src/HOL/Probability/Probability_Mass_Function.thy
author hoelzl
Tue Feb 10 13:50:30 2015 +0100 (2015-02-10)
changeset 59495 03944a830c4a
parent 59494 054f9c9d73ea
child 59496 6faf024a1893
permissions -rw-r--r--
add bind_cond_pmf_cancel
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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/Number_Theory/Binomial"
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  "~~/src/HOL/Library/Multiset"
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begin
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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) countable_support:
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  "countable {x. measure M {x} \<noteq> 0}"
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proof cases
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  assume "measure M (space M) = 0"
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  with bounded_measure measure_le_0_iff have "{x. measure M {x} \<noteq> 0} = {}"
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    by auto
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  then show ?thesis
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    by simp
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next
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  let ?M = "measure M (space M)" and ?m = "\<lambda>x. measure M {x}"
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  assume "?M \<noteq> 0"
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  then have *: "{x. ?m x \<noteq> 0} = (\<Union>n. {x. ?M / Suc n < ?m x})"
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    using reals_Archimedean[of "?m x / ?M" for x]
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    by (auto simp: field_simps not_le[symmetric] measure_nonneg divide_le_0_iff measure_le_0_iff)
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  have **: "\<And>n. finite {x. ?M / Suc n < ?m x}"
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  proof (rule ccontr)
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    fix n assume "infinite {x. ?M / Suc n < ?m x}" (is "infinite ?X")
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    then obtain X where "finite X" "card X = Suc (Suc n)" "X \<subseteq> ?X"
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      by (metis infinite_arbitrarily_large)
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    from this(3) have *: "\<And>x. x \<in> X \<Longrightarrow> ?M / Suc n \<le> ?m x" 
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      by auto
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    { fix x assume "x \<in> X"
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      from `?M \<noteq> 0` *[OF this] have "?m x \<noteq> 0" by (auto simp: field_simps measure_le_0_iff)
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      then have "{x} \<in> sets M" by (auto dest: measure_notin_sets) }
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    note singleton_sets = this
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    have "?M < (\<Sum>x\<in>X. ?M / Suc n)"
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      using `?M \<noteq> 0` 
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      by (simp add: `card X = Suc (Suc n)` real_eq_of_nat[symmetric] real_of_nat_Suc field_simps less_le measure_nonneg)
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    also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"
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      by (rule setsum_mono) fact
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    also have "\<dots> = measure M (\<Union>x\<in>X. {x})"
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      using singleton_sets `finite X`
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      by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def)
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    finally have "?M < measure M (\<Union>x\<in>X. {x})" .
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    moreover have "measure M (\<Union>x\<in>X. {x}) \<le> ?M"
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      using singleton_sets[THEN sets.sets_into_space] by (intro finite_measure_mono) auto
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    ultimately show False by simp
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  qed
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  show ?thesis
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    unfolding * by (intro countable_UN countableI_type countable_finite[OF **])
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qed
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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 {* PMF as measure *}
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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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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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lift_definition map_pmf :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf" is
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  "\<lambda>f M. distr M (count_space UNIV) f"
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proof safe
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  fix M and f :: "'a \<Rightarrow> 'b"
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  let ?D = "distr M (count_space UNIV) f"
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  assume "prob_space M" and [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
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  interpret prob_space M by fact
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  from ae have "AE x in M. measure M (f -` {f x}) \<noteq> 0"
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  proof eventually_elim
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    fix x
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    have "measure M {x} \<le> measure M (f -` {f x})"
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      by (intro finite_measure_mono) auto
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    then show "measure M {x} \<noteq> 0 \<Longrightarrow> measure M (f -` {f x}) \<noteq> 0"
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      using measure_nonneg[of M "{x}"] by auto
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  qed
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  then show "AE x in ?D. measure ?D {x} \<noteq> 0"
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    by (simp add: AE_distr_iff measure_distr measurable_def)
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qed (auto simp: measurable_def prob_space.prob_space_distr)
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declare [[coercion set_pmf]]
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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 sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV"
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  by transfer metis
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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_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 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 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 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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proof -
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  { fix y assume y: "y \<in> M" and P: "AE x in M. P x" "\<not> P y"
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    with P have "AE x in M. x \<noteq> y"
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      by auto
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    with y have False
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      by (simp add: emeasure_pmf_single_eq_zero_iff AE_iff_measurable[OF _ refl]) }
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  then show ?thesis
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    using AE_measure_pmf[of M] by auto
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qed
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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 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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   260
  also have "\<dots> = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space (X \<inter> M))"
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   261
    by (auto intro!: nn_integral_cong_AE split: split_indicator
hoelzl@59000
   262
             simp: pmf.rep_eq measure_pmf.emeasure_eq_measure nn_integral_count_space_indicator
hoelzl@59000
   263
                   AE_count_space set_pmf_iff)
hoelzl@59000
   264
  also have "\<dots> = emeasure M (X \<inter> M)"
hoelzl@59000
   265
    by (rule emeasure_countable_singleton[symmetric]) (auto intro: countable_set_pmf)
hoelzl@59000
   266
  also have "\<dots> = emeasure M X"
hoelzl@59000
   267
    by (auto intro!: emeasure_eq_AE simp: AE_measure_pmf_iff)
hoelzl@59000
   268
  finally show ?thesis
hoelzl@59000
   269
    by (simp add: measure_pmf.emeasure_eq_measure)
hoelzl@59000
   270
qed
hoelzl@59000
   271
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   272
lemma integral_pmf_restrict:
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   273
  "(f::'a \<Rightarrow> 'b::{banach, second_countable_topology}) \<in> borel_measurable (count_space UNIV) \<Longrightarrow>
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   274
    (\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x \<partial>restrict_space M M)"
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   275
  by (auto intro!: integral_cong_AE simp add: integral_restrict_space AE_measure_pmf_iff)
hoelzl@59000
   276
hoelzl@58587
   277
lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1"
hoelzl@58587
   278
proof -
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   279
  have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)"
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   280
    by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf)
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   281
  then show ?thesis
hoelzl@58587
   282
    using measure_pmf.emeasure_space_1 by simp
hoelzl@58587
   283
qed
hoelzl@58587
   284
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   285
lemma in_null_sets_measure_pmfI:
Andreas@59023
   286
  "A \<inter> set_pmf p = {} \<Longrightarrow> A \<in> null_sets (measure_pmf p)"
Andreas@59023
   287
using emeasure_eq_0_AE[where ?P="\<lambda>x. x \<in> A" and M="measure_pmf p"]
Andreas@59023
   288
by(auto simp add: null_sets_def AE_measure_pmf_iff)
Andreas@59023
   289
hoelzl@58587
   290
lemma map_pmf_id[simp]: "map_pmf id = id"
hoelzl@58587
   291
  by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI)
hoelzl@58587
   292
hoelzl@59053
   293
lemma map_pmf_ident[simp]: "map_pmf (\<lambda>x. x) = (\<lambda>x. x)"
hoelzl@59053
   294
  using map_pmf_id unfolding id_def .
hoelzl@59053
   295
hoelzl@58587
   296
lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g"
hoelzl@58587
   297
  by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def) 
hoelzl@58587
   298
hoelzl@59000
   299
lemma map_pmf_comp: "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M"
hoelzl@59000
   300
  using map_pmf_compose[of f g] by (simp add: comp_def)
hoelzl@59000
   301
hoelzl@58587
   302
lemma map_pmf_cong:
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   303
  assumes "p = q"
hoelzl@58587
   304
  shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> map_pmf f p = map_pmf g q"
hoelzl@58587
   305
  unfolding `p = q`[symmetric] measure_pmf_inject[symmetric] map_pmf.rep_eq
hoelzl@58587
   306
  by (auto simp add: emeasure_distr AE_measure_pmf_iff intro!: emeasure_eq_AE measure_eqI)
hoelzl@58587
   307
hoelzl@59002
   308
lemma emeasure_map_pmf[simp]: "emeasure (map_pmf f M) X = emeasure M (f -` X)"
hoelzl@59002
   309
  unfolding map_pmf.rep_eq by (subst emeasure_distr) auto
hoelzl@59002
   310
hoelzl@59002
   311
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@59002
   312
  unfolding map_pmf.rep_eq by (intro nn_integral_distr) auto
hoelzl@59002
   313
Andreas@59023
   314
lemma ereal_pmf_map: "pmf (map_pmf f p) x = (\<integral>\<^sup>+ y. indicator (f -` {x}) y \<partial>measure_pmf p)"
Andreas@59023
   315
proof(transfer fixing: f x)
Andreas@59023
   316
  fix p :: "'b measure"
Andreas@59023
   317
  presume "prob_space p"
Andreas@59023
   318
  then interpret prob_space p .
