(* 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 (in finite_measure) countable_support: (* replace version in pmf *)
"countable {x. measure M {x} \<noteq> 0}"
proof cases
assume "measure M (space M) = 0"
with bounded_measure measure_le_0_iff have "{x. measure M {x} \<noteq> 0} = {}"
by auto
then show ?thesis
by simp
next
let ?M = "measure M (space M)" and ?m = "\<lambda>x. measure M {x}"
assume "?M \<noteq> 0"
then have *: "{x. ?m x \<noteq> 0} = (\<Union>n. {x. ?M / Suc n < ?m x})"
using reals_Archimedean[of "?m x / ?M" for x]
by (auto simp: field_simps not_le[symmetric] measure_nonneg divide_le_0_iff measure_le_0_iff)
have **: "\<And>n. finite {x. ?M / Suc n < ?m x}"
proof (rule ccontr)
fix n assume "infinite {x. ?M / Suc n < ?m x}" (is "infinite ?X")
then obtain X where "finite X" "card X = Suc (Suc n)" "X \<subseteq> ?X"
by (metis infinite_arbitrarily_large)
from this(3) have *: "\<And>x. x \<in> X \<Longrightarrow> ?M / Suc n \<le> ?m x"
by auto
{ fix x assume "x \<in> X"
from `?M \<noteq> 0` *[OF this] have "?m x \<noteq> 0" by (auto simp: field_simps measure_le_0_iff)
then have "{x} \<in> sets M" by (auto dest: measure_notin_sets) }
note singleton_sets = this
have "?M < (\<Sum>x\<in>X. ?M / Suc n)"
using `?M \<noteq> 0`
by (simp add: `card X = Suc (Suc n)` real_eq_of_nat[symmetric] real_of_nat_Suc field_simps less_le measure_nonneg)
also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"
by (rule setsum_mono) fact
also have "\<dots> = measure M (\<Union>x\<in>X. {x})"
using singleton_sets `finite X`
by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def)
finally have "?M < measure M (\<Union>x\<in>X. {x})" .
moreover have "measure M (\<Union>x\<in>X. {x}) \<le> ?M"
using singleton_sets[THEN sets.sets_into_space] by (intro finite_measure_mono) auto
ultimately show False by simp
qed
show ?thesis
unfolding * by (intro countable_UN countableI_type countable_finite[OF **])
qed
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 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 {* PMF as measure *}
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
locale pmf_as_measure
begin
setup_lifting type_definition_pmf
end
context
begin
interpretation pmf_as_measure .
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}" .
lift_definition map_pmf :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf" is
"\<lambda>f M. distr M (count_space UNIV) f"
proof safe
fix M and f :: "'a \<Rightarrow> 'b"
let ?D = "distr M (count_space UNIV) f"
assume "prob_space M" and [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
interpret prob_space M by fact
from ae have "AE x in M. measure M (f -` {f x}) \<noteq> 0"
proof eventually_elim
fix x
have "measure M {x} \<le> measure M (f -` {f x})"
by (intro finite_measure_mono) auto
then show "measure M {x} \<noteq> 0 \<Longrightarrow> measure M (f -` {f x}) \<noteq> 0"
using measure_nonneg[of M "{x}"] by auto
qed
then show "AE x in ?D. measure ?D {x} \<noteq> 0"
by (simp add: AE_distr_iff measure_distr measurable_def)
qed (auto simp: measurable_def prob_space.prob_space_distr)
declare [[coercion set_pmf]]
lemma countable_set_pmf [simp]: "countable (set_pmf p)"
by transfer (metis prob_space.finite_measure finite_measure.countable_support)
lemma sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV"
by transfer metis
lemma sets_measure_pmf_count_space: "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 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 pmf_positive: "x \<in> set_pmf p \<Longrightarrow> 0 < pmf p x"
by transfer (simp add: less_le measure_nonneg)
lemma pmf_nonneg: "0 \<le> pmf p x"
by transfer (simp add: measure_nonneg)
lemma pmf_le_1: "pmf p x \<le> 1"
by (simp add: pmf.rep_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 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"
by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \<longleftrightarrow> (\<forall>y\<in>M. P y)"
proof -
{ fix y assume y: "y \<in> M" and P: "AE x in M. P x" "\<not> P y"
with P have "AE x in M. x \<noteq> y"
by auto
with y have False
by (simp add: emeasure_pmf_single_eq_zero_iff AE_iff_measurable[OF _ refl]) }
then show ?thesis
using AE_measure_pmf[of M] by auto
qed
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 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)
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)
lemma nn_integral_measure_pmf_support:
fixes f :: "'a \<Rightarrow> ereal"
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> ereal"
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)
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)
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)
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 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 map_pmf_id[simp]: "map_pmf id = id"
by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI)
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:
assumes "p = q"
shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> map_pmf f p = map_pmf g q"
unfolding `p = q`[symmetric] measure_pmf_inject[symmetric] map_pmf.rep_eq
by (auto simp add: emeasure_distr AE_measure_pmf_iff intro!: emeasure_eq_AE measure_eqI)
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 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 ereal_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 "ereal (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_set_map:
fixes f :: "'a \<Rightarrow> 'b"
shows "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
proof (rule, transfer, clarsimp simp add: measure_distr measurable_def)
fix f :: "'a \<Rightarrow> 'b" and M :: "'a measure"
assume "prob_space M" and ae: "AE x in M. measure M {x} \<noteq> 0" and [simp]: "sets M = UNIV"
interpret prob_space M by fact
show "{x. measure M (f -` {x}) \<noteq> 0} = f ` {x. measure M {x} \<noteq> 0}"
proof safe
fix x assume "measure M (f -` {x}) \<noteq> 0"
moreover have "measure M (f -` {x}) = measure M {y. f y = x \<and> measure M {y} \<noteq> 0}"
using ae by (intro finite_measure_eq_AE) auto
ultimately have "{y. f y = x \<and> measure M {y} \<noteq> 0} \<noteq> {}"
by (metis measure_empty)
then show "x \<in> f ` {x. measure M {x} \<noteq> 0}"
by auto
next
fix x assume "measure M {x} \<noteq> 0"
then have "0 < measure M {x}"
using measure_nonneg[of M "{x}"] by auto
also have "measure M {x} \<le> measure M (f -` {f x})"
by (intro finite_measure_mono) auto
finally show "measure M (f -` {f x}) = 0 \<Longrightarrow> False"
by simp
qed
qed
lemma set_map_pmf: "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 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
subsection {* PMFs as function *}
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) (ereal \<circ> f)"
proof (intro conjI)
have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
by (simp split: split_indicator)
show "AE x in density (count_space UNIV) (ereal \<circ> f).
measure (density (count_space UNIV) (ereal \<circ> f)) {x} \<noteq> 0"
by (simp add: AE_density nonneg emeasure_density measure_def nn_integral_cmult_indicator)
show "prob_space (density (count_space UNIV) (ereal \<circ> f))"
by default (simp add: emeasure_density prob)
qed simp
lemma pmf_embed_pmf: "pmf embed_pmf x = f x"
proof transfer
have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
by (simp split: split_indicator)
fix x show "measure (density (count_space UNIV) (ereal \<circ> f)) {x} = f x"
by transfer (simp add: measure_def emeasure_density nn_integral_cmult_indicator nonneg)
qed
end
lemma embed_pmf_transfer:
"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"
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. ereal (measure M {x}))"
proof (rule measure_eqI)
fix A :: "'a set"
have "(\<integral>\<^sup>+ x. ereal (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. ereal (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. ereal (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. ereal (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
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 `bi_total A`
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)
end
context
begin
interpretation pmf_as_function .
