src/HOL/Probability/Probability_Mass_Function.thy
author hoelzl
Fri Feb 19 13:40:50 2016 +0100 (2016-02-19)
changeset 62378 85ed00c1fe7c
parent 62324 ae44f16dcea5
child 62390 842917225d56
permissions -rw-r--r--
generalize more theorems to support enat and ennreal
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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: emeasure_le_0_iff) }
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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 ereal_divide': "b \<noteq> 0 \<Longrightarrow> ereal (a / b) = ereal a / ereal b"
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  using ereal_divide[of a b] by simp
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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 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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  by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_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 measure_nonneg)
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lemma pmf_nonneg: "0 \<le> pmf p x"
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  by transfer (simp add: measure_nonneg)
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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 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 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)
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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)
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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] by(simp add: measure_pmf.emeasure_eq_measure)
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lemma nn_integral_measure_pmf_support:
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  fixes f :: "'a \<Rightarrow> ereal"
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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> ereal"
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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)
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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) (auto simp: measure_def emeasure_measure_pmf_finite)
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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)
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  then have "integrable (count_space X) (pmf M) = integrable (count_space (M \<inter> X)) (pmf M)"
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    by (simp add: integrable_iff_bounded pmf_nonneg)
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  then show ?thesis
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    by (simp add: pmf.rep_eq measure_pmf.integrable_measure disjoint_family_on_def)
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qed
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lemma integral_pmf: "(\<integral>x. pmf M x \<partial>count_space X) = measure M X"
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proof -
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  have "(\<integral>x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+x. pmf M x \<partial>count_space X)"
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    by (simp add: pmf_nonneg integrable_pmf nn_integral_eq_integral)
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  also have "\<dots> = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space (X \<inter> M))"
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    by (auto intro!: nn_integral_cong_AE split: split_indicator
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             simp: pmf.rep_eq measure_pmf.emeasure_eq_measure nn_integral_count_space_indicator
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                   AE_count_space set_pmf_iff)
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  also have "\<dots> = emeasure M (X \<inter> M)"
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    by (rule emeasure_countable_singleton[symmetric]) (auto intro: countable_set_pmf)
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  also have "\<dots> = emeasure M X"
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    by (auto intro!: emeasure_eq_AE simp: AE_measure_pmf_iff)
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  finally show ?thesis
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    by (simp add: measure_pmf.emeasure_eq_measure)
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qed
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lemma integral_pmf_restrict:
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   261
  "(f::'a \<Rightarrow> 'b::{banach, second_countable_topology}) \<in> borel_measurable (count_space UNIV) \<Longrightarrow>
hoelzl@59000
   262
    (\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x \<partial>restrict_space M M)"
hoelzl@59000
   263
  by (auto intro!: integral_cong_AE simp add: integral_restrict_space AE_measure_pmf_iff)
hoelzl@59000
   264
hoelzl@58587
   265
lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1"
hoelzl@58587
   266
proof -
hoelzl@58587
   267
  have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)"
hoelzl@58587
   268
    by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf)
hoelzl@58587
   269
  then show ?thesis
hoelzl@58587
   270
    using measure_pmf.emeasure_space_1 by simp
hoelzl@58587
   271
qed
hoelzl@58587
   272
Andreas@59490
   273
lemma emeasure_pmf_UNIV [simp]: "emeasure (measure_pmf M) UNIV = 1"
Andreas@59490
   274
using measure_pmf.emeasure_space_1[of M] by simp
Andreas@59490
   275
Andreas@59023
   276
lemma in_null_sets_measure_pmfI:
Andreas@59023
   277
  "A \<inter> set_pmf p = {} \<Longrightarrow> A \<in> null_sets (measure_pmf p)"
Andreas@59023
   278
using emeasure_eq_0_AE[where ?P="\<lambda>x. x \<in> A" and M="measure_pmf p"]
Andreas@59023
   279
by(auto simp add: null_sets_def AE_measure_pmf_iff)
Andreas@59023
   280
hoelzl@59664
   281
lemma measure_subprob: "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
hoelzl@59664
   282
  by (simp add: space_subprob_algebra subprob_space_measure_pmf)
hoelzl@59664
   283
hoelzl@59664
   284
subsection \<open> Monad Interpretation \<close>
hoelzl@59664
   285
hoelzl@59664
   286
lemma measurable_measure_pmf[measurable]:
hoelzl@59664
   287
  "(\<lambda>x. measure_pmf (M x)) \<in> measurable (count_space UNIV) (subprob_algebra (count_space UNIV))"
hoelzl@59664
   288
  by (auto simp: space_subprob_algebra intro!: prob_space_imp_subprob_space) unfold_locales
hoelzl@59664
   289
hoelzl@59664
   290
lemma bind_measure_pmf_cong:
hoelzl@59664
   291
  assumes "\<And>x. A x \<in> space (subprob_algebra N)" "\<And>x. B x \<in> space (subprob_algebra N)"
hoelzl@59664
   292
  assumes "\<And>i. i \<in> set_pmf x \<Longrightarrow> A i = B i"
hoelzl@59664
   293
  shows "bind (measure_pmf x) A = bind (measure_pmf x) B"
hoelzl@59664
   294
proof (rule measure_eqI)
wenzelm@62026
   295
  show "sets (measure_pmf x \<bind> A) = sets (measure_pmf x \<bind> B)"
hoelzl@59664
   296
    using assms by (subst (1 2) sets_bind) (auto simp: space_subprob_algebra)
hoelzl@59664
   297
next
wenzelm@62026
   298
  fix X assume "X \<in> sets (measure_pmf x \<bind> A)"
hoelzl@59664
   299
  then have X: "X \<in> sets N"
hoelzl@59664
   300
    using assms by (subst (asm) sets_bind) (auto simp: space_subprob_algebra)
wenzelm@62026
   301
  show "emeasure (measure_pmf x \<bind> A) X = emeasure (measure_pmf x \<bind> B) X"
hoelzl@59664
   302
    using assms
hoelzl@59664
   303
    by (subst (1 2) emeasure_bind[where N=N, OF _ _ X])
hoelzl@59664
   304
       (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
hoelzl@59664
   305
qed
hoelzl@59664
   306
hoelzl@59664
   307
lift_definition bind_pmf :: "'a pmf \<Rightarrow> ('a \<Rightarrow> 'b pmf ) \<Rightarrow> 'b pmf" is bind
hoelzl@59664
   308
proof (clarify, intro conjI)
hoelzl@59664
   309
  fix f :: "'a measure" and g :: "'a \<Rightarrow> 'b measure"
hoelzl@59664
   310
  assume "prob_space f"
hoelzl@59664
   311
  then interpret f: prob_space f .
hoelzl@59664
   312
  assume "sets f = UNIV" and ae_f: "AE x in f. measure f {x} \<noteq> 0"
hoelzl@59664
   313
  then have s_f[simp]: "sets f = sets (count_space UNIV)"
hoelzl@59664
   314
    by simp
hoelzl@59664
   315
  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
   316
  then have g: "\<And>x. prob_space (g x)" and s_g[simp]: "\<And>x. sets (g x) = sets (count_space UNIV)"
hoelzl@59664
   317
    and ae_g: "\<And>x. AE y in g x. measure (g x) {y} \<noteq> 0"
hoelzl@59664
   318
    by auto
hoelzl@59664
   319
hoelzl@59664
   320
  have [measurable]: "g \<in> measurable f (subprob_algebra (count_space UNIV))"
hoelzl@59664
   321
    by (auto simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space g)
lp15@59667
   322
wenzelm@62026
   323
  show "prob_space (f \<bind> g)"
hoelzl@59664
   324
    using g by (intro f.prob_space_bind[where S="count_space UNIV"]) auto
wenzelm@62026
   325
  then interpret fg: prob_space "f \<bind> g" .
wenzelm@62026
   326
  show [simp]: "sets (f \<bind> g) = UNIV"
hoelzl@59664
   327
    using sets_eq_imp_space_eq[OF s_f]
hoelzl@59664
   328
    by (subst sets_bind[where N="count_space UNIV"]) auto
wenzelm@62026
   329
  show "AE x in f \<bind> g. measure (f \<bind> g) {x} \<noteq> 0"
hoelzl@59664
   330
    apply (simp add: fg.prob_eq_0 AE_bind[where B="count_space UNIV"])
hoelzl@59664
   331
    using ae_f
hoelzl@59664
   332
    apply eventually_elim
hoelzl@59664
   333
    using ae_g
hoelzl@59664
   334
    apply eventually_elim
hoelzl@59664
   335
    apply (auto dest: AE_measure_singleton)
hoelzl@59664
   336
    done
hoelzl@59664
   337
qed
hoelzl@59664
   338
hoelzl@59664
   339
lemma ereal_pmf_bind: "pmf (bind_pmf N f) i = (\<integral>\<^sup>+x. pmf (f x) i \<partial>measure_pmf N)"
hoelzl@59664
   340
  unfolding pmf.rep_eq bind_pmf.rep_eq
hoelzl@59664
   341
  by (auto simp: measure_pmf.measure_bind[where N="count_space UNIV"] measure_subprob measure_nonneg
hoelzl@59664
   342
           intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])
hoelzl@59664
   343
hoelzl@59664
   344
lemma pmf_bind: "pmf (bind_pmf N f) i = (\<integral>x. pmf (f x) i \<partial>measure_pmf N)"
hoelzl@59664
   345
  using ereal_pmf_bind[of N f i]
hoelzl@59664
   346
  by (subst (asm) nn_integral_eq_integral)
hoelzl@59664
   347
     (auto simp: pmf_nonneg pmf_le_1
hoelzl@59664
   348
           intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])
hoelzl@59664
   349
hoelzl@59664
   350
lemma bind_pmf_const[simp]: "bind_pmf M (\<lambda>x. c) = c"
hoelzl@59664
   351
  by transfer (simp add: bind_const' prob_space_imp_subprob_space)
hoelzl@59664
   352
hoelzl@59665
   353
lemma set_bind_pmf[simp]: "set_pmf (bind_pmf M N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))"
lp15@59667
   354
  unfolding set_pmf_eq ereal_eq_0(1)[symmetric] ereal_pmf_bind
hoelzl@59664
   355
  by (auto simp add: nn_integral_0_iff_AE AE_measure_pmf_iff set_pmf_eq not_le less_le pmf_nonneg)
hoelzl@59664
   356
hoelzl@59664
   357
lemma bind_pmf_cong:
hoelzl@59664
   358
  assumes "p = q"
hoelzl@59664
   359
  shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g"
wenzelm@61808
   360
  unfolding \<open>p = q\<close>[symmetric] measure_pmf_inject[symmetric] bind_pmf.rep_eq
hoelzl@59664
   361
  by (auto simp: AE_measure_pmf_iff Pi_iff space_subprob_algebra subprob_space_measure_pmf
hoelzl@59664
   362
                 sets_bind[where N="count_space UNIV"] emeasure_bind[where N="count_space UNIV"]
hoelzl@59664
   363
           intro!: nn_integral_cong_AE measure_eqI)
hoelzl@59664
   364
hoelzl@59664
   365
lemma bind_pmf_cong_simp:
hoelzl@59664
   366
  "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
   367
  by (simp add: simp_implies_def cong: bind_pmf_cong)
hoelzl@59664
   368
wenzelm@62026
   369
lemma measure_pmf_bind: "measure_pmf (bind_pmf M f) = (measure_pmf M \<bind> (\<lambda>x. measure_pmf (f x)))"
hoelzl@59664
   370
  by transfer simp
hoelzl@59664
   371
hoelzl@59664
   372
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
   373
  using measurable_measure_pmf[of N]
hoelzl@59664
   374
  unfolding measure_pmf_bind
hoelzl@59664
   375
  apply (subst (1 3) nn_integral_max_0[symmetric])
hoelzl@59664
   376
  apply (intro nn_integral_bind[where B="count_space UNIV"])
hoelzl@59664
   377
  apply auto
hoelzl@59664
   378
  done
hoelzl@59664
   379
hoelzl@59664
   380
lemma emeasure_bind_pmf[simp]: "emeasure (bind_pmf M N) X = (\<integral>\<^sup>+x. emeasure (N x) X \<partial>M)"
hoelzl@59664
   381
  using measurable_measure_pmf[of N]
hoelzl@59664
   382
  unfolding measure_pmf_bind
hoelzl@59664
   383
  by (subst emeasure_bind[where N="count_space UNIV"]) auto
lp15@59667
   384
hoelzl@59664
   385
lift_definition return_pmf :: "'a \<Rightarrow> 'a pmf" is "return (count_space UNIV)"
hoelzl@59664
   386
  by (auto intro!: prob_space_return simp: AE_return measure_return)
hoelzl@59664
   387
hoelzl@59664
   388
lemma bind_return_pmf: "bind_pmf (return_pmf x) f = f x"
hoelzl@59664
   389
  by transfer
hoelzl@59664
   390
     (auto intro!: prob_space_imp_subprob_space bind_return[where N="count_space UNIV"]
hoelzl@59664
   391
           simp: space_subprob_algebra)
hoelzl@59664
   392
hoelzl@59665
   393
lemma set_return_pmf[simp]: "set_pmf (return_pmf x) = {x}"
hoelzl@59664
   394
  by transfer (auto simp add: measure_return split: split_indicator)
hoelzl@59664
   395
hoelzl@59664
   396
lemma bind_return_pmf': "bind_pmf N return_pmf = N"
hoelzl@59664
   397
proof (transfer, clarify)
wenzelm@62026
   398
  fix N :: "'a measure" assume "sets N = UNIV" then show "N \<bind> return (count_space UNIV) = N"
hoelzl@59664
   399
    by (subst return_sets_cong[where N=N]) (simp_all add: bind_return')
hoelzl@59664
   400
qed
hoelzl@59664
   401
hoelzl@59664
   402
lemma bind_assoc_pmf: "bind_pmf (bind_pmf A B) C = bind_pmf A (\<lambda>x. bind_pmf (B x) C)"
hoelzl@59664
   403
  by transfer
hoelzl@59664
   404
     (auto intro!: bind_assoc[where N="count_space UNIV" and R="count_space UNIV"]
hoelzl@59664
   405
           simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space)
hoelzl@59664
   406
hoelzl@59664
   407
definition "map_pmf f M = bind_pmf M (\<lambda>x. return_pmf (f x))"
hoelzl@59664
   408
hoelzl@59664
   409
lemma map_bind_pmf: "map_pmf f (bind_pmf M g) = bind_pmf M (\<lambda>x. map_pmf f (g x))"
hoelzl@59664
   410
  by (simp add: map_pmf_def bind_assoc_pmf)
hoelzl@59664
   411
hoelzl@59664
   412
lemma bind_map_pmf: "bind_pmf (map_pmf f M) g = bind_pmf M (\<lambda>x. g (f x))"
hoelzl@59664
   413
  by (simp add: map_pmf_def bind_assoc_pmf bind_return_pmf)
hoelzl@59664
   414
hoelzl@59664
   415
lemma map_pmf_transfer[transfer_rule]:
hoelzl@59664
   416
  "rel_fun op = (rel_fun cr_pmf cr_pmf) (\<lambda>f M. distr M (count_space UNIV) f) map_pmf"
hoelzl@59664
   417
proof -
hoelzl@59664
   418
  have "rel_fun op = (rel_fun pmf_as_measure.cr_pmf pmf_as_measure.cr_pmf)
wenzelm@62026
   419
     (\<lambda>f M. M \<bind> (return (count_space UNIV) o f)) map_pmf"
lp15@59667
   420
    unfolding map_pmf_def[abs_def] comp_def by transfer_prover
hoelzl@59664
   421
  then show ?thesis
hoelzl@59664
   422
    by (force simp: rel_fun_def cr_pmf_def bind_return_distr)
hoelzl@59664
   423
qed
hoelzl@59664
   424
hoelzl@59664
   425
lemma map_pmf_rep_eq:
hoelzl@59664
   426
  "measure_pmf (map_pmf f M) = distr (measure_pmf M) (count_space UNIV) f"
hoelzl@59664
   427
  unfolding map_pmf_def bind_pmf.rep_eq comp_def return_pmf.rep_eq
hoelzl@59664
   428
  using bind_return_distr[of M f "count_space UNIV"] by (simp add: comp_def)
hoelzl@59664
   429
hoelzl@58587
   430
lemma map_pmf_id[simp]: "map_pmf id = id"
hoelzl@58587
   431
  by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI)
hoelzl@58587
   432
hoelzl@59053
   433
lemma map_pmf_ident[simp]: "map_pmf (\<lambda>x. x) = (\<lambda>x. x)"
hoelzl@59053
   434
  using map_pmf_id unfolding id_def .