Andreas@59023
   319
  presume "sets p = UNIV"
Andreas@59023
   320
  then show "ereal (measure (distr p (count_space UNIV) f) {x}) = integral\<^sup>N p (indicator (f -` {x}))"
Andreas@59023
   321
    by(simp add: measure_distr measurable_def emeasure_eq_measure)
Andreas@59023
   322
qed simp_all
Andreas@59023
   323
hoelzl@58587
   324
lemma pmf_set_map: 
hoelzl@58587
   325
  fixes f :: "'a \<Rightarrow> 'b"
hoelzl@58587
   326
  shows "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
hoelzl@58587
   327
proof (rule, transfer, clarsimp simp add: measure_distr measurable_def)
hoelzl@58587
   328
  fix f :: "'a \<Rightarrow> 'b" and M :: "'a measure"
hoelzl@58587
   329
  assume "prob_space M" and ae: "AE x in M. measure M {x} \<noteq> 0" and [simp]: "sets M = UNIV"
hoelzl@58587
   330
  interpret prob_space M by fact
hoelzl@58587
   331
  show "{x. measure M (f -` {x}) \<noteq> 0} = f ` {x. measure M {x} \<noteq> 0}"
hoelzl@58587
   332
  proof safe
hoelzl@58587
   333
    fix x assume "measure M (f -` {x}) \<noteq> 0"
hoelzl@58587
   334
    moreover have "measure M (f -` {x}) = measure M {y. f y = x \<and> measure M {y} \<noteq> 0}"
hoelzl@58587
   335
      using ae by (intro finite_measure_eq_AE) auto
hoelzl@58587
   336
    ultimately have "{y. f y = x \<and> measure M {y} \<noteq> 0} \<noteq> {}"
hoelzl@58587
   337
      by (metis measure_empty)
hoelzl@58587
   338
    then show "x \<in> f ` {x. measure M {x} \<noteq> 0}"
hoelzl@58587
   339
      by auto
hoelzl@58587
   340
  next
hoelzl@58587
   341
    fix x assume "measure M {x} \<noteq> 0"
hoelzl@58587
   342
    then have "0 < measure M {x}"
hoelzl@58587
   343
      using measure_nonneg[of M "{x}"] by auto
hoelzl@58587
   344
    also have "measure M {x} \<le> measure M (f -` {f x})"
hoelzl@58587
   345
      by (intro finite_measure_mono) auto
hoelzl@58587
   346
    finally show "measure M (f -` {f x}) = 0 \<Longrightarrow> False"
hoelzl@58587
   347
      by simp
hoelzl@58587
   348
  qed
hoelzl@58587
   349
qed
hoelzl@58587
   350
hoelzl@59000
   351
lemma set_map_pmf: "set_pmf (map_pmf f M) = f`set_pmf M"
hoelzl@59000
   352
  using pmf_set_map[of f] by (auto simp: comp_def fun_eq_iff)
hoelzl@59000
   353
Andreas@59023
   354
lemma nn_integral_pmf: "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = emeasure (measure_pmf p) A"
Andreas@59023
   355
proof -
Andreas@59023
   356
  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
   357
    by(auto simp add: nn_integral_count_space_indicator indicator_def set_pmf_iff intro: nn_integral_cong)
Andreas@59023
   358
  also have "\<dots> = emeasure (measure_pmf p) (\<Union>x\<in>A \<inter> set_pmf p. {x})"
Andreas@59023
   359
    by(subst emeasure_UN_countable)(auto simp add: emeasure_pmf_single disjoint_family_on_def)
Andreas@59023
   360
  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
   361
    by(rule emeasure_Un_null_set[symmetric])(auto intro: in_null_sets_measure_pmfI)
Andreas@59023
   362
  also have "\<dots> = emeasure (measure_pmf p) A"
Andreas@59023
   363
    by(auto intro: arg_cong2[where f=emeasure])
Andreas@59023
   364
  finally show ?thesis .
Andreas@59023
   365
qed
Andreas@59023
   366
hoelzl@59000
   367
subsection {* PMFs as function *}
hoelzl@59000
   368
hoelzl@58587
   369
context
hoelzl@58587
   370
  fixes f :: "'a \<Rightarrow> real"
hoelzl@58587
   371
  assumes nonneg: "\<And>x. 0 \<le> f x"
hoelzl@58587
   372
  assumes prob: "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
hoelzl@58587
   373
begin
hoelzl@58587
   374
hoelzl@58587
   375
lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ereal \<circ> f)"
hoelzl@58587
   376
proof (intro conjI)
hoelzl@58587
   377
  have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
hoelzl@58587
   378
    by (simp split: split_indicator)
hoelzl@58587
   379
  show "AE x in density (count_space UNIV) (ereal \<circ> f).
hoelzl@58587
   380
    measure (density (count_space UNIV) (ereal \<circ> f)) {x} \<noteq> 0"
hoelzl@59092
   381
    by (simp add: AE_density nonneg measure_def emeasure_density max_def)
hoelzl@58587
   382
  show "prob_space (density (count_space UNIV) (ereal \<circ> f))"
hoelzl@58587
   383
    by default (simp add: emeasure_density prob)
hoelzl@58587
   384
qed simp
hoelzl@58587
   385
hoelzl@58587
   386
lemma pmf_embed_pmf: "pmf embed_pmf x = f x"
hoelzl@58587
   387
proof transfer
hoelzl@58587
   388
  have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
hoelzl@58587
   389
    by (simp split: split_indicator)
hoelzl@58587
   390
  fix x show "measure (density (count_space UNIV) (ereal \<circ> f)) {x} = f x"
hoelzl@59092
   391
    by transfer (simp add: measure_def emeasure_density nonneg max_def)
hoelzl@58587
   392
qed
hoelzl@58587
   393
hoelzl@58587
   394
end
hoelzl@58587
   395
hoelzl@58587
   396
lemma embed_pmf_transfer:
hoelzl@58730
   397
  "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
   398
  by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer)
hoelzl@58587
   399
hoelzl@59000
   400
lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)"
hoelzl@59000
   401
proof (transfer, elim conjE)
hoelzl@59000
   402
  fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
hoelzl@59000
   403
  assume "prob_space M" then interpret prob_space M .
hoelzl@59000
   404
  show "M = density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))"
hoelzl@59000
   405
  proof (rule measure_eqI)
hoelzl@59000
   406
    fix A :: "'a set"
hoelzl@59000
   407
    have "(\<integral>\<^sup>+ x. ereal (measure M {x}) * indicator A x \<partial>count_space UNIV) = 
hoelzl@59000
   408
      (\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)"
hoelzl@59000
   409
      by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator)
hoelzl@59000
   410
    also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))"
hoelzl@59000
   411
      by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space)
hoelzl@59000
   412
    also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})"
hoelzl@59000
   413
      by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support)
hoelzl@59000
   414
         (auto simp: disjoint_family_on_def)
hoelzl@59000
   415
    also have "\<dots> = emeasure M A"
hoelzl@59000
   416
      using ae by (intro emeasure_eq_AE) auto
hoelzl@59000
   417
    finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))) A"
hoelzl@59000
   418
      using emeasure_space_1 by (simp add: emeasure_density)
hoelzl@59000
   419
  qed simp
hoelzl@59000
   420
qed
hoelzl@59000
   421
hoelzl@58587
   422
lemma td_pmf_embed_pmf:
hoelzl@58587
   423
  "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
   424
  unfolding type_definition_def
hoelzl@58587
   425
proof safe
hoelzl@58587
   426
  fix p :: "'a pmf"
hoelzl@58587
   427
  have "(\<integral>\<^sup>+ x. 1 \<partial>measure_pmf p) = 1"
hoelzl@58587
   428
    using measure_pmf.emeasure_space_1[of p] by simp
hoelzl@58587
   429
  then show *: "(\<integral>\<^sup>+ x. ereal (pmf p x) \<partial>count_space UNIV) = 1"
hoelzl@58587
   430
    by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const)
hoelzl@58587
   431
hoelzl@58587
   432
  show "embed_pmf (pmf p) = p"
hoelzl@58587
   433
    by (intro measure_pmf_inject[THEN iffD1])
hoelzl@58587
   434
       (simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def)
hoelzl@58587
   435
next
hoelzl@58587
   436
  fix f :: "'a \<Rightarrow> real" assume "\<forall>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
hoelzl@58587
   437
  then show "pmf (embed_pmf f) = f"
hoelzl@58587
   438
    by (auto intro!: pmf_embed_pmf)
hoelzl@58587
   439
qed (rule pmf_nonneg)
hoelzl@58587
   440
hoelzl@58587
   441
end
hoelzl@58587
   442
hoelzl@58587
   443
locale pmf_as_function
hoelzl@58587
   444
begin
hoelzl@58587
   445
hoelzl@58587
   446
setup_lifting td_pmf_embed_pmf
hoelzl@58587
   447
hoelzl@58730
   448
lemma set_pmf_transfer[transfer_rule]: 
hoelzl@58730
   449
  assumes "bi_total A"
hoelzl@58730
   450
  shows "rel_fun (pcr_pmf A) (rel_set A) (\<lambda>f. {x. f x \<noteq> 0}) set_pmf"  
hoelzl@58730
   451
  using `bi_total A`
hoelzl@58730
   452
  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
   453
     metis+
hoelzl@58730
   454
hoelzl@59000
   455
end
hoelzl@59000
   456
hoelzl@59000
   457
context
hoelzl@59000
   458
begin
hoelzl@59000
   459
hoelzl@59000
   460
interpretation pmf_as_function .
hoelzl@59000
   461
hoelzl@59000
   462
lemma pmf_eqI: "(\<And>i. pmf M i = pmf N i) \<Longrightarrow> M = N"
hoelzl@59000
   463
  by transfer auto
hoelzl@59000
   464
hoelzl@59000
   465
lemma pmf_eq_iff: "M = N \<longleftrightarrow> (\<forall>i. pmf M i = pmf N i)"
hoelzl@59000
   466
  by (auto intro: pmf_eqI)
hoelzl@59000
   467
hoelzl@59000
   468
end
hoelzl@59000
   469
hoelzl@59000
   470
context
hoelzl@59000
   471
begin
hoelzl@59000
   472
hoelzl@59000
   473
interpretation pmf_as_function .