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: "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)
lift_definition geometric_pmf :: "nat pmf" is "\<lambda>n. 1 / 2^Suc n"
proof
note geometric_sums[of "1 / 2"]
note sums_mult[OF this, of "1 / 2"]
from sums_suminf_ereal[OF this]
show "(\<integral>\<^sup>+ x. ereal (1 / 2 ^ Suc x) \<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 / 2^Suc n"
by transfer rule
lemma set_pmf_geometric[simp]: "set_pmf geometric_pmf = UNIV"
by (auto simp: set_pmf_iff)
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. ereal (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_of M"
by (auto simp: set_pmf_iff)
end
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. ereal (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
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[simp]: "emeasure pmf_of_set S = 1"
by (rule measure_pmf.emeasure_eq_1_AE) (auto simp: AE_measure_pmf_iff)
end
end
subsection {* Monad interpretation *}
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_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 \<guillemotright>= A) = sets (measure_pmf x \<guillemotright>= B)"
using assms by (subst (1 2) sets_bind) auto
next
fix X assume "X \<in> sets (measure_pmf x \<guillemotright>= A)"
then have X: "X \<in> sets N"
using assms by (subst (asm) sets_bind) auto
show "emeasure (measure_pmf x \<guillemotright>= A) X = emeasure (measure_pmf x \<guillemotright>= 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
context
begin
interpretation pmf_as_measure .
lift_definition join_pmf :: "'a pmf pmf \<Rightarrow> 'a pmf" is "\<lambda>M. measure_pmf M \<guillemotright>= measure_pmf"
proof (intro conjI)
fix M :: "'a pmf pmf"
have *: "measure_pmf \<in> measurable (measure_pmf M) (subprob_algebra (count_space UNIV))"
using measurable_measure_pmf[of "\<lambda>x. x"] by simp
interpret bind: prob_space "measure_pmf M \<guillemotright>= measure_pmf"
apply (rule measure_pmf.prob_space_bind[OF _ *])
apply (auto intro!: AE_I2)
apply unfold_locales
done
show "prob_space (measure_pmf M \<guillemotright>= measure_pmf)"
by intro_locales
show "sets (measure_pmf M \<guillemotright>= measure_pmf) = UNIV"
by (subst sets_bind[OF *]) auto
have "AE x in measure_pmf M \<guillemotright>= measure_pmf. emeasure (measure_pmf M \<guillemotright>= measure_pmf) {x} \<noteq> 0"
by (auto simp add: AE_bind[OF _ *] AE_measure_pmf_iff emeasure_bind[OF _ *]
nn_integral_0_iff_AE measure_pmf.emeasure_eq_measure measure_le_0_iff set_pmf_iff pmf.rep_eq)
then show "AE x in measure_pmf M \<guillemotright>= measure_pmf. measure (measure_pmf M \<guillemotright>= measure_pmf) {x} \<noteq> 0"
unfolding bind.emeasure_eq_measure by simp
qed
lemma pmf_join: "pmf (join_pmf N) i = (\<integral>M. pmf M i \<partial>measure_pmf N)"
proof (transfer fixing: N i)
have N: "subprob_space (measure_pmf N)"
by (rule prob_space_imp_subprob_space) intro_locales
show "measure (measure_pmf N \<guillemotright>= measure_pmf) {i} = integral\<^sup>L (measure_pmf N) (\<lambda>M. measure M {i})"
using measurable_measure_pmf[of "\<lambda>x. x"]
by (intro subprob_space.measure_bind[where N="count_space UNIV", OF N]) auto
qed (auto simp: Transfer.Rel_def rel_fun_def cr_pmf_def)
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 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_return_pmf: "map_pmf f (return_pmf x) = return_pmf (f x)"
by transfer (simp add: distr_return)
lemma set_return_pmf: "set_pmf (return_pmf x) = {x}"
by transfer (auto simp add: measure_return split: split_indicator)
lemma pmf_return: "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
end
definition "bind_pmf M f = join_pmf (map_pmf f M)"
lemma (in pmf_as_measure) bind_transfer[transfer_rule]:
"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"
proof (auto simp: pmf_as_measure.cr_pmf_def rel_fun_def bind_pmf_def join_pmf.rep_eq map_pmf.rep_eq)
fix M f and g :: "'a \<Rightarrow> 'b pmf" assume "\<forall>x. f x = measure_pmf (g x)"
then have f: "f = (\<lambda>x. measure_pmf (g x))"
by auto
show "measure_pmf M \<guillemotright>= f = distr (measure_pmf M) (count_space UNIV) g \<guillemotright>= measure_pmf"
unfolding f by (subst bind_distr[OF _ measurable_measure_pmf]) auto
qed
lemma pmf_bind: "pmf (bind_pmf N f) i = (\<integral>x. pmf (f x) i \<partial>measure_pmf N)"
by (auto intro!: integral_distr simp: bind_pmf_def pmf_join map_pmf.rep_eq)
lemma bind_return_pmf: "bind_pmf (return_pmf x) f = f x"
unfolding bind_pmf_def map_return_pmf join_return_pmf ..
lemma set_bind_pmf: "set_pmf (bind_pmf M N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))"
apply (simp add: set_eq_iff set_pmf_iff pmf_bind)
apply (subst integral_nonneg_eq_0_iff_AE)
apply (auto simp: pmf_nonneg pmf_le_1 AE_measure_pmf_iff
intro!: measure_pmf.integrable_const_bound[where B=1])
done
lemma measurable_pair_restrict_pmf2:
assumes "countable A"
assumes "\<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"
apply (subst measurable_cong_sets)
apply (rule sets_pair_measure_cong sets_restrict_space_cong sets_measure_pmf_count_space refl)+
apply (simp_all add: restrict_count_space)
apply (subst split_eta[symmetric])
unfolding measurable_split_conv
apply (rule measurable_compose_countable'[OF _ measurable_snd `countable A`])
apply (rule measurable_compose[OF measurable_fst])
apply fact
done
lemma measurable_pair_restrict_pmf1:
assumes "countable A"
assumes "\<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"
apply (subst measurable_cong_sets)
apply (rule sets_pair_measure_cong sets_restrict_space_cong sets_measure_pmf_count_space refl)+
apply (simp_all add: restrict_count_space)
apply (subst split_eta[symmetric])
unfolding measurable_split_conv
apply (rule measurable_compose_countable'[OF _ measurable_fst `countable A`])
apply (rule measurable_compose[OF measurable_snd])
apply fact
done
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
context
begin
interpretation pmf_as_measure .