hoelzl@59053
   435
hoelzl@58587
   436
lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g"
lp15@59667
   437
  by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def)
hoelzl@58587
   438
hoelzl@59000
   439
lemma map_pmf_comp: "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M"
hoelzl@59000
   440
  using map_pmf_compose[of f g] by (simp add: comp_def)
hoelzl@59000
   441
hoelzl@59664
   442
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
   443
  unfolding map_pmf_def by (rule bind_pmf_cong) auto
hoelzl@59664
   444
hoelzl@59664
   445
lemma pmf_set_map: "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
hoelzl@59665
   446
  by (auto simp add: comp_def fun_eq_iff map_pmf_def)
hoelzl@59664
   447
hoelzl@59665
   448
lemma set_map_pmf[simp]: "set_pmf (map_pmf f M) = f`set_pmf M"
hoelzl@59664
   449
  using pmf_set_map[of f] by (auto simp: comp_def fun_eq_iff)
hoelzl@58587
   450
hoelzl@59002
   451
lemma emeasure_map_pmf[simp]: "emeasure (map_pmf f M) X = emeasure M (f -` X)"
hoelzl@59664
   452
  unfolding map_pmf_rep_eq by (subst emeasure_distr) auto
hoelzl@59002
   453
Andreas@61634
   454
lemma measure_map_pmf[simp]: "measure (map_pmf f M) X = measure M (f -` X)"
Andreas@61634
   455
using emeasure_map_pmf[of f M X] by(simp add: measure_pmf.emeasure_eq_measure)
Andreas@61634
   456
hoelzl@59002
   457
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
   458
  unfolding map_pmf_rep_eq by (intro nn_integral_distr) auto
hoelzl@59002
   459
Andreas@59023
   460
lemma ereal_pmf_map: "pmf (map_pmf f p) x = (\<integral>\<^sup>+ y. indicator (f -` {x}) y \<partial>measure_pmf p)"
hoelzl@59664
   461
proof (transfer fixing: f x)
Andreas@59023
   462
  fix p :: "'b measure"
Andreas@59023
   463
  presume "prob_space p"
Andreas@59023
   464
  then interpret prob_space p .
Andreas@59023
   465
  presume "sets p = UNIV"
Andreas@59023
   466
  then show "ereal (measure (distr p (count_space UNIV) f) {x}) = integral\<^sup>N p (indicator (f -` {x}))"
Andreas@59023
   467
    by(simp add: measure_distr measurable_def emeasure_eq_measure)
Andreas@59023
   468
qed simp_all
Andreas@59023
   469
Andreas@59023
   470
lemma nn_integral_pmf: "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = emeasure (measure_pmf p) A"
Andreas@59023
   471
proof -
Andreas@59023
   472
  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
   473
    by(auto simp add: nn_integral_count_space_indicator indicator_def set_pmf_iff intro: nn_integral_cong)
Andreas@59023
   474
  also have "\<dots> = emeasure (measure_pmf p) (\<Union>x\<in>A \<inter> set_pmf p. {x})"
Andreas@59023
   475
    by(subst emeasure_UN_countable)(auto simp add: emeasure_pmf_single disjoint_family_on_def)
Andreas@59023
   476
  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
   477
    by(rule emeasure_Un_null_set[symmetric])(auto intro: in_null_sets_measure_pmfI)
Andreas@59023
   478
  also have "\<dots> = emeasure (measure_pmf p) A"
Andreas@59023
   479
    by(auto intro: arg_cong2[where f=emeasure])
Andreas@59023
   480
  finally show ?thesis .
Andreas@59023
   481
qed
Andreas@59023
   482
Andreas@60068
   483
lemma map_return_pmf [simp]: "map_pmf f (return_pmf x) = return_pmf (f x)"
hoelzl@59664
   484
  by transfer (simp add: distr_return)
hoelzl@59664
   485
hoelzl@59664
   486
lemma map_pmf_const[simp]: "map_pmf (\<lambda>_. c) M = return_pmf c"
hoelzl@59664
   487
  by transfer (auto simp: prob_space.distr_const)
hoelzl@59664
   488
Andreas@60068
   489
lemma pmf_return [simp]: "pmf (return_pmf x) y = indicator {y} x"
hoelzl@59664
   490
  by transfer (simp add: measure_return)
hoelzl@59664
   491
hoelzl@59664
   492
lemma nn_integral_return_pmf[simp]: "0 \<le> f x \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>return_pmf x) = f x"
hoelzl@59664
   493
  unfolding return_pmf.rep_eq by (intro nn_integral_return) auto
hoelzl@59664
   494
hoelzl@59664
   495
lemma emeasure_return_pmf[simp]: "emeasure (return_pmf x) X = indicator X x"
hoelzl@59664
   496
  unfolding return_pmf.rep_eq by (intro emeasure_return) auto
hoelzl@59664
   497
hoelzl@59664
   498
lemma return_pmf_inj[simp]: "return_pmf x = return_pmf y \<longleftrightarrow> x = y"
hoelzl@59664
   499
  by (metis insertI1 set_return_pmf singletonD)
hoelzl@59664
   500
hoelzl@59665
   501
lemma map_pmf_eq_return_pmf_iff:
hoelzl@59665
   502
  "map_pmf f p = return_pmf x \<longleftrightarrow> (\<forall>y \<in> set_pmf p. f y = x)"
hoelzl@59665
   503
proof
hoelzl@59665
   504
  assume "map_pmf f p = return_pmf x"
hoelzl@59665
   505
  then have "set_pmf (map_pmf f p) = set_pmf (return_pmf x)" by simp
hoelzl@59665
   506
  then show "\<forall>y \<in> set_pmf p. f y = x" by auto
hoelzl@59665
   507
next
hoelzl@59665
   508
  assume "\<forall>y \<in> set_pmf p. f y = x"
hoelzl@59665
   509
  then show "map_pmf f p = return_pmf x"
hoelzl@59665
   510
    unfolding map_pmf_const[symmetric, of _ p] by (intro map_pmf_cong) auto
hoelzl@59665
   511
qed
hoelzl@59665
   512
hoelzl@59664
   513
definition "pair_pmf A B = bind_pmf A (\<lambda>x. bind_pmf B (\<lambda>y. return_pmf (x, y)))"
hoelzl@59664
   514
hoelzl@59664
   515
lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b"
hoelzl@59664
   516
  unfolding pair_pmf_def pmf_bind pmf_return
hoelzl@59664
   517
  apply (subst integral_measure_pmf[where A="{b}"])
hoelzl@59664
   518
  apply (auto simp: indicator_eq_0_iff)
hoelzl@59664
   519
  apply (subst integral_measure_pmf[where A="{a}"])
hoelzl@59664
   520
  apply (auto simp: indicator_eq_0_iff setsum_nonneg_eq_0_iff pmf_nonneg)
hoelzl@59664
   521
  done
hoelzl@59664
   522
hoelzl@59665
   523
lemma set_pair_pmf[simp]: "set_pmf (pair_pmf A B) = set_pmf A \<times> set_pmf B"
hoelzl@59664
   524
  unfolding pair_pmf_def set_bind_pmf set_return_pmf by auto
hoelzl@59664
   525
hoelzl@59664
   526
lemma measure_pmf_in_subprob_space[measurable (raw)]:
hoelzl@59664
   527
  "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
hoelzl@59664
   528
  by (simp add: space_subprob_algebra) intro_locales
hoelzl@59664
   529
hoelzl@59664
   530
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
   531
proof -
hoelzl@59664
   532
  have "(\<integral>\<^sup>+x. f x \<partial>pair_pmf A B) = (\<integral>\<^sup>+x. max 0 (f x) * indicator (A \<times> B) x \<partial>pair_pmf A B)"
hoelzl@59664
   533
    by (subst nn_integral_max_0[symmetric])
hoelzl@59665
   534
       (auto simp: AE_measure_pmf_iff intro!: nn_integral_cong_AE)
hoelzl@59664
   535
  also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) * indicator (A \<times> B) (a, b) \<partial>B \<partial>A)"
hoelzl@59664
   536
    by (simp add: pair_pmf_def)
hoelzl@59664
   537
  also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) \<partial>B \<partial>A)"
hoelzl@59664
   538
    by (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
hoelzl@59664
   539
  finally show ?thesis
hoelzl@59664
   540
    unfolding nn_integral_max_0 .
hoelzl@59664
   541
qed
hoelzl@59664
   542
hoelzl@59664
   543
lemma bind_pair_pmf:
hoelzl@59664
   544
  assumes M[measurable]: "M \<in> measurable (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) (subprob_algebra N)"
wenzelm@62026
   545
  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
   546
    (is "?L = ?R")
hoelzl@59664
   547
proof (rule measure_eqI)
hoelzl@59664
   548
  have M'[measurable]: "M \<in> measurable (pair_pmf A B) (subprob_algebra N)"
hoelzl@59664
   549
    using M[THEN measurable_space] by (simp_all add: space_pair_measure)
hoelzl@59664
   550
hoelzl@59664
   551
  note measurable_bind[where N="count_space UNIV", measurable]
hoelzl@59664
   552
  note measure_pmf_in_subprob_space[simp]
hoelzl@59664
   553
hoelzl@59664
   554
  have sets_eq_N: "sets ?L = N"
hoelzl@59664
   555
    by (subst sets_bind[OF sets_kernel[OF M']]) auto
hoelzl@59664
   556
  show "sets ?L = sets ?R"
hoelzl@59664
   557
    using measurable_space[OF M]
hoelzl@59664
   558
    by (simp add: sets_eq_N space_pair_measure space_subprob_algebra)
hoelzl@59664
   559
  fix X assume "X \<in> sets ?L"
hoelzl@59664
   560
  then have X[measurable]: "X \<in> sets N"
hoelzl@59664
   561
    unfolding sets_eq_N .
