(* Title: HOL/Probability/Probability_Mass_Function.thy
Author: Johannes Hölzl, TU München
Author: Andreas Lochbihler, ETH Zurich
*)
section \<open> Probability mass function \<close>
theory Probability_Mass_Function
imports
Giry_Monad
"~~/src/HOL/Library/Multiset"
begin
lemma AE_emeasure_singleton:
assumes x: "emeasure M {x} \<noteq> 0" and ae: "AE x in M. P x" shows "P x"
proof -
from x have x_M: "{x} \<in> sets M"
by (auto intro: emeasure_notin_sets)
from ae obtain N where N: "{x\<in>space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"
by (auto elim: AE_E)
{ assume "\<not> P x"
with x_M[THEN sets.sets_into_space] N have "emeasure M {x} \<le> emeasure M N"
by (intro emeasure_mono) auto
with x N have False
by (auto simp:) }
then show "P x" by auto
qed
lemma AE_measure_singleton: "measure M {x} \<noteq> 0 \<Longrightarrow> AE x in M. P x \<Longrightarrow> P x"
by (metis AE_emeasure_singleton measure_def emeasure_empty measure_empty)
lemma (in finite_measure) AE_support_countable:
assumes [simp]: "sets M = UNIV"
shows "(AE x in M. measure M {x} \<noteq> 0) \<longleftrightarrow> (\<exists>S. countable S \<and> (AE x in M. x \<in> S))"
proof
assume "\<exists>S. countable S \<and> (AE x in M. x \<in> S)"
then obtain S where S[intro]: "countable S" and ae: "AE x in M. x \<in> S"
by auto
then have "emeasure M (\<Union>x\<in>{x\<in>S. emeasure M {x} \<noteq> 0}. {x}) =
(\<integral>\<^sup>+ x. emeasure M {x} * indicator {x\<in>S. emeasure M {x} \<noteq> 0} x \<partial>count_space UNIV)"
by (subst emeasure_UN_countable)
(auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space)
also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} * indicator S x \<partial>count_space UNIV)"
by (auto intro!: nn_integral_cong split: split_indicator)
also have "\<dots> = emeasure M (\<Union>x\<in>S. {x})"
by (subst emeasure_UN_countable)
(auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space)
also have "\<dots> = emeasure M (space M)"
using ae by (intro emeasure_eq_AE) auto
finally have "emeasure M {x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0} = emeasure M (space M)"
by (simp add: emeasure_single_in_space cong: rev_conj_cong)
with finite_measure_compl[of "{x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0}"]
have "AE x in M. x \<in> S \<and> emeasure M {x} \<noteq> 0"
by (intro AE_I[OF order_refl]) (auto simp: emeasure_eq_measure measure_nonneg set_diff_eq cong: conj_cong)
then show "AE x in M. measure M {x} \<noteq> 0"
by (auto simp: emeasure_eq_measure)
qed (auto intro!: exI[of _ "{x. measure M {x} \<noteq> 0}"] countable_support)
subsection \<open> PMF as measure \<close>
typedef 'a pmf = "{M :: 'a measure. prob_space M \<and> sets M = UNIV \<and> (AE x in M. measure M {x} \<noteq> 0)}"
morphisms measure_pmf Abs_pmf
by (intro exI[of _ "uniform_measure (count_space UNIV) {undefined}"])
(auto intro!: prob_space_uniform_measure AE_uniform_measureI)
declare [[coercion measure_pmf]]
lemma prob_space_measure_pmf: "prob_space (measure_pmf p)"
using pmf.measure_pmf[of p] by auto
interpretation measure_pmf: prob_space "measure_pmf M" for M
by (rule prob_space_measure_pmf)
interpretation measure_pmf: subprob_space "measure_pmf M" for M
by (rule prob_space_imp_subprob_space) unfold_locales
lemma subprob_space_measure_pmf: "subprob_space (measure_pmf x)"
by unfold_locales
locale pmf_as_measure
begin
setup_lifting type_definition_pmf
end
context
begin
interpretation pmf_as_measure .
lemma sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV"
by transfer blast
lemma sets_measure_pmf_count_space[measurable_cong]:
"sets (measure_pmf M) = sets (count_space UNIV)"
by simp
lemma space_measure_pmf[simp]: "space (measure_pmf p) = UNIV"
using sets_eq_imp_space_eq[of "measure_pmf p" "count_space UNIV"] by simp
lemma measure_pmf_UNIV [simp]: "measure (measure_pmf p) UNIV = 1"
using measure_pmf.prob_space[of p] by simp
lemma measure_pmf_in_subprob_algebra[measurable (raw)]: "measure_pmf x \<in> space (subprob_algebra (count_space UNIV))"
by (simp add: space_subprob_algebra subprob_space_measure_pmf)
lemma measurable_pmf_measure1[simp]: "measurable (M :: 'a pmf) N = UNIV \<rightarrow> space N"
by (auto simp: measurable_def)
lemma measurable_pmf_measure2[simp]: "measurable N (M :: 'a pmf) = measurable N (count_space UNIV)"
by (intro measurable_cong_sets) simp_all
lemma measurable_pair_restrict_pmf2:
assumes "countable A"
assumes [measurable]: "\<And>y. y \<in> A \<Longrightarrow> (\<lambda>x. f (x, y)) \<in> measurable M L"
shows "f \<in> measurable (M \<Otimes>\<^sub>M restrict_space (measure_pmf N) A) L" (is "f \<in> measurable ?M _")
proof -
have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)"
by (simp add: restrict_count_space)
show ?thesis
by (intro measurable_compose_countable'[where f="\<lambda>a b. f (fst b, a)" and g=snd and I=A,
unfolded prod.collapse] assms)
measurable
qed
lemma measurable_pair_restrict_pmf1:
assumes "countable A"
assumes [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N L"
shows "f \<in> measurable (restrict_space (measure_pmf M) A \<Otimes>\<^sub>M N) L"
proof -
have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)"
by (simp add: restrict_count_space)
show ?thesis
by (intro measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A,
unfolded prod.collapse] assms)
measurable
qed
lift_definition pmf :: "'a pmf \<Rightarrow> 'a \<Rightarrow> real" is "\<lambda>M x. measure M {x}" .
lift_definition set_pmf :: "'a pmf \<Rightarrow> 'a set" is "\<lambda>M. {x. measure M {x} \<noteq> 0}" .
declare [[coercion set_pmf]]
lemma AE_measure_pmf: "AE x in (M::'a pmf). x \<in> M"
by transfer simp
lemma emeasure_pmf_single_eq_zero_iff:
fixes M :: "'a pmf"
shows "emeasure M {y} = 0 \<longleftrightarrow> y \<notin> M"
unfolding set_pmf.rep_eq by (simp add: measure_pmf.emeasure_eq_measure)
lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \<longleftrightarrow> (\<forall>y\<in>M. P y)"
using AE_measure_singleton[of M] AE_measure_pmf[of M]
by (auto simp: set_pmf.rep_eq)
lemma AE_pmfI: "(\<And>y. y \<in> set_pmf M \<Longrightarrow> P y) \<Longrightarrow> almost_everywhere (measure_pmf M) P"
by(simp add: AE_measure_pmf_iff)
lemma countable_set_pmf [simp]: "countable (set_pmf p)"
by transfer (metis prob_space.finite_measure finite_measure.countable_support)
lemma pmf_positive: "x \<in> set_pmf p \<Longrightarrow> 0 < pmf p x"
by transfer (simp add: less_le)
lemma pmf_nonneg[simp]: "0 \<le> pmf p x"
by transfer simp
lemma pmf_not_neg [simp]: "\<not>pmf p x < 0"
by (simp add: not_less pmf_nonneg)
lemma pmf_pos [simp]: "pmf p x \<noteq> 0 \<Longrightarrow> pmf p x > 0"
using pmf_nonneg[of p x] by linarith
lemma pmf_le_1: "pmf p x \<le> 1"
by (simp add: pmf.rep_eq)
lemma set_pmf_not_empty: "set_pmf M \<noteq> {}"
using AE_measure_pmf[of M] by (intro notI) simp
lemma set_pmf_iff: "x \<in> set_pmf M \<longleftrightarrow> pmf M x \<noteq> 0"
by transfer simp
lemma pmf_positive_iff: "0 < pmf p x \<longleftrightarrow> x \<in> set_pmf p"
unfolding less_le by (simp add: set_pmf_iff)
lemma set_pmf_eq: "set_pmf M = {x. pmf M x \<noteq> 0}"
by (auto simp: set_pmf_iff)
lemma set_pmf_eq': "set_pmf p = {x. pmf p x > 0}"
proof safe
fix x assume "x \<in> set_pmf p"
hence "pmf p x \<noteq> 0" by (auto simp: set_pmf_eq)
with pmf_nonneg[of p x] show "pmf p x > 0" by simp
qed (auto simp: set_pmf_eq)
lemma emeasure_pmf_single:
fixes M :: "'a pmf"
shows "emeasure M {x} = pmf M x"
by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
lemma measure_pmf_single: "measure (measure_pmf M) {x} = pmf M x"
using emeasure_pmf_single[of M x] by(simp add: measure_pmf.emeasure_eq_measure pmf_nonneg measure_nonneg)
lemma emeasure_measure_pmf_finite: "finite S \<Longrightarrow> emeasure (measure_pmf M) S = (\<Sum>s\<in>S. pmf M s)"
by (subst emeasure_eq_setsum_singleton) (auto simp: emeasure_pmf_single pmf_nonneg)
lemma measure_measure_pmf_finite: "finite S \<Longrightarrow> measure (measure_pmf M) S = setsum (pmf M) S"
using emeasure_measure_pmf_finite[of S M]
by (simp add: measure_pmf.emeasure_eq_measure measure_nonneg setsum_nonneg pmf_nonneg)
lemma setsum_pmf_eq_1:
assumes "finite A" "set_pmf p \<subseteq> A"
shows "(\<Sum>x\<in>A. pmf p x) = 1"
proof -
have "(\<Sum>x\<in>A. pmf p x) = measure_pmf.prob p A"
by (simp add: measure_measure_pmf_finite assms)
also from assms have "\<dots> = 1"
by (subst measure_pmf.prob_eq_1) (auto simp: AE_measure_pmf_iff)
finally show ?thesis .
qed
lemma nn_integral_measure_pmf_support:
fixes f :: "'a \<Rightarrow> ennreal"
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"
shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>A. f x * pmf M x)"
proof -
have "(\<integral>\<^sup>+x. f x \<partial>M) = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)"
using nn by (intro nn_integral_cong_AE) (auto simp: AE_measure_pmf_iff split: split_indicator)
also have "\<dots> = (\<Sum>x\<in>A. f x * emeasure M {x})"
using assms by (intro nn_integral_indicator_finite) auto
finally show ?thesis
by (simp add: emeasure_measure_pmf_finite)
qed
lemma nn_integral_measure_pmf_finite:
fixes f :: "'a \<Rightarrow> ennreal"
assumes f: "finite (set_pmf M)" and nn: "\<And>x. x \<in> set_pmf M \<Longrightarrow> 0 \<le> f x"
shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>set_pmf M. f x * pmf M x)"
using assms by (intro nn_integral_measure_pmf_support) auto
lemma integrable_measure_pmf_finite:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
shows "finite (set_pmf M) \<Longrightarrow> integrable M f"
by (auto intro!: integrableI_bounded simp: nn_integral_measure_pmf_finite ennreal_mult_less_top)
lemma integral_measure_pmf:
assumes [simp]: "finite A" and "\<And>a. a \<in> set_pmf M \<Longrightarrow> f a \<noteq> 0 \<Longrightarrow> a \<in> A"
shows "(\<integral>x. f x \<partial>measure_pmf M) = (\<Sum>a\<in>A. f a * pmf M a)"
proof -
have "(\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x * indicator A x \<partial>measure_pmf M)"
using assms(2) by (intro integral_cong_AE) (auto split: split_indicator simp: AE_measure_pmf_iff)
also have "\<dots> = (\<Sum>a\<in>A. f a * pmf M a)"
by (subst integral_indicator_finite_real)
(auto simp: measure_def emeasure_measure_pmf_finite pmf_nonneg)
finally show ?thesis .
