src/HOL/Probability/Probability_Mass_Function.thy
author nipkow
Thu Sep 15 11:48:20 2016 +0200 (2016-09-15)
changeset 63882 018998c00003
parent 63680 6e1e8b5abbfa
child 63886 685fb01256af
permissions -rw-r--r--
renamed listsum -> sum_list, listprod ~> prod_list
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(*  Title:      HOL/Probability/Probability_Mass_Function.thy
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    Author:     Johannes Hölzl, TU München
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    Author:     Andreas Lochbihler, ETH Zurich
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*)
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section \<open> Probability mass function \<close>
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theory Probability_Mass_Function
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imports
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  Giry_Monad
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  "~~/src/HOL/Library/Multiset"
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begin
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lemma AE_emeasure_singleton:
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  assumes x: "emeasure M {x} \<noteq> 0" and ae: "AE x in M. P x" shows "P x"
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proof -
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  from x have x_M: "{x} \<in> sets M"
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    by (auto intro: emeasure_notin_sets)
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  from ae obtain N where N: "{x\<in>space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"
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    by (auto elim: AE_E)
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  { assume "\<not> P x"
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    with x_M[THEN sets.sets_into_space] N have "emeasure M {x} \<le> emeasure M N"
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      by (intro emeasure_mono) auto
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    with x N have False
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      by (auto simp:) }
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  then show "P x" by auto
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qed
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lemma AE_measure_singleton: "measure M {x} \<noteq> 0 \<Longrightarrow> AE x in M. P x \<Longrightarrow> P x"
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  by (metis AE_emeasure_singleton measure_def emeasure_empty measure_empty)
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lemma (in finite_measure) AE_support_countable:
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  assumes [simp]: "sets M = UNIV"
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  shows "(AE x in M. measure M {x} \<noteq> 0) \<longleftrightarrow> (\<exists>S. countable S \<and> (AE x in M. x \<in> S))"
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proof
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  assume "\<exists>S. countable S \<and> (AE x in M. x \<in> S)"
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  then obtain S where S[intro]: "countable S" and ae: "AE x in M. x \<in> S"
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    by auto
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  then have "emeasure M (\<Union>x\<in>{x\<in>S. emeasure M {x} \<noteq> 0}. {x}) =
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    (\<integral>\<^sup>+ x. emeasure M {x} * indicator {x\<in>S. emeasure M {x} \<noteq> 0} x \<partial>count_space UNIV)"
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    by (subst emeasure_UN_countable)
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       (auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space)
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  also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} * indicator S x \<partial>count_space UNIV)"
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    by (auto intro!: nn_integral_cong split: split_indicator)
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  also have "\<dots> = emeasure M (\<Union>x\<in>S. {x})"
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    by (subst emeasure_UN_countable)
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       (auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space)
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  also have "\<dots> = emeasure M (space M)"
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    using ae by (intro emeasure_eq_AE) auto
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  finally have "emeasure M {x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0} = emeasure M (space M)"
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    by (simp add: emeasure_single_in_space cong: rev_conj_cong)
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  with finite_measure_compl[of "{x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0}"]
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  have "AE x in M. x \<in> S \<and> emeasure M {x} \<noteq> 0"
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    by (intro AE_I[OF order_refl]) (auto simp: emeasure_eq_measure measure_nonneg set_diff_eq cong: conj_cong)
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  then show "AE x in M. measure M {x} \<noteq> 0"
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    by (auto simp: emeasure_eq_measure)
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qed (auto intro!: exI[of _ "{x. measure M {x} \<noteq> 0}"] countable_support)
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subsection \<open> PMF as measure \<close>
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typedef 'a pmf = "{M :: 'a measure. prob_space M \<and> sets M = UNIV \<and> (AE x in M. measure M {x} \<noteq> 0)}"
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  morphisms measure_pmf Abs_pmf
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  by (intro exI[of _ "uniform_measure (count_space UNIV) {undefined}"])
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     (auto intro!: prob_space_uniform_measure AE_uniform_measureI)
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declare [[coercion measure_pmf]]
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lemma prob_space_measure_pmf: "prob_space (measure_pmf p)"
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  using pmf.measure_pmf[of p] by auto
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interpretation measure_pmf: prob_space "measure_pmf M" for M
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  by (rule prob_space_measure_pmf)
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interpretation measure_pmf: subprob_space "measure_pmf M" for M
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  by (rule prob_space_imp_subprob_space) unfold_locales
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lemma subprob_space_measure_pmf: "subprob_space (measure_pmf x)"
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  by unfold_locales
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locale pmf_as_measure
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begin
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setup_lifting type_definition_pmf
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end
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context
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begin
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interpretation pmf_as_measure .
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lemma sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV"
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  by transfer blast
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lemma sets_measure_pmf_count_space[measurable_cong]:
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  "sets (measure_pmf M) = sets (count_space UNIV)"
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  by simp
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lemma space_measure_pmf[simp]: "space (measure_pmf p) = UNIV"
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  using sets_eq_imp_space_eq[of "measure_pmf p" "count_space UNIV"] by simp
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lemma measure_pmf_UNIV [simp]: "measure (measure_pmf p) UNIV = 1"
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using measure_pmf.prob_space[of p] by simp
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lemma measure_pmf_in_subprob_algebra[measurable (raw)]: "measure_pmf x \<in> space (subprob_algebra (count_space UNIV))"
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  by (simp add: space_subprob_algebra subprob_space_measure_pmf)
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lemma measurable_pmf_measure1[simp]: "measurable (M :: 'a pmf) N = UNIV \<rightarrow> space N"
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  by (auto simp: measurable_def)
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lemma measurable_pmf_measure2[simp]: "measurable N (M :: 'a pmf) = measurable N (count_space UNIV)"
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  by (intro measurable_cong_sets) simp_all
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lemma measurable_pair_restrict_pmf2:
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  assumes "countable A"
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  assumes [measurable]: "\<And>y. y \<in> A \<Longrightarrow> (\<lambda>x. f (x, y)) \<in> measurable M L"
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  shows "f \<in> measurable (M \<Otimes>\<^sub>M restrict_space (measure_pmf N) A) L" (is "f \<in> measurable ?M _")
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proof -
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  have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)"
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    by (simp add: restrict_count_space)
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  show ?thesis
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    by (intro measurable_compose_countable'[where f="\<lambda>a b. f (fst b, a)" and g=snd and I=A,
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                                            unfolded prod.collapse] assms)
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        measurable
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qed
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lemma measurable_pair_restrict_pmf1:
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  assumes "countable A"
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  assumes [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N L"
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  shows "f \<in> measurable (restrict_space (measure_pmf M) A \<Otimes>\<^sub>M N) L"
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proof -
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  have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)"
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    by (simp add: restrict_count_space)
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  show ?thesis
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    by (intro measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A,
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                                            unfolded prod.collapse] assms)
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        measurable
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qed
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lift_definition pmf :: "'a pmf \<Rightarrow> 'a \<Rightarrow> real" is "\<lambda>M x. measure M {x}" .
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lift_definition set_pmf :: "'a pmf \<Rightarrow> 'a set" is "\<lambda>M. {x. measure M {x} \<noteq> 0}" .
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declare [[coercion set_pmf]]
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lemma AE_measure_pmf: "AE x in (M::'a pmf). x \<in> M"
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  by transfer simp
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lemma emeasure_pmf_single_eq_zero_iff:
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  fixes M :: "'a pmf"
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  shows "emeasure M {y} = 0 \<longleftrightarrow> y \<notin> M"
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  unfolding set_pmf.rep_eq by (simp add: measure_pmf.emeasure_eq_measure)
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lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \<longleftrightarrow> (\<forall>y\<in>M. P y)"
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  using AE_measure_singleton[of M] AE_measure_pmf[of M]
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  by (auto simp: set_pmf.rep_eq)
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lemma AE_pmfI: "(\<And>y. y \<in> set_pmf M \<Longrightarrow> P y) \<Longrightarrow> almost_everywhere (measure_pmf M) P"
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by(simp add: AE_measure_pmf_iff)
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lemma countable_set_pmf [simp]: "countable (set_pmf p)"
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  by transfer (metis prob_space.finite_measure finite_measure.countable_support)
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lemma pmf_positive: "x \<in> set_pmf p \<Longrightarrow> 0 < pmf p x"
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  by transfer (simp add: less_le)
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lemma pmf_nonneg[simp]: "0 \<le> pmf p x"
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  by transfer simp
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lemma pmf_not_neg [simp]: "\<not>pmf p x < 0"
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  by (simp add: not_less pmf_nonneg)
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lemma pmf_pos [simp]: "pmf p x \<noteq> 0 \<Longrightarrow> pmf p x > 0"
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  using pmf_nonneg[of p x] by linarith
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lemma pmf_le_1: "pmf p x \<le> 1"
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  by (simp add: pmf.rep_eq)
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lemma set_pmf_not_empty: "set_pmf M \<noteq> {}"
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  using AE_measure_pmf[of M] by (intro notI) simp
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lemma set_pmf_iff: "x \<in> set_pmf M \<longleftrightarrow> pmf M x \<noteq> 0"
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  by transfer simp
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lemma pmf_positive_iff: "0 < pmf p x \<longleftrightarrow> x \<in> set_pmf p"
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  unfolding less_le by (simp add: set_pmf_iff)
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lemma set_pmf_eq: "set_pmf M = {x. pmf M x \<noteq> 0}"
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  by (auto simp: set_pmf_iff)
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lemma set_pmf_eq': "set_pmf p = {x. pmf p x > 0}"
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proof safe
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  fix x assume "x \<in> set_pmf p"
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  hence "pmf p x \<noteq> 0" by (auto simp: set_pmf_eq)
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  with pmf_nonneg[of p x] show "pmf p x > 0" by simp
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qed (auto simp: set_pmf_eq)
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lemma emeasure_pmf_single:
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  fixes M :: "'a pmf"
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  shows "emeasure M {x} = pmf M x"
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  by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
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lemma measure_pmf_single: "measure (measure_pmf M) {x} = pmf M x"
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  using emeasure_pmf_single[of M x] by(simp add: measure_pmf.emeasure_eq_measure pmf_nonneg measure_nonneg)
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lemma emeasure_measure_pmf_finite: "finite S \<Longrightarrow> emeasure (measure_pmf M) S = (\<Sum>s\<in>S. pmf M s)"
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  by (subst emeasure_eq_setsum_singleton) (auto simp: emeasure_pmf_single pmf_nonneg)
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lemma measure_measure_pmf_finite: "finite S \<Longrightarrow> measure (measure_pmf M) S = setsum (pmf M) S"
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  using emeasure_measure_pmf_finite[of S M]
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  by (simp add: measure_pmf.emeasure_eq_measure measure_nonneg setsum_nonneg pmf_nonneg)
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lemma setsum_pmf_eq_1:
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  assumes "finite A" "set_pmf p \<subseteq> A"
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  shows   "(\<Sum>x\<in>A. pmf p x) = 1"
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proof -
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  have "(\<Sum>x\<in>A. pmf p x) = measure_pmf.prob p A"
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    by (simp add: measure_measure_pmf_finite assms)
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  also from assms have "\<dots> = 1"
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    by (subst measure_pmf.prob_eq_1) (auto simp: AE_measure_pmf_iff)
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  finally show ?thesis .
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qed
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lemma nn_integral_measure_pmf_support:
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  fixes f :: "'a \<Rightarrow> ennreal"
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  assumes f: "finite A" and nn: "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x" "\<And>x. x \<in> set_pmf M \<Longrightarrow> x \<notin> A \<Longrightarrow> f x = 0"
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  shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>A. f x * pmf M x)"
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proof -
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  have "(\<integral>\<^sup>+x. f x \<partial>M) = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)"
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    using nn by (intro nn_integral_cong_AE) (auto simp: AE_measure_pmf_iff split: split_indicator)
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  also have "\<dots> = (\<Sum>x\<in>A. f x * emeasure M {x})"
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    using assms by (intro nn_integral_indicator_finite) auto
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  finally show ?thesis
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    by (simp add: emeasure_measure_pmf_finite)
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qed
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lemma nn_integral_measure_pmf_finite:
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  fixes f :: "'a \<Rightarrow> ennreal"
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  assumes f: "finite (set_pmf M)" and nn: "\<And>x. x \<in> set_pmf M \<Longrightarrow> 0 \<le> f x"
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  shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>set_pmf M. f x * pmf M x)"
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  using assms by (intro nn_integral_measure_pmf_support) auto
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lemma integrable_measure_pmf_finite:
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  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
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  shows "finite (set_pmf M) \<Longrightarrow> integrable M f"
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  by (auto intro!: integrableI_bounded simp: nn_integral_measure_pmf_finite ennreal_mult_less_top)
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lemma integral_measure_pmf:
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  assumes [simp]: "finite A" and "\<And>a. a \<in> set_pmf M \<Longrightarrow> f a \<noteq> 0 \<Longrightarrow> a \<in> A"
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  shows "(\<integral>x. f x \<partial>measure_pmf M) = (\<Sum>a\<in>A. f a * pmf M a)"
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proof -
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  have "(\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x * indicator A x \<partial>measure_pmf M)"
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    using assms(2) by (intro integral_cong_AE) (auto split: split_indicator simp: AE_measure_pmf_iff)
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  also have "\<dots> = (\<Sum>a\<in>A. f a * pmf M a)"
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    by (subst integral_indicator_finite_real)
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       (auto simp: measure_def emeasure_measure_pmf_finite pmf_nonneg)
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  finally show ?thesis .
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qed
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lemma integrable_pmf: "integrable (count_space X) (pmf M)"
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proof -
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  have " (\<integral>\<^sup>+ x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+ x. pmf M x \<partial>count_space (M \<inter> X))"
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    by (auto simp add: nn_integral_count_space_indicator set_pmf_iff intro!: nn_integral_cong split: split_indicator)
hoelzl@59000
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  then have "integrable (count_space X) (pmf M) = integrable (count_space (M \<inter> X)) (pmf M)"
hoelzl@59000
   266
    by (simp add: integrable_iff_bounded pmf_nonneg)
hoelzl@59000
   267
  then show ?thesis
Andreas@59023
   268
    by (simp add: pmf.rep_eq measure_pmf.integrable_measure disjoint_family_on_def)
hoelzl@59000
   269
qed
hoelzl@59000
   270
hoelzl@59000
   271
lemma integral_pmf: "(\<integral>x. pmf M x \<partial>count_space X) = measure M X"
hoelzl@59000
   272
proof -
hoelzl@59000
   273
  have "(\<integral>x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+x. pmf M x \<partial>count_space X)"
hoelzl@59000
   274
    by (simp add: pmf_nonneg integrable_pmf nn_integral_eq_integral)
hoelzl@59000
   275
  also have "\<dots> = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space (X \<inter> M))"
hoelzl@59000
   276
    by (auto intro!: nn_integral_cong_AE split: split_indicator
hoelzl@59000
   277
             simp: pmf.rep_eq measure_pmf.emeasure_eq_measure nn_integral_count_space_indicator
hoelzl@59000
   278
                   AE_count_space set_pmf_iff)
hoelzl@59000
   279
  also have "\<dots> = emeasure M (X \<inter> M)"
hoelzl@59000
   280
    by (rule emeasure_countable_singleton[symmetric]) (auto intro: countable_set_pmf)
hoelzl@59000
   281
  also have "\<dots> = emeasure M X"
hoelzl@59000
   282
    by (auto intro!: emeasure_eq_AE simp: AE_measure_pmf_iff)
hoelzl@59000
   283
  finally show ?thesis
hoelzl@62975
   284
    by (simp add: measure_pmf.emeasure_eq_measure measure_nonneg integral_nonneg pmf_nonneg)
hoelzl@59000
   285
qed
hoelzl@59000
   286
hoelzl@59000
   287
lemma integral_pmf_restrict:
hoelzl@59000
   288
  "(f::'a \<Rightarrow> 'b::{banach, second_countable_topology}) \<in> borel_measurable (count_space UNIV) \<Longrightarrow>
hoelzl@59000
   289
    (\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x \<partial>restrict_space M M)"
hoelzl@59000
   290
  by (auto intro!: integral_cong_AE simp add: integral_restrict_space AE_measure_pmf_iff)
hoelzl@59000
   291
hoelzl@58587
   292
lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1"
hoelzl@58587
   293
proof -
hoelzl@58587
   294
  have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)"
hoelzl@58587
   295
    by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf)
hoelzl@58587
   296
  then show ?thesis
hoelzl@58587
   297
    using measure_pmf.emeasure_space_1 by simp
hoelzl@58587
   298
qed
hoelzl@58587
   299
Andreas@59490
   300
lemma emeasure_pmf_UNIV [simp]: "emeasure (measure_pmf M) UNIV = 1"
Andreas@59490
   301
using measure_pmf.emeasure_space_1[of M] by simp
Andreas@59490
   302
Andreas@59023
   303
lemma in_null_sets_measure_pmfI:
Andreas@59023
   304
  "A \<inter> set_pmf p = {} \<Longrightarrow> A \<in> null_sets (measure_pmf p)"
Andreas@59023
   305
using emeasure_eq_0_AE[where ?P="\<lambda>x. x \<in> A" and M="measure_pmf p"]
Andreas@59023
   306
by(auto simp add: null_sets_def AE_measure_pmf_iff)
Andreas@59023
   307
hoelzl@59664
   308
lemma measure_subprob: "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
hoelzl@59664
   309
  by (simp add: space_subprob_algebra subprob_space_measure_pmf)
hoelzl@59664
   310
hoelzl@59664
   311
subsection \<open> Monad Interpretation \<close>
hoelzl@59664
   312
hoelzl@59664
   313
lemma measurable_measure_pmf[measurable]:
hoelzl@59664
   314
  "(\<lambda>x. measure_pmf (M x)) \<in> measurable (count_space UNIV) (subprob_algebra (count_space UNIV))"
hoelzl@59664
   315
  by (auto simp: space_subprob_algebra intro!: prob_space_imp_subprob_space) unfold_locales
hoelzl@59664
   316
hoelzl@59664
   317
lemma bind_measure_pmf_cong:
hoelzl@59664
   318
  assumes "\<And>x. A x \<in> space (subprob_algebra N)" "\<And>x. B x \<in> space (subprob_algebra N)"
hoelzl@59664
   319
  assumes "\<And>i. i \<in> set_pmf x \<Longrightarrow> A i = B i"
hoelzl@59664
   320
  shows "bind (measure_pmf x) A = bind (measure_pmf x) B"
hoelzl@59664
   321
proof (rule measure_eqI)
wenzelm@62026
   322
  show "sets (measure_pmf x \<bind> A) = sets (measure_pmf x \<bind> B)"
hoelzl@59664
   323
    using assms by (subst (1 2) sets_bind) (auto simp: space_subprob_algebra)
hoelzl@59664
   324
next
wenzelm@62026
   325
  fix X assume "X \<in> sets (measure_pmf x \<bind> A)"
hoelzl@59664
   326
  then have X: "X \<in> sets N"
hoelzl@59664
   327
    using assms by (subst (asm) sets_bind) (auto simp: space_subprob_algebra)
wenzelm@62026
   328
  show "emeasure (measure_pmf x \<bind> A) X = emeasure (measure_pmf x \<bind> B) X"
hoelzl@59664
   329
    using assms
hoelzl@59664
   330
    by (subst (1 2) emeasure_bind[where N=N, OF _ _ X])
hoelzl@59664
   331
       (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
hoelzl@59664
   332
qed
hoelzl@59664
   333
hoelzl@59664
   334
lift_definition bind_pmf :: "'a pmf \<Rightarrow> ('a \<Rightarrow> 'b pmf ) \<Rightarrow> 'b pmf" is bind
hoelzl@59664
   335
proof (clarify, intro conjI)
hoelzl@59664
   336
  fix f :: "'a measure" and g :: "'a \<Rightarrow> 'b measure"
hoelzl@59664
   337
  assume "prob_space f"
hoelzl@59664
   338
  then interpret f: prob_space f .
