src/HOL/Probability/Probability_Mass_Function.thy
author wenzelm
Fri, 22 Jul 2016 11:00:43 +0200
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parent 63343 fb5d8a50c641
child 63680 6e1e8b5abbfa
permissions -rw-r--r--
tuned proofs -- avoid unstructured calculation;
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(*  Title:      HOL/Probability/Probability_Mass_Function.thy
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    Author:     Johannes Hölzl, TU München
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    Author:     Andreas Lochbihler, ETH Zurich
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*)
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section \<open> Probability mass function \<close>
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theory Probability_Mass_Function
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imports
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  Giry_Monad
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  "~~/src/HOL/Library/Multiset"
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begin
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lemma AE_emeasure_singleton:
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  assumes x: "emeasure M {x} \<noteq> 0" and ae: "AE x in M. P x" shows "P x"
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proof -
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  from x have x_M: "{x} \<in> sets M"
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    by (auto intro: emeasure_notin_sets)
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  from ae obtain N where N: "{x\<in>space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"
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    by (auto elim: AE_E)
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  { assume "\<not> P x"
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    with x_M[THEN sets.sets_into_space] N have "emeasure M {x} \<le> emeasure M N"
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      by (intro emeasure_mono) auto
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    with x N have False
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      by (auto simp:) }
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  then show "P x" by auto
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qed
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lemma AE_measure_singleton: "measure M {x} \<noteq> 0 \<Longrightarrow> AE x in M. P x \<Longrightarrow> P x"
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  by (metis AE_emeasure_singleton measure_def emeasure_empty measure_empty)
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lemma (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 measure_nonneg set_diff_eq cong: conj_cong)
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  then show "AE x in M. measure M {x} \<noteq> 0"
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    by (auto simp: emeasure_eq_measure)
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qed (auto intro!: exI[of _ "{x. measure M {x} \<noteq> 0}"] countable_support)
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subsection \<open> PMF as measure \<close>
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typedef 'a pmf = "{M :: 'a measure. prob_space M \<and> sets M = UNIV \<and> (AE x in M. measure M {x} \<noteq> 0)}"
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  morphisms measure_pmf Abs_pmf
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  by (intro exI[of _ "uniform_measure (count_space UNIV) {undefined}"])
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     (auto intro!: prob_space_uniform_measure AE_uniform_measureI)
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declare [[coercion measure_pmf]]
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lemma prob_space_measure_pmf: "prob_space (measure_pmf p)"
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  using pmf.measure_pmf[of p] by auto
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interpretation measure_pmf: prob_space "measure_pmf M" for M
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  by (rule prob_space_measure_pmf)
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interpretation measure_pmf: subprob_space "measure_pmf M" for M
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  by (rule prob_space_imp_subprob_space) unfold_locales
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lemma subprob_space_measure_pmf: "subprob_space (measure_pmf x)"
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  by unfold_locales
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locale pmf_as_measure
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begin
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setup_lifting type_definition_pmf
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end
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context
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begin
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interpretation pmf_as_measure .
