author Andreas Lochbihler Wed Feb 11 15:22:37 2015 +0100 (2015-02-11) changeset 59525 dfe6449aecd8 parent 59524 67deb7bed6d3 child 59526 af02440afb4a
more lemmas
```     1.1 --- a/src/HOL/Probability/Giry_Monad.thy	Wed Feb 11 15:04:23 2015 +0100
1.2 +++ b/src/HOL/Probability/Giry_Monad.thy	Wed Feb 11 15:22:37 2015 +0100
1.3 @@ -142,6 +142,20 @@
1.4      "subprob_space (null_measure M) \<longleftrightarrow> space M \<noteq> {}"
1.5    by (auto intro!: subprob_spaceI dest: subprob_space.subprob_not_empty)
1.6
1.7 +lemma subprob_space_restrict_space:
1.8 +  assumes M: "subprob_space M"
1.9 +  and A: "A \<inter> space M \<in> sets M" "A \<inter> space M \<noteq> {}"
1.10 +  shows "subprob_space (restrict_space M A)"
1.11 +proof(rule subprob_spaceI)
1.12 +  have "emeasure (restrict_space M A) (space (restrict_space M A)) = emeasure M (A \<inter> space M)"
1.13 +    using A by(simp add: emeasure_restrict_space space_restrict_space)
1.14 +  also have "\<dots> \<le> 1" by(rule subprob_space.subprob_emeasure_le_1)(rule M)
1.15 +  finally show "emeasure (restrict_space M A) (space (restrict_space M A)) \<le> 1" .
1.16 +next
1.17 +  show "space (restrict_space M A) \<noteq> {}"
1.18 +    using A by(simp add: space_restrict_space)
1.19 +qed
1.20 +
1.21  definition subprob_algebra :: "'a measure \<Rightarrow> 'a measure measure" where
1.22    "subprob_algebra K =
1.23      (\<Squnion>\<^sub>\<sigma> A\<in>sets K. vimage_algebra {M. subprob_space M \<and> sets M = sets K} (\<lambda>M. emeasure M A) borel)"
```
```     2.1 --- a/src/HOL/Probability/Probability_Mass_Function.thy	Wed Feb 11 15:04:23 2015 +0100
2.2 +++ b/src/HOL/Probability/Probability_Mass_Function.thy	Wed Feb 11 15:22:37 2015 +0100
2.3 @@ -502,6 +502,12 @@
2.4    shows "(\<integral>x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
2.5    by (subst integral_measure_pmf[of UNIV]) (auto simp: UNIV_bool)
2.6
2.7 +lemma pmf_bernoulli_half [simp]: "pmf (bernoulli_pmf (1 / 2)) x = 1 / 2"
2.8 +by(cases x) simp_all
2.9 +
2.10 +lemma measure_pmf_bernoulli_half: "measure_pmf (bernoulli_pmf (1 / 2)) = uniform_count_measure UNIV"
2.11 +by(rule measure_eqI)(simp_all add: nn_integral_pmf[symmetric] emeasure_uniform_count_measure nn_integral_count_space_finite sets_uniform_count_measure)
2.12 +
2.13  subsubsection \<open> Geometric Distribution \<close>
2.14
2.15  lift_definition geometric_pmf :: "nat pmf" is "\<lambda>n. 1 / 2^Suc n"
2.16 @@ -639,7 +645,7 @@
2.17    "(\<lambda>x. measure_pmf (M x)) \<in> measurable (count_space UNIV) (subprob_algebra (count_space UNIV))"
2.18    by (auto simp: space_subprob_algebra intro!: prob_space_imp_subprob_space) unfold_locales
2.19
2.20 -lemma bind_pmf_cong:
2.21 +lemma bind_measure_pmf_cong:
2.22    assumes "\<And>x. A x \<in> space (subprob_algebra N)" "\<And>x. B x \<in> space (subprob_algebra N)"