qed
lemma integrable_pmf: "integrable (count_space X) (pmf M)"
proof -
have " (\<integral>\<^sup>+ x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+ x. pmf M x \<partial>count_space (M \<inter> X))"
by (auto simp add: nn_integral_count_space_indicator set_pmf_iff intro!: nn_integral_cong split: split_indicator)
then have "integrable (count_space X) (pmf M) = integrable (count_space (M \<inter> X)) (pmf M)"
by (simp add: integrable_iff_bounded pmf_nonneg)
then show ?thesis
by (simp add: pmf.rep_eq measure_pmf.integrable_measure disjoint_family_on_def)
qed
lemma integral_pmf: "(\<integral>x. pmf M x \<partial>count_space X) = measure M X"
proof -
have "(\<integral>x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+x. pmf M x \<partial>count_space X)"
by (simp add: pmf_nonneg integrable_pmf nn_integral_eq_integral)
also have "\<dots> = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space (X \<inter> M))"
by (auto intro!: nn_integral_cong_AE split: split_indicator
simp: pmf.rep_eq measure_pmf.emeasure_eq_measure nn_integral_count_space_indicator
AE_count_space set_pmf_iff)
also have "\<dots> = emeasure M (X \<inter> M)"
by (rule emeasure_countable_singleton[symmetric]) (auto intro: countable_set_pmf)
also have "\<dots> = emeasure M X"
by (auto intro!: emeasure_eq_AE simp: AE_measure_pmf_iff)
finally show ?thesis
by (simp add: measure_pmf.emeasure_eq_measure measure_nonneg integral_nonneg pmf_nonneg)
qed
lemma integral_pmf_restrict:
"(f::'a \<Rightarrow> 'b::{banach, second_countable_topology}) \<in> borel_measurable (count_space UNIV) \<Longrightarrow>
(\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x \<partial>restrict_space M M)"
by (auto intro!: integral_cong_AE simp add: integral_restrict_space AE_measure_pmf_iff)
lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1"
proof -
have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)"
by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf)
then show ?thesis
using measure_pmf.emeasure_space_1 by simp
qed
lemma emeasure_pmf_UNIV [simp]: "emeasure (measure_pmf M) UNIV = 1"
using measure_pmf.emeasure_space_1[of M] by simp
lemma in_null_sets_measure_pmfI:
"A \<inter> set_pmf p = {} \<Longrightarrow> A \<in> null_sets (measure_pmf p)"
using emeasure_eq_0_AE[where ?P="\<lambda>x. x \<in> A" and M="measure_pmf p"]
by(auto simp add: null_sets_def AE_measure_pmf_iff)
lemma measure_subprob: "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
by (simp add: space_subprob_algebra subprob_space_measure_pmf)
subsection \<open> Monad Interpretation \<close>
lemma measurable_measure_pmf[measurable]:
"(\<lambda>x. measure_pmf (M x)) \<in> measurable (count_space UNIV) (subprob_algebra (count_space UNIV))"
by (auto simp: space_subprob_algebra intro!: prob_space_imp_subprob_space) unfold_locales
lemma bind_measure_pmf_cong:
assumes "\<And>x. A x \<in> space (subprob_algebra N)" "\<And>x. B x \<in> space (subprob_algebra N)"
assumes "\<And>i. i \<in> set_pmf x \<Longrightarrow> A i = B i"
shows "bind (measure_pmf x) A = bind (measure_pmf x) B"
proof (rule measure_eqI)
show "sets (measure_pmf x \<bind> A) = sets (measure_pmf x \<bind> B)"
using assms by (subst (1 2) sets_bind) (auto simp: space_subprob_algebra)
next
fix X assume "X \<in> sets (measure_pmf x \<bind> A)"
then have X: "X \<in> sets N"
using assms by (subst (asm) sets_bind) (auto simp: space_subprob_algebra)
show "emeasure (measure_pmf x \<bind> A) X = emeasure (measure_pmf x \<bind> B) X"
using assms
by (subst (1 2) emeasure_bind[where N=N, OF _ _ X])
(auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
qed
lift_definition bind_pmf :: "'a pmf \<Rightarrow> ('a \<Rightarrow> 'b pmf ) \<Rightarrow> 'b pmf" is bind
proof (clarify, intro conjI)
fix f :: "'a measure" and g :: "'a \<Rightarrow> 'b measure"
assume "prob_space f"
then interpret f: prob_space f .
assume "sets f = UNIV" and ae_f: "AE x in f. measure f {x} \<noteq> 0"
then have s_f[simp]: "sets f = sets (count_space UNIV)"
by simp
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)"
then have g: "\<And>x. prob_space (g x)" and s_g[simp]: "\<And>x. sets (g x) = sets (count_space UNIV)"
and ae_g: "\<And>x. AE y in g x. measure (g x) {y} \<noteq> 0"
by auto
have [measurable]: "g \<in> measurable f (subprob_algebra (count_space UNIV))"
by (auto simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space g)
show "prob_space (f \<bind> g)"
using g by (intro f.prob_space_bind[where S="count_space UNIV"]) auto
then interpret fg: prob_space "f \<bind> g" .
show [simp]: "sets (f \<bind> g) = UNIV"
using sets_eq_imp_space_eq[OF s_f]
by (subst sets_bind[where N="count_space UNIV"]) auto
show "AE x in f \<bind> g. measure (f \<bind> g) {x} \<noteq> 0"
apply (simp add: fg.prob_eq_0 AE_bind[where B="count_space UNIV"])
using ae_f
apply eventually_elim
using ae_g
apply eventually_elim
apply (auto dest: AE_measure_singleton)
done
qed
adhoc_overloading Monad_Syntax.bind bind_pmf
lemma ennreal_pmf_bind: "pmf (bind_pmf N f) i = (\<integral>\<^sup>+x. pmf (f x) i \<partial>measure_pmf N)"
unfolding pmf.rep_eq bind_pmf.rep_eq
by (auto simp: measure_pmf.measure_bind[where N="count_space UNIV"] measure_subprob measure_nonneg
intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])
lemma pmf_bind: "pmf (bind_pmf N f) i = (\<integral>x. pmf (f x) i \<partial>measure_pmf N)"
using ennreal_pmf_bind[of N f i]
by (subst (asm) nn_integral_eq_integral)
(auto simp: pmf_nonneg pmf_le_1 pmf_nonneg integral_nonneg
intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])
lemma bind_pmf_const[simp]: "bind_pmf M (\<lambda>x. c) = c"
by transfer (simp add: bind_const' prob_space_imp_subprob_space)
lemma set_bind_pmf[simp]: "set_pmf (bind_pmf M N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))"
proof -
have "set_pmf (bind_pmf M N) = {x. ennreal (pmf (bind_pmf M N) x) \<noteq> 0}"
by (simp add: set_pmf_eq pmf_nonneg)
also have "\<dots> = (\<Union>M\<in>set_pmf M. set_pmf (N M))"
unfolding ennreal_pmf_bind
by (subst nn_integral_0_iff_AE) (auto simp: AE_measure_pmf_iff pmf_nonneg set_pmf_eq)
finally show ?thesis .
qed
lemma bind_pmf_cong [fundef_cong]:
assumes "p = q"
shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g"
unfolding \<open>p = q\<close>[symmetric] measure_pmf_inject[symmetric] bind_pmf.rep_eq
by (auto simp: AE_measure_pmf_iff Pi_iff space_subprob_algebra subprob_space_measure_pmf
sets_bind[where N="count_space UNIV"] emeasure_bind[where N="count_space UNIV"]
intro!: nn_integral_cong_AE measure_eqI)
lemma bind_pmf_cong_simp:
"p = q \<Longrightarrow> (\<And>x. x \<in> set_pmf q =simp=> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g"
by (simp add: simp_implies_def cong: bind_pmf_cong)
lemma measure_pmf_bind: "measure_pmf (bind_pmf M f) = (measure_pmf M \<bind> (\<lambda>x. measure_pmf (f x)))"
by transfer simp
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)"
using measurable_measure_pmf[of N]
unfolding measure_pmf_bind
apply (intro nn_integral_bind[where B="count_space UNIV"])
apply auto
done
lemma emeasure_bind_pmf[simp]: "emeasure (bind_pmf M N) X = (\<integral>\<^sup>+x. emeasure (N x) X \<partial>M)"
using measurable_measure_pmf[of N]
unfolding measure_pmf_bind
by (subst emeasure_bind[where N="count_space UNIV"]) auto
lift_definition return_pmf :: "'a \<Rightarrow> 'a pmf" is "return (count_space UNIV)"
by (auto intro!: prob_space_return simp: AE_return measure_return)
lemma bind_return_pmf: "bind_pmf (return_pmf x) f = f x"
by transfer
(auto intro!: prob_space_imp_subprob_space bind_return[where N="count_space UNIV"]
simp: space_subprob_algebra)
lemma set_return_pmf[simp]: "set_pmf (return_pmf x) = {x}"
by transfer (auto simp add: measure_return split: split_indicator)
lemma bind_return_pmf': "bind_pmf N return_pmf = N"
proof (transfer, clarify)
fix N :: "'a measure" assume "sets N = UNIV" then show "N \<bind> return (count_space UNIV) = N"
by (subst return_sets_cong[where N=N]) (simp_all add: bind_return')
qed
lemma bind_assoc_pmf: "bind_pmf (bind_pmf A B) C = bind_pmf A (\<lambda>x. bind_pmf (B x) C)"
by transfer
(auto intro!: bind_assoc[where N="count_space UNIV" and R="count_space UNIV"]
simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space)
definition "map_pmf f M = bind_pmf M (\<lambda>x. return_pmf (f x))"
lemma map_bind_pmf: "map_pmf f (bind_pmf M g) = bind_pmf M (\<lambda>x. map_pmf f (g x))"
by (simp add: map_pmf_def bind_assoc_pmf)
lemma bind_map_pmf: "bind_pmf (map_pmf f M) g = bind_pmf M (\<lambda>x. g (f x))"
by (simp add: map_pmf_def bind_assoc_pmf bind_return_pmf)
lemma map_pmf_transfer[transfer_rule]:
"rel_fun op = (rel_fun cr_pmf cr_pmf) (\<lambda>f M. distr M (count_space UNIV) f) map_pmf"
proof -
have "rel_fun op = (rel_fun pmf_as_measure.cr_pmf pmf_as_measure.cr_pmf)
(\<lambda>f M. M \<bind> (return (count_space UNIV) o f)) map_pmf"
unfolding map_pmf_def[abs_def] comp_def by transfer_prover
then show ?thesis
by (force simp: rel_fun_def cr_pmf_def bind_return_distr)
qed
lemma map_pmf_rep_eq:
"measure_pmf (map_pmf f M) = distr (measure_pmf M) (count_space UNIV) f"
unfolding map_pmf_def bind_pmf.rep_eq comp_def return_pmf.rep_eq
using bind_return_distr[of M f "count_space UNIV"] by (simp add: comp_def)
lemma map_pmf_id[simp]: "map_pmf id = id"
by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI)
lemma map_pmf_ident[simp]: "map_pmf (\<lambda>x. x) = (\<lambda>x. x)"
using map_pmf_id unfolding id_def .
lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g"
by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def)
lemma map_pmf_comp: "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M"
using map_pmf_compose[of f g] by (simp add: comp_def)
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"
unfolding map_pmf_def by (rule bind_pmf_cong) auto
lemma pmf_set_map: "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
by (auto simp add: comp_def fun_eq_iff map_pmf_def)
lemma set_map_pmf[simp]: "set_pmf (map_pmf f M) = f`set_pmf M"
using pmf_set_map[of f] by (auto simp: comp_def fun_eq_iff)
lemma emeasure_map_pmf[simp]: "emeasure (map_pmf f M) X = emeasure M (f -` X)"
unfolding map_pmf_rep_eq by (subst emeasure_distr) auto
lemma measure_map_pmf[simp]: "measure (map_pmf f M) X = measure M (f -` X)"
using emeasure_map_pmf[of f M X] by(simp add: measure_pmf.emeasure_eq_measure measure_nonneg)
lemma nn_integral_map_pmf[simp]: "(\<integral>\<^sup>+x. f x \<partial>map_pmf g M) = (\<integral>\<^sup>+x. f (g x) \<partial>M)"
unfolding map_pmf_rep_eq by (intro nn_integral_distr) auto
lemma ennreal_pmf_map: "pmf (map_pmf f p) x = (\<integral>\<^sup>+ y. indicator (f -` {x}) y \<partial>measure_pmf p)"
proof (transfer fixing: f x)
fix p :: "'b measure"
presume "prob_space p"
then interpret prob_space p .
presume "sets p = UNIV"
then show "ennreal (measure (distr p (count_space UNIV) f) {x}) = integral\<^sup>N p (indicator (f -` {x}))"
by(simp add: measure_distr measurable_def emeasure_eq_measure)
qed simp_all
lemma pmf_map: "pmf (map_pmf f p) x = measure p (f -` {x})"
proof (transfer fixing: f x)
fix p :: "'b measure"
presume "prob_space p"
then interpret prob_space p .
presume "sets p = UNIV"
then show "measure (distr p (count_space UNIV) f) {x} = measure p (f -` {x})"
by(simp add: measure_distr measurable_def emeasure_eq_measure)
qed simp_all
lemma nn_integral_pmf: "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = emeasure (measure_pmf p) A"
proof -
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))"
by(auto simp add: nn_integral_count_space_indicator indicator_def set_pmf_iff intro: nn_integral_cong)
also have "\<dots> = emeasure (measure_pmf p) (\<Union>x\<in>A \<inter> set_pmf p. {x})"
by(subst emeasure_UN_countable)(auto simp add: emeasure_pmf_single disjoint_family_on_def)
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})"
by(rule emeasure_Un_null_set[symmetric])(auto intro: in_null_sets_measure_pmfI)
also have "\<dots> = emeasure (measure_pmf p) A"
by(auto intro: arg_cong2[where f=emeasure])
finally show ?thesis .
