author hoelzl Thu, 20 Oct 2016 18:42:01 +0200 changeset 64321 95be866e49fc parent 64320 ba194424b895 child 64323 20d15328b248
HOL-Probability: generalize theorems about cumulative distribution function
```--- a/src/HOL/Probability/Distribution_Functions.thy	Thu Oct 20 18:41:59 2016 +0200
+++ b/src/HOL/Probability/Distribution_Functions.thy	Thu Oct 20 18:42:01 2016 +0200
@@ -36,11 +36,11 @@

locale finite_borel_measure = finite_measure M for M :: "real measure" +
-  assumes M_super_borel: "sets borel \<subseteq> sets M"
+  assumes M_is_borel: "sets M = sets borel"
begin

lemma sets_M[intro]: "a \<in> sets borel \<Longrightarrow> a \<in> sets M"
-  using M_super_borel by auto
+  using M_is_borel by auto

lemma cdf_diff_eq:
assumes "x < y"
@@ -57,7 +57,7 @@
unfolding cdf_def by (auto intro!: finite_measure_mono)

lemma borel_UNIV: "space M = UNIV"
- by (metis in_mono sets.sets_into_space space_in_borel top_le M_super_borel)
+ by (metis in_mono sets.sets_into_space space_in_borel top_le M_is_borel)

lemma cdf_nonneg: "cdf M x \<ge> 0"
unfolding cdf_def by (rule measure_nonneg)
@@ -142,11 +142,17 @@
end

locale real_distribution = prob_space M for M :: "real measure" +
-  assumes events_eq_borel [simp, measurable_cong]: "sets M = sets borel" and space_eq_univ [simp]: "space M = UNIV"
+  assumes events_eq_borel [simp, measurable_cong]: "sets M = sets borel"
begin

+lemma finite_borel_measure_M: "finite_borel_measure M"
+  by standard auto
+
sublocale finite_borel_measure M
-  by standard auto
+  by (rule finite_borel_measure_M)
+
+lemma space_eq_univ [simp]: "space M = UNIV"
+  using events_eq_borel[THEN sets_eq_imp_space_eq] by simp

lemma cdf_bounded_prob: "\<And>x. cdf M x \<le> 1"
by (subst prob_space [symmetric], rule cdf_bounded)
@@ -167,20 +173,23 @@
"random_variable borel X \<Longrightarrow> real_distribution (distr M borel X)"
unfolding real_distribution_def real_distribution_axioms_def by (auto intro!: prob_space_distr)

-subsection \<open>uniqueness\<close>
+subsection \<open>Uniqueness\<close>

-lemma (in real_distribution) emeasure_Ioc:
+lemma (in finite_borel_measure) emeasure_Ioc:
assumes "a \<le> b" shows "emeasure M {a <.. b} = cdf M b - cdf M a"
proof -
have "{a <.. b} = {..b} - {..a}"
by auto
-  with \<open>a \<le> b\<close> show ?thesis
+  moreover have "{..x} \<in> sets M" for x
+    using atMost_borel[of x] M_is_borel by auto
+  moreover note \<open>a \<le> b\<close>
+  ultimately show ?thesis
by (simp add: emeasure_eq_measure finite_measure_Diff cdf_def)
qed

-lemma cdf_unique:
+lemma cdf_unique':
fixes M1 M2
-  assumes "real_distribution M1" and "real_distribution M2"
+  assumes "finite_borel_measure M1" and "finite_borel_measure M2"
assumes "cdf M1 = cdf M2"
shows "M1 = M2"
proof (rule measure_eqI_generator_eq[where \<Omega>=UNIV])
@@ -188,14 +197,56 @@
then obtain a b where Xeq: "X = {a<..b}" by auto
then show "emeasure M1 X = emeasure M2 X"
by (cases "a \<le> b")
-       (simp_all add: assms(1,2)[THEN real_distribution.emeasure_Ioc] assms(3))
