src/HOL/Probability/Probability_Mass_Function.thy
author hoelzl
Fri Dec 05 12:06:18 2014 +0100 (2014-12-05)
changeset 59092 d469103c0737
parent 59053 43e07797269b
child 59093 2b106e58a177
permissions -rw-r--r--
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
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(*  Title:      HOL/Probability/Probability_Mass_Function.thy
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    Author:     Johannes Hölzl, TU München 
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    Author:     Andreas Lochbihler, ETH Zurich
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*)
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section \<open> Probability mass function \<close>
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theory Probability_Mass_Function
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imports
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  Giry_Monad
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  "~~/src/HOL/Library/Multiset"
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begin
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lemma bind_return'': "sets M = sets N \<Longrightarrow> M \<guillemotright>= return N = M"
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   by (cases "space M = {}")
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      (simp_all add: bind_empty space_empty[symmetric] bind_nonempty join_return'
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                cong: subprob_algebra_cong)
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lemma (in prob_space) distr_const[simp]:
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  "c \<in> space N \<Longrightarrow> distr M N (\<lambda>x. c) = return N c"
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  by (rule measure_eqI) (auto simp: emeasure_distr emeasure_space_1)
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lemma (in finite_measure) countable_support:
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  "countable {x. measure M {x} \<noteq> 0}"
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proof cases
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  assume "measure M (space M) = 0"
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  with bounded_measure measure_le_0_iff have "{x. measure M {x} \<noteq> 0} = {}"
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    by auto
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  then show ?thesis
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    by simp
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next
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  let ?M = "measure M (space M)" and ?m = "\<lambda>x. measure M {x}"
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  assume "?M \<noteq> 0"
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  then have *: "{x. ?m x \<noteq> 0} = (\<Union>n. {x. ?M / Suc n < ?m x})"
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    using reals_Archimedean[of "?m x / ?M" for x]
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    by (auto simp: field_simps not_le[symmetric] measure_nonneg divide_le_0_iff measure_le_0_iff)
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  have **: "\<And>n. finite {x. ?M / Suc n < ?m x}"
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  proof (rule ccontr)
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    fix n assume "infinite {x. ?M / Suc n < ?m x}" (is "infinite ?X")
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    then obtain X where "finite X" "card X = Suc (Suc n)" "X \<subseteq> ?X"
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      by (metis infinite_arbitrarily_large)
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    from this(3) have *: "\<And>x. x \<in> X \<Longrightarrow> ?M / Suc n \<le> ?m x" 
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      by auto
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    { fix x assume "x \<in> X"
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      from `?M \<noteq> 0` *[OF this] have "?m x \<noteq> 0" by (auto simp: field_simps measure_le_0_iff)
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      then have "{x} \<in> sets M" by (auto dest: measure_notin_sets) }
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    note singleton_sets = this
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    have "?M < (\<Sum>x\<in>X. ?M / Suc n)"
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      using `?M \<noteq> 0` 
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      by (simp add: `card X = Suc (Suc n)` real_eq_of_nat[symmetric] real_of_nat_Suc field_simps less_le measure_nonneg)
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    also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"
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      by (rule setsum_mono) fact
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    also have "\<dots> = measure M (\<Union>x\<in>X. {x})"
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      using singleton_sets `finite X`
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      by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def)
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    finally have "?M < measure M (\<Union>x\<in>X. {x})" .
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    moreover have "measure M (\<Union>x\<in>X. {x}) \<le> ?M"
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      using singleton_sets[THEN sets.sets_into_space] by (intro finite_measure_mono) auto
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    ultimately show False by simp
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  qed
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  show ?thesis
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    unfolding * by (intro countable_UN countableI_type countable_finite[OF **])
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qed
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lemma (in finite_measure) AE_support_countable:
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  assumes [simp]: "sets M = UNIV"
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  shows "(AE x in M. measure M {x} \<noteq> 0) \<longleftrightarrow> (\<exists>S. countable S \<and> (AE x in M. x \<in> S))"
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proof
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  assume "\<exists>S. countable S \<and> (AE x in M. x \<in> S)"
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  then obtain S where S[intro]: "countable S" and ae: "AE x in M. x \<in> S"
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    by auto
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  then have "emeasure M (\<Union>x\<in>{x\<in>S. emeasure M {x} \<noteq> 0}. {x}) = 
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    (\<integral>\<^sup>+ x. emeasure M {x} * indicator {x\<in>S. emeasure M {x} \<noteq> 0} x \<partial>count_space UNIV)"
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    by (subst emeasure_UN_countable)
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       (auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space)
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  also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} * indicator S x \<partial>count_space UNIV)"
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    by (auto intro!: nn_integral_cong split: split_indicator)
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  also have "\<dots> = emeasure M (\<Union>x\<in>S. {x})"
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    by (subst emeasure_UN_countable)
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       (auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space)
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  also have "\<dots> = emeasure M (space M)"
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    using ae by (intro emeasure_eq_AE) auto
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  finally have "emeasure M {x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0} = emeasure M (space M)"
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    by (simp add: emeasure_single_in_space cong: rev_conj_cong)
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  with finite_measure_compl[of "{x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0}"]
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  have "AE x in M. x \<in> S \<and> emeasure M {x} \<noteq> 0"
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    by (intro AE_I[OF order_refl]) (auto simp: emeasure_eq_measure set_diff_eq cong: conj_cong)
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  then show "AE x in M. measure M {x} \<noteq> 0"
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    by (auto simp: emeasure_eq_measure)
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qed (auto intro!: exI[of _ "{x. measure M {x} \<noteq> 0}"] countable_support)
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subsection {* PMF as measure *}
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typedef 'a pmf = "{M :: 'a measure. prob_space M \<and> sets M = UNIV \<and> (AE x in M. measure M {x} \<noteq> 0)}"
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  morphisms measure_pmf Abs_pmf
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  by (intro exI[of _ "uniform_measure (count_space UNIV) {undefined}"])
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     (auto intro!: prob_space_uniform_measure AE_uniform_measureI)
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declare [[coercion measure_pmf]]
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lemma prob_space_measure_pmf: "prob_space (measure_pmf p)"
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  using pmf.measure_pmf[of p] by auto
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interpretation measure_pmf!: prob_space "measure_pmf M" for M
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  by (rule prob_space_measure_pmf)
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interpretation measure_pmf!: subprob_space "measure_pmf M" for M
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  by (rule prob_space_imp_subprob_space) unfold_locales
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lemma subprob_space_measure_pmf: "subprob_space (measure_pmf x)"
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  by unfold_locales
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locale pmf_as_measure
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begin
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setup_lifting type_definition_pmf
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end
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context
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begin
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interpretation pmf_as_measure .
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lift_definition pmf :: "'a pmf \<Rightarrow> 'a \<Rightarrow> real" is "\<lambda>M x. measure M {x}" .
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lift_definition set_pmf :: "'a pmf \<Rightarrow> 'a set" is "\<lambda>M. {x. measure M {x} \<noteq> 0}" .
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lift_definition map_pmf :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf" is
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  "\<lambda>f M. distr M (count_space UNIV) f"
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proof safe
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  fix M and f :: "'a \<Rightarrow> 'b"
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  let ?D = "distr M (count_space UNIV) f"
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  assume "prob_space M" and [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
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  interpret prob_space M by fact
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  from ae have "AE x in M. measure M (f -` {f x}) \<noteq> 0"
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  proof eventually_elim
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    fix x
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    have "measure M {x} \<le> measure M (f -` {f x})"
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      by (intro finite_measure_mono) auto
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    then show "measure M {x} \<noteq> 0 \<Longrightarrow> measure M (f -` {f x}) \<noteq> 0"
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      using measure_nonneg[of M "{x}"] by auto
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  qed
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  then show "AE x in ?D. measure ?D {x} \<noteq> 0"
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    by (simp add: AE_distr_iff measure_distr measurable_def)
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qed (auto simp: measurable_def prob_space.prob_space_distr)
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declare [[coercion set_pmf]]
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lemma countable_set_pmf [simp]: "countable (set_pmf p)"
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  by transfer (metis prob_space.finite_measure finite_measure.countable_support)
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lemma sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV"
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  by transfer metis
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lemma sets_measure_pmf_count_space[measurable_cong]:
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  "sets (measure_pmf M) = sets (count_space UNIV)"
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  by simp
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lemma space_measure_pmf[simp]: "space (measure_pmf p) = UNIV"
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  using sets_eq_imp_space_eq[of "measure_pmf p" "count_space UNIV"] by simp
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lemma measure_pmf_in_subprob_algebra[measurable (raw)]: "measure_pmf x \<in> space (subprob_algebra (count_space UNIV))"
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  by (simp add: space_subprob_algebra subprob_space_measure_pmf)
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lemma measurable_pmf_measure1[simp]: "measurable (M :: 'a pmf) N = UNIV \<rightarrow> space N"
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  by (auto simp: measurable_def)
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lemma measurable_pmf_measure2[simp]: "measurable N (M :: 'a pmf) = measurable N (count_space UNIV)"
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  by (intro measurable_cong_sets) simp_all
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lemma pmf_positive: "x \<in> set_pmf p \<Longrightarrow> 0 < pmf p x"
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  by transfer (simp add: less_le measure_nonneg)
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lemma pmf_nonneg: "0 \<le> pmf p x"
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  by transfer (simp add: measure_nonneg)
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lemma pmf_le_1: "pmf p x \<le> 1"
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  by (simp add: pmf.rep_eq)
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lemma emeasure_pmf_single:
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  fixes M :: "'a pmf"
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  shows "emeasure M {x} = pmf M x"
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  by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
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lemma AE_measure_pmf: "AE x in (M::'a pmf). x \<in> M"
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  by transfer simp
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lemma emeasure_pmf_single_eq_zero_iff:
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  fixes M :: "'a pmf"
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  shows "emeasure M {y} = 0 \<longleftrightarrow> y \<notin> M"
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  by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
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lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \<longleftrightarrow> (\<forall>y\<in>M. P y)"
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proof -
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  { fix y assume y: "y \<in> M" and P: "AE x in M. P x" "\<not> P y"
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    with P have "AE x in M. x \<noteq> y"
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      by auto
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    with y have False
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      by (simp add: emeasure_pmf_single_eq_zero_iff AE_iff_measurable[OF _ refl]) }
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  then show ?thesis
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    using AE_measure_pmf[of M] by auto
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qed
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lemma set_pmf_not_empty: "set_pmf M \<noteq> {}"
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  using AE_measure_pmf[of M] by (intro notI) simp
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lemma set_pmf_iff: "x \<in> set_pmf M \<longleftrightarrow> pmf M x \<noteq> 0"
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  by transfer simp
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lemma emeasure_measure_pmf_finite: "finite S \<Longrightarrow> emeasure (measure_pmf M) S = (\<Sum>s\<in>S. pmf M s)"
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  by (subst emeasure_eq_setsum_singleton) (auto simp: emeasure_pmf_single)
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lemma measure_measure_pmf_finite: "finite S \<Longrightarrow> measure (measure_pmf M) S = setsum (pmf M) S"
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using emeasure_measure_pmf_finite[of S M]
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by(simp add: measure_pmf.emeasure_eq_measure)
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lemma nn_integral_measure_pmf_support:
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  fixes f :: "'a \<Rightarrow> ereal"
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  assumes f: "finite A" and nn: "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x" "\<And>x. x \<in> set_pmf M \<Longrightarrow> x \<notin> A \<Longrightarrow> f x = 0"
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  shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>A. f x * pmf M x)"
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proof -
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  have "(\<integral>\<^sup>+x. f x \<partial>M) = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)"
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    using nn by (intro nn_integral_cong_AE) (auto simp: AE_measure_pmf_iff split: split_indicator)
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  also have "\<dots> = (\<Sum>x\<in>A. f x * emeasure M {x})"
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    using assms by (intro nn_integral_indicator_finite) auto
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  finally show ?thesis
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    by (simp add: emeasure_measure_pmf_finite)
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qed
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lemma nn_integral_measure_pmf_finite:
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  fixes f :: "'a \<Rightarrow> ereal"
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  assumes f: "finite (set_pmf M)" and nn: "\<And>x. x \<in> set_pmf M \<Longrightarrow> 0 \<le> f x"
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  shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>set_pmf M. f x * pmf M x)"
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  using assms by (intro nn_integral_measure_pmf_support) auto
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lemma integrable_measure_pmf_finite:
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  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
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  shows "finite (set_pmf M) \<Longrightarrow> integrable M f"
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  by (auto intro!: integrableI_bounded simp: nn_integral_measure_pmf_finite)
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lemma integral_measure_pmf:
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  assumes [simp]: "finite A" and "\<And>a. a \<in> set_pmf M \<Longrightarrow> f a \<noteq> 0 \<Longrightarrow> a \<in> A"
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  shows "(\<integral>x. f x \<partial>measure_pmf M) = (\<Sum>a\<in>A. f a * pmf M a)"
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proof -
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  have "(\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x * indicator A x \<partial>measure_pmf M)"
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    using assms(2) by (intro integral_cong_AE) (auto split: split_indicator simp: AE_measure_pmf_iff)
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  also have "\<dots> = (\<Sum>a\<in>A. f a * pmf M a)"
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    by (subst integral_indicator_finite_real) (auto simp: measure_def emeasure_measure_pmf_finite)
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  finally show ?thesis .