hoelzl@59664
   339
  assume "sets f = UNIV" and ae_f: "AE x in f. measure f {x} \<noteq> 0"
hoelzl@59664
   340
  then have s_f[simp]: "sets f = sets (count_space UNIV)"
hoelzl@59664
   341
    by simp
hoelzl@59664
   342
  assume g: "\<And>x. prob_space (g x) \<and> sets (g x) = UNIV \<and> (AE y in g x. measure (g x) {y} \<noteq> 0)"
hoelzl@59664
   343
  then have g: "\<And>x. prob_space (g x)" and s_g[simp]: "\<And>x. sets (g x) = sets (count_space UNIV)"
hoelzl@59664
   344
    and ae_g: "\<And>x. AE y in g x. measure (g x) {y} \<noteq> 0"
hoelzl@59664
   345
    by auto
hoelzl@59664
   346
hoelzl@59664
   347
  have [measurable]: "g \<in> measurable f (subprob_algebra (count_space UNIV))"
hoelzl@59664
   348
    by (auto simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space g)
lp15@59667
   349
wenzelm@62026
   350
  show "prob_space (f \<bind> g)"
hoelzl@59664
   351
    using g by (intro f.prob_space_bind[where S="count_space UNIV"]) auto
wenzelm@62026
   352
  then interpret fg: prob_space "f \<bind> g" .
wenzelm@62026
   353
  show [simp]: "sets (f \<bind> g) = UNIV"
hoelzl@59664
   354
    using sets_eq_imp_space_eq[OF s_f]
hoelzl@59664
   355
    by (subst sets_bind[where N="count_space UNIV"]) auto
wenzelm@62026
   356
  show "AE x in f \<bind> g. measure (f \<bind> g) {x} \<noteq> 0"
hoelzl@59664
   357
    apply (simp add: fg.prob_eq_0 AE_bind[where B="count_space UNIV"])
hoelzl@59664
   358
    using ae_f
hoelzl@59664
   359
    apply eventually_elim
hoelzl@59664
   360
    using ae_g
hoelzl@59664
   361
    apply eventually_elim
hoelzl@59664
   362
    apply (auto dest: AE_measure_singleton)
hoelzl@59664
   363
    done
hoelzl@59664
   364
qed
hoelzl@59664
   365
eberlm@63099
   366
adhoc_overloading Monad_Syntax.bind bind_pmf
eberlm@63099
   367
hoelzl@62975
   368
lemma ennreal_pmf_bind: "pmf (bind_pmf N f) i = (\<integral>\<^sup>+x. pmf (f x) i \<partial>measure_pmf N)"
hoelzl@59664
   369
  unfolding pmf.rep_eq bind_pmf.rep_eq
hoelzl@59664
   370
  by (auto simp: measure_pmf.measure_bind[where N="count_space UNIV"] measure_subprob measure_nonneg
hoelzl@59664
   371
           intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])
hoelzl@59664
   372
hoelzl@59664
   373
lemma pmf_bind: "pmf (bind_pmf N f) i = (\<integral>x. pmf (f x) i \<partial>measure_pmf N)"
hoelzl@62975
   374
  using ennreal_pmf_bind[of N f i]
hoelzl@59664
   375
  by (subst (asm) nn_integral_eq_integral)
hoelzl@62975
   376
     (auto simp: pmf_nonneg pmf_le_1 pmf_nonneg integral_nonneg
hoelzl@59664
   377
           intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])
hoelzl@59664
   378
hoelzl@59664
   379
lemma bind_pmf_const[simp]: "bind_pmf M (\<lambda>x. c) = c"
hoelzl@59664
   380
  by transfer (simp add: bind_const' prob_space_imp_subprob_space)
hoelzl@59664
   381
hoelzl@59665
   382
lemma set_bind_pmf[simp]: "set_pmf (bind_pmf M N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))"
hoelzl@62975
   383
proof -
hoelzl@62975
   384
  have "set_pmf (bind_pmf M N) = {x. ennreal (pmf (bind_pmf M N) x) \<noteq> 0}"
hoelzl@62975
   385
    by (simp add: set_pmf_eq pmf_nonneg)
hoelzl@62975
   386
  also have "\<dots> = (\<Union>M\<in>set_pmf M. set_pmf (N M))"
hoelzl@62975
   387
    unfolding ennreal_pmf_bind
hoelzl@62975
   388
    by (subst nn_integral_0_iff_AE) (auto simp: AE_measure_pmf_iff pmf_nonneg set_pmf_eq)
hoelzl@62975
   389
  finally show ?thesis .
hoelzl@62975
   390
qed
hoelzl@59664
   391
eberlm@63099
   392
lemma bind_pmf_cong [fundef_cong]:
hoelzl@59664
   393
  assumes "p = q"
hoelzl@59664
   394
  shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g"
wenzelm@61808
   395
  unfolding \<open>p = q\<close>[symmetric] measure_pmf_inject[symmetric] bind_pmf.rep_eq
hoelzl@59664
   396
  by (auto simp: AE_measure_pmf_iff Pi_iff space_subprob_algebra subprob_space_measure_pmf
hoelzl@59664
   397
                 sets_bind[where N="count_space UNIV"] emeasure_bind[where N="count_space UNIV"]
hoelzl@59664
   398
           intro!: nn_integral_cong_AE measure_eqI)
hoelzl@59664
   399
hoelzl@59664
   400
lemma bind_pmf_cong_simp:
hoelzl@59664
   401
  "p = q \<Longrightarrow> (\<And>x. x \<in> set_pmf q =simp=> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g"
hoelzl@59664
   402
  by (simp add: simp_implies_def cong: bind_pmf_cong)
hoelzl@59664
   403
wenzelm@62026
   404
lemma measure_pmf_bind: "measure_pmf (bind_pmf M f) = (measure_pmf M \<bind> (\<lambda>x. measure_pmf (f x)))"
hoelzl@59664
   405
  by transfer simp
hoelzl@59664
   406
hoelzl@59664
   407
lemma nn_integral_bind_pmf[simp]: "(\<integral>\<^sup>+x. f x \<partial>bind_pmf M N) = (\<integral>\<^sup>+x. \<integral>\<^sup>+y. f y \<partial>N x \<partial>M)"
hoelzl@59664
   408
  using measurable_measure_pmf[of N]
hoelzl@59664
   409
  unfolding measure_pmf_bind
hoelzl@59664
   410
  apply (intro nn_integral_bind[where B="count_space UNIV"])
hoelzl@59664
   411
  apply auto
hoelzl@59664
   412
  done
hoelzl@59664
   413
hoelzl@59664
   414
lemma emeasure_bind_pmf[simp]: "emeasure (bind_pmf M N) X = (\<integral>\<^sup>+x. emeasure (N x) X \<partial>M)"
hoelzl@59664
   415
  using measurable_measure_pmf[of N]
hoelzl@59664
   416
  unfolding measure_pmf_bind
hoelzl@59664
   417
  by (subst emeasure_bind[where N="count_space UNIV"]) auto
lp15@59667
   418
hoelzl@59664
   419
lift_definition return_pmf :: "'a \<Rightarrow> 'a pmf" is "return (count_space UNIV)"
hoelzl@59664
   420
  by (auto intro!: prob_space_return simp: AE_return measure_return)
hoelzl@59664
   421
hoelzl@59664
   422
lemma bind_return_pmf: "bind_pmf (return_pmf x) f = f x"
hoelzl@59664
   423
  by transfer
hoelzl@59664
   424
     (auto intro!: prob_space_imp_subprob_space bind_return[where N="count_space UNIV"]
hoelzl@59664
   425
           simp: space_subprob_algebra)
hoelzl@59664
   426
hoelzl@59665
   427
lemma set_return_pmf[simp]: "set_pmf (return_pmf x) = {x}"
hoelzl@59664
   428
  by transfer (auto simp add: measure_return split: split_indicator)
hoelzl@59664
   429
hoelzl@59664
   430
lemma bind_return_pmf': "bind_pmf N return_pmf = N"
hoelzl@59664
   431
proof (transfer, clarify)
wenzelm@62026
   432
  fix N :: "'a measure" assume "sets N = UNIV" then show "N \<bind> return (count_space UNIV) = N"
hoelzl@59664
   433
    by (subst return_sets_cong[where N=N]) (simp_all add: bind_return')
hoelzl@59664
   434
qed
hoelzl@59664
   435
hoelzl@59664
   436
lemma bind_assoc_pmf: "bind_pmf (bind_pmf A B) C = bind_pmf A (\<lambda>x. bind_pmf (B x) C)"
hoelzl@59664
   437
  by transfer
hoelzl@59664
   438
     (auto intro!: bind_assoc[where N="count_space UNIV" and R="count_space UNIV"]
hoelzl@59664
   439
           simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space)
hoelzl@59664
   440
hoelzl@59664
   441
definition "map_pmf f M = bind_pmf M (\<lambda>x. return_pmf (f x))"
hoelzl@59664
   442
hoelzl@59664
   443
lemma map_bind_pmf: "map_pmf f (bind_pmf M g) = bind_pmf M (\<lambda>x. map_pmf f (g x))"
hoelzl@59664
   444
  by (simp add: map_pmf_def bind_assoc_pmf)
hoelzl@59664
   445
hoelzl@59664
   446
lemma bind_map_pmf: "bind_pmf (map_pmf f M) g = bind_pmf M (\<lambda>x. g (f x))"
hoelzl@59664
   447
  by (simp add: map_pmf_def bind_assoc_pmf bind_return_pmf)
hoelzl@59664
   448
hoelzl@59664
   449
lemma map_pmf_transfer[transfer_rule]:
hoelzl@59664
   450
  "rel_fun op = (rel_fun cr_pmf cr_pmf) (\<lambda>f M. distr M (count_space UNIV) f) map_pmf"
hoelzl@59664
   451
proof -
hoelzl@59664
   452
  have "rel_fun op = (rel_fun pmf_as_measure.cr_pmf pmf_as_measure.cr_pmf)
wenzelm@62026
   453
     (\<lambda>f M. M \<bind> (return (count_space UNIV) o f)) map_pmf"
lp15@59667
   454
    unfolding map_pmf_def[abs_def] comp_def by transfer_prover
hoelzl@59664
   455
  then show ?thesis
hoelzl@59664
   456
    by (force simp: rel_fun_def cr_pmf_def bind_return_distr)
hoelzl@59664
   457
qed
hoelzl@59664
   458
hoelzl@59664
   459
lemma map_pmf_rep_eq:
hoelzl@59664
   460
  "measure_pmf (map_pmf f M) = distr (measure_pmf M) (count_space UNIV) f"
hoelzl@59664
   461
  unfolding map_pmf_def bind_pmf.rep_eq comp_def return_pmf.rep_eq
hoelzl@59664
   462
  using bind_return_distr[of M f "count_space UNIV"] by (simp add: comp_def)
hoelzl@59664
   463
hoelzl@58587
   464
lemma map_pmf_id[simp]: "map_pmf id = id"
hoelzl@58587
   465
  by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI)
hoelzl@58587
   466
hoelzl@59053
   467
lemma map_pmf_ident[simp]: "map_pmf (\<lambda>x. x) = (\<lambda>x. x)"
hoelzl@59053
   468
  using map_pmf_id unfolding id_def .
hoelzl@59053
   469
hoelzl@58587
   470
lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g"
lp15@59667
   471
  by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def)
hoelzl@58587
   472
hoelzl@59000
   473
lemma map_pmf_comp: "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M"
hoelzl@59000
   474
  using map_pmf_compose[of f g] by (simp add: comp_def)
hoelzl@59000
   475
hoelzl@59664
   476
lemma map_pmf_cong: "p = q \<Longrightarrow> (\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> map_pmf f p = map_pmf g q"
hoelzl@59664
   477
  unfolding map_pmf_def by (rule bind_pmf_cong) auto
hoelzl@59664
   478
hoelzl@59664
   479
lemma pmf_set_map: "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
hoelzl@59665
   480
  by (auto simp add: comp_def fun_eq_iff map_pmf_def)
hoelzl@59664
   481
hoelzl@59665
   482
lemma set_map_pmf[simp]: "set_pmf (map_pmf f M) = f`set_pmf M"
hoelzl@59664
   483
  using pmf_set_map[of f] by (auto simp: comp_def fun_eq_iff)
hoelzl@58587
   484
hoelzl@59002
   485
lemma emeasure_map_pmf[simp]: "emeasure (map_pmf f M) X = emeasure M (f -` X)"
hoelzl@59664
   486
  unfolding map_pmf_rep_eq by (subst emeasure_distr) auto
hoelzl@59002
   487
Andreas@61634
   488
lemma measure_map_pmf[simp]: "measure (map_pmf f M) X = measure M (f -` X)"
hoelzl@62975
   489
using emeasure_map_pmf[of f M X] by(simp add: measure_pmf.emeasure_eq_measure measure_nonneg)
Andreas@61634
   490
hoelzl@59002
   491
lemma nn_integral_map_pmf[simp]: "(\<integral>\<^sup>+x. f x \<partial>map_pmf g M) = (\<integral>\<^sup>+x. f (g x) \<partial>M)"
hoelzl@59664
   492
  unfolding map_pmf_rep_eq by (intro nn_integral_distr) auto
hoelzl@59002
   493
hoelzl@62975
   494
lemma ennreal_pmf_map: "pmf (map_pmf f p) x = (\<integral>\<^sup>+ y. indicator (f -` {x}) y \<partial>measure_pmf p)"
hoelzl@59664
   495
proof (transfer fixing: f x)
Andreas@59023
   496
  fix p :: "'b measure"
Andreas@59023
   497
  presume "prob_space p"
Andreas@59023
   498
  then interpret prob_space p .
Andreas@59023
   499
  presume "sets p = UNIV"
hoelzl@62975
   500
  then show "ennreal (measure (distr p (count_space UNIV) f) {x}) = integral\<^sup>N p (indicator (f -` {x}))"
hoelzl@62975
   501
    by(simp add: measure_distr measurable_def emeasure_eq_measure)
hoelzl@62975
   502
qed simp_all
hoelzl@62975
   503
hoelzl@62975
   504
lemma pmf_map: "pmf (map_pmf f p) x = measure p (f -` {x})"
hoelzl@62975
   505
proof (transfer fixing: f x)
hoelzl@62975
   506
  fix p :: "'b measure"
hoelzl@62975
   507
  presume "prob_space p"
hoelzl@62975
   508
  then interpret prob_space p .
hoelzl@62975
   509
  presume "sets p = UNIV"
hoelzl@62975
   510
  then show "measure (distr p (count_space UNIV) f) {x} = measure p (f -` {x})"
Andreas@59023
   511
    by(simp add: measure_distr measurable_def emeasure_eq_measure)
Andreas@59023
   512
qed simp_all
Andreas@59023
   513
Andreas@59023
   514
lemma nn_integral_pmf: "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = emeasure (measure_pmf p) A"
Andreas@59023
   515
proof -
Andreas@59023
   516
  have "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = (\<integral>\<^sup>+ x. pmf p x \<partial>count_space (A \<inter> set_pmf p))"
Andreas@59023
   517
    by(auto simp add: nn_integral_count_space_indicator indicator_def set_pmf_iff intro: nn_integral_cong)
Andreas@59023
   518
  also have "\<dots> = emeasure (measure_pmf p) (\<Union>x\<in>A \<inter> set_pmf p. {x})"
Andreas@59023
   519
    by(subst emeasure_UN_countable)(auto simp add: emeasure_pmf_single disjoint_family_on_def)
Andreas@59023
   520
  also have "\<dots> = emeasure (measure_pmf p) ((\<Union>x\<in>A \<inter> set_pmf p. {x}) \<union> {x. x \<in> A \<and> x \<notin> set_pmf p})"
Andreas@59023
   521
    by(rule emeasure_Un_null_set[symmetric])(auto intro: in_null_sets_measure_pmfI)
Andreas@59023
   522
  also have "\<dots> = emeasure (measure_pmf p) A"
Andreas@59023
   523
    by(auto intro: arg_cong2[where f=emeasure])
Andreas@59023
   524
  finally show ?thesis .
Andreas@59023
   525
qed
Andreas@59023
   526
hoelzl@62975
   527
lemma integral_map_pmf[simp]:
hoelzl@62975
   528
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
hoelzl@62975
   529
  shows "integral\<^sup>L (map_pmf g p) f = integral\<^sup>L p (\<lambda>x. f (g x))"
hoelzl@62975
   530
  by (simp add: integral_distr map_pmf_rep_eq)
hoelzl@62975
   531
Andreas@60068
   532
lemma map_return_pmf [simp]: "map_pmf f (return_pmf x) = return_pmf (f x)"
hoelzl@59664
   533
  by transfer (simp add: distr_return)
hoelzl@59664
   534
hoelzl@59664
   535
lemma map_pmf_const[simp]: "map_pmf (\<lambda>_. c) M = return_pmf c"
hoelzl@59664
   536
  by transfer (auto simp: prob_space.distr_const)
hoelzl@59664
   537
Andreas@60068
   538
lemma pmf_return [simp]: "pmf (return_pmf x) y = indicator {y} x"
hoelzl@59664
   539
  by transfer (simp add: measure_return)
hoelzl@59664
   540
hoelzl@59664
   541
lemma nn_integral_return_pmf[simp]: "0 \<le> f x \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>return_pmf x) = f x"
hoelzl@59664
   542
  unfolding return_pmf.rep_eq by (intro nn_integral_return) auto
hoelzl@59664
   543
hoelzl@59664
   544
lemma emeasure_return_pmf[simp]: "emeasure (return_pmf x) X = indicator X x"
hoelzl@59664
   545
  unfolding return_pmf.rep_eq by (intro emeasure_return) auto
hoelzl@59664
   546
eberlm@63099
   547
lemma measure_return_pmf [simp]: "measure_pmf.prob (return_pmf x) A = indicator A x"
eberlm@63099
   548
proof -
eberlm@63099
   549
  have "ennreal (measure_pmf.prob (return_pmf x) A) = 
eberlm@63099
   550
          emeasure (measure_pmf (return_pmf x)) A"
eberlm@63099
   551
    by (simp add: measure_pmf.emeasure_eq_measure)
eberlm@63099
   552
  also have "\<dots> = ennreal (indicator A x)" by (simp add: ennreal_indicator)
eberlm@63099
   553
  finally show ?thesis by simp
eberlm@63099
   554
qed
eberlm@63099
   555
hoelzl@59664
   556
lemma return_pmf_inj[simp]: "return_pmf x = return_pmf y \<longleftrightarrow> x = y"
hoelzl@59664
   557
  by (metis insertI1 set_return_pmf singletonD)
hoelzl@59664
   558
hoelzl@59665
   559
lemma map_pmf_eq_return_pmf_iff:
hoelzl@59665
   560
  "map_pmf f p = return_pmf x \<longleftrightarrow> (\<forall>y \<in> set_pmf p. f y = x)"
hoelzl@59665
   561
proof
hoelzl@59665
   562
  assume "map_pmf f p = return_pmf x"
hoelzl@59665
   563
  then have "set_pmf (map_pmf f p) = set_pmf (return_pmf x)" by simp
hoelzl@59665
   564
  then show "\<forall>y \<in> set_pmf p. f y = x" by auto
hoelzl@59665
   565
next
hoelzl@59665
   566
  assume "\<forall>y \<in> set_pmf p. f y = x"
hoelzl@59665
   567
  then show "map_pmf f p = return_pmf x"
hoelzl@59665
   568
    unfolding map_pmf_const[symmetric, of _ p] by (intro map_pmf_cong) auto
hoelzl@59665
   569
qed
hoelzl@59665
   570
hoelzl@59664
   571
definition "pair_pmf A B = bind_pmf A (\<lambda>x. bind_pmf B (\<lambda>y. return_pmf (x, y)))"
hoelzl@59664
   572
hoelzl@59664
   573
lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b"
hoelzl@59664
   574
  unfolding pair_pmf_def pmf_bind pmf_return
hoelzl@59664
   575
  apply (subst integral_measure_pmf[where A="{b}"])
hoelzl@59664
   576
  apply (auto simp: indicator_eq_0_iff)
hoelzl@59664
   577
  apply (subst integral_measure_pmf[where A="{a}"])
hoelzl@59664
   578
  apply (auto simp: indicator_eq_0_iff setsum_nonneg_eq_0_iff pmf_nonneg)
hoelzl@59664
   579
  done
hoelzl@59664
   580
hoelzl@59665
   581
lemma set_pair_pmf[simp]: "set_pmf (pair_pmf A B) = set_pmf A \<times> set_pmf B"
hoelzl@59664
   582
  unfolding pair_pmf_def set_bind_pmf set_return_pmf by auto
hoelzl@59664
   583
hoelzl@59664
   584
lemma measure_pmf_in_subprob_space[measurable (raw)]:
hoelzl@59664
   585
  "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
hoelzl@59664
   586
  by (simp add: space_subprob_algebra) intro_locales
hoelzl@59664
   587
hoelzl@59664
   588
lemma nn_integral_pair_pmf': "(\<integral>\<^sup>+x. f x \<partial>pair_pmf A B) = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. f (a, b) \<partial>B \<partial>A)"
hoelzl@59664
   589
proof -
hoelzl@62975
   590
  have "(\<integral>\<^sup>+x. f x \<partial>pair_pmf A B) = (\<integral>\<^sup>+x. f x * indicator (A \<times> B) x \<partial>pair_pmf A B)"
hoelzl@62975
   591
    by (auto simp: AE_measure_pmf_iff intro!: nn_integral_cong_AE)
hoelzl@62975
   592
  also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. f (a, b) * indicator (A \<times> B) (a, b) \<partial>B \<partial>A)"
hoelzl@59664
   593
    by (simp add: pair_pmf_def)
hoelzl@62975
   594
  also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. f (a, b) \<partial>B \<partial>A)"
hoelzl@59664
   595
    by (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
hoelzl@62975
   596
  finally show ?thesis .