qed
lemma integral_map_pmf[simp]:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
shows "integral\<^sup>L (map_pmf g p) f = integral\<^sup>L p (\<lambda>x. f (g x))"
by (simp add: integral_distr map_pmf_rep_eq)
lemma map_return_pmf [simp]: "map_pmf f (return_pmf x) = return_pmf (f x)"
by transfer (simp add: distr_return)
lemma map_pmf_const[simp]: "map_pmf (\<lambda>_. c) M = return_pmf c"
by transfer (auto simp: prob_space.distr_const)
lemma pmf_return [simp]: "pmf (return_pmf x) y = indicator {y} x"
by transfer (simp add: measure_return)
lemma nn_integral_return_pmf[simp]: "0 \<le> f x \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>return_pmf x) = f x"
unfolding return_pmf.rep_eq by (intro nn_integral_return) auto
lemma emeasure_return_pmf[simp]: "emeasure (return_pmf x) X = indicator X x"
unfolding return_pmf.rep_eq by (intro emeasure_return) auto
lemma measure_return_pmf [simp]: "measure_pmf.prob (return_pmf x) A = indicator A x"
proof -
have "ennreal (measure_pmf.prob (return_pmf x) A) =
emeasure (measure_pmf (return_pmf x)) A"
by (simp add: measure_pmf.emeasure_eq_measure)
also have "\<dots> = ennreal (indicator A x)" by (simp add: ennreal_indicator)
finally show ?thesis by simp
qed
lemma return_pmf_inj[simp]: "return_pmf x = return_pmf y \<longleftrightarrow> x = y"
by (metis insertI1 set_return_pmf singletonD)
lemma map_pmf_eq_return_pmf_iff:
"map_pmf f p = return_pmf x \<longleftrightarrow> (\<forall>y \<in> set_pmf p. f y = x)"
proof
assume "map_pmf f p = return_pmf x"
then have "set_pmf (map_pmf f p) = set_pmf (return_pmf x)" by simp
then show "\<forall>y \<in> set_pmf p. f y = x" by auto
next
assume "\<forall>y \<in> set_pmf p. f y = x"
then show "map_pmf f p = return_pmf x"
unfolding map_pmf_const[symmetric, of _ p] by (intro map_pmf_cong) auto
qed
definition "pair_pmf A B = bind_pmf A (\<lambda>x. bind_pmf B (\<lambda>y. return_pmf (x, y)))"
lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b"
unfolding pair_pmf_def pmf_bind pmf_return
apply (subst integral_measure_pmf[where A="{b}"])
apply (auto simp: indicator_eq_0_iff)
apply (subst integral_measure_pmf[where A="{a}"])
apply (auto simp: indicator_eq_0_iff setsum_nonneg_eq_0_iff pmf_nonneg)
done
lemma set_pair_pmf[simp]: "set_pmf (pair_pmf A B) = set_pmf A \<times> set_pmf B"
unfolding pair_pmf_def set_bind_pmf set_return_pmf by auto
lemma measure_pmf_in_subprob_space[measurable (raw)]:
"measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
by (simp add: space_subprob_algebra) intro_locales
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)"
proof -
have "(\<integral>\<^sup>+x. f x \<partial>pair_pmf A B) = (\<integral>\<^sup>+x. f x * indicator (A \<times> B) x \<partial>pair_pmf A B)"
by (auto simp: AE_measure_pmf_iff intro!: nn_integral_cong_AE)
also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. f (a, b) * indicator (A \<times> B) (a, b) \<partial>B \<partial>A)"
by (simp add: pair_pmf_def)
also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. f (a, b) \<partial>B \<partial>A)"
by (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
finally show ?thesis .
qed
lemma bind_pair_pmf:
assumes M[measurable]: "M \<in> measurable (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) (subprob_algebra N)"
shows "measure_pmf (pair_pmf A B) \<bind> M = (measure_pmf A \<bind> (\<lambda>x. measure_pmf B \<bind> (\<lambda>y. M (x, y))))"
(is "?L = ?R")
proof (rule measure_eqI)
have M'[measurable]: "M \<in> measurable (pair_pmf A B) (subprob_algebra N)"
using M[THEN measurable_space] by (simp_all add: space_pair_measure)
note measurable_bind[where N="count_space UNIV", measurable]
note measure_pmf_in_subprob_space[simp]
have sets_eq_N: "sets ?L = N"
by (subst sets_bind[OF sets_kernel[OF M']]) auto
show "sets ?L = sets ?R"
using measurable_space[OF M]
by (simp add: sets_eq_N space_pair_measure space_subprob_algebra)
fix X assume "X \<in> sets ?L"
then have X[measurable]: "X \<in> sets N"
unfolding sets_eq_N .
then show "emeasure ?L X = emeasure ?R X"
apply (simp add: emeasure_bind[OF _ M' X])
apply (simp add: nn_integral_bind[where B="count_space UNIV"] pair_pmf_def measure_pmf_bind[of A]
nn_integral_measure_pmf_finite)
apply (subst emeasure_bind[OF _ _ X])
apply measurable
apply (subst emeasure_bind[OF _ _ X])
apply measurable
done
qed
lemma map_fst_pair_pmf: "map_pmf fst (pair_pmf A B) = A"
by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
lemma map_snd_pair_pmf: "map_pmf snd (pair_pmf A B) = B"
by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
lemma nn_integral_pmf':
"inj_on f A \<Longrightarrow> (\<integral>\<^sup>+x. pmf p (f x) \<partial>count_space A) = emeasure p (f ` A)"
by (subst nn_integral_bij_count_space[where g=f and B="f`A"])
(auto simp: bij_betw_def nn_integral_pmf)
lemma pmf_le_0_iff[simp]: "pmf M p \<le> 0 \<longleftrightarrow> pmf M p = 0"
using pmf_nonneg[of M p] by arith
lemma min_pmf_0[simp]: "min (pmf M p) 0 = 0" "min 0 (pmf M p) = 0"
using pmf_nonneg[of M p] by arith+
lemma pmf_eq_0_set_pmf: "pmf M p = 0 \<longleftrightarrow> p \<notin> set_pmf M"
unfolding set_pmf_iff by simp
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"
by (auto simp: pmf.rep_eq map_pmf_rep_eq measure_distr AE_measure_pmf_iff inj_onD
intro!: measure_pmf.finite_measure_eq_AE)
lemma pmf_map_inj': "inj f \<Longrightarrow> pmf (map_pmf f M) (f x) = pmf M x"
apply(cases "x \<in> set_pmf M")
apply(simp add: pmf_map_inj[OF subset_inj_on])
apply(simp add: pmf_eq_0_set_pmf[symmetric])
apply(auto simp add: pmf_eq_0_set_pmf dest: injD)
done
lemma pmf_map_outside: "x \<notin> f ` set_pmf M \<Longrightarrow> pmf (map_pmf f M) x = 0"
unfolding pmf_eq_0_set_pmf by simp
subsection \<open> PMFs as function \<close>
context
fixes f :: "'a \<Rightarrow> real"
assumes nonneg: "\<And>x. 0 \<le> f x"
assumes prob: "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
begin
lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ennreal \<circ> f)"
proof (intro conjI)
have *[simp]: "\<And>x y. ennreal (f y) * indicator {x} y = ennreal (f x) * indicator {x} y"
by (simp split: split_indicator)
show "AE x in density (count_space UNIV) (ennreal \<circ> f).
measure (density (count_space UNIV) (ennreal \<circ> f)) {x} \<noteq> 0"
by (simp add: AE_density nonneg measure_def emeasure_density max_def)
show "prob_space (density (count_space UNIV) (ennreal \<circ> f))"
by standard (simp add: emeasure_density prob)
qed simp
lemma pmf_embed_pmf: "pmf embed_pmf x = f x"
proof transfer
have *[simp]: "\<And>x y. ennreal (f y) * indicator {x} y = ennreal (f x) * indicator {x} y"
by (simp split: split_indicator)
fix x show "measure (density (count_space UNIV) (ennreal \<circ> f)) {x} = f x"
by transfer (simp add: measure_def emeasure_density nonneg max_def)
qed
lemma set_embed_pmf: "set_pmf embed_pmf = {x. f x \<noteq> 0}"
by(auto simp add: set_pmf_eq pmf_embed_pmf)
end
lemma embed_pmf_transfer:
"rel_fun (eq_onp (\<lambda>f. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ennreal (f x) \<partial>count_space UNIV) = 1)) pmf_as_measure.cr_pmf (\<lambda>f. density (count_space UNIV) (ennreal \<circ> f)) embed_pmf"
by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer)
lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)"
proof (transfer, elim conjE)
fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
assume "prob_space M" then interpret prob_space M .
show "M = density (count_space UNIV) (\<lambda>x. ennreal (measure M {x}))"
proof (rule measure_eqI)
fix A :: "'a set"
have "(\<integral>\<^sup>+ x. ennreal (measure M {x}) * indicator A x \<partial>count_space UNIV) =
(\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)"
by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator)
also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))"
by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space)
also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})"
by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support)
(auto simp: disjoint_family_on_def)
also have "\<dots> = emeasure M A"
using ae by (intro emeasure_eq_AE) auto
finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ennreal (measure M {x}))) A"
using emeasure_space_1 by (simp add: emeasure_density)
qed simp
qed
lemma td_pmf_embed_pmf:
"type_definition pmf embed_pmf {f::'a \<Rightarrow> real. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ennreal (f x) \<partial>count_space UNIV) = 1}"
unfolding type_definition_def
proof safe
fix p :: "'a pmf"
have "(\<integral>\<^sup>+ x. 1 \<partial>measure_pmf p) = 1"
using measure_pmf.emeasure_space_1[of p] by simp
then show *: "(\<integral>\<^sup>+ x. ennreal (pmf p x) \<partial>count_space UNIV) = 1"
by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const)
show "embed_pmf (pmf p) = p"
by (intro measure_pmf_inject[THEN iffD1])
(simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def)
next
fix f :: "'a \<Rightarrow> real" assume "\<forall>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
then show "pmf (embed_pmf f) = f"
by (auto intro!: pmf_embed_pmf)
qed (rule pmf_nonneg)
end
lemma nn_integral_measure_pmf: "(\<integral>\<^sup>+ x. f x \<partial>measure_pmf p) = \<integral>\<^sup>+ x. ennreal (pmf p x) * f x \<partial>count_space UNIV"
by(simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg)
locale pmf_as_function
begin
setup_lifting td_pmf_embed_pmf
lemma set_pmf_transfer[transfer_rule]:
assumes "bi_total A"
shows "rel_fun (pcr_pmf A) (rel_set A) (\<lambda>f. {x. f x \<noteq> 0}) set_pmf"
using \<open>bi_total A\<close>
by (auto simp: pcr_pmf_def cr_pmf_def rel_fun_def rel_set_def bi_total_def Bex_def set_pmf_iff)
metis+
end
context
begin
interpretation pmf_as_function .
lemma pmf_eqI: "(\<And>i. pmf M i = pmf N i) \<Longrightarrow> M = N"
by transfer auto
lemma pmf_eq_iff: "M = N \<longleftrightarrow> (\<forall>i. pmf M i = pmf N i)"
by (auto intro: pmf_eqI)
lemma pmf_neq_exists_less:
assumes "M \<noteq> N"
shows "\<exists>x. pmf M x < pmf N x"
proof (rule ccontr)
assume "\<not>(\<exists>x. pmf M x < pmf N x)"
hence ge: "pmf M x \<ge> pmf N x" for x by (auto simp: not_less)
from assms obtain x where "pmf M x \<noteq> pmf N x" by (auto simp: pmf_eq_iff)
with ge[of x] have gt: "pmf M x > pmf N x" by simp
have "1 = measure (measure_pmf M) UNIV" by simp
also have "\<dots> = measure (measure_pmf N) {x} + measure (measure_pmf N) (UNIV - {x})"
by (subst measure_pmf.finite_measure_Union [symmetric]) simp_all
also from gt have "measure (measure_pmf N) {x} < measure (measure_pmf M) {x}"
by (simp add: measure_pmf_single)
also have "measure (measure_pmf N) (UNIV - {x}) \<le> measure (measure_pmf M) (UNIV - {x})"
by (subst (1 2) integral_pmf [symmetric])
(intro integral_mono integrable_pmf, simp_all add: ge)
also have "measure (measure_pmf M) {x} + \<dots> = 1"
by (subst measure_pmf.finite_measure_Union [symmetric]) simp_all
finally show False by simp_all
qed
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))"
unfolding pmf_eq_iff pmf_bind
proof
fix i
interpret B: prob_space "restrict_space B B"
by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
(auto simp: AE_measure_pmf_iff)
interpret A: prob_space "restrict_space A A"
by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
(auto simp: AE_measure_pmf_iff)
interpret AB: pair_prob_space "restrict_space A A" "restrict_space B B"
by unfold_locales
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)"
by (rule integral_cong) (auto intro!: integral_pmf_restrict)
also have "\<dots> = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>restrict_space A A)"
by (intro integral_pmf_restrict B.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
countable_set_pmf borel_measurable_count_space)
also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>restrict_space B B)"
by (rule AB.Fubini_integral[symmetric])
(auto intro!: AB.integrable_const_bound[where B=1] measurable_pair_restrict_pmf2
simp: pmf_nonneg pmf_le_1 measurable_restrict_space1)
also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>B)"
by (intro integral_pmf_restrict[symmetric] A.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
countable_set_pmf borel_measurable_count_space)
also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>A \<partial>B)"
by (rule integral_cong) (auto intro!: integral_pmf_restrict[symmetric])
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)" .