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qed
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lemma integrable_pmf: "integrable (count_space X) (pmf M)"
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proof -
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  have " (\<integral>\<^sup>+ x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+ x. pmf M x \<partial>count_space (M \<inter> X))"
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    by (auto simp add: nn_integral_count_space_indicator set_pmf_iff intro!: nn_integral_cong split: split_indicator)
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  then have "integrable (count_space X) (pmf M) = integrable (count_space (M \<inter> X)) (pmf M)"
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    by (simp add: integrable_iff_bounded pmf_nonneg)
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  then show ?thesis
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    by (simp add: pmf.rep_eq measure_pmf.integrable_measure disjoint_family_on_def)
hoelzl@59000
   261
qed
hoelzl@59000
   262
hoelzl@59000
   263
lemma integral_pmf: "(\<integral>x. pmf M x \<partial>count_space X) = measure M X"
hoelzl@59000
   264
proof -
hoelzl@59000
   265
  have "(\<integral>x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+x. pmf M x \<partial>count_space X)"
hoelzl@59000
   266
    by (simp add: pmf_nonneg integrable_pmf nn_integral_eq_integral)
hoelzl@59000
   267
  also have "\<dots> = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space (X \<inter> M))"
hoelzl@59000
   268
    by (auto intro!: nn_integral_cong_AE split: split_indicator
hoelzl@59000
   269
             simp: pmf.rep_eq measure_pmf.emeasure_eq_measure nn_integral_count_space_indicator
hoelzl@59000
   270
                   AE_count_space set_pmf_iff)
hoelzl@59000
   271
  also have "\<dots> = emeasure M (X \<inter> M)"
hoelzl@59000
   272
    by (rule emeasure_countable_singleton[symmetric]) (auto intro: countable_set_pmf)
hoelzl@59000
   273
  also have "\<dots> = emeasure M X"
hoelzl@59000
   274
    by (auto intro!: emeasure_eq_AE simp: AE_measure_pmf_iff)
hoelzl@59000
   275
  finally show ?thesis
hoelzl@59000
   276
    by (simp add: measure_pmf.emeasure_eq_measure)
hoelzl@59000
   277
qed
hoelzl@59000
   278
hoelzl@59000
   279
lemma integral_pmf_restrict:
hoelzl@59000
   280
  "(f::'a \<Rightarrow> 'b::{banach, second_countable_topology}) \<in> borel_measurable (count_space UNIV) \<Longrightarrow>
hoelzl@59000
   281
    (\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x \<partial>restrict_space M M)"
hoelzl@59000
   282
  by (auto intro!: integral_cong_AE simp add: integral_restrict_space AE_measure_pmf_iff)
hoelzl@59000
   283
hoelzl@58587
   284
lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1"
hoelzl@58587
   285
proof -
hoelzl@58587
   286
  have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)"
hoelzl@58587
   287
    by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf)
hoelzl@58587
   288
  then show ?thesis
hoelzl@58587
   289
    using measure_pmf.emeasure_space_1 by simp
hoelzl@58587
   290
qed
hoelzl@58587
   291
Andreas@59023
   292
lemma in_null_sets_measure_pmfI:
Andreas@59023
   293
  "A \<inter> set_pmf p = {} \<Longrightarrow> A \<in> null_sets (measure_pmf p)"
Andreas@59023
   294
using emeasure_eq_0_AE[where ?P="\<lambda>x. x \<in> A" and M="measure_pmf p"]
Andreas@59023
   295
by(auto simp add: null_sets_def AE_measure_pmf_iff)
Andreas@59023
   296
hoelzl@58587
   297
lemma map_pmf_id[simp]: "map_pmf id = id"
hoelzl@58587
   298
  by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI)
hoelzl@58587
   299
hoelzl@59053
   300
lemma map_pmf_ident[simp]: "map_pmf (\<lambda>x. x) = (\<lambda>x. x)"
hoelzl@59053
   301
  using map_pmf_id unfolding id_def .
hoelzl@59053
   302
hoelzl@58587
   303
lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g"
hoelzl@58587
   304
  by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def) 
hoelzl@58587
   305
hoelzl@59000
   306
lemma map_pmf_comp: "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M"
hoelzl@59000
   307
  using map_pmf_compose[of f g] by (simp add: comp_def)
hoelzl@59000
   308
hoelzl@58587
   309
lemma map_pmf_cong:
hoelzl@58587
   310
  assumes "p = q"
hoelzl@58587
   311
  shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> map_pmf f p = map_pmf g q"
hoelzl@58587
   312
  unfolding `p = q`[symmetric] measure_pmf_inject[symmetric] map_pmf.rep_eq
hoelzl@58587
   313
  by (auto simp add: emeasure_distr AE_measure_pmf_iff intro!: emeasure_eq_AE measure_eqI)
hoelzl@58587
   314
hoelzl@59002
   315
lemma emeasure_map_pmf[simp]: "emeasure (map_pmf f M) X = emeasure M (f -` X)"
hoelzl@59002
   316
  unfolding map_pmf.rep_eq by (subst emeasure_distr) auto
hoelzl@59002
   317
hoelzl@59002
   318
lemma nn_integral_map_pmf[simp]: "(\<integral>\<^sup>+x. f x \<partial>map_pmf g M) = (\<integral>\<^sup>+x. f (g x) \<partial>M)"
hoelzl@59002
   319
  unfolding map_pmf.rep_eq by (intro nn_integral_distr) auto
hoelzl@59002
   320
Andreas@59023
   321
lemma ereal_pmf_map: "pmf (map_pmf f p) x = (\<integral>\<^sup>+ y. indicator (f -` {x}) y \<partial>measure_pmf p)"
Andreas@59023
   322
proof(transfer fixing: f x)
Andreas@59023
   323
  fix p :: "'b measure"
Andreas@59023
   324
  presume "prob_space p"
Andreas@59023
   325
  then interpret prob_space p .
Andreas@59023
   326
  presume "sets p = UNIV"
Andreas@59023
   327
  then show "ereal (measure (distr p (count_space UNIV) f) {x}) = integral\<^sup>N p (indicator (f -` {x}))"
Andreas@59023
   328
    by(simp add: measure_distr measurable_def emeasure_eq_measure)
Andreas@59023
   329
qed simp_all
Andreas@59023
   330
hoelzl@58587
   331
lemma pmf_set_map: 
hoelzl@58587
   332
  fixes f :: "'a \<Rightarrow> 'b"
hoelzl@58587
   333
  shows "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
hoelzl@58587
   334
proof (rule, transfer, clarsimp simp add: measure_distr measurable_def)
hoelzl@58587
   335
  fix f :: "'a \<Rightarrow> 'b" and M :: "'a measure"
hoelzl@58587
   336
  assume "prob_space M" and ae: "AE x in M. measure M {x} \<noteq> 0" and [simp]: "sets M = UNIV"
hoelzl@58587
   337
  interpret prob_space M by fact
hoelzl@58587
   338
  show "{x. measure M (f -` {x}) \<noteq> 0} = f ` {x. measure M {x} \<noteq> 0}"
hoelzl@58587
   339
  proof safe
hoelzl@58587
   340
    fix x assume "measure M (f -` {x}) \<noteq> 0"
hoelzl@58587
   341
    moreover have "measure M (f -` {x}) = measure M {y. f y = x \<and> measure M {y} \<noteq> 0}"
hoelzl@58587
   342
      using ae by (intro finite_measure_eq_AE) auto
hoelzl@58587
   343
    ultimately have "{y. f y = x \<and> measure M {y} \<noteq> 0} \<noteq> {}"
hoelzl@58587
   344
      by (metis measure_empty)
hoelzl@58587
   345
    then show "x \<in> f ` {x. measure M {x} \<noteq> 0}"
hoelzl@58587
   346
      by auto
hoelzl@58587
   347
  next
hoelzl@58587
   348
    fix x assume "measure M {x} \<noteq> 0"
hoelzl@58587
   349
    then have "0 < measure M {x}"
hoelzl@58587
   350
      using measure_nonneg[of M "{x}"] by auto
hoelzl@58587
   351
    also have "measure M {x} \<le> measure M (f -` {f x})"
hoelzl@58587
   352
      by (intro finite_measure_mono) auto
hoelzl@58587
   353
    finally show "measure M (f -` {f x}) = 0 \<Longrightarrow> False"
hoelzl@58587
   354
      by simp
hoelzl@58587
   355
  qed
hoelzl@58587
   356
qed
hoelzl@58587
   357
hoelzl@59000
   358
lemma set_map_pmf: "set_pmf (map_pmf f M) = f`set_pmf M"
hoelzl@59000
   359
  using pmf_set_map[of f] by (auto simp: comp_def fun_eq_iff)
hoelzl@59000
   360
Andreas@59023
   361
lemma nn_integral_pmf: "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = emeasure (measure_pmf p) A"
Andreas@59023
   362
proof -
Andreas@59023
   363
  have "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = (\<integral>\<^sup>+ x. pmf p x \<partial>count_space (A \<inter> set_pmf p))"
Andreas@59023
   364
    by(auto simp add: nn_integral_count_space_indicator indicator_def set_pmf_iff intro: nn_integral_cong)
Andreas@59023
   365
  also have "\<dots> = emeasure (measure_pmf p) (\<Union>x\<in>A \<inter> set_pmf p. {x})"
Andreas@59023
   366
    by(subst emeasure_UN_countable)(auto simp add: emeasure_pmf_single disjoint_family_on_def)
Andreas@59023
   367
  also have "\<dots> = emeasure (measure_pmf p) ((\<Union>x\<in>A \<inter> set_pmf p. {x}) \<union> {x. x \<in> A \<and> x \<notin> set_pmf p})"
Andreas@59023
   368
    by(rule emeasure_Un_null_set[symmetric])(auto intro: in_null_sets_measure_pmfI)
Andreas@59023
   369
  also have "\<dots> = emeasure (measure_pmf p) A"
Andreas@59023
   370
    by(auto intro: arg_cong2[where f=emeasure])
Andreas@59023
   371
  finally show ?thesis .
Andreas@59023
   372
qed
Andreas@59023
   373
hoelzl@59000
   374
subsection {* PMFs as function *}
hoelzl@59000
   375
hoelzl@58587
   376
context
hoelzl@58587
   377
  fixes f :: "'a \<Rightarrow> real"
hoelzl@58587
   378
  assumes nonneg: "\<And>x. 0 \<le> f x"
hoelzl@58587
   379
  assumes prob: "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
hoelzl@58587
   380
begin
hoelzl@58587
   381
hoelzl@58587
   382
lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ereal \<circ> f)"
hoelzl@58587
   383
proof (intro conjI)
hoelzl@58587
   384
  have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
hoelzl@58587
   385
    by (simp split: split_indicator)
hoelzl@58587
   386
  show "AE x in density (count_space UNIV) (ereal \<circ> f).
hoelzl@58587
   387
    measure (density (count_space UNIV) (ereal \<circ> f)) {x} \<noteq> 0"
hoelzl@59092
   388
    by (simp add: AE_density nonneg measure_def emeasure_density max_def)
hoelzl@58587
   389
  show "prob_space (density (count_space UNIV) (ereal \<circ> f))"
hoelzl@58587
   390
    by default (simp add: emeasure_density prob)
hoelzl@58587
   391
qed simp
hoelzl@58587
   392
hoelzl@58587
   393
lemma pmf_embed_pmf: "pmf embed_pmf x = f x"
hoelzl@58587
   394
proof transfer
hoelzl@58587
   395
  have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
hoelzl@58587
   396
    by (simp split: split_indicator)
hoelzl@58587
   397
  fix x show "measure (density (count_space UNIV) (ereal \<circ> f)) {x} = f x"
hoelzl@59092
   398
    by transfer (simp add: measure_def emeasure_density nonneg max_def)
hoelzl@58587
   399
qed
hoelzl@58587
   400
hoelzl@58587
   401
end
hoelzl@58587
   402
hoelzl@58587
   403
lemma embed_pmf_transfer:
hoelzl@58730
   404
  "rel_fun (eq_onp (\<lambda>f. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ereal (f x) \<partial>count_space UNIV) = 1)) pmf_as_measure.cr_pmf (\<lambda>f. density (count_space UNIV) (ereal \<circ> f)) embed_pmf"
hoelzl@58587
   405
  by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer)
hoelzl@58587
   406
hoelzl@59000
   407
lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)"
hoelzl@59000
   408
proof (transfer, elim conjE)
hoelzl@59000
   409
  fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
hoelzl@59000
   410
  assume "prob_space M" then interpret prob_space M .