+       (simp_all add: assms(1,2)[THEN finite_borel_measure.emeasure_Ioc] assms(3))
next
show "(\<Union>i. {- real (i::nat)<..real i}) = UNIV"
by (rule UN_Ioc_eq_UNIV)
-qed (auto simp: real_distribution.emeasure_Ioc[OF assms(1)]
-  assms(1,2)[THEN real_distribution.events_eq_borel] borel_sigma_sets_Ioc
+qed (auto simp: finite_borel_measure.emeasure_Ioc[OF assms(1)]
+  assms(1,2)[THEN finite_borel_measure.M_is_borel] borel_sigma_sets_Ioc
Int_stable_def)

+lemma cdf_unique:
+  "real_distribution M1 \<Longrightarrow> real_distribution M2 \<Longrightarrow> cdf M1 = cdf M2 \<Longrightarrow> M1 = M2"
+  using cdf_unique'[of M1 M2] by (simp add: real_distribution.finite_borel_measure_M)
+
+lemma
+  fixes F :: "real \<Rightarrow> real"
+  assumes nondecF : "\<And> x y. x \<le> y \<Longrightarrow> F x \<le> F y"
+    and right_cont_F : "\<And>a. continuous (at_right a) F"
+    and lim_F_at_bot : "(F \<longlongrightarrow> 0) at_bot"
+    and lim_F_at_top : "(F \<longlongrightarrow> m) at_top"
+    and m: "0 \<le> m"
+  shows interval_measure_UNIV: "emeasure (interval_measure F) UNIV = m"
+    and finite_borel_measure_interval_measure: "finite_borel_measure (interval_measure F)"
+proof -
+  let ?F = "interval_measure F"
+  { have "ennreal (m - 0) = (SUP i::nat. ennreal (F (real i) - F (- real i)))"
+      by (intro LIMSEQ_unique[OF _ LIMSEQ_SUP] tendsto_ennrealI tendsto_intros
+                lim_F_at_bot[THEN filterlim_compose] lim_F_at_top[THEN filterlim_compose]
+                lim_F_at_bot[THEN filterlim_compose] filterlim_real_sequentially
+                filterlim_uminus_at_top[THEN iffD1])
+         (auto simp: incseq_def nondecF intro!: diff_mono)
+    also have "\<dots> = (SUP i::nat. emeasure ?F {- real i<..real i})"
+      by (subst emeasure_interval_measure_Ioc) (simp_all add: nondecF right_cont_F)
+    also have "\<dots> = emeasure ?F (\<Union>i::nat. {- real i<..real i})"
+      by (rule SUP_emeasure_incseq) (auto simp: incseq_def)
+    also have "(\<Union>i. {- real (i::nat)<..real i}) = space ?F"
+    finally have "emeasure ?F (space ?F) = m"
+      by simp }
+  note * = this
+  then show "emeasure (interval_measure F) UNIV = m"
+    by simp
+
+  interpret finite_measure ?F
+  proof
+    show "emeasure ?F (space ?F) \<noteq> \<infinity>"
+      using * by simp
+  qed
+  show "finite_borel_measure (interval_measure F)"
+    proof qed simp_all
+qed
+
lemma real_distribution_interval_measure:
fixes F :: "real \<Rightarrow> real"
assumes nondecF : "\<And> x y. x \<le> y \<Longrightarrow> F x \<le> F y" and
@@ -206,53 +257,47 @@
proof -
let ?F = "interval_measure F"
interpret prob_space ?F
-  proof
-    have "ennreal (1 - 0) = (SUP i::nat. ennreal (F (real i) - F (- real i)))"
-      by (intro LIMSEQ_unique[OF _ LIMSEQ_SUP] tendsto_ennrealI tendsto_intros
-                lim_F_at_bot[THEN filterlim_compose] lim_F_at_top[THEN filterlim_compose]
-                lim_F_at_bot[THEN filterlim_compose] filterlim_real_sequentially