lemma measure_pmf_bind: "measure_pmf (bind_pmf M f) = (measure_pmf M \<guillemotright>= (\<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 (subst (1 3) nn_integral_max_0[symmetric])
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
lemma bind_return_pmf': "bind_pmf N return_pmf = N"
proof (transfer, clarify)
fix N :: "'a measure" assume "sets N = UNIV" then show "N \<guillemotright>= return (count_space UNIV) = N"
by (subst return_sets_cong[where N=N]) (simp_all add: bind_return')
qed
lemma bind_return_pmf'': "bind_pmf N (\<lambda>x. return_pmf (f x)) = map_pmf f N"
proof (transfer, clarify)
fix N :: "'b measure" and f :: "'b \<Rightarrow> 'a" assume "prob_space N" "sets N = UNIV"
then show "N \<guillemotright>= (\<lambda>x. return (count_space UNIV) (f x)) = distr N (count_space UNIV) f"
by (subst bind_return_distr[symmetric])
(auto simp: prob_space.not_empty measurable_def comp_def)
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)
end
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: "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 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) \<guillemotright>= M = (measure_pmf A \<guillemotright>= (\<lambda>x. measure_pmf B \<guillemotright>= (\<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)
have sets_eq_N: "sets ?L = N"
by (simp add: sets_bind[OF M'])
show "sets ?L = sets ?R"
unfolding sets_eq_N
apply (subst sets_bind[where N=N])
apply (rule measurable_bind)
apply (rule measurable_compose[OF _ measurable_measure_pmf])
apply measurable
apply (auto intro!: sets_pair_measure_cong sets_measure_pmf_count_space)
done
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])
unfolding pair_pmf_def measure_pmf_bind[of A]
apply (subst nn_integral_bind)
apply (rule measurable_compose[OF M' measurable_emeasure_subprob_algebra, OF X])
apply (subst measurable_cong_sets[OF sets_measure_pmf_count_space refl])
apply (subst subprob_algebra_cong[OF sets_measure_pmf_count_space])
apply measurable
unfolding measure_pmf_bind
apply (subst nn_integral_bind)
apply (rule measurable_compose[OF M' measurable_emeasure_subprob_algebra, OF X])
apply (subst measurable_cong_sets[OF sets_measure_pmf_count_space refl])
apply (subst subprob_algebra_cong[OF sets_measure_pmf_count_space])
apply measurable
apply (simp add: nn_integral_measure_pmf_finite set_return_pmf emeasure_nonneg pmf_return one_ereal_def[symmetric])
apply (subst emeasure_bind[OF _ _ X])
apply simp
apply (rule measurable_bind[where N="count_space UNIV"])
apply (rule measurable_compose[OF _ measurable_measure_pmf])
apply measurable
apply (rule sets_pair_measure_cong sets_measure_pmf_count_space refl)+
apply (subst measurable_cong_sets[OF sets_pair_measure_cong[OF sets_measure_pmf_count_space refl] refl])
apply simp
apply (subst emeasure_bind[OF _ _ X])
apply simp
apply (rule measurable_compose[OF _ M])
apply (auto simp: space_pair_measure)
done
qed
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"
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)
{ fix p :: "'s pmf"
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 inj_on_to_nat_on)
also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq"
by (metis Field_natLeq card_of_least natLeq_Well_order)
finally show "(card_of (set_pmf p), natLeq) \<in> ordLeq" . }
show "\<And>R. rel_pmf R =
(BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf fst))\<inverse>\<inverse> OO
BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf snd)"
by (auto simp add: fun_eq_iff BNF_Def.Grp_def OO_def rel_pmf.simps)
{ fix p :: "'a pmf" and f :: "'a \<Rightarrow> 'b" and g x
assume p: "\<And>z. z \<in> set_pmf p \<Longrightarrow> f z = g z"
and x: "x \<in> set_pmf p"
thus "f x = g x" by simp }
fix R :: "'a => 'b \<Rightarrow> bool" and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
{ 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
have support_subset: "set_pmf pq O set_pmf qr \<subseteq> set_pmf p \<times> set_pmf r"
by(auto simp add: p r set_map_pmf intro: rev_image_eqI)
let ?A = "\<lambda>y. {x. (x, y) \<in> set_pmf pq}"
and ?B = "\<lambda>y. {z. (y, z) \<in> set_pmf qr}"
def ppp \<equiv> "\<lambda>A. \<lambda>f :: 'a \<Rightarrow> real. \<lambda>n. if n \<in> to_nat_on A ` A then f (from_nat_into A n) else 0"
have [simp]: "\<And>A f n. (\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x) \<Longrightarrow> 0 \<le> ppp A f n"
"\<And>A f n x. \<lbrakk> x \<in> A; countable A \<rbrakk> \<Longrightarrow> ppp A f (to_nat_on A x) = f x"
"\<And>A f n. n \<notin> to_nat_on A ` A \<Longrightarrow> ppp A f n = 0"
by(auto simp add: ppp_def intro: from_nat_into)
def rrr \<equiv> "\<lambda>A. \<lambda>f :: 'c \<Rightarrow> real. \<lambda>n. if n \<in> to_nat_on A ` A then f (from_nat_into A n) else 0"
have [simp]: "\<And>A f n. (\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x) \<Longrightarrow> 0 \<le> rrr A f n"
"\<And>A f n x. \<lbrakk> x \<in> A; countable A \<rbrakk> \<Longrightarrow> rrr A f (to_nat_on A x) = f x"
"\<And>A f n. n \<notin> to_nat_on A ` A \<Longrightarrow> rrr A f n = 0"
by(auto simp add: rrr_def intro: from_nat_into)
def pp \<equiv> "\<lambda>y. ppp (?A y) (\<lambda>x. pmf pq (x, y))"
and rr \<equiv> "\<lambda>y. rrr (?B y) (\<lambda>z. pmf qr (y, z))"
have pos_p [simp]: "\<And>y n. 0 \<le> pp y n"
and pos_r [simp]: "\<And>y n. 0 \<le> rr y n"
by(simp_all add: pmf_nonneg pp_def rr_def)
{ fix y n
have "pp y n \<le> 0 \<longleftrightarrow> pp y n = 0" "\<not> 0 < pp y n \<longleftrightarrow> pp y n = 0"
and "min (pp y n) 0 = 0" "min 0 (pp y n) = 0"
using pos_p[of y n] by(auto simp del: pos_p) }
note pp_convs [simp] = this
{ fix y n
have "rr y n \<le> 0 \<longleftrightarrow> rr y n = 0" "\<not> 0 < rr y n \<longleftrightarrow> rr y n = 0"
and "min (rr y n) 0 = 0" "min 0 (rr y n) = 0"
using pos_r[of y n] by(auto simp del: pos_r) }
note r_convs [simp] = this
have "\<And>y. ?A y \<subseteq> set_pmf p" by(auto simp add: p set_map_pmf intro: rev_image_eqI)
then have [simp]: "\<And>y. countable (?A y)" by(rule countable_subset) simp
have "\<And>y. ?B y \<subseteq> set_pmf r" by(auto simp add: r set_map_pmf intro: rev_image_eqI)
then have [simp]: "\<And>y. countable (?B y)" by(rule countable_subset) simp
let ?P = "\<lambda>y. to_nat_on (?A y)"
and ?R = "\<lambda>y. to_nat_on (?B y)"
have eq: "\<And>y. (\<integral>\<^sup>+ x. pp y x \<partial>count_space UNIV) = \<integral>\<^sup>+ z. rr y z \<partial>count_space UNIV"
proof -
fix y
have "(\<integral>\<^sup>+ x. pp y x \<partial>count_space UNIV) = (\<integral>\<^sup>+ x. pp y x \<partial>count_space (?P y ` ?A y))"
by(auto simp add: pp_def nn_integral_count_space_indicator indicator_def intro!: nn_integral_cong)
also have "\<dots> = (\<integral>\<^sup>+ x. pp y (?P y x) \<partial>count_space (?A y))"
by(intro nn_integral_bij_count_space[symmetric] inj_on_imp_bij_betw inj_on_to_nat_on) simp
also have "\<dots> = (\<integral>\<^sup>+ x. pmf pq (x, y) \<partial>count_space (?A y))"
by(rule nn_integral_cong)(simp add: pp_def)
also have "\<dots> = \<integral>\<^sup>+ x. emeasure (measure_pmf pq) {(x, y)} \<partial>count_space (?A y)"
by(simp add: emeasure_pmf_single)
also have "\<dots> = emeasure (measure_pmf pq) (\<Union>x\<in>?A y. {(x, y)})"
by(subst emeasure_UN_countable)(simp_all add: disjoint_family_on_def)
also have "\<dots> = emeasure (measure_pmf pq) ((\<Union>x\<in>?A y. {(x, y)}) \<union> {(x, y'). x \<notin> ?A y \<and> y' = y})"
by(rule emeasure_Un_null_set[symmetric])+
(auto simp add: q set_map_pmf split_beta intro!: in_null_sets_measure_pmfI intro: rev_image_eqI)
also have "\<dots> = emeasure (measure_pmf pq) (snd -` {y})"
by(rule arg_cong2[where f=emeasure])+auto
also have "\<dots> = pmf q y" by(simp add: q ereal_pmf_map)
also have "\<dots> = emeasure (measure_pmf qr) (fst -` {y})"
by(simp add: q' ereal_pmf_map)
also have "\<dots> = emeasure (measure_pmf qr) ((\<Union>z\<in>?B y. {(y, z)}) \<union> {(y', z). z \<notin> ?B y \<and> y' = y})"
by(rule arg_cong2[where f=emeasure])+auto
also have "\<dots> = emeasure (measure_pmf qr) (\<Union>z\<in>?B y. {(y, z)})"
by(rule emeasure_Un_null_set)
(auto simp add: q' set_map_pmf split_beta intro!: in_null_sets_measure_pmfI intro: rev_image_eqI)
also have "\<dots> = \<integral>\<^sup>+ z. emeasure (measure_pmf qr) {(y, z)} \<partial>count_space (?B y)"
by(subst emeasure_UN_countable)(simp_all add: disjoint_family_on_def)
also have "\<dots> = (\<integral>\<^sup>+ z. pmf qr (y, z) \<partial>count_space (?B y))"
by(simp add: emeasure_pmf_single)
also have "\<dots> = (\<integral>\<^sup>+ z. rr y (?R y z) \<partial>count_space (?B y))"
by(rule nn_integral_cong)(simp add: rr_def)
also have "\<dots> = (\<integral>\<^sup>+ z. rr y z \<partial>count_space (?R y ` ?B y))"
by(intro nn_integral_bij_count_space inj_on_imp_bij_betw inj_on_to_nat_on) simp
also have "\<dots> = \<integral>\<^sup>+ z. rr y z \<partial>count_space UNIV"
by(auto simp add: rr_def nn_integral_count_space_indicator indicator_def intro!: nn_integral_cong)
finally show "?thesis y" .
