. emeasure M K)"
1.58 -      and outer: "emeasure M B = (INF U:{U. B \<subseteq> U \<and> open U}. emeasure M U)" by auto
1.59 +    case (basic B)
1.60 +    { case 1 from basic closed_in_D show ?case by auto }
1.61 +    { case 2 from basic closed_in_D show ?case by auto }
1.62 +  next
1.63 +    case (compl B)
1.64 +    note inner = compl(2) and outer = compl(3)
1.65 +    from compl have [simp]: "B \<in> sets M" by (auto simp: sb borel_eq_closed)
1.66 +    case 2
1.67      have "M (space M - B) = M (space M) - emeasure M B" by (auto simp: emeasure_compl)
1.68      also have "\<dots> = (INF K:{K. K \<subseteq> B \<and> compact K}. M (space M) -  M K)"
1.69        unfolding inner by (subst INFI_ereal_cminus) force+
1.70 @@ -315,7 +313,7 @@
1.71      also have "\<dots> \<ge> (INF U:{U. U \<subseteq> B \<and> closed U}. M (space M - U))"
1.72        by (rule INF_superset_mono) (auto simp add: compact_imp_closed)
1.73      also have "(INF U:{U. U \<subseteq> B \<and> closed U}. M (space M - U)) =
1.74 -      (INF U:{U. space M - B \<subseteq> U \<and> open U}. emeasure M U)"
1.75 +        (INF U:{U. space M - B \<subseteq> U \<and> open U}. emeasure M U)"
1.76        by (subst INF_image[of "\<lambda>u. space M - u", symmetric])
1.77           (rule INF_cong, auto simp add: sU intro!: INF_cong)
1.78      finally have
1.79 @@ -323,41 +321,39 @@
1.80      moreover have
1.81        "(INF U:{U. space M - B \<subseteq> U \<and> open U}. emeasure M U) \<ge> emeasure M (space M - B)"
1.82        by (auto simp: sb sU intro!: INF_greatest emeasure_mono)
1.83 -    ultimately have "?outer (space M - B)" by simp
1.84 -    moreover
1.85 -    {
1.86 -      have "M (space M - B) = M (space M) - emeasure M B" by (auto simp: emeasure_compl)
1.87 -      also have "\<dots> = (SUP U: {U. B \<subseteq> U \<and> open U}. M (space M) -  M U)"
1.88 -        unfolding outer by (subst SUPR_ereal_cminus) auto
1.89 -      also have "\<dots> = (SUP U:{U. B \<subseteq> U \<and> open U}. M (space M - U))"
1.90 -        by (rule SUP_cong) (auto simp add: emeasure_compl sb compact_imp_closed)
1.91 -      also have "\<dots> = (SUP K:{K. K \<subseteq> space M - B \<and> closed K}. emeasure M K)"
1.92 -        by (subst SUP_image[of "\<lambda>u. space M - u", symmetric])
1.93 -           (rule SUP_cong, auto simp: sU)
1.94 -      also have "\<dots> = (SUP K:{K. K \<subseteq> space M - B \<and> compact K}. emeasure M K)"
1.95 -      proof (safe intro!: antisym SUP_least)
1.96 -        fix K assume "closed K" "K \<subseteq> space M - B"
1.97 -        from closed_in_D[OF `closed K`]
1.98 -        have K_inner: "emeasure M K = (SUP K:{Ka. Ka \<subseteq> K \<and> compact Ka}. emeasure M K)" by simp
1.99 -        show "emeasure M K \<le> (SUP K:{K. K \<subseteq> space M - B \<and> compact K}. emeasure M K)"
1.100 -          unfolding K_inner using `K \<subseteq> space M - B`
1.101 -          by (auto intro!: SUP_upper SUP_least)
1.102 -      qed (fastforce intro!: SUP_least SUP_upper simp: compact_imp_closed)
1.103 -      finally have "?inner (space M - B)" .
1.104 -    } hence "?inner (space M - B)" .
1.105 -    ultimately show "space M - B \<in> ?D" by auto
1.106 +    ultimately show ?case by (auto intro!: antisym simp: sets_eq_imp_space_eq[OF sb])
1.107 +
1.108 +    case 1
1.109 +    have "M (space M - B) = M (space M) - emeasure M B" by (auto simp: emeasure_compl)
1.110 +    also have "\<dots> = (SUP U: {U. B \<subseteq> U \<and> open U}. M (space M) -  M U)"
1.111 +      unfolding outer by (subst SUPR_ereal_cminus) auto
1.112 +    also have "\<dots> = (SUP U:{U. B \<subseteq> U \<and> open U}. M (space M - U))"
1.113 +      by (rule SUP_cong) (auto simp add: emeasure_compl sb compact_imp_closed)
1.114 +    also have "\<dots> = (SUP K:{K. K \<subseteq> space M - B \<and> closed K}. emeasure M K)"
1.115 +      by (subst SUP_image[of "\<lambda>u. space M - u", symmetric])
1.116 +         (rule SUP_cong, auto simp: sU)
1.117 +    also have "\<dots> = (SUP K:{K. K \<subseteq> space M - B \<and> compact K}. emeasure M K)"
1.118 +    proof (safe intro!: antisym SUP_least)
1.119 +      fix K assume "closed K" "K \<subseteq> space M - B"
1.120 +      from closed_in_D[OF `closed K`]
1.121 +      have K_inner: "emeasure M K = (SUP K:{Ka. Ka \<subseteq> K \<and> compact Ka}. emeasure M K)" by simp
1.122 +      show "emeasure M K \<le> (SUP K:{K. K \<subseteq> space M - B \<and> compact K}. emeasure M K)"
1.123 +        unfolding K_inner using `K \<subseteq> space M - B`