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lemma sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV"
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  by transfer blast
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lemma sets_measure_pmf_count_space[measurable_cong]:
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  "sets (measure_pmf M) = sets (count_space UNIV)"
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  by simp
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lemma space_measure_pmf[simp]: "space (measure_pmf p) = UNIV"
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  using sets_eq_imp_space_eq[of "measure_pmf p" "count_space UNIV"] by simp
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lemma measure_pmf_UNIV [simp]: "measure (measure_pmf p) UNIV = 1"
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using measure_pmf.prob_space[of p] by simp
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lemma measure_pmf_in_subprob_algebra[measurable (raw)]: "measure_pmf x \<in> space (subprob_algebra (count_space UNIV))"
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  by (simp add: space_subprob_algebra subprob_space_measure_pmf)
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lemma measurable_pmf_measure1[simp]: "measurable (M :: 'a pmf) N = UNIV \<rightarrow> space N"
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  by (auto simp: measurable_def)
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lemma measurable_pmf_measure2[simp]: "measurable N (M :: 'a pmf) = measurable N (count_space UNIV)"
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  by (intro measurable_cong_sets) simp_all
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lemma measurable_pair_restrict_pmf2:
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  assumes "countable A"
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  assumes [measurable]: "\<And>y. y \<in> A \<Longrightarrow> (\<lambda>x. f (x, y)) \<in> measurable M L"
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  shows "f \<in> measurable (M \<Otimes>\<^sub>M restrict_space (measure_pmf N) A) L" (is "f \<in> measurable ?M _")
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proof -
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  have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)"
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    by (simp add: restrict_count_space)
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  show ?thesis
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    by (intro measurable_compose_countable'[where f="\<lambda>a b. f (fst b, a)" and g=snd and I=A,
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                                            unfolded prod.collapse] assms)
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        measurable
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qed
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lemma measurable_pair_restrict_pmf1:
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  assumes "countable A"
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  assumes [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N L"
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  shows "f \<in> measurable (restrict_space (measure_pmf M) A \<Otimes>\<^sub>M N) L"
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proof -
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  have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)"
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    by (simp add: restrict_count_space)
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  show ?thesis
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    by (intro measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A,
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                                            unfolded prod.collapse] assms)
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        measurable
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qed
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diff changeset
   142
lift_definition pmf :: "'a pmf \<Rightarrow> 'a \<Rightarrow> real" is "\<lambda>M x. measure M {x}" .
224741ede5ae build pmf's on bind
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parents: 59557
diff changeset
   143
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diff changeset
   144
lift_definition set_pmf :: "'a pmf \<Rightarrow> 'a set" is "\<lambda>M. {x. measure M {x} \<noteq> 0}" .
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parents: 59557
diff changeset
   145
declare [[coercion set_pmf]]
58587
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parents:
diff changeset
   146
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
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   147
lemma AE_measure_pmf: "AE x in (M::'a pmf). x \<in> M"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   148
  by transfer simp
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   149
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   150
lemma emeasure_pmf_single_eq_zero_iff:
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hoelzl
parents:
diff changeset
   151
  fixes M :: "'a pmf"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   152
  shows "emeasure M {y} = 0 \<longleftrightarrow> y \<notin> M"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   153
  unfolding set_pmf.rep_eq by (simp add: measure_pmf.emeasure_eq_measure)
58587
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   154
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   155
lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \<longleftrightarrow> (\<forall>y\<in>M. P y)"
59664
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parents: 59557
diff changeset
   156
  using AE_measure_singleton[of M] AE_measure_pmf[of M]
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   157
  by (auto simp: set_pmf.rep_eq)
224741ede5ae build pmf's on bind
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parents: 59557
diff changeset
   158
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Andreas Lochbihler
parents: 61610
diff changeset
   159
lemma AE_pmfI: "(\<And>y. y \<in> set_pmf M \<Longrightarrow> P y) \<Longrightarrow> almost_everywhere (measure_pmf M) P"
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Andreas Lochbihler
parents: 61610