qed
def assign_aux \<equiv> "\<lambda>y remainder start weight z.
if z < start then 0
else if z = start then min weight remainder
else if remainder + setsum (rr y) {Suc start ..<z} < weight then min (weight - remainder - setsum (rr y) {Suc start..<z}) (rr y z) else 0"
hence assign_aux_alt_def: "\<And>y remainder start weight z. assign_aux y remainder start weight z =
(if z < start then 0
else if z = start then min weight remainder
else if remainder + setsum (rr y) {Suc start ..<z} < weight then min (weight - remainder - setsum (rr y) {Suc start..<z}) (rr y z) else 0)"
by simp
{ fix y and remainder :: real and start and weight :: real
assume weight_nonneg: "0 \<le> weight"
let ?assign_aux = "assign_aux y remainder start weight"
{ fix z
have "setsum ?assign_aux {..<z} =
(if z \<le> start then 0 else if remainder + setsum (rr y) {Suc start..<z} < weight then remainder + setsum (rr y) {Suc start..<z} else weight)"
proof(induction z)
case (Suc z) show ?case
by(auto simp add: Suc.IH assign_aux_alt_def[where z=z] not_less)(metis add.commute add.left_commute add_increasing pos_r)
qed(auto simp add: assign_aux_def) }
note setsum_start_assign_aux = this
moreover {
assume remainder_nonneg: "0 \<le> remainder"
have [simp]: "\<And>z. 0 \<le> ?assign_aux z"
by(simp add: assign_aux_def weight_nonneg remainder_nonneg)
moreover have "\<And>z. \<lbrakk> rr y z = 0; remainder \<le> rr y start \<rbrakk> \<Longrightarrow> ?assign_aux z = 0"
using remainder_nonneg weight_nonneg
by(auto simp add: assign_aux_def min_def)
moreover have "(\<integral>\<^sup>+ z. ?assign_aux z \<partial>count_space UNIV) =
min weight (\<integral>\<^sup>+ z. (if z < start then 0 else if z = start then remainder else rr y z) \<partial>count_space UNIV)"
(is "?lhs = ?rhs" is "_ = min _ (\<integral>\<^sup>+ y. ?f y \<partial>_)")
proof -
have "?lhs = (SUP n. \<Sum>z<n. ereal (?assign_aux z))"
by(simp add: nn_integral_count_space_nat suminf_ereal_eq_SUP)
also have "\<dots> = (SUP n. min weight (\<Sum>z<n. ?f z))"
proof(rule arg_cong2[where f=SUPREMUM] ext refl)+
fix n
have "(\<Sum>z<n. ereal (?assign_aux z)) = min weight ((if n > start then remainder else 0) + setsum ?f {Suc start..<n})"
using weight_nonneg remainder_nonneg by(simp add: setsum_start_assign_aux min_def)
also have "\<dots> = min weight (setsum ?f {start..<n})"
by(simp add: setsum_head_upt_Suc)
also have "\<dots> = min weight (setsum ?f {..<n})"
by(intro arg_cong2[where f=min] setsum.mono_neutral_left) auto
finally show "(\<Sum>z<n. ereal (?assign_aux z)) = \<dots>" .
qed
also have "\<dots> = min weight (SUP n. setsum ?f {..<n})"
unfolding inf_min[symmetric] by(subst inf_SUP) simp
also have "\<dots> = ?rhs"
by(simp add: nn_integral_count_space_nat suminf_ereal_eq_SUP remainder_nonneg)
finally show ?thesis .
qed
moreover note calculation }
moreover note calculation }
note setsum_start_assign_aux = this(1)
and assign_aux_nonneg [simp] = this(2)
and assign_aux_eq_0_outside = this(3)
and nn_integral_assign_aux = this(4)
{ fix y and remainder :: real and start target
have "setsum (rr y) {Suc start..<target} \<ge> 0" by(simp add: setsum_nonneg)
moreover assume "0 \<le> remainder"
ultimately have "assign_aux y remainder start 0 target = 0"
by(auto simp add: assign_aux_def min_def) }
note assign_aux_weight_0 [simp] = this
def find_start \<equiv> "\<lambda>y weight. if \<exists>n. weight \<le> setsum (rr y) {..n} then Some (LEAST n. weight \<le> setsum (rr y) {..n}) else None"
have find_start_eq_Some_above:
"\<And>y weight n. find_start y weight = Some n \<Longrightarrow> weight \<le> setsum (rr y) {..n}"
by(drule sym)(auto simp add: find_start_def split: split_if_asm intro: LeastI)
{ fix y weight n
assume find_start: "find_start y weight = Some n"
and weight: "0 \<le> weight"
have "setsum (rr y) {..n} \<le> rr y n + weight"
proof(rule ccontr)
assume "\<not> ?thesis"
hence "rr y n + weight < setsum (rr y) {..n}" by simp
moreover with weight obtain n' where "n = Suc n'" by(cases n) auto
ultimately have "weight \<le> setsum (rr y) {..n'}" by simp
hence "(LEAST n. weight \<le> setsum (rr y) {..n}) \<le> n'" by(rule Least_le)
moreover from find_start have "n = (LEAST n. weight \<le> setsum (rr y) {..n})"
by(auto simp add: find_start_def split: split_if_asm)
ultimately show False using \<open>n = Suc n'\<close> by auto
qed }
note find_start_eq_Some_least = this
have find_start_0 [simp]: "\<And>y. find_start y 0 = Some 0"
by(auto simp add: find_start_def intro!: exI[where x=0])
{ fix y and weight :: real
assume "weight < \<integral>\<^sup>+ z. rr y z \<partial>count_space UNIV"
also have "(\<integral>\<^sup>+ z. rr y z \<partial>count_space UNIV) = (SUP n. \<Sum>z<n. ereal (rr y z))"
by(simp add: nn_integral_count_space_nat suminf_ereal_eq_SUP)
finally obtain n where "weight < (\<Sum>z<n. rr y z)" by(auto simp add: less_SUP_iff)
hence "weight \<in> dom (find_start y)"
by(auto simp add: find_start_def)(meson atMost_iff finite_atMost lessThan_iff less_imp_le order_trans pos_r setsum_mono3 subsetI) }
note in_dom_find_startI = this
{ fix y and w w' :: real and m
let ?m' = "LEAST m. w' \<le> setsum (rr y) {..m}"
assume "w' \<le> w"
also assume "find_start y w = Some m"
hence "w \<le> setsum (rr y) {..m}" by(rule find_start_eq_Some_above)
finally have "find_start y w' = Some ?m'" by(auto simp add: find_start_def)
moreover from \<open>w' \<le> setsum (rr y) {..m}\<close> have "?m' \<le> m" by(rule Least_le)
ultimately have "\<exists>m'. find_start y w' = Some m' \<and> m' \<le> m" by blast }
note find_start_mono = this[rotated]
def assign \<equiv> "\<lambda>y x z. let used = setsum (pp y) {..<x}
in case find_start y used of None \<Rightarrow> 0
| Some start \<Rightarrow> assign_aux y (setsum (rr y) {..start} - used) start (pp y x) z"
hence assign_alt_def: "\<And>y x z. assign y x z =
(let used = setsum (pp y) {..<x}
in case find_start y used of None \<Rightarrow> 0
| Some start \<Rightarrow> assign_aux y (setsum (rr y) {..start} - used) start (pp y x) z)"
by simp
have assign_nonneg [simp]: "\<And>y x z. 0 \<le> assign y x z"
by(simp add: assign_def diff_le_iff find_start_eq_Some_above split: option.split)
have assign_eq_0_outside: "\<And>y x z. \<lbrakk> pp y x = 0 \<or> rr y z = 0 \<rbrakk> \<Longrightarrow> assign y x z = 0"
by(auto simp add: assign_def assign_aux_eq_0_outside diff_le_iff find_start_eq_Some_above find_start_eq_Some_least setsum_nonneg split: option.split)
{ fix y x z
have "(\<Sum>n<Suc x. assign y n z) =