hoelzl@59000
   411
  show "M = density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))"
hoelzl@59000
   412
  proof (rule measure_eqI)
hoelzl@59000
   413
    fix A :: "'a set"
hoelzl@59000
   414
    have "(\<integral>\<^sup>+ x. ereal (measure M {x}) * indicator A x \<partial>count_space UNIV) = 
hoelzl@59000
   415
      (\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)"
hoelzl@59000
   416
      by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator)
hoelzl@59000
   417
    also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))"
hoelzl@59000
   418
      by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space)
hoelzl@59000
   419
    also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})"
hoelzl@59000
   420
      by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support)
hoelzl@59000
   421
         (auto simp: disjoint_family_on_def)
hoelzl@59000
   422
    also have "\<dots> = emeasure M A"
hoelzl@59000
   423
      using ae by (intro emeasure_eq_AE) auto
hoelzl@59000
   424
    finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))) A"
hoelzl@59000
   425
      using emeasure_space_1 by (simp add: emeasure_density)
hoelzl@59000
   426
  qed simp
hoelzl@59000
   427
qed
hoelzl@59000
   428
hoelzl@58587
   429
lemma td_pmf_embed_pmf:
hoelzl@58587
   430
  "type_definition pmf embed_pmf {f::'a \<Rightarrow> real. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ereal (f x) \<partial>count_space UNIV) = 1}"
hoelzl@58587
   431
  unfolding type_definition_def
hoelzl@58587
   432
proof safe
hoelzl@58587
   433
  fix p :: "'a pmf"
hoelzl@58587
   434
  have "(\<integral>\<^sup>+ x. 1 \<partial>measure_pmf p) = 1"
hoelzl@58587
   435
    using measure_pmf.emeasure_space_1[of p] by simp
hoelzl@58587
   436
  then show *: "(\<integral>\<^sup>+ x. ereal (pmf p x) \<partial>count_space UNIV) = 1"
hoelzl@58587
   437
    by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const)
hoelzl@58587
   438
hoelzl@58587
   439
  show "embed_pmf (pmf p) = p"
hoelzl@58587
   440
    by (intro measure_pmf_inject[THEN iffD1])
hoelzl@58587
   441
       (simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def)
hoelzl@58587
   442
next
hoelzl@58587
   443
  fix f :: "'a \<Rightarrow> real" assume "\<forall>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
hoelzl@58587
   444
  then show "pmf (embed_pmf f) = f"
hoelzl@58587
   445
    by (auto intro!: pmf_embed_pmf)
hoelzl@58587
   446
qed (rule pmf_nonneg)
hoelzl@58587
   447
hoelzl@58587
   448
end
hoelzl@58587
   449
hoelzl@58587
   450
locale pmf_as_function
hoelzl@58587
   451
begin
hoelzl@58587
   452
hoelzl@58587
   453
setup_lifting td_pmf_embed_pmf
hoelzl@58587
   454
hoelzl@58730
   455
lemma set_pmf_transfer[transfer_rule]: 
hoelzl@58730
   456
  assumes "bi_total A"
hoelzl@58730
   457
  shows "rel_fun (pcr_pmf A) (rel_set A) (\<lambda>f. {x. f x \<noteq> 0}) set_pmf"  
hoelzl@58730
   458
  using `bi_total A`
hoelzl@58730
   459
  by (auto simp: pcr_pmf_def cr_pmf_def rel_fun_def rel_set_def bi_total_def Bex_def set_pmf_iff)
hoelzl@58730
   460
     metis+
hoelzl@58730
   461
hoelzl@59000
   462
end
hoelzl@59000
   463
hoelzl@59000
   464
context
hoelzl@59000
   465
begin
hoelzl@59000
   466
hoelzl@59000
   467
interpretation pmf_as_function .
hoelzl@59000
   468
hoelzl@59000
   469
lemma pmf_eqI: "(\<And>i. pmf M i = pmf N i) \<Longrightarrow> M = N"
hoelzl@59000
   470
  by transfer auto
hoelzl@59000
   471
hoelzl@59000
   472
lemma pmf_eq_iff: "M = N \<longleftrightarrow> (\<forall>i. pmf M i = pmf N i)"
hoelzl@59000
   473
  by (auto intro: pmf_eqI)
hoelzl@59000
   474
hoelzl@59000
   475
end
hoelzl@59000
   476
hoelzl@59000
   477
context
hoelzl@59000
   478
begin
hoelzl@59000
   479
hoelzl@59000
   480
interpretation pmf_as_function .
hoelzl@59000
   481
hoelzl@59000
   482
lift_definition bernoulli_pmf :: "real \<Rightarrow> bool pmf" is
hoelzl@59000
   483
  "\<lambda>p b. ((\<lambda>p. if b then p else 1 - p) \<circ> min 1 \<circ> max 0) p"
hoelzl@59000
   484
  by (auto simp: nn_integral_count_space_finite[where A="{False, True}"] UNIV_bool
hoelzl@59000
   485
           split: split_max split_min)
hoelzl@59000
   486
hoelzl@59000
   487
lemma pmf_bernoulli_True[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) True = p"
hoelzl@59000
   488
  by transfer simp
hoelzl@59000
   489
hoelzl@59000
   490
lemma pmf_bernoulli_False[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) False = 1 - p"
hoelzl@59000
   491
  by transfer simp
hoelzl@59000
   492
hoelzl@59000
   493
lemma set_pmf_bernoulli: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (bernoulli_pmf p) = UNIV"
hoelzl@59000
   494
  by (auto simp add: set_pmf_iff UNIV_bool)
hoelzl@59000
   495
hoelzl@59002
   496
lemma nn_integral_bernoulli_pmf[simp]: 
hoelzl@59002
   497
  assumes [simp]: "0 \<le> p" "p \<le> 1" "\<And>x. 0 \<le> f x"
hoelzl@59002
   498
  shows "(\<integral>\<^sup>+x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
hoelzl@59002
   499
  by (subst nn_integral_measure_pmf_support[of UNIV])
hoelzl@59002
   500
     (auto simp: UNIV_bool field_simps)
hoelzl@59002
   501
hoelzl@59002
   502
lemma integral_bernoulli_pmf[simp]: 
hoelzl@59002
   503
  assumes [simp]: "0 \<le> p" "p \<le> 1"
hoelzl@59002
   504
  shows "(\<integral>x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
hoelzl@59002
   505
  by (subst integral_measure_pmf[of UNIV]) (auto simp: UNIV_bool)
hoelzl@59002
   506
hoelzl@59000
   507
lift_definition geometric_pmf :: "nat pmf" is "\<lambda>n. 1 / 2^Suc n"
hoelzl@59000
   508
proof
hoelzl@59000
   509
  note geometric_sums[of "1 / 2"]
hoelzl@59000
   510
  note sums_mult[OF this, of "1 / 2"]
hoelzl@59000
   511
  from sums_suminf_ereal[OF this]
hoelzl@59000
   512
  show "(\<integral>\<^sup>+ x. ereal (1 / 2 ^ Suc x) \<partial>count_space UNIV) = 1"
hoelzl@59000
   513
    by (simp add: nn_integral_count_space_nat field_simps)
hoelzl@59000
   514
qed simp
hoelzl@59000
   515
hoelzl@59000
   516
lemma pmf_geometric[simp]: "pmf geometric_pmf n = 1 / 2^Suc n"
hoelzl@59000
   517
  by transfer rule
hoelzl@59000
   518
hoelzl@59002
   519
lemma set_pmf_geometric[simp]: "set_pmf geometric_pmf = UNIV"
hoelzl@59000
   520
  by (auto simp: set_pmf_iff)
hoelzl@59000
   521
hoelzl@59000
   522
context
hoelzl@59000
   523
  fixes M :: "'a multiset" assumes M_not_empty: "M \<noteq> {#}"
hoelzl@59000
   524
begin
hoelzl@59000
   525
hoelzl@59000
   526
lift_definition pmf_of_multiset :: "'a pmf" is "\<lambda>x. count M x / size M"
hoelzl@59000
   527
proof
hoelzl@59000
   528
  show "(\<integral>\<^sup>+ x. ereal (real (count M x) / real (size M)) \<partial>count_space UNIV) = 1"  
hoelzl@59000
   529
    using M_not_empty
hoelzl@59000
   530
    by (simp add: zero_less_divide_iff nn_integral_count_space nonempty_has_size
hoelzl@59000
   531
                  setsum_divide_distrib[symmetric])
hoelzl@59000
   532
       (auto simp: size_multiset_overloaded_eq intro!: setsum.cong)
hoelzl@59000
   533
qed simp
hoelzl@59000
   534
hoelzl@59000
   535
lemma pmf_of_multiset[simp]: "pmf pmf_of_multiset x = count M x / size M"
hoelzl@59000
   536
  by transfer rule
hoelzl@59000
   537
hoelzl@59000
   538
lemma set_pmf_of_multiset[simp]: "set_pmf pmf_of_multiset = set_of M"
hoelzl@59000
   539
  by (auto simp: set_pmf_iff)
hoelzl@59000
   540
hoelzl@59000
   541
end
hoelzl@59000
   542
hoelzl@59000
   543
context
hoelzl@59000
   544
  fixes S :: "'a set" assumes S_not_empty: "S \<noteq> {}" and S_finite: "finite S"
hoelzl@59000
   545
begin
hoelzl@59000
   546
hoelzl@59000
   547
lift_definition pmf_of_set :: "'a pmf" is "\<lambda>x. indicator S x / card S"
hoelzl@59000
   548
proof
hoelzl@59000
   549
  show "(\<integral>\<^sup>+ x. ereal (indicator S x / real (card S)) \<partial>count_space UNIV) = 1"  
hoelzl@59000
   550
    using S_not_empty S_finite by (subst nn_integral_count_space'[of S]) auto
hoelzl@59000
   551
qed simp
hoelzl@59000
   552
hoelzl@59000
   553
lemma pmf_of_set[simp]: "pmf pmf_of_set x = indicator S x / card S"
hoelzl@59000
   554
  by transfer rule
hoelzl@59000
   555
hoelzl@59000
   556
lemma set_pmf_of_set[simp]: "set_pmf pmf_of_set = S"
hoelzl@59000
   557
  using S_finite S_not_empty by (auto simp: set_pmf_iff)
hoelzl@59000
   558
hoelzl@59002
   559
lemma emeasure_pmf_of_set[simp]: "emeasure pmf_of_set S = 1"
hoelzl@59002
   560
  by (rule measure_pmf.emeasure_eq_1_AE) (auto simp: AE_measure_pmf_iff)
hoelzl@59002
   561
hoelzl@59000
   562
end
hoelzl@59000
   563
hoelzl@59000
   564
end
hoelzl@59000
   565
hoelzl@59000
   566
subsection {* Monad interpretation *}
hoelzl@59000
   567
hoelzl@59000
   568
lemma measurable_measure_pmf[measurable]:
hoelzl@59000
   569
  "(\<lambda>x. measure_pmf (M x)) \<in> measurable (count_space UNIV) (subprob_algebra (count_space UNIV))"
hoelzl@59000
   570
  by (auto simp: space_subprob_algebra intro!: prob_space_imp_subprob_space) unfold_locales
hoelzl@59000
   571
hoelzl@59000
   572
lemma bind_pmf_cong:
hoelzl@59000
   573
  assumes "\<And>x. A x \<in> space (subprob_algebra N)" "\<And>x. B x \<in> space (subprob_algebra N)"
hoelzl@59000
   574
  assumes "\<And>i. i \<in> set_pmf x \<Longrightarrow> A i = B i"
hoelzl@59000
   575
  shows "bind (measure_pmf x) A = bind (measure_pmf x) B"
hoelzl@59000
   576
proof (rule measure_eqI)
hoelzl@59000
   577
  show "sets (measure_pmf x \<guillemotright>= A) = sets (measure_pmf x \<guillemotright>= B)"
hoelzl@59048
   578
    using assms by (subst (1 2) sets_bind) (auto simp: space_subprob_algebra)
hoelzl@59000
   579
next
hoelzl@59000
   580
  fix X assume "X \<in> sets (measure_pmf x \<guillemotright>= A)"
hoelzl@59000
   581
  then have X: "X \<in> sets N"
hoelzl@59048
   582
    using assms by (subst (asm) sets_bind) (auto simp: space_subprob_algebra)
hoelzl@59000
   583
  show "emeasure (measure_pmf x \<guillemotright>= A) X = emeasure (measure_pmf x \<guillemotright>= B) X"
hoelzl@59000
   584
    using assms
hoelzl@59000
   585
    by (subst (1 2) emeasure_bind[where N=N, OF _ _ X])
hoelzl@59000
   586
       (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
hoelzl@59000
   587
qed
hoelzl@59000
   588
hoelzl@59000
   589
context
hoelzl@59000
   590
begin
hoelzl@59000
   591
hoelzl@59000
   592
interpretation pmf_as_measure .