hoelzl@59664
   562
  then show "emeasure ?L X = emeasure ?R X"
hoelzl@59664
   563
    apply (simp add: emeasure_bind[OF _ M' X])
hoelzl@59664
   564
    apply (simp add: nn_integral_bind[where B="count_space UNIV"] pair_pmf_def measure_pmf_bind[of A]
Andreas@60068
   565
                     nn_integral_measure_pmf_finite emeasure_nonneg one_ereal_def[symmetric])
hoelzl@59664
   566
    apply (subst emeasure_bind[OF _ _ X])
hoelzl@59664
   567
    apply measurable
hoelzl@59664
   568
    apply (subst emeasure_bind[OF _ _ X])
hoelzl@59664
   569
    apply measurable
hoelzl@59664
   570
    done
hoelzl@59664
   571
qed
hoelzl@59664
   572
hoelzl@59664
   573
lemma map_fst_pair_pmf: "map_pmf fst (pair_pmf A B) = A"
hoelzl@59664
   574
  by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
hoelzl@59664
   575
hoelzl@59664
   576
lemma map_snd_pair_pmf: "map_pmf snd (pair_pmf A B) = B"
hoelzl@59664
   577
  by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
hoelzl@59664
   578
hoelzl@59664
   579
lemma nn_integral_pmf':
hoelzl@59664
   580
  "inj_on f A \<Longrightarrow> (\<integral>\<^sup>+x. pmf p (f x) \<partial>count_space A) = emeasure p (f ` A)"
hoelzl@59664
   581
  by (subst nn_integral_bij_count_space[where g=f and B="f`A"])
hoelzl@59664
   582
     (auto simp: bij_betw_def nn_integral_pmf)
hoelzl@59664
   583
hoelzl@59664
   584
lemma pmf_le_0_iff[simp]: "pmf M p \<le> 0 \<longleftrightarrow> pmf M p = 0"
hoelzl@59664
   585
  using pmf_nonneg[of M p] by simp
hoelzl@59664
   586
hoelzl@59664
   587
lemma min_pmf_0[simp]: "min (pmf M p) 0 = 0" "min 0 (pmf M p) = 0"
hoelzl@59664
   588
  using pmf_nonneg[of M p] by simp_all
hoelzl@59664
   589
hoelzl@59664
   590
lemma pmf_eq_0_set_pmf: "pmf M p = 0 \<longleftrightarrow> p \<notin> set_pmf M"
hoelzl@59664
   591
  unfolding set_pmf_iff by simp
hoelzl@59664
   592
hoelzl@59664
   593
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
   594
  by (auto simp: pmf.rep_eq map_pmf_rep_eq measure_distr AE_measure_pmf_iff inj_onD
hoelzl@59664
   595
           intro!: measure_pmf.finite_measure_eq_AE)
hoelzl@59664
   596
Andreas@60068
   597
lemma pmf_map_inj': "inj f \<Longrightarrow> pmf (map_pmf f M) (f x) = pmf M x"
Andreas@60068
   598
apply(cases "x \<in> set_pmf M")
Andreas@60068
   599
 apply(simp add: pmf_map_inj[OF subset_inj_on])
Andreas@60068
   600
apply(simp add: pmf_eq_0_set_pmf[symmetric])
Andreas@60068
   601
apply(auto simp add: pmf_eq_0_set_pmf dest: injD)
Andreas@60068
   602
done
Andreas@60068
   603
Andreas@60068
   604
lemma pmf_map_outside: "x \<notin> f ` set_pmf M \<Longrightarrow> pmf (map_pmf f M) x = 0"
Andreas@60068
   605
unfolding pmf_eq_0_set_pmf by simp
Andreas@60068
   606
hoelzl@59664
   607
subsection \<open> PMFs as function \<close>
hoelzl@59000
   608
hoelzl@58587
   609
context
hoelzl@58587
   610
  fixes f :: "'a \<Rightarrow> real"
hoelzl@58587
   611
  assumes nonneg: "\<And>x. 0 \<le> f x"
hoelzl@58587
   612
  assumes prob: "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
hoelzl@58587
   613
begin
hoelzl@58587
   614
hoelzl@58587
   615
lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ereal \<circ> f)"
hoelzl@58587
   616
proof (intro conjI)
hoelzl@58587
   617
  have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
hoelzl@58587
   618
    by (simp split: split_indicator)
hoelzl@58587
   619
  show "AE x in density (count_space UNIV) (ereal \<circ> f).
hoelzl@58587
   620
    measure (density (count_space UNIV) (ereal \<circ> f)) {x} \<noteq> 0"
hoelzl@59092
   621
    by (simp add: AE_density nonneg measure_def emeasure_density max_def)
hoelzl@58587
   622
  show "prob_space (density (count_space UNIV) (ereal \<circ> f))"
wenzelm@61169
   623
    by standard (simp add: emeasure_density prob)
hoelzl@58587
   624
qed simp
hoelzl@58587
   625
hoelzl@58587
   626
lemma pmf_embed_pmf: "pmf embed_pmf x = f x"
hoelzl@58587
   627
proof transfer
hoelzl@58587
   628
  have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
hoelzl@58587
   629
    by (simp split: split_indicator)
hoelzl@58587
   630
  fix x show "measure (density (count_space UNIV) (ereal \<circ> f)) {x} = f x"
hoelzl@59092
   631
    by transfer (simp add: measure_def emeasure_density nonneg max_def)
hoelzl@58587
   632
qed
hoelzl@58587
   633
Andreas@60068
   634
lemma set_embed_pmf: "set_pmf embed_pmf = {x. f x \<noteq> 0}"
Andreas@60068
   635
by(auto simp add: set_pmf_eq assms pmf_embed_pmf)
Andreas@60068
   636
hoelzl@58587
   637
end
hoelzl@58587
   638
hoelzl@58587
   639
lemma embed_pmf_transfer:
hoelzl@58730
   640
  "rel_fun (eq_onp (\<lambda>f. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ereal (f x) \<partial>count_space UNIV) = 1)) pmf_as_measure.cr_pmf (\<lambda>f. density (count_space UNIV) (ereal \<circ> f)) embed_pmf"
hoelzl@58587
   641
  by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer)
hoelzl@58587
   642
hoelzl@59000
   643
lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)"
hoelzl@59000
   644
proof (transfer, elim conjE)
hoelzl@59000
   645
  fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
hoelzl@59000
   646
  assume "prob_space M" then interpret prob_space M .
hoelzl@59000
   647
  show "M = density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))"
hoelzl@59000
   648
  proof (rule measure_eqI)
hoelzl@59000
   649
    fix A :: "'a set"
lp15@59667
   650
    have "(\<integral>\<^sup>+ x. ereal (measure M {x}) * indicator A x \<partial>count_space UNIV) =
hoelzl@59000
   651
      (\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)"
hoelzl@59000
   652
      by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator)
hoelzl@59000
   653
    also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))"
hoelzl@59000
   654
      by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space)
hoelzl@59000
   655
    also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})"
hoelzl@59000
   656
      by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support)
hoelzl@59000
   657
         (auto simp: disjoint_family_on_def)
hoelzl@59000
   658
    also have "\<dots> = emeasure M A"
hoelzl@59000
   659
      using ae by (intro emeasure_eq_AE) auto
hoelzl@59000
   660
    finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))) A"
hoelzl@59000
   661
      using emeasure_space_1 by (simp add: emeasure_density)
hoelzl@59000
   662
  qed simp
hoelzl@59000
   663
qed
hoelzl@59000
   664
hoelzl@58587
   665
lemma td_pmf_embed_pmf:
hoelzl@58587
   666
  "type_definition pmf embed_pmf {f::'a \<Rightarrow> real. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ereal (f x) \<partial>count_space UNIV) = 1}"
hoelzl@58587
   667
  unfolding type_definition_def
hoelzl@58587
   668
proof safe
hoelzl@58587
   669
  fix p :: "'a pmf"
hoelzl@58587
   670
  have "(\<integral>\<^sup>+ x. 1 \<partial>measure_pmf p) = 1"
hoelzl@58587
   671
    using measure_pmf.emeasure_space_1[of p] by simp
hoelzl@58587
   672
  then show *: "(\<integral>\<^sup>+ x. ereal (pmf p x) \<partial>count_space UNIV) = 1"
hoelzl@58587
   673
    by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const)
hoelzl@58587
   674
hoelzl@58587
   675
  show "embed_pmf (pmf p) = p"
hoelzl@58587
   676
    by (intro measure_pmf_inject[THEN iffD1])
hoelzl@58587
   677
       (simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def)
hoelzl@58587
   678
next
hoelzl@58587
   679
  fix f :: "'a \<Rightarrow> real" assume "\<forall>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
hoelzl@58587
   680
  then show "pmf (embed_pmf f) = f"
hoelzl@58587
   681
    by (auto intro!: pmf_embed_pmf)
hoelzl@58587
   682
qed (rule pmf_nonneg)
hoelzl@58587
   683
hoelzl@58587
   684
end
hoelzl@58587
   685
Andreas@60068
   686
lemma nn_integral_measure_pmf: "(\<integral>\<^sup>+ x. f x \<partial>measure_pmf p) = \<integral>\<^sup>+ x. ereal (pmf p x) * f x \<partial>count_space UNIV"
Andreas@60068
   687
by(simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg)
Andreas@60068
   688
hoelzl@58587
   689
locale pmf_as_function
hoelzl@58587
   690
begin
hoelzl@58587
   691
hoelzl@58587
   692
setup_lifting td_pmf_embed_pmf
hoelzl@58587
   693
lp15@59667
   694
lemma set_pmf_transfer[transfer_rule]:
hoelzl@58730
   695
  assumes "bi_total A"
lp15@59667
   696
  shows "rel_fun (pcr_pmf A) (rel_set A) (\<lambda>f. {x. f x \<noteq> 0}) set_pmf"
wenzelm@61808
   697
  using \<open>bi_total A\<close>
hoelzl@58730
   698
  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
   699
     metis+
hoelzl@58730
   700
hoelzl@59000
   701
end
hoelzl@59000
   702
hoelzl@59000
   703
context
hoelzl@59000
   704
begin
hoelzl@59000
   705
hoelzl@59000
   706
interpretation pmf_as_function .
hoelzl@59000
   707
hoelzl@59000
   708
lemma pmf_eqI: "(\<And>i. pmf M i = pmf N i) \<Longrightarrow> M = N"
hoelzl@59000
   709
  by transfer auto
hoelzl@59000
   710
hoelzl@59000
   711
lemma pmf_eq_iff: "M = N \<longleftrightarrow> (\<forall>i. pmf M i = pmf N i)"
hoelzl@59000
   712
  by (auto intro: pmf_eqI)
hoelzl@59000
   713
hoelzl@59664
   714
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
   715
  unfolding pmf_eq_iff pmf_bind
hoelzl@59664
   716
proof
hoelzl@59664
   717
  fix i
hoelzl@59664
   718
  interpret B: prob_space "restrict_space B B"
hoelzl@59664
   719
    by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
hoelzl@59664
   720
       (auto simp: AE_measure_pmf_iff)
hoelzl@59664
   721
  interpret A: prob_space "restrict_space A A"
hoelzl@59664
   722
    by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
hoelzl@59664
   723
       (auto simp: AE_measure_pmf_iff)
hoelzl@59664
   724
hoelzl@59664
   725
  interpret AB: pair_prob_space "restrict_space A A" "restrict_space B B"
hoelzl@59664
   726
    by unfold_locales
hoelzl@59664
   727
hoelzl@59664
   728
  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
   729
    by (rule integral_cong) (auto intro!: integral_pmf_restrict)
hoelzl@59664
   730
  also have "\<dots> = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>restrict_space A A)"
hoelzl@59664
   731
    by (intro integral_pmf_restrict B.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
hoelzl@59664
   732
              countable_set_pmf borel_measurable_count_space)
hoelzl@59664
   733
  also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>restrict_space B B)"
hoelzl@59664
   734
    by (rule AB.Fubini_integral[symmetric])
hoelzl@59664
   735
       (auto intro!: AB.integrable_const_bound[where B=1] measurable_pair_restrict_pmf2
hoelzl@59664
   736
             simp: pmf_nonneg pmf_le_1 measurable_restrict_space1)
hoelzl@59664
   737
  also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>B)"
hoelzl@59664
   738
    by (intro integral_pmf_restrict[symmetric] A.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
hoelzl@59664
   739
              countable_set_pmf borel_measurable_count_space)
hoelzl@59664
   740
  also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>A \<partial>B)"
hoelzl@59664
   741
    by (rule integral_cong) (auto intro!: integral_pmf_restrict[symmetric])
hoelzl@59664
   742
  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
   743
qed
hoelzl@59664
   744
hoelzl@59664
   745
lemma pair_map_pmf1: "pair_pmf (map_pmf f A) B = map_pmf (apfst f) (pair_pmf A B)"
hoelzl@59664
   746
proof (safe intro!: pmf_eqI)
hoelzl@59664
   747
  fix a :: "'a" and b :: "'b"
hoelzl@59664
   748
  have [simp]: "\<And>c d. indicator (apfst f -` {(a, b)}) (c, d) = indicator (f -` {a}) c * (indicator {b} d::ereal)"
hoelzl@59664
   749
    by (auto split: split_indicator)
hoelzl@59664
   750
hoelzl@59664
   751
  have "ereal (pmf (pair_pmf (map_pmf f A) B) (a, b)) =
hoelzl@59664
   752
         ereal (pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b))"
hoelzl@59664
   753
    unfolding pmf_pair ereal_pmf_map
hoelzl@59664
   754
    by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_multc pmf_nonneg
hoelzl@59664
   755
                  emeasure_map_pmf[symmetric] del: emeasure_map_pmf)
hoelzl@59664
   756
  then show "pmf (pair_pmf (map_pmf f A) B) (a, b) = pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b)"
hoelzl@59664
   757
    by simp
hoelzl@59664
   758
qed
hoelzl@59664
   759
hoelzl@59664
   760
lemma pair_map_pmf2: "pair_pmf A (map_pmf f B) = map_pmf (apsnd f) (pair_pmf A B)"
hoelzl@59664
   761
proof (safe intro!: pmf_eqI)
hoelzl@59664
   762
  fix a :: "'a" and b :: "'b"
hoelzl@59664
   763
  have [simp]: "\<And>c d. indicator (apsnd f -` {(a, b)}) (c, d) = indicator {a} c * (indicator (f -` {b}) d::ereal)"
hoelzl@59664
   764
    by (auto split: split_indicator)
hoelzl@59664
   765
hoelzl@59664
   766
  have "ereal (pmf (pair_pmf A (map_pmf f B)) (a, b)) =
hoelzl@59664
   767
         ereal (pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b))"
hoelzl@59664
   768
    unfolding pmf_pair ereal_pmf_map
hoelzl@59664
   769
    by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_cmult nn_integral_multc pmf_nonneg