qed
lemma pair_map_pmf1: "pair_pmf (map_pmf f A) B = map_pmf (apfst f) (pair_pmf A B)"
proof (safe intro!: pmf_eqI)
fix a :: "'a" and b :: "'b"
have [simp]: "\<And>c d. indicator (apfst f -` {(a, b)}) (c, d) = indicator (f -` {a}) c * (indicator {b} d::ennreal)"
by (auto split: split_indicator)
have "ennreal (pmf (pair_pmf (map_pmf f A) B) (a, b)) =
ennreal (pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b))"
unfolding pmf_pair ennreal_pmf_map
by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_multc pmf_nonneg
emeasure_map_pmf[symmetric] ennreal_mult del: emeasure_map_pmf)
then show "pmf (pair_pmf (map_pmf f A) B) (a, b) = pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b)"
by (simp add: pmf_nonneg)
qed
lemma pair_map_pmf2: "pair_pmf A (map_pmf f B) = map_pmf (apsnd f) (pair_pmf A B)"
proof (safe intro!: pmf_eqI)
fix a :: "'a" and b :: "'b"
have [simp]: "\<And>c d. indicator (apsnd f -` {(a, b)}) (c, d) = indicator {a} c * (indicator (f -` {b}) d::ennreal)"
by (auto split: split_indicator)
have "ennreal (pmf (pair_pmf A (map_pmf f B)) (a, b)) =
ennreal (pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b))"
unfolding pmf_pair ennreal_pmf_map
by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_cmult nn_integral_multc pmf_nonneg
emeasure_map_pmf[symmetric] ennreal_mult del: emeasure_map_pmf)
then show "pmf (pair_pmf A (map_pmf f B)) (a, b) = pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b)"
by (simp add: pmf_nonneg)
qed
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)"
by (simp add: pair_map_pmf2 pair_map_pmf1 map_pmf_comp split_beta')
end
lemma pair_return_pmf1: "pair_pmf (return_pmf x) y = map_pmf (Pair x) y"
by(simp add: pair_pmf_def bind_return_pmf map_pmf_def)
lemma pair_return_pmf2: "pair_pmf x (return_pmf y) = map_pmf (\<lambda>x. (x, y)) x"
by(simp add: pair_pmf_def bind_return_pmf map_pmf_def)
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))"
by(simp add: pair_pmf_def bind_return_pmf map_pmf_def bind_assoc_pmf)
lemma pair_commute_pmf: "pair_pmf x y = map_pmf (\<lambda>(x, y). (y, x)) (pair_pmf y x)"
unfolding pair_pmf_def by(subst bind_commute_pmf)(simp add: map_pmf_def bind_assoc_pmf bind_return_pmf)
lemma set_pmf_subset_singleton: "set_pmf p \<subseteq> {x} \<longleftrightarrow> p = return_pmf x"
proof(intro iffI pmf_eqI)
fix i
assume x: "set_pmf p \<subseteq> {x}"
hence *: "set_pmf p = {x}" using set_pmf_not_empty[of p] by auto
have "ennreal (pmf p x) = \<integral>\<^sup>+ i. indicator {x} i \<partial>p" by(simp add: emeasure_pmf_single)
also have "\<dots> = \<integral>\<^sup>+ i. 1 \<partial>p" by(rule nn_integral_cong_AE)(simp add: AE_measure_pmf_iff * )
also have "\<dots> = 1" by simp
finally show "pmf p i = pmf (return_pmf x) i" using x
by(auto split: split_indicator simp add: pmf_eq_0_set_pmf)
qed auto
lemma bind_eq_return_pmf:
"bind_pmf p f = return_pmf x \<longleftrightarrow> (\<forall>y\<in>set_pmf p. f y = return_pmf x)"
(is "?lhs \<longleftrightarrow> ?rhs")
proof(intro iffI strip)
fix y
assume y: "y \<in> set_pmf p"
assume "?lhs"
hence "set_pmf (bind_pmf p f) = {x}" by simp
hence "(\<Union>y\<in>set_pmf p. set_pmf (f y)) = {x}" by simp
hence "set_pmf (f y) \<subseteq> {x}" using y by auto
thus "f y = return_pmf x" by(simp add: set_pmf_subset_singleton)
next
assume *: ?rhs
show ?lhs
proof(rule pmf_eqI)
fix i
have "ennreal (pmf (bind_pmf p f) i) = \<integral>\<^sup>+ y. ennreal (pmf (f y) i) \<partial>p"
by (simp add: ennreal_pmf_bind)
also have "\<dots> = \<integral>\<^sup>+ y. ennreal (pmf (return_pmf x) i) \<partial>p"
by(rule nn_integral_cong_AE)(simp add: AE_measure_pmf_iff * )
also have "\<dots> = ennreal (pmf (return_pmf x) i)"
by simp
finally show "pmf (bind_pmf p f) i = pmf (return_pmf x) i"
by (simp add: pmf_nonneg)
qed
qed
lemma pmf_False_conv_True: "pmf p False = 1 - pmf p True"
proof -
have "pmf p False + pmf p True = measure p {False} + measure p {True}"
by(simp add: measure_pmf_single)
also have "\<dots> = measure p ({False} \<union> {True})"
by(subst measure_pmf.finite_measure_Union) simp_all
also have "{False} \<union> {True} = space p" by auto
finally show ?thesis by simp
qed
lemma pmf_True_conv_False: "pmf p True = 1 - pmf p False"
by(simp add: pmf_False_conv_True)
subsection \<open> Conditional Probabilities \<close>
lemma measure_pmf_zero_iff: "measure (measure_pmf p) s = 0 \<longleftrightarrow> set_pmf p \<inter> s = {}"
by (subst measure_pmf.prob_eq_0) (auto simp: AE_measure_pmf_iff)
context
fixes p :: "'a pmf" and s :: "'a set"
assumes not_empty: "set_pmf p \<inter> s \<noteq> {}"
begin
interpretation pmf_as_measure .
lemma emeasure_measure_pmf_not_zero: "emeasure (measure_pmf p) s \<noteq> 0"
proof
assume "emeasure (measure_pmf p) s = 0"
then have "AE x in measure_pmf p. x \<notin> s"
by (rule AE_I[rotated]) auto
with not_empty show False
by (auto simp: AE_measure_pmf_iff)
qed
lemma measure_measure_pmf_not_zero: "measure (measure_pmf p) s \<noteq> 0"
using emeasure_measure_pmf_not_zero by (simp add: measure_pmf.emeasure_eq_measure measure_nonneg)
lift_definition cond_pmf :: "'a pmf" is
"uniform_measure (measure_pmf p) s"
proof (intro conjI)
show "prob_space (uniform_measure (measure_pmf p) s)"
by (intro prob_space_uniform_measure) (auto simp: emeasure_measure_pmf_not_zero)
show "AE x in uniform_measure (measure_pmf p) s. measure (uniform_measure (measure_pmf p) s) {x} \<noteq> 0"
by (simp add: emeasure_measure_pmf_not_zero measure_measure_pmf_not_zero AE_uniform_measure
AE_measure_pmf_iff set_pmf.rep_eq less_top[symmetric])
qed simp
lemma pmf_cond: "pmf cond_pmf x = (if x \<in> s then pmf p x / measure p s else 0)"
by transfer (simp add: emeasure_measure_pmf_not_zero pmf.rep_eq)
lemma set_cond_pmf[simp]: "set_pmf cond_pmf = set_pmf p \<inter> s"
by (auto simp add: set_pmf_iff pmf_cond measure_measure_pmf_not_zero split: if_split_asm)
end
lemma measure_pmf_posI: "x \<in> set_pmf p \<Longrightarrow> x \<in> A \<Longrightarrow> measure_pmf.prob p A > 0"
using measure_measure_pmf_not_zero[of p A] by (subst zero_less_measure_iff) blast
lemma cond_map_pmf:
assumes "set_pmf p \<inter> f -` s \<noteq> {}"
shows "cond_pmf (map_pmf f p) s = map_pmf f (cond_pmf p (f -` s))"
proof -
have *: "set_pmf (map_pmf f p) \<inter> s \<noteq> {}"
using assms by auto
{ fix x
have "ennreal (pmf (map_pmf f (cond_pmf p (f -` s))) x) =
emeasure p (f -` s \<inter> f -` {x}) / emeasure p (f -` s)"
unfolding ennreal_pmf_map cond_pmf.rep_eq[OF assms] by (simp add: nn_integral_uniform_measure)
also have "f -` s \<inter> f -` {x} = (if x \<in> s then f -` {x} else {})"
by auto
also have "emeasure p (if x \<in> s then f -` {x} else {}) / emeasure p (f -` s) =
ennreal (pmf (cond_pmf (map_pmf f p) s) x)"
using measure_measure_pmf_not_zero[OF *]
by (simp add: pmf_cond[OF *] ennreal_pmf_map measure_pmf.emeasure_eq_measure
divide_ennreal pmf_nonneg measure_nonneg zero_less_measure_iff pmf_map)
finally have "ennreal (pmf (cond_pmf (map_pmf f p) s) x) = ennreal (pmf (map_pmf f (cond_pmf p (f -` s))) x)"
by simp }
then show ?thesis
by (intro pmf_eqI) (simp add: pmf_nonneg)
qed
lemma bind_cond_pmf_cancel:
assumes [simp]: "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}"
assumes [simp]: "\<And>y. y \<in> set_pmf q \<Longrightarrow> set_pmf p \<inter> {x. R x y} \<noteq> {}"
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}"
shows "bind_pmf p (\<lambda>x. cond_pmf q {y. R x y}) = q"
proof (rule pmf_eqI)
fix i
have "ennreal (pmf (bind_pmf p (\<lambda>x. cond_pmf q {y. R x y})) i) =
(\<integral>\<^sup>+x. ennreal (pmf q i / measure p {x. R x i}) * ennreal (indicator {x. R x i} x) \<partial>p)"
by (auto simp add: ennreal_pmf_bind AE_measure_pmf_iff pmf_cond pmf_eq_0_set_pmf pmf_nonneg measure_nonneg
intro!: nn_integral_cong_AE)
also have "\<dots> = (pmf q i * measure p {x. R x i}) / measure p {x. R x i}"
by (simp add: pmf_nonneg measure_nonneg zero_ennreal_def[symmetric] ennreal_indicator
nn_integral_cmult measure_pmf.emeasure_eq_measure ennreal_mult[symmetric])
also have "\<dots> = pmf q i"
by (cases "pmf q i = 0")
(simp_all add: pmf_eq_0_set_pmf measure_measure_pmf_not_zero pmf_nonneg)
finally show "pmf (bind_pmf p (\<lambda>x. cond_pmf q {y. R x y})) i = pmf q i"
by (simp add: pmf_nonneg)
qed
subsection \<open> Relator \<close>
inductive rel_pmf :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf \<Rightarrow> bool"
for R p q
where
"\<lbrakk> \<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y;
map_pmf fst pq = p; map_pmf snd pq = q \<rbrakk>
\<Longrightarrow> rel_pmf R p q"
lemma rel_pmfI:
assumes R: "rel_set R (set_pmf p) (set_pmf q)"
assumes eq: "\<And>x y. x \<in> set_pmf p \<Longrightarrow> y \<in> set_pmf q \<Longrightarrow> R x y \<Longrightarrow>
measure p {x. R x y} = measure q {y. R x y}"
shows "rel_pmf R p q"
proof
let ?pq = "bind_pmf p (\<lambda>x. bind_pmf (cond_pmf q {y. R x y}) (\<lambda>y. return_pmf (x, y)))"
have "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}"
using R by (auto simp: rel_set_def)
then show "\<And>x y. (x, y) \<in> set_pmf ?pq \<Longrightarrow> R x y"
by auto
show "map_pmf fst ?pq = p"
by (simp add: map_bind_pmf bind_return_pmf')
show "map_pmf snd ?pq = q"
using R eq
apply (simp add: bind_cond_pmf_cancel map_bind_pmf bind_return_pmf')
apply (rule bind_cond_pmf_cancel)
apply (auto simp: rel_set_def)
done
qed
lemma rel_pmf_imp_rel_set: "rel_pmf R p q \<Longrightarrow> rel_set R (set_pmf p) (set_pmf q)"
by (force simp add: rel_pmf.simps rel_set_def)
lemma rel_pmfD_measure:
assumes rel_R: "rel_pmf R p q" and R: "\<And>a b. R a b \<Longrightarrow> R a y \<longleftrightarrow> R x b"
assumes "x \<in> set_pmf p" "y \<in> set_pmf q"
shows "measure p {x. R x y} = measure q {y. R x y}"
proof -
from rel_R obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
and eq: "p = map_pmf fst pq" "q = map_pmf snd pq"
by (auto elim: rel_pmf.cases)
have "measure p {x. R x y} = measure pq {x. R (fst x) y}"
by (simp add: eq map_pmf_rep_eq measure_distr)
also have "\<dots> = measure pq {y. R x (snd y)}"
by (intro measure_pmf.finite_measure_eq_AE)
(auto simp: AE_measure_pmf_iff R dest!: pq)
also have "\<dots> = measure q {y. R x y}"
by (simp add: eq map_pmf_rep_eq measure_distr)
finally show "measure p {x. R x y} = measure q {y. R x y}" .