hoelzl@59000
   474
hoelzl@59093
   475
subsubsection \<open> Bernoulli Distribution \<close>
hoelzl@59093
   476
hoelzl@59000
   477
lift_definition bernoulli_pmf :: "real \<Rightarrow> bool pmf" is
hoelzl@59000
   478
  "\<lambda>p b. ((\<lambda>p. if b then p else 1 - p) \<circ> min 1 \<circ> max 0) p"
hoelzl@59000
   479
  by (auto simp: nn_integral_count_space_finite[where A="{False, True}"] UNIV_bool
hoelzl@59000
   480
           split: split_max split_min)
hoelzl@59000
   481
hoelzl@59000
   482
lemma pmf_bernoulli_True[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) True = p"
hoelzl@59000
   483
  by transfer simp
hoelzl@59000
   484
hoelzl@59000
   485
lemma pmf_bernoulli_False[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) False = 1 - p"
hoelzl@59000
   486
  by transfer simp
hoelzl@59000
   487
hoelzl@59000
   488
lemma set_pmf_bernoulli: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (bernoulli_pmf p) = UNIV"
hoelzl@59000
   489
  by (auto simp add: set_pmf_iff UNIV_bool)
hoelzl@59000
   490
hoelzl@59002
   491
lemma nn_integral_bernoulli_pmf[simp]: 
hoelzl@59002
   492
  assumes [simp]: "0 \<le> p" "p \<le> 1" "\<And>x. 0 \<le> f x"
hoelzl@59002
   493
  shows "(\<integral>\<^sup>+x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
hoelzl@59002
   494
  by (subst nn_integral_measure_pmf_support[of UNIV])
hoelzl@59002
   495
     (auto simp: UNIV_bool field_simps)
hoelzl@59002
   496
hoelzl@59002
   497
lemma integral_bernoulli_pmf[simp]: 
hoelzl@59002
   498
  assumes [simp]: "0 \<le> p" "p \<le> 1"
hoelzl@59002
   499
  shows "(\<integral>x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
hoelzl@59002
   500
  by (subst integral_measure_pmf[of UNIV]) (auto simp: UNIV_bool)
hoelzl@59002
   501
hoelzl@59093
   502
subsubsection \<open> Geometric Distribution \<close>
hoelzl@59093
   503
hoelzl@59000
   504
lift_definition geometric_pmf :: "nat pmf" is "\<lambda>n. 1 / 2^Suc n"
hoelzl@59000
   505
proof
hoelzl@59000
   506
  note geometric_sums[of "1 / 2"]
hoelzl@59000
   507
  note sums_mult[OF this, of "1 / 2"]
hoelzl@59000
   508
  from sums_suminf_ereal[OF this]
hoelzl@59000
   509
  show "(\<integral>\<^sup>+ x. ereal (1 / 2 ^ Suc x) \<partial>count_space UNIV) = 1"
hoelzl@59000
   510
    by (simp add: nn_integral_count_space_nat field_simps)
hoelzl@59000
   511
qed simp
hoelzl@59000
   512
hoelzl@59000
   513
lemma pmf_geometric[simp]: "pmf geometric_pmf n = 1 / 2^Suc n"
hoelzl@59000
   514
  by transfer rule
hoelzl@59000
   515
hoelzl@59002
   516
lemma set_pmf_geometric[simp]: "set_pmf geometric_pmf = UNIV"
hoelzl@59000
   517
  by (auto simp: set_pmf_iff)
hoelzl@59000
   518
hoelzl@59093
   519
subsubsection \<open> Uniform Multiset Distribution \<close>
hoelzl@59093
   520
hoelzl@59000
   521
context
hoelzl@59000
   522
  fixes M :: "'a multiset" assumes M_not_empty: "M \<noteq> {#}"
hoelzl@59000
   523
begin
hoelzl@59000
   524
hoelzl@59000
   525
lift_definition pmf_of_multiset :: "'a pmf" is "\<lambda>x. count M x / size M"
hoelzl@59000
   526
proof
hoelzl@59000
   527
  show "(\<integral>\<^sup>+ x. ereal (real (count M x) / real (size M)) \<partial>count_space UNIV) = 1"  
hoelzl@59000
   528
    using M_not_empty
hoelzl@59000
   529
    by (simp add: zero_less_divide_iff nn_integral_count_space nonempty_has_size
hoelzl@59000
   530
                  setsum_divide_distrib[symmetric])
hoelzl@59000
   531
       (auto simp: size_multiset_overloaded_eq intro!: setsum.cong)
hoelzl@59000
   532
qed simp
hoelzl@59000
   533
hoelzl@59000
   534
lemma pmf_of_multiset[simp]: "pmf pmf_of_multiset x = count M x / size M"
hoelzl@59000
   535
  by transfer rule
hoelzl@59000
   536
hoelzl@59000
   537
lemma set_pmf_of_multiset[simp]: "set_pmf pmf_of_multiset = set_of M"
hoelzl@59000
   538
  by (auto simp: set_pmf_iff)
hoelzl@59000
   539
hoelzl@59000
   540
end
hoelzl@59000
   541
hoelzl@59093
   542
subsubsection \<open> Uniform Distribution \<close>
hoelzl@59093
   543
hoelzl@59000
   544
context
hoelzl@59000
   545
  fixes S :: "'a set" assumes S_not_empty: "S \<noteq> {}" and S_finite: "finite S"
hoelzl@59000
   546
begin
hoelzl@59000
   547
hoelzl@59000
   548
lift_definition pmf_of_set :: "'a pmf" is "\<lambda>x. indicator S x / card S"
hoelzl@59000
   549
proof
hoelzl@59000
   550
  show "(\<integral>\<^sup>+ x. ereal (indicator S x / real (card S)) \<partial>count_space UNIV) = 1"  
hoelzl@59000
   551
    using S_not_empty S_finite by (subst nn_integral_count_space'[of S]) auto
hoelzl@59000
   552
qed simp
hoelzl@59000
   553
hoelzl@59000
   554
lemma pmf_of_set[simp]: "pmf pmf_of_set x = indicator S x / card S"
hoelzl@59000
   555
  by transfer rule
hoelzl@59000
   556
hoelzl@59000
   557
lemma set_pmf_of_set[simp]: "set_pmf pmf_of_set = S"
hoelzl@59000
   558
  using S_finite S_not_empty by (auto simp: set_pmf_iff)
hoelzl@59000
   559
hoelzl@59002
   560
lemma emeasure_pmf_of_set[simp]: "emeasure pmf_of_set S = 1"
hoelzl@59002
   561
  by (rule measure_pmf.emeasure_eq_1_AE) (auto simp: AE_measure_pmf_iff)
hoelzl@59002
   562
hoelzl@59000
   563
end
hoelzl@59000
   564
hoelzl@59093
   565
subsubsection \<open> Poisson Distribution \<close>
hoelzl@59093
   566
hoelzl@59093
   567
context
hoelzl@59093
   568
  fixes rate :: real assumes rate_pos: "0 < rate"
hoelzl@59093
   569
begin
hoelzl@59093
   570
hoelzl@59093
   571
lift_definition poisson_pmf :: "nat pmf" is "\<lambda>k. rate ^ k / fact k * exp (-rate)"
hoelzl@59093
   572
proof
hoelzl@59093
   573
  (* Proof by Manuel Eberl *)
hoelzl@59093
   574
hoelzl@59093
   575
  have summable: "summable (\<lambda>x::nat. rate ^ x / fact x)" using summable_exp
hoelzl@59093
   576
    by (simp add: field_simps field_divide_inverse[symmetric])
hoelzl@59093
   577
  have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x * exp (-rate) \<partial>count_space UNIV) =
hoelzl@59093
   578
          exp (-rate) * (\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV)"
hoelzl@59093
   579
    by (simp add: field_simps nn_integral_cmult[symmetric])
hoelzl@59093
   580
  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
   581
    by (simp_all add: nn_integral_count_space_nat suminf_ereal summable suminf_ereal_finite)
hoelzl@59093
   582
  also have "... = exp rate" unfolding exp_def
hoelzl@59093
   583
    by (simp add: field_simps field_divide_inverse[symmetric] transfer_int_nat_factorial)
hoelzl@59093
   584
  also have "ereal (exp (-rate)) * ereal (exp rate) = 1"
hoelzl@59093
   585
    by (simp add: mult_exp_exp)
hoelzl@59093
   586
  finally show "(\<integral>\<^sup>+ x. ereal (rate ^ x / real (fact x) * exp (- rate)) \<partial>count_space UNIV) = 1" .
hoelzl@59093
   587
qed (simp add: rate_pos[THEN less_imp_le])
hoelzl@59093
   588
hoelzl@59093
   589
lemma pmf_poisson[simp]: "pmf poisson_pmf k = rate ^ k / fact k * exp (-rate)"
hoelzl@59093
   590
  by transfer rule
hoelzl@59093
   591
hoelzl@59093
   592
lemma set_pmf_poisson[simp]: "set_pmf poisson_pmf = UNIV"
hoelzl@59093
   593
  using rate_pos by (auto simp: set_pmf_iff)
hoelzl@59093
   594
hoelzl@59000
   595
end
hoelzl@59000
   596
hoelzl@59093
   597
subsubsection \<open> Binomial Distribution \<close>
hoelzl@59093
   598
hoelzl@59093
   599
context
hoelzl@59093
   600
  fixes n :: nat and p :: real assumes p_nonneg: "0 \<le> p" and p_le_1: "p \<le> 1"
hoelzl@59093
   601
begin
hoelzl@59093
   602
hoelzl@59093
   603
lift_definition binomial_pmf :: "nat pmf" is "\<lambda>k. (n choose k) * p^k * (1 - p)^(n - k)"
hoelzl@59093
   604
proof
hoelzl@59093
   605
  have "(\<integral>\<^sup>+k. ereal (real (n choose k) * p ^ k * (1 - p) ^ (n - k)) \<partial>count_space UNIV) =
hoelzl@59093
   606
    ereal (\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k))"
hoelzl@59093
   607
    using p_le_1 p_nonneg by (subst nn_integral_count_space') auto
hoelzl@59093
   608
  also have "(\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k)) = (p + (1 - p)) ^ n"
hoelzl@59093
   609
    by (subst binomial_ring) (simp add: atLeast0AtMost real_of_nat_def)
hoelzl@59093
   610
  finally show "(\<integral>\<^sup>+ x. ereal (real (n choose x) * p ^ x * (1 - p) ^ (n - x)) \<partial>count_space UNIV) = 1"
hoelzl@59093
   611
    by simp
hoelzl@59093
   612
qed (insert p_nonneg p_le_1, simp)
hoelzl@59093
   613
hoelzl@59093
   614
lemma pmf_binomial[simp]: "pmf binomial_pmf k = (n choose k) * p^k * (1 - p)^(n - k)"
hoelzl@59093
   615
  by transfer rule
hoelzl@59093
   616
hoelzl@59093
   617
lemma set_pmf_binomial_eq: "set_pmf binomial_pmf = (if p = 0 then {0} else if p = 1 then {n} else {.. n})"
hoelzl@59093
   618
  using p_nonneg p_le_1 unfolding set_eq_iff set_pmf_iff pmf_binomial by (auto simp: set_pmf_iff)
hoelzl@59093
   619
hoelzl@59093
   620
end
hoelzl@59093
   621
hoelzl@59093
   622
end
hoelzl@59093
   623
hoelzl@59093
   624
lemma set_pmf_binomial_0[simp]: "set_pmf (binomial_pmf n 0) = {0}"
hoelzl@59093
   625
  by (simp add: set_pmf_binomial_eq)
hoelzl@59093
   626
hoelzl@59093
   627
lemma set_pmf_binomial_1[simp]: "set_pmf (binomial_pmf n 1) = {n}"
hoelzl@59093
   628
  by (simp add: set_pmf_binomial_eq)
hoelzl@59093
   629
hoelzl@59093
   630
lemma set_pmf_binomial[simp]: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (binomial_pmf n p) = {..n}"
hoelzl@59093
   631
  by (simp add: set_pmf_binomial_eq)
hoelzl@59093
   632
hoelzl@59093
   633
subsection \<open> Monad Interpretation \<close>
hoelzl@59000
   634
hoelzl@59000
   635
lemma measurable_measure_pmf[measurable]:
hoelzl@59000
   636
  "(\<lambda>x. measure_pmf (M x)) \<in> measurable (count_space UNIV) (subprob_algebra (count_space UNIV))"
hoelzl@59000
   637
  by (auto simp: space_subprob_algebra intro!: prob_space_imp_subprob_space) unfold_locales
hoelzl@59000
   638
hoelzl@59000
   639
lemma bind_pmf_cong:
hoelzl@59000
   640
  assumes "\<And>x. A x \<in> space (subprob_algebra N)" "\<And>x. B x \<in> space (subprob_algebra N)"
hoelzl@59000
   641
  assumes "\<And>i. i \<in> set_pmf x \<Longrightarrow> A i = B i"
hoelzl@59000
   642
  shows "bind (measure_pmf x) A = bind (measure_pmf x) B"
hoelzl@59000
   643
proof (rule measure_eqI)
hoelzl@59000
   644
  show "sets (measure_pmf x \<guillemotright>= A) = sets (measure_pmf x \<guillemotright>= B)"
hoelzl@59048
   645
    using assms by (subst (1 2) sets_bind) (auto simp: space_subprob_algebra)
hoelzl@59000
   646
next
hoelzl@59000
   647
  fix X assume "X \<in> sets (measure_pmf x \<guillemotright>= A)"
hoelzl@59000
   648
  then have X: "X \<in> sets N"
hoelzl@59048
   649
    using assms by (subst (asm) sets_bind) (auto simp: space_subprob_algebra)
hoelzl@59000
   650
  show "emeasure (measure_pmf x \<guillemotright>= A) X = emeasure (measure_pmf x \<guillemotright>= B) X"
hoelzl@59000
   651
    using assms
hoelzl@59000
   652
    by (subst (1 2) emeasure_bind[where N=N, OF _ _ X])
hoelzl@59000
   653
       (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
hoelzl@59000
   654
qed
hoelzl@59000
   655
hoelzl@59000
   656
context
hoelzl@59000
   657
begin
hoelzl@59000
   658
hoelzl@59000
   659
interpretation pmf_as_measure .