(case find_start y (setsum (pp y) {..<x}) of None \<Rightarrow> rr y z
| Some m \<Rightarrow> if z < m then rr y z
else min (rr y z) (max 0 (setsum (pp y) {..<x} + pp y x - setsum (rr y) {..<z})))"
(is "?lhs x = ?rhs x")
proof(induction x)
case 0 thus ?case
by(auto simp add: assign_def assign_aux_def setsum_head_upt_Suc atLeast0LessThan[symmetric] not_less field_simps max_def)
next
case (Suc x)
have "?lhs (Suc x) = ?lhs x + assign y (Suc x) z" by simp
also have "?lhs x = ?rhs x" by(rule Suc.IH)
also have "?rhs x + assign y (Suc x) z = ?rhs (Suc x)"
proof(cases "find_start y (setsum (pp y) {..<Suc x})")
case None
thus ?thesis
by(auto split: option.split simp add: assign_def min_def max_def diff_le_iff setsum_nonneg not_le field_simps)
(metis add.commute add_increasing find_start_def lessThan_Suc_atMost less_imp_le option.distinct(1) setsum_lessThan_Suc)+
next
case (Some m)
have [simp]: "setsum (rr y) {..m} = rr y m + setsum (rr y) {..<m}"
by(simp add: ivl_disj_un(2)[symmetric])
from Some obtain m' where m': "find_start y (setsum (pp y) {..<x}) = Some m'" "m' \<le> m"
by(auto dest: find_start_mono[where w'2="setsum (pp y) {..<x}"])
moreover {
assume "z < m"
then have "setsum (rr y) {..z} \<le> setsum (rr y) {..<m}"
by(auto intro: setsum_mono3)
also have "\<dots> \<le> setsum (pp y) {..<Suc x}" using find_start_eq_Some_least[OF Some]
by(simp add: ivl_disj_un(2)[symmetric] setsum_nonneg)
finally have "rr y z \<le> max 0 (setsum (pp y) {..<x} + pp y x - setsum (rr y) {..<z})"
by(auto simp add: ivl_disj_un(2)[symmetric] max_def diff_le_iff simp del: r_convs)
} moreover {
assume "m \<le> z"
have "setsum (pp y) {..<Suc x} \<le> setsum (rr y) {..m}"
using Some by(rule find_start_eq_Some_above)
also have "\<dots> \<le> setsum (rr y) {..<Suc z}" using \<open>m \<le> z\<close> by(intro setsum_mono3) auto
finally have "max 0 (setsum (pp y) {..<x} + pp y x - setsum (rr y) {..<z}) \<le> rr y z" by simp
moreover have "z \<noteq> m \<Longrightarrow> setsum (rr y) {..m} + setsum (rr y) {Suc m..<z} = setsum (rr y) {..<z}"
using \<open>m \<le> z\<close>
by(subst ivl_disj_un(8)[where l="Suc m", symmetric])
(simp_all add: setsum_Un ivl_disj_un(2)[symmetric] setsum.neutral)
moreover note calculation
} moreover {
assume "m < z"
have "setsum (pp y) {..<Suc x} \<le> setsum (rr y) {..m}"
using Some by(rule find_start_eq_Some_above)
also have "\<dots> \<le> setsum (rr y) {..<z}" using \<open>m < z\<close> by(intro setsum_mono3) auto
finally have "max 0 (setsum (pp y) {..<Suc x} - setsum (rr y) {..<z}) = 0" by simp }
moreover have "setsum (pp y) {..<Suc x} \<ge> setsum (rr y) {..<m}"
using find_start_eq_Some_least[OF Some]
by(simp add: setsum_nonneg ivl_disj_un(2)[symmetric])
moreover hence "setsum (pp y) {..<Suc (Suc x)} \<ge> setsum (rr y) {..<m}"
by(fastforce intro: order_trans)
ultimately show ?thesis using Some
by(auto simp add: assign_def assign_aux_def Let_def field_simps max_def)
qed
finally show ?case .
qed }
note setsum_assign = this
have nn_integral_assign1: "\<And>y z. (\<integral>\<^sup>+ x. assign y x z \<partial>count_space UNIV) = rr y z"
proof -
fix y z
have "(\<integral>\<^sup>+ x. assign y x z \<partial>count_space UNIV) = (SUP n. ereal (\<Sum>x<n. assign y x z))"
by(simp add: nn_integral_count_space_nat suminf_ereal_eq_SUP)
also have "\<dots> = rr y z"
proof(rule antisym)
show "(SUP n. ereal (\<Sum>x<n. assign y x z)) \<le> rr y z"
proof(rule SUP_least)
fix n
show "ereal (\<Sum>x<n. (assign y x z)) \<le> rr y z"
using setsum_assign[of y z "n - 1"]
by(cases n)(simp_all split: option.split)
qed
show "rr y z \<le> (SUP n. ereal (\<Sum>x<n. assign y x z))"
proof(cases "setsum (rr y) {..z} < \<integral>\<^sup>+ x. pp y x \<partial>count_space UNIV")
case True
then obtain n where "setsum (rr y) {..z} < setsum (pp y) {..<n}"
by(auto simp add: nn_integral_count_space_nat suminf_ereal_eq_SUP less_SUP_iff)
moreover have "\<And>k. k < z \<Longrightarrow> setsum (rr y) {..k} \<le> setsum (rr y) {..<z}"
by(auto intro: setsum_mono3)
ultimately have "rr y z \<le> (\<Sum>x<Suc n. assign y x z)"
by(subst setsum_assign)(auto split: option.split dest!: find_start_eq_Some_above simp add: ivl_disj_un(2)[symmetric] add.commute add_increasing le_diff_eq le_max_iff_disj)
also have "\<dots> \<le> (SUP n. ereal (\<Sum>x<n. assign y x z))"
by(rule SUP_upper) simp
finally show ?thesis by simp
next
case False
have "setsum (rr y) {..z} = \<integral>\<^sup>+ z. rr y z \<partial>count_space {..z}"
by(simp add: nn_integral_count_space_finite max_def)
also have "\<dots> \<le> \<integral>\<^sup>+ z. rr y z \<partial>count_space UNIV"
by(auto simp add: nn_integral_count_space_indicator indicator_def intro: nn_integral_mono)
also have "\<dots> = \<integral>\<^sup>+ x. pp y x \<partial>count_space UNIV" by(simp add: eq)
finally have *: "setsum (rr y) {..z} = \<dots>" using False by simp
also have "\<dots> = (SUP n. ereal (\<Sum>x<n. pp y x))"
by(simp add: nn_integral_count_space_nat suminf_ereal_eq_SUP)
also have "\<dots> \<le> (SUP n. ereal (\<Sum>x<n. assign y x z)) + setsum (rr y) {..<z}"
proof(rule SUP_least)
fix n
have "setsum (pp y) {..<n} = \<integral>\<^sup>+ x. pp y x \<partial>count_space {..<n}"
by(simp add: nn_integral_count_space_finite max_def)
also have "\<dots> \<le> \<integral>\<^sup>+ x. pp y x \<partial>count_space UNIV"
by(auto simp add: nn_integral_count_space_indicator indicator_def intro: nn_integral_mono)
also have "\<dots> = setsum (rr y) {..z}" using * by simp
finally obtain k where k: "find_start y (setsum (pp y) {..<n}) = Some k"
by(fastforce simp add: find_start_def)
with \<open>ereal (setsum (pp y) {..<n}) \<le> setsum (rr y) {..z}\<close>
have "k \<le> z" by(auto simp add: find_start_def split: split_if_asm intro: Least_le)
then have "setsum (pp y) {..<n} - setsum (rr y) {..<z} \<le> ereal (\<Sum>x<Suc n. assign y x z)"
using \<open>ereal (setsum (pp y) {..<n}) \<le> setsum (rr y) {..z}\<close>
by(subst setsum_assign)(auto simp add: field_simps max_def k ivl_disj_un(2)[symmetric], metis le_add_same_cancel2 max.bounded_iff max_def pos_p)
also have "\<dots> \<le> (SUP n. ereal (\<Sum>x<n. assign y x z))"
by(rule SUP_upper) simp
finally show "ereal (\<Sum>x<n. pp y x) \<le> \<dots> + setsum (rr y) {..<z}"
by(simp add: ereal_minus(1)[symmetric] ereal_minus_le del: ereal_minus(1))
qed
finally show ?thesis
by(simp add: ivl_disj_un(2)[symmetric] plus_ereal.simps(1)[symmetric] ereal_add_le_add_iff2 del: plus_ereal.simps(1))
qed
qed
finally show "?thesis y z" .