diff changeset
   160
by(simp add: AE_measure_pmf_iff)
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   161
59664
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parents: 59557
diff changeset
   162
lemma countable_set_pmf [simp]: "countable (set_pmf p)"
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parents: 59557
diff changeset
   163
  by transfer (metis prob_space.finite_measure finite_measure.countable_support)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   164
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diff changeset
   165
lemma pmf_positive: "x \<in> set_pmf p \<Longrightarrow> 0 < pmf p x"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
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diff changeset
   166
  by transfer (simp add: less_le)
59664
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hoelzl
parents: 59557
diff changeset
   167
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   168
lemma pmf_nonneg[simp]: "0 \<le> pmf p x"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   169
  by transfer simp
63099
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eberlm
parents: 63092
diff changeset
   170
  
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eberlm
parents: 63092
diff changeset
   171
lemma pmf_not_neg [simp]: "\<not>pmf p x < 0"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   172
  by (simp add: not_less pmf_nonneg)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   173
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   174
lemma pmf_pos [simp]: "pmf p x \<noteq> 0 \<Longrightarrow> pmf p x > 0"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   175
  using pmf_nonneg[of p x] by linarith
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   176
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diff changeset
   177
lemma pmf_le_1: "pmf p x \<le> 1"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   178
  by (simp add: pmf.rep_eq)
58587
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   179
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   180
lemma set_pmf_not_empty: "set_pmf M \<noteq> {}"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   181
  using AE_measure_pmf[of M] by (intro notI) simp
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   182
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   183
lemma set_pmf_iff: "x \<in> set_pmf M \<longleftrightarrow> pmf M x \<noteq> 0"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   184
  by transfer simp
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   185
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   186
lemma pmf_positive_iff: "0 < pmf p x \<longleftrightarrow> x \<in> set_pmf p"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   187
  unfolding less_le by (simp add: set_pmf_iff)
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   188
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   189
lemma set_pmf_eq: "set_pmf M = {x. pmf M x \<noteq> 0}"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   190
  by (auto simp: set_pmf_iff)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   191
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   192
lemma set_pmf_eq': "set_pmf p = {x. pmf p x > 0}"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   193
proof safe
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   194
  fix x assume "x \<in> set_pmf p"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   195
  hence "pmf p x \<noteq> 0" by (auto simp: set_pmf_eq)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   196
  with pmf_nonneg[of p x] show "pmf p x > 0" by simp
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   197
qed (auto simp: set_pmf_eq)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   198
59664
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hoelzl
parents: 59557
diff changeset
   199
lemma emeasure_pmf_single:
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   200
  fixes M :: "'a pmf"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   201
  shows "emeasure M {x} = pmf M x"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   202
  by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   203
60068
ef2123db900c add various lemmas about pmfs
Andreas Lochbihler
parents: 59731
diff changeset
   204
lemma measure_pmf_single: "measure (measure_pmf M) {x} = pmf M x"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   205
  using emeasure_pmf_single[of M x] by(simp add: measure_pmf.emeasure_eq_measure pmf_nonneg measure_nonneg)
60068
ef2123db900c add various lemmas about pmfs
Andreas Lochbihler
parents: 59731
diff changeset
   206
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   207
lemma emeasure_measure_pmf_finite: "finite S \<Longrightarrow> emeasure (measure_pmf M) S = (\<Sum>s\<in>S. pmf M s)"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   208
  by (subst emeasure_eq_setsum_singleton) (auto simp: emeasure_pmf_single pmf_nonneg)
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   209
59023
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   210
lemma measure_measure_pmf_finite: "finite S \<Longrightarrow> measure (measure_pmf M) S = setsum (pmf M) S"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   211
  using emeasure_measure_pmf_finite[of S M]
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   212
  by (simp add: measure_pmf.emeasure_eq_measure measure_nonneg setsum_nonneg pmf_nonneg)
59023
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   213
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   214
lemma setsum_pmf_eq_1:
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   215
  assumes "finite A" "set_pmf p \<subseteq> A"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   216
  shows   "(\<Sum>x\<in>A. pmf p x) = 1"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   217
proof -
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   218
  have "(\<Sum>x\<in>A. pmf p x) = measure_pmf.prob p A"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   219
    by (simp add: measure_measure_pmf_finite assms)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   220
  also from assms have "\<dots> = 1"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   221
    by (subst measure_pmf.prob_eq_1) (auto simp: AE_measure_pmf_iff)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   222
  finally show ?thesis .