hoelzl@59000
   660
hoelzl@59000
   661
lift_definition join_pmf :: "'a pmf pmf \<Rightarrow> 'a pmf" is "\<lambda>M. measure_pmf M \<guillemotright>= measure_pmf"
hoelzl@59000
   662
proof (intro conjI)
hoelzl@59000
   663
  fix M :: "'a pmf pmf"
hoelzl@59000
   664
hoelzl@59000
   665
  interpret bind: prob_space "measure_pmf M \<guillemotright>= measure_pmf"
hoelzl@59048
   666
    apply (intro measure_pmf.prob_space_bind[where S="count_space UNIV"] AE_I2)
hoelzl@59048
   667
    apply (auto intro!: subprob_space_measure_pmf simp: space_subprob_algebra)
hoelzl@59000
   668
    apply unfold_locales
hoelzl@59000
   669
    done
hoelzl@59000
   670
  show "prob_space (measure_pmf M \<guillemotright>= measure_pmf)"
hoelzl@59000
   671
    by intro_locales
hoelzl@59000
   672
  show "sets (measure_pmf M \<guillemotright>= measure_pmf) = UNIV"
hoelzl@59048
   673
    by (subst sets_bind) auto
hoelzl@59000
   674
  have "AE x in measure_pmf M \<guillemotright>= measure_pmf. emeasure (measure_pmf M \<guillemotright>= measure_pmf) {x} \<noteq> 0"
hoelzl@59048
   675
    by (auto simp: AE_bind[where B="count_space UNIV"] measure_pmf_in_subprob_algebra
hoelzl@59048
   676
                   emeasure_bind[where N="count_space UNIV"] AE_measure_pmf_iff nn_integral_0_iff_AE
hoelzl@59048
   677
                   measure_pmf.emeasure_eq_measure measure_le_0_iff set_pmf_iff pmf.rep_eq)
hoelzl@59000
   678
  then show "AE x in measure_pmf M \<guillemotright>= measure_pmf. measure (measure_pmf M \<guillemotright>= measure_pmf) {x} \<noteq> 0"
hoelzl@59000
   679
    unfolding bind.emeasure_eq_measure by simp
hoelzl@59000
   680
qed
hoelzl@59000
   681
hoelzl@59000
   682
lemma pmf_join: "pmf (join_pmf N) i = (\<integral>M. pmf M i \<partial>measure_pmf N)"
hoelzl@59000
   683
proof (transfer fixing: N i)
hoelzl@59000
   684
  have N: "subprob_space (measure_pmf N)"
hoelzl@59000
   685
    by (rule prob_space_imp_subprob_space) intro_locales
hoelzl@59000
   686
  show "measure (measure_pmf N \<guillemotright>= measure_pmf) {i} = integral\<^sup>L (measure_pmf N) (\<lambda>M. measure M {i})"
hoelzl@59000
   687
    using measurable_measure_pmf[of "\<lambda>x. x"]
hoelzl@59000
   688
    by (intro subprob_space.measure_bind[where N="count_space UNIV", OF N]) auto
hoelzl@59000
   689
qed (auto simp: Transfer.Rel_def rel_fun_def cr_pmf_def)
hoelzl@59000
   690
hoelzl@59493
   691
lemma ereal_pmf_join: "ereal (pmf (join_pmf N) i) = (\<integral>\<^sup>+M. pmf M i \<partial>measure_pmf N)"
hoelzl@59493
   692
  unfolding pmf_join
hoelzl@59493
   693
  by (intro nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])
hoelzl@59493
   694
     (auto simp: pmf_le_1 pmf_nonneg)
hoelzl@59493
   695
Andreas@59024
   696
lemma set_pmf_join_pmf: "set_pmf (join_pmf f) = (\<Union>p\<in>set_pmf f. set_pmf p)"
Andreas@59024
   697
apply(simp add: set_eq_iff set_pmf_iff pmf_join)
Andreas@59024
   698
apply(subst integral_nonneg_eq_0_iff_AE)
Andreas@59024
   699
apply(auto simp add: pmf_le_1 pmf_nonneg AE_measure_pmf_iff intro!: measure_pmf.integrable_const_bound[where B=1])
Andreas@59024
   700
done
Andreas@59024
   701
hoelzl@59000
   702
lift_definition return_pmf :: "'a \<Rightarrow> 'a pmf" is "return (count_space UNIV)"
hoelzl@59000
   703
  by (auto intro!: prob_space_return simp: AE_return measure_return)
hoelzl@59000
   704
hoelzl@59000
   705
lemma join_return_pmf: "join_pmf (return_pmf M) = M"
hoelzl@59000
   706
  by (simp add: integral_return pmf_eq_iff pmf_join return_pmf.rep_eq)
hoelzl@59000
   707
hoelzl@59000
   708
lemma map_return_pmf: "map_pmf f (return_pmf x) = return_pmf (f x)"
hoelzl@59000
   709
  by transfer (simp add: distr_return)
hoelzl@59000
   710
hoelzl@59052
   711
lemma map_pmf_const[simp]: "map_pmf (\<lambda>_. c) M = return_pmf c"
hoelzl@59052
   712
  by transfer (auto simp: prob_space.distr_const)
hoelzl@59052
   713
hoelzl@59002
   714
lemma set_return_pmf: "set_pmf (return_pmf x) = {x}"
hoelzl@59000
   715
  by transfer (auto simp add: measure_return split: split_indicator)
hoelzl@59000
   716
hoelzl@59000
   717
lemma pmf_return: "pmf (return_pmf x) y = indicator {y} x"
hoelzl@59000
   718
  by transfer (simp add: measure_return)
hoelzl@59000
   719
hoelzl@59002
   720
lemma nn_integral_return_pmf[simp]: "0 \<le> f x \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>return_pmf x) = f x"
hoelzl@59002
   721
  unfolding return_pmf.rep_eq by (intro nn_integral_return) auto
hoelzl@59002
   722
hoelzl@59002
   723
lemma emeasure_return_pmf[simp]: "emeasure (return_pmf x) X = indicator X x"
hoelzl@59002
   724
  unfolding return_pmf.rep_eq by (intro emeasure_return) auto
hoelzl@59002
   725
hoelzl@59000
   726
end
hoelzl@59000
   727
hoelzl@59475
   728
lemma return_pmf_inj[simp]: "return_pmf x = return_pmf y \<longleftrightarrow> x = y"
hoelzl@59475
   729
  by (metis insertI1 set_return_pmf singletonD)
hoelzl@59475
   730
hoelzl@59000
   731
definition "bind_pmf M f = join_pmf (map_pmf f M)"
hoelzl@59000
   732
hoelzl@59000
   733
lemma (in pmf_as_measure) bind_transfer[transfer_rule]:
hoelzl@59000
   734
  "rel_fun pmf_as_measure.cr_pmf (rel_fun (rel_fun op = pmf_as_measure.cr_pmf) pmf_as_measure.cr_pmf) op \<guillemotright>= bind_pmf"
hoelzl@59000
   735
proof (auto simp: pmf_as_measure.cr_pmf_def rel_fun_def bind_pmf_def join_pmf.rep_eq map_pmf.rep_eq)
hoelzl@59000
   736
  fix M f and g :: "'a \<Rightarrow> 'b pmf" assume "\<forall>x. f x = measure_pmf (g x)"
hoelzl@59000
   737
  then have f: "f = (\<lambda>x. measure_pmf (g x))"
hoelzl@59000
   738
    by auto
hoelzl@59000
   739
  show "measure_pmf M \<guillemotright>= f = distr (measure_pmf M) (count_space UNIV) g \<guillemotright>= measure_pmf"
hoelzl@59000
   740
    unfolding f by (subst bind_distr[OF _ measurable_measure_pmf]) auto
hoelzl@59000
   741
qed
hoelzl@59000
   742
hoelzl@59493
   743
lemma ereal_pmf_bind: "pmf (bind_pmf N f) i = (\<integral>\<^sup>+x. pmf (f x) i \<partial>measure_pmf N)"
hoelzl@59493
   744
  by (auto intro!: nn_integral_distr simp: bind_pmf_def ereal_pmf_join map_pmf.rep_eq)
hoelzl@59493
   745
hoelzl@59000
   746
lemma pmf_bind: "pmf (bind_pmf N f) i = (\<integral>x. pmf (f x) i \<partial>measure_pmf N)"
hoelzl@59000
   747
  by (auto intro!: integral_distr simp: bind_pmf_def pmf_join map_pmf.rep_eq)
hoelzl@59000
   748
hoelzl@59000
   749
lemma bind_return_pmf: "bind_pmf (return_pmf x) f = f x"
hoelzl@59000
   750
  unfolding bind_pmf_def map_return_pmf join_return_pmf ..