qed
{ fix y x
have "(\<integral>\<^sup>+ z. assign y x z \<partial>count_space UNIV) = pp y x"
proof(cases "setsum (pp y) {..<x} = \<integral>\<^sup>+ x. pp y x \<partial>count_space UNIV")
case False
let ?used = "setsum (pp y) {..<x}"
have "?used = \<integral>\<^sup>+ x. pp y x \<partial>count_space {..<x}"
by(simp add: nn_integral_count_space_finite max_def)
also have "\<dots> \<le> \<integral>\<^sup>+ x. pp y x \<partial>count_space UNIV"
by(auto simp add: nn_integral_count_space_indicator indicator_def intro!: nn_integral_mono)
finally have "?used < \<dots>" using False by auto
also note eq finally have "?used \<in> dom (find_start y)" by(rule in_dom_find_startI)
then obtain k where k: "find_start y ?used = Some k" by auto
let ?f = "\<lambda>z. if z < k then 0 else if z = k then setsum (rr y) {..k} - ?used else rr y z"
let ?g = "\<lambda>x'. if x' < x then 0 else pp y x'"
have "pp y x = ?g x" by simp
also have "?g x \<le> \<integral>\<^sup>+ x'. ?g x' \<partial>count_space UNIV" by(rule nn_integral_ge_point) simp
also {
have "?used = \<integral>\<^sup>+ x. pp y x \<partial>count_space {..<x}"
by(simp add: nn_integral_count_space_finite max_def)
also have "\<dots> = \<integral>\<^sup>+ x'. (if x' < x then pp y x' else 0) \<partial>count_space UNIV"
by(simp add: nn_integral_count_space_indicator indicator_def if_distrib zero_ereal_def cong: if_cong)
also have "(\<integral>\<^sup>+ x'. ?g x' \<partial>count_space UNIV) + \<dots> = \<integral>\<^sup>+ x. pp y x \<partial>count_space UNIV"
by(subst nn_integral_add[symmetric])(auto intro: nn_integral_cong)
also note calculation }
ultimately have "ereal (pp y x) + ?used \<le> \<integral>\<^sup>+ x. pp y x \<partial>count_space UNIV"
by (metis (no_types, lifting) ereal_add_mono order_refl)
also note eq
also have "(\<integral>\<^sup>+ z. rr y z \<partial>count_space UNIV) = (\<integral>\<^sup>+ z. ?f z \<partial>count_space UNIV) + (\<integral>\<^sup>+ z. (if z < k then rr y z else if z = k then ?used - setsum (rr y) {..<k} else 0) \<partial>count_space UNIV)"
using k by(subst nn_integral_add[symmetric])(auto intro!: nn_integral_cong simp add: ivl_disj_un(2)[symmetric] setsum_nonneg dest: find_start_eq_Some_least find_start_eq_Some_above)
also have "(\<integral>\<^sup>+ z. (if z < k then rr y z else if z = k then ?used - setsum (rr y) {..<k} else 0) \<partial>count_space UNIV) =
(\<integral>\<^sup>+ z. (if z < k then rr y z else if z = k then ?used - setsum (rr y) {..<k} else 0) \<partial>count_space {..k})"
by(auto simp add: nn_integral_count_space_indicator indicator_def intro: nn_integral_cong)
also have "\<dots> = ?used"
using k by(auto simp add: nn_integral_count_space_finite max_def ivl_disj_un(2)[symmetric] diff_le_iff setsum_nonneg dest: find_start_eq_Some_least)
finally have "pp y x \<le> (\<integral>\<^sup>+ z. ?f z \<partial>count_space UNIV)"
by(cases "\<integral>\<^sup>+ z. ?f z \<partial>count_space UNIV") simp_all
then show ?thesis using k
by(simp add: assign_def nn_integral_assign_aux diff_le_iff find_start_eq_Some_above min_def)
next
case True
have "setsum (pp y) {..x} = \<integral>\<^sup>+ x. pp y x \<partial>count_space {..x}"
by(simp add: nn_integral_count_space_finite max_def)
also have "\<dots> \<le> \<integral>\<^sup>+ x. pp y x \<partial>count_space UNIV"
by(auto simp add: nn_integral_count_space_indicator indicator_def intro: nn_integral_mono)
also have "\<dots> = setsum (pp y) {..<x}" by(simp add: True)
finally have "pp y x = 0" by(simp add: ivl_disj_un(2)[symmetric] eq_iff del: pp_convs)
thus ?thesis
by(cases "find_start y (setsum (pp y) {..<x})")(simp_all add: assign_def diff_le_iff find_start_eq_Some_above)
qed }
note nn_integral_assign2 = this
let ?f = "\<lambda>y x z. if x \<in> ?A y \<and> z \<in> ?B y then assign y (?P y x) (?R y z) else 0"
def f \<equiv> "\<lambda>y x z. ereal (?f y x z)"
have pos: "\<And>y x z. 0 \<le> f y x z" by(simp add: f_def)
{ fix y x z
have "f y x z \<le> 0 \<longleftrightarrow> f y x z = 0" using pos[of y x z] by simp }
note f [simp] = this
have support:
"\<And>x y z. (x, y) \<notin> set_pmf pq \<Longrightarrow> f y x z = 0"
"\<And>x y z. (y, z) \<notin> set_pmf qr \<Longrightarrow> f y x z = 0"
by(auto simp add: f_def)
from pos support have support':
"\<And>x z. x \<notin> set_pmf p \<Longrightarrow> (\<integral>\<^sup>+ y. f y x z \<partial>count_space UNIV) = 0"
"\<And>x z. z \<notin> set_pmf r \<Longrightarrow> (\<integral>\<^sup>+ y. f y x z \<partial>count_space UNIV) = 0"
and support'':
"\<And>x y z. x \<notin> set_pmf p \<Longrightarrow> f y x z = 0"
"\<And>x y z. y \<notin> set_pmf q \<Longrightarrow> f y x z = 0"
"\<And>x y z. z \<notin> set_pmf r \<Longrightarrow> f y x z = 0"
by(auto simp add: nn_integral_0_iff_AE AE_count_space p q r set_map_pmf image_iff)(metis fst_conv snd_conv)+
have f_x: "\<And>y z. (\<integral>\<^sup>+ x. f y x z \<partial>count_space (set_pmf p)) = pmf qr (y, z)"
proof(case_tac "z \<in> ?B y")
fix y z
assume z: "z \<in> ?B y"
have "(\<integral>\<^sup>+ x. f y x z \<partial>count_space (set_pmf p)) = (\<integral>\<^sup>+ x. ?f y x z \<partial>count_space (?A y))"
using support''(1)[of _ y z]
by(fastforce simp add: f_def nn_integral_count_space_indicator indicator_def intro!: nn_integral_cong)
also have "\<dots> = \<integral>\<^sup>+ x. assign y (?P y x) (?R y z) \<partial>count_space (?A y)"
using z by(intro nn_integral_cong) simp
also have "\<dots> = \<integral>\<^sup>+ x. assign y x (?R y z) \<partial>count_space (?P y ` ?A y)"
by(intro nn_integral_bij_count_space inj_on_imp_bij_betw inj_on_to_nat_on) simp
also have "\<dots> = \<integral>\<^sup>+ x. assign y x (?R y z) \<partial>count_space UNIV"
by(auto simp add: nn_integral_count_space_indicator indicator_def assign_eq_0_outside pp_def intro!: nn_integral_cong)
also have "\<dots> = rr y (?R y z)" by(rule nn_integral_assign1)
also have "\<dots> = pmf qr (y, z)" using z by(simp add: rr_def)
finally show "?thesis y z" .