1.124 +        by (auto intro!: SUP_upper SUP_least)
1.125 +    qed (fastforce intro!: SUP_least SUP_upper simp: compact_imp_closed)
1.126 +    finally show ?case by (auto intro!: antisym simp: sets_eq_imp_space_eq[OF sb])
1.127    next
1.128 -    fix D :: "nat \<Rightarrow> _"
1.129 -    assume "range D \<subseteq> ?D" hence "range D \<subseteq> sets M" by auto
1.130 -    moreover assume "disjoint_family D"
1.131 -    ultimately have M[symmetric]: "(\<Sum>i. M (D i)) = M (\<Union>i. D i)" by (rule suminf_emeasure)
1.132 +    case (union D)
1.133 +    then have "range D \<subseteq> sets M" by (auto simp: sb borel_eq_closed)
1.134 +    with union have M[symmetric]: "(\<Sum>i. M (D i)) = M (\<Union>i. D i)" by (intro suminf_emeasure)
1.135      also have "(\<lambda>n. \<Sum>i\<in>{0..<n}. M (D i)) ----> (\<Sum>i. M (D i))"
1.136        by (intro summable_sumr_LIMSEQ_suminf summable_ereal_pos emeasure_nonneg)
1.137      finally have measure_LIMSEQ: "(\<lambda>n. \<Sum>i = 0..<n. measure M (D i)) ----> measure M (\<Union>i. D i)"
1.139      have "(\<Union>i. D i) \<in> sets M" using `range D \<subseteq> sets M` by auto
1.140 -    moreover
1.141 -    hence "?inner (\<Union>i. D i)"
1.142 +
1.143 +    case 1
1.144 +    show ?case
1.145      proof (rule approx_inner)
1.146        fix e::real assume "e > 0"
1.147        with measure_LIMSEQ
1.148 @@ -379,7 +375,7 @@
1.149          fix i
1.150          from `0 < e` have "0 < e/(2*Suc n0)" by (auto intro: divide_pos_pos)
1.151          have "emeasure M (D i) = (SUP K:{K. K \<subseteq> (D i) \<and> compact K}. emeasure M K)"
1.152 -          using `range D \<subseteq> ?D` by blast
1.153 +          using union by blast
1.154          from SUP_approx_ereal[OF `0 < e/(2*Suc n0)` this]
1.155          show "\<exists>K. K \<subseteq> D i \<and> compact K \<and> emeasure M (D i) \<le> emeasure M K + e/(2*Suc n0)"
1.156            by (auto simp: emeasure_eq_measure)
1.157 @@ -410,8 +406,9 @@
1.158        ultimately
1.159        have "?K\<subseteq>(\<Union>i. D i) \<and> compact ?K \<and> emeasure M (\<Union>i. D i) \<le> emeasure M ?K + ereal e" by simp
1.160        thus "\<exists>K\<subseteq>\<Union>i. D i. compact K \<and> emeasure M (\<Union>i. D i) \<le> emeasure M K + ereal e" ..
1.161 -    qed
1.162 -    moreover have "?outer (\<Union>i. D i)"
1.163 +    qed fact
1.164 +    case 2
1.165 +    show ?case
1.166      proof (rule approx_outer[OF `(\<Union>i. D i) \<in> sets M`])
1.167        fix e::real assume "e > 0"
1.168        have "\<forall>i::nat. \<exists>U. D i \<subseteq> U \<and> open U \<and> e/(2 powr Suc i) > emeasure M U - emeasure M (D i)"
1.169 @@ -419,7 +416,7 @@
1.170          fix i::nat
1.171          from `0 < e` have "0 < e/(2 powr Suc i)" by (auto intro: divide_pos_pos)
1.172          have "emeasure M (D i) = (INF U:{U. (D i) \<subseteq> U \<and> open U}. emeasure M U)"
1.173 -          using `range D \<subseteq> ?D` by blast
1.174 +          using union by blast
1.175          from INF_approx_ereal[OF `0 < e/(2 powr Suc i)` this]
1.176          show "\<exists>U. D i \<subseteq> U \<and> open U \<and> e/(2 powr Suc i) > emeasure M U - emeasure M (D i)"
1.177            by (auto simp: emeasure_eq_measure)
1.178 @@ -455,20 +452,7 @@
1.179        have "(\<Union>i. D i) \<subseteq> ?U \<and> open ?U \<and> emeasure M ?U \<le> emeasure M (\<Union>i. D i) + ereal e" by simp
1.180        thus "\<exists>B. (\<Union>i. D i) \<subseteq> B \<and> open B \<and> emeasure M B \<le> emeasure M (\<Union>i. D i) + ereal e" ..
1.181      qed
1.182 -    ultimately show "(\<Union>i. D i) \<in> ?D" by safe
1.183    qed
1.184 -  have "sets borel = sigma_sets (space M) (Collect closed)" by (simp add: borel_eq_closed sU)
1.185 -  also have "\<dots> = dynkin (space M) (Collect closed)"
1.186 -  proof (rule sigma_eq_dynkin)
1.187 -    show "Collect closed \<subseteq> Pow (space M)" using Sigma_Algebra.sets_into_space by (auto simp: sU)
1.188 -    show "Int_stable (Collect closed)" by (auto simp: Int_stable_def)
1.189 -  qed
1.190 -  also have "\<dots> \<subseteq> ?D" using closed_in_D
1.191 -    by (intro dynkin.dynkin_subset) (auto simp add: compact_imp_closed sb)
1.192 -  finally have "sets borel \<subseteq> ?D" .
1.193 -  moreover have "?D \<subseteq> sets borel" by (auto simp: sb)
1.194 -  ultimately have "sets borel = ?D" by simp
1.195 -  with assms show "?inner B" and "?outer B" by auto
1.196  qed
1.197
1.198  end
```