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author | eberlm <eberlm@in.tum.de> |

Tue, 04 Apr 2017 08:57:21 +0200 | |

changeset 65395 | 7504569a73c7 |

parent 65354 | 4ff2ba82d668 |

child 65396 | b42167902f57 |

moved material from AFP to distribution

--- a/src/HOL/Analysis/Harmonic_Numbers.thy Mon Apr 03 22:18:56 2017 +0200 +++ b/src/HOL/Analysis/Harmonic_Numbers.thy Tue Apr 04 08:57:21 2017 +0200 @@ -37,6 +37,9 @@ lemma of_real_harm: "of_real (harm n) = harm n" unfolding harm_def by simp +lemma abs_harm [simp]: "(abs (harm n) :: real) = harm n" + using harm_nonneg[of n] by (rule abs_of_nonneg) + lemma norm_harm: "norm (harm n) = harm n" by (subst of_real_harm [symmetric]) (simp add: harm_nonneg) @@ -91,6 +94,15 @@ finally show "ln (real (Suc n) + 1) \<le> harm (Suc n)" by - simp qed (simp_all add: harm_def) +lemma harm_at_top: "filterlim (harm :: nat \<Rightarrow> real) at_top sequentially" +proof (rule filterlim_at_top_mono) + show "eventually (\<lambda>n. harm n \<ge> ln (real (Suc n))) at_top" + using ln_le_harm by (intro always_eventually allI) (simp_all add: add_ac) + show "filterlim (\<lambda>n. ln (real (Suc n))) at_top sequentially" + by (intro filterlim_compose[OF ln_at_top] filterlim_compose[OF filterlim_real_sequentially] + filterlim_Suc) +qed + subsection \<open>The Euler--Mascheroni constant\<close>

--- a/src/HOL/Probability/Probability_Mass_Function.thy Mon Apr 03 22:18:56 2017 +0200 +++ b/src/HOL/Probability/Probability_Mass_Function.thy Tue Apr 04 08:57:21 2017 +0200 @@ -663,6 +663,7 @@ lemma measurable_set_pmf[measurable]: "Measurable.pred (count_space UNIV) (\<lambda>x. x \<in> set_pmf M)" by simp + subsection \<open> PMFs as function \<close> context @@ -754,6 +755,39 @@ apply (subst lebesgue_integral_count_space_finite_support) apply (auto intro!: finite_subset[OF _ \<open>finite A\<close>] sum.mono_neutral_left simp: pmf_eq_0_set_pmf) done + +lemma expectation_return_pmf [simp]: + fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" + shows "measure_pmf.expectation (return_pmf x) f = f x" + by (subst integral_measure_pmf[of "{x}"]) simp_all + +lemma pmf_expectation_bind: + fixes p :: "'a pmf" and f :: "'a \<Rightarrow> 'b pmf" + and h :: "'b \<Rightarrow> 'c::{banach, second_countable_topology}" + assumes "finite A" "\<And>x. x \<in> A \<Longrightarrow> finite (set_pmf (f x))" "set_pmf p \<subseteq> A" + shows "measure_pmf.expectation (p \<bind> f) h = + (\<Sum>a\<in>A. pmf p a *\<^sub>R measure_pmf.expectation (f a) h)" +proof - + have "measure_pmf.expectation (p \<bind> f) h = (\<Sum>a\<in>(\<Union>x\<in>A. set_pmf (f x)). pmf (p \<bind> f) a *\<^sub>R h a)" + using assms by (intro integral_measure_pmf) auto + also have "\<dots> = (\<Sum>x\<in>(\<Union>x\<in>A. set_pmf (f x)). (\<Sum>a\<in>A. (pmf p a * pmf (f a) x) *\<^sub>R h x))" + proof (intro sum.cong refl, goal_cases) + case (1 x) + thus ?case + by (subst pmf_bind, subst integral_measure_pmf[of A]) + (insert assms, auto simp: scaleR_sum_left) + qed + also have "\<dots> = (\<Sum>j\<in>A. pmf p j *\<^sub>R (\<Sum>i\<in>(\<Union>x\<in>A. set_pmf (f x)). pmf (f j) i *\<^sub>R h i))" + by (subst sum.commute) (simp add: scaleR_sum_right) + also have "\<dots> = (\<Sum>j\<in>A. pmf p j *\<^sub>R measure_pmf.expectation (f j) h)" + proof (intro sum.cong refl, goal_cases) + case (1 x) + thus ?case + by (subst integral_measure_pmf[of "(\<Union>x\<in>A. set_pmf (f x))"]) + (insert assms, auto simp: scaleR_sum_left) + qed + finally show ?thesis . +qed lemma continuous_on_LINT_pmf: -- \<open>This is dominated convergence!?\<close> fixes f :: "'i \<Rightarrow> 'a::topological_space \<Rightarrow> 'b::{banach, second_countable_topology}" @@ -1725,6 +1759,14 @@ by (simp add: measure_nonneg measure_pmf.emeasure_eq_measure) end + +lemma pmf_expectation_bind_pmf_of_set: + fixes A :: "'a set" and f :: "'a \<Rightarrow> 'b pmf" + and h :: "'b \<Rightarrow> 'c::{banach, second_countable_topology}" + assumes "A \<noteq> {}" "finite A" "\<And>x. x \<in> A \<Longrightarrow> finite (set_pmf (f x))" + shows "measure_pmf.expectation (pmf_of_set A \<bind> f) h = + (\<Sum>a\<in>A. measure_pmf.expectation (f a) h /\<^sub>R real (card A))" + using assms by (subst pmf_expectation_bind[of A]) (auto simp: divide_simps) lemma map_pmf_of_set: assumes "finite A" "A \<noteq> {}" @@ -1773,6 +1815,16 @@ qed qed +lemma map_pmf_of_set_bij_betw: + assumes "bij_betw f A B" "A \<noteq> {}" "finite A" + shows "map_pmf f (pmf_of_set A) = pmf_of_set B" +proof - + have "map_pmf f (pmf_of_set A) = pmf_of_set (f ` A)" + by (intro map_pmf_of_set_inj assms bij_betw_imp_inj_on[OF assms(1)]) + also from assms have "f ` A = B" by (simp add: bij_betw_def) + finally show ?thesis . +qed + text \<open> Choosing an element uniformly at random from the union of a disjoint family of finite non-empty sets with the same size is the same as first choosing a set