hoelzl@59000
   751
hoelzl@59052
   752
lemma join_eq_bind_pmf: "join_pmf M = bind_pmf M id"
hoelzl@59052
   753
  by (simp add: bind_pmf_def)
hoelzl@59052
   754
hoelzl@59052
   755
lemma bind_pmf_const[simp]: "bind_pmf M (\<lambda>x. c) = c"
hoelzl@59052
   756
  unfolding bind_pmf_def map_pmf_const join_return_pmf ..
hoelzl@59052
   757
hoelzl@59002
   758
lemma set_bind_pmf: "set_pmf (bind_pmf M N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))"
hoelzl@59002
   759
  apply (simp add: set_eq_iff set_pmf_iff pmf_bind)
hoelzl@59002
   760
  apply (subst integral_nonneg_eq_0_iff_AE)
hoelzl@59002
   761
  apply (auto simp: pmf_nonneg pmf_le_1 AE_measure_pmf_iff
hoelzl@59002
   762
              intro!: measure_pmf.integrable_const_bound[where B=1])
hoelzl@59002
   763
  done
hoelzl@59002
   764
hoelzl@59425
   765
hoelzl@59002
   766
lemma measurable_pair_restrict_pmf2:
hoelzl@59002
   767
  assumes "countable A"
hoelzl@59425
   768
  assumes [measurable]: "\<And>y. y \<in> A \<Longrightarrow> (\<lambda>x. f (x, y)) \<in> measurable M L"
hoelzl@59425
   769
  shows "f \<in> measurable (M \<Otimes>\<^sub>M restrict_space (measure_pmf N) A) L" (is "f \<in> measurable ?M _")
hoelzl@59425
   770
proof -
hoelzl@59425
   771
  have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)"
hoelzl@59425
   772
    by (simp add: restrict_count_space)
hoelzl@59425
   773
hoelzl@59425
   774
  show ?thesis
hoelzl@59425
   775
    by (intro measurable_compose_countable'[where f="\<lambda>a b. f (fst b, a)" and g=snd and I=A,
hoelzl@59425
   776
                                            unfolded pair_collapse] assms)
hoelzl@59425
   777
        measurable
hoelzl@59425
   778
qed
hoelzl@59002
   779
hoelzl@59002
   780
lemma measurable_pair_restrict_pmf1:
hoelzl@59002
   781
  assumes "countable A"
hoelzl@59425
   782
  assumes [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N L"
hoelzl@59002
   783
  shows "f \<in> measurable (restrict_space (measure_pmf M) A \<Otimes>\<^sub>M N) L"
hoelzl@59425
   784
proof -
hoelzl@59425
   785
  have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)"
hoelzl@59425
   786
    by (simp add: restrict_count_space)
hoelzl@59425
   787
hoelzl@59425
   788
  show ?thesis
hoelzl@59425
   789
    by (intro measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A,
hoelzl@59425
   790
                                            unfolded pair_collapse] assms)
hoelzl@59425
   791
        measurable
hoelzl@59425
   792
qed
hoelzl@59002
   793
                                
hoelzl@59000
   794
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@59000
   795
  unfolding pmf_eq_iff pmf_bind
hoelzl@59000
   796
proof
hoelzl@59000
   797
  fix i
hoelzl@59000
   798
  interpret B: prob_space "restrict_space B B"
hoelzl@59000
   799
    by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
hoelzl@59000
   800
       (auto simp: AE_measure_pmf_iff)
hoelzl@59000
   801
  interpret A: prob_space "restrict_space A A"
hoelzl@59000
   802
    by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
hoelzl@59000
   803
       (auto simp: AE_measure_pmf_iff)
hoelzl@59000
   804
hoelzl@59000
   805
  interpret AB: pair_prob_space "restrict_space A A" "restrict_space B B"
hoelzl@59000
   806
    by unfold_locales
hoelzl@59000
   807
hoelzl@59000
   808
  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@59000
   809
    by (rule integral_cong) (auto intro!: integral_pmf_restrict)
hoelzl@59000
   810
  also have "\<dots> = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>restrict_space A A)"
hoelzl@59002
   811
    by (intro integral_pmf_restrict B.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
hoelzl@59002
   812
              countable_set_pmf borel_measurable_count_space)
hoelzl@59000
   813
  also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>restrict_space B B)"
hoelzl@59002
   814
    by (rule AB.Fubini_integral[symmetric])
hoelzl@59002
   815
       (auto intro!: AB.integrable_const_bound[where B=1] measurable_pair_restrict_pmf2
Andreas@59023
   816
             simp: pmf_nonneg pmf_le_1 measurable_restrict_space1)
hoelzl@59000
   817
  also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>B)"
hoelzl@59002
   818
    by (intro integral_pmf_restrict[symmetric] A.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
hoelzl@59002
   819
              countable_set_pmf borel_measurable_count_space)
hoelzl@59000
   820
  also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>A \<partial>B)"
hoelzl@59000
   821
    by (rule integral_cong) (auto intro!: integral_pmf_restrict[symmetric])
hoelzl@59000
   822
  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@59000
   823
qed
hoelzl@59000
   824
hoelzl@59000
   825
hoelzl@59000
   826
context
hoelzl@59000
   827
begin
hoelzl@59000
   828
hoelzl@59000
   829
interpretation pmf_as_measure .
hoelzl@59000
   830
hoelzl@59002
   831
lemma measure_pmf_bind: "measure_pmf (bind_pmf M f) = (measure_pmf M \<guillemotright>= (\<lambda>x. measure_pmf (f x)))"
hoelzl@59002
   832
  by transfer simp
hoelzl@59002
   833
hoelzl@59002
   834
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@59002
   835
  using measurable_measure_pmf[of N]
hoelzl@59002
   836
  unfolding measure_pmf_bind
hoelzl@59002
   837
  apply (subst (1 3) nn_integral_max_0[symmetric])
hoelzl@59002
   838
  apply (intro nn_integral_bind[where B="count_space UNIV"])
hoelzl@59002
   839
  apply auto
hoelzl@59002
   840
  done
hoelzl@59002
   841
hoelzl@59002
   842
lemma emeasure_bind_pmf[simp]: "emeasure (bind_pmf M N) X = (\<integral>\<^sup>+x. emeasure (N x) X \<partial>M)"
hoelzl@59002
   843
  using measurable_measure_pmf[of N]
hoelzl@59002
   844
  unfolding measure_pmf_bind
hoelzl@59002
   845
  by (subst emeasure_bind[where N="count_space UNIV"]) auto
hoelzl@59002
   846
hoelzl@59000
   847
lemma bind_return_pmf': "bind_pmf N return_pmf = N"
hoelzl@59000
   848
proof (transfer, clarify)
hoelzl@59000
   849
  fix N :: "'a measure" assume "sets N = UNIV" then show "N \<guillemotright>= return (count_space UNIV) = N"
hoelzl@59000
   850
    by (subst return_sets_cong[where N=N]) (simp_all add: bind_return')
hoelzl@59000
   851
qed
hoelzl@59000
   852
hoelzl@59000
   853
lemma bind_return_pmf'': "bind_pmf N (\<lambda>x. return_pmf (f x)) = map_pmf f N"
hoelzl@59000
   854
proof (transfer, clarify)
hoelzl@59000
   855
  fix N :: "'b measure" and f :: "'b \<Rightarrow> 'a" assume "prob_space N" "sets N = UNIV"
hoelzl@59000
   856
  then show "N \<guillemotright>= (\<lambda>x. return (count_space UNIV) (f x)) = distr N (count_space UNIV) f"
hoelzl@59000
   857
    by (subst bind_return_distr[symmetric])
hoelzl@59000
   858
       (auto simp: prob_space.not_empty measurable_def comp_def)
hoelzl@59000
   859
qed
hoelzl@59000
   860
hoelzl@59000
   861
lemma bind_assoc_pmf: "bind_pmf (bind_pmf A B) C = bind_pmf A (\<lambda>x. bind_pmf (B x) C)"
hoelzl@59000
   862
  by transfer
hoelzl@59000
   863
     (auto intro!: bind_assoc[where N="count_space UNIV" and R="count_space UNIV"]
hoelzl@59000
   864
           simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space)
hoelzl@59000
   865
hoelzl@59000
   866
end
hoelzl@59000
   867
hoelzl@59493
   868
lemma map_bind_pmf: "map_pmf f (bind_pmf M g) = bind_pmf M (\<lambda>x. map_pmf f (g x))"
hoelzl@59493
   869
  unfolding bind_return_pmf''[symmetric] bind_assoc_pmf[of M] ..