qed(auto simp add: f_def zero_ereal_def[symmetric] set_pmf_iff)
have f_z: "\<And>x y. (\<integral>\<^sup>+ z. f y x z \<partial>count_space (set_pmf r)) = pmf pq (x, y)"
proof(case_tac "x \<in> ?A y")
fix x y
assume x: "x \<in> ?A y"
have "(\<integral>\<^sup>+ z. f y x z \<partial>count_space (set_pmf r)) = (\<integral>\<^sup>+ z. ?f y x z \<partial>count_space (?B y))"
using support''(3)[of _ y x]
by(fastforce simp add: f_def nn_integral_count_space_indicator indicator_def intro!: nn_integral_cong)
also have "\<dots> = \<integral>\<^sup>+ z. assign y (?P y x) (?R y z) \<partial>count_space (?B y)"
using x by(intro nn_integral_cong) simp
also have "\<dots> = \<integral>\<^sup>+ z. assign y (?P y x) z \<partial>count_space (?R y ` ?B y)"
by(intro nn_integral_bij_count_space inj_on_imp_bij_betw inj_on_to_nat_on) simp
also have "\<dots> = \<integral>\<^sup>+ z. assign y (?P y x) z \<partial>count_space UNIV"
by(auto simp add: nn_integral_count_space_indicator indicator_def assign_eq_0_outside rr_def intro!: nn_integral_cong)
also have "\<dots> = pp y (?P y x)" by(rule nn_integral_assign2)
also have "\<dots> = pmf pq (x, y)" using x by(simp add: pp_def)
finally show "?thesis x y" .
qed(auto simp add: f_def zero_ereal_def[symmetric] set_pmf_iff)
let ?pr = "\<lambda>(x, z). \<integral>\<^sup>+ y. f y x z \<partial>count_space UNIV"
have pr_pos: "\<And>xz. 0 \<le> ?pr xz"
by(auto simp add: nn_integral_nonneg)
have pr': "?pr = (\<lambda>(x, z). \<integral>\<^sup>+ y. f y x z \<partial>count_space (set_pmf q))"
by(auto simp add: fun_eq_iff nn_integral_count_space_indicator indicator_def support'' intro: nn_integral_cong)
have "(\<integral>\<^sup>+ xz. ?pr xz \<partial>count_space UNIV) = (\<integral>\<^sup>+ xz. ?pr xz * indicator (set_pmf p \<times> set_pmf r) xz \<partial>count_space UNIV)"
by(rule nn_integral_cong)(auto simp add: indicator_def support' intro: ccontr)
also have "\<dots> = (\<integral>\<^sup>+ xz. ?pr xz \<partial>count_space (set_pmf p \<times> set_pmf r))"
by(simp add: nn_integral_count_space_indicator)
also have "\<dots> = (\<integral>\<^sup>+ xz. ?pr xz \<partial>(count_space (set_pmf p) \<Otimes>\<^sub>M count_space (set_pmf r)))"
by(simp add: pair_measure_countable)
also have "\<dots> = (\<integral>\<^sup>+ (x, z). \<integral>\<^sup>+ y. f y x z \<partial>count_space (set_pmf q) \<partial>(count_space (set_pmf p) \<Otimes>\<^sub>M count_space (set_pmf r)))"
by(simp add: pr')
also have "\<dots> = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ z. \<integral>\<^sup>+ y. f y x z \<partial>count_space (set_pmf q) \<partial>count_space (set_pmf r) \<partial>count_space (set_pmf p))"
by(subst sigma_finite_measure.nn_integral_fst[symmetric, OF sigma_finite_measure_count_space_countable])(simp_all add: pair_measure_countable)
also have "\<dots> = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. \<integral>\<^sup>+ z. f y x z \<partial>count_space (set_pmf r) \<partial>count_space (set_pmf q) \<partial>count_space (set_pmf p))"
by(subst (2) pair_sigma_finite.Fubini')(simp_all add: pair_sigma_finite.intro sigma_finite_measure_count_space_countable pair_measure_countable)
also have "\<dots> = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. pmf pq (x, y) \<partial>count_space (set_pmf q) \<partial>count_space (set_pmf p))"
by(simp add: f_z)
also have "\<dots> = (\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. pmf pq (x, y) \<partial>count_space (set_pmf p) \<partial>count_space (set_pmf q))"
by(subst pair_sigma_finite.Fubini')(simp_all add: pair_sigma_finite.intro sigma_finite_measure_count_space_countable pair_measure_countable)
also have "\<dots> = (\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. emeasure (measure_pmf pq) {(x, y)} \<partial>count_space (set_pmf p) \<partial>count_space (set_pmf q))"
by(simp add: emeasure_pmf_single)
also have "\<dots> = (\<integral>\<^sup>+ y. emeasure (measure_pmf pq) (\<Union>x\<in>set_pmf p. {(x, y)}) \<partial>count_space (set_pmf q))"
by(subst emeasure_UN_countable)(simp_all add: disjoint_family_on_def)
also have "\<dots> = (\<integral>\<^sup>+ y. emeasure (measure_pmf pq) ((\<Union>x\<in>set_pmf p. {(x, y)}) \<union> {(x, y'). x \<notin> set_pmf p \<and> y' = y}) \<partial>count_space (set_pmf q))"
by(rule nn_integral_cong emeasure_Un_null_set[symmetric])+
(auto simp add: p set_map_pmf split_beta intro!: in_null_sets_measure_pmfI intro: rev_image_eqI)
also have "\<dots> = (\<integral>\<^sup>+ y. emeasure (measure_pmf pq) (snd -` {y}) \<partial>count_space (set_pmf q))"
by(rule nn_integral_cong arg_cong2[where f=emeasure])+auto
also have "\<dots> = (\<integral>\<^sup>+ y. pmf q y \<partial>count_space (set_pmf q))"
by(simp add: ereal_pmf_map q)
also have "\<dots> = (\<integral>\<^sup>+ y. pmf q y \<partial>count_space UNIV)"
by(auto simp add: nn_integral_count_space_indicator indicator_def set_pmf_iff intro: nn_integral_cong)
also have "\<dots> = 1"
by(subst nn_integral_pmf)(simp add: measure_pmf.emeasure_eq_1_AE)
finally have pr_prob: "(\<integral>\<^sup>+ xz. ?pr xz \<partial>count_space UNIV) = 1" .