hoelzl@59664
   597
qed
hoelzl@59664
   598
hoelzl@59664
   599
lemma bind_pair_pmf:
hoelzl@59664
   600
  assumes M[measurable]: "M \<in> measurable (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) (subprob_algebra N)"
wenzelm@62026
   601
  shows "measure_pmf (pair_pmf A B) \<bind> M = (measure_pmf A \<bind> (\<lambda>x. measure_pmf B \<bind> (\<lambda>y. M (x, y))))"
hoelzl@59664
   602
    (is "?L = ?R")
hoelzl@59664
   603
proof (rule measure_eqI)
hoelzl@59664
   604
  have M'[measurable]: "M \<in> measurable (pair_pmf A B) (subprob_algebra N)"
hoelzl@59664
   605
    using M[THEN measurable_space] by (simp_all add: space_pair_measure)
hoelzl@59664
   606
hoelzl@59664
   607
  note measurable_bind[where N="count_space UNIV", measurable]
hoelzl@59664
   608
  note measure_pmf_in_subprob_space[simp]
hoelzl@59664
   609
hoelzl@59664
   610
  have sets_eq_N: "sets ?L = N"
hoelzl@59664
   611
    by (subst sets_bind[OF sets_kernel[OF M']]) auto
hoelzl@59664
   612
  show "sets ?L = sets ?R"
hoelzl@59664
   613
    using measurable_space[OF M]
hoelzl@59664
   614
    by (simp add: sets_eq_N space_pair_measure space_subprob_algebra)
hoelzl@59664
   615
  fix X assume "X \<in> sets ?L"
hoelzl@59664
   616
  then have X[measurable]: "X \<in> sets N"
hoelzl@59664
   617
    unfolding sets_eq_N .
hoelzl@59664
   618
  then show "emeasure ?L X = emeasure ?R X"
hoelzl@59664
   619
    apply (simp add: emeasure_bind[OF _ M' X])
hoelzl@59664
   620
    apply (simp add: nn_integral_bind[where B="count_space UNIV"] pair_pmf_def measure_pmf_bind[of A]
hoelzl@62975
   621
                     nn_integral_measure_pmf_finite)
hoelzl@59664
   622
    apply (subst emeasure_bind[OF _ _ X])
hoelzl@59664
   623
    apply measurable
hoelzl@59664
   624
    apply (subst emeasure_bind[OF _ _ X])
hoelzl@59664
   625
    apply measurable
hoelzl@59664
   626
    done
hoelzl@59664
   627
qed
hoelzl@59664
   628
hoelzl@59664
   629
lemma map_fst_pair_pmf: "map_pmf fst (pair_pmf A B) = A"
hoelzl@59664
   630
  by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
hoelzl@59664
   631
hoelzl@59664
   632
lemma map_snd_pair_pmf: "map_pmf snd (pair_pmf A B) = B"
hoelzl@59664
   633
  by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
hoelzl@59664
   634
hoelzl@59664
   635
lemma nn_integral_pmf':
hoelzl@59664
   636
  "inj_on f A \<Longrightarrow> (\<integral>\<^sup>+x. pmf p (f x) \<partial>count_space A) = emeasure p (f ` A)"
hoelzl@59664
   637
  by (subst nn_integral_bij_count_space[where g=f and B="f`A"])
hoelzl@59664
   638
     (auto simp: bij_betw_def nn_integral_pmf)
hoelzl@59664
   639
hoelzl@59664
   640
lemma pmf_le_0_iff[simp]: "pmf M p \<le> 0 \<longleftrightarrow> pmf M p = 0"
hoelzl@62975
   641
  using pmf_nonneg[of M p] by arith
hoelzl@59664
   642
hoelzl@59664
   643
lemma min_pmf_0[simp]: "min (pmf M p) 0 = 0" "min 0 (pmf M p) = 0"
hoelzl@62975
   644
  using pmf_nonneg[of M p] by arith+
hoelzl@59664
   645
hoelzl@59664
   646
lemma pmf_eq_0_set_pmf: "pmf M p = 0 \<longleftrightarrow> p \<notin> set_pmf M"
hoelzl@59664
   647
  unfolding set_pmf_iff by simp
hoelzl@59664
   648
hoelzl@59664
   649
lemma pmf_map_inj: "inj_on f (set_pmf M) \<Longrightarrow> x \<in> set_pmf M \<Longrightarrow> pmf (map_pmf f M) (f x) = pmf M x"
hoelzl@59664
   650
  by (auto simp: pmf.rep_eq map_pmf_rep_eq measure_distr AE_measure_pmf_iff inj_onD
hoelzl@59664
   651
           intro!: measure_pmf.finite_measure_eq_AE)
hoelzl@59664
   652
Andreas@60068
   653
lemma pmf_map_inj': "inj f \<Longrightarrow> pmf (map_pmf f M) (f x) = pmf M x"
Andreas@60068
   654
apply(cases "x \<in> set_pmf M")
Andreas@60068
   655
 apply(simp add: pmf_map_inj[OF subset_inj_on])
Andreas@60068
   656
apply(simp add: pmf_eq_0_set_pmf[symmetric])
Andreas@60068
   657
apply(auto simp add: pmf_eq_0_set_pmf dest: injD)
Andreas@60068
   658
done
Andreas@60068
   659
Andreas@60068
   660
lemma pmf_map_outside: "x \<notin> f ` set_pmf M \<Longrightarrow> pmf (map_pmf f M) x = 0"
Andreas@60068
   661
unfolding pmf_eq_0_set_pmf by simp
Andreas@60068
   662
hoelzl@59664
   663
subsection \<open> PMFs as function \<close>
hoelzl@59000
   664
hoelzl@58587
   665
context
hoelzl@58587
   666
  fixes f :: "'a \<Rightarrow> real"
hoelzl@58587
   667
  assumes nonneg: "\<And>x. 0 \<le> f x"
hoelzl@58587
   668
  assumes prob: "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
hoelzl@58587
   669
begin
hoelzl@58587
   670
hoelzl@62975
   671
lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ennreal \<circ> f)"
hoelzl@58587
   672
proof (intro conjI)
hoelzl@62975
   673
  have *[simp]: "\<And>x y. ennreal (f y) * indicator {x} y = ennreal (f x) * indicator {x} y"
hoelzl@58587
   674
    by (simp split: split_indicator)
hoelzl@62975
   675
  show "AE x in density (count_space UNIV) (ennreal \<circ> f).
hoelzl@62975
   676
    measure (density (count_space UNIV) (ennreal \<circ> f)) {x} \<noteq> 0"
hoelzl@59092
   677
    by (simp add: AE_density nonneg measure_def emeasure_density max_def)
hoelzl@62975
   678
  show "prob_space (density (count_space UNIV) (ennreal \<circ> f))"
wenzelm@61169
   679
    by standard (simp add: emeasure_density prob)
hoelzl@58587
   680
qed simp
hoelzl@58587
   681
hoelzl@58587
   682
lemma pmf_embed_pmf: "pmf embed_pmf x = f x"
hoelzl@58587
   683
proof transfer
hoelzl@62975
   684
  have *[simp]: "\<And>x y. ennreal (f y) * indicator {x} y = ennreal (f x) * indicator {x} y"
hoelzl@58587
   685
    by (simp split: split_indicator)
hoelzl@62975
   686
  fix x show "measure (density (count_space UNIV) (ennreal \<circ> f)) {x} = f x"
hoelzl@59092
   687
    by transfer (simp add: measure_def emeasure_density nonneg max_def)
hoelzl@58587
   688
qed
hoelzl@58587
   689
Andreas@60068
   690
lemma set_embed_pmf: "set_pmf embed_pmf = {x. f x \<noteq> 0}"
wenzelm@63092
   691
by(auto simp add: set_pmf_eq pmf_embed_pmf)
Andreas@60068
   692
hoelzl@58587
   693
end
hoelzl@58587
   694
hoelzl@58587
   695
lemma embed_pmf_transfer:
hoelzl@62975
   696
  "rel_fun (eq_onp (\<lambda>f. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ennreal (f x) \<partial>count_space UNIV) = 1)) pmf_as_measure.cr_pmf (\<lambda>f. density (count_space UNIV) (ennreal \<circ> f)) embed_pmf"
hoelzl@58587
   697
  by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer)
hoelzl@58587
   698
hoelzl@59000
   699
lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)"
hoelzl@59000
   700
proof (transfer, elim conjE)
hoelzl@59000
   701
  fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
hoelzl@59000
   702
  assume "prob_space M" then interpret prob_space M .
hoelzl@62975
   703
  show "M = density (count_space UNIV) (\<lambda>x. ennreal (measure M {x}))"
hoelzl@59000
   704
  proof (rule measure_eqI)
hoelzl@59000
   705
    fix A :: "'a set"
hoelzl@62975
   706
    have "(\<integral>\<^sup>+ x. ennreal (measure M {x}) * indicator A x \<partial>count_space UNIV) =
hoelzl@59000
   707
      (\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)"
hoelzl@59000
   708
      by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator)
hoelzl@59000
   709
    also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))"
hoelzl@59000
   710
      by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space)
hoelzl@59000
   711
    also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})"
hoelzl@59000
   712
      by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support)
hoelzl@59000
   713
         (auto simp: disjoint_family_on_def)
hoelzl@59000
   714
    also have "\<dots> = emeasure M A"
hoelzl@59000
   715
      using ae by (intro emeasure_eq_AE) auto
hoelzl@62975
   716
    finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ennreal (measure M {x}))) A"
hoelzl@59000
   717
      using emeasure_space_1 by (simp add: emeasure_density)
hoelzl@59000
   718
  qed simp
hoelzl@59000
   719
qed
hoelzl@59000
   720
hoelzl@58587
   721
lemma td_pmf_embed_pmf:
hoelzl@62975
   722
  "type_definition pmf embed_pmf {f::'a \<Rightarrow> real. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ennreal (f x) \<partial>count_space UNIV) = 1}"
hoelzl@58587
   723
  unfolding type_definition_def
hoelzl@58587
   724
proof safe
hoelzl@58587
   725
  fix p :: "'a pmf"
hoelzl@58587
   726
  have "(\<integral>\<^sup>+ x. 1 \<partial>measure_pmf p) = 1"
hoelzl@58587
   727
    using measure_pmf.emeasure_space_1[of p] by simp
hoelzl@62975
   728
  then show *: "(\<integral>\<^sup>+ x. ennreal (pmf p x) \<partial>count_space UNIV) = 1"
hoelzl@58587
   729
    by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const)
hoelzl@58587
   730
hoelzl@58587
   731
  show "embed_pmf (pmf p) = p"
hoelzl@58587
   732
    by (intro measure_pmf_inject[THEN iffD1])
hoelzl@58587
   733
       (simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def)
hoelzl@58587
   734
next
hoelzl@58587
   735
  fix f :: "'a \<Rightarrow> real" assume "\<forall>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
hoelzl@58587
   736
  then show "pmf (embed_pmf f) = f"
hoelzl@58587
   737
    by (auto intro!: pmf_embed_pmf)
hoelzl@58587
   738
qed (rule pmf_nonneg)
hoelzl@58587
   739
hoelzl@58587
   740
end
hoelzl@58587
   741
hoelzl@62975
   742
lemma nn_integral_measure_pmf: "(\<integral>\<^sup>+ x. f x \<partial>measure_pmf p) = \<integral>\<^sup>+ x. ennreal (pmf p x) * f x \<partial>count_space UNIV"
Andreas@60068
   743
by(simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg)
Andreas@60068
   744
hoelzl@58587
   745
locale pmf_as_function
hoelzl@58587
   746
begin
hoelzl@58587
   747
hoelzl@58587
   748
setup_lifting td_pmf_embed_pmf
hoelzl@58587
   749
lp15@59667
   750
lemma set_pmf_transfer[transfer_rule]:
hoelzl@58730
   751
  assumes "bi_total A"
lp15@59667
   752
  shows "rel_fun (pcr_pmf A) (rel_set A) (\<lambda>f. {x. f x \<noteq> 0}) set_pmf"
wenzelm@61808
   753
  using \<open>bi_total A\<close>
hoelzl@58730
   754
  by (auto simp: pcr_pmf_def cr_pmf_def rel_fun_def rel_set_def bi_total_def Bex_def set_pmf_iff)
hoelzl@58730
   755
     metis+
hoelzl@58730
   756
hoelzl@59000
   757
end
hoelzl@59000
   758
hoelzl@59000
   759
context
hoelzl@59000
   760
begin
hoelzl@59000
   761
hoelzl@59000
   762
interpretation pmf_as_function .
hoelzl@59000
   763
hoelzl@59000
   764
lemma pmf_eqI: "(\<And>i. pmf M i = pmf N i) \<Longrightarrow> M = N"
hoelzl@59000
   765
  by transfer auto
hoelzl@59000
   766
hoelzl@59000
   767
lemma pmf_eq_iff: "M = N \<longleftrightarrow> (\<forall>i. pmf M i = pmf N i)"
hoelzl@59000
   768
  by (auto intro: pmf_eqI)
hoelzl@59000
   769
eberlm@63099
   770
lemma pmf_neq_exists_less:
eberlm@63099
   771
  assumes "M \<noteq> N"
eberlm@63099
   772
  shows   "\<exists>x. pmf M x < pmf N x"
eberlm@63099
   773
proof (rule ccontr)
eberlm@63099
   774
  assume "\<not>(\<exists>x. pmf M x < pmf N x)"
eberlm@63099
   775
  hence ge: "pmf M x \<ge> pmf N x" for x by (auto simp: not_less)
eberlm@63099
   776
  from assms obtain x where "pmf M x \<noteq> pmf N x" by (auto simp: pmf_eq_iff)
eberlm@63099
   777
  with ge[of x] have gt: "pmf M x > pmf N x" by simp
eberlm@63099
   778
  have "1 = measure (measure_pmf M) UNIV" by simp
eberlm@63099
   779
  also have "\<dots> = measure (measure_pmf N) {x} + measure (measure_pmf N) (UNIV - {x})"
eberlm@63099
   780
    by (subst measure_pmf.finite_measure_Union [symmetric]) simp_all
eberlm@63099
   781
  also from gt have "measure (measure_pmf N) {x} < measure (measure_pmf M) {x}" 
eberlm@63099
   782
    by (simp add: measure_pmf_single)
eberlm@63099
   783
  also have "measure (measure_pmf N) (UNIV - {x}) \<le> measure (measure_pmf M) (UNIV - {x})"
eberlm@63099
   784
    by (subst (1 2) integral_pmf [symmetric]) 
eberlm@63099
   785
       (intro integral_mono integrable_pmf, simp_all add: ge)
eberlm@63099
   786
  also have "measure (measure_pmf M) {x} + \<dots> = 1"
eberlm@63099
   787
    by (subst measure_pmf.finite_measure_Union [symmetric]) simp_all
eberlm@63099
   788
  finally show False by simp_all
eberlm@63099
   789
qed
eberlm@63099
   790
hoelzl@59664
   791
lemma bind_commute_pmf: "bind_pmf A (\<lambda>x. bind_pmf B (C x)) = bind_pmf B (\<lambda>y. bind_pmf A (\<lambda>x. C x y))"
hoelzl@59664
   792
  unfolding pmf_eq_iff pmf_bind
hoelzl@59664
   793
proof
hoelzl@59664
   794
  fix i
hoelzl@59664
   795
  interpret B: prob_space "restrict_space B B"
hoelzl@59664
   796
    by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
hoelzl@59664
   797
       (auto simp: AE_measure_pmf_iff)
hoelzl@59664
   798
  interpret A: prob_space "restrict_space A A"
hoelzl@59664
   799
    by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
hoelzl@59664
   800
       (auto simp: AE_measure_pmf_iff)
hoelzl@59664
   801
hoelzl@59664
   802
  interpret AB: pair_prob_space "restrict_space A A" "restrict_space B B"
hoelzl@59664
   803
    by unfold_locales
hoelzl@59664
   804
hoelzl@59664
   805
  have "(\<integral> x. \<integral> y. pmf (C x y) i \<partial>B \<partial>A) = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>A)"
hoelzl@59664
   806
    by (rule integral_cong) (auto intro!: integral_pmf_restrict)
hoelzl@59664
   807
  also have "\<dots> = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>restrict_space A A)"
hoelzl@59664
   808
    by (intro integral_pmf_restrict B.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
hoelzl@59664
   809
              countable_set_pmf borel_measurable_count_space)
hoelzl@59664
   810
  also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>restrict_space B B)"
hoelzl@59664
   811
    by (rule AB.Fubini_integral[symmetric])
hoelzl@59664
   812
       (auto intro!: AB.integrable_const_bound[where B=1] measurable_pair_restrict_pmf2
hoelzl@59664
   813
             simp: pmf_nonneg pmf_le_1 measurable_restrict_space1)
hoelzl@59664
   814
  also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>B)"
hoelzl@59664
   815
    by (intro integral_pmf_restrict[symmetric] A.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
hoelzl@59664
   816
              countable_set_pmf borel_measurable_count_space)
hoelzl@59664
   817
  also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>A \<partial>B)"
hoelzl@59664
   818
    by (rule integral_cong) (auto intro!: integral_pmf_restrict[symmetric])
hoelzl@59664
   819
  finally show "(\<integral> x. \<integral> y. pmf (C x y) i \<partial>B \<partial>A) = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>A \<partial>B)" .