qed
lemma rel_pmf_measureD:
assumes "rel_pmf R p q"
shows "measure (measure_pmf p) A \<le> measure (measure_pmf q) {y. \<exists>x\<in>A. R x y}" (is "?lhs \<le> ?rhs")
using assms
proof cases
fix pq
assume R: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
and p[symmetric]: "map_pmf fst pq = p"
and q[symmetric]: "map_pmf snd pq = q"
have "?lhs = measure (measure_pmf pq) (fst -` A)" by(simp add: p)
also have "\<dots> \<le> measure (measure_pmf pq) {y. \<exists>x\<in>A. R x (snd y)}"
by(rule measure_pmf.finite_measure_mono_AE)(auto 4 3 simp add: AE_measure_pmf_iff dest: R)
also have "\<dots> = ?rhs" by(simp add: q)
finally show ?thesis .
qed
lemma rel_pmf_iff_measure:
assumes "symp R" "transp R"
shows "rel_pmf R p q \<longleftrightarrow>
rel_set R (set_pmf p) (set_pmf q) \<and>
(\<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})"
by (safe intro!: rel_pmf_imp_rel_set rel_pmfI)
(auto intro!: rel_pmfD_measure dest: sympD[OF \<open>symp R\<close>] transpD[OF \<open>transp R\<close>])
lemma quotient_rel_set_disjoint:
"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 = {}"
using in_quotient_imp_closed[of UNIV "{(x, y). R x y}" C]
by (auto 0 0 simp: equivp_equiv rel_set_def set_eq_iff elim: equivpE)
(blast dest: equivp_symp)+
lemma quotientD: "equiv X R \<Longrightarrow> A \<in> X // R \<Longrightarrow> x \<in> A \<Longrightarrow> A = R `` {x}"
by (metis Image_singleton_iff equiv_class_eq_iff quotientE)
lemma rel_pmf_iff_equivp:
assumes "equivp R"
shows "rel_pmf R p q \<longleftrightarrow> (\<forall>C\<in>UNIV // {(x, y). R x y}. measure p C = measure q C)"
(is "_ \<longleftrightarrow> (\<forall>C\<in>_//?R. _)")
proof (subst rel_pmf_iff_measure, safe)
show "symp R" "transp R"
using assms by (auto simp: equivp_reflp_symp_transp)
next
fix C assume C: "C \<in> UNIV // ?R" and R: "rel_set R (set_pmf p) (set_pmf q)"
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}"
show "measure p C = measure q C"
proof (cases "p \<inter> C = {}")
case True
then have "q \<inter> C = {}"
using quotient_rel_set_disjoint[OF assms C R] by simp
with True show ?thesis
unfolding measure_pmf_zero_iff[symmetric] by simp
next
case False
then have "q \<inter> C \<noteq> {}"
using quotient_rel_set_disjoint[OF assms C R] by simp
with False obtain x y where in_set: "x \<in> set_pmf p" "y \<in> set_pmf q" and in_C: "x \<in> C" "y \<in> C"
by auto
then have "R x y"
using in_quotient_imp_in_rel[of UNIV ?R C x y] C assms
by (simp add: equivp_equiv)
with in_set eq have "measure p {x. R x y} = measure q {y. R x y}"
by auto
moreover have "{y. R x y} = C"
using assms \<open>x \<in> C\<close> C quotientD[of UNIV ?R C x] by (simp add: equivp_equiv)
moreover have "{x. R x y} = C"
using assms \<open>y \<in> C\<close> C quotientD[of UNIV "?R" C y] sympD[of R]
by (auto simp add: equivp_equiv elim: equivpE)
ultimately show ?thesis
by auto
qed
next
assume eq: "\<forall>C\<in>UNIV // ?R. measure p C = measure q C"
show "rel_set R (set_pmf p) (set_pmf q)"
unfolding rel_set_def
proof safe
fix x assume x: "x \<in> set_pmf p"
have "{y. R x y} \<in> UNIV // ?R"
by (auto simp: quotient_def)
with eq have *: "measure q {y. R x y} = measure p {y. R x y}"
by auto
have "measure q {y. R x y} \<noteq> 0"
using x assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp)
then show "\<exists>y\<in>set_pmf q. R x y"
unfolding measure_pmf_zero_iff by auto
next
fix y assume y: "y \<in> set_pmf q"
have "{x. R x y} \<in> UNIV // ?R"
using assms by (auto simp: quotient_def dest: equivp_symp)
with eq have *: "measure p {x. R x y} = measure q {x. R x y}"
by auto
have "measure p {x. R x y} \<noteq> 0"
using y assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp)
then show "\<exists>x\<in>set_pmf p. R x y"
unfolding measure_pmf_zero_iff by auto
qed
fix x y assume "x \<in> set_pmf p" "y \<in> set_pmf q" "R x y"
have "{y. R x y} \<in> UNIV // ?R" "{x. R x y} = {y. R x y}"
using assms \<open>R x y\<close> by (auto simp: quotient_def dest: equivp_symp equivp_transp)
with eq show "measure p {x. R x y} = measure q {y. R x y}"
by auto
qed
bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: rel_pmf
proof -
show "map_pmf id = id" by (rule map_pmf_id)
show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose)
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"
by (intro map_pmf_cong refl)
show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
by (rule pmf_set_map)
show "(card_of (set_pmf p), natLeq) \<in> ordLeq" for p :: "'s pmf"
proof -
have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq"
by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"])
(auto intro: countable_set_pmf)
also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq"
by (metis Field_natLeq card_of_least natLeq_Well_order)
finally show ?thesis .
qed
show "\<And>R. rel_pmf R = (\<lambda>x y. \<exists>z. set_pmf z \<subseteq> {(x, y). R x y} \<and>
map_pmf fst z = x \<and> map_pmf snd z = y)"
by (auto simp add: fun_eq_iff rel_pmf.simps)
show "rel_pmf R OO rel_pmf S \<le> rel_pmf (R OO S)"
for R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
proof -
{ fix p q r
assume pq: "rel_pmf R p q"
and qr:"rel_pmf S q r"
from pq obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto
from qr obtain qr where qr: "\<And>y z. (y, z) \<in> set_pmf qr \<Longrightarrow> S y z"
and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto
define pr where "pr =
bind_pmf pq (\<lambda>xy. bind_pmf (cond_pmf qr {yz. fst yz = snd xy})
(\<lambda>yz. return_pmf (fst xy, snd yz)))"
have pr_welldefined: "\<And>y. y \<in> q \<Longrightarrow> qr \<inter> {yz. fst yz = y} \<noteq> {}"
by (force simp: q')
have "rel_pmf (R OO S) p r"
proof (rule rel_pmf.intros)
fix x z assume "(x, z) \<in> pr"
then have "\<exists>y. (x, y) \<in> pq \<and> (y, z) \<in> qr"
by (auto simp: q pr_welldefined pr_def split_beta)
with pq qr show "(R OO S) x z"
by blast
next
have "map_pmf snd pr = map_pmf snd (bind_pmf q (\<lambda>y. cond_pmf qr {yz. fst yz = y}))"
by (simp add: pr_def q split_beta bind_map_pmf map_pmf_def[symmetric] map_bind_pmf map_pmf_comp)
then show "map_pmf snd pr = r"
unfolding r q' bind_map_pmf by (subst (asm) bind_cond_pmf_cancel) (auto simp: eq_commute)
qed (simp add: pr_def map_bind_pmf split_beta map_pmf_def[symmetric] p map_pmf_comp)
}
then show ?thesis
by(auto simp add: le_fun_def)
qed
qed (fact natLeq_card_order natLeq_cinfinite)+
lemma map_pmf_idI: "(\<And>x. x \<in> set_pmf p \<Longrightarrow> f x = x) \<Longrightarrow> map_pmf f p = p"
by(simp cong: pmf.map_cong)
lemma rel_pmf_conj[simp]:
"rel_pmf (\<lambda>x y. P \<and> Q x y) x y \<longleftrightarrow> P \<and> rel_pmf Q x y"
"rel_pmf (\<lambda>x y. Q x y \<and> P) x y \<longleftrightarrow> P \<and> rel_pmf Q x y"
using set_pmf_not_empty by (fastforce simp: pmf.in_rel subset_eq)+
lemma rel_pmf_top[simp]: "rel_pmf top = top"
by (auto simp: pmf.in_rel[abs_def] fun_eq_iff map_fst_pair_pmf map_snd_pair_pmf
intro: exI[of _ "pair_pmf x y" for x y])
lemma rel_pmf_return_pmf1: "rel_pmf R (return_pmf x) M \<longleftrightarrow> (\<forall>a\<in>M. R x a)"
proof safe
fix a assume "a \<in> M" "rel_pmf R (return_pmf x) M"
then obtain pq where *: "\<And>a b. (a, b) \<in> set_pmf pq \<Longrightarrow> R a b"
and eq: "return_pmf x = map_pmf fst pq" "M = map_pmf snd pq"
by (force elim: rel_pmf.cases)
moreover have "set_pmf (return_pmf x) = {x}"
by simp
with \<open>a \<in> M\<close> have "(x, a) \<in> pq"
by (force simp: eq)
with * show "R x a"
by auto
qed (auto intro!: rel_pmf.intros[where pq="pair_pmf (return_pmf x) M"]
simp: map_fst_pair_pmf map_snd_pair_pmf)
lemma rel_pmf_return_pmf2: "rel_pmf R M (return_pmf x) \<longleftrightarrow> (\<forall>a\<in>M. R a x)"
by (subst pmf.rel_flip[symmetric]) (simp add: rel_pmf_return_pmf1)
lemma rel_return_pmf[simp]: "rel_pmf R (return_pmf x1) (return_pmf x2) = R x1 x2"
unfolding rel_pmf_return_pmf2 set_return_pmf by simp
lemma rel_pmf_False[simp]: "rel_pmf (\<lambda>x y. False) x y = False"
unfolding pmf.in_rel fun_eq_iff using set_pmf_not_empty by fastforce
lemma rel_pmf_rel_prod:
"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'"
proof safe
assume "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
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"
and eq: "map_pmf fst pq = pair_pmf A A'" "map_pmf snd pq = pair_pmf B B'"
by (force elim: rel_pmf.cases)
show "rel_pmf R A B"
proof (rule rel_pmf.intros)
let ?f = "\<lambda>(a, b). (fst a, fst b)"
have [simp]: "(\<lambda>x. fst (?f x)) = fst o fst" "(\<lambda>x. snd (?f x)) = fst o snd"
by auto
show "map_pmf fst (map_pmf ?f pq) = A"
by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
show "map_pmf snd (map_pmf ?f pq) = B"
by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
fix a b assume "(a, b) \<in> set_pmf (map_pmf ?f pq)"
then obtain c d where "((a, c), (b, d)) \<in> set_pmf pq"
by auto
from pq[OF this] show "R a b" ..
qed
show "rel_pmf S A' B'"
proof (rule rel_pmf.intros)
let ?f = "\<lambda>(a, b). (snd a, snd b)"
have [simp]: "(\<lambda>x. fst (?f x)) = snd o fst" "(\<lambda>x. snd (?f x)) = snd o snd"
by auto
show "map_pmf fst (map_pmf ?f pq) = A'"
by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
show "map_pmf snd (map_pmf ?f pq) = B'"
by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
fix c d assume "(c, d) \<in> set_pmf (map_pmf ?f pq)"
then obtain a b where "((a, c), (b, d)) \<in> set_pmf pq"
by auto
from pq[OF this] show "S c d" ..