hoelzl@59000
   593
hoelzl@59000
   594
lift_definition join_pmf :: "'a pmf pmf \<Rightarrow> 'a pmf" is "\<lambda>M. measure_pmf M \<guillemotright>= measure_pmf"
hoelzl@59000
   595
proof (intro conjI)
hoelzl@59000
   596
  fix M :: "'a pmf pmf"
hoelzl@59000
   597
hoelzl@59000
   598
  interpret bind: prob_space "measure_pmf M \<guillemotright>= measure_pmf"
hoelzl@59048
   599
    apply (intro measure_pmf.prob_space_bind[where S="count_space UNIV"] AE_I2)
hoelzl@59048
   600
    apply (auto intro!: subprob_space_measure_pmf simp: space_subprob_algebra)
hoelzl@59000
   601
    apply unfold_locales
hoelzl@59000
   602
    done
hoelzl@59000
   603
  show "prob_space (measure_pmf M \<guillemotright>= measure_pmf)"
hoelzl@59000
   604
    by intro_locales
hoelzl@59000
   605
  show "sets (measure_pmf M \<guillemotright>= measure_pmf) = UNIV"
hoelzl@59048
   606
    by (subst sets_bind) auto
hoelzl@59000
   607
  have "AE x in measure_pmf M \<guillemotright>= measure_pmf. emeasure (measure_pmf M \<guillemotright>= measure_pmf) {x} \<noteq> 0"
hoelzl@59048
   608
    by (auto simp: AE_bind[where B="count_space UNIV"] measure_pmf_in_subprob_algebra
hoelzl@59048
   609
                   emeasure_bind[where N="count_space UNIV"] AE_measure_pmf_iff nn_integral_0_iff_AE
hoelzl@59048
   610
                   measure_pmf.emeasure_eq_measure measure_le_0_iff set_pmf_iff pmf.rep_eq)
hoelzl@59000
   611
  then show "AE x in measure_pmf M \<guillemotright>= measure_pmf. measure (measure_pmf M \<guillemotright>= measure_pmf) {x} \<noteq> 0"
hoelzl@59000
   612
    unfolding bind.emeasure_eq_measure by simp
hoelzl@59000
   613
qed
hoelzl@59000
   614
hoelzl@59000
   615
lemma pmf_join: "pmf (join_pmf N) i = (\<integral>M. pmf M i \<partial>measure_pmf N)"
hoelzl@59000
   616
proof (transfer fixing: N i)
hoelzl@59000
   617
  have N: "subprob_space (measure_pmf N)"
hoelzl@59000
   618
    by (rule prob_space_imp_subprob_space) intro_locales
hoelzl@59000
   619
  show "measure (measure_pmf N \<guillemotright>= measure_pmf) {i} = integral\<^sup>L (measure_pmf N) (\<lambda>M. measure M {i})"
hoelzl@59000
   620
    using measurable_measure_pmf[of "\<lambda>x. x"]
hoelzl@59000
   621
    by (intro subprob_space.measure_bind[where N="count_space UNIV", OF N]) auto
hoelzl@59000
   622
qed (auto simp: Transfer.Rel_def rel_fun_def cr_pmf_def)
hoelzl@59000
   623
Andreas@59024
   624
lemma set_pmf_join_pmf: "set_pmf (join_pmf f) = (\<Union>p\<in>set_pmf f. set_pmf p)"
Andreas@59024
   625
apply(simp add: set_eq_iff set_pmf_iff pmf_join)
Andreas@59024
   626
apply(subst integral_nonneg_eq_0_iff_AE)
Andreas@59024
   627
apply(auto simp add: pmf_le_1 pmf_nonneg AE_measure_pmf_iff intro!: measure_pmf.integrable_const_bound[where B=1])
Andreas@59024
   628
done
Andreas@59024
   629
hoelzl@59000
   630
lift_definition return_pmf :: "'a \<Rightarrow> 'a pmf" is "return (count_space UNIV)"
hoelzl@59000
   631
  by (auto intro!: prob_space_return simp: AE_return measure_return)
hoelzl@59000
   632
hoelzl@59000
   633
lemma join_return_pmf: "join_pmf (return_pmf M) = M"
hoelzl@59000
   634
  by (simp add: integral_return pmf_eq_iff pmf_join return_pmf.rep_eq)
hoelzl@59000
   635
hoelzl@59000
   636
lemma map_return_pmf: "map_pmf f (return_pmf x) = return_pmf (f x)"
hoelzl@59000
   637
  by transfer (simp add: distr_return)
hoelzl@59000
   638
hoelzl@59052
   639
lemma map_pmf_const[simp]: "map_pmf (\<lambda>_. c) M = return_pmf c"
hoelzl@59052
   640
  by transfer (auto simp: prob_space.distr_const)
hoelzl@59052
   641
hoelzl@59002
   642
lemma set_return_pmf: "set_pmf (return_pmf x) = {x}"
hoelzl@59000
   643
  by transfer (auto simp add: measure_return split: split_indicator)
hoelzl@59000
   644
hoelzl@59000
   645
lemma pmf_return: "pmf (return_pmf x) y = indicator {y} x"
hoelzl@59000
   646
  by transfer (simp add: measure_return)
hoelzl@59000
   647
hoelzl@59002
   648
lemma nn_integral_return_pmf[simp]: "0 \<le> f x \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>return_pmf x) = f x"
hoelzl@59002
   649
  unfolding return_pmf.rep_eq by (intro nn_integral_return) auto
hoelzl@59002
   650
hoelzl@59002
   651
lemma emeasure_return_pmf[simp]: "emeasure (return_pmf x) X = indicator X x"
hoelzl@59002
   652
  unfolding return_pmf.rep_eq by (intro emeasure_return) auto
hoelzl@59002
   653
hoelzl@59000
   654
end
hoelzl@59000
   655
hoelzl@59000
   656
definition "bind_pmf M f = join_pmf (map_pmf f M)"
hoelzl@59000
   657
hoelzl@59000
   658
lemma (in pmf_as_measure) bind_transfer[transfer_rule]:
hoelzl@59000
   659
  "rel_fun pmf_as_measure.cr_pmf (rel_fun (rel_fun op = pmf_as_measure.cr_pmf) pmf_as_measure.cr_pmf) op \<guillemotright>= bind_pmf"
hoelzl@59000
   660
proof (auto simp: pmf_as_measure.cr_pmf_def rel_fun_def bind_pmf_def join_pmf.rep_eq map_pmf.rep_eq)
hoelzl@59000
   661
  fix M f and g :: "'a \<Rightarrow> 'b pmf" assume "\<forall>x. f x = measure_pmf (g x)"
hoelzl@59000
   662
  then have f: "f = (\<lambda>x. measure_pmf (g x))"
hoelzl@59000
   663
    by auto
hoelzl@59000
   664
  show "measure_pmf M \<guillemotright>= f = distr (measure_pmf M) (count_space UNIV) g \<guillemotright>= measure_pmf"
hoelzl@59000
   665
    unfolding f by (subst bind_distr[OF _ measurable_measure_pmf]) auto
hoelzl@59000
   666
qed
hoelzl@59000
   667
hoelzl@59000
   668
lemma pmf_bind: "pmf (bind_pmf N f) i = (\<integral>x. pmf (f x) i \<partial>measure_pmf N)"
hoelzl@59000
   669
  by (auto intro!: integral_distr simp: bind_pmf_def pmf_join map_pmf.rep_eq)
hoelzl@59000
   670
hoelzl@59000
   671
lemma bind_return_pmf: "bind_pmf (return_pmf x) f = f x"
hoelzl@59000
   672
  unfolding bind_pmf_def map_return_pmf join_return_pmf ..
hoelzl@59000
   673
hoelzl@59052
   674
lemma join_eq_bind_pmf: "join_pmf M = bind_pmf M id"
hoelzl@59052
   675
  by (simp add: bind_pmf_def)
hoelzl@59052
   676
hoelzl@59052
   677
lemma bind_pmf_const[simp]: "bind_pmf M (\<lambda>x. c) = c"
hoelzl@59052
   678
  unfolding bind_pmf_def map_pmf_const join_return_pmf ..
hoelzl@59052
   679
hoelzl@59002
   680
lemma set_bind_pmf: "set_pmf (bind_pmf M N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))"
hoelzl@59002
   681
  apply (simp add: set_eq_iff set_pmf_iff pmf_bind)
hoelzl@59002
   682
  apply (subst integral_nonneg_eq_0_iff_AE)
hoelzl@59002
   683
  apply (auto simp: pmf_nonneg pmf_le_1 AE_measure_pmf_iff
hoelzl@59002
   684
              intro!: measure_pmf.integrable_const_bound[where B=1])
hoelzl@59002
   685
  done
hoelzl@59002
   686
hoelzl@59002
   687
lemma measurable_pair_restrict_pmf2:
hoelzl@59002
   688
  assumes "countable A"
hoelzl@59002
   689
  assumes "\<And>y. y \<in> A \<Longrightarrow> (\<lambda>x. f (x, y)) \<in> measurable M L"
hoelzl@59002
   690
  shows "f \<in> measurable (M \<Otimes>\<^sub>M restrict_space (measure_pmf N) A) L"
hoelzl@59002
   691
  apply (subst measurable_cong_sets)
hoelzl@59002
   692
  apply (rule sets_pair_measure_cong sets_restrict_space_cong sets_measure_pmf_count_space refl)+
hoelzl@59002
   693
  apply (simp_all add: restrict_count_space)
hoelzl@59002
   694
  apply (subst split_eta[symmetric])
hoelzl@59002
   695
  unfolding measurable_split_conv
hoelzl@59002
   696
  apply (rule measurable_compose_countable'[OF _ measurable_snd `countable A`])
hoelzl@59002
   697
  apply (rule measurable_compose[OF measurable_fst])
hoelzl@59002
   698
  apply fact
hoelzl@59002
   699
  done
hoelzl@59002
   700
hoelzl@59002
   701
lemma measurable_pair_restrict_pmf1:
hoelzl@59002
   702
  assumes "countable A"
hoelzl@59002
   703
  assumes "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N L"
hoelzl@59002
   704
  shows "f \<in> measurable (restrict_space (measure_pmf M) A \<Otimes>\<^sub>M N) L"
hoelzl@59002
   705
  apply (subst measurable_cong_sets)
hoelzl@59002
   706
  apply (rule sets_pair_measure_cong sets_restrict_space_cong sets_measure_pmf_count_space refl)+
hoelzl@59002
   707
  apply (simp_all add: restrict_count_space)
hoelzl@59002
   708
  apply (subst split_eta[symmetric])
hoelzl@59002
   709
  unfolding measurable_split_conv
hoelzl@59002
   710
  apply (rule measurable_compose_countable'[OF _ measurable_fst `countable A`])
hoelzl@59002
   711
  apply (rule measurable_compose[OF measurable_snd])
hoelzl@59002
   712
  apply fact
hoelzl@59002
   713
  done
hoelzl@59002
   714
                                
hoelzl@59000
   715
lemma bind_commute_pmf: "bind_pmf A (\<lambda>x. bind_pmf B (C x)) = bind_pmf B (\<lambda>y. bind_pmf A (\<lambda>x. C x y))"
hoelzl@59000
   716
  unfolding pmf_eq_iff pmf_bind
hoelzl@59000
   717
proof
hoelzl@59000
   718
  fix i
hoelzl@59000
   719
  interpret B: prob_space "restrict_space B B"
hoelzl@59000
   720
    by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
hoelzl@59000
   721
       (auto simp: AE_measure_pmf_iff)
hoelzl@59000
   722
  interpret A: prob_space "restrict_space A A"
hoelzl@59000
   723
    by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
hoelzl@59000
   724
       (auto simp: AE_measure_pmf_iff)
hoelzl@59000
   725
hoelzl@59000
   726
  interpret AB: pair_prob_space "restrict_space A A" "restrict_space B B"
hoelzl@59000
   727
    by unfold_locales
hoelzl@59000
   728
hoelzl@59000
   729
  have "(\<integral> x. \<integral> y. pmf (C x y) i \<partial>B \<partial>A) = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>A)"
hoelzl@59000
   730
    by (rule integral_cong) (auto intro!: integral_pmf_restrict)
hoelzl@59000
   731
  also have "\<dots> = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>restrict_space A A)"
hoelzl@59002
   732
    by (intro integral_pmf_restrict B.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
hoelzl@59002
   733
              countable_set_pmf borel_measurable_count_space)
hoelzl@59000
   734
  also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>restrict_space B B)"
hoelzl@59002
   735
    by (rule AB.Fubini_integral[symmetric])
hoelzl@59002
   736
       (auto intro!: AB.integrable_const_bound[where B=1] measurable_pair_restrict_pmf2
Andreas@59023
   737
             simp: pmf_nonneg pmf_le_1 measurable_restrict_space1)
hoelzl@59000
   738
  also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>B)"
hoelzl@59002
   739
    by (intro integral_pmf_restrict[symmetric] A.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
hoelzl@59002
   740
              countable_set_pmf borel_measurable_count_space)
hoelzl@59000
   741
  also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>A \<partial>B)"
hoelzl@59000
   742
    by (rule integral_cong) (auto intro!: integral_pmf_restrict[symmetric])
hoelzl@59000
   743
  finally show "(\<integral> x. \<integral> y. pmf (C x y) i \<partial>B \<partial>A) = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>A \<partial>B)" .