hoelzl@59664
   820
qed
hoelzl@59664
   821
hoelzl@59664
   822
lemma pair_map_pmf1: "pair_pmf (map_pmf f A) B = map_pmf (apfst f) (pair_pmf A B)"
hoelzl@59664
   823
proof (safe intro!: pmf_eqI)
hoelzl@59664
   824
  fix a :: "'a" and b :: "'b"
hoelzl@62975
   825
  have [simp]: "\<And>c d. indicator (apfst f -` {(a, b)}) (c, d) = indicator (f -` {a}) c * (indicator {b} d::ennreal)"
hoelzl@59664
   826
    by (auto split: split_indicator)
hoelzl@59664
   827
hoelzl@62975
   828
  have "ennreal (pmf (pair_pmf (map_pmf f A) B) (a, b)) =
hoelzl@62975
   829
         ennreal (pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b))"
hoelzl@62975
   830
    unfolding pmf_pair ennreal_pmf_map
hoelzl@59664
   831
    by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_multc pmf_nonneg
hoelzl@62975
   832
                  emeasure_map_pmf[symmetric] ennreal_mult del: emeasure_map_pmf)
hoelzl@59664
   833
  then show "pmf (pair_pmf (map_pmf f A) B) (a, b) = pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b)"
hoelzl@62975
   834
    by (simp add: pmf_nonneg)
hoelzl@59664
   835
qed
hoelzl@59664
   836
hoelzl@59664
   837
lemma pair_map_pmf2: "pair_pmf A (map_pmf f B) = map_pmf (apsnd f) (pair_pmf A B)"
hoelzl@59664
   838
proof (safe intro!: pmf_eqI)
hoelzl@59664
   839
  fix a :: "'a" and b :: "'b"
hoelzl@62975
   840
  have [simp]: "\<And>c d. indicator (apsnd f -` {(a, b)}) (c, d) = indicator {a} c * (indicator (f -` {b}) d::ennreal)"
hoelzl@59664
   841
    by (auto split: split_indicator)
hoelzl@59664
   842
hoelzl@62975
   843
  have "ennreal (pmf (pair_pmf A (map_pmf f B)) (a, b)) =
hoelzl@62975
   844
         ennreal (pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b))"
hoelzl@62975
   845
    unfolding pmf_pair ennreal_pmf_map
hoelzl@59664
   846
    by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_cmult nn_integral_multc pmf_nonneg
hoelzl@62975
   847
                  emeasure_map_pmf[symmetric] ennreal_mult del: emeasure_map_pmf)
hoelzl@59664
   848
  then show "pmf (pair_pmf A (map_pmf f B)) (a, b) = pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b)"
hoelzl@62975
   849
    by (simp add: pmf_nonneg)
hoelzl@59664
   850
qed
hoelzl@59664
   851
hoelzl@59664
   852
lemma map_pair: "map_pmf (\<lambda>(a, b). (f a, g b)) (pair_pmf A B) = pair_pmf (map_pmf f A) (map_pmf g B)"
hoelzl@59664
   853
  by (simp add: pair_map_pmf2 pair_map_pmf1 map_pmf_comp split_beta')
hoelzl@59664
   854
hoelzl@59000
   855
end
hoelzl@59000
   856
Andreas@61634
   857
lemma pair_return_pmf1: "pair_pmf (return_pmf x) y = map_pmf (Pair x) y"
Andreas@61634
   858
by(simp add: pair_pmf_def bind_return_pmf map_pmf_def)
Andreas@61634
   859
Andreas@61634
   860
lemma pair_return_pmf2: "pair_pmf x (return_pmf y) = map_pmf (\<lambda>x. (x, y)) x"
Andreas@61634
   861
by(simp add: pair_pmf_def bind_return_pmf map_pmf_def)
Andreas@61634
   862
Andreas@61634
   863
lemma pair_pair_pmf: "pair_pmf (pair_pmf u v) w = map_pmf (\<lambda>(x, (y, z)). ((x, y), z)) (pair_pmf u (pair_pmf v w))"
Andreas@61634
   864
by(simp add: pair_pmf_def bind_return_pmf map_pmf_def bind_assoc_pmf)
Andreas@61634
   865
Andreas@61634
   866
lemma pair_commute_pmf: "pair_pmf x y = map_pmf (\<lambda>(x, y). (y, x)) (pair_pmf y x)"
Andreas@61634
   867
unfolding pair_pmf_def by(subst bind_commute_pmf)(simp add: map_pmf_def bind_assoc_pmf bind_return_pmf)
Andreas@61634
   868
Andreas@61634
   869
lemma set_pmf_subset_singleton: "set_pmf p \<subseteq> {x} \<longleftrightarrow> p = return_pmf x"
Andreas@61634
   870
proof(intro iffI pmf_eqI)
Andreas@61634
   871
  fix i
Andreas@61634
   872
  assume x: "set_pmf p \<subseteq> {x}"
Andreas@61634
   873
  hence *: "set_pmf p = {x}" using set_pmf_not_empty[of p] by auto
hoelzl@62975
   874
  have "ennreal (pmf p x) = \<integral>\<^sup>+ i. indicator {x} i \<partial>p" by(simp add: emeasure_pmf_single)
Andreas@61634
   875
  also have "\<dots> = \<integral>\<^sup>+ i. 1 \<partial>p" by(rule nn_integral_cong_AE)(simp add: AE_measure_pmf_iff * )
Andreas@61634
   876
  also have "\<dots> = 1" by simp
Andreas@61634
   877
  finally show "pmf p i = pmf (return_pmf x) i" using x
Andreas@61634
   878
    by(auto split: split_indicator simp add: pmf_eq_0_set_pmf)
Andreas@61634
   879
qed auto
Andreas@61634
   880
Andreas@61634
   881
lemma bind_eq_return_pmf:
Andreas@61634
   882
  "bind_pmf p f = return_pmf x \<longleftrightarrow> (\<forall>y\<in>set_pmf p. f y = return_pmf x)"
Andreas@61634
   883
  (is "?lhs \<longleftrightarrow> ?rhs")
Andreas@61634
   884
proof(intro iffI strip)
Andreas@61634
   885
  fix y
Andreas@61634
   886
  assume y: "y \<in> set_pmf p"
Andreas@61634
   887
  assume "?lhs"
Andreas@61634
   888
  hence "set_pmf (bind_pmf p f) = {x}" by simp
Andreas@61634
   889
  hence "(\<Union>y\<in>set_pmf p. set_pmf (f y)) = {x}" by simp
Andreas@61634
   890
  hence "set_pmf (f y) \<subseteq> {x}" using y by auto
Andreas@61634
   891
  thus "f y = return_pmf x" by(simp add: set_pmf_subset_singleton)
Andreas@61634
   892
next
Andreas@61634
   893
  assume *: ?rhs
Andreas@61634
   894
  show ?lhs
Andreas@61634
   895
  proof(rule pmf_eqI)
Andreas@61634
   896
    fix i
hoelzl@62975
   897
    have "ennreal (pmf (bind_pmf p f) i) = \<integral>\<^sup>+ y. ennreal (pmf (f y) i) \<partial>p"
hoelzl@62975
   898
      by (simp add: ennreal_pmf_bind)
hoelzl@62975
   899
    also have "\<dots> = \<integral>\<^sup>+ y. ennreal (pmf (return_pmf x) i) \<partial>p"
Andreas@61634
   900
      by(rule nn_integral_cong_AE)(simp add: AE_measure_pmf_iff * )
hoelzl@62975
   901
    also have "\<dots> = ennreal (pmf (return_pmf x) i)"
hoelzl@62975
   902
      by simp
hoelzl@62975
   903
    finally show "pmf (bind_pmf p f) i = pmf (return_pmf x) i"
hoelzl@62975
   904
      by (simp add: pmf_nonneg)
Andreas@61634
   905
  qed
Andreas@61634
   906
qed
Andreas@61634
   907
Andreas@61634
   908
lemma pmf_False_conv_True: "pmf p False = 1 - pmf p True"
Andreas@61634
   909
proof -
Andreas@61634
   910
  have "pmf p False + pmf p True = measure p {False} + measure p {True}"
Andreas@61634
   911
    by(simp add: measure_pmf_single)
Andreas@61634
   912
  also have "\<dots> = measure p ({False} \<union> {True})"
Andreas@61634
   913
    by(subst measure_pmf.finite_measure_Union) simp_all
Andreas@61634
   914
  also have "{False} \<union> {True} = space p" by auto
Andreas@61634
   915
  finally show ?thesis by simp
Andreas@61634
   916
qed
Andreas@61634
   917
Andreas@61634
   918
lemma pmf_True_conv_False: "pmf p True = 1 - pmf p False"
Andreas@61634
   919
by(simp add: pmf_False_conv_True)
Andreas@61634
   920
hoelzl@59664
   921
subsection \<open> Conditional Probabilities \<close>
hoelzl@59664
   922
hoelzl@59670
   923
lemma measure_pmf_zero_iff: "measure (measure_pmf p) s = 0 \<longleftrightarrow> set_pmf p \<inter> s = {}"
hoelzl@59670
   924
  by (subst measure_pmf.prob_eq_0) (auto simp: AE_measure_pmf_iff)
hoelzl@59670
   925
hoelzl@59664
   926
context
hoelzl@59664
   927
  fixes p :: "'a pmf" and s :: "'a set"
hoelzl@59664
   928
  assumes not_empty: "set_pmf p \<inter> s \<noteq> {}"
hoelzl@59664
   929
begin
hoelzl@59664
   930
hoelzl@59664
   931
interpretation pmf_as_measure .
hoelzl@59664
   932
hoelzl@59664
   933
lemma emeasure_measure_pmf_not_zero: "emeasure (measure_pmf p) s \<noteq> 0"
hoelzl@59664
   934
proof
hoelzl@59664
   935
  assume "emeasure (measure_pmf p) s = 0"
hoelzl@59664
   936
  then have "AE x in measure_pmf p. x \<notin> s"
hoelzl@59664
   937
    by (rule AE_I[rotated]) auto
hoelzl@59664
   938
  with not_empty show False
hoelzl@59664
   939
    by (auto simp: AE_measure_pmf_iff)
hoelzl@59664
   940
qed
hoelzl@59664
   941
hoelzl@59664
   942
lemma measure_measure_pmf_not_zero: "measure (measure_pmf p) s \<noteq> 0"
hoelzl@62975
   943
  using emeasure_measure_pmf_not_zero by (simp add: measure_pmf.emeasure_eq_measure measure_nonneg)
hoelzl@59664
   944
hoelzl@59664
   945
lift_definition cond_pmf :: "'a pmf" is
hoelzl@59664
   946
  "uniform_measure (measure_pmf p) s"
hoelzl@59664
   947
proof (intro conjI)
hoelzl@59664
   948
  show "prob_space (uniform_measure (measure_pmf p) s)"
hoelzl@59664
   949
    by (intro prob_space_uniform_measure) (auto simp: emeasure_measure_pmf_not_zero)
hoelzl@59664
   950
  show "AE x in uniform_measure (measure_pmf p) s. measure (uniform_measure (measure_pmf p) s) {x} \<noteq> 0"
hoelzl@59664
   951
    by (simp add: emeasure_measure_pmf_not_zero measure_measure_pmf_not_zero AE_uniform_measure
hoelzl@62975
   952
                  AE_measure_pmf_iff set_pmf.rep_eq less_top[symmetric])
hoelzl@59664
   953
qed simp
hoelzl@59664
   954
hoelzl@59664
   955
lemma pmf_cond: "pmf cond_pmf x = (if x \<in> s then pmf p x / measure p s else 0)"
hoelzl@59664
   956
  by transfer (simp add: emeasure_measure_pmf_not_zero pmf.rep_eq)
hoelzl@59664
   957
hoelzl@59665
   958
lemma set_cond_pmf[simp]: "set_pmf cond_pmf = set_pmf p \<inter> s"
nipkow@62390
   959
  by (auto simp add: set_pmf_iff pmf_cond measure_measure_pmf_not_zero split: if_split_asm)
hoelzl@59664
   960
hoelzl@59664
   961
end
hoelzl@59664
   962
eberlm@63099
   963
lemma measure_pmf_posI: "x \<in> set_pmf p \<Longrightarrow> x \<in> A \<Longrightarrow> measure_pmf.prob p A > 0"
eberlm@63099
   964
  using measure_measure_pmf_not_zero[of p A] by (subst zero_less_measure_iff) blast
eberlm@63099
   965
hoelzl@59664
   966
lemma cond_map_pmf:
hoelzl@59664
   967
  assumes "set_pmf p \<inter> f -` s \<noteq> {}"
hoelzl@59664
   968
  shows "cond_pmf (map_pmf f p) s = map_pmf f (cond_pmf p (f -` s))"
hoelzl@59664
   969
proof -
hoelzl@59664
   970
  have *: "set_pmf (map_pmf f p) \<inter> s \<noteq> {}"
hoelzl@59665
   971
    using assms by auto
hoelzl@59664
   972
  { fix x
hoelzl@62975
   973
    have "ennreal (pmf (map_pmf f (cond_pmf p (f -` s))) x) =
hoelzl@59664
   974
      emeasure p (f -` s \<inter> f -` {x}) / emeasure p (f -` s)"
hoelzl@62975
   975
      unfolding ennreal_pmf_map cond_pmf.rep_eq[OF assms] by (simp add: nn_integral_uniform_measure)
hoelzl@59664
   976
    also have "f -` s \<inter> f -` {x} = (if x \<in> s then f -` {x} else {})"
hoelzl@59664
   977
      by auto
hoelzl@59664
   978
    also have "emeasure p (if x \<in> s then f -` {x} else {}) / emeasure p (f -` s) =
hoelzl@62975
   979
      ennreal (pmf (cond_pmf (map_pmf f p) s) x)"
hoelzl@59664
   980
      using measure_measure_pmf_not_zero[OF *]
hoelzl@62975
   981
      by (simp add: pmf_cond[OF *] ennreal_pmf_map measure_pmf.emeasure_eq_measure
hoelzl@62975
   982
                    divide_ennreal pmf_nonneg measure_nonneg zero_less_measure_iff pmf_map)
hoelzl@62975
   983
    finally have "ennreal (pmf (cond_pmf (map_pmf f p) s) x) = ennreal (pmf (map_pmf f (cond_pmf p (f -` s))) x)"
hoelzl@59664
   984
      by simp }
hoelzl@59664
   985
  then show ?thesis
hoelzl@62975
   986
    by (intro pmf_eqI) (simp add: pmf_nonneg)
hoelzl@59664
   987
qed
hoelzl@59664
   988
hoelzl@59664
   989
lemma bind_cond_pmf_cancel:
hoelzl@59670
   990
  assumes [simp]: "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}"
hoelzl@59670
   991
  assumes [simp]: "\<And>y. y \<in> set_pmf q \<Longrightarrow> set_pmf p \<inter> {x. R x y} \<noteq> {}"
hoelzl@59670
   992
  assumes [simp]: "\<And>x y. x \<in> set_pmf p \<Longrightarrow> y \<in> set_pmf q \<Longrightarrow> R x y \<Longrightarrow> measure q {y. R x y} = measure p {x. R x y}"
hoelzl@59670
   993
  shows "bind_pmf p (\<lambda>x. cond_pmf q {y. R x y}) = q"
hoelzl@59664
   994
proof (rule pmf_eqI)
hoelzl@59670
   995
  fix i
hoelzl@62975
   996
  have "ennreal (pmf (bind_pmf p (\<lambda>x. cond_pmf q {y. R x y})) i) =
hoelzl@62975
   997
    (\<integral>\<^sup>+x. ennreal (pmf q i / measure p {x. R x i}) * ennreal (indicator {x. R x i} x) \<partial>p)"
hoelzl@62975
   998
    by (auto simp add: ennreal_pmf_bind AE_measure_pmf_iff pmf_cond pmf_eq_0_set_pmf pmf_nonneg measure_nonneg
hoelzl@62975
   999
             intro!: nn_integral_cong_AE)
hoelzl@59670
  1000
  also have "\<dots> = (pmf q i * measure p {x. R x i}) / measure p {x. R x i}"
hoelzl@62975
  1001
    by (simp add: pmf_nonneg measure_nonneg zero_ennreal_def[symmetric] ennreal_indicator
hoelzl@62975
  1002
                  nn_integral_cmult measure_pmf.emeasure_eq_measure ennreal_mult[symmetric])
hoelzl@59670
  1003
  also have "\<dots> = pmf q i"
hoelzl@62975
  1004
    by (cases "pmf q i = 0")
hoelzl@62975
  1005
       (simp_all add: pmf_eq_0_set_pmf measure_measure_pmf_not_zero pmf_nonneg)
hoelzl@59670
  1006
  finally show "pmf (bind_pmf p (\<lambda>x. cond_pmf q {y. R x y})) i = pmf q i"
hoelzl@62975
  1007
    by (simp add: pmf_nonneg)
hoelzl@59664
  1008
qed
hoelzl@59664
  1009
hoelzl@59664
  1010
subsection \<open> Relator \<close>
hoelzl@59664
  1011
hoelzl@59664
  1012
inductive rel_pmf :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf \<Rightarrow> bool"
hoelzl@59664
  1013
for R p q
hoelzl@59664
  1014
where
lp15@59667
  1015
  "\<lbrakk> \<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y;
hoelzl@59664
  1016
     map_pmf fst pq = p; map_pmf snd pq = q \<rbrakk>
hoelzl@59664
  1017
  \<Longrightarrow> rel_pmf R p q"
hoelzl@59664
  1018
hoelzl@59681
  1019
lemma rel_pmfI:
hoelzl@59681
  1020
  assumes R: "rel_set R (set_pmf p) (set_pmf q)"
hoelzl@59681
  1021
  assumes eq: "\<And>x y. x \<in> set_pmf p \<Longrightarrow> y \<in> set_pmf q \<Longrightarrow> R x y \<Longrightarrow>
hoelzl@59681
  1022
    measure p {x. R x y} = measure q {y. R x y}"
hoelzl@59681
  1023
  shows "rel_pmf R p q"
hoelzl@59681
  1024
proof
hoelzl@59681
  1025
  let ?pq = "bind_pmf p (\<lambda>x. bind_pmf (cond_pmf q {y. R x y}) (\<lambda>y. return_pmf (x, y)))"
hoelzl@59681
  1026
  have "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}"
hoelzl@59681
  1027
    using R by (auto simp: rel_set_def)
hoelzl@59681
  1028
  then show "\<And>x y. (x, y) \<in> set_pmf ?pq \<Longrightarrow> R x y"
hoelzl@59681
  1029
    by auto
hoelzl@59681
  1030
  show "map_pmf fst ?pq = p"
Andreas@60068
  1031
    by (simp add: map_bind_pmf bind_return_pmf')
hoelzl@59681
  1032
hoelzl@59681
  1033
  show "map_pmf snd ?pq = q"
hoelzl@59681
  1034
    using R eq
Andreas@60068
  1035
    apply (simp add: bind_cond_pmf_cancel map_bind_pmf bind_return_pmf')
hoelzl@59681
  1036
    apply (rule bind_cond_pmf_cancel)
hoelzl@59681
  1037
    apply (auto simp: rel_set_def)
hoelzl@59681
  1038
    done
hoelzl@59681
  1039
qed
hoelzl@59681
  1040
hoelzl@59681
  1041
lemma rel_pmf_imp_rel_set: "rel_pmf R p q \<Longrightarrow> rel_set R (set_pmf p) (set_pmf q)"
hoelzl@59681
  1042
  by (force simp add: rel_pmf.simps rel_set_def)
hoelzl@59681
  1043
hoelzl@59681
  1044
lemma rel_pmfD_measure:
hoelzl@59681
  1045
  assumes rel_R: "rel_pmf R p q" and R: "\<And>a b. R a b \<Longrightarrow> R a y \<longleftrightarrow> R x b"
hoelzl@59681
  1046
  assumes "x \<in> set_pmf p" "y \<in> set_pmf q"
hoelzl@59681
  1047
  shows "measure p {x. R x y} = measure q {y. R x y}"
hoelzl@59681
  1048
proof -
hoelzl@59681
  1049
  from rel_R obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
hoelzl@59681
  1050
    and eq: "p = map_pmf fst pq" "q = map_pmf snd pq"
hoelzl@59681
  1051
    by (auto elim: rel_pmf.cases)
hoelzl@59681
  1052
  have "measure p {x. R x y} = measure pq {x. R (fst x) y}"
hoelzl@59681
  1053
    by (simp add: eq map_pmf_rep_eq measure_distr)
hoelzl@59681
  1054
  also have "\<dots> = measure pq {y. R x (snd y)}"
hoelzl@59681
  1055
    by (intro measure_pmf.finite_measure_eq_AE)
hoelzl@59681
  1056
       (auto simp: AE_measure_pmf_iff R dest!: pq)
hoelzl@59681
  1057
  also have "\<dots> = measure q {y. R x y}"
hoelzl@59681
  1058
    by (simp add: eq map_pmf_rep_eq measure_distr)
hoelzl@59681
  1059
  finally show "measure p {x. R x y} = measure q {y. R x y}" .
hoelzl@59681
  1060
qed
hoelzl@59681
  1061
Andreas@61634
  1062
lemma rel_pmf_measureD:
Andreas@61634
  1063
  assumes "rel_pmf R p q"
Andreas@61634
  1064
  shows "measure (measure_pmf p) A \<le> measure (measure_pmf q) {y. \<exists>x\<in>A. R x y}" (is "?lhs \<le> ?rhs")
Andreas@61634
  1065
using assms
Andreas@61634
  1066
proof cases
Andreas@61634
  1067
  fix pq
Andreas@61634
  1068
  assume R: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
Andreas@61634
  1069
    and p[symmetric]: "map_pmf fst pq = p"
Andreas@61634
  1070
    and q[symmetric]: "map_pmf snd pq = q"
Andreas@61634
  1071
  have "?lhs = measure (measure_pmf pq) (fst -` A)" by(simp add: p)
Andreas@61634
  1072
  also have "\<dots> \<le> measure (measure_pmf pq) {y. \<exists>x\<in>A. R x (snd y)}"
Andreas@61634
  1073
    by(rule measure_pmf.finite_measure_mono_AE)(auto 4 3 simp add: AE_measure_pmf_iff dest: R)
Andreas@61634
  1074
  also have "\<dots> = ?rhs" by(simp add: q)
Andreas@61634
  1075
  finally show ?thesis .
