1.1 --- a/src/HOL/Probability/Probability_Mass_Function.thy Tue Nov 03 11:20:21 2015 +0100
1.2 +++ b/src/HOL/Probability/Probability_Mass_Function.thy Tue Nov 10 14:18:41 2015 +0000
1.3 @@ -59,7 +59,7 @@
1.4 note singleton_sets = this
1.5 have "?M < (\<Sum>x\<in>X. ?M / Suc n)"
1.6 using `?M \<noteq> 0`
1.7 - by (simp add: `card X = Suc (Suc n)` real_eq_of_nat[symmetric] real_of_nat_Suc field_simps less_le measure_nonneg)
1.8 + by (simp add: `card X = Suc (Suc n)` of_nat_Suc field_simps less_le measure_nonneg)
1.9 also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"
1.10 by (rule setsum_mono) fact
1.11 also have "\<dots> = measure M (\<Union>x\<in>X. {x})"
1.12 @@ -956,7 +956,7 @@
1.13
1.14 lemma quotient_rel_set_disjoint:
1.15 "equivp R \<Longrightarrow> C \<in> UNIV // {(x, y). R x y} \<Longrightarrow> rel_set R A B \<Longrightarrow> A \<inter> C = {} \<longleftrightarrow> B \<inter> C = {}"
1.16 - using in_quotient_imp_closed[of UNIV "{(x, y). R x y}" C]
1.17 + using in_quotient_imp_closed[of UNIV "{(x, y). R x y}" C]
1.18 by (auto 0 0 simp: equivp_equiv rel_set_def set_eq_iff elim: equivpE)
1.19 (blast dest: equivp_symp)+
1.20
1.21 @@ -973,17 +973,17 @@
1.22 next
1.23 fix C assume C: "C \<in> UNIV // ?R" and R: "rel_set R (set_pmf p) (set_pmf q)"
1.24 assume eq: "\<forall>x\<in>set_pmf p. \<forall>y\<in>set_pmf q. R x y \<longrightarrow> measure p {x. R x y} = measure q {y. R x y}"
1.25 -
1.26 +
1.27 show "measure p C = measure q C"
1.28 proof cases
1.29 assume "p \<inter> C = {}"
1.30 - moreover then have "q \<inter> C = {}"
1.31 + moreover then have "q \<inter> C = {}"
1.32 using quotient_rel_set_disjoint[OF assms C R] by simp
1.33 ultimately show ?thesis
1.34 unfolding measure_pmf_zero_iff[symmetric] by simp
1.35 next
1.36 assume "p \<inter> C \<noteq> {}"
1.37 - moreover then have "q \<inter> C \<noteq> {}"
1.38 + moreover then have "q \<inter> C \<noteq> {}"
1.39 using quotient_rel_set_disjoint[OF assms C R] by simp
1.40 ultimately obtain x y where in_set: "x \<in> set_pmf p" "y \<in> set_pmf q" and in_C: "x \<in> C" "y \<in> C"
1.41 by auto
1.42 @@ -1068,11 +1068,11 @@
1.43 and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto
1.44 from qr obtain qr where qr: "\<And>y z. (y, z) \<in> set_pmf qr \<Longrightarrow> S y z"
1.45 and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto
1.46 -
1.47 +
1.48 def pr \<equiv> "bind_pmf pq (\<lambda>xy. bind_pmf (cond_pmf qr {yz. fst yz = snd xy}) (\<lambda>yz. return_pmf (fst xy, snd yz)))"
1.49 have pr_welldefined: "\<And>y. y \<in> q \<Longrightarrow> qr \<inter> {yz. fst yz = y} \<noteq> {}"
1.50 by (force simp: q')
1.51 -
1.52 +
1.53 have "rel_pmf (R OO S) p r"
1.54 proof (rule rel_pmf.intros)
1.55 fix x z assume "(x, z) \<in> pr"
1.56 @@ -1283,7 +1283,7 @@
1.57 proof (subst rel_pmf_iff_equivp, safe)
1.58 show "equivp (inf R R\<inverse>\<inverse>)"
1.59 using trans refl by (auto simp: equivp_reflp_symp_transp intro: sympI transpI reflpI dest: transpD reflpD)
1.60 -
1.61 +
1.62 fix C assume "C \<in> UNIV // {(x, y). inf R R\<inverse>\<inverse> x y}"
1.63 then obtain x where C: "C = {y. R x y \<and> R y x}"
1.64 by (auto elim: quotientE)
1.65 @@ -1399,7 +1399,7 @@
1.66 end
1.67
1.68 lemma set_pmf_geometric: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (geometric_pmf p) = UNIV"
1.69 - by (auto simp: set_pmf_iff)
1.70 + by (auto simp: set_pmf_iff)
1.71
1.72 subsubsection \<open> Uniform Multiset Distribution \<close>
1.73
1.74 @@ -1445,7 +1445,7 @@
1.75 lemma emeasure_pmf_of_set[simp]: "emeasure pmf_of_set S = 1"
1.76 by (rule measure_pmf.emeasure_eq_1_AE) (auto simp: AE_measure_pmf_iff)
1.77
1.78 -lemma nn_integral_pmf_of_set':
1.79 +lemma nn_integral_pmf_of_set':
1.80 "(\<And>x. x \<in> S \<Longrightarrow> f x \<ge> 0) \<Longrightarrow> nn_integral (measure_pmf pmf_of_set) f = setsum f S / card S"
1.81 apply(subst nn_integral_measure_pmf_finite, simp_all add: S_finite)
1.82 apply(simp add: setsum_ereal_left_distrib[symmetric])
1.83 @@ -1453,7 +1453,7 @@
1.84 apply(simp add: ereal_times_divide_eq one_ereal_def[symmetric])
1.85 done
1.86
1.87 -lemma nn_integral_pmf_of_set:
1.88 +lemma nn_integral_pmf_of_set:
1.89 "nn_integral (measure_pmf pmf_of_set) f = setsum (max 0 \<circ> f) S / card S"
1.90 apply(subst nn_integral_max_0[symmetric])
1.91 apply(subst nn_integral_pmf_of_set')
1.92 @@ -1476,7 +1476,7 @@
1.93 lemma pmf_of_set_singleton: "pmf_of_set {x} = return_pmf x"
1.94 by(rule pmf_eqI)(simp add: indicator_def)
1.95
1.96 -lemma map_pmf_of_set_inj:
1.97 +lemma map_pmf_of_set_inj:
1.98 assumes f: "inj_on f A"
1.99 and [simp]: "A \<noteq> {}" "finite A"
1.100 shows "map_pmf f (pmf_of_set A) = pmf_of_set (f ` A)" (is "?lhs = ?rhs")
1.101 @@ -1540,7 +1540,7 @@
1.102 ereal (\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k))"
1.103 using p_le_1 p_nonneg by (subst nn_integral_count_space') auto
1.104 also have "(\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k)) = (p + (1 - p)) ^ n"
1.105 - by (subst binomial_ring) (simp add: atLeast0AtMost real_of_nat_def)
1.106 + by (subst binomial_ring) (simp add: atLeast0AtMost)
1.107 finally show "(\<integral>\<^sup>+ x. ereal (real (n choose x) * p ^ x * (1 - p) ^ (n - x)) \<partial>count_space UNIV) = 1"
1.108 by simp
1.109 qed (insert p_nonneg p_le_1, simp)