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   223
qed
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   224
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   225
lemma nn_integral_measure_pmf_support:
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   226
  fixes f :: "'a \<Rightarrow> ennreal"
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   227
  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"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   228
  shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>A. f x * pmf M x)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   229
proof -
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   230
  have "(\<integral>\<^sup>+x. f x \<partial>M) = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   231
    using nn by (intro nn_integral_cong_AE) (auto simp: AE_measure_pmf_iff split: split_indicator)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   232
  also have "\<dots> = (\<Sum>x\<in>A. f x * emeasure M {x})"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   233
    using assms by (intro nn_integral_indicator_finite) auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   234
  finally show ?thesis
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   235
    by (simp add: emeasure_measure_pmf_finite)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   236
qed
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   237
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   238
lemma nn_integral_measure_pmf_finite:
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   239
  fixes f :: "'a \<Rightarrow> ennreal"
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   240
  assumes f: "finite (set_pmf M)" and nn: "\<And>x. x \<in> set_pmf M \<Longrightarrow> 0 \<le> f x"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   241
  shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>set_pmf M. f x * pmf M x)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   242
  using assms by (intro nn_integral_measure_pmf_support) auto
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   243
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   244
lemma integrable_measure_pmf_finite:
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   245
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   246
  shows "finite (set_pmf M) \<Longrightarrow> integrable M f"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   247
  by (auto intro!: integrableI_bounded simp: nn_integral_measure_pmf_finite ennreal_mult_less_top)
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   248
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   249
lemma integral_measure_pmf:
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   250
  assumes [simp]: "finite A" and "\<And>a. a \<in> set_pmf M \<Longrightarrow> f a \<noteq> 0 \<Longrightarrow> a \<in> A"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   251
  shows "(\<integral>x. f x \<partial>measure_pmf M) = (\<Sum>a\<in>A. f a * pmf M a)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   252
proof -
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   253
  have "(\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x * indicator A x \<partial>measure_pmf M)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   254
    using assms(2) by (intro integral_cong_AE) (auto split: split_indicator simp: AE_measure_pmf_iff)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   255
  also have "\<dots> = (\<Sum>a\<in>A. f a * pmf M a)"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   256
    by (subst integral_indicator_finite_real)
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   257
       (auto simp: measure_def emeasure_measure_pmf_finite pmf_nonneg)
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   258
  finally show ?thesis .
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   259
qed
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   260
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   261
lemma integrable_pmf: "integrable (count_space X) (pmf M)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   262
proof -
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   263
  have " (\<integral>\<^sup>+ x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+ x. pmf M x \<partial>count_space (M \<inter> X))"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   264
    by (auto simp add: nn_integral_count_space_indicator set_pmf_iff intro!: nn_integral_cong split: split_indicator)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   265
  then have "integrable (count_space X) (pmf M) = integrable (count_space (M \<inter> X)) (pmf M)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   266
    by (simp add: integrable_iff_bounded pmf_nonneg)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   267
  then show ?thesis
59023
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   268
    by (simp add: pmf.rep_eq measure_pmf.integrable_measure disjoint_family_on_def)
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   269
qed
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   270
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   271
lemma integral_pmf: "(\<integral>x. pmf M x \<partial>count_space X) = measure M X"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   272
proof -
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   273
  have "(\<integral>x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+x. pmf M x \<partial>count_space X)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   274
    by (simp add: pmf_nonneg integrable_pmf nn_integral_eq_integral)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   275
  also have "\<dots> = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space (X \<inter> M))"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   276
    by (auto intro!: nn_integral_cong_AE split: split_indicator
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   277
             simp: pmf.rep_eq measure_pmf.emeasure_eq_measure nn_integral_count_space_indicator
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   278
                   AE_count_space set_pmf_iff)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   279