qed
next
assume "rel_pmf R A B" "rel_pmf S A' B'"
then obtain Rpq Spq
where Rpq: "\<And>a b. (a, b) \<in> set_pmf Rpq \<Longrightarrow> R a b"
"map_pmf fst Rpq = A" "map_pmf snd Rpq = B"
and Spq: "\<And>a b. (a, b) \<in> set_pmf Spq \<Longrightarrow> S a b"
"map_pmf fst Spq = A'" "map_pmf snd Spq = B'"
by (force elim: rel_pmf.cases)
let ?f = "(\<lambda>((a, c), (b, d)). ((a, b), (c, d)))"
let ?pq = "map_pmf ?f (pair_pmf Rpq Spq)"
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))"
by auto
show "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
by (rule rel_pmf.intros[where pq="?pq"])
(auto simp: map_snd_pair_pmf map_fst_pair_pmf map_pmf_comp Rpq Spq
map_pair)
qed
lemma rel_pmf_reflI:
assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> P x x"
shows "rel_pmf P p p"
by (rule rel_pmf.intros[where pq="map_pmf (\<lambda>x. (x, x)) p"])
(auto simp add: pmf.map_comp o_def assms)
lemma rel_pmf_bij_betw:
assumes f: "bij_betw f (set_pmf p) (set_pmf q)"
and eq: "\<And>x. x \<in> set_pmf p \<Longrightarrow> pmf p x = pmf q (f x)"
shows "rel_pmf (\<lambda>x y. f x = y) p q"
proof(rule rel_pmf.intros)
let ?pq = "map_pmf (\<lambda>x. (x, f x)) p"
show "map_pmf fst ?pq = p" by(simp add: pmf.map_comp o_def)
have "map_pmf f p = q"
proof(rule pmf_eqI)
fix i
show "pmf (map_pmf f p) i = pmf q i"
proof(cases "i \<in> set_pmf q")
case True
with f obtain j where "i = f j" "j \<in> set_pmf p"
by(auto simp add: bij_betw_def image_iff)
thus ?thesis using f by(simp add: bij_betw_def pmf_map_inj eq)
next
case False thus ?thesis
by(subst pmf_map_outside)(auto simp add: set_pmf_iff eq[symmetric])
qed
qed
then show "map_pmf snd ?pq = q" by(simp add: pmf.map_comp o_def)
qed auto
context
begin
interpretation pmf_as_measure .
definition "join_pmf M = bind_pmf M (\<lambda>x. x)"
lemma bind_eq_join_pmf: "bind_pmf M f = join_pmf (map_pmf f M)"
unfolding join_pmf_def bind_map_pmf ..
lemma join_eq_bind_pmf: "join_pmf M = bind_pmf M id"
by (simp add: join_pmf_def id_def)
lemma pmf_join: "pmf (join_pmf N) i = (\<integral>M. pmf M i \<partial>measure_pmf N)"
unfolding join_pmf_def pmf_bind ..
lemma ennreal_pmf_join: "ennreal (pmf (join_pmf N) i) = (\<integral>\<^sup>+M. pmf M i \<partial>measure_pmf N)"
unfolding join_pmf_def ennreal_pmf_bind ..
lemma set_pmf_join_pmf[simp]: "set_pmf (join_pmf f) = (\<Union>p\<in>set_pmf f. set_pmf p)"
by (simp add: join_pmf_def)
lemma join_return_pmf: "join_pmf (return_pmf M) = M"
by (simp add: integral_return pmf_eq_iff pmf_join return_pmf.rep_eq)
lemma map_join_pmf: "map_pmf f (join_pmf AA) = join_pmf (map_pmf (map_pmf f) AA)"
by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf)
lemma join_map_return_pmf: "join_pmf (map_pmf return_pmf A) = A"
by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
end
lemma rel_pmf_joinI:
assumes "rel_pmf (rel_pmf P) p q"
shows "rel_pmf P (join_pmf p) (join_pmf q)"
proof -
from assms obtain pq where p: "p = map_pmf fst pq"
and q: "q = map_pmf snd pq"
and P: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> rel_pmf P x y"
by cases auto
from P obtain PQ
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"
and x: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf fst (PQ x y) = x"
and y: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf snd (PQ x y) = y"
by(metis rel_pmf.simps)
let ?r = "bind_pmf pq (\<lambda>(x, y). PQ x y)"
have "\<And>a b. (a, b) \<in> set_pmf ?r \<Longrightarrow> P a b" by (auto intro: PQ)
moreover have "map_pmf fst ?r = join_pmf p" "map_pmf snd ?r = join_pmf q"
by (simp_all add: p q x y join_pmf_def map_bind_pmf bind_map_pmf split_def cong: bind_pmf_cong)
ultimately show ?thesis ..
qed
lemma rel_pmf_bindI:
assumes pq: "rel_pmf R p q"
and fg: "\<And>x y. R x y \<Longrightarrow> rel_pmf P (f x) (g y)"
shows "rel_pmf P (bind_pmf p f) (bind_pmf q g)"
unfolding bind_eq_join_pmf
by (rule rel_pmf_joinI)
(auto simp add: pmf.rel_map intro: pmf.rel_mono[THEN le_funD, THEN le_funD, THEN le_boolD, THEN mp, OF _ pq] fg)
text \<open>
Proof that @{const rel_pmf} preserves orders.
Antisymmetry proof follows Thm. 1 in N. Saheb-Djahromi, Cpo's of measures for nondeterminism,
Theoretical Computer Science 12(1):19--37, 1980,
\<^url>\<open>http://dx.doi.org/10.1016/0304-3975(80)90003-1\<close>
\<close>
lemma
assumes *: "rel_pmf R p q"
and refl: "reflp R" and trans: "transp R"
shows measure_Ici: "measure p {y. R x y} \<le> measure q {y. R x y}" (is ?thesis1)
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)
proof -
from * obtain pq
where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
and p: "p = map_pmf fst pq"
and q: "q = map_pmf snd pq"
by cases auto
show ?thesis1 ?thesis2 unfolding p q map_pmf_rep_eq using refl trans
by(auto 4 3 simp add: measure_distr reflpD AE_measure_pmf_iff intro!: measure_pmf.finite_measure_mono_AE dest!: pq elim: transpE)
qed
lemma rel_pmf_inf:
fixes p q :: "'a pmf"
assumes 1: "rel_pmf R p q"
assumes 2: "rel_pmf R q p"
and refl: "reflp R" and trans: "transp R"
shows "rel_pmf (inf R R\<inverse>\<inverse>) p q"
proof (subst rel_pmf_iff_equivp, safe)
show "equivp (inf R R\<inverse>\<inverse>)"
using trans refl by (auto simp: equivp_reflp_symp_transp intro: sympI transpI reflpI dest: transpD reflpD)
fix C assume "C \<in> UNIV // {(x, y). inf R R\<inverse>\<inverse> x y}"
then obtain x where C: "C = {y. R x y \<and> R y x}"
by (auto elim: quotientE)
let ?R = "\<lambda>x y. R x y \<and> R y x"
let ?\<mu>R = "\<lambda>y. measure q {x. ?R x y}"
have "measure p {y. ?R x y} = measure p ({y. R x y} - {y. R x y \<and> \<not> R y x})"
by(auto intro!: arg_cong[where f="measure p"])
also have "\<dots> = measure p {y. R x y} - measure p {y. R x y \<and> \<not> R y x}"
by (rule measure_pmf.finite_measure_Diff) auto
also have "measure p {y. R x y \<and> \<not> R y x} = measure q {y. R x y \<and> \<not> R y x}"
using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ioi)
also have "measure p {y. R x y} = measure q {y. R x y}"
using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ici)
also have "measure q {y. R x y} - measure q {y. R x y \<and> \<not> R y x} =
measure q ({y. R x y} - {y. R x y \<and> \<not> R y x})"
by(rule measure_pmf.finite_measure_Diff[symmetric]) auto
also have "\<dots> = ?\<mu>R x"
by(auto intro!: arg_cong[where f="measure q"])
finally show "measure p C = measure q C"
by (simp add: C conj_commute)
qed
lemma rel_pmf_antisym:
fixes p q :: "'a pmf"
assumes 1: "rel_pmf R p q"
assumes 2: "rel_pmf R q p"
and refl: "reflp R" and trans: "transp R" and antisym: "antisymP R"
shows "p = q"
proof -
from 1 2 refl trans have "rel_pmf (inf R R\<inverse>\<inverse>) p q" by(rule rel_pmf_inf)
also have "inf R R\<inverse>\<inverse> = op ="
using refl antisym by (auto intro!: ext simp add: reflpD dest: antisymD)
finally show ?thesis unfolding pmf.rel_eq .
qed
lemma reflp_rel_pmf: "reflp R \<Longrightarrow> reflp (rel_pmf R)"
by(blast intro: reflpI rel_pmf_reflI reflpD)
lemma antisymP_rel_pmf:
"\<lbrakk> reflp R; transp R; antisymP R \<rbrakk>
\<Longrightarrow> antisymP (rel_pmf R)"
by(rule antisymI)(blast intro: rel_pmf_antisym)
lemma transp_rel_pmf:
assumes "transp R"
shows "transp (rel_pmf R)"
proof (rule transpI)
fix x y z
assume "rel_pmf R x y" and "rel_pmf R y z"
hence "rel_pmf (R OO R) x z" by (simp add: pmf.rel_compp relcompp.relcompI)
thus "rel_pmf R x z"
using assms by (metis (no_types) pmf.rel_mono rev_predicate2D transp_relcompp_less_eq)
qed
subsection \<open> Distributions \<close>
context
begin
interpretation pmf_as_function .