hoelzl@59493
   870
hoelzl@59493
   871
lemma bind_map_pmf: "bind_pmf (map_pmf f M) g = bind_pmf M (\<lambda>x. g (f x))"
hoelzl@59493
   872
  unfolding bind_return_pmf''[symmetric] bind_assoc_pmf bind_return_pmf ..
hoelzl@59493
   873
hoelzl@59052
   874
lemma map_join_pmf: "map_pmf f (join_pmf AA) = join_pmf (map_pmf (map_pmf f) AA)"
hoelzl@59052
   875
  unfolding bind_pmf_def[symmetric]
hoelzl@59052
   876
  unfolding bind_return_pmf''[symmetric] join_eq_bind_pmf bind_assoc_pmf
hoelzl@59052
   877
  by (simp add: bind_return_pmf'')
hoelzl@59052
   878
hoelzl@59000
   879
definition "pair_pmf A B = bind_pmf A (\<lambda>x. bind_pmf B (\<lambda>y. return_pmf (x, y)))"
hoelzl@59000
   880
hoelzl@59000
   881
lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b"
hoelzl@59000
   882
  unfolding pair_pmf_def pmf_bind pmf_return
hoelzl@59000
   883
  apply (subst integral_measure_pmf[where A="{b}"])
hoelzl@59000
   884
  apply (auto simp: indicator_eq_0_iff)
hoelzl@59000
   885
  apply (subst integral_measure_pmf[where A="{a}"])
hoelzl@59000
   886
  apply (auto simp: indicator_eq_0_iff setsum_nonneg_eq_0_iff pmf_nonneg)
hoelzl@59000
   887
  done
hoelzl@59000
   888
hoelzl@59002
   889
lemma set_pair_pmf: "set_pmf (pair_pmf A B) = set_pmf A \<times> set_pmf B"
hoelzl@59002
   890
  unfolding pair_pmf_def set_bind_pmf set_return_pmf by auto
hoelzl@59002
   891
hoelzl@59048
   892
lemma measure_pmf_in_subprob_space[measurable (raw)]:
hoelzl@59048
   893
  "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
hoelzl@59048
   894
  by (simp add: space_subprob_algebra) intro_locales
hoelzl@59048
   895
hoelzl@59134
   896
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@59134
   897
proof -
hoelzl@59134
   898
  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@59134
   899
    by (subst nn_integral_max_0[symmetric])
hoelzl@59134
   900
       (auto simp: AE_measure_pmf_iff set_pair_pmf intro!: nn_integral_cong_AE)
hoelzl@59134
   901
  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@59134
   902
    by (simp add: pair_pmf_def)
hoelzl@59134
   903
  also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) \<partial>B \<partial>A)"
hoelzl@59134
   904
    by (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
hoelzl@59134
   905
  finally show ?thesis
hoelzl@59134
   906
    unfolding nn_integral_max_0 .
hoelzl@59134
   907
qed
hoelzl@59134
   908
hoelzl@59134
   909
lemma pair_map_pmf1: "pair_pmf (map_pmf f A) B = map_pmf (apfst f) (pair_pmf A B)"
hoelzl@59134
   910
proof (safe intro!: pmf_eqI)
hoelzl@59134
   911
  fix a :: "'a" and b :: "'b"
hoelzl@59134
   912
  have [simp]: "\<And>c d. indicator (apfst f -` {(a, b)}) (c, d) = indicator (f -` {a}) c * (indicator {b} d::ereal)"
hoelzl@59134
   913
    by (auto split: split_indicator)
hoelzl@59134
   914
hoelzl@59134
   915
  have "ereal (pmf (pair_pmf (map_pmf f A) B) (a, b)) =
hoelzl@59134
   916
         ereal (pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b))"
hoelzl@59134
   917
    unfolding pmf_pair ereal_pmf_map
hoelzl@59134
   918
    by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_multc pmf_nonneg
hoelzl@59134
   919
                  emeasure_map_pmf[symmetric] del: emeasure_map_pmf)
hoelzl@59134
   920
  then show "pmf (pair_pmf (map_pmf f A) B) (a, b) = pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b)"
hoelzl@59134
   921
    by simp
hoelzl@59134
   922
qed
hoelzl@59134
   923
hoelzl@59134
   924
lemma pair_map_pmf2: "pair_pmf A (map_pmf f B) = map_pmf (apsnd f) (pair_pmf A B)"
hoelzl@59134
   925
proof (safe intro!: pmf_eqI)
hoelzl@59134
   926
  fix a :: "'a" and b :: "'b"
hoelzl@59134
   927
  have [simp]: "\<And>c d. indicator (apsnd f -` {(a, b)}) (c, d) = indicator {a} c * (indicator (f -` {b}) d::ereal)"
hoelzl@59134
   928
    by (auto split: split_indicator)
hoelzl@59134
   929
hoelzl@59134
   930
  have "ereal (pmf (pair_pmf A (map_pmf f B)) (a, b)) =
hoelzl@59134
   931
         ereal (pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b))"
hoelzl@59134
   932
    unfolding pmf_pair ereal_pmf_map
hoelzl@59134
   933
    by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_cmult nn_integral_multc pmf_nonneg
hoelzl@59134
   934
                  emeasure_map_pmf[symmetric] del: emeasure_map_pmf)
hoelzl@59134
   935
  then show "pmf (pair_pmf A (map_pmf f B)) (a, b) = pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b)"
hoelzl@59134
   936
    by simp
hoelzl@59134
   937
qed
hoelzl@59134
   938
hoelzl@59134
   939
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@59134
   940
  by (simp add: pair_map_pmf2 pair_map_pmf1 map_pmf_comp split_beta')
hoelzl@59134
   941
hoelzl@59000
   942
lemma bind_pair_pmf:
hoelzl@59000
   943
  assumes M[measurable]: "M \<in> measurable (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) (subprob_algebra N)"
hoelzl@59000
   944
  shows "measure_pmf (pair_pmf A B) \<guillemotright>= M = (measure_pmf A \<guillemotright>= (\<lambda>x. measure_pmf B \<guillemotright>= (\<lambda>y. M (x, y))))"
hoelzl@59000
   945
    (is "?L = ?R")
hoelzl@59000
   946
proof (rule measure_eqI)
hoelzl@59000
   947
  have M'[measurable]: "M \<in> measurable (pair_pmf A B) (subprob_algebra N)"
hoelzl@59000
   948
    using M[THEN measurable_space] by (simp_all add: space_pair_measure)
hoelzl@59000
   949
hoelzl@59048
   950
  note measurable_bind[where N="count_space UNIV", measurable]
hoelzl@59048
   951
  note measure_pmf_in_subprob_space[simp]
hoelzl@59048
   952
hoelzl@59000
   953
  have sets_eq_N: "sets ?L = N"
hoelzl@59048
   954
    by (subst sets_bind[OF sets_kernel[OF M']]) auto
hoelzl@59000
   955
  show "sets ?L = sets ?R"
hoelzl@59048
   956
    using measurable_space[OF M]
hoelzl@59048
   957
    by (simp add: sets_eq_N space_pair_measure space_subprob_algebra)
hoelzl@59000
   958
  fix X assume "X \<in> sets ?L"
hoelzl@59000
   959
  then have X[measurable]: "X \<in> sets N"
hoelzl@59000
   960
    unfolding sets_eq_N .
hoelzl@59000
   961
  then show "emeasure ?L X = emeasure ?R X"
hoelzl@59000
   962
    apply (simp add: emeasure_bind[OF _ M' X])
hoelzl@59048
   963
    apply (simp add: nn_integral_bind[where B="count_space UNIV"] pair_pmf_def measure_pmf_bind[of A]
hoelzl@59048
   964
      nn_integral_measure_pmf_finite set_return_pmf emeasure_nonneg pmf_return one_ereal_def[symmetric])
hoelzl@59048
   965
    apply (subst emeasure_bind[OF _ _ X])
hoelzl@59000
   966
    apply measurable
hoelzl@59000
   967
    apply (subst emeasure_bind[OF _ _ X])
hoelzl@59000
   968
    apply measurable
hoelzl@59000
   969
    done
hoelzl@59000
   970
qed
hoelzl@59000
   971
hoelzl@59052
   972
lemma join_map_return_pmf: "join_pmf (map_pmf return_pmf A) = A"
hoelzl@59052
   973
  unfolding bind_pmf_def[symmetric] bind_return_pmf' ..
hoelzl@59052
   974
hoelzl@59052
   975
lemma map_fst_pair_pmf: "map_pmf fst (pair_pmf A B) = A"
hoelzl@59052
   976
  by (simp add: pair_pmf_def bind_return_pmf''[symmetric] bind_assoc_pmf bind_return_pmf bind_return_pmf')
hoelzl@59052
   977
hoelzl@59052
   978
lemma map_snd_pair_pmf: "map_pmf snd (pair_pmf A B) = B"
hoelzl@59052
   979
  by (simp add: pair_pmf_def bind_return_pmf''[symmetric] bind_assoc_pmf bind_return_pmf bind_return_pmf')
hoelzl@59052
   980
hoelzl@59053
   981
lemma nn_integral_pmf':
hoelzl@59053
   982
  "inj_on f A \<Longrightarrow> (\<integral>\<^sup>+x. pmf p (f x) \<partial>count_space A) = emeasure p (f ` A)"
hoelzl@59053
   983
  by (subst nn_integral_bij_count_space[where g=f and B="f`A"])
hoelzl@59053
   984
     (auto simp: bij_betw_def nn_integral_pmf)
hoelzl@59053
   985
hoelzl@59053
   986
lemma pmf_le_0_iff[simp]: "pmf M p \<le> 0 \<longleftrightarrow> pmf M p = 0"
hoelzl@59053
   987
  using pmf_nonneg[of M p] by simp
hoelzl@59053
   988
hoelzl@59053
   989
lemma min_pmf_0[simp]: "min (pmf M p) 0 = 0" "min 0 (pmf M p) = 0"
hoelzl@59053
   990
  using pmf_nonneg[of M p] by simp_all
hoelzl@59053
   991
hoelzl@59053
   992
lemma pmf_eq_0_set_pmf: "pmf M p = 0 \<longleftrightarrow> p \<notin> set_pmf M"
hoelzl@59053
   993
  unfolding set_pmf_iff by simp
hoelzl@59053
   994
hoelzl@59053
   995
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@59053
   996
  by (auto simp: pmf.rep_eq map_pmf.rep_eq measure_distr AE_measure_pmf_iff inj_onD
hoelzl@59053
   997
           intro!: measure_pmf.finite_measure_eq_AE)
hoelzl@59053
   998
hoelzl@59493
   999
subsection \<open> Conditional Probabilities \<close>
hoelzl@59493
  1000
hoelzl@59493
  1001
context
hoelzl@59493
  1002
  fixes p :: "'a pmf" and s :: "'a set"
hoelzl@59493
  1003
  assumes not_empty: "set_pmf p \<inter> s \<noteq> {}"
hoelzl@59493
  1004
begin
hoelzl@59493
  1005
hoelzl@59493
  1006
interpretation pmf_as_measure .