hoelzl@59664
   770
                  emeasure_map_pmf[symmetric] del: emeasure_map_pmf)
hoelzl@59664
   771
  then show "pmf (pair_pmf A (map_pmf f B)) (a, b) = pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b)"
hoelzl@59664
   772
    by simp
hoelzl@59664
   773
qed
hoelzl@59664
   774
hoelzl@59664
   775
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
   776
  by (simp add: pair_map_pmf2 pair_map_pmf1 map_pmf_comp split_beta')
hoelzl@59664
   777
hoelzl@59000
   778
end
hoelzl@59000
   779
Andreas@61634
   780
lemma pair_return_pmf1: "pair_pmf (return_pmf x) y = map_pmf (Pair x) y"
Andreas@61634
   781
by(simp add: pair_pmf_def bind_return_pmf map_pmf_def)
Andreas@61634
   782
Andreas@61634
   783
lemma pair_return_pmf2: "pair_pmf x (return_pmf y) = map_pmf (\<lambda>x. (x, y)) x"
Andreas@61634
   784
by(simp add: pair_pmf_def bind_return_pmf map_pmf_def)
Andreas@61634
   785
Andreas@61634
   786
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
   787
by(simp add: pair_pmf_def bind_return_pmf map_pmf_def bind_assoc_pmf)
Andreas@61634
   788
Andreas@61634
   789
lemma pair_commute_pmf: "pair_pmf x y = map_pmf (\<lambda>(x, y). (y, x)) (pair_pmf y x)"
Andreas@61634
   790
unfolding pair_pmf_def by(subst bind_commute_pmf)(simp add: map_pmf_def bind_assoc_pmf bind_return_pmf)
Andreas@61634
   791
Andreas@61634
   792
lemma set_pmf_subset_singleton: "set_pmf p \<subseteq> {x} \<longleftrightarrow> p = return_pmf x"
Andreas@61634
   793
proof(intro iffI pmf_eqI)
Andreas@61634
   794
  fix i
Andreas@61634
   795
  assume x: "set_pmf p \<subseteq> {x}"
Andreas@61634
   796
  hence *: "set_pmf p = {x}" using set_pmf_not_empty[of p] by auto
Andreas@61634
   797
  have "ereal (pmf p x) = \<integral>\<^sup>+ i. indicator {x} i \<partial>p" by(simp add: emeasure_pmf_single)
Andreas@61634
   798
  also have "\<dots> = \<integral>\<^sup>+ i. 1 \<partial>p" by(rule nn_integral_cong_AE)(simp add: AE_measure_pmf_iff * )
Andreas@61634
   799
  also have "\<dots> = 1" by simp
Andreas@61634
   800
  finally show "pmf p i = pmf (return_pmf x) i" using x
Andreas@61634
   801
    by(auto split: split_indicator simp add: pmf_eq_0_set_pmf)
Andreas@61634
   802
qed auto
Andreas@61634
   803
Andreas@61634
   804
lemma bind_eq_return_pmf:
Andreas@61634
   805
  "bind_pmf p f = return_pmf x \<longleftrightarrow> (\<forall>y\<in>set_pmf p. f y = return_pmf x)"
Andreas@61634
   806
  (is "?lhs \<longleftrightarrow> ?rhs")
Andreas@61634
   807
proof(intro iffI strip)
Andreas@61634
   808
  fix y
Andreas@61634
   809
  assume y: "y \<in> set_pmf p"
Andreas@61634
   810
  assume "?lhs"
Andreas@61634
   811
  hence "set_pmf (bind_pmf p f) = {x}" by simp
Andreas@61634
   812
  hence "(\<Union>y\<in>set_pmf p. set_pmf (f y)) = {x}" by simp
Andreas@61634
   813
  hence "set_pmf (f y) \<subseteq> {x}" using y by auto
Andreas@61634
   814
  thus "f y = return_pmf x" by(simp add: set_pmf_subset_singleton)
Andreas@61634
   815
next
Andreas@61634
   816
  assume *: ?rhs
Andreas@61634
   817
  show ?lhs
Andreas@61634
   818
  proof(rule pmf_eqI)
Andreas@61634
   819
    fix i
Andreas@61634
   820
    have "ereal (pmf (bind_pmf p f) i) = \<integral>\<^sup>+ y. ereal (pmf (f y) i) \<partial>p" by(simp add: ereal_pmf_bind)
Andreas@61634
   821
    also have "\<dots> = \<integral>\<^sup>+ y. ereal (pmf (return_pmf x) i) \<partial>p"
Andreas@61634
   822
      by(rule nn_integral_cong_AE)(simp add: AE_measure_pmf_iff * )
Andreas@61634
   823
    also have "\<dots> = ereal (pmf (return_pmf x) i)" by simp
Andreas@61634
   824
    finally show "pmf (bind_pmf p f) i = pmf (return_pmf x) i" by simp
Andreas@61634
   825
  qed
Andreas@61634
   826
qed
Andreas@61634
   827
Andreas@61634
   828
lemma pmf_False_conv_True: "pmf p False = 1 - pmf p True"
Andreas@61634
   829
proof -
Andreas@61634
   830
  have "pmf p False + pmf p True = measure p {False} + measure p {True}"
Andreas@61634
   831
    by(simp add: measure_pmf_single)
Andreas@61634
   832
  also have "\<dots> = measure p ({False} \<union> {True})"
Andreas@61634
   833
    by(subst measure_pmf.finite_measure_Union) simp_all
Andreas@61634
   834
  also have "{False} \<union> {True} = space p" by auto
Andreas@61634
   835
  finally show ?thesis by simp
Andreas@61634
   836
qed
Andreas@61634
   837
Andreas@61634
   838
lemma pmf_True_conv_False: "pmf p True = 1 - pmf p False"
Andreas@61634
   839
by(simp add: pmf_False_conv_True)
Andreas@61634
   840
hoelzl@59664
   841
subsection \<open> Conditional Probabilities \<close>
hoelzl@59664
   842
hoelzl@59670
   843
lemma measure_pmf_zero_iff: "measure (measure_pmf p) s = 0 \<longleftrightarrow> set_pmf p \<inter> s = {}"
hoelzl@59670
   844
  by (subst measure_pmf.prob_eq_0) (auto simp: AE_measure_pmf_iff)
hoelzl@59670
   845
hoelzl@59664
   846
context
hoelzl@59664
   847
  fixes p :: "'a pmf" and s :: "'a set"
hoelzl@59664
   848
  assumes not_empty: "set_pmf p \<inter> s \<noteq> {}"
hoelzl@59664
   849
begin
hoelzl@59664
   850
hoelzl@59664
   851
interpretation pmf_as_measure .
hoelzl@59664
   852
hoelzl@59664
   853
lemma emeasure_measure_pmf_not_zero: "emeasure (measure_pmf p) s \<noteq> 0"
hoelzl@59664
   854
proof
hoelzl@59664
   855
  assume "emeasure (measure_pmf p) s = 0"
hoelzl@59664
   856
  then have "AE x in measure_pmf p. x \<notin> s"
hoelzl@59664
   857
    by (rule AE_I[rotated]) auto
hoelzl@59664
   858
  with not_empty show False
hoelzl@59664
   859
    by (auto simp: AE_measure_pmf_iff)
hoelzl@59664
   860
qed
hoelzl@59664
   861
hoelzl@59664
   862
lemma measure_measure_pmf_not_zero: "measure (measure_pmf p) s \<noteq> 0"
hoelzl@59664
   863
  using emeasure_measure_pmf_not_zero unfolding measure_pmf.emeasure_eq_measure by simp
hoelzl@59664
   864
hoelzl@59664
   865
lift_definition cond_pmf :: "'a pmf" is
hoelzl@59664
   866
  "uniform_measure (measure_pmf p) s"
hoelzl@59664
   867
proof (intro conjI)
hoelzl@59664
   868
  show "prob_space (uniform_measure (measure_pmf p) s)"
hoelzl@59664
   869
    by (intro prob_space_uniform_measure) (auto simp: emeasure_measure_pmf_not_zero)
hoelzl@59664
   870
  show "AE x in uniform_measure (measure_pmf p) s. measure (uniform_measure (measure_pmf p) s) {x} \<noteq> 0"
hoelzl@59664
   871
    by (simp add: emeasure_measure_pmf_not_zero measure_measure_pmf_not_zero AE_uniform_measure
hoelzl@59664
   872
                  AE_measure_pmf_iff set_pmf.rep_eq)
hoelzl@59664
   873
qed simp
hoelzl@59664
   874
hoelzl@59664
   875
lemma pmf_cond: "pmf cond_pmf x = (if x \<in> s then pmf p x / measure p s else 0)"
hoelzl@59664
   876
  by transfer (simp add: emeasure_measure_pmf_not_zero pmf.rep_eq)
hoelzl@59664
   877
hoelzl@59665
   878
lemma set_cond_pmf[simp]: "set_pmf cond_pmf = set_pmf p \<inter> s"
hoelzl@59664
   879
  by (auto simp add: set_pmf_iff pmf_cond measure_measure_pmf_not_zero split: split_if_asm)
hoelzl@59664
   880
hoelzl@59664
   881
end
hoelzl@59664
   882
hoelzl@59664
   883
lemma cond_map_pmf:
hoelzl@59664
   884
  assumes "set_pmf p \<inter> f -` s \<noteq> {}"
hoelzl@59664
   885
  shows "cond_pmf (map_pmf f p) s = map_pmf f (cond_pmf p (f -` s))"
hoelzl@59664
   886
proof -
hoelzl@59664
   887
  have *: "set_pmf (map_pmf f p) \<inter> s \<noteq> {}"
hoelzl@59665
   888
    using assms by auto
hoelzl@59664
   889
  { fix x
hoelzl@59664
   890
    have "ereal (pmf (map_pmf f (cond_pmf p (f -` s))) x) =
hoelzl@59664
   891
      emeasure p (f -` s \<inter> f -` {x}) / emeasure p (f -` s)"
hoelzl@59664
   892
      unfolding ereal_pmf_map cond_pmf.rep_eq[OF assms] by (simp add: nn_integral_uniform_measure)
hoelzl@59664
   893
    also have "f -` s \<inter> f -` {x} = (if x \<in> s then f -` {x} else {})"
hoelzl@59664
   894
      by auto
hoelzl@59664
   895
    also have "emeasure p (if x \<in> s then f -` {x} else {}) / emeasure p (f -` s) =
hoelzl@59664
   896
      ereal (pmf (cond_pmf (map_pmf f p) s) x)"
hoelzl@59664
   897
      using measure_measure_pmf_not_zero[OF *]
hoelzl@59664
   898
      by (simp add: pmf_cond[OF *] ereal_divide' ereal_pmf_map measure_pmf.emeasure_eq_measure[symmetric]
hoelzl@59664
   899
               del: ereal_divide)
hoelzl@59664
   900
    finally have "ereal (pmf (cond_pmf (map_pmf f p) s) x) = ereal (pmf (map_pmf f (cond_pmf p (f -` s))) x)"
hoelzl@59664
   901
      by simp }
hoelzl@59664
   902
  then show ?thesis
hoelzl@59664
   903
    by (intro pmf_eqI) simp
hoelzl@59664
   904
qed
hoelzl@59664
   905
hoelzl@59664
   906
lemma bind_cond_pmf_cancel:
hoelzl@59670
   907
  assumes [simp]: "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}"
hoelzl@59670
   908
  assumes [simp]: "\<And>y. y \<in> set_pmf q \<Longrightarrow> set_pmf p \<inter> {x. R x y} \<noteq> {}"
hoelzl@59670
   909
  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
   910
  shows "bind_pmf p (\<lambda>x. cond_pmf q {y. R x y}) = q"
hoelzl@59664
   911
proof (rule pmf_eqI)
hoelzl@59670
   912
  fix i
hoelzl@59670
   913
  have "ereal (pmf (bind_pmf p (\<lambda>x. cond_pmf q {y. R x y})) i) =
hoelzl@59670
   914
    (\<integral>\<^sup>+x. ereal (pmf q i / measure p {x. R x i}) * ereal (indicator {x. R x i} x) \<partial>p)"
hoelzl@59670
   915
    by (auto simp add: ereal_pmf_bind AE_measure_pmf_iff pmf_cond pmf_eq_0_set_pmf intro!: nn_integral_cong_AE)
hoelzl@59670
   916
  also have "\<dots> = (pmf q i * measure p {x. R x i}) / measure p {x. R x i}"
hoelzl@59670
   917
    by (simp add: pmf_nonneg measure_nonneg zero_ereal_def[symmetric] ereal_indicator
hoelzl@59670
   918
                  nn_integral_cmult measure_pmf.emeasure_eq_measure)
hoelzl@59670
   919
  also have "\<dots> = pmf q i"
hoelzl@59670
   920
    by (cases "pmf q i = 0") (simp_all add: pmf_eq_0_set_pmf measure_measure_pmf_not_zero)
hoelzl@59670
   921
  finally show "pmf (bind_pmf p (\<lambda>x. cond_pmf q {y. R x y})) i = pmf q i"
hoelzl@59670
   922
    by simp
hoelzl@59664
   923
qed
hoelzl@59664
   924
hoelzl@59664
   925
subsection \<open> Relator \<close>
hoelzl@59664
   926
hoelzl@59664
   927
inductive rel_pmf :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf \<Rightarrow> bool"
hoelzl@59664
   928
for R p q
hoelzl@59664
   929
where
lp15@59667
   930
  "\<lbrakk> \<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y;
hoelzl@59664
   931
     map_pmf fst pq = p; map_pmf snd pq = q \<rbrakk>
hoelzl@59664
   932
  \<Longrightarrow> rel_pmf R p q"
hoelzl@59664
   933
hoelzl@59681
   934
lemma rel_pmfI:
hoelzl@59681
   935
  assumes R: "rel_set R (set_pmf p) (set_pmf q)"
hoelzl@59681
   936
  assumes eq: "\<And>x y. x \<in> set_pmf p \<Longrightarrow> y \<in> set_pmf q \<Longrightarrow> R x y \<Longrightarrow>
hoelzl@59681
   937
    measure p {x. R x y} = measure q {y. R x y}"
hoelzl@59681
   938
  shows "rel_pmf R p q"
hoelzl@59681
   939
proof
hoelzl@59681
   940
  let ?pq = "bind_pmf p (\<lambda>x. bind_pmf (cond_pmf q {y. R x y}) (\<lambda>y. return_pmf (x, y)))"
hoelzl@59681
   941
  have "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}"
hoelzl@59681
   942
    using R by (auto simp: rel_set_def)
hoelzl@59681
   943
  then show "\<And>x y. (x, y) \<in> set_pmf ?pq \<Longrightarrow> R x y"
hoelzl@59681
   944
    by auto
hoelzl@59681
   945
  show "map_pmf fst ?pq = p"
Andreas@60068
   946
    by (simp add: map_bind_pmf bind_return_pmf')
hoelzl@59681
   947
hoelzl@59681
   948
  show "map_pmf snd ?pq = q"
hoelzl@59681
   949
    using R eq
Andreas@60068
   950
    apply (simp add: bind_cond_pmf_cancel map_bind_pmf bind_return_pmf')
hoelzl@59681
   951
    apply (rule bind_cond_pmf_cancel)
hoelzl@59681
   952
    apply (auto simp: rel_set_def)
hoelzl@59681
   953
    done
hoelzl@59681
   954
qed
hoelzl@59681
   955
hoelzl@59681
   956
lemma rel_pmf_imp_rel_set: "rel_pmf R p q \<Longrightarrow> rel_set R (set_pmf p) (set_pmf q)"
hoelzl@59681
   957
  by (force simp add: rel_pmf.simps rel_set_def)
hoelzl@59681
   958
hoelzl@59681
   959
lemma rel_pmfD_measure:
hoelzl@59681
   960
  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
   961
  assumes "x \<in> set_pmf p" "y \<in> set_pmf q"
hoelzl@59681
   962
  shows "measure p {x. R x y} = measure q {y. R x y}"
hoelzl@59681
   963
proof -
hoelzl@59681
   964
  from rel_R obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
hoelzl@59681
   965
    and eq: "p = map_pmf fst pq" "q = map_pmf snd pq"
hoelzl@59681
   966
    by (auto elim: rel_pmf.cases)
hoelzl@59681
   967
  have "measure p {x. R x y} = measure pq {x. R (fst x) y}"
hoelzl@59681
   968
    by (simp add: eq map_pmf_rep_eq measure_distr)
hoelzl@59681
   969
  also have "\<dots> = measure pq {y. R x (snd y)}"
hoelzl@59681
   970
    by (intro measure_pmf.finite_measure_eq_AE)
hoelzl@59681
   971
       (auto simp: AE_measure_pmf_iff R dest!: pq)
hoelzl@59681
   972
  also have "\<dots> = measure q {y. R x y}"
hoelzl@59681
   973
    by (simp add: eq map_pmf_rep_eq measure_distr)
hoelzl@59681
   974
  finally show "measure p {x. R x y} = measure q {y. R x y}" .