subsubsection \<open> Bernoulli Distribution \<close>
lift_definition bernoulli_pmf :: "real \<Rightarrow> bool pmf" is
"\<lambda>p b. ((\<lambda>p. if b then p else 1 - p) \<circ> min 1 \<circ> max 0) p"
by (auto simp: nn_integral_count_space_finite[where A="{False, True}"] UNIV_bool
split: split_max split_min)
lemma pmf_bernoulli_True[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) True = p"
by transfer simp
lemma pmf_bernoulli_False[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) False = 1 - p"
by transfer simp
lemma set_pmf_bernoulli[simp]: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (bernoulli_pmf p) = UNIV"
by (auto simp add: set_pmf_iff UNIV_bool)
lemma nn_integral_bernoulli_pmf[simp]:
assumes [simp]: "0 \<le> p" "p \<le> 1" "\<And>x. 0 \<le> f x"
shows "(\<integral>\<^sup>+x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
by (subst nn_integral_measure_pmf_support[of UNIV])
(auto simp: UNIV_bool field_simps)
lemma integral_bernoulli_pmf[simp]:
assumes [simp]: "0 \<le> p" "p \<le> 1"
shows "(\<integral>x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
by (subst integral_measure_pmf[of UNIV]) (auto simp: UNIV_bool)
lemma pmf_bernoulli_half [simp]: "pmf (bernoulli_pmf (1 / 2)) x = 1 / 2"
by(cases x) simp_all
lemma measure_pmf_bernoulli_half: "measure_pmf (bernoulli_pmf (1 / 2)) = uniform_count_measure UNIV"
by (rule measure_eqI)
(simp_all add: nn_integral_pmf[symmetric] emeasure_uniform_count_measure ennreal_divide_numeral[symmetric]
nn_integral_count_space_finite sets_uniform_count_measure divide_ennreal_def mult_ac
ennreal_of_nat_eq_real_of_nat)
subsubsection \<open> Geometric Distribution \<close>
context
fixes p :: real assumes p[arith]: "0 < p" "p \<le> 1"
begin
lift_definition geometric_pmf :: "nat pmf" is "\<lambda>n. (1 - p)^n * p"
proof
have "(\<Sum>i. ennreal (p * (1 - p) ^ i)) = ennreal (p * (1 / (1 - (1 - p))))"
by (intro suminf_ennreal_eq sums_mult geometric_sums) auto
then show "(\<integral>\<^sup>+ x. ennreal ((1 - p)^x * p) \<partial>count_space UNIV) = 1"
by (simp add: nn_integral_count_space_nat field_simps)
qed simp
lemma pmf_geometric[simp]: "pmf geometric_pmf n = (1 - p)^n * p"
by transfer rule
end
lemma set_pmf_geometric: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (geometric_pmf p) = UNIV"
by (auto simp: set_pmf_iff)
subsubsection \<open> Uniform Multiset Distribution \<close>
context
fixes M :: "'a multiset" assumes M_not_empty: "M \<noteq> {#}"
begin
lift_definition pmf_of_multiset :: "'a pmf" is "\<lambda>x. count M x / size M"
proof
show "(\<integral>\<^sup>+ x. ennreal (real (count M x) / real (size M)) \<partial>count_space UNIV) = 1"
using M_not_empty
by (simp add: zero_less_divide_iff nn_integral_count_space nonempty_has_size
setsum_divide_distrib[symmetric])
(auto simp: size_multiset_overloaded_eq intro!: setsum.cong)
qed simp
lemma pmf_of_multiset[simp]: "pmf pmf_of_multiset x = count M x / size M"
by transfer rule
lemma set_pmf_of_multiset[simp]: "set_pmf pmf_of_multiset = set_mset M"
by (auto simp: set_pmf_iff)
end
subsubsection \<open> Uniform Distribution \<close>
context
fixes S :: "'a set" assumes S_not_empty: "S \<noteq> {}" and S_finite: "finite S"
begin
lift_definition pmf_of_set :: "'a pmf" is "\<lambda>x. indicator S x / card S"
proof
show "(\<integral>\<^sup>+ x. ennreal (indicator S x / real (card S)) \<partial>count_space UNIV) = 1"
using S_not_empty S_finite
by (subst nn_integral_count_space'[of S])
(auto simp: ennreal_of_nat_eq_real_of_nat ennreal_mult[symmetric])
qed simp
lemma pmf_of_set[simp]: "pmf pmf_of_set x = indicator S x / card S"
by transfer rule
lemma set_pmf_of_set[simp]: "set_pmf pmf_of_set = S"
using S_finite S_not_empty by (auto simp: set_pmf_iff)
lemma emeasure_pmf_of_set_space[simp]: "emeasure pmf_of_set S = 1"
by (rule measure_pmf.emeasure_eq_1_AE) (auto simp: AE_measure_pmf_iff)
lemma nn_integral_pmf_of_set: "nn_integral (measure_pmf pmf_of_set) f = setsum f S / card S"
by (subst nn_integral_measure_pmf_finite)
(simp_all add: setsum_left_distrib[symmetric] card_gt_0_iff S_not_empty S_finite divide_ennreal_def
divide_ennreal[symmetric] ennreal_of_nat_eq_real_of_nat[symmetric] ennreal_times_divide)
lemma integral_pmf_of_set: "integral\<^sup>L (measure_pmf pmf_of_set) f = setsum f S / card S"
by (subst integral_measure_pmf[of S]) (auto simp: S_finite setsum_divide_distrib)
lemma emeasure_pmf_of_set: "emeasure (measure_pmf pmf_of_set) A = card (S \<inter> A) / card S"
by (subst nn_integral_indicator[symmetric], simp)
(simp add: S_finite S_not_empty card_gt_0_iff indicator_def setsum.If_cases divide_ennreal
ennreal_of_nat_eq_real_of_nat nn_integral_pmf_of_set)
lemma measure_pmf_of_set: "measure (measure_pmf pmf_of_set) A = card (S \<inter> A) / card S"
using emeasure_pmf_of_set[of A]
by (simp add: measure_nonneg measure_pmf.emeasure_eq_measure)
end
lemma map_pmf_of_set:
assumes "finite A" "A \<noteq> {}"
shows "map_pmf f (pmf_of_set A) = pmf_of_multiset (image_mset f (mset_set A))"
(is "?lhs = ?rhs")
proof (intro pmf_eqI)
fix x
from assms have "ennreal (pmf ?lhs x) = ennreal (pmf ?rhs x)"
by (subst ennreal_pmf_map)
(simp_all add: emeasure_pmf_of_set mset_set_empty_iff count_image_mset Int_commute)
thus "pmf ?lhs x = pmf ?rhs x" by simp
qed
lemma pmf_bind_pmf_of_set:
assumes "A \<noteq> {}" "finite A"
shows "pmf (bind_pmf (pmf_of_set A) f) x =
(\<Sum>xa\<in>A. pmf (f xa) x) / real_of_nat (card A)" (is "?lhs = ?rhs")
proof -
from assms have "card A > 0" by auto
with assms have "ennreal ?lhs = ennreal ?rhs"
by (subst ennreal_pmf_bind)
(simp_all add: nn_integral_pmf_of_set max_def pmf_nonneg divide_ennreal [symmetric]
setsum_nonneg ennreal_of_nat_eq_real_of_nat)
thus ?thesis by (subst (asm) ennreal_inj) (auto intro!: setsum_nonneg divide_nonneg_nonneg)
qed
lemma pmf_of_set_singleton: "pmf_of_set {x} = return_pmf x"
by(rule pmf_eqI)(simp add: indicator_def)
lemma map_pmf_of_set_inj:
assumes f: "inj_on f A"
and [simp]: "A \<noteq> {}" "finite A"
shows "map_pmf f (pmf_of_set A) = pmf_of_set (f ` A)" (is "?lhs = ?rhs")
proof(rule pmf_eqI)
fix i
show "pmf ?lhs i = pmf ?rhs i"
proof(cases "i \<in> f ` A")
case True
then obtain i' where "i = f i'" "i' \<in> A" by auto
thus ?thesis using f by(simp add: card_image pmf_map_inj)
next
case False
hence "pmf ?lhs i = 0" by(simp add: pmf_eq_0_set_pmf set_map_pmf)
moreover have "pmf ?rhs i = 0" using False by simp
ultimately show ?thesis by simp
qed
qed
text \<open>
Choosing an element uniformly at random from the union of a disjoint family
of finite non-empty sets with the same size is the same as first choosing a set
from the family uniformly at random and then choosing an element from the chosen set
uniformly at random.
\<close>
lemma pmf_of_set_UN:
assumes "finite (UNION A f)" "A \<noteq> {}" "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> {}"
"\<And>x. x \<in> A \<Longrightarrow> card (f x) = n" "disjoint_family_on f A"
shows "pmf_of_set (UNION A f) = do {x \<leftarrow> pmf_of_set A; pmf_of_set (f x)}"
(is "?lhs = ?rhs")
proof (intro pmf_eqI)
fix x
from assms have [simp]: "finite A"
using infinite_disjoint_family_imp_infinite_UNION[of A f] by blast
from assms have "ereal (pmf (pmf_of_set (UNION A f)) x) =
ereal (indicator (\<Union>x\<in>A. f x) x / real (card (\<Union>x\<in>A. f x)))"
by (subst pmf_of_set) auto
also from assms have "card (\<Union>x\<in>A. f x) = card A * n"
by (subst card_UN_disjoint) (auto simp: disjoint_family_on_def)
also from assms
have "indicator (\<Union>x\<in>A. f x) x / real \<dots> =
indicator (\<Union>x\<in>A. f x) x / (n * real (card A))"
by (simp add: setsum_divide_distrib [symmetric] mult_ac)
also from assms have "indicator (\<Union>x\<in>A. f x) x = (\<Sum>y\<in>A. indicator (f y) x)"
by (intro indicator_UN_disjoint) simp_all
also from assms have "ereal ((\<Sum>y\<in>A. indicator (f y) x) / (real n * real (card A))) =
ereal (pmf ?rhs x)"
by (subst pmf_bind_pmf_of_set) (simp_all add: setsum_divide_distrib)
finally show "pmf ?lhs x = pmf ?rhs x" by simp
qed
lemma bernoulli_pmf_half_conv_pmf_of_set: "bernoulli_pmf (1 / 2) = pmf_of_set UNIV"
by (rule pmf_eqI) simp_all
subsubsection \<open> Poisson Distribution \<close>
context
fixes rate :: real assumes rate_pos: "0 < rate"
begin
lift_definition poisson_pmf :: "nat pmf" is "\<lambda>k. rate ^ k / fact k * exp (-rate)"
proof (* by Manuel Eberl *)
have summable: "summable (\<lambda>x::nat. rate ^ x / fact x)" using summable_exp
by (simp add: field_simps divide_inverse [symmetric])
have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x * exp (-rate) \<partial>count_space UNIV) =
exp (-rate) * (\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV)"
by (simp add: field_simps nn_integral_cmult[symmetric] ennreal_mult'[symmetric])
also from rate_pos have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV) = (\<Sum>x. rate ^ x / fact x)"
by (simp_all add: nn_integral_count_space_nat suminf_ennreal summable ennreal_suminf_neq_top)
also have "... = exp rate" unfolding exp_def
by (simp add: field_simps divide_inverse [symmetric])
also have "ennreal (exp (-rate)) * ennreal (exp rate) = 1"
by (simp add: mult_exp_exp ennreal_mult[symmetric])
finally show "(\<integral>\<^sup>+ x. ennreal (rate ^ x / (fact x) * exp (- rate)) \<partial>count_space UNIV) = 1" .
qed (simp add: rate_pos[THEN less_imp_le])
lemma pmf_poisson[simp]: "pmf poisson_pmf k = rate ^ k / fact k * exp (-rate)"
by transfer rule
lemma set_pmf_poisson[simp]: "set_pmf poisson_pmf = UNIV"
using rate_pos by (auto simp: set_pmf_iff)
end
subsubsection \<open> Binomial Distribution \<close>
context
fixes n :: nat and p :: real assumes p_nonneg: "0 \<le> p" and p_le_1: "p \<le> 1"
begin
lift_definition binomial_pmf :: "nat pmf" is "\<lambda>k. (n choose k) * p^k * (1 - p)^(n - k)"
proof
have "(\<integral>\<^sup>+k. ennreal (real (n choose k) * p ^ k * (1 - p) ^ (n - k)) \<partial>count_space UNIV) =
ennreal (\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k))"
using p_le_1 p_nonneg by (subst nn_integral_count_space') auto
also have "(\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k)) = (p + (1 - p)) ^ n"
by (subst binomial_ring) (simp add: atLeast0AtMost)
finally show "(\<integral>\<^sup>+ x. ennreal (real (n choose x) * p ^ x * (1 - p) ^ (n - x)) \<partial>count_space UNIV) = 1"
by simp
qed (insert p_nonneg p_le_1, simp)
lemma pmf_binomial[simp]: "pmf binomial_pmf k = (n choose k) * p^k * (1 - p)^(n - k)"
by transfer rule
lemma set_pmf_binomial_eq: "set_pmf binomial_pmf = (if p = 0 then {0} else if p = 1 then {n} else {.. n})"
using p_nonneg p_le_1 unfolding set_eq_iff set_pmf_iff pmf_binomial by (auto simp: set_pmf_iff)
end
end
lemma set_pmf_binomial_0[simp]: "set_pmf (binomial_pmf n 0) = {0}"
by (simp add: set_pmf_binomial_eq)
lemma set_pmf_binomial_1[simp]: "set_pmf (binomial_pmf n 1) = {n}"
by (simp add: set_pmf_binomial_eq)
lemma set_pmf_binomial[simp]: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (binomial_pmf n p) = {..n}"
by (simp add: set_pmf_binomial_eq)
context includes lifting_syntax
begin
lemma bind_pmf_parametric [transfer_rule]:
"(rel_pmf A ===> (A ===> rel_pmf B) ===> rel_pmf B) bind_pmf bind_pmf"
by(blast intro: rel_pmf_bindI dest: rel_funD)
lemma return_pmf_parametric [transfer_rule]: "(A ===> rel_pmf A) return_pmf return_pmf"
by(rule rel_funI) simp
end