Andreas@61634
  1076
qed
Andreas@61634
  1077
hoelzl@59681
  1078
lemma rel_pmf_iff_measure:
hoelzl@59681
  1079
  assumes "symp R" "transp R"
hoelzl@59681
  1080
  shows "rel_pmf R p q \<longleftrightarrow>
hoelzl@59681
  1081
    rel_set R (set_pmf p) (set_pmf q) \<and>
hoelzl@59681
  1082
    (\<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})"
hoelzl@59681
  1083
  by (safe intro!: rel_pmf_imp_rel_set rel_pmfI)
hoelzl@59681
  1084
     (auto intro!: rel_pmfD_measure dest: sympD[OF \<open>symp R\<close>] transpD[OF \<open>transp R\<close>])
hoelzl@59681
  1085
hoelzl@59681
  1086
lemma quotient_rel_set_disjoint:
hoelzl@59681
  1087
  "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 = {}"
lp15@61609
  1088
  using in_quotient_imp_closed[of UNIV "{(x, y). R x y}" C]
hoelzl@59681
  1089
  by (auto 0 0 simp: equivp_equiv rel_set_def set_eq_iff elim: equivpE)
hoelzl@59681
  1090
     (blast dest: equivp_symp)+
hoelzl@59681
  1091
hoelzl@59681
  1092
lemma quotientD: "equiv X R \<Longrightarrow> A \<in> X // R \<Longrightarrow> x \<in> A \<Longrightarrow> A = R `` {x}"
hoelzl@59681
  1093
  by (metis Image_singleton_iff equiv_class_eq_iff quotientE)
hoelzl@59681
  1094
hoelzl@59681
  1095
lemma rel_pmf_iff_equivp:
hoelzl@59681
  1096
  assumes "equivp R"
hoelzl@59681
  1097
  shows "rel_pmf R p q \<longleftrightarrow> (\<forall>C\<in>UNIV // {(x, y). R x y}. measure p C = measure q C)"
hoelzl@59681
  1098
    (is "_ \<longleftrightarrow>   (\<forall>C\<in>_//?R. _)")
hoelzl@59681
  1099
proof (subst rel_pmf_iff_measure, safe)
hoelzl@59681
  1100
  show "symp R" "transp R"
hoelzl@59681
  1101
    using assms by (auto simp: equivp_reflp_symp_transp)
hoelzl@59681
  1102
next
hoelzl@59681
  1103
  fix C assume C: "C \<in> UNIV // ?R" and R: "rel_set R (set_pmf p) (set_pmf q)"
hoelzl@59681
  1104
  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}"
lp15@61609
  1105
hoelzl@59681
  1106
  show "measure p C = measure q C"
wenzelm@63540
  1107
  proof (cases "p \<inter> C = {}")
wenzelm@63540
  1108
    case True
wenzelm@63540
  1109
    then have "q \<inter> C = {}"
hoelzl@59681
  1110
      using quotient_rel_set_disjoint[OF assms C R] by simp
wenzelm@63540
  1111
    with True show ?thesis
hoelzl@59681
  1112
      unfolding measure_pmf_zero_iff[symmetric] by simp
hoelzl@59681
  1113
  next
wenzelm@63540
  1114
    case False
wenzelm@63540
  1115
    then have "q \<inter> C \<noteq> {}"
hoelzl@59681
  1116
      using quotient_rel_set_disjoint[OF assms C R] by simp
wenzelm@63540
  1117
    with False 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"
hoelzl@59681
  1118
      by auto
hoelzl@59681
  1119
    then have "R x y"
hoelzl@59681
  1120
      using in_quotient_imp_in_rel[of UNIV ?R C x y] C assms
hoelzl@59681
  1121
      by (simp add: equivp_equiv)
hoelzl@59681
  1122
    with in_set eq have "measure p {x. R x y} = measure q {y. R x y}"
hoelzl@59681
  1123
      by auto
hoelzl@59681
  1124
    moreover have "{y. R x y} = C"
wenzelm@61808
  1125
      using assms \<open>x \<in> C\<close> C quotientD[of UNIV ?R C x] by (simp add: equivp_equiv)
hoelzl@59681
  1126
    moreover have "{x. R x y} = C"
wenzelm@61808
  1127
      using assms \<open>y \<in> C\<close> C quotientD[of UNIV "?R" C y] sympD[of R]
hoelzl@59681
  1128
      by (auto simp add: equivp_equiv elim: equivpE)
hoelzl@59681
  1129
    ultimately show ?thesis
hoelzl@59681
  1130
      by auto
hoelzl@59681
  1131
  qed
hoelzl@59681
  1132
next
hoelzl@59681
  1133
  assume eq: "\<forall>C\<in>UNIV // ?R. measure p C = measure q C"
hoelzl@59681
  1134
  show "rel_set R (set_pmf p) (set_pmf q)"
hoelzl@59681
  1135
    unfolding rel_set_def
hoelzl@59681
  1136
  proof safe
hoelzl@59681
  1137
    fix x assume x: "x \<in> set_pmf p"
hoelzl@59681
  1138
    have "{y. R x y} \<in> UNIV // ?R"
hoelzl@59681
  1139
      by (auto simp: quotient_def)
hoelzl@59681
  1140
    with eq have *: "measure q {y. R x y} = measure p {y. R x y}"
hoelzl@59681
  1141
      by auto
hoelzl@59681
  1142
    have "measure q {y. R x y} \<noteq> 0"
hoelzl@59681
  1143
      using x assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp)
hoelzl@59681
  1144
    then show "\<exists>y\<in>set_pmf q. R x y"
hoelzl@59681
  1145
      unfolding measure_pmf_zero_iff by auto
hoelzl@59681
  1146
  next
hoelzl@59681
  1147
    fix y assume y: "y \<in> set_pmf q"
hoelzl@59681
  1148
    have "{x. R x y} \<in> UNIV // ?R"
hoelzl@59681
  1149
      using assms by (auto simp: quotient_def dest: equivp_symp)
hoelzl@59681
  1150
    with eq have *: "measure p {x. R x y} = measure q {x. R x y}"
hoelzl@59681
  1151
      by auto
hoelzl@59681
  1152
    have "measure p {x. R x y} \<noteq> 0"
hoelzl@59681
  1153
      using y assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp)
hoelzl@59681
  1154
    then show "\<exists>x\<in>set_pmf p. R x y"
hoelzl@59681
  1155
      unfolding measure_pmf_zero_iff by auto
hoelzl@59681
  1156
  qed
hoelzl@59681
  1157
hoelzl@59681
  1158
  fix x y assume "x \<in> set_pmf p" "y \<in> set_pmf q" "R x y"
hoelzl@59681
  1159
  have "{y. R x y} \<in> UNIV // ?R" "{x. R x y} = {y. R x y}"
wenzelm@61808
  1160
    using assms \<open>R x y\<close> by (auto simp: quotient_def dest: equivp_symp equivp_transp)
hoelzl@59681
  1161
  with eq show "measure p {x. R x y} = measure q {y. R x y}"
hoelzl@59681
  1162
    by auto
hoelzl@59681
  1163
qed
hoelzl@59681
  1164
hoelzl@59664
  1165
bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: rel_pmf
hoelzl@59664
  1166
proof -
hoelzl@59664
  1167
  show "map_pmf id = id" by (rule map_pmf_id)
lp15@59667
  1168
  show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose)
hoelzl@59664
  1169
  show "\<And>f g::'a \<Rightarrow> 'b. \<And>p. (\<And>x. x \<in> set_pmf p \<Longrightarrow> f x = g x) \<Longrightarrow> map_pmf f p = map_pmf g p"
hoelzl@59664
  1170
    by (intro map_pmf_cong refl)
hoelzl@59664
  1171
hoelzl@59664
  1172
  show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
hoelzl@59664
  1173
    by (rule pmf_set_map)
hoelzl@59664
  1174
wenzelm@60595
  1175
  show "(card_of (set_pmf p), natLeq) \<in> ordLeq" for p :: "'s pmf"
wenzelm@60595
  1176
  proof -
hoelzl@59664
  1177
    have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq"
hoelzl@59664
  1178
      by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"])
hoelzl@59664
  1179
         (auto intro: countable_set_pmf)
hoelzl@59664
  1180
    also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq"
hoelzl@59664
  1181
      by (metis Field_natLeq card_of_least natLeq_Well_order)
wenzelm@60595
  1182
    finally show ?thesis .
wenzelm@60595
  1183
  qed
hoelzl@59664
  1184
traytel@62324
  1185
  show "\<And>R. rel_pmf R = (\<lambda>x y. \<exists>z. set_pmf z \<subseteq> {(x, y). R x y} \<and>
traytel@62324
  1186
    map_pmf fst z = x \<and> map_pmf snd z = y)"
traytel@62324
  1187
     by (auto simp add: fun_eq_iff rel_pmf.simps)
hoelzl@59664
  1188
wenzelm@60595
  1189
  show "rel_pmf R OO rel_pmf S \<le> rel_pmf (R OO S)"
wenzelm@60595
  1190
    for R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
wenzelm@60595
  1191
  proof -
wenzelm@60595
  1192
    { fix p q r
wenzelm@60595
  1193
      assume pq: "rel_pmf R p q"
wenzelm@60595
  1194
        and qr:"rel_pmf S q r"
wenzelm@60595
  1195
      from pq obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
wenzelm@60595
  1196
        and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto
wenzelm@60595
  1197
      from qr obtain qr where qr: "\<And>y z. (y, z) \<in> set_pmf qr \<Longrightarrow> S y z"
wenzelm@60595
  1198
        and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto
lp15@61609
  1199
wenzelm@63040
  1200
      define pr where "pr =
wenzelm@63040
  1201
        bind_pmf pq (\<lambda>xy. bind_pmf (cond_pmf qr {yz. fst yz = snd xy})
wenzelm@63040
  1202
          (\<lambda>yz. return_pmf (fst xy, snd yz)))"
wenzelm@60595
  1203
      have pr_welldefined: "\<And>y. y \<in> q \<Longrightarrow> qr \<inter> {yz. fst yz = y} \<noteq> {}"
wenzelm@60595
  1204
        by (force simp: q')
lp15@61609
  1205
wenzelm@60595
  1206
      have "rel_pmf (R OO S) p r"
wenzelm@60595
  1207
      proof (rule rel_pmf.intros)
wenzelm@60595
  1208
        fix x z assume "(x, z) \<in> pr"
wenzelm@60595
  1209
        then have "\<exists>y. (x, y) \<in> pq \<and> (y, z) \<in> qr"
wenzelm@60595
  1210
          by (auto simp: q pr_welldefined pr_def split_beta)
wenzelm@60595
  1211
        with pq qr show "(R OO S) x z"
wenzelm@60595
  1212
          by blast
wenzelm@60595
  1213
      next
wenzelm@60595
  1214
        have "map_pmf snd pr = map_pmf snd (bind_pmf q (\<lambda>y. cond_pmf qr {yz. fst yz = y}))"
wenzelm@60595
  1215
          by (simp add: pr_def q split_beta bind_map_pmf map_pmf_def[symmetric] map_bind_pmf map_pmf_comp)
wenzelm@60595
  1216
        then show "map_pmf snd pr = r"
wenzelm@60595
  1217
          unfolding r q' bind_map_pmf by (subst (asm) bind_cond_pmf_cancel) (auto simp: eq_commute)
wenzelm@60595
  1218
      qed (simp add: pr_def map_bind_pmf split_beta map_pmf_def[symmetric] p map_pmf_comp)
wenzelm@60595
  1219
    }
wenzelm@60595
  1220
    then show ?thesis
wenzelm@60595
  1221
      by(auto simp add: le_fun_def)
wenzelm@60595
  1222
  qed
hoelzl@59664
  1223
qed (fact natLeq_card_order natLeq_cinfinite)+
hoelzl@59664
  1224
Andreas@61634
  1225
lemma map_pmf_idI: "(\<And>x. x \<in> set_pmf p \<Longrightarrow> f x = x) \<Longrightarrow> map_pmf f p = p"
Andreas@61634
  1226
by(simp cong: pmf.map_cong)
Andreas@61634
  1227
hoelzl@59665
  1228
lemma rel_pmf_conj[simp]:
hoelzl@59665
  1229
  "rel_pmf (\<lambda>x y. P \<and> Q x y) x y \<longleftrightarrow> P \<and> rel_pmf Q x y"
hoelzl@59665
  1230
  "rel_pmf (\<lambda>x y. Q x y \<and> P) x y \<longleftrightarrow> P \<and> rel_pmf Q x y"
hoelzl@59665
  1231
  using set_pmf_not_empty by (fastforce simp: pmf.in_rel subset_eq)+
hoelzl@59665
  1232
hoelzl@59665
  1233
lemma rel_pmf_top[simp]: "rel_pmf top = top"
hoelzl@59665
  1234
  by (auto simp: pmf.in_rel[abs_def] fun_eq_iff map_fst_pair_pmf map_snd_pair_pmf
hoelzl@59665
  1235
           intro: exI[of _ "pair_pmf x y" for x y])
hoelzl@59665
  1236
hoelzl@59664
  1237
lemma rel_pmf_return_pmf1: "rel_pmf R (return_pmf x) M \<longleftrightarrow> (\<forall>a\<in>M. R x a)"
hoelzl@59664
  1238
proof safe
hoelzl@59664
  1239
  fix a assume "a \<in> M" "rel_pmf R (return_pmf x) M"
hoelzl@59664
  1240
  then obtain pq where *: "\<And>a b. (a, b) \<in> set_pmf pq \<Longrightarrow> R a b"
hoelzl@59664
  1241
    and eq: "return_pmf x = map_pmf fst pq" "M = map_pmf snd pq"
hoelzl@59664
  1242
    by (force elim: rel_pmf.cases)
hoelzl@59664
  1243
  moreover have "set_pmf (return_pmf x) = {x}"
hoelzl@59665
  1244
    by simp
wenzelm@61808
  1245
  with \<open>a \<in> M\<close> have "(x, a) \<in> pq"
hoelzl@59665
  1246
    by (force simp: eq)
hoelzl@59664
  1247
  with * show "R x a"
hoelzl@59664
  1248
    by auto
hoelzl@59664
  1249
qed (auto intro!: rel_pmf.intros[where pq="pair_pmf (return_pmf x) M"]
hoelzl@59665
  1250
          simp: map_fst_pair_pmf map_snd_pair_pmf)
hoelzl@59664
  1251
hoelzl@59664
  1252
lemma rel_pmf_return_pmf2: "rel_pmf R M (return_pmf x) \<longleftrightarrow> (\<forall>a\<in>M. R a x)"
hoelzl@59664
  1253
  by (subst pmf.rel_flip[symmetric]) (simp add: rel_pmf_return_pmf1)
hoelzl@59664
  1254
hoelzl@59664
  1255
lemma rel_return_pmf[simp]: "rel_pmf R (return_pmf x1) (return_pmf x2) = R x1 x2"
hoelzl@59664
  1256
  unfolding rel_pmf_return_pmf2 set_return_pmf by simp
hoelzl@59664
  1257
hoelzl@59664
  1258
lemma rel_pmf_False[simp]: "rel_pmf (\<lambda>x y. False) x y = False"
hoelzl@59664
  1259
  unfolding pmf.in_rel fun_eq_iff using set_pmf_not_empty by fastforce
hoelzl@59664
  1260
hoelzl@59664
  1261
lemma rel_pmf_rel_prod:
hoelzl@59664
  1262
  "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B') \<longleftrightarrow> rel_pmf R A B \<and> rel_pmf S A' B'"
hoelzl@59664
  1263
proof safe
hoelzl@59664
  1264
  assume "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
hoelzl@59664
  1265
  then obtain pq where pq: "\<And>a b c d. ((a, c), (b, d)) \<in> set_pmf pq \<Longrightarrow> R a b \<and> S c d"
hoelzl@59664
  1266
    and eq: "map_pmf fst pq = pair_pmf A A'" "map_pmf snd pq = pair_pmf B B'"
hoelzl@59664
  1267
    by (force elim: rel_pmf.cases)
hoelzl@59664
  1268
  show "rel_pmf R A B"
hoelzl@59664
  1269
  proof (rule rel_pmf.intros)
hoelzl@59664
  1270
    let ?f = "\<lambda>(a, b). (fst a, fst b)"
hoelzl@59664
  1271
    have [simp]: "(\<lambda>x. fst (?f x)) = fst o fst" "(\<lambda>x. snd (?f x)) = fst o snd"
hoelzl@59664
  1272
      by auto
hoelzl@59664
  1273
hoelzl@59664
  1274
    show "map_pmf fst (map_pmf ?f pq) = A"
hoelzl@59664
  1275
      by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
hoelzl@59664
  1276
    show "map_pmf snd (map_pmf ?f pq) = B"
hoelzl@59664
  1277
      by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
hoelzl@59664
  1278
hoelzl@59664
  1279
    fix a b assume "(a, b) \<in> set_pmf (map_pmf ?f pq)"
hoelzl@59664
  1280
    then obtain c d where "((a, c), (b, d)) \<in> set_pmf pq"
hoelzl@59665
  1281
      by auto
hoelzl@59664
  1282
    from pq[OF this] show "R a b" ..
hoelzl@59664
  1283
  qed
hoelzl@59664
  1284
  show "rel_pmf S A' B'"
hoelzl@59664
  1285
  proof (rule rel_pmf.intros)
hoelzl@59664
  1286
    let ?f = "\<lambda>(a, b). (snd a, snd b)"
hoelzl@59664
  1287
    have [simp]: "(\<lambda>x. fst (?f x)) = snd o fst" "(\<lambda>x. snd (?f x)) = snd o snd"
hoelzl@59664
  1288
      by auto
hoelzl@59664
  1289
hoelzl@59664
  1290
    show "map_pmf fst (map_pmf ?f pq) = A'"
hoelzl@59664
  1291
      by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
hoelzl@59664
  1292
    show "map_pmf snd (map_pmf ?f pq) = B'"
hoelzl@59664
  1293
      by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
hoelzl@59664
  1294
hoelzl@59664
  1295
    fix c d assume "(c, d) \<in> set_pmf (map_pmf ?f pq)"
hoelzl@59664
  1296
    then obtain a b where "((a, c), (b, d)) \<in> set_pmf pq"
hoelzl@59665
  1297
      by auto
hoelzl@59664
  1298
    from pq[OF this] show "S c d" ..