  also have "\<dots> = emeasure M (X \<inter> M)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   280
    by (rule emeasure_countable_singleton[symmetric]) (auto intro: countable_set_pmf)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   281
  also have "\<dots> = emeasure M X"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   282
    by (auto intro!: emeasure_eq_AE simp: AE_measure_pmf_iff)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   283
  finally show ?thesis
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   284
    by (simp add: measure_pmf.emeasure_eq_measure measure_nonneg integral_nonneg pmf_nonneg)
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   285
qed
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   286
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   287
lemma integral_pmf_restrict:
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   288
  "(f::'a \<Rightarrow> 'b::{banach, second_countable_topology}) \<in> borel_measurable (count_space UNIV) \<Longrightarrow>
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   289
    (\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x \<partial>restrict_space M M)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   290
  by (auto intro!: integral_cong_AE simp add: integral_restrict_space AE_measure_pmf_iff)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   291
58587
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   292
lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   293
proof -
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   294
  have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   295
    by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   296
  then show ?thesis
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   297
    using measure_pmf.emeasure_space_1 by simp
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   298
qed
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   299
59490
f71732294f29 tune proof
Andreas Lochbihler
parents: 59475
diff changeset
   300
lemma emeasure_pmf_UNIV [simp]: "emeasure (measure_pmf M) UNIV = 1"
f71732294f29 tune proof
Andreas Lochbihler
parents: 59475
diff changeset
   301
using measure_pmf.emeasure_space_1[of M] by simp
f71732294f29 tune proof
Andreas Lochbihler
parents: 59475
diff changeset
   302
59023
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   303
lemma in_null_sets_measure_pmfI:
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   304
  "A \<inter> set_pmf p = {} \<Longrightarrow> A \<in> null_sets (measure_pmf p)"
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   305
using emeasure_eq_0_AE[where ?P="\<lambda>x. x \<in> A" and M="measure_pmf p"]
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   306
by(auto simp add: null_sets_def AE_measure_pmf_iff)
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   307
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   308
lemma measure_subprob: "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   309
  by (simp add: space_subprob_algebra subprob_space_measure_pmf)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   310
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   311
subsection \<open> Monad Interpretation \<close>
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   312
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   313
lemma measurable_measure_pmf[measurable]:
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   314
  "(\<lambda>x. measure_pmf (M x)) \<in> measurable (count_space UNIV) (subprob_algebra (count_space UNIV))"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   315
  by (auto simp: space_subprob_algebra intro!: prob_space_imp_subprob_space) unfold_locales
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   316
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   317
lemma bind_measure_pmf_cong:
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   318
  assumes "\<And>x. A x \<in> space (subprob_algebra N)" "\<And>x. B x \<in> space (subprob_algebra N)"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   319
  assumes "\<And>i. i \<in> set_pmf x \<Longrightarrow> A i = B i"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   320
  shows "bind (measure_pmf x) A = bind (measure_pmf x) B"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   321
proof (rule measure_eqI)
62026
ea3b1b0413b4 more symbols;
wenzelm
parents: 61808
diff changeset
   322
  show "sets (measure_pmf x \<bind> A) = sets (measure_pmf x \<bind> B)"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   323
    using assms by (subst (1 2) sets_bind) (auto simp: space_subprob_algebra)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   324
next
62026
ea3b1b0413b4 more symbols;
wenzelm
parents: 61808
diff changeset
   325
  fix X assume "X \<in> sets (measure_pmf x \<bind> A)"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   326
  then have X: "X \<in> sets N"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   327
    using assms by (subst (asm) sets_bind) (auto simp: space_subprob_algebra)
62026
ea3b1b0413b4 more symbols;
wenzelm
parents: 61808
diff changeset
   328
  show "emeasure (measure_pmf x \<bind> A) X = emeasure (measure_pmf x \<bind> B) X"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   329
    using assms
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   330
    by (subst (1 2) emeasure_bind[where N=N, OF _ _ X])
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   331
       (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   332
qed
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   333
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   334
lift_definition bind_pmf :: "'a pmf \<Rightarrow> ('a \<Rightarrow> 'b pmf ) \<Rightarrow> 'b pmf" is bind
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   335
proof (clarify, intro conjI)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   336
  fix f :: "'a measure" and g :: "'a \<Rightarrow> 'b measure"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   337
  assume "prob_space f"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   338
  then interpret f: prob_space f .