hoelzl@59000
   744
qed
hoelzl@59000
   745
hoelzl@59000
   746
hoelzl@59000
   747
context
hoelzl@59000
   748
begin
hoelzl@59000
   749
hoelzl@59000
   750
interpretation pmf_as_measure .
hoelzl@59000
   751
hoelzl@59002
   752
lemma measure_pmf_bind: "measure_pmf (bind_pmf M f) = (measure_pmf M \<guillemotright>= (\<lambda>x. measure_pmf (f x)))"
hoelzl@59002
   753
  by transfer simp
hoelzl@59002
   754
hoelzl@59002
   755
lemma nn_integral_bind_pmf[simp]: "(\<integral>\<^sup>+x. f x \<partial>bind_pmf M N) = (\<integral>\<^sup>+x. \<integral>\<^sup>+y. f y \<partial>N x \<partial>M)"
hoelzl@59002
   756
  using measurable_measure_pmf[of N]
hoelzl@59002
   757
  unfolding measure_pmf_bind
hoelzl@59002
   758
  apply (subst (1 3) nn_integral_max_0[symmetric])
hoelzl@59002
   759
  apply (intro nn_integral_bind[where B="count_space UNIV"])
hoelzl@59002
   760
  apply auto
hoelzl@59002
   761
  done
hoelzl@59002
   762
hoelzl@59002
   763
lemma emeasure_bind_pmf[simp]: "emeasure (bind_pmf M N) X = (\<integral>\<^sup>+x. emeasure (N x) X \<partial>M)"
hoelzl@59002
   764
  using measurable_measure_pmf[of N]
hoelzl@59002
   765
  unfolding measure_pmf_bind
hoelzl@59002
   766
  by (subst emeasure_bind[where N="count_space UNIV"]) auto
hoelzl@59002
   767
hoelzl@59000
   768
lemma bind_return_pmf': "bind_pmf N return_pmf = N"
hoelzl@59000
   769
proof (transfer, clarify)
hoelzl@59000
   770
  fix N :: "'a measure" assume "sets N = UNIV" then show "N \<guillemotright>= return (count_space UNIV) = N"
hoelzl@59000
   771
    by (subst return_sets_cong[where N=N]) (simp_all add: bind_return')
hoelzl@59000
   772
qed
hoelzl@59000
   773
hoelzl@59000
   774
lemma bind_return_pmf'': "bind_pmf N (\<lambda>x. return_pmf (f x)) = map_pmf f N"
hoelzl@59000
   775
proof (transfer, clarify)
hoelzl@59000
   776
  fix N :: "'b measure" and f :: "'b \<Rightarrow> 'a" assume "prob_space N" "sets N = UNIV"
hoelzl@59000
   777
  then show "N \<guillemotright>= (\<lambda>x. return (count_space UNIV) (f x)) = distr N (count_space UNIV) f"
hoelzl@59000
   778
    by (subst bind_return_distr[symmetric])
hoelzl@59000
   779
       (auto simp: prob_space.not_empty measurable_def comp_def)
hoelzl@59000
   780
qed
hoelzl@59000
   781
hoelzl@59000
   782
lemma bind_assoc_pmf: "bind_pmf (bind_pmf A B) C = bind_pmf A (\<lambda>x. bind_pmf (B x) C)"
hoelzl@59000
   783
  by transfer
hoelzl@59000
   784
     (auto intro!: bind_assoc[where N="count_space UNIV" and R="count_space UNIV"]
hoelzl@59000
   785
           simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space)
hoelzl@59000
   786
hoelzl@59000
   787
end
hoelzl@59000
   788
hoelzl@59052
   789
lemma map_join_pmf: "map_pmf f (join_pmf AA) = join_pmf (map_pmf (map_pmf f) AA)"
hoelzl@59052
   790
  unfolding bind_pmf_def[symmetric]
hoelzl@59052
   791
  unfolding bind_return_pmf''[symmetric] join_eq_bind_pmf bind_assoc_pmf
hoelzl@59052
   792
  by (simp add: bind_return_pmf'')
hoelzl@59052
   793
hoelzl@59000
   794
definition "pair_pmf A B = bind_pmf A (\<lambda>x. bind_pmf B (\<lambda>y. return_pmf (x, y)))"
hoelzl@59000
   795
hoelzl@59000
   796
lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b"
hoelzl@59000
   797
  unfolding pair_pmf_def pmf_bind pmf_return
hoelzl@59000
   798
  apply (subst integral_measure_pmf[where A="{b}"])
hoelzl@59000
   799
  apply (auto simp: indicator_eq_0_iff)
hoelzl@59000
   800
  apply (subst integral_measure_pmf[where A="{a}"])
hoelzl@59000
   801
  apply (auto simp: indicator_eq_0_iff setsum_nonneg_eq_0_iff pmf_nonneg)
hoelzl@59000
   802
  done
hoelzl@59000
   803
hoelzl@59002
   804
lemma set_pair_pmf: "set_pmf (pair_pmf A B) = set_pmf A \<times> set_pmf B"
hoelzl@59002
   805
  unfolding pair_pmf_def set_bind_pmf set_return_pmf by auto
hoelzl@59002
   806
hoelzl@59048
   807
lemma measure_pmf_in_subprob_space[measurable (raw)]:
hoelzl@59048
   808
  "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
hoelzl@59048
   809
  by (simp add: space_subprob_algebra) intro_locales
hoelzl@59048
   810
hoelzl@59000
   811
lemma bind_pair_pmf:
hoelzl@59000
   812
  assumes M[measurable]: "M \<in> measurable (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) (subprob_algebra N)"
hoelzl@59000
   813
  shows "measure_pmf (pair_pmf A B) \<guillemotright>= M = (measure_pmf A \<guillemotright>= (\<lambda>x. measure_pmf B \<guillemotright>= (\<lambda>y. M (x, y))))"
hoelzl@59000
   814
    (is "?L = ?R")
hoelzl@59000
   815
proof (rule measure_eqI)
hoelzl@59000
   816
  have M'[measurable]: "M \<in> measurable (pair_pmf A B) (subprob_algebra N)"
hoelzl@59000
   817
    using M[THEN measurable_space] by (simp_all add: space_pair_measure)
hoelzl@59000
   818
hoelzl@59048
   819
  note measurable_bind[where N="count_space UNIV", measurable]
hoelzl@59048
   820
  note measure_pmf_in_subprob_space[simp]
hoelzl@59048
   821
hoelzl@59000
   822
  have sets_eq_N: "sets ?L = N"
hoelzl@59048
   823
    by (subst sets_bind[OF sets_kernel[OF M']]) auto
hoelzl@59000
   824
  show "sets ?L = sets ?R"
hoelzl@59048
   825
    using measurable_space[OF M]
hoelzl@59048
   826
    by (simp add: sets_eq_N space_pair_measure space_subprob_algebra)
hoelzl@59000
   827
  fix X assume "X \<in> sets ?L"
hoelzl@59000
   828
  then have X[measurable]: "X \<in> sets N"
hoelzl@59000
   829
    unfolding sets_eq_N .
hoelzl@59000
   830
  then show "emeasure ?L X = emeasure ?R X"
hoelzl@59000
   831
    apply (simp add: emeasure_bind[OF _ M' X])
hoelzl@59048
   832
    apply (simp add: nn_integral_bind[where B="count_space UNIV"] pair_pmf_def measure_pmf_bind[of A]
hoelzl@59048
   833
      nn_integral_measure_pmf_finite set_return_pmf emeasure_nonneg pmf_return one_ereal_def[symmetric])
hoelzl@59048
   834
    apply (subst emeasure_bind[OF _ _ X])
hoelzl@59000
   835
    apply measurable
hoelzl@59000
   836
    apply (subst emeasure_bind[OF _ _ X])
hoelzl@59000
   837
    apply measurable
hoelzl@59000
   838
    done
hoelzl@59000
   839
qed
hoelzl@59000
   840
hoelzl@59052
   841
lemma join_map_return_pmf: "join_pmf (map_pmf return_pmf A) = A"
hoelzl@59052
   842
  unfolding bind_pmf_def[symmetric] bind_return_pmf' ..
hoelzl@59052
   843
hoelzl@59052
   844
lemma map_fst_pair_pmf: "map_pmf fst (pair_pmf A B) = A"
hoelzl@59052
   845
  by (simp add: pair_pmf_def bind_return_pmf''[symmetric] bind_assoc_pmf bind_return_pmf bind_return_pmf')
hoelzl@59052
   846
hoelzl@59052
   847
lemma map_snd_pair_pmf: "map_pmf snd (pair_pmf A B) = B"
hoelzl@59052
   848
  by (simp add: pair_pmf_def bind_return_pmf''[symmetric] bind_assoc_pmf bind_return_pmf bind_return_pmf')
hoelzl@59052
   849
hoelzl@59053
   850
lemma nn_integral_pmf':
hoelzl@59053
   851
  "inj_on f A \<Longrightarrow> (\<integral>\<^sup>+x. pmf p (f x) \<partial>count_space A) = emeasure p (f ` A)"
hoelzl@59053
   852
  by (subst nn_integral_bij_count_space[where g=f and B="f`A"])
hoelzl@59053
   853
     (auto simp: bij_betw_def nn_integral_pmf)
hoelzl@59053
   854
hoelzl@59053
   855
lemma pmf_le_0_iff[simp]: "pmf M p \<le> 0 \<longleftrightarrow> pmf M p = 0"
hoelzl@59053
   856
  using pmf_nonneg[of M p] by simp
hoelzl@59053
   857
hoelzl@59053
   858
lemma min_pmf_0[simp]: "min (pmf M p) 0 = 0" "min 0 (pmf M p) = 0"
hoelzl@59053
   859
  using pmf_nonneg[of M p] by simp_all
hoelzl@59053
   860
hoelzl@59053
   861
lemma pmf_eq_0_set_pmf: "pmf M p = 0 \<longleftrightarrow> p \<notin> set_pmf M"
hoelzl@59053
   862
  unfolding set_pmf_iff by simp
hoelzl@59053
   863
hoelzl@59053
   864
lemma pmf_map_inj: "inj_on f (set_pmf M) \<Longrightarrow> x \<in> set_pmf M \<Longrightarrow> pmf (map_pmf f M) (f x) = pmf M x"
hoelzl@59053
   865
  by (auto simp: pmf.rep_eq map_pmf.rep_eq measure_distr AE_measure_pmf_iff inj_onD
hoelzl@59053
   866
           intro!: measure_pmf.finite_measure_eq_AE)
hoelzl@59053
   867
Andreas@59023
   868
inductive rel_pmf :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf \<Rightarrow> bool"
Andreas@59023
   869
for R p q
Andreas@59023
   870
where
Andreas@59023
   871
  "\<lbrakk> \<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y; 
Andreas@59023
   872
     map_pmf fst pq = p; map_pmf snd pq = q \<rbrakk>
Andreas@59023
   873
  \<Longrightarrow> rel_pmf R p q"
hoelzl@58587
   874
Andreas@59023
   875
bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: rel_pmf
hoelzl@58587
   876
proof -
hoelzl@58587
   877
  show "map_pmf id = id" by (rule map_pmf_id)
hoelzl@58587
   878
  show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose) 
hoelzl@58587
   879
  show "\<And>f g::'a \<Rightarrow> 'b. \<And>p. (\<And>x. x \<in> set_pmf p \<Longrightarrow> f x = g x) \<Longrightarrow> map_pmf f p = map_pmf g p"
Andreas@59023
   880
    by (intro map_pmf_cong refl)
hoelzl@58587
   881
hoelzl@58587
   882
  show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
hoelzl@58587
   883
    by (rule pmf_set_map)
hoelzl@58587
   884
hoelzl@58587
   885
  { fix p :: "'s pmf"
hoelzl@58587
   886
    have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq"
hoelzl@58587
   887
      by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"])
hoelzl@59053
   888
         (auto intro: countable_set_pmf)
hoelzl@58587
   889
    also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq"
hoelzl@58587
   890
      by (metis Field_natLeq card_of_least natLeq_Well_order)
hoelzl@58587
   891
    finally show "(card_of (set_pmf p), natLeq) \<in> ordLeq" . }
hoelzl@58587
   892
Andreas@59023
   893
  show "\<And>R. rel_pmf R =
Andreas@59023
   894
         (BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf fst))\<inverse>\<inverse> OO
Andreas@59023
   895
         BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf snd)"
Andreas@59023
   896
     by (auto simp add: fun_eq_iff BNF_Def.Grp_def OO_def rel_pmf.simps)
Andreas@59023
   897
Andreas@59023
   898
  { fix p :: "'a pmf" and f :: "'a \<Rightarrow> 'b" and g x
Andreas@59023
   899
    assume p: "\<And>z. z \<in> set_pmf p \<Longrightarrow> f z = g z"
Andreas@59023
   900
      and x: "x \<in> set_pmf p"
Andreas@59023
   901
    thus "f x = g x" by simp }
Andreas@59023
   902
Andreas@59023
   903
  fix R :: "'a => 'b \<Rightarrow> bool" and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
Andreas@59023
   904
  { fix p q r
Andreas@59023
   905
    assume pq: "rel_pmf R p q"
Andreas@59023
   906
      and qr:"rel_pmf S q r"
Andreas@59023
   907
    from pq obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
Andreas@59023
   908
      and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto
Andreas@59023
   909
    from qr obtain qr where qr: "\<And>y z. (y, z) \<in> set_pmf qr \<Longrightarrow> S y z"
Andreas@59023
   910
      and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto
Andreas@59023
   911
hoelzl@59053
   912
    note pmf_nonneg[intro, simp]
Andreas@59023
   913
hoelzl@59053
   914
    def A \<equiv> "\<lambda>y. {x. (x, y) \<in> set_pmf pq}"
hoelzl@59053
   915
    then have "\<And>y. A y \<subseteq> set_pmf p" by (auto simp add: p set_map_pmf intro: rev_image_eqI)
hoelzl@59053
   916
    then have [simp]: "\<And>y. countable (A y)" by (rule countable_subset) simp