hoelzl@59664
  1299
  qed
hoelzl@59664
  1300
next
hoelzl@59664
  1301
  assume "rel_pmf R A B" "rel_pmf S A' B'"
hoelzl@59664
  1302
  then obtain Rpq Spq
hoelzl@59664
  1303
    where Rpq: "\<And>a b. (a, b) \<in> set_pmf Rpq \<Longrightarrow> R a b"
hoelzl@59664
  1304
        "map_pmf fst Rpq = A" "map_pmf snd Rpq = B"
hoelzl@59664
  1305
      and Spq: "\<And>a b. (a, b) \<in> set_pmf Spq \<Longrightarrow> S a b"
hoelzl@59664
  1306
        "map_pmf fst Spq = A'" "map_pmf snd Spq = B'"
hoelzl@59664
  1307
    by (force elim: rel_pmf.cases)
hoelzl@59664
  1308
hoelzl@59664
  1309
  let ?f = "(\<lambda>((a, c), (b, d)). ((a, b), (c, d)))"
hoelzl@59664
  1310
  let ?pq = "map_pmf ?f (pair_pmf Rpq Spq)"
hoelzl@59664
  1311
  have [simp]: "(\<lambda>x. fst (?f x)) = (\<lambda>(a, b). (fst a, fst b))" "(\<lambda>x. snd (?f x)) = (\<lambda>(a, b). (snd a, snd b))"
hoelzl@59664
  1312
    by auto
hoelzl@59664
  1313
hoelzl@59664
  1314
  show "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
hoelzl@59664
  1315
    by (rule rel_pmf.intros[where pq="?pq"])
hoelzl@59665
  1316
       (auto simp: map_snd_pair_pmf map_fst_pair_pmf map_pmf_comp Rpq Spq
hoelzl@59664
  1317
                   map_pair)
hoelzl@59664
  1318
qed
hoelzl@59664
  1319
lp15@59667
  1320
lemma rel_pmf_reflI:
hoelzl@59664
  1321
  assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> P x x"
hoelzl@59664
  1322
  shows "rel_pmf P p p"
hoelzl@59665
  1323
  by (rule rel_pmf.intros[where pq="map_pmf (\<lambda>x. (x, x)) p"])
hoelzl@59665
  1324
     (auto simp add: pmf.map_comp o_def assms)
hoelzl@59664
  1325
Andreas@61634
  1326
lemma rel_pmf_bij_betw:
Andreas@61634
  1327
  assumes f: "bij_betw f (set_pmf p) (set_pmf q)"
Andreas@61634
  1328
  and eq: "\<And>x. x \<in> set_pmf p \<Longrightarrow> pmf p x = pmf q (f x)"
Andreas@61634
  1329
  shows "rel_pmf (\<lambda>x y. f x = y) p q"
Andreas@61634
  1330
proof(rule rel_pmf.intros)
Andreas@61634
  1331
  let ?pq = "map_pmf (\<lambda>x. (x, f x)) p"
Andreas@61634
  1332
  show "map_pmf fst ?pq = p" by(simp add: pmf.map_comp o_def)
Andreas@61634
  1333
Andreas@61634
  1334
  have "map_pmf f p = q"
Andreas@61634
  1335
  proof(rule pmf_eqI)
Andreas@61634
  1336
    fix i
Andreas@61634
  1337
    show "pmf (map_pmf f p) i = pmf q i"
Andreas@61634
  1338
    proof(cases "i \<in> set_pmf q")
Andreas@61634
  1339
      case True
Andreas@61634
  1340
      with f obtain j where "i = f j" "j \<in> set_pmf p"
Andreas@61634
  1341
        by(auto simp add: bij_betw_def image_iff)
Andreas@61634
  1342
      thus ?thesis using f by(simp add: bij_betw_def pmf_map_inj eq)
Andreas@61634
  1343
    next
Andreas@61634
  1344
      case False thus ?thesis
Andreas@61634
  1345
        by(subst pmf_map_outside)(auto simp add: set_pmf_iff eq[symmetric])
Andreas@61634
  1346
    qed
Andreas@61634
  1347
  qed
Andreas@61634
  1348
  then show "map_pmf snd ?pq = q" by(simp add: pmf.map_comp o_def)
Andreas@61634
  1349
qed auto
Andreas@61634
  1350
hoelzl@59664
  1351
context
hoelzl@59664
  1352
begin
hoelzl@59664
  1353
hoelzl@59664
  1354
interpretation pmf_as_measure .
hoelzl@59664
  1355
hoelzl@59664
  1356
definition "join_pmf M = bind_pmf M (\<lambda>x. x)"
hoelzl@59664
  1357
hoelzl@59664
  1358
lemma bind_eq_join_pmf: "bind_pmf M f = join_pmf (map_pmf f M)"
hoelzl@59664
  1359
  unfolding join_pmf_def bind_map_pmf ..
hoelzl@59664
  1360
hoelzl@59664
  1361
lemma join_eq_bind_pmf: "join_pmf M = bind_pmf M id"
hoelzl@59664
  1362
  by (simp add: join_pmf_def id_def)
hoelzl@59664
  1363
hoelzl@59664
  1364
lemma pmf_join: "pmf (join_pmf N) i = (\<integral>M. pmf M i \<partial>measure_pmf N)"
hoelzl@59664
  1365
  unfolding join_pmf_def pmf_bind ..
hoelzl@59664
  1366
hoelzl@62975
  1367
lemma ennreal_pmf_join: "ennreal (pmf (join_pmf N) i) = (\<integral>\<^sup>+M. pmf M i \<partial>measure_pmf N)"
hoelzl@62975
  1368
  unfolding join_pmf_def ennreal_pmf_bind ..
hoelzl@59664
  1369
hoelzl@59665
  1370
lemma set_pmf_join_pmf[simp]: "set_pmf (join_pmf f) = (\<Union>p\<in>set_pmf f. set_pmf p)"
hoelzl@59665
  1371
  by (simp add: join_pmf_def)
hoelzl@59664
  1372
hoelzl@59664
  1373
lemma join_return_pmf: "join_pmf (return_pmf M) = M"
hoelzl@59664
  1374
  by (simp add: integral_return pmf_eq_iff pmf_join return_pmf.rep_eq)
hoelzl@59664
  1375
hoelzl@59664
  1376
lemma map_join_pmf: "map_pmf f (join_pmf AA) = join_pmf (map_pmf (map_pmf f) AA)"
hoelzl@59664
  1377
  by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf)
hoelzl@59664
  1378
hoelzl@59664
  1379
lemma join_map_return_pmf: "join_pmf (map_pmf return_pmf A) = A"
hoelzl@59664
  1380
  by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
hoelzl@59664
  1381
hoelzl@59664
  1382
end
hoelzl@59664
  1383
hoelzl@59664
  1384
lemma rel_pmf_joinI:
hoelzl@59664
  1385
  assumes "rel_pmf (rel_pmf P) p q"
hoelzl@59664
  1386
  shows "rel_pmf P (join_pmf p) (join_pmf q)"
hoelzl@59664
  1387
proof -
hoelzl@59664
  1388
  from assms obtain pq where p: "p = map_pmf fst pq"
hoelzl@59664
  1389
    and q: "q = map_pmf snd pq"
hoelzl@59664
  1390
    and P: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> rel_pmf P x y"
hoelzl@59664
  1391
    by cases auto
lp15@59667
  1392
  from P obtain PQ
hoelzl@59664
  1393
    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"
hoelzl@59664
  1394
    and x: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf fst (PQ x y) = x"
hoelzl@59664
  1395
    and y: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf snd (PQ x y) = y"
hoelzl@59664
  1396
    by(metis rel_pmf.simps)
hoelzl@59664
  1397
hoelzl@59664
  1398
  let ?r = "bind_pmf pq (\<lambda>(x, y). PQ x y)"
hoelzl@59665
  1399
  have "\<And>a b. (a, b) \<in> set_pmf ?r \<Longrightarrow> P a b" by (auto intro: PQ)
hoelzl@59664
  1400
  moreover have "map_pmf fst ?r = join_pmf p" "map_pmf snd ?r = join_pmf q"
hoelzl@59664
  1401
    by (simp_all add: p q x y join_pmf_def map_bind_pmf bind_map_pmf split_def cong: bind_pmf_cong)
hoelzl@59664
  1402
  ultimately show ?thesis ..
hoelzl@59664
  1403
qed
hoelzl@59664
  1404
hoelzl@59664
  1405
lemma rel_pmf_bindI:
hoelzl@59664
  1406
  assumes pq: "rel_pmf R p q"
hoelzl@59664
  1407
  and fg: "\<And>x y. R x y \<Longrightarrow> rel_pmf P (f x) (g y)"
hoelzl@59664
  1408
  shows "rel_pmf P (bind_pmf p f) (bind_pmf q g)"
hoelzl@59664
  1409
  unfolding bind_eq_join_pmf
hoelzl@59664
  1410
  by (rule rel_pmf_joinI)
hoelzl@59664
  1411
     (auto simp add: pmf.rel_map intro: pmf.rel_mono[THEN le_funD, THEN le_funD, THEN le_boolD, THEN mp, OF _ pq] fg)
hoelzl@59664
  1412
wenzelm@61808
  1413
text \<open>
hoelzl@59664
  1414
  Proof that @{const rel_pmf} preserves orders.
lp15@59667
  1415
  Antisymmetry proof follows Thm. 1 in N. Saheb-Djahromi, Cpo's of measures for nondeterminism,
lp15@59667
  1416
  Theoretical Computer Science 12(1):19--37, 1980,
wenzelm@63680
  1417
  \<^url>\<open>http://dx.doi.org/10.1016/0304-3975(80)90003-1\<close>
wenzelm@61808
  1418
\<close>
hoelzl@59664
  1419
lp15@59667
  1420
lemma
hoelzl@59664
  1421
  assumes *: "rel_pmf R p q"
hoelzl@59664
  1422
  and refl: "reflp R" and trans: "transp R"
hoelzl@59664
  1423
  shows measure_Ici: "measure p {y. R x y} \<le> measure q {y. R x y}" (is ?thesis1)
hoelzl@59664
  1424
  and measure_Ioi: "measure p {y. R x y \<and> \<not> R y x} \<le> measure q {y. R x y \<and> \<not> R y x}" (is ?thesis2)
hoelzl@59664
  1425
proof -
hoelzl@59664
  1426
  from * obtain pq
hoelzl@59664
  1427
    where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
hoelzl@59664
  1428
    and p: "p = map_pmf fst pq"
hoelzl@59664
  1429
    and q: "q = map_pmf snd pq"
hoelzl@59664
  1430
    by cases auto
hoelzl@59664
  1431
  show ?thesis1 ?thesis2 unfolding p q map_pmf_rep_eq using refl trans
hoelzl@59664
  1432
    by(auto 4 3 simp add: measure_distr reflpD AE_measure_pmf_iff intro!: measure_pmf.finite_measure_mono_AE dest!: pq elim: transpE)
hoelzl@59664
  1433
qed
hoelzl@59664
  1434
hoelzl@59664
  1435
lemma rel_pmf_inf:
hoelzl@59664
  1436
  fixes p q :: "'a pmf"
hoelzl@59664
  1437
  assumes 1: "rel_pmf R p q"
hoelzl@59664
  1438
  assumes 2: "rel_pmf R q p"
hoelzl@59664
  1439
  and refl: "reflp R" and trans: "transp R"
hoelzl@59664
  1440
  shows "rel_pmf (inf R R\<inverse>\<inverse>) p q"
hoelzl@59681
  1441
proof (subst rel_pmf_iff_equivp, safe)
hoelzl@59681
  1442
  show "equivp (inf R R\<inverse>\<inverse>)"
hoelzl@59681
  1443
    using trans refl by (auto simp: equivp_reflp_symp_transp intro: sympI transpI reflpI dest: transpD reflpD)
lp15@61609
  1444
hoelzl@59681
  1445
  fix C assume "C \<in> UNIV // {(x, y). inf R R\<inverse>\<inverse> x y}"
hoelzl@59681
  1446
  then obtain x where C: "C = {y. R x y \<and> R y x}"
hoelzl@59681
  1447
    by (auto elim: quotientE)
hoelzl@59681
  1448
hoelzl@59670
  1449
  let ?R = "\<lambda>x y. R x y \<and> R y x"
hoelzl@59670
  1450
  let ?\<mu>R = "\<lambda>y. measure q {x. ?R x y}"
hoelzl@59681
  1451
  have "measure p {y. ?R x y} = measure p ({y. R x y} - {y. R x y \<and> \<not> R y x})"
hoelzl@59681
  1452
    by(auto intro!: arg_cong[where f="measure p"])
hoelzl@59681
  1453
  also have "\<dots> = measure p {y. R x y} - measure p {y. R x y \<and> \<not> R y x}"
hoelzl@59681
  1454
    by (rule measure_pmf.finite_measure_Diff) auto
hoelzl@59681
  1455
  also have "measure p {y. R x y \<and> \<not> R y x} = measure q {y. R x y \<and> \<not> R y x}"
hoelzl@59681
  1456
    using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ioi)
hoelzl@59681
  1457
  also have "measure p {y. R x y} = measure q {y. R x y}"
hoelzl@59681
  1458
    using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ici)
hoelzl@59681
  1459
  also have "measure q {y. R x y} - measure q {y. R x y \<and> \<not> R y x} =
hoelzl@59681
  1460
    measure q ({y. R x y} - {y. R x y \<and> \<not> R y x})"
hoelzl@59681
  1461
    by(rule measure_pmf.finite_measure_Diff[symmetric]) auto
hoelzl@59681
  1462
  also have "\<dots> = ?\<mu>R x"
hoelzl@59681
  1463
    by(auto intro!: arg_cong[where f="measure q"])
hoelzl@59681
  1464
  finally show "measure p C = measure q C"
hoelzl@59681
  1465
    by (simp add: C conj_commute)
hoelzl@59664
  1466
qed
hoelzl@59664
  1467
hoelzl@59664
  1468
lemma rel_pmf_antisym:
hoelzl@59664
  1469
  fixes p q :: "'a pmf"
hoelzl@59664
  1470
  assumes 1: "rel_pmf R p q"
hoelzl@59664
  1471
  assumes 2: "rel_pmf R q p"
hoelzl@59664
  1472
  and refl: "reflp R" and trans: "transp R" and antisym: "antisymP R"
hoelzl@59664
  1473
  shows "p = q"
hoelzl@59664
  1474
proof -
hoelzl@59664
  1475
  from 1 2 refl trans have "rel_pmf (inf R R\<inverse>\<inverse>) p q" by(rule rel_pmf_inf)
hoelzl@59664
  1476
  also have "inf R R\<inverse>\<inverse> = op ="
hoelzl@59665
  1477
    using refl antisym by (auto intro!: ext simp add: reflpD dest: antisymD)
hoelzl@59664
  1478
  finally show ?thesis unfolding pmf.rel_eq .
hoelzl@59664
  1479
qed
hoelzl@59664
  1480
hoelzl@59664
  1481
lemma reflp_rel_pmf: "reflp R \<Longrightarrow> reflp (rel_pmf R)"
hoelzl@59664
  1482
by(blast intro: reflpI rel_pmf_reflI reflpD)
hoelzl@59664
  1483
hoelzl@59664
  1484
lemma antisymP_rel_pmf:
hoelzl@59664
  1485
  "\<lbrakk> reflp R; transp R; antisymP R \<rbrakk>
hoelzl@59664
  1486
  \<Longrightarrow> antisymP (rel_pmf R)"
hoelzl@59664
  1487
by(rule antisymI)(blast intro: rel_pmf_antisym)
hoelzl@59664
  1488
hoelzl@59664
  1489
lemma transp_rel_pmf:
hoelzl@59664
  1490
  assumes "transp R"
hoelzl@59664
  1491
  shows "transp (rel_pmf R)"
hoelzl@59664
  1492
proof (rule transpI)
hoelzl@59664
  1493
  fix x y z
hoelzl@59664
  1494
  assume "rel_pmf R x y" and "rel_pmf R y z"
hoelzl@59664
  1495
  hence "rel_pmf (R OO R) x z" by (simp add: pmf.rel_compp relcompp.relcompI)
hoelzl@59664
  1496
  thus "rel_pmf R x z"
hoelzl@59664
  1497
    using assms by (metis (no_types) pmf.rel_mono rev_predicate2D transp_relcompp_less_eq)
hoelzl@59664
  1498
qed
hoelzl@59664
  1499
hoelzl@59664
  1500
subsection \<open> Distributions \<close>
hoelzl@59664
  1501
hoelzl@59000
  1502
context
hoelzl@59000
  1503
begin
hoelzl@59000
  1504
hoelzl@59000
  1505
interpretation pmf_as_function .