hoelzl@59681
   975
qed
hoelzl@59681
   976
Andreas@61634
   977
lemma rel_pmf_measureD:
Andreas@61634
   978
  assumes "rel_pmf R p q"
Andreas@61634
   979
  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
   980
using assms
Andreas@61634
   981
proof cases
Andreas@61634
   982
  fix pq
Andreas@61634
   983
  assume R: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
Andreas@61634
   984
    and p[symmetric]: "map_pmf fst pq = p"
Andreas@61634
   985
    and q[symmetric]: "map_pmf snd pq = q"
Andreas@61634
   986
  have "?lhs = measure (measure_pmf pq) (fst -` A)" by(simp add: p)
Andreas@61634
   987
  also have "\<dots> \<le> measure (measure_pmf pq) {y. \<exists>x\<in>A. R x (snd y)}"
Andreas@61634
   988
    by(rule measure_pmf.finite_measure_mono_AE)(auto 4 3 simp add: AE_measure_pmf_iff dest: R)
Andreas@61634
   989
  also have "\<dots> = ?rhs" by(simp add: q)
Andreas@61634
   990
  finally show ?thesis .
Andreas@61634
   991
qed
Andreas@61634
   992
hoelzl@59681
   993
lemma rel_pmf_iff_measure:
hoelzl@59681
   994
  assumes "symp R" "transp R"
hoelzl@59681
   995
  shows "rel_pmf R p q \<longleftrightarrow>
hoelzl@59681
   996
    rel_set R (set_pmf p) (set_pmf q) \<and>
hoelzl@59681
   997
    (\<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
   998
  by (safe intro!: rel_pmf_imp_rel_set rel_pmfI)
hoelzl@59681
   999
     (auto intro!: rel_pmfD_measure dest: sympD[OF \<open>symp R\<close>] transpD[OF \<open>transp R\<close>])
hoelzl@59681
  1000
hoelzl@59681
  1001
lemma quotient_rel_set_disjoint:
hoelzl@59681
  1002
  "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
  1003
  using in_quotient_imp_closed[of UNIV "{(x, y). R x y}" C]
hoelzl@59681
  1004
  by (auto 0 0 simp: equivp_equiv rel_set_def set_eq_iff elim: equivpE)
hoelzl@59681
  1005
     (blast dest: equivp_symp)+
hoelzl@59681
  1006
hoelzl@59681
  1007
lemma quotientD: "equiv X R \<Longrightarrow> A \<in> X // R \<Longrightarrow> x \<in> A \<Longrightarrow> A = R `` {x}"
hoelzl@59681
  1008
  by (metis Image_singleton_iff equiv_class_eq_iff quotientE)
hoelzl@59681
  1009
hoelzl@59681
  1010
lemma rel_pmf_iff_equivp:
hoelzl@59681
  1011
  assumes "equivp R"
hoelzl@59681
  1012
  shows "rel_pmf R p q \<longleftrightarrow> (\<forall>C\<in>UNIV // {(x, y). R x y}. measure p C = measure q C)"
hoelzl@59681
  1013
    (is "_ \<longleftrightarrow>   (\<forall>C\<in>_//?R. _)")
hoelzl@59681
  1014
proof (subst rel_pmf_iff_measure, safe)
hoelzl@59681
  1015
  show "symp R" "transp R"
hoelzl@59681
  1016
    using assms by (auto simp: equivp_reflp_symp_transp)
hoelzl@59681
  1017
next
hoelzl@59681
  1018
  fix C assume C: "C \<in> UNIV // ?R" and R: "rel_set R (set_pmf p) (set_pmf q)"
hoelzl@59681
  1019
  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
  1020
hoelzl@59681
  1021
  show "measure p C = measure q C"
hoelzl@59681
  1022
  proof cases
hoelzl@59681
  1023
    assume "p \<inter> C = {}"
lp15@61609
  1024
    moreover then have "q \<inter> C = {}"
hoelzl@59681
  1025
      using quotient_rel_set_disjoint[OF assms C R] by simp
hoelzl@59681
  1026
    ultimately show ?thesis
hoelzl@59681
  1027
      unfolding measure_pmf_zero_iff[symmetric] by simp
hoelzl@59681
  1028
  next
hoelzl@59681
  1029
    assume "p \<inter> C \<noteq> {}"
lp15@61609
  1030
    moreover then have "q \<inter> C \<noteq> {}"
hoelzl@59681
  1031
      using quotient_rel_set_disjoint[OF assms C R] by simp
hoelzl@59681
  1032
    ultimately obtain x y where in_set: "x \<in> set_pmf p" "y \<in> set_pmf q" and in_C: "x \<in> C" "y \<in> C"
hoelzl@59681
  1033
      by auto
hoelzl@59681
  1034
    then have "R x y"
hoelzl@59681
  1035
      using in_quotient_imp_in_rel[of UNIV ?R C x y] C assms
hoelzl@59681
  1036
      by (simp add: equivp_equiv)
hoelzl@59681
  1037
    with in_set eq have "measure p {x. R x y} = measure q {y. R x y}"
hoelzl@59681
  1038
      by auto
hoelzl@59681
  1039
    moreover have "{y. R x y} = C"
wenzelm@61808
  1040
      using assms \<open>x \<in> C\<close> C quotientD[of UNIV ?R C x] by (simp add: equivp_equiv)
hoelzl@59681
  1041
    moreover have "{x. R x y} = C"
wenzelm@61808
  1042
      using assms \<open>y \<in> C\<close> C quotientD[of UNIV "?R" C y] sympD[of R]
hoelzl@59681
  1043
      by (auto simp add: equivp_equiv elim: equivpE)
hoelzl@59681
  1044
    ultimately show ?thesis
hoelzl@59681
  1045
      by auto
hoelzl@59681
  1046
  qed
hoelzl@59681
  1047
next
hoelzl@59681
  1048
  assume eq: "\<forall>C\<in>UNIV // ?R. measure p C = measure q C"
hoelzl@59681
  1049
  show "rel_set R (set_pmf p) (set_pmf q)"
hoelzl@59681
  1050
    unfolding rel_set_def
hoelzl@59681
  1051
  proof safe
hoelzl@59681
  1052
    fix x assume x: "x \<in> set_pmf p"
hoelzl@59681
  1053
    have "{y. R x y} \<in> UNIV // ?R"
hoelzl@59681
  1054
      by (auto simp: quotient_def)
hoelzl@59681
  1055
    with eq have *: "measure q {y. R x y} = measure p {y. R x y}"
hoelzl@59681
  1056
      by auto
hoelzl@59681
  1057
    have "measure q {y. R x y} \<noteq> 0"
hoelzl@59681
  1058
      using x assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp)
hoelzl@59681
  1059
    then show "\<exists>y\<in>set_pmf q. R x y"
hoelzl@59681
  1060
      unfolding measure_pmf_zero_iff by auto
hoelzl@59681
  1061
  next
hoelzl@59681
  1062
    fix y assume y: "y \<in> set_pmf q"
hoelzl@59681
  1063
    have "{x. R x y} \<in> UNIV // ?R"
hoelzl@59681
  1064
      using assms by (auto simp: quotient_def dest: equivp_symp)
hoelzl@59681
  1065
    with eq have *: "measure p {x. R x y} = measure q {x. R x y}"
hoelzl@59681
  1066
      by auto
hoelzl@59681
  1067
    have "measure p {x. R x y} \<noteq> 0"
hoelzl@59681
  1068
      using y assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp)
hoelzl@59681
  1069
    then show "\<exists>x\<in>set_pmf p. R x y"
hoelzl@59681
  1070
      unfolding measure_pmf_zero_iff by auto
hoelzl@59681
  1071
  qed
hoelzl@59681
  1072
hoelzl@59681
  1073
  fix x y assume "x \<in> set_pmf p" "y \<in> set_pmf q" "R x y"
hoelzl@59681
  1074
  have "{y. R x y} \<in> UNIV // ?R" "{x. R x y} = {y. R x y}"
wenzelm@61808
  1075
    using assms \<open>R x y\<close> by (auto simp: quotient_def dest: equivp_symp equivp_transp)
hoelzl@59681
  1076
  with eq show "measure p {x. R x y} = measure q {y. R x y}"
hoelzl@59681
  1077
    by auto
hoelzl@59681
  1078
qed
hoelzl@59681
  1079
hoelzl@59664
  1080
bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: rel_pmf
hoelzl@59664
  1081
proof -
hoelzl@59664
  1082
  show "map_pmf id = id" by (rule map_pmf_id)
lp15@59667
  1083
  show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose)
hoelzl@59664
  1084
  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
  1085
    by (intro map_pmf_cong refl)
hoelzl@59664
  1086
hoelzl@59664
  1087
  show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
hoelzl@59664
  1088
    by (rule pmf_set_map)
hoelzl@59664
  1089
wenzelm@60595
  1090
  show "(card_of (set_pmf p), natLeq) \<in> ordLeq" for p :: "'s pmf"
wenzelm@60595
  1091
  proof -
hoelzl@59664
  1092
    have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq"
hoelzl@59664
  1093
      by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"])
hoelzl@59664
  1094
         (auto intro: countable_set_pmf)
hoelzl@59664
  1095
    also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq"
hoelzl@59664
  1096
      by (metis Field_natLeq card_of_least natLeq_Well_order)
wenzelm@60595
  1097
    finally show ?thesis .