--- a/src/HOL/Probability/Random_Permutations.thy Mon Apr 03 22:18:56 2017 +0200 +++ b/src/HOL/Probability/Random_Permutations.thy Tue Apr 04 08:57:21 2017 +0200 @@ -176,4 +176,56 @@ using assms by (simp add: fold_bind_random_permutation_altdef bind_assoc_pmf fold_random_permutation_fold bind_return_pmf map_pmf_def) +text \<open> + The following useful lemma allows us to swap partitioning a set w.\,r.\,t.\ a + predicate and drawing a random permutation of that set. +\<close> +lemma partition_random_permutations: + assumes "finite A" + shows "map_pmf (partition P) (pmf_of_set (permutations_of_set A)) = + pair_pmf (pmf_of_set (permutations_of_set {x\<in>A. P x})) + (pmf_of_set (permutations_of_set {x\<in>A. \<not>P x}))" (is "?lhs = ?rhs") +proof (rule pmf_eqI, clarify, goal_cases) + case (1 xs ys) + show ?case + proof (cases "xs \<in> permutations_of_set {x\<in>A. P x} \<and> ys \<in> permutations_of_set {x\<in>A. \<not>P x}") + case True + let ?n1 = "card {x\<in>A. P x}" and ?n2 = "card {x\<in>A. \<not>P x}" + have card_eq: "card A = ?n1 + ?n2" + proof - + have "?n1 + ?n2 = card ({x\<in>A. P x} \<union> {x\<in>A. \<not>P x})" + using assms by (intro card_Un_disjoint [symmetric]) auto + also have "{x\<in>A. P x} \<union> {x\<in>A. \<not>P x} = A" by blast + finally show ?thesis .. + qed + + from True have lengths [simp]: "length xs = ?n1" "length ys = ?n2" + by (auto intro!: length_finite_permutations_of_set) + have "pmf ?lhs (xs, ys) = + real (card (permutations_of_set A \<inter> partition P -` {(xs, ys)})) / fact (card A)" + using assms by (auto simp: pmf_map measure_pmf_of_set) + also have "partition P -` {(xs, ys)} = shuffle xs ys" + using True by (intro inv_image_partition) (auto simp: permutations_of_set_def) + also have "permutations_of_set A \<inter> shuffle xs ys = shuffle xs ys" + using True distinct_disjoint_shuffle[of xs ys] + by (auto simp: permutations_of_set_def dest: set_shuffle) + also have "card (shuffle xs ys) = length xs + length ys choose length xs" + using True by (intro card_disjoint_shuffle) (auto simp: permutations_of_set_def) + also have "length xs + length ys = card A" by (simp add: card_eq) + also have "real (card A choose length xs) = fact (card A) / (fact ?n1 * fact (card A - ?n1))" + by (subst binomial_fact) (auto intro!: card_mono assms) + also have "\<dots> / fact (card A) = 1 / (fact ?n1 * fact ?n2)" + by (simp add: divide_simps card_eq) + also have "\<dots> = pmf ?rhs (xs, ys)" using True assms by (simp add: pmf_pair) + finally show ?thesis . + next + case False + hence *: "xs \<notin> permutations_of_set {x\<in>A. P x} \<or> ys \<notin> permutations_of_set {x\<in>A. \<not>P x}" by blast + hence eq: "permutations_of_set A \<inter> (partition P -` {(xs, ys)}) = {}" + by (auto simp: o_def permutations_of_set_def) + from * show ?thesis + by (elim disjE) (insert assms eq, simp_all add: pmf_pair pmf_map measure_pmf_of_set) + qed +qed + end