hoelzl@59053
   917
    have A: "\<And>x y. (x, y) \<in> set_pmf pq \<longleftrightarrow> x \<in> A y"
hoelzl@59053
   918
      by (simp add: A_def)
Andreas@59023
   919
hoelzl@59053
   920
    let ?P = "\<lambda>y. to_nat_on (A y)"
hoelzl@59053
   921
    def pp \<equiv> "map_pmf (\<lambda>(x, y). (y, ?P y x)) pq"
hoelzl@59053
   922
    let ?pp = "\<lambda>y x. pmf pp (y, x)"
hoelzl@59053
   923
    { fix x y have "x \<in> A y \<Longrightarrow> pmf pp (y, ?P y x) = pmf pq (x, y)"
hoelzl@59053
   924
        unfolding pp_def
hoelzl@59053
   925
        by (intro pmf_map_inj[of "\<lambda>(x, y). (y, ?P y x)" pq "(x, y)", simplified])
hoelzl@59053
   926
           (auto simp: inj_on_def A) }
hoelzl@59053
   927
    note pmf_pp = this
Andreas@59023
   928
hoelzl@59053
   929
    def B \<equiv> "\<lambda>y. {z. (y, z) \<in> set_pmf qr}"
hoelzl@59053
   930
    then have "\<And>y. B y \<subseteq> set_pmf r" by (auto simp add: r set_map_pmf intro: rev_image_eqI)
hoelzl@59053
   931
    then have [simp]: "\<And>y. countable (B y)" by (rule countable_subset) simp
hoelzl@59053
   932
    have B: "\<And>y z. (y, z) \<in> set_pmf qr \<longleftrightarrow> z \<in> B y"
hoelzl@59053
   933
      by (simp add: B_def)
Andreas@59023
   934
hoelzl@59053
   935
    let ?R = "\<lambda>y. to_nat_on (B y)"
hoelzl@59053
   936
    def rr \<equiv> "map_pmf (\<lambda>(y, z). (y, ?R y z)) qr"
hoelzl@59053
   937
    let ?rr = "\<lambda>y z. pmf rr (y, z)"
hoelzl@59053
   938
    { fix y z have "z \<in> B y \<Longrightarrow> pmf rr (y, ?R y z) = pmf qr (y, z)"
hoelzl@59053
   939
        unfolding rr_def
hoelzl@59053
   940
        by (intro pmf_map_inj[of "\<lambda>(y, z). (y, ?R y z)" qr "(y, z)", simplified])
hoelzl@59053
   941
           (auto simp: inj_on_def B) }
hoelzl@59053
   942
    note pmf_rr = this
Andreas@59023
   943
hoelzl@59053
   944
    have eq: "\<And>y. (\<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV) = (\<integral>\<^sup>+ z. ?rr y z \<partial>count_space UNIV)"
Andreas@59023
   945
    proof -
Andreas@59023
   946
      fix y
hoelzl@59053
   947
      have "(\<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV) = pmf q y"
hoelzl@59053
   948
        by (simp add: nn_integral_pmf' inj_on_def pp_def q)
hoelzl@59053
   949
           (auto simp add: ereal_pmf_map intro!: arg_cong2[where f=emeasure])
hoelzl@59053
   950
      also have "\<dots> = (\<integral>\<^sup>+ x. ?rr y x \<partial>count_space UNIV)"
hoelzl@59053
   951
        by (simp add: nn_integral_pmf' inj_on_def rr_def q')
hoelzl@59053
   952
           (auto simp add: ereal_pmf_map intro!: arg_cong2[where f=emeasure])
Andreas@59023
   953
      finally show "?thesis y" .
Andreas@59023
   954
    qed
hoelzl@58587
   955
Andreas@59023
   956
    def assign_aux \<equiv> "\<lambda>y remainder start weight z.
Andreas@59023
   957
       if z < start then 0
Andreas@59023
   958
       else if z = start then min weight remainder
hoelzl@59053
   959
       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"
Andreas@59023
   960
    hence assign_aux_alt_def: "\<And>y remainder start weight z. assign_aux y remainder start weight z = 
Andreas@59023
   961
       (if z < start then 0
Andreas@59023
   962
        else if z = start then min weight remainder
hoelzl@59053
   963
        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)"
Andreas@59023
   964
       by simp
Andreas@59023
   965
    { fix y and remainder :: real and start and weight :: real
Andreas@59023
   966
      assume weight_nonneg: "0 \<le> weight"
Andreas@59023
   967
      let ?assign_aux = "assign_aux y remainder start weight"
Andreas@59023
   968
      { fix z
Andreas@59023
   969
        have "setsum ?assign_aux {..<z} =
hoelzl@59053
   970
           (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)"
Andreas@59023
   971
        proof(induction z)
Andreas@59023
   972
          case (Suc z) show ?case
hoelzl@59053
   973
            by (auto simp add: Suc.IH assign_aux_alt_def[where z=z] not_less)
hoelzl@59053
   974
               (metis add.commute add.left_commute add_increasing pmf_nonneg)
Andreas@59023
   975
        qed(auto simp add: assign_aux_def) }
Andreas@59023
   976
      note setsum_start_assign_aux = this
Andreas@59023
   977
      moreover {
Andreas@59023
   978
        assume remainder_nonneg: "0 \<le> remainder"
Andreas@59023
   979
        have [simp]: "\<And>z. 0 \<le> ?assign_aux z"
Andreas@59023
   980
          by(simp add: assign_aux_def weight_nonneg remainder_nonneg)
hoelzl@59053
   981
        moreover have "\<And>z. \<lbrakk> ?rr y z = 0; remainder \<le> ?rr y start \<rbrakk> \<Longrightarrow> ?assign_aux z = 0"
Andreas@59023
   982
          using remainder_nonneg weight_nonneg
Andreas@59023
   983
          by(auto simp add: assign_aux_def min_def)
Andreas@59023
   984
        moreover have "(\<integral>\<^sup>+ z. ?assign_aux z \<partial>count_space UNIV) = 
hoelzl@59053
   985
          min weight (\<integral>\<^sup>+ z. (if z < start then 0 else if z = start then remainder else ?rr y z) \<partial>count_space UNIV)"
Andreas@59023
   986
          (is "?lhs = ?rhs" is "_ = min _ (\<integral>\<^sup>+ y. ?f y \<partial>_)")
Andreas@59023
   987
        proof -
Andreas@59023
   988
          have "?lhs = (SUP n. \<Sum>z<n. ereal (?assign_aux z))"
Andreas@59023
   989
            by(simp add: nn_integral_count_space_nat suminf_ereal_eq_SUP)
Andreas@59023
   990
          also have "\<dots> = (SUP n. min weight (\<Sum>z<n. ?f z))"
Andreas@59023
   991
          proof(rule arg_cong2[where f=SUPREMUM] ext refl)+
Andreas@59023
   992
            fix n
Andreas@59023
   993
            have "(\<Sum>z<n. ereal (?assign_aux z)) = min weight ((if n > start then remainder else 0) + setsum ?f {Suc start..<n})"
Andreas@59023
   994
              using weight_nonneg remainder_nonneg by(simp add: setsum_start_assign_aux min_def)
Andreas@59023
   995
            also have "\<dots> = min weight (setsum ?f {start..<n})"
Andreas@59023
   996
              by(simp add: setsum_head_upt_Suc)
Andreas@59023
   997
            also have "\<dots> = min weight (setsum ?f {..<n})"
Andreas@59023
   998
              by(intro arg_cong2[where f=min] setsum.mono_neutral_left) auto
Andreas@59023
   999
            finally show "(\<Sum>z<n. ereal (?assign_aux z)) = \<dots>" .
Andreas@59023
  1000
          qed
Andreas@59023
  1001
          also have "\<dots> = min weight (SUP n. setsum ?f {..<n})"
Andreas@59023
  1002
            unfolding inf_min[symmetric] by(subst inf_SUP) simp
Andreas@59023
  1003
          also have "\<dots> = ?rhs"
Andreas@59023
  1004
            by(simp add: nn_integral_count_space_nat suminf_ereal_eq_SUP remainder_nonneg)
Andreas@59023
  1005
          finally show ?thesis .
Andreas@59023
  1006
        qed
Andreas@59023
  1007
        moreover note calculation }
Andreas@59023
  1008
      moreover note calculation }
Andreas@59023
  1009
    note setsum_start_assign_aux = this(1)
Andreas@59023
  1010
      and assign_aux_nonneg [simp] = this(2)
Andreas@59023
  1011
      and assign_aux_eq_0_outside = this(3)
Andreas@59023
  1012
      and nn_integral_assign_aux = this(4)
Andreas@59023
  1013
    { fix y and remainder :: real and start target
hoelzl@59053
  1014
      have "setsum (?rr y) {Suc start..<target} \<ge> 0" by (simp add: setsum_nonneg)
Andreas@59023
  1015
      moreover assume "0 \<le> remainder"
Andreas@59023
  1016
      ultimately have "assign_aux y remainder start 0 target = 0"
Andreas@59023
  1017
        by(auto simp add: assign_aux_def min_def) }
Andreas@59023
  1018
    note assign_aux_weight_0 [simp] = this
Andreas@59023
  1019
hoelzl@59053
  1020
    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"
Andreas@59023
  1021
    have find_start_eq_Some_above:
hoelzl@59053
  1022
      "\<And>y weight n. find_start y weight = Some n \<Longrightarrow> weight \<le> setsum (?rr y) {..n}"
Andreas@59023
  1023
      by(drule sym)(auto simp add: find_start_def split: split_if_asm intro: LeastI)
Andreas@59023
  1024
    { fix y weight n
Andreas@59023
  1025
      assume find_start: "find_start y weight = Some n"
Andreas@59023
  1026
      and weight: "0 \<le> weight"
hoelzl@59053
  1027
      have "setsum (?rr y) {..n} \<le> ?rr y n + weight"
Andreas@59023
  1028
      proof(rule ccontr)
Andreas@59023
  1029
        assume "\<not> ?thesis"
hoelzl@59053
  1030
        hence "?rr y n + weight < setsum (?rr y) {..n}" by simp
Andreas@59023
  1031
        moreover with weight obtain n' where "n = Suc n'" by(cases n) auto
hoelzl@59053
  1032
        ultimately have "weight \<le> setsum (?rr y) {..n'}" by simp
hoelzl@59053
  1033
        hence "(LEAST n. weight \<le> setsum (?rr y) {..n}) \<le> n'" by(rule Least_le)
hoelzl@59053
  1034
        moreover from find_start have "n = (LEAST n. weight \<le> setsum (?rr y) {..n})"
Andreas@59023
  1035
          by(auto simp add: find_start_def split: split_if_asm)
Andreas@59023
  1036
        ultimately show False using \<open>n = Suc n'\<close> by auto
Andreas@59023
  1037
      qed }
Andreas@59023
  1038
    note find_start_eq_Some_least = this
Andreas@59023
  1039
    have find_start_0 [simp]: "\<And>y. find_start y 0 = Some 0"
Andreas@59023
  1040
      by(auto simp add: find_start_def intro!: exI[where x=0])
Andreas@59023
  1041
    { fix y and weight :: real
hoelzl@59053
  1042
      assume "weight < \<integral>\<^sup>+ z. ?rr y z \<partial>count_space UNIV"
hoelzl@59053
  1043
      also have "(\<integral>\<^sup>+ z. ?rr y z \<partial>count_space UNIV) = (SUP n. \<Sum>z<n. ereal (?rr y z))"
Andreas@59023
  1044
        by(simp add: nn_integral_count_space_nat suminf_ereal_eq_SUP)
hoelzl@59053
  1045
      finally obtain n where "weight < (\<Sum>z<n. ?rr y z)" by(auto simp add: less_SUP_iff)
Andreas@59023
  1046
      hence "weight \<in> dom (find_start y)"
hoelzl@59053
  1047
        by(auto simp add: find_start_def)(meson atMost_iff finite_atMost lessThan_iff less_imp_le order_trans pmf_nonneg setsum_mono3 subsetI) }
Andreas@59023
  1048
    note in_dom_find_startI = this
Andreas@59023
  1049
    { fix y and w w' :: real and m
hoelzl@59053
  1050
      let ?m' = "LEAST m. w' \<le> setsum (?rr y) {..m}"
Andreas@59023
  1051
      assume "w' \<le> w"
Andreas@59023
  1052
      also  assume "find_start y w = Some m"
hoelzl@59053
  1053
      hence "w \<le> setsum (?rr y) {..m}" by(rule find_start_eq_Some_above)
Andreas@59023
  1054
      finally have "find_start y w' = Some ?m'" by(auto simp add: find_start_def)
hoelzl@59053
  1055
      moreover from \<open>w' \<le> setsum (?rr y) {..m}\<close> have "?m' \<le> m" by(rule Least_le)
Andreas@59023
  1056
      ultimately have "\<exists>m'. find_start y w' = Some m' \<and> m' \<le> m" by blast }
Andreas@59023
  1057
    note find_start_mono = this[rotated]
Andreas@59023
  1058
hoelzl@59053
  1059
    def assign \<equiv> "\<lambda>y x z. let used = setsum (?pp y) {..<x}
Andreas@59023
  1060
      in case find_start y used of None \<Rightarrow> 0
hoelzl@59053
  1061
         | Some start \<Rightarrow> assign_aux y (setsum (?rr y) {..start} - used) start (?pp y x) z"
Andreas@59023
  1062