wenzelm@60595
  1098
  qed
hoelzl@59664
  1099
traytel@62324
  1100
  show "\<And>R. rel_pmf R = (\<lambda>x y. \<exists>z. set_pmf z \<subseteq> {(x, y). R x y} \<and>
traytel@62324
  1101
    map_pmf fst z = x \<and> map_pmf snd z = y)"
traytel@62324
  1102
     by (auto simp add: fun_eq_iff rel_pmf.simps)
hoelzl@59664
  1103
wenzelm@60595
  1104
  show "rel_pmf R OO rel_pmf S \<le> rel_pmf (R OO S)"
wenzelm@60595
  1105
    for R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
wenzelm@60595
  1106
  proof -
wenzelm@60595
  1107
    { fix p q r
wenzelm@60595
  1108
      assume pq: "rel_pmf R p q"
wenzelm@60595
  1109
        and qr:"rel_pmf S q r"
wenzelm@60595
  1110
      from pq obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
wenzelm@60595
  1111
        and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto
wenzelm@60595
  1112
      from qr obtain qr where qr: "\<And>y z. (y, z) \<in> set_pmf qr \<Longrightarrow> S y z"
wenzelm@60595
  1113
        and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto
lp15@61609
  1114
wenzelm@60595
  1115
      def pr \<equiv> "bind_pmf pq (\<lambda>xy. bind_pmf (cond_pmf qr {yz. fst yz = snd xy}) (\<lambda>yz. return_pmf (fst xy, snd yz)))"
wenzelm@60595
  1116
      have pr_welldefined: "\<And>y. y \<in> q \<Longrightarrow> qr \<inter> {yz. fst yz = y} \<noteq> {}"
wenzelm@60595
  1117
        by (force simp: q')
lp15@61609
  1118
wenzelm@60595
  1119
      have "rel_pmf (R OO S) p r"
wenzelm@60595
  1120
      proof (rule rel_pmf.intros)
wenzelm@60595
  1121
        fix x z assume "(x, z) \<in> pr"
wenzelm@60595
  1122
        then have "\<exists>y. (x, y) \<in> pq \<and> (y, z) \<in> qr"
wenzelm@60595
  1123
          by (auto simp: q pr_welldefined pr_def split_beta)
wenzelm@60595
  1124
        with pq qr show "(R OO S) x z"
wenzelm@60595
  1125
          by blast
wenzelm@60595
  1126
      next
wenzelm@60595
  1127
        have "map_pmf snd pr = map_pmf snd (bind_pmf q (\<lambda>y. cond_pmf qr {yz. fst yz = y}))"
wenzelm@60595
  1128
          by (simp add: pr_def q split_beta bind_map_pmf map_pmf_def[symmetric] map_bind_pmf map_pmf_comp)
wenzelm@60595
  1129
        then show "map_pmf snd pr = r"
wenzelm@60595
  1130
          unfolding r q' bind_map_pmf by (subst (asm) bind_cond_pmf_cancel) (auto simp: eq_commute)
wenzelm@60595
  1131
      qed (simp add: pr_def map_bind_pmf split_beta map_pmf_def[symmetric] p map_pmf_comp)
wenzelm@60595
  1132
    }
wenzelm@60595
  1133
    then show ?thesis
wenzelm@60595
  1134
      by(auto simp add: le_fun_def)
wenzelm@60595
  1135
  qed
hoelzl@59664
  1136
qed (fact natLeq_card_order natLeq_cinfinite)+
hoelzl@59664
  1137
Andreas@61634
  1138
lemma map_pmf_idI: "(\<And>x. x \<in> set_pmf p \<Longrightarrow> f x = x) \<Longrightarrow> map_pmf f p = p"
Andreas@61634
  1139
by(simp cong: pmf.map_cong)
Andreas@61634
  1140
hoelzl@59665
  1141
lemma rel_pmf_conj[simp]:
hoelzl@59665
  1142
  "rel_pmf (\<lambda>x y. P \<and> Q x y) x y \<longleftrightarrow> P \<and> rel_pmf Q x y"
hoelzl@59665
  1143
  "rel_pmf (\<lambda>x y. Q x y \<and> P) x y \<longleftrightarrow> P \<and> rel_pmf Q x y"
hoelzl@59665
  1144
  using set_pmf_not_empty by (fastforce simp: pmf.in_rel subset_eq)+
hoelzl@59665
  1145
hoelzl@59665
  1146
lemma rel_pmf_top[simp]: "rel_pmf top = top"
hoelzl@59665
  1147
  by (auto simp: pmf.in_rel[abs_def] fun_eq_iff map_fst_pair_pmf map_snd_pair_pmf
hoelzl@59665
  1148
           intro: exI[of _ "pair_pmf x y" for x y])
hoelzl@59665
  1149
hoelzl@59664
  1150
lemma rel_pmf_return_pmf1: "rel_pmf R (return_pmf x) M \<longleftrightarrow> (\<forall>a\<in>M. R x a)"
hoelzl@59664
  1151
proof safe
hoelzl@59664
  1152
  fix a assume "a \<in> M" "rel_pmf R (return_pmf x) M"
hoelzl@59664
  1153
  then obtain pq where *: "\<And>a b. (a, b) \<in> set_pmf pq \<Longrightarrow> R a b"
hoelzl@59664
  1154
    and eq: "return_pmf x = map_pmf fst pq" "M = map_pmf snd pq"
hoelzl@59664
  1155
    by (force elim: rel_pmf.cases)
hoelzl@59664
  1156
  moreover have "set_pmf (return_pmf x) = {x}"
hoelzl@59665
  1157
    by simp
wenzelm@61808
  1158
  with \<open>a \<in> M\<close> have "(x, a) \<in> pq"
hoelzl@59665
  1159
    by (force simp: eq)
hoelzl@59664
  1160
  with * show "R x a"
hoelzl@59664
  1161
    by auto
hoelzl@59664
  1162
qed (auto intro!: rel_pmf.intros[where pq="pair_pmf (return_pmf x) M"]
hoelzl@59665
  1163
          simp: map_fst_pair_pmf map_snd_pair_pmf)
hoelzl@59664
  1164
hoelzl@59664
  1165
lemma rel_pmf_return_pmf2: "rel_pmf R M (return_pmf x) \<longleftrightarrow> (\<forall>a\<in>M. R a x)"
hoelzl@59664
  1166
  by (subst pmf.rel_flip[symmetric]) (simp add: rel_pmf_return_pmf1)
hoelzl@59664
  1167
hoelzl@59664
  1168
lemma rel_return_pmf[simp]: "rel_pmf R (return_pmf x1) (return_pmf x2) = R x1 x2"
hoelzl@59664
  1169
  unfolding rel_pmf_return_pmf2 set_return_pmf by simp
hoelzl@59664
  1170
hoelzl@59664
  1171
lemma rel_pmf_False[simp]: "rel_pmf (\<lambda>x y. False) x y = False"
hoelzl@59664
  1172
  unfolding pmf.in_rel fun_eq_iff using set_pmf_not_empty by fastforce
hoelzl@59664
  1173
hoelzl@59664
  1174
lemma rel_pmf_rel_prod:
hoelzl@59664
  1175
  "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
  1176
proof safe
hoelzl@59664
  1177
  assume "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
hoelzl@59664
  1178
  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
  1179
    and eq: "map_pmf fst pq = pair_pmf A A'" "map_pmf snd pq = pair_pmf B B'"
hoelzl@59664
  1180
    by (force elim: rel_pmf.cases)
hoelzl@59664
  1181
  show "rel_pmf R A B"
hoelzl@59664
  1182
  proof (rule rel_pmf.intros)
hoelzl@59664
  1183
    let ?f = "\<lambda>(a, b). (fst a, fst b)"
hoelzl@59664
  1184
    have [simp]: "(\<lambda>x. fst (?f x)) = fst o fst" "(\<lambda>x. snd (?f x)) = fst o snd"
hoelzl@59664
  1185
      by auto
hoelzl@59664
  1186
hoelzl@59664
  1187
    show "map_pmf fst (map_pmf ?f pq) = A"
hoelzl@59664
  1188
      by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
hoelzl@59664
  1189
    show "map_pmf snd (map_pmf ?f pq) = B"
hoelzl@59664
  1190
      by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
hoelzl@59664
  1191
hoelzl@59664
  1192
    fix a b assume "(a, b) \<in> set_pmf (map_pmf ?f pq)"
hoelzl@59664
  1193
    then obtain c d where "((a, c), (b, d)) \<in> set_pmf pq"
hoelzl@59665
  1194
      by auto
hoelzl@59664
  1195
    from pq[OF this] show "R a b" ..
hoelzl@59664
  1196
  qed
hoelzl@59664
  1197
  show "rel_pmf S A' B'"
hoelzl@59664
  1198
  proof (rule rel_pmf.intros)
hoelzl@59664
  1199
    let ?f = "\<lambda>(a, b). (snd a, snd b)"
hoelzl@59664
  1200
    have [simp]: "(\<lambda>x. fst (?f x)) = snd o fst" "(\<lambda>x. snd (?f x)) = snd o snd"
hoelzl@59664
  1201
      by auto
hoelzl@59664
  1202
hoelzl@59664
  1203
    show "map_pmf fst (map_pmf ?f pq) = A'"
hoelzl@59664
  1204
      by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
hoelzl@59664
  1205
    show "map_pmf snd (map_pmf ?f pq) = B'"
hoelzl@59664
  1206
      by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
hoelzl@59664
  1207
hoelzl@59664
  1208
    fix c d assume "(c, d) \<in> set_pmf (map_pmf ?f pq)"
hoelzl@59664
  1209
    then obtain a b where "((a, c), (b, d)) \<in> set_pmf pq"
hoelzl@59665
  1210
      by auto
hoelzl@59664
  1211
    from pq[OF this] show "S c d" ..
hoelzl@59664
  1212
  qed
hoelzl@59664
  1213
next
hoelzl@59664
  1214
  assume "rel_pmf R A B" "rel_pmf S A' B'"
hoelzl@59664
  1215
  then obtain Rpq Spq
hoelzl@59664
  1216
    where Rpq: "\<And>a b. (a, b) \<in> set_pmf Rpq \<Longrightarrow> R a b"
hoelzl@59664
  1217
        "map_pmf fst Rpq = A" "map_pmf snd Rpq = B"
hoelzl@59664
  1218
      and Spq: "\<And>a b. (a, b) \<in> set_pmf Spq \<Longrightarrow> S a b"
hoelzl@59664
  1219
        "map_pmf fst Spq = A'" "map_pmf snd Spq = B'"
hoelzl@59664
  1220
    by (force elim: rel_pmf.cases)
hoelzl@59664
  1221
hoelzl@59664
  1222
  let ?f = "(\<lambda>((a, c), (b, d)). ((a, b), (c, d)))"
hoelzl@59664
  1223
  let ?pq = "map_pmf ?f (pair_pmf Rpq Spq)"
hoelzl@59664
  1224
  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
  1225
    by auto
hoelzl@59664
  1226
hoelzl@59664
  1227
  show "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
hoelzl@59664
  1228
    by (rule rel_pmf.intros[where pq="?pq"])
hoelzl@59665
  1229
       (auto simp: map_snd_pair_pmf map_fst_pair_pmf map_pmf_comp Rpq Spq
hoelzl@59664
  1230
                   map_pair)
hoelzl@59664
  1231
qed
hoelzl@59664
  1232
lp15@59667
  1233
lemma rel_pmf_reflI:
hoelzl@59664
  1234
  assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> P x x"
hoelzl@59664
  1235
  shows "rel_pmf P p p"
hoelzl@59665
  1236
  by (rule rel_pmf.intros[where pq="map_pmf (\<lambda>x. (x, x)) p"])
hoelzl@59665
  1237
     (auto simp add: pmf.map_comp o_def assms)
hoelzl@59664
  1238
Andreas@61634
  1239
lemma rel_pmf_bij_betw:
Andreas@61634
  1240
  assumes f: "bij_betw f (set_pmf p) (set_pmf q)"
Andreas@61634
  1241
  and eq: "\<And>x. x \<in> set_pmf p \<Longrightarrow> pmf p x = pmf q (f x)"
Andreas@61634
  1242
  shows "rel_pmf (\<lambda>x y. f x = y) p q"
Andreas@61634
  1243
proof(rule rel_pmf.intros)
Andreas@61634
  1244
  let ?pq = "map_pmf (\<lambda>x. (x, f x)) p"
Andreas@61634
  1245
  show "map_pmf fst ?pq = p" by(simp add: pmf.map_comp o_def)
Andreas@61634
  1246
Andreas@61634
  1247
  have "map_pmf f p = q"
Andreas@61634
  1248
  proof(rule pmf_eqI)
Andreas@61634
  1249
    fix i
Andreas@61634
  1250
    show "pmf (map_pmf f p) i = pmf q i"
Andreas@61634
  1251
    proof(cases "i \<in> set_pmf q")
Andreas@61634
  1252
      case True
Andreas@61634
  1253
      with f obtain j where "i = f j" "j \<in> set_pmf p"
Andreas@61634
  1254
        by(auto simp add: bij_betw_def image_iff)
Andreas@61634
  1255
      thus ?thesis using f by(simp add: bij_betw_def pmf_map_inj eq)
Andreas@61634
  1256
    next
Andreas@61634
  1257
      case False thus ?thesis
Andreas@61634
  1258
        by(subst pmf_map_outside)(auto simp add: set_pmf_iff eq[symmetric])
Andreas@61634
  1259
    qed
Andreas@61634
  1260
  qed
Andreas@61634
  1261
  then show "map_pmf snd ?pq = q" by(simp add: pmf.map_comp o_def)
Andreas@61634
  1262
qed auto
Andreas@61634
  1263
hoelzl@59664
  1264
context
hoelzl@59664
  1265
begin
hoelzl@59664
  1266
hoelzl@59664
  1267
interpretation pmf_as_measure .
hoelzl@59664
  1268
hoelzl@59664
  1269
definition "join_pmf M = bind_pmf M (\<lambda>x. x)"
hoelzl@59664
  1270
hoelzl@59664
  1271
lemma bind_eq_join_pmf: "bind_pmf M f = join_pmf (map_pmf f M)"
hoelzl@59664
  1272
  unfolding join_pmf_def bind_map_pmf ..
hoelzl@59664
  1273
hoelzl@59664
  1274
lemma join_eq_bind_pmf: "join_pmf M = bind_pmf M id"
hoelzl@59664
  1275
  by (simp add: join_pmf_def id_def)
hoelzl@59664
  1276
hoelzl@59664
  1277
lemma pmf_join: "pmf (join_pmf N) i = (\<integral>M. pmf M i \<partial>measure_pmf N)"
hoelzl@59664
  1278
  unfolding join_pmf_def pmf_bind ..
hoelzl@59664
  1279
hoelzl@59664
  1280
lemma ereal_pmf_join: "ereal (pmf (join_pmf N) i) = (\<integral>\<^sup>+M. pmf M i \<partial>measure_pmf N)"
hoelzl@59664
  1281
  unfolding join_pmf_def ereal_pmf_bind ..
hoelzl@59664
  1282
hoelzl@59665
  1283
lemma set_pmf_join_pmf[simp]: "set_pmf (join_pmf f) = (\<Union>p\<in>set_pmf f. set_pmf p)"
hoelzl@59665
  1284
  by (simp add: join_pmf_def)
hoelzl@59664
  1285
hoelzl@59664
  1286
lemma join_return_pmf: "join_pmf (return_pmf M) = M"
hoelzl@59664
  1287
  by (simp add: integral_return pmf_eq_iff pmf_join return_pmf.rep_eq)
hoelzl@59664
  1288
hoelzl@59664
  1289
lemma map_join_pmf: "map_pmf f (join_pmf AA) = join_pmf (map_pmf (map_pmf f) AA)"
hoelzl@59664
  1290
  by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf)
hoelzl@59664
  1291
hoelzl@59664
  1292
lemma join_map_return_pmf: "join_pmf (map_pmf return_pmf A) = A"
hoelzl@59664
  1293
  by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
hoelzl@59664
  1294
hoelzl@59664
  1295
end
hoelzl@59664
  1296
hoelzl@59664
  1297
lemma rel_pmf_joinI:
hoelzl@59664
  1298
  assumes "rel_pmf (rel_pmf P) p q"
hoelzl@59664
  1299
  shows "rel_pmf P (join_pmf p) (join_pmf q)"
hoelzl@59664
  1300
proof -
hoelzl@59664
  1301
  from assms obtain pq where p: "p = map_pmf fst pq"
hoelzl@59664
  1302
    and q: "q = map_pmf snd pq"
hoelzl@59664
  1303
    and P: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> rel_pmf P x y"
hoelzl@59664
  1304
    by cases auto
lp15@59667
  1305
  from P obtain PQ
hoelzl@59664
  1306
    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
  1307
    and x: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf fst (PQ x y) = x"
hoelzl@59664
  1308
    and y: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf snd (PQ x y) = y"
hoelzl@59664
  1309
    by(metis rel_pmf.simps)
hoelzl@59664
  1310
hoelzl@59664
  1311
  let ?r = "bind_pmf pq (\<lambda>(x, y). PQ x y)"
hoelzl@59665
  1312
  have "\<And>a b. (a, b) \<in> set_pmf ?r \<Longrightarrow> P a b" by (auto intro: PQ)
hoelzl@59664
  1313
  moreover have "map_pmf fst ?r = join_pmf p" "map_pmf snd ?r = join_pmf q"
hoelzl@59664
  1314
    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
  1315
  ultimately show ?thesis ..