-                filterlim_uminus_at_top[THEN iffD1])
-         (auto simp: incseq_def nondecF intro!: diff_mono)
-    also have "\<dots> = (SUP i::nat. emeasure ?F {- real i<..real i})"
-      by (subst emeasure_interval_measure_Ioc) (simp_all add: nondecF right_cont_F)
-    also have "\<dots> = emeasure ?F (\<Union>i::nat. {- real i<..real i})"
-      by (rule SUP_emeasure_incseq) (auto simp: incseq_def)
-    also have "(\<Union>i. {- real (i::nat)<..real i}) = space ?F"
-    finally show "emeasure ?F (space ?F) = 1"
-  qed
+    proof qed (use interval_measure_UNIV[OF assms] in simp)
show ?thesis
proof qed simp_all
qed

-lemma cdf_interval_measure:
+lemma
fixes F :: "real \<Rightarrow> real"
assumes nondecF : "\<And> x y. x \<le> y \<Longrightarrow> F x \<le> F y" and
right_cont_F : "\<And>a. continuous (at_right a) F" and
-    lim_F_at_bot : "(F \<longlongrightarrow> 0) at_bot" and
-    lim_F_at_top : "(F \<longlongrightarrow> 1) at_top"
-  shows "cdf (interval_measure F) = F"
+    lim_F_at_bot : "(F \<longlongrightarrow> 0) at_bot"
+  shows emeasure_interval_measure_Iic: "emeasure (interval_measure F) {.. x} = F x"
+    and measure_interval_measure_Iic: "measure (interval_measure F) {.. x} = F x"
unfolding cdf_def
-proof (intro ext)
-  interpret real_distribution "interval_measure F"
-    by (rule real_distribution_interval_measure) fact+
-  fix x
-  have "F x - 0 = measure (interval_measure F) (\<Union>i::nat. {-real i <.. x})"
-  proof (intro LIMSEQ_unique[OF _ finite_Lim_measure_incseq])
+proof -
+  have F_nonneg[simp]: "0 \<le> F y" for y
+    using lim_F_at_bot by (rule tendsto_upperbound) (auto simp: eventually_at_bot_linorder nondecF intro!: exI[of _ y])
+
+  have "emeasure (interval_measure F) (\<Union>i::nat. {-real i <.. x}) = F x - ennreal 0"
+  proof (intro LIMSEQ_unique[OF Lim_emeasure_incseq])
have "(\<lambda>i. F x - F (- real i)) \<longlonglongrightarrow> F x - 0"
by (intro tendsto_intros lim_F_at_bot[THEN filterlim_compose] filterlim_real_sequentially
filterlim_uminus_at_top[THEN iffD1])
-    then show "(\<lambda>i. measure (interval_measure F) {- real i<..x}) \<longlonglongrightarrow> F x - 0"
-      apply (rule filterlim_cong[OF refl refl, THEN iffD1, rotated])
+    from tendsto_ennrealI[OF this]
+    show "(\<lambda>i. emeasure (interval_measure F) {- real i<..x}) \<longlonglongrightarrow> F x - ennreal 0"
+      apply (rule filterlim_cong[THEN iffD1, rotated 3])
+        apply simp
+       apply simp
apply (rule eventually_sequentiallyI[where c="nat (ceiling (- x))"])
-      apply (simp add: measure_interval_measure_Ioc right_cont_F nondecF)
+      apply (simp add: emeasure_interval_measure_Ioc right_cont_F nondecF)
done
qed (auto simp: incseq_def)
also have "(\<Union>i::nat. {-real i <.. x}) = {..x}"
by auto (metis minus_minus neg_less_iff_less reals_Archimedean2)
-  finally show "measure (interval_measure F) {..x} = F x"
+  finally show "emeasure (interval_measure F) {..x} = F x"
by simp
+  then show "measure (interval_measure F) {..x} = F x"