    hence assign_alt_def: "\<And>y x z. assign y x z = 
hoelzl@59053
  1063
      (let used = setsum (?pp y) {..<x}
Andreas@59023
  1064
       in case find_start y used of None \<Rightarrow> 0
hoelzl@59053
  1065
          | Some start \<Rightarrow> assign_aux y (setsum (?rr y) {..start} - used) start (?pp y x) z)"
Andreas@59023
  1066
      by simp
Andreas@59023
  1067
    have assign_nonneg [simp]: "\<And>y x z. 0 \<le> assign y x z"
hoelzl@59053
  1068
      by(simp add: assign_def diff_le_iff find_start_eq_Some_above Let_def split: option.split)
hoelzl@59053
  1069
    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"
hoelzl@59053
  1070
      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 Let_def split: option.split)
Andreas@59023
  1071
Andreas@59023
  1072
    { fix y x z
Andreas@59023
  1073
      have "(\<Sum>n<Suc x. assign y n z) =
hoelzl@59053
  1074
            (case find_start y (setsum (?pp y) {..<x}) of None \<Rightarrow> ?rr y z
hoelzl@59053
  1075
             | Some m \<Rightarrow> if z < m then ?rr y z 
hoelzl@59053
  1076
                         else min (?rr y z) (max 0 (setsum (?pp y) {..<x} + ?pp y x - setsum (?rr y) {..<z})))"
Andreas@59023
  1077
        (is "?lhs x = ?rhs x")
Andreas@59023
  1078
      proof(induction x)
Andreas@59023
  1079
        case 0 thus ?case 
Andreas@59023
  1080
          by(auto simp add: assign_def assign_aux_def setsum_head_upt_Suc atLeast0LessThan[symmetric] not_less field_simps max_def)
Andreas@59023
  1081
      next
Andreas@59023
  1082
        case (Suc x)
Andreas@59023
  1083
        have "?lhs (Suc x) = ?lhs x + assign y (Suc x) z" by simp
Andreas@59023
  1084
        also have "?lhs x = ?rhs x" by(rule Suc.IH)
Andreas@59023
  1085
        also have "?rhs x + assign y (Suc x) z = ?rhs (Suc x)"
hoelzl@59053
  1086
        proof(cases "find_start y (setsum (?pp y) {..<Suc x})")
Andreas@59023
  1087
          case None
Andreas@59023
  1088
          thus ?thesis
Andreas@59023
  1089
            by(auto split: option.split simp add: assign_def min_def max_def diff_le_iff setsum_nonneg not_le field_simps)
Andreas@59023
  1090
              (metis add.commute add_increasing find_start_def lessThan_Suc_atMost less_imp_le option.distinct(1) setsum_lessThan_Suc)+
hoelzl@59053
  1091
        next 
Andreas@59023
  1092
          case (Some m)
hoelzl@59053
  1093
          have [simp]: "setsum (?rr y) {..m} = ?rr y m + setsum (?rr y) {..<m}"
Andreas@59023
  1094
            by(simp add: ivl_disj_un(2)[symmetric])
hoelzl@59053
  1095
          from Some obtain m' where m': "find_start y (setsum (?pp y) {..<x}) = Some m'" "m' \<le> m"
hoelzl@59053
  1096
            by(auto dest: find_start_mono[where w'2="setsum (?pp y) {..<x}"])
Andreas@59023
  1097
          moreover {
Andreas@59023
  1098
            assume "z < m"
hoelzl@59053
  1099
            then have "setsum (?rr y) {..z} \<le> setsum (?rr y) {..<m}"
Andreas@59023
  1100
              by(auto intro: setsum_mono3)
hoelzl@59053
  1101
            also have "\<dots> \<le> setsum (?pp y) {..<Suc x}" using find_start_eq_Some_least[OF Some]
Andreas@59023
  1102
              by(simp add: ivl_disj_un(2)[symmetric] setsum_nonneg)
hoelzl@59053
  1103
            finally have "?rr y z \<le> max 0 (setsum (?pp y) {..<x} + ?pp y x - setsum (?rr y) {..<z})"
hoelzl@59053
  1104
              by(auto simp add: ivl_disj_un(2)[symmetric] max_def diff_le_iff simp del: pmf_le_0_iff)
Andreas@59023
  1105
          } moreover {
Andreas@59023
  1106
            assume "m \<le> z"
hoelzl@59053
  1107
            have "setsum (?pp y) {..<Suc x} \<le> setsum (?rr y) {..m}"
Andreas@59023
  1108
              using Some by(rule find_start_eq_Some_above)
hoelzl@59053
  1109
            also have "\<dots> \<le> setsum (?rr y) {..<Suc z}" using \<open>m \<le> z\<close> by(intro setsum_mono3) auto
hoelzl@59053
  1110
            finally have "max 0 (setsum (?pp y) {..<x} + ?pp y x - setsum (?rr y) {..<z}) \<le> ?rr y z" by simp
hoelzl@59053
  1111
            moreover have "z \<noteq> m \<Longrightarrow> setsum (?rr y) {..m} + setsum (?rr y) {Suc m..<z} = setsum (?rr y) {..<z}"
Andreas@59023
  1112
              using \<open>m \<le> z\<close>
Andreas@59023
  1113
              by(subst ivl_disj_un(8)[where l="Suc m", symmetric])
Andreas@59023
  1114
                (simp_all add: setsum_Un ivl_disj_un(2)[symmetric] setsum.neutral)
Andreas@59023
  1115
            moreover note calculation
Andreas@59023
  1116
          } moreover {
Andreas@59023
  1117
            assume "m < z"
hoelzl@59053
  1118
            have "setsum (?pp y) {..<Suc x} \<le> setsum (?rr y) {..m}"
Andreas@59023
  1119
              using Some by(rule find_start_eq_Some_above)
hoelzl@59053
  1120
            also have "\<dots> \<le> setsum (?rr y) {..<z}" using \<open>m < z\<close> by(intro setsum_mono3) auto
hoelzl@59053
  1121
            finally have "max 0 (setsum (?pp y) {..<Suc x} - setsum (?rr y) {..<z}) = 0" by simp }
hoelzl@59053
  1122
          moreover have "setsum (?pp y) {..<Suc x} \<ge> setsum (?rr y) {..<m}"
Andreas@59023
  1123
            using find_start_eq_Some_least[OF Some]
Andreas@59023
  1124
            by(simp add: setsum_nonneg ivl_disj_un(2)[symmetric])
hoelzl@59053
  1125
          moreover hence "setsum (?pp y) {..<Suc (Suc x)} \<ge> setsum (?rr y) {..<m}"
Andreas@59023
  1126
            by(fastforce intro: order_trans)
Andreas@59023
  1127
          ultimately show ?thesis using Some
Andreas@59023
  1128
            by(auto simp add: assign_def assign_aux_def Let_def field_simps max_def)
Andreas@59023
  1129
        qed
Andreas@59023
  1130
        finally show ?case .
Andreas@59023
  1131
      qed }
Andreas@59023
  1132
    note setsum_assign = this
hoelzl@58587
  1133
hoelzl@59053
  1134
    have nn_integral_assign1: "\<And>y z. (\<integral>\<^sup>+ x. assign y x z \<partial>count_space UNIV) = ?rr y z"
Andreas@59023
  1135
    proof -
Andreas@59023
  1136
      fix y z
Andreas@59023
  1137
      have "(\<integral>\<^sup>+ x. assign y x z \<partial>count_space UNIV) = (SUP n. ereal (\<Sum>x<n. assign y x z))"
Andreas@59023
  1138
        by(simp add: nn_integral_count_space_nat suminf_ereal_eq_SUP)
hoelzl@59053
  1139
      also have "\<dots> = ?rr y z"
Andreas@59023
  1140
      proof(rule antisym)
hoelzl@59053
  1141
        show "(SUP n. ereal (\<Sum>x<n. assign y x z)) \<le> ?rr y z"
Andreas@59023
  1142
        proof(rule SUP_least)
Andreas@59023
  1143
          fix n
hoelzl@59053
  1144
          show "ereal (\<Sum>x<n. (assign y x z)) \<le> ?rr y z"
Andreas@59023
  1145
            using setsum_assign[of y z "n - 1"]
Andreas@59023
  1146
            by(cases n)(simp_all split: option.split)
Andreas@59023
  1147
        qed
hoelzl@59053
  1148
        show "?rr y z \<le> (SUP n. ereal (\<Sum>x<n. assign y x z))"
hoelzl@59053
  1149
        proof(cases "setsum (?rr y) {..z} < \<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV")
Andreas@59023
  1150
          case True
hoelzl@59053
  1151
          then obtain n where "setsum (?rr y) {..z} < setsum (?pp y) {..<n}"
Andreas@59023
  1152
            by(auto simp add: nn_integral_count_space_nat suminf_ereal_eq_SUP less_SUP_iff)
hoelzl@59053
  1153
          moreover have "\<And>k. k < z \<Longrightarrow> setsum (?rr y) {..k} \<le> setsum (?rr y) {..<z}"
Andreas@59023
  1154
            by(auto intro: setsum_mono3)
hoelzl@59053
  1155
          ultimately have "?rr y z \<le> (\<Sum>x<Suc n. assign y x z)"
Andreas@59023
  1156
            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)
Andreas@59023
  1157
          also have "\<dots> \<le> (SUP n. ereal (\<Sum>x<n. assign y x z))" 
Andreas@59023
  1158
            by(rule SUP_upper) simp
Andreas@59023
  1159
          finally show ?thesis by simp
Andreas@59023
  1160
        next
Andreas@59023
  1161
          case False
hoelzl@59053
  1162
          have "setsum (?rr y) {..z} = \<integral>\<^sup>+ z. ?rr y z \<partial>count_space {..z}"
Andreas@59023
  1163
            by(simp add: nn_integral_count_space_finite max_def)
hoelzl@59053
  1164
          also have "\<dots> \<le> \<integral>\<^sup>+ z. ?rr y z \<partial>count_space UNIV"
Andreas@59023
  1165
            by(auto simp add: nn_integral_count_space_indicator indicator_def intro: nn_integral_mono)
hoelzl@59053
  1166
          also have "\<dots> = \<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV" by(simp add: eq)
hoelzl@59053
  1167
          finally have *: "setsum (?rr y) {..z} = \<dots>" using False by simp
hoelzl@59053
  1168
          also have "\<dots> = (SUP n. ereal (\<Sum>x<n. ?pp y x))"
Andreas@59023
  1169
            by(simp add: nn_integral_count_space_nat suminf_ereal_eq_SUP)
hoelzl@59053
  1170
          also have "\<dots> \<le> (SUP n. ereal (\<Sum>x<n. assign y x z)) + setsum (?rr y) {..<z}"
Andreas@59023
  1171
          proof(rule SUP_least)
Andreas@59023
  1172
            fix n
hoelzl@59053
  1173
            have "setsum (?pp y) {..<n} = \<integral>\<^sup>+ x. ?pp y x \<partial>count_space {..<n}"
Andreas@59023
  1174
              by(simp add: nn_integral_count_space_finite max_def)
hoelzl@59053
  1175
            also have "\<dots> \<le> \<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV"
Andreas@59023
  1176
              by(auto simp add: nn_integral_count_space_indicator indicator_def intro: nn_integral_mono)
hoelzl@59053
  1177
            also have "\<dots> = setsum (?rr y) {..z}" using * by simp
hoelzl@59053
  1178
            finally obtain k where k: "find_start y (setsum (?pp y) {..<n}) = Some k"
Andreas@59023
  1179
              by(fastforce simp add: find_start_def)
hoelzl@59053
  1180
            with \<open>ereal (setsum (?pp y) {..<n}) \<le> setsum (?rr y) {..z}\<close>
Andreas@59023
  1181
            have "k \<le> z" by(auto simp add: find_start_def split: split_if_asm intro: Least_le)
hoelzl@59053
  1182
            then have "setsum (?pp y) {..<n} - setsum (?rr y) {..<z} \<le> ereal (\<Sum>x<Suc n. assign y x z)"
hoelzl@59053
  1183
              using \<open>ereal (setsum (?pp y) {..<n}) \<le> setsum (?rr y) {..z}\<close>
hoelzl@59053
  1184
              apply (subst setsum_assign)
hoelzl@59053
  1185
              apply (auto simp add: field_simps max_def k ivl_disj_un(2)[symmetric])
hoelzl@59053
  1186
              apply (meson add_increasing le_cases pmf_nonneg)
hoelzl@59053
  1187
              done
Andreas@59023
  1188
            also have "\<dots> \<le> (SUP n. ereal (\<Sum>x<n. assign y x z))"
Andreas@59023
  1189
              by(rule SUP_upper) simp
hoelzl@59053
  1190
            finally show "ereal (\<Sum>x<n. ?pp y x) \<le> \<dots> + setsum (?rr y) {..<z}" 
Andreas@59023
  1191
              by(simp add: ereal_minus(1)[symmetric] ereal_minus_le del: ereal_minus(1))
Andreas@59023
  1192
          qed
Andreas@59023
  1193
          finally show ?thesis
Andreas@59023
  1194
            by(simp add: ivl_disj_un(2)[symmetric] plus_ereal.simps(1)[symmetric] ereal_add_le_add_iff2 del: plus_ereal.simps(1))
Andreas@59023
  1195
        qed
Andreas@59023
  1196
      qed
Andreas@59023
  1197
      finally show "?thesis y z" .