hoelzl@59664
  1316
qed
hoelzl@59664
  1317
hoelzl@59664
  1318
lemma rel_pmf_bindI:
hoelzl@59664
  1319
  assumes pq: "rel_pmf R p q"
hoelzl@59664
  1320
  and fg: "\<And>x y. R x y \<Longrightarrow> rel_pmf P (f x) (g y)"
hoelzl@59664
  1321
  shows "rel_pmf P (bind_pmf p f) (bind_pmf q g)"
hoelzl@59664
  1322
  unfolding bind_eq_join_pmf
hoelzl@59664
  1323
  by (rule rel_pmf_joinI)
hoelzl@59664
  1324
     (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
  1325
wenzelm@61808
  1326
text \<open>
hoelzl@59664
  1327
  Proof that @{const rel_pmf} preserves orders.
lp15@59667
  1328
  Antisymmetry proof follows Thm. 1 in N. Saheb-Djahromi, Cpo's of measures for nondeterminism,
lp15@59667
  1329
  Theoretical Computer Science 12(1):19--37, 1980,
hoelzl@59664
  1330
  @{url "http://dx.doi.org/10.1016/0304-3975(80)90003-1"}
wenzelm@61808
  1331
\<close>
hoelzl@59664
  1332
lp15@59667
  1333
lemma
hoelzl@59664
  1334
  assumes *: "rel_pmf R p q"
hoelzl@59664
  1335
  and refl: "reflp R" and trans: "transp R"
hoelzl@59664
  1336
  shows measure_Ici: "measure p {y. R x y} \<le> measure q {y. R x y}" (is ?thesis1)
hoelzl@59664
  1337
  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
  1338
proof -
hoelzl@59664
  1339
  from * obtain pq
hoelzl@59664
  1340
    where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
hoelzl@59664
  1341
    and p: "p = map_pmf fst pq"
hoelzl@59664
  1342
    and q: "q = map_pmf snd pq"
hoelzl@59664
  1343
    by cases auto
hoelzl@59664
  1344
  show ?thesis1 ?thesis2 unfolding p q map_pmf_rep_eq using refl trans
hoelzl@59664
  1345
    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
  1346
qed
hoelzl@59664
  1347
hoelzl@59664
  1348
lemma rel_pmf_inf:
hoelzl@59664
  1349
  fixes p q :: "'a pmf"
hoelzl@59664
  1350
  assumes 1: "rel_pmf R p q"
hoelzl@59664
  1351
  assumes 2: "rel_pmf R q p"
hoelzl@59664
  1352
  and refl: "reflp R" and trans: "transp R"
hoelzl@59664
  1353
  shows "rel_pmf (inf R R\<inverse>\<inverse>) p q"
hoelzl@59681
  1354
proof (subst rel_pmf_iff_equivp, safe)
hoelzl@59681
  1355
  show "equivp (inf R R\<inverse>\<inverse>)"
hoelzl@59681
  1356
    using trans refl by (auto simp: equivp_reflp_symp_transp intro: sympI transpI reflpI dest: transpD reflpD)
lp15@61609
  1357
hoelzl@59681
  1358
  fix C assume "C \<in> UNIV // {(x, y). inf R R\<inverse>\<inverse> x y}"
hoelzl@59681
  1359
  then obtain x where C: "C = {y. R x y \<and> R y x}"
hoelzl@59681
  1360
    by (auto elim: quotientE)
hoelzl@59681
  1361
hoelzl@59670
  1362
  let ?R = "\<lambda>x y. R x y \<and> R y x"
hoelzl@59670
  1363
  let ?\<mu>R = "\<lambda>y. measure q {x. ?R x y}"
hoelzl@59681
  1364
  have "measure p {y. ?R x y} = measure p ({y. R x y} - {y. R x y \<and> \<not> R y x})"
hoelzl@59681
  1365
    by(auto intro!: arg_cong[where f="measure p"])
hoelzl@59681
  1366
  also have "\<dots> = measure p {y. R x y} - measure p {y. R x y \<and> \<not> R y x}"
hoelzl@59681
  1367
    by (rule measure_pmf.finite_measure_Diff) auto
hoelzl@59681
  1368
  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
  1369
    using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ioi)
hoelzl@59681
  1370
  also have "measure p {y. R x y} = measure q {y. R x y}"
hoelzl@59681
  1371
    using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ici)
hoelzl@59681
  1372
  also have "measure q {y. R x y} - measure q {y. R x y \<and> \<not> R y x} =
hoelzl@59681
  1373
    measure q ({y. R x y} - {y. R x y \<and> \<not> R y x})"
hoelzl@59681
  1374
    by(rule measure_pmf.finite_measure_Diff[symmetric]) auto
hoelzl@59681
  1375
  also have "\<dots> = ?\<mu>R x"
hoelzl@59681
  1376
    by(auto intro!: arg_cong[where f="measure q"])
hoelzl@59681
  1377
  finally show "measure p C = measure q C"
hoelzl@59681
  1378
    by (simp add: C conj_commute)
hoelzl@59664
  1379
qed
hoelzl@59664
  1380
hoelzl@59664
  1381
lemma rel_pmf_antisym:
hoelzl@59664
  1382
  fixes p q :: "'a pmf"
hoelzl@59664
  1383
  assumes 1: "rel_pmf R p q"
hoelzl@59664
  1384
  assumes 2: "rel_pmf R q p"
hoelzl@59664
  1385
  and refl: "reflp R" and trans: "transp R" and antisym: "antisymP R"
hoelzl@59664
  1386
  shows "p = q"
hoelzl@59664
  1387
proof -
hoelzl@59664
  1388
  from 1 2 refl trans have "rel_pmf (inf R R\<inverse>\<inverse>) p q" by(rule rel_pmf_inf)
hoelzl@59664
  1389
  also have "inf R R\<inverse>\<inverse> = op ="
hoelzl@59665
  1390
    using refl antisym by (auto intro!: ext simp add: reflpD dest: antisymD)
hoelzl@59664
  1391
  finally show ?thesis unfolding pmf.rel_eq .
hoelzl@59664
  1392
qed
hoelzl@59664
  1393
hoelzl@59664
  1394
lemma reflp_rel_pmf: "reflp R \<Longrightarrow> reflp (rel_pmf R)"
hoelzl@59664
  1395
by(blast intro: reflpI rel_pmf_reflI reflpD)
hoelzl@59664
  1396
hoelzl@59664
  1397
lemma antisymP_rel_pmf:
hoelzl@59664
  1398
  "\<lbrakk> reflp R; transp R; antisymP R \<rbrakk>
hoelzl@59664
  1399
  \<Longrightarrow> antisymP (rel_pmf R)"
hoelzl@59664
  1400
by(rule antisymI)(blast intro: rel_pmf_antisym)
hoelzl@59664
  1401
hoelzl@59664
  1402
lemma transp_rel_pmf:
hoelzl@59664
  1403
  assumes "transp R"
hoelzl@59664
  1404
  shows "transp (rel_pmf R)"
hoelzl@59664
  1405
proof (rule transpI)
hoelzl@59664
  1406
  fix x y z
hoelzl@59664
  1407
  assume "rel_pmf R x y" and "rel_pmf R y z"
hoelzl@59664
  1408
  hence "rel_pmf (R OO R) x z" by (simp add: pmf.rel_compp relcompp.relcompI)
hoelzl@59664
  1409
  thus "rel_pmf R x z"
hoelzl@59664
  1410
    using assms by (metis (no_types) pmf.rel_mono rev_predicate2D transp_relcompp_less_eq)
hoelzl@59664
  1411
qed
hoelzl@59664
  1412
hoelzl@59664
  1413
subsection \<open> Distributions \<close>
hoelzl@59664
  1414
hoelzl@59000
  1415
context
hoelzl@59000
  1416
begin
hoelzl@59000
  1417
hoelzl@59000
  1418
interpretation pmf_as_function .
hoelzl@59000
  1419
hoelzl@59093
  1420
subsubsection \<open> Bernoulli Distribution \<close>
hoelzl@59093
  1421
hoelzl@59000
  1422
lift_definition bernoulli_pmf :: "real \<Rightarrow> bool pmf" is
hoelzl@59000
  1423
  "\<lambda>p b. ((\<lambda>p. if b then p else 1 - p) \<circ> min 1 \<circ> max 0) p"
hoelzl@59000
  1424
  by (auto simp: nn_integral_count_space_finite[where A="{False, True}"] UNIV_bool
hoelzl@59000
  1425
           split: split_max split_min)
hoelzl@59000
  1426
hoelzl@59000
  1427
lemma pmf_bernoulli_True[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) True = p"
hoelzl@59000
  1428
  by transfer simp
hoelzl@59000
  1429
hoelzl@59000
  1430
lemma pmf_bernoulli_False[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) False = 1 - p"
hoelzl@59000
  1431
  by transfer simp
hoelzl@59000
  1432
hoelzl@59000
  1433
lemma set_pmf_bernoulli: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (bernoulli_pmf p) = UNIV"
hoelzl@59000
  1434
  by (auto simp add: set_pmf_iff UNIV_bool)
hoelzl@59000
  1435
lp15@59667
  1436
lemma nn_integral_bernoulli_pmf[simp]:
hoelzl@59002
  1437
  assumes [simp]: "0 \<le> p" "p \<le> 1" "\<And>x. 0 \<le> f x"
hoelzl@59002
  1438
  shows "(\<integral>\<^sup>+x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
hoelzl@59002
  1439
  by (subst nn_integral_measure_pmf_support[of UNIV])
hoelzl@59002
  1440
     (auto simp: UNIV_bool field_simps)
hoelzl@59002
  1441
lp15@59667
  1442
lemma integral_bernoulli_pmf[simp]:
hoelzl@59002
  1443
  assumes [simp]: "0 \<le> p" "p \<le> 1"
hoelzl@59002
  1444
  shows "(\<integral>x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
hoelzl@59002
  1445
  by (subst integral_measure_pmf[of UNIV]) (auto simp: UNIV_bool)
hoelzl@59002
  1446
Andreas@59525
  1447
lemma pmf_bernoulli_half [simp]: "pmf (bernoulli_pmf (1 / 2)) x = 1 / 2"
Andreas@59525
  1448
by(cases x) simp_all
Andreas@59525
  1449
Andreas@59525
  1450
lemma measure_pmf_bernoulli_half: "measure_pmf (bernoulli_pmf (1 / 2)) = uniform_count_measure UNIV"
Andreas@59525
  1451
by(rule measure_eqI)(simp_all add: nn_integral_pmf[symmetric] emeasure_uniform_count_measure nn_integral_count_space_finite sets_uniform_count_measure)
Andreas@59525
  1452
hoelzl@59093
  1453
subsubsection \<open> Geometric Distribution \<close>
hoelzl@59093
  1454
hoelzl@60602
  1455
context
hoelzl@60602
  1456
  fixes p :: real assumes p[arith]: "0 < p" "p \<le> 1"
hoelzl@60602
  1457
begin
hoelzl@60602
  1458
hoelzl@60602
  1459
lift_definition geometric_pmf :: "nat pmf" is "\<lambda>n. (1 - p)^n * p"
hoelzl@59000
  1460
proof
hoelzl@60602
  1461
  have "(\<Sum>i. ereal (p * (1 - p) ^ i)) = ereal (p * (1 / (1 - (1 - p))))"
hoelzl@60602
  1462
    by (intro sums_suminf_ereal sums_mult geometric_sums) auto
hoelzl@60602
  1463
  then show "(\<integral>\<^sup>+ x. ereal ((1 - p)^x * p) \<partial>count_space UNIV) = 1"
hoelzl@59000
  1464
    by (simp add: nn_integral_count_space_nat field_simps)
hoelzl@59000
  1465
qed simp
hoelzl@59000
  1466
hoelzl@60602
  1467
lemma pmf_geometric[simp]: "pmf geometric_pmf n = (1 - p)^n * p"
hoelzl@59000
  1468
  by transfer rule
hoelzl@59000
  1469
hoelzl@60602
  1470
end
hoelzl@60602
  1471
hoelzl@60602
  1472
lemma set_pmf_geometric: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (geometric_pmf p) = UNIV"
lp15@61609
  1473
  by (auto simp: set_pmf_iff)
hoelzl@59000
  1474
hoelzl@59093
  1475
subsubsection \<open> Uniform Multiset Distribution \<close>
hoelzl@59093
  1476
hoelzl@59000
  1477
context
hoelzl@59000
  1478
  fixes M :: "'a multiset" assumes M_not_empty: "M \<noteq> {#}"
hoelzl@59000
  1479
begin
hoelzl@59000
  1480
hoelzl@59000
  1481
lift_definition pmf_of_multiset :: "'a pmf" is "\<lambda>x. count M x / size M"
hoelzl@59000
  1482
proof
lp15@59667
  1483
  show "(\<integral>\<^sup>+ x. ereal (real (count M x) / real (size M)) \<partial>count_space UNIV) = 1"
hoelzl@59000
  1484
    using M_not_empty
hoelzl@59000
  1485
    by (simp add: zero_less_divide_iff nn_integral_count_space nonempty_has_size
hoelzl@59000
  1486
                  setsum_divide_distrib[symmetric])
hoelzl@59000
  1487
       (auto simp: size_multiset_overloaded_eq intro!: setsum.cong)
hoelzl@59000
  1488
qed simp
hoelzl@59000
  1489
hoelzl@59000
  1490
lemma pmf_of_multiset[simp]: "pmf pmf_of_multiset x = count M x / size M"