primrec replicate_pmf :: "nat \<Rightarrow> 'a pmf \<Rightarrow> 'a list pmf" where
"replicate_pmf 0 _ = return_pmf []"
| "replicate_pmf (Suc n) p = do {x \<leftarrow> p; xs \<leftarrow> replicate_pmf n p; return_pmf (x#xs)}"
lemma replicate_pmf_1: "replicate_pmf 1 p = map_pmf (\<lambda>x. [x]) p"
by (simp add: map_pmf_def bind_return_pmf)
lemma set_replicate_pmf:
"set_pmf (replicate_pmf n p) = {xs\<in>lists (set_pmf p). length xs = n}"
by (induction n) (auto simp: length_Suc_conv)
lemma replicate_pmf_distrib:
"replicate_pmf (m + n) p =
do {xs \<leftarrow> replicate_pmf m p; ys \<leftarrow> replicate_pmf n p; return_pmf (xs @ ys)}"
by (induction m) (simp_all add: bind_return_pmf bind_return_pmf' bind_assoc_pmf)
lemma power_diff':
assumes "b \<le> a"
shows "x ^ (a - b) = (if x = 0 \<and> a = b then 1 else x ^ a / (x::'a::field) ^ b)"
proof (cases "x = 0")
case True
with assms show ?thesis by (cases "a - b") simp_all
qed (insert assms, simp_all add: power_diff)
lemma binomial_pmf_Suc:
assumes "p \<in> {0..1}"
shows "binomial_pmf (Suc n) p =
do {b \<leftarrow> bernoulli_pmf p;
k \<leftarrow> binomial_pmf n p;
return_pmf ((if b then 1 else 0) + k)}" (is "_ = ?rhs")
proof (intro pmf_eqI)
fix k
have A: "indicator {Suc a} (Suc b) = indicator {a} b" for a b
by (simp add: indicator_def)
show "pmf (binomial_pmf (Suc n) p) k = pmf ?rhs k"
by (cases k; cases "k > n")
(insert assms, auto simp: pmf_bind measure_pmf_single A divide_simps algebra_simps
not_less less_eq_Suc_le [symmetric] power_diff')
qed
lemma binomial_pmf_0: "p \<in> {0..1} \<Longrightarrow> binomial_pmf 0 p = return_pmf 0"
by (rule pmf_eqI) (simp_all add: indicator_def)
lemma binomial_pmf_altdef:
assumes "p \<in> {0..1}"
shows "binomial_pmf n p = map_pmf (length \<circ> filter id) (replicate_pmf n (bernoulli_pmf p))"
by (induction n)
(insert assms, auto simp: binomial_pmf_Suc map_pmf_def bind_return_pmf bind_assoc_pmf
bind_return_pmf' binomial_pmf_0 intro!: bind_pmf_cong)
subsection \<open>PMFs from assiciation lists\<close>
definition pmf_of_list ::" ('a \<times> real) list \<Rightarrow> 'a pmf" where
"pmf_of_list xs = embed_pmf (\<lambda>x. listsum (map snd (filter (\<lambda>z. fst z = x) xs)))"
definition pmf_of_list_wf where
"pmf_of_list_wf xs \<longleftrightarrow> (\<forall>x\<in>set (map snd xs) . x \<ge> 0) \<and> listsum (map snd xs) = 1"
lemma pmf_of_list_wfI:
"(\<And>x. x \<in> set (map snd xs) \<Longrightarrow> x \<ge> 0) \<Longrightarrow> listsum (map snd xs) = 1 \<Longrightarrow> pmf_of_list_wf xs"
unfolding pmf_of_list_wf_def by simp
context
begin
private lemma pmf_of_list_aux:
assumes "\<And>x. x \<in> set (map snd xs) \<Longrightarrow> x \<ge> 0"
assumes "listsum (map snd xs) = 1"
shows "(\<integral>\<^sup>+ x. ennreal (listsum (map snd [z\<leftarrow>xs . fst z = x])) \<partial>count_space UNIV) = 1"
proof -
have "(\<integral>\<^sup>+ x. ennreal (listsum (map snd (filter (\<lambda>z. fst z = x) xs))) \<partial>count_space UNIV) =
(\<integral>\<^sup>+ x. ennreal (listsum (map (\<lambda>(x',p). indicator {x'} x * p) xs)) \<partial>count_space UNIV)"
by (intro nn_integral_cong ennreal_cong, subst listsum_map_filter') (auto intro: listsum_cong)
also have "\<dots> = (\<Sum>(x',p)\<leftarrow>xs. (\<integral>\<^sup>+ x. ennreal (indicator {x'} x * p) \<partial>count_space UNIV))"
using assms(1)
proof (induction xs)
case (Cons x xs)
from Cons.prems have "snd x \<ge> 0" by simp
moreover have "b \<ge> 0" if "(a,b) \<in> set xs" for a b
using Cons.prems[of b] that by force
ultimately have "(\<integral>\<^sup>+ y. ennreal (\<Sum>(x', p)\<leftarrow>x # xs. indicator {x'} y * p) \<partial>count_space UNIV) =
(\<integral>\<^sup>+ y. ennreal (indicator {fst x} y * snd x) +
ennreal (\<Sum>(x', p)\<leftarrow>xs. indicator {x'} y * p) \<partial>count_space UNIV)"
by (intro nn_integral_cong, subst ennreal_plus [symmetric])
(auto simp: case_prod_unfold indicator_def intro!: listsum_nonneg)
also have "\<dots> = (\<integral>\<^sup>+ y. ennreal (indicator {fst x} y * snd x) \<partial>count_space UNIV) +
(\<integral>\<^sup>+ y. ennreal (\<Sum>(x', p)\<leftarrow>xs. indicator {x'} y * p) \<partial>count_space UNIV)"
by (intro nn_integral_add)
(force intro!: listsum_nonneg AE_I2 intro: Cons simp: indicator_def)+
also have "(\<integral>\<^sup>+ y. ennreal (\<Sum>(x', p)\<leftarrow>xs. indicator {x'} y * p) \<partial>count_space UNIV) =
(\<Sum>(x', p)\<leftarrow>xs. (\<integral>\<^sup>+ y. ennreal (indicator {x'} y * p) \<partial>count_space UNIV))"
using Cons(1) by (intro Cons) simp_all
finally show ?case by (simp add: case_prod_unfold)
qed simp
also have "\<dots> = (\<Sum>(x',p)\<leftarrow>xs. ennreal p * (\<integral>\<^sup>+ x. indicator {x'} x \<partial>count_space UNIV))"
using assms(1)
by (intro listsum_cong, simp only: case_prod_unfold, subst nn_integral_cmult [symmetric])
(auto intro!: assms(1) simp: max_def times_ereal.simps [symmetric] mult_ac ereal_indicator
simp del: times_ereal.simps)+
also from assms have "\<dots> = listsum (map snd xs)" by (simp add: case_prod_unfold listsum_ennreal)
also have "\<dots> = 1" using assms(2) by simp
finally show ?thesis .
qed
lemma pmf_pmf_of_list:
assumes "pmf_of_list_wf xs"
shows "pmf (pmf_of_list xs) x = listsum (map snd (filter (\<lambda>z. fst z = x) xs))"
using assms pmf_of_list_aux[of xs] unfolding pmf_of_list_def pmf_of_list_wf_def
by (subst pmf_embed_pmf) (auto intro!: listsum_nonneg)
end
lemma set_pmf_of_list:
assumes "pmf_of_list_wf xs"
shows "set_pmf (pmf_of_list xs) \<subseteq> set (map fst xs)"
proof clarify
fix x assume A: "x \<in> set_pmf (pmf_of_list xs)"
show "x \<in> set (map fst xs)"
proof (rule ccontr)
assume "x \<notin> set (map fst xs)"
hence "[z\<leftarrow>xs . fst z = x] = []" by (auto simp: filter_empty_conv)
with A assms show False by (simp add: pmf_pmf_of_list set_pmf_eq)
qed
qed
lemma finite_set_pmf_of_list:
assumes "pmf_of_list_wf xs"
shows "finite (set_pmf (pmf_of_list xs))"
using assms by (rule finite_subset[OF set_pmf_of_list]) simp_all
lemma emeasure_Int_set_pmf:
"emeasure (measure_pmf p) (A \<inter> set_pmf p) = emeasure (measure_pmf p) A"
by (rule emeasure_eq_AE) (auto simp: AE_measure_pmf_iff)
lemma measure_Int_set_pmf:
"measure (measure_pmf p) (A \<inter> set_pmf p) = measure (measure_pmf p) A"
using emeasure_Int_set_pmf[of p A] by (simp add: Sigma_Algebra.measure_def)
lemma emeasure_pmf_of_list:
assumes "pmf_of_list_wf xs"
shows "emeasure (pmf_of_list xs) A = ennreal (listsum (map snd (filter (\<lambda>x. fst x \<in> A) xs)))"
proof -
have "emeasure (pmf_of_list xs) A = nn_integral (measure_pmf (pmf_of_list xs)) (indicator A)"
by simp
also from assms
have "\<dots> = (\<Sum>x\<in>set_pmf (pmf_of_list xs) \<inter> A. ennreal (listsum (map snd [z\<leftarrow>xs . fst z = x])))"
by (subst nn_integral_measure_pmf_finite) (simp_all add: finite_set_pmf_of_list pmf_pmf_of_list)
also from assms
have "\<dots> = ennreal (\<Sum>x\<in>set_pmf (pmf_of_list xs) \<inter> A. listsum (map snd [z\<leftarrow>xs . fst z = x]))"
by (subst setsum_ennreal) (auto simp: pmf_of_list_wf_def intro!: listsum_nonneg)
also have "\<dots> = ennreal (\<Sum>x\<in>set_pmf (pmf_of_list xs) \<inter> A.
indicator A x * pmf (pmf_of_list xs) x)" (is "_ = ennreal ?S")
using assms by (intro ennreal_cong setsum.cong) (auto simp: pmf_pmf_of_list)
also have "?S = (\<Sum>x\<in>set_pmf (pmf_of_list xs). indicator A x * pmf (pmf_of_list xs) x)"
using assms by (intro setsum.mono_neutral_left set_pmf_of_list finite_set_pmf_of_list) auto
also have "\<dots> = (\<Sum>x\<in>set (map fst xs). indicator A x * pmf (pmf_of_list xs) x)"
using assms by (intro setsum.mono_neutral_left set_pmf_of_list) (auto simp: set_pmf_eq)
also have "\<dots> = (\<Sum>x\<in>set (map fst xs). indicator A x *
listsum (map snd (filter (\<lambda>z. fst z = x) xs)))"
using assms by (simp add: pmf_pmf_of_list)
also have "\<dots> = (\<Sum>x\<in>set (map fst xs). listsum (map snd (filter (\<lambda>z. fst z = x \<and> x \<in> A) xs)))"
by (intro setsum.cong) (auto simp: indicator_def)
also have "\<dots> = (\<Sum>x\<in>set (map fst xs). (\<Sum>xa = 0..<length xs.
if fst (xs ! xa) = x \<and> x \<in> A then snd (xs ! xa) else 0))"
by (intro setsum.cong refl, subst listsum_map_filter', subst listsum_setsum_nth) simp
also have "\<dots> = (\<Sum>xa = 0..<length xs. (\<Sum>x\<in>set (map fst xs).
if fst (xs ! xa) = x \<and> x \<in> A then snd (xs ! xa) else 0))"
by (rule setsum.commute)
also have "\<dots> = (\<Sum>xa = 0..<length xs. if fst (xs ! xa) \<in> A then
(\<Sum>x\<in>set (map fst xs). if x = fst (xs ! xa) then snd (xs ! xa) else 0) else 0)"
by (auto intro!: setsum.cong setsum.neutral)
also have "\<dots> = (\<Sum>xa = 0..<length xs. if fst (xs ! xa) \<in> A then snd (xs ! xa) else 0)"
by (intro setsum.cong refl) (simp_all add: setsum.delta)
also have "\<dots> = listsum (map snd (filter (\<lambda>x. fst x \<in> A) xs))"
by (subst listsum_map_filter', subst listsum_setsum_nth) simp_all
finally show ?thesis .
qed
lemma measure_pmf_of_list:
assumes "pmf_of_list_wf xs"
shows "measure (pmf_of_list xs) A = listsum (map snd (filter (\<lambda>x. fst x \<in> A) xs))"
using assms unfolding pmf_of_list_wf_def Sigma_Algebra.measure_def
by (subst emeasure_pmf_of_list [OF assms], subst enn2real_ennreal) (auto intro!: listsum_nonneg)
(* TODO Move? *)
lemma listsum_nonneg_eq_zero_iff:
fixes xs :: "'a :: linordered_ab_group_add list"
shows "(\<And>x. x \<in> set xs \<Longrightarrow> x \<ge> 0) \<Longrightarrow> listsum xs = 0 \<longleftrightarrow> set xs \<subseteq> {0}"
proof (induction xs)
case (Cons x xs)
from Cons.prems have "listsum (x#xs) = 0 \<longleftrightarrow> x = 0 \<and> listsum xs = 0"
unfolding listsum_simps by (subst add_nonneg_eq_0_iff) (auto intro: listsum_nonneg)
with Cons.IH Cons.prems show ?case by simp
qed simp_all
lemma listsum_filter_nonzero:
"listsum (filter (\<lambda>x. x \<noteq> 0) xs) = listsum xs"
by (induction xs) simp_all
(* END MOVE *)
lemma set_pmf_of_list_eq:
assumes "pmf_of_list_wf xs" "\<And>x. x \<in> snd ` set xs \<Longrightarrow> x > 0"
shows "set_pmf (pmf_of_list xs) = fst ` set xs"
proof
{
fix x assume A: "x \<in> fst ` set xs" and B: "x \<notin> set_pmf (pmf_of_list xs)"
then obtain y where y: "(x, y) \<in> set xs" by auto
from B have "listsum (map snd [z\<leftarrow>xs. fst z = x]) = 0"
by (simp add: pmf_pmf_of_list[OF assms(1)] set_pmf_eq)
moreover from y have "y \<in> snd ` {xa \<in> set xs. fst xa = x}" by force
ultimately have "y = 0" using assms(1)
by (subst (asm) listsum_nonneg_eq_zero_iff) (auto simp: pmf_of_list_wf_def)
with assms(2) y have False by force
}
thus "fst ` set xs \<subseteq> set_pmf (pmf_of_list xs)" by blast
qed (insert set_pmf_of_list[OF assms(1)], simp_all)
lemma pmf_of_list_remove_zeros:
assumes "pmf_of_list_wf xs"
defines "xs' \<equiv> filter (\<lambda>z. snd z \<noteq> 0) xs"
shows "pmf_of_list_wf xs'" "pmf_of_list xs' = pmf_of_list xs"
proof -
have "map snd [z\<leftarrow>xs . snd z \<noteq> 0] = filter (\<lambda>x. x \<noteq> 0) (map snd xs)"
by (induction xs) simp_all
with assms(1) show wf: "pmf_of_list_wf xs'"
by (auto simp: pmf_of_list_wf_def xs'_def listsum_filter_nonzero)
have "listsum (map snd [z\<leftarrow>xs' . fst z = i]) = listsum (map snd [z\<leftarrow>xs . fst z = i])" for i
unfolding xs'_def by (induction xs) simp_all
with assms(1) wf show "pmf_of_list xs' = pmf_of_list xs"
by (intro pmf_eqI) (simp_all add: pmf_pmf_of_list)
qed
end