Andreas@59023
  1198
    qed
Andreas@59023
  1199
Andreas@59023
  1200
    { fix y x
hoelzl@59053
  1201
      have "(\<integral>\<^sup>+ z. assign y x z \<partial>count_space UNIV) = ?pp y x"
hoelzl@59053
  1202
      proof(cases "setsum (?pp y) {..<x} = \<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV")
Andreas@59023
  1203
        case False
hoelzl@59053
  1204
        let ?used = "setsum (?pp y) {..<x}"
hoelzl@59053
  1205
        have "?used = \<integral>\<^sup>+ x. ?pp y x \<partial>count_space {..<x}"
Andreas@59023
  1206
          by(simp add: nn_integral_count_space_finite max_def)
hoelzl@59053
  1207
        also have "\<dots> \<le> \<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV"
Andreas@59023
  1208
          by(auto simp add: nn_integral_count_space_indicator indicator_def intro!: nn_integral_mono)
Andreas@59023
  1209
        finally have "?used < \<dots>" using False by auto
Andreas@59023
  1210
        also note eq finally have "?used \<in> dom (find_start y)" by(rule in_dom_find_startI)
Andreas@59023
  1211
        then obtain k where k: "find_start y ?used = Some k" by auto
hoelzl@59053
  1212
        let ?f = "\<lambda>z. if z < k then 0 else if z = k then setsum (?rr y) {..k} - ?used else ?rr y z"
hoelzl@59053
  1213
        let ?g = "\<lambda>x'. if x' < x then 0 else ?pp y x'"
hoelzl@59053
  1214
        have "?pp y x = ?g x" by simp
Andreas@59023
  1215
        also have "?g x \<le> \<integral>\<^sup>+ x'. ?g x' \<partial>count_space UNIV" by(rule nn_integral_ge_point) simp
Andreas@59023
  1216
        also {
hoelzl@59053
  1217
          have "?used = \<integral>\<^sup>+ x. ?pp y x \<partial>count_space {..<x}"
Andreas@59023
  1218
            by(simp add: nn_integral_count_space_finite max_def)
hoelzl@59053
  1219
          also have "\<dots> = \<integral>\<^sup>+ x'. (if x' < x then ?pp y x' else 0) \<partial>count_space UNIV"
hoelzl@59053
  1220
            by(simp add: nn_integral_count_space_indicator indicator_def if_distrib zero_ereal_def cong del: if_cong)
hoelzl@59053
  1221
          also have "(\<integral>\<^sup>+ x'. ?g x' \<partial>count_space UNIV) + \<dots> = \<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV"
Andreas@59023
  1222
            by(subst nn_integral_add[symmetric])(auto intro: nn_integral_cong)
Andreas@59023
  1223
          also note calculation }
hoelzl@59053
  1224
        ultimately have "ereal (?pp y x) + ?used \<le> \<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV"
Andreas@59023
  1225
          by (metis (no_types, lifting) ereal_add_mono order_refl)
Andreas@59023
  1226
        also note eq
hoelzl@59053
  1227
        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)"
Andreas@59023
  1228
          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)
hoelzl@59053
  1229
        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) =
hoelzl@59053
  1230
          (\<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})"
Andreas@59023
  1231
          by(auto simp add: nn_integral_count_space_indicator indicator_def intro: nn_integral_cong)
Andreas@59023
  1232
        also have "\<dots> = ?used" 
Andreas@59023
  1233
          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)
hoelzl@59053
  1234
        finally have "?pp y x \<le> (\<integral>\<^sup>+ z. ?f z \<partial>count_space UNIV)"
Andreas@59023
  1235
          by(cases "\<integral>\<^sup>+ z. ?f z \<partial>count_space UNIV") simp_all
Andreas@59023
  1236
        then show ?thesis using k
Andreas@59023
  1237
          by(simp add: assign_def nn_integral_assign_aux diff_le_iff find_start_eq_Some_above min_def)
Andreas@59023
  1238
      next
Andreas@59023
  1239
        case True
hoelzl@59053
  1240
        have "setsum (?pp y) {..x} = \<integral>\<^sup>+ x. ?pp y x \<partial>count_space {..x}"
Andreas@59023
  1241
          by(simp add: nn_integral_count_space_finite max_def)
hoelzl@59053
  1242
        also have "\<dots> \<le> \<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV"
Andreas@59023
  1243
          by(auto simp add: nn_integral_count_space_indicator indicator_def intro: nn_integral_mono)
hoelzl@59053
  1244
        also have "\<dots> = setsum (?pp y) {..<x}" by(simp add: True)
hoelzl@59053
  1245
        finally have "?pp y x = 0" by(simp add: ivl_disj_un(2)[symmetric] eq_iff del: pmf_le_0_iff)
Andreas@59023
  1246
        thus ?thesis
hoelzl@59053
  1247
          by(cases "find_start y (setsum (?pp y) {..<x})")(simp_all add: assign_def diff_le_iff find_start_eq_Some_above)
Andreas@59023
  1248
      qed }
Andreas@59023
  1249
    note nn_integral_assign2 = this
Andreas@59023
  1250
hoelzl@59053
  1251
    def a \<equiv> "embed_pmf (\<lambda>(y, x, z). assign y x z)"
Andreas@59023
  1252
    { fix y x z
hoelzl@59053
  1253
      have "assign y x z = pmf a (y, x, z)"
hoelzl@59053
  1254
        unfolding a_def
hoelzl@59053
  1255
      proof (subst pmf_embed_pmf)
hoelzl@59053
  1256
        have "(\<integral>\<^sup>+ x. ereal ((\<lambda>(y, x, z). assign y x z) x) \<partial>count_space UNIV) =
hoelzl@59053
  1257
          (\<integral>\<^sup>+ x. ereal ((\<lambda>(y, x, z). assign y x z) x) \<partial>(count_space ((\<lambda>((y, x), z). (y, x, z)) ` (pp \<times> UNIV))))"
hoelzl@59053
  1258
          by (force simp add: nn_integral_count_space_indicator pmf_eq_0_set_pmf split: split_indicator
hoelzl@59053
  1259
                    intro!: nn_integral_cong assign_eq_0_outside)
hoelzl@59053
  1260
        also have "\<dots> = (\<integral>\<^sup>+ x. ereal ((\<lambda>((y, x), z). assign y x z) x) \<partial>(count_space (pp \<times> UNIV)))"
hoelzl@59053
  1261
          by (subst nn_integral_bij_count_space[OF inj_on_imp_bij_betw, symmetric])
hoelzl@59053
  1262
             (auto simp: inj_on_def intro!: nn_integral_cong)
hoelzl@59053
  1263
        also have "\<dots> = (\<integral>\<^sup>+ y. \<integral>\<^sup>+z. ereal ((\<lambda>((y, x), z). assign y x z) (y, z)) \<partial>count_space UNIV \<partial>count_space pp)"
hoelzl@59053
  1264
          by (subst sigma_finite_measure.nn_integral_fst)
hoelzl@59053
  1265
             (auto simp: pair_measure_countable sigma_finite_measure_count_space_countable)
hoelzl@59053
  1266
        also have "\<dots> = (\<integral>\<^sup>+ z. ?pp (fst z) (snd z) \<partial>count_space pp)"
hoelzl@59053
  1267
          by (subst nn_integral_assign2[symmetric]) (auto intro!: nn_integral_cong)
hoelzl@59053
  1268
        finally show "(\<integral>\<^sup>+ x. ereal ((\<lambda>(y, x, z). assign y x z) x) \<partial>count_space UNIV) = 1"
hoelzl@59053
  1269
          by (simp add: nn_integral_pmf emeasure_pmf)
hoelzl@59053
  1270
      qed auto }
hoelzl@59053
  1271
    note a = this
Andreas@59023
  1272
hoelzl@59053
  1273
    def pr \<equiv> "map_pmf (\<lambda>(y, x, z). (from_nat_into (A y) x, from_nat_into (B y) z)) a"
Andreas@59023
  1274
hoelzl@59053
  1275
    have "rel_pmf (R OO S) p r"
Andreas@59023
  1276
    proof
hoelzl@59053
  1277
      have pp_eq: "pp = map_pmf (\<lambda>(y, x, z). (y, x)) a"
hoelzl@59053
  1278
      proof (rule pmf_eqI)
hoelzl@59053
  1279
        fix i
hoelzl@59053
  1280
        show "pmf pp i = pmf (map_pmf (\<lambda>(y, x, z). (y, x)) a) i"
hoelzl@59053
  1281
          using nn_integral_assign2[of "fst i" "snd i", symmetric]
hoelzl@59053
  1282
          by (auto simp add: a nn_integral_pmf' inj_on_def ereal.inject[symmetric] ereal_pmf_map 
hoelzl@59053
  1283
                   simp del: ereal.inject intro!: arg_cong2[where f=emeasure])
hoelzl@59053
  1284
      qed
hoelzl@59053
  1285
      moreover have pq_eq: "pq = map_pmf (\<lambda>(y, x). (from_nat_into (A y) x, y)) pp"
hoelzl@59053
  1286
        by (simp add: pp_def map_pmf_comp split_beta A[symmetric] cong: map_pmf_cong)
hoelzl@59053
  1287
      ultimately show "map_pmf fst pr = p"
hoelzl@59053
  1288
        unfolding p pr_def by (simp add: map_pmf_comp split_beta)
hoelzl@59053
  1289
hoelzl@59053
  1290
      have rr_eq: "rr = map_pmf (\<lambda>(y, x, z). (y, z)) a"
hoelzl@59053
  1291
      proof (rule pmf_eqI)
hoelzl@59053
  1292
        fix i show "pmf rr i = pmf (map_pmf (\<lambda>(y, x, z). (y, z)) a) i"
hoelzl@59053
  1293
          using nn_integral_assign1[of "fst i" "snd i", symmetric]
hoelzl@59053
  1294
          by (auto simp add: a nn_integral_pmf' inj_on_def ereal.inject[symmetric] ereal_pmf_map 
hoelzl@59053
  1295
                   simp del: ereal.inject intro!: arg_cong2[where f=emeasure])
hoelzl@59053
  1296
      qed
hoelzl@59053
  1297
      moreover have qr_eq: "qr = map_pmf (\<lambda>(y, z). (y, from_nat_into (B y) z)) rr"
hoelzl@59053
  1298
        by (simp add: rr_def map_pmf_comp split_beta B[symmetric] cong: map_pmf_cong)
hoelzl@59053
  1299
      ultimately show "map_pmf snd pr = r"
hoelzl@59053
  1300
        unfolding r pr_def by (simp add: map_pmf_comp split_beta)
hoelzl@59053
  1301
hoelzl@59053
  1302
      fix x z assume "(x, z) \<in> set_pmf pr"
hoelzl@59053
  1303
      then have "\<exists>y. (x, y) \<in> set_pmf pq \<and> (y, z) \<in> set_pmf qr"
hoelzl@59053
  1304
        by (force simp add: pp_eq pq_eq rr_eq qr_eq set_map_pmf pr_def image_image)
hoelzl@59053
  1305
      with pq qr show "(R OO S) x z"
hoelzl@59053
  1306
        by blast
hoelzl@59053
  1307
    qed }
Andreas@59023
  1308
  then show "rel_pmf R OO rel_pmf S \<le> rel_pmf (R OO S)"
Andreas@59023
  1309
    by(auto simp add: le_fun_def)
Andreas@59023
  1310
qed (fact natLeq_card_order natLeq_cinfinite)+
hoelzl@58587
  1311
hoelzl@58587
  1312
end
hoelzl@58587
  1313