have pr_bounded: "\<And>xz. ?pr xz \<noteq> \<infinity>"
proof -
fix xz
have "?pr xz \<le> \<integral>\<^sup>+ xz. ?pr xz \<partial>count_space UNIV"
by(rule nn_integral_ge_point) simp
also have "\<dots> = 1" by(fact pr_prob)
finally show "?thesis xz" by auto
qed
def pr \<equiv> "embed_pmf (real \<circ> ?pr)"
have pmf_pr: "\<And>xz. pmf pr xz = real (?pr xz)" using pr_pos pr_prob
unfolding pr_def by(subst pmf_embed_pmf)(auto simp add: real_of_ereal_pos ereal_real pr_bounded)
have set_pmf_pr_subset: "set_pmf pr \<subseteq> set_pmf pq O set_pmf qr"
proof
fix xz :: "'a \<times> 'c"
obtain x z where xz: "xz = (x, z)" by(cases xz)
assume "xz \<in> set_pmf pr"
with xz have "pmf pr (x, z) \<noteq> 0" by(simp add: set_pmf_iff)
hence "\<exists>y. f y x z \<noteq> 0" by(rule contrapos_np)(simp add: pmf_pr)
then obtain y where y: "f y x z \<noteq> 0" ..
then have "(x, y) \<in> set_pmf pq" "(y, z) \<in> set_pmf qr"
using support by fastforce+
then show "xz \<in> set_pmf pq O set_pmf qr" using xz by auto
qed
hence "\<And>x z. (x, z) \<in> set_pmf pr \<Longrightarrow> (R OO S) x z" using pq qr by blast
moreover
have "map_pmf fst pr = p"
proof(rule pmf_eqI)
fix x
have "pmf (map_pmf fst pr) x = emeasure (measure_pmf pr) (fst -` {x})"
by(simp add: ereal_pmf_map)
also have "\<dots> = \<integral>\<^sup>+ xz. pmf pr xz \<partial>count_space (fst -` {x})"
by(simp add: nn_integral_pmf)
also have "\<dots> = \<integral>\<^sup>+ xz. ?pr xz \<partial>count_space (fst -` {x})"
by(simp add: pmf_pr ereal_real pr_bounded pr_pos)
also have "\<dots> = \<integral>\<^sup>+ xz. ?pr xz \<partial>count_space {x} \<Otimes>\<^sub>M count_space (set_pmf r)"
by(auto simp add: nn_integral_count_space_indicator indicator_def support' pair_measure_countable intro!: nn_integral_cong)
also have "\<dots> = \<integral>\<^sup>+ z. \<integral>\<^sup>+ x. ?pr (x, z) \<partial>count_space {x} \<partial>count_space (set_pmf r)"
by(subst pair_sigma_finite.nn_integral_snd[symmetric])(simp_all add: pair_measure_countable pair_sigma_finite.intro sigma_finite_measure_count_space_countable)
also have "\<dots> = \<integral>\<^sup>+ z. ?pr (x, z) \<partial>count_space (set_pmf r)"
using pr_pos by(clarsimp simp add: nn_integral_count_space_finite max_def)
also have "\<dots> = \<integral>\<^sup>+ z. \<integral>\<^sup>+ y. f y x z \<partial>count_space (set_pmf q) \<partial>count_space (set_pmf r)"
by(simp add: pr')
also have "\<dots> = \<integral>\<^sup>+ y. \<integral>\<^sup>+ z. f y x z \<partial>count_space (set_pmf r) \<partial>count_space (set_pmf q)"
by(subst pair_sigma_finite.Fubini')(simp_all add: pair_sigma_finite.intro sigma_finite_measure_count_space_countable pair_measure_countable)
also have "\<dots> = \<integral>\<^sup>+ y. pmf pq (x, y) \<partial>count_space (set_pmf q)"
by(simp add: f_z)
also have "\<dots> = \<integral>\<^sup>+ y. emeasure (measure_pmf pq) {(x, y)} \<partial>count_space (set_pmf q)"
by(simp add: emeasure_pmf_single)
also have "\<dots> = emeasure (measure_pmf pq) (\<Union>y\<in>set_pmf q. {(x, y)})"
by(subst emeasure_UN_countable)(simp_all add: disjoint_family_on_def)
also have "\<dots> = emeasure (measure_pmf pq) ((\<Union>y\<in>set_pmf q. {(x, y)}) \<union> {(x', y). y \<notin> set_pmf q \<and> x' = x})"
by(rule emeasure_Un_null_set[symmetric])+
(auto simp add: q set_map_pmf split_beta intro!: in_null_sets_measure_pmfI intro: rev_image_eqI)
also have "\<dots> = emeasure (measure_pmf pq) (fst -` {x})"
by(rule arg_cong2[where f=emeasure])+auto
also have "\<dots> = pmf p x" by(simp add: ereal_pmf_map p)
finally show "pmf (map_pmf fst pr) x = pmf p x" by simp
qed
moreover
have "map_pmf snd pr = r"
proof(rule pmf_eqI)
fix z
have "pmf (map_pmf snd pr) z = emeasure (measure_pmf pr) (snd -` {z})"
by(simp add: ereal_pmf_map)
also have "\<dots> = \<integral>\<^sup>+ xz. pmf pr xz \<partial>count_space (snd -` {z})"
by(simp add: nn_integral_pmf)
also have "\<dots> = \<integral>\<^sup>+ xz. ?pr xz \<partial>count_space (snd -` {z})"
by(simp add: pmf_pr ereal_real pr_bounded pr_pos)
also have "\<dots> = \<integral>\<^sup>+ xz. ?pr xz \<partial>count_space (set_pmf p) \<Otimes>\<^sub>M count_space {z}"
by(auto simp add: nn_integral_count_space_indicator indicator_def support' pair_measure_countable intro!: nn_integral_cong)
also have "\<dots> = \<integral>\<^sup>+ x. \<integral>\<^sup>+ z. ?pr (x, z) \<partial>count_space {z} \<partial>count_space (set_pmf p)"
by(subst sigma_finite_measure.nn_integral_fst[symmetric])(simp_all add: pair_measure_countable sigma_finite_measure_count_space_countable)
also have "\<dots> = \<integral>\<^sup>+ x. ?pr (x, z) \<partial>count_space (set_pmf p)"
using pr_pos by(clarsimp simp add: nn_integral_count_space_finite max_def)
also have "\<dots> = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f y x z \<partial>count_space (set_pmf q) \<partial>count_space (set_pmf p)"
by(simp add: pr')
also have "\<dots> = \<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f y x z \<partial>count_space (set_pmf p) \<partial>count_space (set_pmf q)"
by(subst pair_sigma_finite.Fubini')(simp_all add: pair_sigma_finite.intro sigma_finite_measure_count_space_countable pair_measure_countable)
also have "\<dots> = \<integral>\<^sup>+ y. pmf qr (y, z) \<partial>count_space (set_pmf q)"
by(simp add: f_x)
also have "\<dots> = \<integral>\<^sup>+ y. emeasure (measure_pmf qr) {(y, z)} \<partial>count_space (set_pmf q)"
by(simp add: emeasure_pmf_single)
also have "\<dots> = emeasure (measure_pmf qr) (\<Union>y\<in>set_pmf q. {(y, z)})"
by(subst emeasure_UN_countable)(simp_all add: disjoint_family_on_def)
also have "\<dots> = emeasure (measure_pmf qr) ((\<Union>y\<in>set_pmf q. {(y, z)}) \<union> {(y, z'). y \<notin> set_pmf q \<and> z' = z})"
by(rule emeasure_Un_null_set[symmetric])+
(auto simp add: q' set_map_pmf split_beta intro!: in_null_sets_measure_pmfI intro: rev_image_eqI)
also have "\<dots> = emeasure (measure_pmf qr) (snd -` {z})"
by(rule arg_cong2[where f=emeasure])+auto
also have "\<dots> = pmf r z" by(simp add: ereal_pmf_map r)
finally show "pmf (map_pmf snd pr) z = pmf r z" by simp
qed
ultimately have "rel_pmf (R OO S) p r" .. }
then show "rel_pmf R OO rel_pmf S \<le> rel_pmf (R OO S)"
by(auto simp add: le_fun_def)
qed (fact natLeq_card_order natLeq_cinfinite)+
end