hoelzl@59000
  1506
hoelzl@59093
  1507
subsubsection \<open> Bernoulli Distribution \<close>
hoelzl@59093
  1508
hoelzl@59000
  1509
lift_definition bernoulli_pmf :: "real \<Rightarrow> bool pmf" is
hoelzl@59000
  1510
  "\<lambda>p b. ((\<lambda>p. if b then p else 1 - p) \<circ> min 1 \<circ> max 0) p"
hoelzl@59000
  1511
  by (auto simp: nn_integral_count_space_finite[where A="{False, True}"] UNIV_bool
hoelzl@59000
  1512
           split: split_max split_min)
hoelzl@59000
  1513
hoelzl@59000
  1514
lemma pmf_bernoulli_True[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) True = p"
hoelzl@59000
  1515
  by transfer simp
hoelzl@59000
  1516
hoelzl@59000
  1517
lemma pmf_bernoulli_False[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) False = 1 - p"
hoelzl@59000
  1518
  by transfer simp
hoelzl@59000
  1519
hoelzl@62975
  1520
lemma set_pmf_bernoulli[simp]: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (bernoulli_pmf p) = UNIV"
hoelzl@59000
  1521
  by (auto simp add: set_pmf_iff UNIV_bool)
hoelzl@59000
  1522
lp15@59667
  1523
lemma nn_integral_bernoulli_pmf[simp]:
hoelzl@59002
  1524
  assumes [simp]: "0 \<le> p" "p \<le> 1" "\<And>x. 0 \<le> f x"
hoelzl@59002
  1525
  shows "(\<integral>\<^sup>+x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
hoelzl@59002
  1526
  by (subst nn_integral_measure_pmf_support[of UNIV])
hoelzl@59002
  1527
     (auto simp: UNIV_bool field_simps)
hoelzl@59002
  1528
lp15@59667
  1529
lemma integral_bernoulli_pmf[simp]:
hoelzl@59002
  1530
  assumes [simp]: "0 \<le> p" "p \<le> 1"
hoelzl@59002
  1531
  shows "(\<integral>x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
hoelzl@59002
  1532
  by (subst integral_measure_pmf[of UNIV]) (auto simp: UNIV_bool)
hoelzl@59002
  1533
Andreas@59525
  1534
lemma pmf_bernoulli_half [simp]: "pmf (bernoulli_pmf (1 / 2)) x = 1 / 2"
Andreas@59525
  1535
by(cases x) simp_all
Andreas@59525
  1536
Andreas@59525
  1537
lemma measure_pmf_bernoulli_half: "measure_pmf (bernoulli_pmf (1 / 2)) = uniform_count_measure UNIV"
hoelzl@62975
  1538
  by (rule measure_eqI)
hoelzl@62975
  1539
     (simp_all add: nn_integral_pmf[symmetric] emeasure_uniform_count_measure ennreal_divide_numeral[symmetric]
hoelzl@62975
  1540
                    nn_integral_count_space_finite sets_uniform_count_measure divide_ennreal_def mult_ac
hoelzl@62975
  1541
                    ennreal_of_nat_eq_real_of_nat)
Andreas@59525
  1542
hoelzl@59093
  1543
subsubsection \<open> Geometric Distribution \<close>
hoelzl@59093
  1544
hoelzl@60602
  1545
context
hoelzl@60602
  1546
  fixes p :: real assumes p[arith]: "0 < p" "p \<le> 1"
hoelzl@60602
  1547
begin
hoelzl@60602
  1548
hoelzl@60602
  1549
lift_definition geometric_pmf :: "nat pmf" is "\<lambda>n. (1 - p)^n * p"
hoelzl@59000
  1550
proof
hoelzl@62975
  1551
  have "(\<Sum>i. ennreal (p * (1 - p) ^ i)) = ennreal (p * (1 / (1 - (1 - p))))"
hoelzl@62975
  1552
    by (intro suminf_ennreal_eq sums_mult geometric_sums) auto
hoelzl@62975
  1553
  then show "(\<integral>\<^sup>+ x. ennreal ((1 - p)^x * p) \<partial>count_space UNIV) = 1"
hoelzl@59000
  1554
    by (simp add: nn_integral_count_space_nat field_simps)
hoelzl@59000
  1555
qed simp
hoelzl@59000
  1556
hoelzl@60602
  1557
lemma pmf_geometric[simp]: "pmf geometric_pmf n = (1 - p)^n * p"
hoelzl@59000
  1558
  by transfer rule
hoelzl@59000
  1559
hoelzl@60602
  1560
end
hoelzl@60602
  1561
hoelzl@60602
  1562
lemma set_pmf_geometric: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (geometric_pmf p) = UNIV"
lp15@61609
  1563
  by (auto simp: set_pmf_iff)
hoelzl@59000
  1564
hoelzl@59093
  1565
subsubsection \<open> Uniform Multiset Distribution \<close>
hoelzl@59093
  1566
hoelzl@59000
  1567
context
hoelzl@59000
  1568
  fixes M :: "'a multiset" assumes M_not_empty: "M \<noteq> {#}"
hoelzl@59000
  1569
begin
hoelzl@59000
  1570
hoelzl@59000
  1571
lift_definition pmf_of_multiset :: "'a pmf" is "\<lambda>x. count M x / size M"
hoelzl@59000
  1572
proof
hoelzl@62975
  1573
  show "(\<integral>\<^sup>+ x. ennreal (real (count M x) / real (size M)) \<partial>count_space UNIV) = 1"
hoelzl@59000
  1574
    using M_not_empty
hoelzl@59000
  1575
    by (simp add: zero_less_divide_iff nn_integral_count_space nonempty_has_size
hoelzl@59000
  1576
                  setsum_divide_distrib[symmetric])
hoelzl@59000
  1577
       (auto simp: size_multiset_overloaded_eq intro!: setsum.cong)
hoelzl@59000
  1578
qed simp
hoelzl@59000
  1579
hoelzl@59000
  1580
lemma pmf_of_multiset[simp]: "pmf pmf_of_multiset x = count M x / size M"
hoelzl@59000
  1581
  by transfer rule
hoelzl@59000
  1582
nipkow@60495
  1583
lemma set_pmf_of_multiset[simp]: "set_pmf pmf_of_multiset = set_mset M"
hoelzl@59000
  1584
  by (auto simp: set_pmf_iff)
hoelzl@59000
  1585
hoelzl@59000
  1586
end
hoelzl@59000
  1587
hoelzl@59093
  1588
subsubsection \<open> Uniform Distribution \<close>
hoelzl@59093
  1589
hoelzl@59000
  1590
context
hoelzl@59000
  1591
  fixes S :: "'a set" assumes S_not_empty: "S \<noteq> {}" and S_finite: "finite S"
hoelzl@59000
  1592
begin
hoelzl@59000
  1593
hoelzl@59000
  1594
lift_definition pmf_of_set :: "'a pmf" is "\<lambda>x. indicator S x / card S"
hoelzl@59000
  1595
proof
hoelzl@62975
  1596
  show "(\<integral>\<^sup>+ x. ennreal (indicator S x / real (card S)) \<partial>count_space UNIV) = 1"
hoelzl@62975
  1597
    using S_not_empty S_finite
hoelzl@62975
  1598
    by (subst nn_integral_count_space'[of S])
hoelzl@62975
  1599
       (auto simp: ennreal_of_nat_eq_real_of_nat ennreal_mult[symmetric])
hoelzl@59000
  1600
qed simp
hoelzl@59000
  1601
hoelzl@59000
  1602
lemma pmf_of_set[simp]: "pmf pmf_of_set x = indicator S x / card S"
hoelzl@59000
  1603
  by transfer rule
hoelzl@59000
  1604
hoelzl@59000
  1605
lemma set_pmf_of_set[simp]: "set_pmf pmf_of_set = S"
hoelzl@59000
  1606
  using S_finite S_not_empty by (auto simp: set_pmf_iff)
hoelzl@59000
  1607
Andreas@61634
  1608
lemma emeasure_pmf_of_set_space[simp]: "emeasure pmf_of_set S = 1"
hoelzl@59002
  1609
  by (rule measure_pmf.emeasure_eq_1_AE) (auto simp: AE_measure_pmf_iff)
hoelzl@59002
  1610
hoelzl@62975
  1611
lemma nn_integral_pmf_of_set: "nn_integral (measure_pmf pmf_of_set) f = setsum f S / card S"
hoelzl@62975
  1612
  by (subst nn_integral_measure_pmf_finite)
hoelzl@62975
  1613
     (simp_all add: setsum_left_distrib[symmetric] card_gt_0_iff S_not_empty S_finite divide_ennreal_def
hoelzl@62975
  1614
                divide_ennreal[symmetric] ennreal_of_nat_eq_real_of_nat[symmetric] ennreal_times_divide)
Andreas@60068
  1615
hoelzl@62975
  1616
lemma integral_pmf_of_set: "integral\<^sup>L (measure_pmf pmf_of_set) f = setsum f S / card S"
hoelzl@62975
  1617
  by (subst integral_measure_pmf[of S]) (auto simp: S_finite setsum_divide_distrib)
Andreas@60068
  1618
hoelzl@62975
  1619
lemma emeasure_pmf_of_set: "emeasure (measure_pmf pmf_of_set) A = card (S \<inter> A) / card S"
hoelzl@62975
  1620
  by (subst nn_integral_indicator[symmetric], simp)
hoelzl@62975
  1621
     (simp add: S_finite S_not_empty card_gt_0_iff indicator_def setsum.If_cases divide_ennreal
hoelzl@62975
  1622
                ennreal_of_nat_eq_real_of_nat nn_integral_pmf_of_set)
Andreas@60068
  1623
hoelzl@62975
  1624
lemma measure_pmf_of_set: "measure (measure_pmf pmf_of_set) A = card (S \<inter> A) / card S"
wenzelm@63092
  1625
  using emeasure_pmf_of_set[of A]
hoelzl@62975
  1626
  by (simp add: measure_nonneg measure_pmf.emeasure_eq_measure)
Andreas@61634
  1627
hoelzl@59000
  1628
end
hoelzl@59000
  1629
eberlm@63099
  1630
lemma map_pmf_of_set:
eberlm@63099
  1631
  assumes "finite A" "A \<noteq> {}"
eberlm@63099
  1632
  shows   "map_pmf f (pmf_of_set A) = pmf_of_multiset (image_mset f (mset_set A))" 
eberlm@63099
  1633
    (is "?lhs = ?rhs")
eberlm@63099
  1634
proof (intro pmf_eqI)
eberlm@63099
  1635
  fix x
eberlm@63099
  1636
  from assms have "ennreal (pmf ?lhs x) = ennreal (pmf ?rhs x)"
eberlm@63099
  1637
    by (subst ennreal_pmf_map)
eberlm@63099
  1638
       (simp_all add: emeasure_pmf_of_set mset_set_empty_iff count_image_mset Int_commute)
eberlm@63099
  1639
  thus "pmf ?lhs x = pmf ?rhs x" by simp
eberlm@63099
  1640
qed
eberlm@63099
  1641
eberlm@63099
  1642
lemma pmf_bind_pmf_of_set:
eberlm@63099
  1643
  assumes "A \<noteq> {}" "finite A"
eberlm@63099
  1644
  shows   "pmf (bind_pmf (pmf_of_set A) f) x = 
eberlm@63099
  1645
             (\<Sum>xa\<in>A. pmf (f xa) x) / real_of_nat (card A)" (is "?lhs = ?rhs")
eberlm@63099
  1646
proof -
eberlm@63099
  1647
  from assms have "card A > 0" by auto
eberlm@63099
  1648
  with assms have "ennreal ?lhs = ennreal ?rhs"
eberlm@63099
  1649
    by (subst ennreal_pmf_bind) 
eberlm@63099
  1650
       (simp_all add: nn_integral_pmf_of_set max_def pmf_nonneg divide_ennreal [symmetric] 
eberlm@63099
  1651
        setsum_nonneg ennreal_of_nat_eq_real_of_nat)
eberlm@63099
  1652
  thus ?thesis by (subst (asm) ennreal_inj) (auto intro!: setsum_nonneg divide_nonneg_nonneg)
eberlm@63099
  1653
qed
eberlm@63099
  1654
Andreas@60068
  1655
lemma pmf_of_set_singleton: "pmf_of_set {x} = return_pmf x"
Andreas@60068
  1656
by(rule pmf_eqI)(simp add: indicator_def)
Andreas@60068
  1657
lp15@61609
  1658
lemma map_pmf_of_set_inj:
Andreas@60068
  1659
  assumes f: "inj_on f A"
Andreas@60068
  1660
  and [simp]: "A \<noteq> {}" "finite A"
Andreas@60068
  1661
  shows "map_pmf f (pmf_of_set A) = pmf_of_set (f ` A)" (is "?lhs = ?rhs")
Andreas@60068
  1662
proof(rule pmf_eqI)
Andreas@60068
  1663
  fix i
Andreas@60068
  1664
  show "pmf ?lhs i = pmf ?rhs i"
Andreas@60068
  1665
  proof(cases "i \<in> f ` A")
Andreas@60068
  1666
    case True
Andreas@60068
  1667
    then obtain i' where "i = f i'" "i' \<in> A" by auto
Andreas@60068
  1668
    thus ?thesis using f by(simp add: card_image pmf_map_inj)
Andreas@60068
  1669
  next
Andreas@60068
  1670
    case False
Andreas@60068
  1671
    hence "pmf ?lhs i = 0" by(simp add: pmf_eq_0_set_pmf set_map_pmf)
Andreas@60068
  1672
    moreover have "pmf ?rhs i = 0" using False by simp
Andreas@60068
  1673
    ultimately show ?thesis by simp
Andreas@60068
  1674
  qed
Andreas@60068
  1675
qed
Andreas@60068
  1676
eberlm@63099
  1677
text \<open>
eberlm@63099
  1678
  Choosing an element uniformly at random from the union of a disjoint family 
eberlm@63099
  1679
  of finite non-empty sets with the same size is the same as first choosing a set 
eberlm@63099
  1680
  from the family uniformly at random and then choosing an element from the chosen set 
eberlm@63099
  1681
  uniformly at random.  
eberlm@63099
  1682
\<close>
eberlm@63099
  1683
lemma pmf_of_set_UN:
eberlm@63099
  1684
  assumes "finite (UNION A f)" "A \<noteq> {}" "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> {}"
eberlm@63099
  1685
          "\<And>x. x \<in> A \<Longrightarrow> card (f x) = n" "disjoint_family_on f A"
eberlm@63099
  1686
  shows   "pmf_of_set (UNION A f) = do {x \<leftarrow> pmf_of_set A; pmf_of_set (f x)}"
eberlm@63099
  1687
            (is "?lhs = ?rhs")
eberlm@63099
  1688
proof (intro pmf_eqI)
eberlm@63099
  1689
  fix x
eberlm@63099
  1690
  from assms have [simp]: "finite A"
eberlm@63099
  1691
    using infinite_disjoint_family_imp_infinite_UNION[of A f] by blast
eberlm@63099
  1692
  from assms have "ereal (pmf (pmf_of_set (UNION A f)) x) =
eberlm@63099
  1693
    ereal (indicator (\<Union>x\<in>A. f x) x / real (card (\<Union>x\<in>A. f x)))"
eberlm@63099
  1694
    by (subst pmf_of_set) auto
eberlm@63099
  1695
  also from assms have "card (\<Union>x\<in>A. f x) = card A * n"
eberlm@63099
  1696
    by (subst card_UN_disjoint) (auto simp: disjoint_family_on_def)
eberlm@63099
  1697
  also from assms 
eberlm@63099
  1698
    have "indicator (\<Union>x\<in>A. f x) x / real \<dots> = 
eberlm@63099
  1699
              indicator (\<Union>x\<in>A. f x) x / (n * real (card A))"
eberlm@63099
  1700
      by (simp add: setsum_divide_distrib [symmetric] mult_ac)
eberlm@63099
  1701
  also from assms have "indicator (\<Union>x\<in>A. f x) x = (\<Sum>y\<in>A. indicator (f y) x)"
eberlm@63099
  1702
    by (intro indicator_UN_disjoint) simp_all
eberlm@63099
  1703
  also from assms have "ereal ((\<Sum>y\<in>A. indicator (f y) x) / (real n * real (card A))) =
eberlm@63099
  1704
                          ereal (pmf ?rhs x)"
eberlm@63099
  1705
    by (subst pmf_bind_pmf_of_set) (simp_all add: setsum_divide_distrib)
eberlm@63099
  1706
  finally show "pmf ?lhs x = pmf ?rhs x" by simp
eberlm@63099
  1707
qed
eberlm@63099
  1708
Andreas@60068
  1709
lemma bernoulli_pmf_half_conv_pmf_of_set: "bernoulli_pmf (1 / 2) = pmf_of_set UNIV"
hoelzl@62975
  1710
  by (rule pmf_eqI) simp_all
Andreas@61634
  1711
hoelzl@59093
  1712
subsubsection \<open> Poisson Distribution \<close>
hoelzl@59093
  1713
hoelzl@59093
  1714
context
hoelzl@59093
  1715
  fixes rate :: real assumes rate_pos: "0 < rate"
hoelzl@59093
  1716
begin
hoelzl@59093
  1717
hoelzl@59093
  1718
lift_definition poisson_pmf :: "nat pmf" is "\<lambda>k. rate ^ k / fact k * exp (-rate)"
lp15@59730
  1719
proof  (* by Manuel Eberl *)
hoelzl@59093
  1720
  have summable: "summable (\<lambda>x::nat. rate ^ x / fact x)" using summable_exp
haftmann@59557
  1721
    by (simp add: field_simps divide_inverse [symmetric])
hoelzl@59093
  1722
  have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x * exp (-rate) \<partial>count_space UNIV) =
hoelzl@59093
  1723
          exp (-rate) * (\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV)"
hoelzl@62975
  1724
    by (simp add: field_simps nn_integral_cmult[symmetric] ennreal_mult'[symmetric])
hoelzl@59093
  1725
  also from rate_pos have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV) = (\<Sum>x. rate ^ x / fact x)"
hoelzl@62975
  1726
    by (simp_all add: nn_integral_count_space_nat suminf_ennreal summable ennreal_suminf_neq_top)
hoelzl@59093
  1727
  also have "... = exp rate" unfolding exp_def
lp15@59730
  1728
    by (simp add: field_simps divide_inverse [symmetric])
hoelzl@62975
  1729
  also have "ennreal (exp (-rate)) * ennreal (exp rate) = 1"
hoelzl@62975
  1730
    by (simp add: mult_exp_exp ennreal_mult[symmetric])
hoelzl@62975
  1731
  finally show "(\<integral>\<^sup>+ x. ennreal (rate ^ x / (fact x) * exp (- rate)) \<partial>count_space UNIV) = 1" .
hoelzl@59093
  1732
qed (simp add: rate_pos[THEN less_imp_le])
hoelzl@59093
  1733
hoelzl@59093
  1734
lemma pmf_poisson[simp]: "pmf poisson_pmf k = rate ^ k / fact k * exp (-rate)"
hoelzl@59093
  1735
  by transfer rule
hoelzl@59093
  1736
hoelzl@59093
  1737
lemma set_pmf_poisson[simp]: "set_pmf poisson_pmf = UNIV"
hoelzl@59093
  1738
  using rate_pos by (auto simp: set_pmf_iff)
hoelzl@59093
  1739
hoelzl@59000
  1740
end
hoelzl@59000
  1741
hoelzl@59093
  1742
subsubsection \<open> Binomial Distribution \<close>
hoelzl@59093
  1743
hoelzl@59093
  1744
context
hoelzl@59093
  1745
  fixes n :: nat and p :: real assumes p_nonneg: "0 \<le> p" and p_le_1: "p \<le> 1"
hoelzl@59093
  1746
begin
hoelzl@59093
  1747
hoelzl@59093
  1748
lift_definition binomial_pmf :: "nat pmf" is "\<lambda>k. (n choose k) * p^k * (1 - p)^(n - k)"
hoelzl@59093
  1749
proof
hoelzl@62975
  1750
  have "(\<integral>\<^sup>+k. ennreal (real (n choose k) * p ^ k * (1 - p) ^ (n - k)) \<partial>count_space UNIV) =
hoelzl@62975
  1751
    ennreal (\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k))"
hoelzl@59093
  1752
    using p_le_1 p_nonneg by (subst nn_integral_count_space') auto
hoelzl@59093
  1753
  also have "(\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k)) = (p + (1 - p)) ^ n"
lp15@61609
  1754
    by (subst binomial_ring) (simp add: atLeast0AtMost)
hoelzl@62975
  1755
  finally show "(\<integral>\<^sup>+ x. ennreal (real (n choose x) * p ^ x * (1 - p) ^ (n - x)) \<partial>count_space UNIV) = 1"
hoelzl@59093
  1756
    by simp
hoelzl@59093
  1757
qed (insert p_nonneg p_le_1, simp)
hoelzl@59093
  1758
hoelzl@59093
  1759
lemma pmf_binomial[simp]: "pmf binomial_pmf k = (n choose k) * p^k * (1 - p)^(n - k)"
hoelzl@59093
  1760
  by transfer rule
hoelzl@59093
  1761
hoelzl@59093
  1762
lemma set_pmf_binomial_eq: "set_pmf binomial_pmf = (if p = 0 then {0} else if p = 1 then {n} else {.. n})"
hoelzl@59093
  1763
  using p_nonneg p_le_1 unfolding set_eq_iff set_pmf_iff pmf_binomial by (auto simp: set_pmf_iff)
hoelzl@59093
  1764
hoelzl@59093
  1765
end
hoelzl@59093
  1766
hoelzl@59093
  1767
end
hoelzl@59093
  1768
hoelzl@59093
  1769
lemma set_pmf_binomial_0[simp]: "set_pmf (binomial_pmf n 0) = {0}"
hoelzl@59093
  1770
  by (simp add: set_pmf_binomial_eq)
hoelzl@59093
  1771
hoelzl@59093
  1772
lemma set_pmf_binomial_1[simp]: "set_pmf (binomial_pmf n 1) = {n}"
hoelzl@59093
  1773
  by (simp add: set_pmf_binomial_eq)
hoelzl@59093
  1774
hoelzl@59093
  1775
lemma set_pmf_binomial[simp]: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (binomial_pmf n p) = {..n}"
hoelzl@59093
  1776
  by (simp add: set_pmf_binomial_eq)
hoelzl@59093
  1777
wenzelm@63343
  1778
context includes lifting_syntax
wenzelm@63343
  1779
begin
Andreas@61634
  1780
Andreas@61634
  1781
lemma bind_pmf_parametric [transfer_rule]:
Andreas@61634
  1782