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   339
  assume "sets f = UNIV" and ae_f: "AE x in f. measure f {x} \<noteq> 0"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   340
  then have s_f[simp]: "sets f = sets (count_space UNIV)"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   341
    by simp
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   342
  assume g: "\<And>x. prob_space (g x) \<and> sets (g x) = UNIV \<and> (AE y in g x. measure (g x) {y} \<noteq> 0)"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   343
  then have g: "\<And>x. prob_space (g x)" and s_g[simp]: "\<And>x. sets (g x) = sets (count_space UNIV)"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   344
    and ae_g: "\<And>x. AE y in g x. measure (g x) {y} \<noteq> 0"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   345
    by auto
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   346
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   347
  have [measurable]: "g \<in> measurable f (subprob_algebra (count_space UNIV))"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   348
    by (auto simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space g)
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59665
diff changeset
   349
62026
ea3b1b0413b4 more symbols;
wenzelm
parents: 61808
diff changeset
   350
  show "prob_space (f \<bind> g)"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   351
    using g by (intro f.prob_space_bind[where S="count_space UNIV"]) auto
62026
ea3b1b0413b4 more symbols;
wenzelm
parents: 61808
diff changeset
   352
  then interpret fg: prob_space "f \<bind> g" .
ea3b1b0413b4 more symbols;
wenzelm
parents: 61808
diff changeset
   353
  show [simp]: "sets (f \<bind> g) = UNIV"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   354
    using sets_eq_imp_space_eq[OF s_f]
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   355
    by (subst sets_bind[where N="count_space UNIV"]) auto
62026
ea3b1b0413b4 more symbols;
wenzelm
parents: 61808
diff changeset
   356
  show "AE x in f \<bind> g. measure (f \<bind> g) {x} \<noteq> 0"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   357
    apply (simp add: fg.prob_eq_0 AE_bind[where B="count_space UNIV"])
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   358
    using ae_f
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   359
    apply eventually_elim
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   360
    using ae_g
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   361
    apply eventually_elim
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   362
    apply (auto dest: AE_measure_singleton)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   363
    done
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   364
qed
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   365
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   366
adhoc_overloading Monad_Syntax.bind bind_pmf
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   367
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   368
lemma ennreal_pmf_bind: "pmf (bind_pmf N f) i = (\<integral>\<^sup>+x. pmf (f x) i \<partial>measure_pmf N)"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   369
  unfolding pmf.rep_eq bind_pmf.rep_eq
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   370
  by (auto simp: measure_pmf.measure_bind[where N="count_space UNIV"] measure_subprob measure_nonneg
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   371
           intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   372
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   373
lemma pmf_bind: "pmf (bind_pmf N f) i = (\<integral>x. pmf (f x) i \<partial>measure_pmf N)"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   374
  using ennreal_pmf_bind[of N f i]
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   375
  by (subst (asm) nn_integral_eq_integral)
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   376
     (auto simp: pmf_nonneg pmf_le_1 pmf_nonneg integral_nonneg
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   377
           intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   378
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   379
lemma bind_pmf_const[simp]: "bind_pmf M (\<lambda>x. c) = c"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   380
  by transfer (simp add: bind_const' prob_space_imp_subprob_space)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   381
59665
37adca7fd48f add set_pmf lemmas to simpset
hoelzl
parents: 59664
diff changeset
   382
lemma set_bind_pmf[simp]: "set_pmf (bind_pmf M N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   383
proof -
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   384
  have "set_pmf (bind_pmf M N) = {x. ennreal (pmf (bind_pmf M N) x) \<noteq> 0}"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   385
    by (simp add: set_pmf_eq pmf_nonneg)
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   386
  also have "\<dots> = (\<Union>M\<in>set_pmf M. set_pmf (N M))"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   387
    unfolding ennreal_pmf_bind
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   388
    by (subst nn_integral_0_iff_AE) (auto simp: AE_measure_pmf_iff pmf_nonneg set_pmf_eq)
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   389
  finally show ?thesis .