hoelzl@59000
  1491
  by transfer rule
hoelzl@59000
  1492
nipkow@60495
  1493
lemma set_pmf_of_multiset[simp]: "set_pmf pmf_of_multiset = set_mset M"
hoelzl@59000
  1494
  by (auto simp: set_pmf_iff)
hoelzl@59000
  1495
hoelzl@59000
  1496
end
hoelzl@59000
  1497
hoelzl@59093
  1498
subsubsection \<open> Uniform Distribution \<close>
hoelzl@59093
  1499
hoelzl@59000
  1500
context
hoelzl@59000
  1501
  fixes S :: "'a set" assumes S_not_empty: "S \<noteq> {}" and S_finite: "finite S"
hoelzl@59000
  1502
begin
hoelzl@59000
  1503
hoelzl@59000
  1504
lift_definition pmf_of_set :: "'a pmf" is "\<lambda>x. indicator S x / card S"
hoelzl@59000
  1505
proof
lp15@59667
  1506
  show "(\<integral>\<^sup>+ x. ereal (indicator S x / real (card S)) \<partial>count_space UNIV) = 1"
hoelzl@59000
  1507
    using S_not_empty S_finite by (subst nn_integral_count_space'[of S]) auto
hoelzl@59000
  1508
qed simp
hoelzl@59000
  1509
hoelzl@59000
  1510
lemma pmf_of_set[simp]: "pmf pmf_of_set x = indicator S x / card S"
hoelzl@59000
  1511
  by transfer rule
hoelzl@59000
  1512
hoelzl@59000
  1513
lemma set_pmf_of_set[simp]: "set_pmf pmf_of_set = S"
hoelzl@59000
  1514
  using S_finite S_not_empty by (auto simp: set_pmf_iff)
hoelzl@59000
  1515
Andreas@61634
  1516
lemma emeasure_pmf_of_set_space[simp]: "emeasure pmf_of_set S = 1"
hoelzl@59002
  1517
  by (rule measure_pmf.emeasure_eq_1_AE) (auto simp: AE_measure_pmf_iff)
hoelzl@59002
  1518
lp15@61609
  1519
lemma nn_integral_pmf_of_set':
Andreas@60068
  1520
  "(\<And>x. x \<in> S \<Longrightarrow> f x \<ge> 0) \<Longrightarrow> nn_integral (measure_pmf pmf_of_set) f = setsum f S / card S"
Andreas@60068
  1521
apply(subst nn_integral_measure_pmf_finite, simp_all add: S_finite)
Andreas@60068
  1522
apply(simp add: setsum_ereal_left_distrib[symmetric])
Andreas@60068
  1523
apply(subst ereal_divide', simp add: S_not_empty S_finite)
Andreas@60068
  1524
apply(simp add: ereal_times_divide_eq one_ereal_def[symmetric])
Andreas@60068
  1525
done
Andreas@60068
  1526
lp15@61609
  1527
lemma nn_integral_pmf_of_set:
Andreas@60068
  1528
  "nn_integral (measure_pmf pmf_of_set) f = setsum (max 0 \<circ> f) S / card S"
Andreas@60068
  1529
apply(subst nn_integral_max_0[symmetric])
Andreas@60068
  1530
apply(subst nn_integral_pmf_of_set')
Andreas@60068
  1531
apply simp_all
Andreas@60068
  1532
done
Andreas@60068
  1533
Andreas@60068
  1534
lemma integral_pmf_of_set:
Andreas@60068
  1535
  "integral\<^sup>L (measure_pmf pmf_of_set) f = setsum f S / card S"
Andreas@60068
  1536
apply(simp add: real_lebesgue_integral_def integrable_measure_pmf_finite nn_integral_pmf_of_set S_finite)
Andreas@60068
  1537
apply(subst real_of_ereal_minus')
Andreas@60068
  1538
 apply(simp add: ereal_max_0 S_finite del: ereal_max)
Andreas@60068
  1539
apply(simp add: ereal_max_0 S_finite S_not_empty del: ereal_max)
Andreas@60068
  1540
apply(simp add: field_simps S_finite S_not_empty)
Andreas@60068
  1541
apply(subst setsum.distrib[symmetric])
Andreas@60068
  1542
apply(rule setsum.cong; simp_all)
Andreas@60068
  1543
done
Andreas@60068
  1544
Andreas@61634
  1545
lemma emeasure_pmf_of_set:
Andreas@61634
  1546
  "emeasure (measure_pmf pmf_of_set) A = card (S \<inter> A) / card S"
Andreas@61634
  1547
apply(subst nn_integral_indicator[symmetric], simp)
Andreas@61634
  1548
apply(subst nn_integral_pmf_of_set)
Andreas@61634
  1549
apply(simp add: o_def max_def ereal_indicator[symmetric] S_not_empty S_finite real_of_nat_indicator[symmetric] of_nat_setsum[symmetric] setsum_indicator_eq_card del: of_nat_setsum)
Andreas@61634
  1550
done
Andreas@61634
  1551
hoelzl@59000
  1552
end
hoelzl@59000
  1553
Andreas@60068
  1554
lemma pmf_of_set_singleton: "pmf_of_set {x} = return_pmf x"
Andreas@60068
  1555
by(rule pmf_eqI)(simp add: indicator_def)
Andreas@60068
  1556
lp15@61609
  1557
lemma map_pmf_of_set_inj:
Andreas@60068
  1558
  assumes f: "inj_on f A"
Andreas@60068
  1559
  and [simp]: "A \<noteq> {}" "finite A"
Andreas@60068
  1560
  shows "map_pmf f (pmf_of_set A) = pmf_of_set (f ` A)" (is "?lhs = ?rhs")
Andreas@60068
  1561
proof(rule pmf_eqI)
Andreas@60068
  1562
  fix i
Andreas@60068
  1563
  show "pmf ?lhs i = pmf ?rhs i"
Andreas@60068
  1564
  proof(cases "i \<in> f ` A")
Andreas@60068
  1565
    case True
Andreas@60068
  1566
    then obtain i' where "i = f i'" "i' \<in> A" by auto
Andreas@60068
  1567
    thus ?thesis using f by(simp add: card_image pmf_map_inj)
Andreas@60068
  1568
  next
Andreas@60068
  1569
    case False
Andreas@60068
  1570
    hence "pmf ?lhs i = 0" by(simp add: pmf_eq_0_set_pmf set_map_pmf)
Andreas@60068
  1571
    moreover have "pmf ?rhs i = 0" using False by simp
Andreas@60068
  1572
    ultimately show ?thesis by simp
Andreas@60068
  1573
  qed
Andreas@60068
  1574
qed
Andreas@60068
  1575
Andreas@60068
  1576
lemma bernoulli_pmf_half_conv_pmf_of_set: "bernoulli_pmf (1 / 2) = pmf_of_set UNIV"
Andreas@60068
  1577
by(rule pmf_eqI) simp_all
Andreas@60068
  1578
Andreas@61634
  1579
Andreas@61634
  1580
Andreas@61634
  1581
lemma measure_pmf_of_set:
Andreas@61634
  1582
  assumes "S \<noteq> {}" "finite S"
Andreas@61634
  1583
  shows "measure (measure_pmf (pmf_of_set S)) A = card (S \<inter> A) / card S"
Andreas@61634
  1584
using emeasure_pmf_of_set[OF assms, of A]
Andreas@61634
  1585
unfolding measure_pmf.emeasure_eq_measure by simp
Andreas@61634
  1586
hoelzl@59093
  1587
subsubsection \<open> Poisson Distribution \<close>
hoelzl@59093
  1588
hoelzl@59093
  1589
context
hoelzl@59093
  1590
  fixes rate :: real assumes rate_pos: "0 < rate"
hoelzl@59093
  1591
begin
hoelzl@59093
  1592
hoelzl@59093
  1593
lift_definition poisson_pmf :: "nat pmf" is "\<lambda>k. rate ^ k / fact k * exp (-rate)"
lp15@59730
  1594
proof  (* by Manuel Eberl *)
hoelzl@59093
  1595
  have summable: "summable (\<lambda>x::nat. rate ^ x / fact x)" using summable_exp
haftmann@59557
  1596
    by (simp add: field_simps divide_inverse [symmetric])
hoelzl@59093
  1597
  have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x * exp (-rate) \<partial>count_space UNIV) =
hoelzl@59093
  1598
          exp (-rate) * (\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV)"
hoelzl@59093
  1599
    by (simp add: field_simps nn_integral_cmult[symmetric])
hoelzl@59093
  1600
  also from rate_pos have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV) = (\<Sum>x. rate ^ x / fact x)"
hoelzl@59093
  1601
    by (simp_all add: nn_integral_count_space_nat suminf_ereal summable suminf_ereal_finite)
hoelzl@59093
  1602
  also have "... = exp rate" unfolding exp_def
lp15@59730
  1603
    by (simp add: field_simps divide_inverse [symmetric])
hoelzl@59093
  1604
  also have "ereal (exp (-rate)) * ereal (exp rate) = 1"
hoelzl@59093
  1605
    by (simp add: mult_exp_exp)
lp15@59730
  1606
  finally show "(\<integral>\<^sup>+ x. ereal (rate ^ x / (fact x) * exp (- rate)) \<partial>count_space UNIV) = 1" .
hoelzl@59093
  1607
qed (simp add: rate_pos[THEN less_imp_le])
hoelzl@59093
  1608
hoelzl@59093
  1609
lemma pmf_poisson[simp]: "pmf poisson_pmf k = rate ^ k / fact k * exp (-rate)"
hoelzl@59093
  1610
  by transfer rule
hoelzl@59093
  1611
hoelzl@59093
  1612
lemma set_pmf_poisson[simp]: "set_pmf poisson_pmf = UNIV"
hoelzl@59093
  1613
  using rate_pos by (auto simp: set_pmf_iff)
hoelzl@59093
  1614
hoelzl@59000
  1615
end
hoelzl@59000
  1616
hoelzl@59093
  1617
subsubsection \<open> Binomial Distribution \<close>
hoelzl@59093
  1618
hoelzl@59093
  1619
context
hoelzl@59093
  1620
  fixes n :: nat and p :: real assumes p_nonneg: "0 \<le> p" and p_le_1: "p \<le> 1"
hoelzl@59093
  1621
begin
hoelzl@59093
  1622
hoelzl@59093
  1623
lift_definition binomial_pmf :: "nat pmf" is "\<lambda>k. (n choose k) * p^k * (1 - p)^(n - k)"
hoelzl@59093
  1624
proof
hoelzl@59093
  1625
  have "(\<integral>\<^sup>+k. ereal (real (n choose k) * p ^ k * (1 - p) ^ (n - k)) \<partial>count_space UNIV) =
hoelzl@59093
  1626
    ereal (\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k))"
hoelzl@59093
  1627
    using p_le_1 p_nonneg by (subst nn_integral_count_space') auto
hoelzl@59093
  1628
  also have "(\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k)) = (p + (1 - p)) ^ n"
lp15@61609
  1629
    by (subst binomial_ring) (simp add: atLeast0AtMost)
hoelzl@59093
  1630
  finally show "(\<integral>\<^sup>+ x. ereal (real (n choose x) * p ^ x * (1 - p) ^ (n - x)) \<partial>count_space UNIV) = 1"
hoelzl@59093
  1631
    by simp
hoelzl@59093
  1632
qed (insert p_nonneg p_le_1, simp)
hoelzl@59093
  1633
hoelzl@59093
  1634
lemma pmf_binomial[simp]: "pmf binomial_pmf k = (n choose k) * p^k * (1 - p)^(n - k)"
hoelzl@59093
  1635
  by transfer rule
hoelzl@59093
  1636
hoelzl@59093
  1637
lemma set_pmf_binomial_eq: "set_pmf binomial_pmf = (if p = 0 then {0} else if p = 1 then {n} else {.. n})"
hoelzl@59093
  1638
  using p_nonneg p_le_1 unfolding set_eq_iff set_pmf_iff pmf_binomial by (auto simp: set_pmf_iff)
hoelzl@59093
  1639
hoelzl@59093
  1640
end
hoelzl@59093
  1641
hoelzl@59093
  1642
end
hoelzl@59093
  1643
hoelzl@59093
  1644
lemma set_pmf_binomial_0[simp]: "set_pmf (binomial_pmf n 0) = {0}"
hoelzl@59093
  1645
  by (simp add: set_pmf_binomial_eq)
hoelzl@59093
  1646
hoelzl@59093
  1647
lemma set_pmf_binomial_1[simp]: "set_pmf (binomial_pmf n 1) = {n}"
hoelzl@59093
  1648
  by (simp add: set_pmf_binomial_eq)
hoelzl@59093
  1649
hoelzl@59093
  1650
lemma set_pmf_binomial[simp]: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (binomial_pmf n p) = {..n}"
hoelzl@59093
  1651
  by (simp add: set_pmf_binomial_eq)
hoelzl@59093
  1652
Andreas@61634
  1653
context begin interpretation lifting_syntax .
Andreas@61634
  1654
Andreas@61634
  1655
lemma bind_pmf_parametric [transfer_rule]:
Andreas@61634
  1656
  "(rel_pmf A ===> (A ===> rel_pmf B) ===> rel_pmf B) bind_pmf bind_pmf"
Andreas@61634
  1657
by(blast intro: rel_pmf_bindI dest: rel_funD)
Andreas@61634
  1658
Andreas@61634
  1659
lemma return_pmf_parametric [transfer_rule]: "(A ===> rel_pmf A) return_pmf return_pmf"
Andreas@61634
  1660
by(rule rel_funI) simp
Andreas@61634
  1661
hoelzl@59000
  1662
end
Andreas@61634
  1663
Andreas@61634
  1664
end