2.23    assumes "\<And>i. i \<in> set_pmf x \<Longrightarrow> A i = B i"
2.24    shows "bind (measure_pmf x) A = bind (measure_pmf x) B"
2.25 @@ -879,6 +885,16 @@
2.26    unfolding bind_return_pmf''[symmetric] join_eq_bind_pmf bind_assoc_pmf
2.28
2.29 +lemma bind_pmf_cong:
2.30 +  "\<lbrakk> p = q; \<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x \<rbrakk>
2.31 +  \<Longrightarrow> bind_pmf p f = bind_pmf q g"
2.32 +by(simp add: bind_pmf_def cong: map_pmf_cong)
2.33 +
2.34 +lemma bind_pmf_cong_simp:
2.35 +  "\<lbrakk> p = q; \<And>x. x \<in> set_pmf q =simp=> f x = g x \<rbrakk>
2.36 +  \<Longrightarrow> bind_pmf p f = bind_pmf q g"
2.37 +by(simp add: simp_implies_def cong: bind_pmf_cong)
2.38 +
2.39  definition "pair_pmf A B = bind_pmf A (\<lambda>x. bind_pmf B (\<lambda>y. return_pmf (x, y)))"
2.40
2.41  lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b"
2.42 @@ -1235,5 +1251,38 @@
2.43                     map_pair)
2.44  qed
2.45
2.46 +lemma rel_pmf_reflI:
2.47 +  assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> P x x"
2.48 +  shows "rel_pmf P p p"
2.49 +by(rule rel_pmf.intros[where pq="map_pmf (\<lambda>x. (x, x)) p"])(auto simp add: pmf.map_comp o_def set_map_pmf assms)
2.50 +
2.51 +lemma rel_pmf_joinI:
2.52 +  assumes "rel_pmf (rel_pmf P) p q"
2.53 +  shows "rel_pmf P (join_pmf p) (join_pmf q)"
2.54 +proof -
2.55 +  from assms obtain pq where p: "p = map_pmf fst pq"
2.56 +    and q: "q = map_pmf snd pq"
2.57 +    and P: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> rel_pmf P x y"
2.58 +    by cases auto
2.59 +  from P obtain PQ
2.60 +    where PQ: "\<And>x y a b. \<lbrakk> (x, y) \<in> set_pmf pq; (a, b) \<in> set_pmf (PQ x y) \<rbrakk> \<Longrightarrow> P a b"
2.61 +    and x: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf fst (PQ x y) = x"
2.62 +    and y: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf snd (PQ x y) = y"
2.63 +    by(metis rel_pmf.simps)
2.64 +
2.65 +  let ?r = "bind_pmf pq (\<lambda>(x, y). PQ x y)"
2.66 +  have "\<And>a b. (a, b) \<in> set_pmf ?r \<Longrightarrow> P a b" by(auto simp add: set_bind_pmf intro: PQ)
2.67 +  moreover have "map_pmf fst ?r = join_pmf p" "map_pmf snd ?r = join_pmf q"
2.68 +    by(simp_all add: bind_pmf_def map_join_pmf pmf.map_comp o_def split_def p q x y cong: pmf.map_cong)
2.69 +  ultimately show ?thesis ..
2.70 +qed
2.71 +
2.72 +lemma rel_pmf_bindI:
2.73 +  assumes pq: "rel_pmf R p q"
2.74 +  and fg: "\<And>x y. R x y \<Longrightarrow> rel_pmf P (f x) (g y)"
2.75 +  shows "rel_pmf P (bind_pmf p f) (bind_pmf q g)"
2.76 +unfolding bind_pmf_def
2.77 +by(rule rel_pmf_joinI)(auto simp add: pmf.rel_map intro: pmf.rel_mono[THEN le_funD, THEN le_funD, THEN le_boolD, THEN mp, OF _ pq] fg)
2.78 +
2.79  end
2.80
```