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   390
qed
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   391
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   392
lemma bind_pmf_cong [fundef_cong]:
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   393
  assumes "p = q"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   394
  shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61634
diff changeset
   395
  unfolding \<open>p = q\<close>[symmetric] measure_pmf_inject[symmetric] bind_pmf.rep_eq
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   396
  by (auto simp: AE_measure_pmf_iff Pi_iff space_subprob_algebra subprob_space_measure_pmf
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   397
                 sets_bind[where N="count_space UNIV"] emeasure_bind[where N="count_space UNIV"]
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   398
           intro!: nn_integral_cong_AE measure_eqI)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   399
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   400
lemma bind_pmf_cong_simp:
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   401
  "p = q \<Longrightarrow> (\<And>x. x \<in> set_pmf q =simp=> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   402
  by (simp add: simp_implies_def cong: bind_pmf_cong)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   403
62026
ea3b1b0413b4 more symbols;
wenzelm
parents: 61808
diff changeset
   404
lemma measure_pmf_bind: "measure_pmf (bind_pmf M f) = (measure_pmf M \<bind> (\<lambda>x. measure_pmf (f x)))"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   405
  by transfer simp
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   406
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   407
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)"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   408
  using measurable_measure_pmf[of N]
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   409
  unfolding measure_pmf_bind
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   410
  apply (intro nn_integral_bind[where B="count_space UNIV"])
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   411
  apply auto
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   412
  done
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   413
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   414
lemma emeasure_bind_pmf[simp]: "emeasure (bind_pmf M N) X = (\<integral>\<^sup>+x. emeasure (N x) X \<partial>M)"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   415
  using measurable_measure_pmf[of N]
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   416
  unfolding measure_pmf_bind
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   417
  by (subst emeasure_bind[where N="count_space UNIV"]) auto
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59665
diff changeset
   418
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   419
lift_definition return_pmf :: "'a \<Rightarrow> 'a pmf" is "return (count_space UNIV)"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   420
  by (auto intro!: prob_space_return simp: AE_return measure_return)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   421
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   422
lemma bind_return_pmf: "bind_pmf (return_pmf x) f = f x"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   423
  by transfer
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   424
     (auto intro!: prob_space_imp_subprob_space bind_return[where N="count_space UNIV"]
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   425
           simp: space_subprob_algebra)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   426
59665
37adca7fd48f add set_pmf lemmas to simpset
hoelzl
parents: 59664
diff changeset
   427
lemma set_return_pmf[simp]: "set_pmf (return_pmf x) = {x}"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   428
  by transfer (auto simp add: measure_return split: split_indicator)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   429
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   430
lemma bind_return_pmf': "bind_pmf N return_pmf = N"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   431
proof (transfer, clarify)
62026
ea3b1b0413b4 more symbols;
wenzelm
parents: 61808
diff changeset
   432
  fix N :: "'a measure" assume "sets N = UNIV" then show "N \<bind> return (count_space UNIV) = N"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   433
    by (subst return_sets_cong[where N=N]) (simp_all add: bind_return')
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   434
qed
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   435
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   436
lemma bind_assoc_pmf: "bind_pmf (bind_pmf A B) C = bind_pmf A (\<lambda>x. bind_pmf (B x) C)"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   437
  by transfer
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   438
     (auto intro!: bind_assoc[where N="count_space UNIV" and R="count_space UNIV"]
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   439
           simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   440
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   441
definition "map_pmf f M = bind_pmf M (\<lambda>x. return_pmf (f x))"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   442
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   443
lemma map_bind_pmf: "map_pmf f (bind_pmf M g) = bind_pmf M (\<lambda>x. map_pmf f (g x))"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   444
  by (simp add: map_pmf_def bind_assoc_pmf)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   445
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   446
lemma bind_map_pmf: "bind_pmf (map_pmf f M) g = bind_pmf M (\<lambda>x. g (f x))"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   447
  by (simp add: map_pmf_def bind_assoc_pmf bind_return_pmf)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   448
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   449
lemma map_pmf_transfer[transfer_rule]:
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   450
  "rel_fun op = (rel_fun cr_pmf cr_pmf) (\<lambda>f M. distr M (count_space UNIV) f) map_pmf"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   451
proof -
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   452
  have "rel_fun op = (rel_fun pmf_as_measure.cr_pmf pmf_as_measure.cr_pmf)
62026
ea3b1b0413b4 more symbols;
wenzelm
parents: 61808
diff changeset
   453
     (\<lambda>f M. M \<bind> (return (count_space UNIV) o f)) map_pmf"
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59665
diff changeset
   454
    unfolding map_pmf_def[abs_def] comp_def by transfer_prover
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   455
  then show ?thesis
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   456
    by (force simp: rel_fun_def cr_pmf_def bind_return_distr)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   457
qed
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   458
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   459
lemma map_pmf_rep_eq:
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   460
  "measure_pmf (map_pmf f M) = distr (measure_pmf M) (count_space UNIV) f"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   461
  unfolding map_pmf_def bind_pmf.rep_eq comp_def return_pmf.rep_eq
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   462
  using bind_return_distr[of M f "count_space UNIV"] by (simp add: comp_def)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   463
58587
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   464
lemma map_pmf_id[simp]: "map_pmf id = id"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   465
  by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   466
59053
43e07797269b tuned proof that pmfs are bnfs
hoelzl
parents: 59052
diff changeset
   467
lemma map_pmf_ident[simp]: "map_pmf (\<lambda>x. x) = (\<lambda>x. x)"
43e07797269b tuned proof that pmfs are bnfs
hoelzl
parents: 59052
diff changeset
   468
  using map_pmf_id unfolding id_def .
