--- a/src/HOL/Probability/Probability_Mass_Function.thy Sun Jun 03 19:06:56 2018 +0200
+++ b/src/HOL/Probability/Probability_Mass_Function.thy Wed Jun 06 11:12:37 2018 +0200
@@ -2028,7 +2028,7 @@
private lemma pmf_of_list_aux:
assumes "\<And>x. x \<in> set (map snd xs) \<Longrightarrow> x \<ge> 0"
assumes "sum_list (map snd xs) = 1"
- shows "(\<integral>\<^sup>+ x. ennreal (sum_list (map snd (filter (\<lambda>z. fst z = x) xs))) \<partial>count_space UNIV) = 1"
+ shows "(\<integral>\<^sup>+ x. ennreal (sum_list (map snd [z\<leftarrow>xs . fst z = x])) \<partial>count_space UNIV) = 1"
proof -
have "(\<integral>\<^sup>+ x. ennreal (sum_list (map snd (filter (\<lambda>z. fst z = x) xs))) \<partial>count_space UNIV) =
(\<integral>\<^sup>+ x. ennreal (sum_list (map (\<lambda>(x',p). indicator {x'} x * p) xs)) \<partial>count_space UNIV)"
@@ -2083,7 +2083,7 @@
show "x \<in> set (map fst xs)"
proof (rule ccontr)
assume "x \<notin> set (map fst xs)"
- hence "filter (\<lambda>z. fst z = x) xs = []" by (auto simp: filter_empty_conv)
+ hence "[z\<leftarrow>xs . fst z = x] = []" by (auto simp: filter_empty_conv)
with A assms show False by (simp add: pmf_pmf_of_list set_pmf_eq)
qed
qed
@@ -2122,10 +2122,10 @@
have "emeasure (pmf_of_list xs) A = nn_integral (measure_pmf (pmf_of_list xs)) (indicator A)"
by simp
also from assms
- have "\<dots> = (\<Sum>x\<in>set_pmf (pmf_of_list xs) \<inter> A. ennreal (sum_list (map snd (filter (\<lambda>z. fst z = x) xs))))"
+ have "\<dots> = (\<Sum>x\<in>set_pmf (pmf_of_list xs) \<inter> A. ennreal (sum_list (map snd [z\<leftarrow>xs . fst z = x])))"
by (subst nn_integral_measure_pmf_finite) (simp_all add: finite_set_pmf_of_list pmf_pmf_of_list Int_def)
also from assms
- have "\<dots> = ennreal (\<Sum>x\<in>set_pmf (pmf_of_list xs) \<inter> A. sum_list (map snd (filter (\<lambda>z. fst z = x) xs)))"
+ have "\<dots> = ennreal (\<Sum>x\<in>set_pmf (pmf_of_list xs) \<inter> A. sum_list (map snd [z\<leftarrow>xs . fst z = x]))"
by (subst sum_ennreal) (auto simp: pmf_of_list_wf_def intro!: sum_list_nonneg)
also have "\<dots> = ennreal (\<Sum>x\<in>set_pmf (pmf_of_list xs) \<inter> A.
indicator A x * pmf (pmf_of_list xs) x)" (is "_ = ennreal ?S")
@@ -2184,7 +2184,7 @@
{
fix x assume A: "x \<in> fst ` set xs" and B: "x \<notin> set_pmf (pmf_of_list xs)"
then obtain y where y: "(x, y) \<in> set xs" by auto
- from B have "sum_list (map snd (filter (\<lambda>z. fst z = x) xs)) = 0"
+ from B have "sum_list (map snd [z\<leftarrow>xs. fst z = x]) = 0"
by (simp add: pmf_pmf_of_list[OF assms(1)] set_pmf_eq)
moreover from y have "y \<in> snd ` {xa \<in> set xs. fst xa = x}" by force
ultimately have "y = 0" using assms(1)
@@ -2199,11 +2199,11 @@
defines "xs' \<equiv> filter (\<lambda>z. snd z \<noteq> 0) xs"
shows "pmf_of_list_wf xs'" "pmf_of_list xs' = pmf_of_list xs"
proof -
- have "map snd (filter (\<lambda>z. snd z \<noteq> 0) xs) = filter (\<lambda>x. x \<noteq> 0) (map snd xs)"
+ have "map snd [z\<leftarrow>xs . snd z \<noteq> 0] = filter (\<lambda>x. x \<noteq> 0) (map snd xs)"
by (induction xs) simp_all
with assms(1) show wf: "pmf_of_list_wf xs'"
by (auto simp: pmf_of_list_wf_def xs'_def sum_list_filter_nonzero)
- have "sum_list (map snd (filter (\<lambda>z. fst z = i) xs')) = sum_list (map snd (filter (\<lambda>z. fst z = i) xs))" for i
+ have "sum_list (map snd [z\<leftarrow>xs' . fst z = i]) = sum_list (map snd [z\<leftarrow>xs . fst z = i])" for i
unfolding xs'_def by (induction xs) simp_all
with assms(1) wf show "pmf_of_list xs' = pmf_of_list xs"
by (intro pmf_eqI) (simp_all add: pmf_pmf_of_list)