  "(rel_pmf A ===> (A ===> rel_pmf B) ===> rel_pmf B) bind_pmf bind_pmf"
Andreas@61634
  1783
by(blast intro: rel_pmf_bindI dest: rel_funD)
Andreas@61634
  1784
Andreas@61634
  1785
lemma return_pmf_parametric [transfer_rule]: "(A ===> rel_pmf A) return_pmf return_pmf"
Andreas@61634
  1786
by(rule rel_funI) simp
Andreas@61634
  1787
hoelzl@59000
  1788
end
Andreas@61634
  1789
eberlm@63099
  1790
eberlm@63194
  1791
primrec replicate_pmf :: "nat \<Rightarrow> 'a pmf \<Rightarrow> 'a list pmf" where
eberlm@63194
  1792
  "replicate_pmf 0 _ = return_pmf []"
eberlm@63194
  1793
| "replicate_pmf (Suc n) p = do {x \<leftarrow> p; xs \<leftarrow> replicate_pmf n p; return_pmf (x#xs)}"
eberlm@63194
  1794
eberlm@63194
  1795
lemma replicate_pmf_1: "replicate_pmf 1 p = map_pmf (\<lambda>x. [x]) p"
eberlm@63194
  1796
  by (simp add: map_pmf_def bind_return_pmf)
eberlm@63194
  1797
  
eberlm@63194
  1798
lemma set_replicate_pmf: 
eberlm@63194
  1799
  "set_pmf (replicate_pmf n p) = {xs\<in>lists (set_pmf p). length xs = n}"
eberlm@63194
  1800
  by (induction n) (auto simp: length_Suc_conv)
eberlm@63194
  1801
eberlm@63194
  1802
lemma replicate_pmf_distrib:
eberlm@63194
  1803
  "replicate_pmf (m + n) p = 
eberlm@63194
  1804
     do {xs \<leftarrow> replicate_pmf m p; ys \<leftarrow> replicate_pmf n p; return_pmf (xs @ ys)}"
eberlm@63194
  1805
  by (induction m) (simp_all add: bind_return_pmf bind_return_pmf' bind_assoc_pmf)
eberlm@63194
  1806
eberlm@63194
  1807
lemma power_diff': 
eberlm@63194
  1808
  assumes "b \<le> a"
eberlm@63194
  1809
  shows   "x ^ (a - b) = (if x = 0 \<and> a = b then 1 else x ^ a / (x::'a::field) ^ b)"
eberlm@63194
  1810
proof (cases "x = 0")
eberlm@63194
  1811
  case True
eberlm@63194
  1812
  with assms show ?thesis by (cases "a - b") simp_all
eberlm@63194
  1813
qed (insert assms, simp_all add: power_diff)
eberlm@63194
  1814
eberlm@63194
  1815
  
eberlm@63194
  1816
lemma binomial_pmf_Suc:
eberlm@63194
  1817
  assumes "p \<in> {0..1}"
eberlm@63194
  1818
  shows   "binomial_pmf (Suc n) p = 
eberlm@63194
  1819
             do {b \<leftarrow> bernoulli_pmf p; 
eberlm@63194
  1820
                 k \<leftarrow> binomial_pmf n p; 
eberlm@63194
  1821
                 return_pmf ((if b then 1 else 0) + k)}" (is "_ = ?rhs")
eberlm@63194
  1822
proof (intro pmf_eqI)
eberlm@63194
  1823
  fix k
eberlm@63194
  1824
  have A: "indicator {Suc a} (Suc b) = indicator {a} b" for a b
eberlm@63194
  1825
    by (simp add: indicator_def)
eberlm@63194
  1826
  show "pmf (binomial_pmf (Suc n) p) k = pmf ?rhs k"
eberlm@63194
  1827
    by (cases k; cases "k > n")
eberlm@63194
  1828
       (insert assms, auto simp: pmf_bind measure_pmf_single A divide_simps algebra_simps
eberlm@63194
  1829
          not_less less_eq_Suc_le [symmetric] power_diff')
eberlm@63194
  1830
qed
eberlm@63194
  1831
eberlm@63194
  1832
lemma binomial_pmf_0: "p \<in> {0..1} \<Longrightarrow> binomial_pmf 0 p = return_pmf 0"
eberlm@63194
  1833
  by (rule pmf_eqI) (simp_all add: indicator_def)
eberlm@63194
  1834
eberlm@63194
  1835
lemma binomial_pmf_altdef:
eberlm@63194
  1836
  assumes "p \<in> {0..1}"
eberlm@63194
  1837
  shows   "binomial_pmf n p = map_pmf (length \<circ> filter id) (replicate_pmf n (bernoulli_pmf p))"
eberlm@63194
  1838
  by (induction n) 
eberlm@63194
  1839
     (insert assms, auto simp: binomial_pmf_Suc map_pmf_def bind_return_pmf bind_assoc_pmf 
eberlm@63194
  1840
        bind_return_pmf' binomial_pmf_0 intro!: bind_pmf_cong)
eberlm@63194
  1841
eberlm@63194
  1842
eberlm@63099
  1843
subsection \<open>PMFs from assiciation lists\<close>
eberlm@63099
  1844
eberlm@63099
  1845
definition pmf_of_list ::" ('a \<times> real) list \<Rightarrow> 'a pmf" where 
nipkow@63882
  1846
  "pmf_of_list xs = embed_pmf (\<lambda>x. sum_list (map snd (filter (\<lambda>z. fst z = x) xs)))"
eberlm@63099
  1847
eberlm@63099
  1848
definition pmf_of_list_wf where
nipkow@63882
  1849
  "pmf_of_list_wf xs \<longleftrightarrow> (\<forall>x\<in>set (map snd xs) . x \<ge> 0) \<and> sum_list (map snd xs) = 1"
eberlm@63099
  1850
eberlm@63099
  1851
lemma pmf_of_list_wfI:
nipkow@63882
  1852
  "(\<And>x. x \<in> set (map snd xs) \<Longrightarrow> x \<ge> 0) \<Longrightarrow> sum_list (map snd xs) = 1 \<Longrightarrow> pmf_of_list_wf xs"
eberlm@63099
  1853
  unfolding pmf_of_list_wf_def by simp
eberlm@63099
  1854
eberlm@63099
  1855
context
eberlm@63099
  1856
begin
eberlm@63099
  1857
eberlm@63099
  1858
private lemma pmf_of_list_aux:
eberlm@63099
  1859
  assumes "\<And>x. x \<in> set (map snd xs) \<Longrightarrow> x \<ge> 0"
nipkow@63882
  1860
  assumes "sum_list (map snd xs) = 1"
nipkow@63882
  1861
  shows "(\<integral>\<^sup>+ x. ennreal (sum_list (map snd [z\<leftarrow>xs . fst z = x])) \<partial>count_space UNIV) = 1"
eberlm@63099
  1862
proof -
nipkow@63882
  1863
  have "(\<integral>\<^sup>+ x. ennreal (sum_list (map snd (filter (\<lambda>z. fst z = x) xs))) \<partial>count_space UNIV) =
nipkow@63882
  1864
            (\<integral>\<^sup>+ x. ennreal (sum_list (map (\<lambda>(x',p). indicator {x'} x * p) xs)) \<partial>count_space UNIV)"
nipkow@63882
  1865
    by (intro nn_integral_cong ennreal_cong, subst sum_list_map_filter') (auto intro: sum_list_cong)
eberlm@63099
  1866
  also have "\<dots> = (\<Sum>(x',p)\<leftarrow>xs. (\<integral>\<^sup>+ x. ennreal (indicator {x'} x * p) \<partial>count_space UNIV))"
eberlm@63099
  1867
    using assms(1)
eberlm@63099
  1868
  proof (induction xs)
eberlm@63099
  1869
    case (Cons x xs)
eberlm@63099
  1870
    from Cons.prems have "snd x \<ge> 0" by simp
eberlm@63099
  1871
    moreover have "b \<ge> 0" if "(a,b) \<in> set xs" for a b
eberlm@63099
  1872
      using Cons.prems[of b] that by force
eberlm@63099
  1873
    ultimately have "(\<integral>\<^sup>+ y. ennreal (\<Sum>(x', p)\<leftarrow>x # xs. indicator {x'} y * p) \<partial>count_space UNIV) = 
eberlm@63099
  1874
            (\<integral>\<^sup>+ y. ennreal (indicator {fst x} y * snd x) + 
eberlm@63099
  1875
            ennreal (\<Sum>(x', p)\<leftarrow>xs. indicator {x'} y * p) \<partial>count_space UNIV)"
eberlm@63099
  1876
      by (intro nn_integral_cong, subst ennreal_plus [symmetric]) 
nipkow@63882
  1877
         (auto simp: case_prod_unfold indicator_def intro!: sum_list_nonneg)
eberlm@63099
  1878
    also have "\<dots> = (\<integral>\<^sup>+ y. ennreal (indicator {fst x} y * snd x) \<partial>count_space UNIV) + 
eberlm@63099
  1879
                      (\<integral>\<^sup>+ y. ennreal (\<Sum>(x', p)\<leftarrow>xs. indicator {x'} y * p) \<partial>count_space UNIV)"
eberlm@63099
  1880
      by (intro nn_integral_add)
nipkow@63882
  1881
         (force intro!: sum_list_nonneg AE_I2 intro: Cons simp: indicator_def)+
eberlm@63099
  1882
    also have "(\<integral>\<^sup>+ y. ennreal (\<Sum>(x', p)\<leftarrow>xs. indicator {x'} y * p) \<partial>count_space UNIV) =
eberlm@63099
  1883
               (\<Sum>(x', p)\<leftarrow>xs. (\<integral>\<^sup>+ y. ennreal (indicator {x'} y * p) \<partial>count_space UNIV))"
eberlm@63099
  1884
      using Cons(1) by (intro Cons) simp_all
eberlm@63099
  1885
    finally show ?case by (simp add: case_prod_unfold)
eberlm@63099
  1886
  qed simp
eberlm@63099
  1887
  also have "\<dots> = (\<Sum>(x',p)\<leftarrow>xs. ennreal p * (\<integral>\<^sup>+ x. indicator {x'} x \<partial>count_space UNIV))"
eberlm@63099
  1888
    using assms(1)
nipkow@63882
  1889
    by (intro sum_list_cong, simp only: case_prod_unfold, subst nn_integral_cmult [symmetric])
eberlm@63099
  1890
       (auto intro!: assms(1) simp: max_def times_ereal.simps [symmetric] mult_ac ereal_indicator
eberlm@63099
  1891
             simp del: times_ereal.simps)+
nipkow@63882
  1892
  also from assms have "\<dots> = sum_list (map snd xs)" by (simp add: case_prod_unfold sum_list_ennreal)
eberlm@63099
  1893
  also have "\<dots> = 1" using assms(2) by simp
eberlm@63099
  1894
  finally show ?thesis .
eberlm@63099
  1895
qed
eberlm@63099
  1896
eberlm@63099
  1897
lemma pmf_pmf_of_list:
eberlm@63099
  1898
  assumes "pmf_of_list_wf xs"
nipkow@63882
  1899
  shows   "pmf (pmf_of_list xs) x = sum_list (map snd (filter (\<lambda>z. fst z = x) xs))"
eberlm@63099
  1900
  using assms pmf_of_list_aux[of xs] unfolding pmf_of_list_def pmf_of_list_wf_def
nipkow@63882
  1901
  by (subst pmf_embed_pmf) (auto intro!: sum_list_nonneg)
eberlm@63099
  1902
Andreas@61634
  1903
end
eberlm@63099
  1904
eberlm@63099
  1905
lemma set_pmf_of_list:
eberlm@63099
  1906
  assumes "pmf_of_list_wf xs"
eberlm@63099
  1907
  shows   "set_pmf (pmf_of_list xs) \<subseteq> set (map fst xs)"
eberlm@63099
  1908
proof clarify
eberlm@63099
  1909
  fix x assume A: "x \<in> set_pmf (pmf_of_list xs)"
eberlm@63099
  1910
  show "x \<in> set (map fst xs)"
eberlm@63099
  1911
  proof (rule ccontr)
eberlm@63099
  1912
    assume "x \<notin> set (map fst xs)"
eberlm@63099
  1913
    hence "[z\<leftarrow>xs . fst z = x] = []" by (auto simp: filter_empty_conv)
eberlm@63099
  1914
    with A assms show False by (simp add: pmf_pmf_of_list set_pmf_eq)
eberlm@63099
  1915
  qed
eberlm@63099
  1916
qed
eberlm@63099
  1917
eberlm@63099
  1918
lemma finite_set_pmf_of_list:
eberlm@63099
  1919
  assumes "pmf_of_list_wf xs"
eberlm@63099
  1920
  shows   "finite (set_pmf (pmf_of_list xs))"
eberlm@63099
  1921
  using assms by (rule finite_subset[OF set_pmf_of_list]) simp_all
eberlm@63099
  1922
eberlm@63099
  1923
lemma emeasure_Int_set_pmf:
eberlm@63099
  1924
  "emeasure (measure_pmf p) (A \<inter> set_pmf p) = emeasure (measure_pmf p) A"
eberlm@63099
  1925
  by (rule emeasure_eq_AE) (auto simp: AE_measure_pmf_iff)
eberlm@63099
  1926
eberlm@63099
  1927
lemma measure_Int_set_pmf:
eberlm@63099
  1928
  "measure (measure_pmf p) (A \<inter> set_pmf p) = measure (measure_pmf p) A"
eberlm@63099
  1929
  using emeasure_Int_set_pmf[of p A] by (simp add: Sigma_Algebra.measure_def)
eberlm@63099
  1930
eberlm@63099
  1931
lemma emeasure_pmf_of_list:
eberlm@63099
  1932
  assumes "pmf_of_list_wf xs"
nipkow@63882
  1933
  shows   "emeasure (pmf_of_list xs) A = ennreal (sum_list (map snd (filter (\<lambda>x. fst x \<in> A) xs)))"
eberlm@63099
  1934
proof -
eberlm@63099
  1935
  have "emeasure (pmf_of_list xs) A = nn_integral (measure_pmf (pmf_of_list xs)) (indicator A)"
eberlm@63099
  1936
    by simp
eberlm@63099
  1937
  also from assms 
nipkow@63882
  1938
    have "\<dots> = (\<Sum>x\<in>set_pmf (pmf_of_list xs) \<inter> A. ennreal (sum_list (map snd [z\<leftarrow>xs . fst z = x])))"
eberlm@63099
  1939
    by (subst nn_integral_measure_pmf_finite) (simp_all add: finite_set_pmf_of_list pmf_pmf_of_list)
eberlm@63099
  1940
  also from assms 
nipkow@63882
  1941
    have "\<dots> = ennreal (\<Sum>x\<in>set_pmf (pmf_of_list xs) \<inter> A. sum_list (map snd [z\<leftarrow>xs . fst z = x]))"
nipkow@63882
  1942
    by (subst setsum_ennreal) (auto simp: pmf_of_list_wf_def intro!: sum_list_nonneg)
eberlm@63099
  1943
  also have "\<dots> = ennreal (\<Sum>x\<in>set_pmf (pmf_of_list xs) \<inter> A. 
eberlm@63099
  1944
      indicator A x * pmf (pmf_of_list xs) x)" (is "_ = ennreal ?S")
eberlm@63099
  1945
    using assms by (intro ennreal_cong setsum.cong) (auto simp: pmf_pmf_of_list)
eberlm@63099
  1946
  also have "?S = (\<Sum>x\<in>set_pmf (pmf_of_list xs). indicator A x * pmf (pmf_of_list xs) x)"
eberlm@63099
  1947
    using assms by (intro setsum.mono_neutral_left set_pmf_of_list finite_set_pmf_of_list) auto
eberlm@63099
  1948
  also have "\<dots> = (\<Sum>x\<in>set (map fst xs). indicator A x * pmf (pmf_of_list xs) x)"
eberlm@63099
  1949
    using assms by (intro setsum.mono_neutral_left set_pmf_of_list) (auto simp: set_pmf_eq)
eberlm@63099
  1950
  also have "\<dots> = (\<Sum>x\<in>set (map fst xs). indicator A x * 
nipkow@63882
  1951
                      sum_list (map snd (filter (\<lambda>z. fst z = x) xs)))"
eberlm@63099
  1952
    using assms by (simp add: pmf_pmf_of_list)
nipkow@63882
  1953
  also have "\<dots> = (\<Sum>x\<in>set (map fst xs). sum_list (map snd (filter (\<lambda>z. fst z = x \<and> x \<in> A) xs)))"
eberlm@63099
  1954
    by (intro setsum.cong) (auto simp: indicator_def)
eberlm@63099
  1955
  also have "\<dots> = (\<Sum>x\<in>set (map fst xs). (\<Sum>xa = 0..<length xs.
eberlm@63099
  1956
                     if fst (xs ! xa) = x \<and> x \<in> A then snd (xs ! xa) else 0))"
nipkow@63882
  1957
    by (intro setsum.cong refl, subst sum_list_map_filter', subst sum_list_setsum_nth) simp
eberlm@63099
  1958
  also have "\<dots> = (\<Sum>xa = 0..<length xs. (\<Sum>x\<in>set (map fst xs). 
eberlm@63099
  1959
                     if fst (xs ! xa) = x \<and> x \<in> A then snd (xs ! xa) else 0))"
eberlm@63099
  1960
    by (rule setsum.commute)
eberlm@63099
  1961
  also have "\<dots> = (\<Sum>xa = 0..<length xs. if fst (xs ! xa) \<in> A then 
eberlm@63099
  1962
                     (\<Sum>x\<in>set (map fst xs). if x = fst (xs ! xa) then snd (xs ! xa) else 0) else 0)"
eberlm@63099
  1963
    by (auto intro!: setsum.cong setsum.neutral)
eberlm@63099
  1964
  also have "\<dots> = (\<Sum>xa = 0..<length xs. if fst (xs ! xa) \<in> A then snd (xs ! xa) else 0)"
eberlm@63099
  1965
    by (intro setsum.cong refl) (simp_all add: setsum.delta)
nipkow@63882
  1966
  also have "\<dots> = sum_list (map snd (filter (\<lambda>x. fst x \<in> A) xs))"
nipkow@63882
  1967
    by (subst sum_list_map_filter', subst sum_list_setsum_nth) simp_all
eberlm@63099
  1968
  finally show ?thesis . 
eberlm@63099
  1969
qed
eberlm@63099
  1970
eberlm@63099
  1971
lemma measure_pmf_of_list:
eberlm@63099
  1972
  assumes "pmf_of_list_wf xs"
nipkow@63882
  1973
  shows   "measure (pmf_of_list xs) A = sum_list (map snd (filter (\<lambda>x. fst x \<in> A) xs))"
eberlm@63099
  1974
  using assms unfolding pmf_of_list_wf_def Sigma_Algebra.measure_def
nipkow@63882
  1975
  by (subst emeasure_pmf_of_list [OF assms], subst enn2real_ennreal) (auto intro!: sum_list_nonneg)
eberlm@63099
  1976
eberlm@63194
  1977
(* TODO Move? *)
nipkow@63882
  1978
lemma sum_list_nonneg_eq_zero_iff:
eberlm@63194
  1979
  fixes xs :: "'a :: linordered_ab_group_add list"
nipkow@63882
  1980
  shows "(\<And>x. x \<in> set xs \<Longrightarrow> x \<ge> 0) \<Longrightarrow> sum_list xs = 0 \<longleftrightarrow> set xs \<subseteq> {0}"
eberlm@63194
  1981
proof (induction xs)
eberlm@63194
  1982
  case (Cons x xs)
nipkow@63882
  1983
  from Cons.prems have "sum_list (x#xs) = 0 \<longleftrightarrow> x = 0 \<and> sum_list xs = 0"
nipkow@63882
  1984
    unfolding sum_list_simps by (subst add_nonneg_eq_0_iff) (auto intro: sum_list_nonneg)
eberlm@63194
  1985
  with Cons.IH Cons.prems show ?case by simp
eberlm@63194
  1986
qed simp_all
eberlm@63194
  1987
nipkow@63882
  1988
lemma sum_list_filter_nonzero:
nipkow@63882
  1989
  "sum_list (filter (\<lambda>x. x \<noteq> 0) xs) = sum_list xs"
eberlm@63194
  1990
  by (induction xs) simp_all
eberlm@63194
  1991
(* END MOVE *)
eberlm@63194
  1992
  
eberlm@63194
  1993
lemma set_pmf_of_list_eq:
eberlm@63194
  1994
  assumes "pmf_of_list_wf xs" "\<And>x. x \<in> snd ` set xs \<Longrightarrow> x > 0"
eberlm@63194
  1995
  shows   "set_pmf (pmf_of_list xs) = fst ` set xs"
eberlm@63194
  1996
proof
eberlm@63194
  1997
  {
eberlm@63194
  1998
    fix x assume A: "x \<in> fst ` set xs" and B: "x \<notin> set_pmf (pmf_of_list xs)"
eberlm@63194
  1999
    then obtain y where y: "(x, y) \<in> set xs" by auto
nipkow@63882
  2000
    from B have "sum_list (map snd [z\<leftarrow>xs. fst z = x]) = 0"
eberlm@63194
  2001
      by (simp add: pmf_pmf_of_list[OF assms(1)] set_pmf_eq)
eberlm@63194
  2002
    moreover from y have "y \<in> snd ` {xa \<in> set xs. fst xa = x}" by force
eberlm@63194
  2003
    ultimately have "y = 0" using assms(1) 
nipkow@63882
  2004
      by (subst (asm) sum_list_nonneg_eq_zero_iff) (auto simp: pmf_of_list_wf_def)
eberlm@63194
  2005
    with assms(2) y have False by force
eberlm@63194
  2006
  }
eberlm@63194
  2007
  thus "fst ` set xs \<subseteq> set_pmf (pmf_of_list xs)" by blast
eberlm@63194
  2008
qed (insert set_pmf_of_list[OF assms(1)], simp_all)
eberlm@63194
  2009
  
eberlm@63194
  2010
lemma pmf_of_list_remove_zeros:
eberlm@63194
  2011
  assumes "pmf_of_list_wf xs"
eberlm@63194
  2012
  defines "xs' \<equiv> filter (\<lambda>z. snd z \<noteq> 0) xs"
eberlm@63194
  2013
  shows   "pmf_of_list_wf xs'" "pmf_of_list xs' = pmf_of_list xs"
eberlm@63194
  2014
proof -
eberlm@63194
  2015
  have "map snd [z\<leftarrow>xs . snd z \<noteq> 0] = filter (\<lambda>x. x \<noteq> 0) (map snd xs)"
eberlm@63194
  2016
    by (induction xs) simp_all
eberlm@63194
  2017
  with assms(1) show wf: "pmf_of_list_wf xs'"
nipkow@63882
  2018
    by (auto simp: pmf_of_list_wf_def xs'_def sum_list_filter_nonzero)
nipkow@63882
  2019
  have "sum_list (map snd [z\<leftarrow>xs' . fst z = i]) = sum_list (map snd [z\<leftarrow>xs . fst z = i])" for i
eberlm@63194
  2020
    unfolding xs'_def by (induction xs) simp_all
eberlm@63194
  2021
  with assms(1) wf show "pmf_of_list xs' = pmf_of_list xs"
eberlm@63194
  2022
    by (intro pmf_eqI) (simp_all add: pmf_pmf_of_list)
eberlm@63194
  2023
qed
eberlm@63194
  2024
eberlm@63099
  2025
end