hoelzl@59493
  1007
hoelzl@59493
  1008
lemma emeasure_measure_pmf_not_zero: "emeasure (measure_pmf p) s \<noteq> 0"
hoelzl@59493
  1009
proof
hoelzl@59493
  1010
  assume "emeasure (measure_pmf p) s = 0"
hoelzl@59493
  1011
  then have "AE x in measure_pmf p. x \<notin> s"
hoelzl@59493
  1012
    by (rule AE_I[rotated]) auto
hoelzl@59493
  1013
  with not_empty show False
hoelzl@59493
  1014
    by (auto simp: AE_measure_pmf_iff)
hoelzl@59493
  1015
qed
hoelzl@59493
  1016
hoelzl@59493
  1017
lemma measure_measure_pmf_not_zero: "measure (measure_pmf p) s \<noteq> 0"
hoelzl@59493
  1018
  using emeasure_measure_pmf_not_zero unfolding measure_pmf.emeasure_eq_measure by simp
hoelzl@59493
  1019
hoelzl@59493
  1020
lift_definition cond_pmf :: "'a pmf" is
hoelzl@59493
  1021
  "uniform_measure (measure_pmf p) s"
hoelzl@59493
  1022
proof (intro conjI)
hoelzl@59493
  1023
  show "prob_space (uniform_measure (measure_pmf p) s)"
hoelzl@59493
  1024
    by (intro prob_space_uniform_measure) (auto simp: emeasure_measure_pmf_not_zero)
hoelzl@59493
  1025
  show "AE x in uniform_measure (measure_pmf p) s. measure (uniform_measure (measure_pmf p) s) {x} \<noteq> 0"
hoelzl@59493
  1026
    by (simp add: emeasure_measure_pmf_not_zero measure_measure_pmf_not_zero AE_uniform_measure
hoelzl@59493
  1027
                  AE_measure_pmf_iff set_pmf.rep_eq)
hoelzl@59493
  1028
qed simp
hoelzl@59493
  1029
hoelzl@59493
  1030
lemma pmf_cond: "pmf cond_pmf x = (if x \<in> s then pmf p x / measure p s else 0)"
hoelzl@59493
  1031
  by transfer (simp add: emeasure_measure_pmf_not_zero pmf.rep_eq)
hoelzl@59493
  1032
hoelzl@59493
  1033
lemma set_cond_pmf: "set_pmf cond_pmf = set_pmf p \<inter> s"
hoelzl@59493
  1034
  by (auto simp add: set_pmf_iff pmf_cond measure_measure_pmf_not_zero split: split_if_asm)
hoelzl@59493
  1035
hoelzl@59493
  1036
end
hoelzl@59493
  1037
hoelzl@59494
  1038
lemma cond_map_pmf:
hoelzl@59494
  1039
  assumes "set_pmf p \<inter> f -` s \<noteq> {}"
hoelzl@59494
  1040
  shows "cond_pmf (map_pmf f p) s = map_pmf f (cond_pmf p (f -` s))"
hoelzl@59494
  1041
proof -
hoelzl@59494
  1042
  have *: "set_pmf (map_pmf f p) \<inter> s \<noteq> {}"
hoelzl@59494
  1043
    using assms by (simp add: set_map_pmf) auto
hoelzl@59494
  1044
  { fix x
hoelzl@59494
  1045
    have "ereal (pmf (map_pmf f (cond_pmf p (f -` s))) x) =
hoelzl@59494
  1046
      emeasure p (f -` s \<inter> f -` {x}) / emeasure p (f -` s)"
hoelzl@59494
  1047
      unfolding ereal_pmf_map cond_pmf.rep_eq[OF assms] by (simp add: nn_integral_uniform_measure)
hoelzl@59494
  1048
    also have "f -` s \<inter> f -` {x} = (if x \<in> s then f -` {x} else {})"
hoelzl@59494
  1049
      by auto
hoelzl@59494
  1050
    also have "emeasure p (if x \<in> s then f -` {x} else {}) / emeasure p (f -` s) =
hoelzl@59494
  1051
      ereal (pmf (cond_pmf (map_pmf f p) s) x)"
hoelzl@59494
  1052
      using measure_measure_pmf_not_zero[OF *]
hoelzl@59494
  1053
      by (simp add: pmf_cond[OF *] ereal_divide' ereal_pmf_map measure_pmf.emeasure_eq_measure[symmetric]
hoelzl@59494
  1054
               del: ereal_divide)
hoelzl@59494
  1055
    finally have "ereal (pmf (cond_pmf (map_pmf f p) s) x) = ereal (pmf (map_pmf f (cond_pmf p (f -` s))) x)"
hoelzl@59494
  1056
      by simp }
hoelzl@59494
  1057
  then show ?thesis
hoelzl@59494
  1058
    by (intro pmf_eqI) simp
hoelzl@59494
  1059
qed
hoelzl@59494
  1060
hoelzl@59495
  1061
lemma bind_cond_pmf_cancel:
hoelzl@59495
  1062
  assumes in_S: "\<And>x. x \<in> set_pmf p \<Longrightarrow> x \<in> S x"
hoelzl@59495
  1063
  assumes S_eq: "\<And>x y. x \<in> S y \<Longrightarrow> S x = S y"
hoelzl@59495
  1064
  shows "bind_pmf p (\<lambda>x. cond_pmf p (S x)) = p"
hoelzl@59495
  1065
proof (rule pmf_eqI)
hoelzl@59495
  1066
  have [simp]: "\<And>x. x \<in> p \<Longrightarrow> p \<inter> (S x) \<noteq> {}"
hoelzl@59495
  1067
    using in_S by auto
hoelzl@59495
  1068
  fix z
hoelzl@59495
  1069
  have pmf_le: "pmf p z \<le> measure p (S z)"
hoelzl@59495
  1070
  proof cases
hoelzl@59495
  1071
    assume "z \<in> p" from in_S[OF this] show ?thesis
hoelzl@59495
  1072
      by (auto intro!: measure_pmf.finite_measure_mono simp: pmf.rep_eq)
hoelzl@59495
  1073
  qed (simp add: set_pmf_iff measure_nonneg)
hoelzl@59495
  1074
hoelzl@59495
  1075
  have "ereal (pmf (bind_pmf p (\<lambda>x. cond_pmf p (S x))) z) =
hoelzl@59495
  1076
    (\<integral>\<^sup>+ x. ereal (pmf p z / measure p (S z)) * indicator (S z) x \<partial>p)"
hoelzl@59495
  1077
    by (subst ereal_pmf_bind)
hoelzl@59495
  1078
       (auto intro!: nn_integral_cong_AE dest!: S_eq split: split_indicator
hoelzl@59495
  1079
             simp: AE_measure_pmf_iff pmf_cond pmf_eq_0_set_pmf in_S)
hoelzl@59495
  1080
  also have "\<dots> = pmf p z"
hoelzl@59495
  1081
    using pmf_le pmf_nonneg[of p z]
hoelzl@59495
  1082
    by (subst nn_integral_cmult) (simp_all add: measure_nonneg measure_pmf.emeasure_eq_measure)
hoelzl@59495
  1083
  finally show "pmf (bind_pmf p (\<lambda>x. cond_pmf p (S x))) z = pmf p z"
hoelzl@59495
  1084
    by simp
hoelzl@59495
  1085
qed
hoelzl@59495
  1086
Andreas@59023
  1087
inductive rel_pmf :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf \<Rightarrow> bool"
Andreas@59023
  1088
for R p q
Andreas@59023
  1089
where
Andreas@59023
  1090
  "\<lbrakk> \<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y; 
Andreas@59023
  1091
     map_pmf fst pq = p; map_pmf snd pq = q \<rbrakk>
Andreas@59023
  1092
  \<Longrightarrow> rel_pmf R p q"
hoelzl@58587
  1093
Andreas@59023
  1094
bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: rel_pmf
hoelzl@58587
  1095
proof -
hoelzl@58587
  1096
  show "map_pmf id = id" by (rule map_pmf_id)
hoelzl@58587
  1097
  show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose) 
hoelzl@58587
  1098
  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"
Andreas@59023
  1099
    by (intro map_pmf_cong refl)
hoelzl@58587
  1100
hoelzl@58587
  1101
  show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
hoelzl@58587
  1102
    by (rule pmf_set_map)
hoelzl@58587
  1103
hoelzl@58587
  1104
  { fix p :: "'s pmf"
hoelzl@58587
  1105
    have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq"
hoelzl@58587
  1106
      by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"])
hoelzl@59053
  1107
         (auto intro: countable_set_pmf)
hoelzl@58587
  1108
    also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq"
hoelzl@58587
  1109
      by (metis Field_natLeq card_of_least natLeq_Well_order)
hoelzl@58587
  1110
    finally show "(card_of (set_pmf p), natLeq) \<in> ordLeq" . }
hoelzl@58587
  1111
Andreas@59023
  1112
  show "\<And>R. rel_pmf R =
Andreas@59023
  1113
         (BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf fst))\<inverse>\<inverse> OO
Andreas@59023
  1114
         BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf snd)"
Andreas@59023
  1115
     by (auto simp add: fun_eq_iff BNF_Def.Grp_def OO_def rel_pmf.simps)
Andreas@59023
  1116
Andreas@59023
  1117
  { fix p :: "'a pmf" and f :: "'a \<Rightarrow> 'b" and g x
Andreas@59023
  1118
    assume p: "\<And>z. z \<in> set_pmf p \<Longrightarrow> f z = g z"
Andreas@59023
  1119
      and x: "x \<in> set_pmf p"
Andreas@59023
  1120
    thus "f x = g x" by simp }
Andreas@59023
  1121
Andreas@59023
  1122
  fix R :: "'a => 'b \<Rightarrow> bool" and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
Andreas@59023
  1123
  { fix p q r
Andreas@59023
  1124
    assume pq: "rel_pmf R p q"
Andreas@59023
  1125
      and qr:"rel_pmf S q r"
Andreas@59023
  1126
    from pq obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
Andreas@59023
  1127
      and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto
Andreas@59023
  1128
    from qr obtain qr where qr: "\<And>y z. (y, z) \<in> set_pmf qr \<Longrightarrow> S y z"
Andreas@59023
  1129
      and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto
Andreas@59023
  1130
hoelzl@59493
  1131
    def pr \<equiv> "bind_pmf pq (\<lambda>(x, y). bind_pmf (cond_pmf qr {(y', z). y' = y}) (\<lambda>(y', z). return_pmf (x, z)))"
hoelzl@59493
  1132
    have pr_welldefined: "\<And>y. y \<in> q \<Longrightarrow> qr \<inter> {(y', z). y' = y} \<noteq> {}"
hoelzl@59493
  1133
      by (force simp: q' set_map_pmf)
Andreas@59023
  1134
hoelzl@59053
  1135
    have "rel_pmf (R OO S) p r"
hoelzl@59493
  1136
    proof (rule rel_pmf.intros)
hoelzl@59493
  1137
      fix x z assume "(x, z) \<in> pr"
hoelzl@59493
  1138
      then have "\<exists>y. (x, y) \<in> pq \<and> (y, z) \<in> qr"
hoelzl@59493
  1139
        by (auto simp: q pr_welldefined pr_def set_bind_pmf split_beta set_return_pmf set_cond_pmf set_map_pmf)
hoelzl@59053
  1140
      with pq qr show "(R OO S) x z"
hoelzl@59053
  1141
        by blast
hoelzl@59493
  1142
    next
hoelzl@59495
  1143
      have "map_pmf snd pr = map_pmf snd (bind_pmf q (\<lambda>y. cond_pmf qr {(y', z). y' = y}))"
hoelzl@59495
  1144
        by (simp add: pr_def q split_beta bind_map_pmf bind_return_pmf'' map_bind_pmf map_return_pmf)
hoelzl@59493
  1145
      then show "map_pmf snd pr = r"
hoelzl@59495
  1146
        unfolding r q' bind_map_pmf by (subst (asm) bind_cond_pmf_cancel) auto
hoelzl@59493
  1147
    qed (simp add: pr_def map_bind_pmf split_beta map_return_pmf bind_return_pmf'' p) }
Andreas@59023
  1148
  then show "rel_pmf R OO rel_pmf S \<le> rel_pmf (R OO S)"
Andreas@59023
  1149
    by(auto simp add: le_fun_def)
Andreas@59023
  1150
qed (fact natLeq_card_order natLeq_cinfinite)+
hoelzl@58587
  1151
hoelzl@59134
  1152
lemma rel_pmf_return_pmf1: "rel_pmf R (return_pmf x) M \<longleftrightarrow> (\<forall>a\<in>M. R x a)"
hoelzl@59134
  1153
proof safe
hoelzl@59134
  1154
  fix a assume "a \<in> M" "rel_pmf R (return_pmf x) M"
hoelzl@59134
  1155
  then obtain pq where *: "\<And>a b. (a, b) \<in> set_pmf pq \<Longrightarrow> R a b"
hoelzl@59134
  1156
    and eq: "return_pmf x = map_pmf fst pq" "M = map_pmf snd pq"
hoelzl@59134
  1157
    by (force elim: rel_pmf.cases)
hoelzl@59134
  1158
  moreover have "set_pmf (return_pmf x) = {x}"
hoelzl@59134
  1159
    by (simp add: set_return_pmf)
hoelzl@59134
  1160
  with `a \<in> M` have "(x, a) \<in> pq"
hoelzl@59134
  1161
    by (force simp: eq set_map_pmf)
hoelzl@59134
  1162
  with * show "R x a"
hoelzl@59134
  1163
    by auto
hoelzl@59134
  1164
qed (auto intro!: rel_pmf.intros[where pq="pair_pmf (return_pmf x) M"]
hoelzl@59134
  1165
          simp: map_fst_pair_pmf map_snd_pair_pmf set_pair_pmf set_return_pmf)
hoelzl@59134
  1166
hoelzl@59134
  1167
lemma rel_pmf_return_pmf2: "rel_pmf R M (return_pmf x) \<longleftrightarrow> (\<forall>a\<in>M. R a x)"
hoelzl@59134
  1168
  by (subst pmf.rel_flip[symmetric]) (simp add: rel_pmf_return_pmf1)
hoelzl@59134
  1169
hoelzl@59475
  1170
lemma rel_return_pmf[simp]: "rel_pmf R (return_pmf x1) (return_pmf x2) = R x1 x2"
hoelzl@59475
  1171
  unfolding rel_pmf_return_pmf2 set_return_pmf by simp
hoelzl@59475
  1172
hoelzl@59475
  1173
lemma rel_pmf_False[simp]: "rel_pmf (\<lambda>x y. False) x y = False"
hoelzl@59475
  1174
  unfolding pmf.in_rel fun_eq_iff using set_pmf_not_empty by fastforce
hoelzl@59475
  1175
hoelzl@59134
  1176
lemma rel_pmf_rel_prod:
hoelzl@59134
  1177
  "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@59134
  1178
proof safe
hoelzl@59134
  1179
  assume "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
hoelzl@59134
  1180
  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@59134
  1181
    and eq: "map_pmf fst pq = pair_pmf A A'" "map_pmf snd pq = pair_pmf B B'"
hoelzl@59134
  1182
    by (force elim: rel_pmf.cases)
hoelzl@59134
  1183
  show "rel_pmf R A B"
hoelzl@59134
  1184
  proof (rule rel_pmf.intros)
hoelzl@59134
  1185
    let ?f = "\<lambda>(a, b). (fst a, fst b)"
hoelzl@59134
  1186
    have [simp]: "(\<lambda>x. fst (?f x)) = fst o fst" "(\<lambda>x. snd (?f x)) = fst o snd"
hoelzl@59134
  1187
      by auto
hoelzl@59134
  1188
hoelzl@59134
  1189
    show "map_pmf fst (map_pmf ?f pq) = A"
hoelzl@59134
  1190
      by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
hoelzl@59134
  1191
    show "map_pmf snd (map_pmf ?f pq) = B"
hoelzl@59134
  1192
      by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
hoelzl@59134
  1193
hoelzl@59134
  1194
    fix a b assume "(a, b) \<in> set_pmf (map_pmf ?f pq)"
hoelzl@59134
  1195
    then obtain c d where "((a, c), (b, d)) \<in> set_pmf pq"
hoelzl@59134
  1196
      by (auto simp: set_map_pmf)
hoelzl@59134
  1197
    from pq[OF this] show "R a b" ..
hoelzl@59134
  1198
  qed
hoelzl@59134
  1199
  show "rel_pmf S A' B'"
hoelzl@59134
  1200
  proof (rule rel_pmf.intros)
hoelzl@59134
  1201
    let ?f = "\<lambda>(a, b). (snd a, snd b)"
hoelzl@59134
  1202
    have [simp]: "(\<lambda>x. fst (?f x)) = snd o fst" "(\<lambda>x. snd (?f x)) = snd o snd"
hoelzl@59134
  1203
      by auto
hoelzl@59134
  1204
hoelzl@59134
  1205
    show "map_pmf fst (map_pmf ?f pq) = A'"
hoelzl@59134
  1206
      by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
hoelzl@59134
  1207
    show "map_pmf snd (map_pmf ?f pq) = B'"
hoelzl@59134
  1208
      by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
hoelzl@59134
  1209
hoelzl@59134
  1210
    fix c d assume "(c, d) \<in> set_pmf (map_pmf ?f pq)"
hoelzl@59134
  1211
    then obtain a b where "((a, c), (b, d)) \<in> set_pmf pq"
hoelzl@59134
  1212
      by (auto simp: set_map_pmf)
hoelzl@59134
  1213
    from pq[OF this] show "S c d" ..
hoelzl@59134
  1214
  qed
hoelzl@59134
  1215
next
hoelzl@59134
  1216
  assume "rel_pmf R A B" "rel_pmf S A' B'"
hoelzl@59134
  1217
  then obtain Rpq Spq
hoelzl@59134
  1218
    where Rpq: "\<And>a b. (a, b) \<in> set_pmf Rpq \<Longrightarrow> R a b"
hoelzl@59134
  1219
        "map_pmf fst Rpq = A" "map_pmf snd Rpq = B"
hoelzl@59134
  1220
      and Spq: "\<And>a b. (a, b) \<in> set_pmf Spq \<Longrightarrow> S a b"
hoelzl@59134
  1221
        "map_pmf fst Spq = A'" "map_pmf snd Spq = B'"
hoelzl@59134
  1222
    by (force elim: rel_pmf.cases)
hoelzl@59134
  1223
hoelzl@59134
  1224
  let ?f = "(\<lambda>((a, c), (b, d)). ((a, b), (c, d)))"
hoelzl@59134
  1225
  let ?pq = "map_pmf ?f (pair_pmf Rpq Spq)"
hoelzl@59134
  1226
  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@59134
  1227
    by auto
hoelzl@59134
  1228
hoelzl@59134
  1229
  show "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
hoelzl@59134
  1230
    by (rule rel_pmf.intros[where pq="?pq"])
hoelzl@59134
  1231
       (auto simp: map_snd_pair_pmf map_fst_pair_pmf set_pair_pmf set_map_pmf map_pmf_comp Rpq Spq
hoelzl@59134
  1232
                   map_pair)
hoelzl@59134
  1233
qed
hoelzl@59134
  1234
hoelzl@58587
  1235
end
hoelzl@58587
  1236