43e07797269b tuned proof that pmfs are bnfs
hoelzl
parents: 59052
diff changeset
   469
58587
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   470
lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g"
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59665
diff changeset
   471
  by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def)
58587
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   472
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   473
lemma map_pmf_comp: "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   474
  using map_pmf_compose[of f g] by (simp add: comp_def)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   475
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   476
lemma map_pmf_cong: "p = q \<Longrightarrow> (\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> map_pmf f p = map_pmf g q"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   477
  unfolding map_pmf_def by (rule bind_pmf_cong) auto
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   478
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   479
lemma pmf_set_map: "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
59665
37adca7fd48f add set_pmf lemmas to simpset
hoelzl
parents: 59664
diff changeset
   480
  by (auto simp add: comp_def fun_eq_iff map_pmf_def)
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   481
59665
37adca7fd48f add set_pmf lemmas to simpset
hoelzl
parents: 59664
diff changeset
   482
lemma set_map_pmf[simp]: "set_pmf (map_pmf f M) = f`set_pmf M"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   483
  using pmf_set_map[of f] by (auto simp: comp_def fun_eq_iff)
58587
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   484
59002
2c8b2fb54b88 cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents: 59000
diff changeset
   485
lemma emeasure_map_pmf[simp]: "emeasure (map_pmf f M) X = emeasure M (f -` X)"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   486
  unfolding map_pmf_rep_eq by (subst emeasure_distr) auto
59002
2c8b2fb54b88 cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents: 59000
diff changeset
   487
61634
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   488
lemma measure_map_pmf[simp]: "measure (map_pmf f M) X = measure M (f -` X)"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   489
using emeasure_map_pmf[of f M X] by(simp add: measure_pmf.emeasure_eq_measure measure_nonneg)
61634
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   490
59002
2c8b2fb54b88 cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents: 59000
diff changeset
   491
lemma nn_integral_map_pmf[simp]: "(\<integral>\<^sup>+x. f x \<partial>map_pmf g M) = (\<integral>\<^sup>+x. f (g x) \<partial>M)"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   492
  unfolding map_pmf_rep_eq by (intro nn_integral_distr) auto
59002
2c8b2fb54b88 cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents: 59000
diff changeset
   493
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   494
lemma ennreal_pmf_map: "pmf (map_pmf f p) x = (\<integral>\<^sup>+ y. indicator (f -` {x}) y \<partial>measure_pmf p)"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   495
proof (transfer fixing: f x)
59023
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   496
  fix p :: "'b measure"
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   497
  presume "prob_space p"
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   498
  then interpret prob_space p .
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   499
  presume "sets p = UNIV"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   500
  then show "ennreal (measure (distr p (count_space UNIV) f) {x}) = integral\<^sup>N p (indicator (f -` {x}))"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   501
    by(simp add: measure_distr measurable_def emeasure_eq_measure)
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   502
qed simp_all
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   503
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   504
lemma pmf_map: "pmf (map_pmf f p) x = measure p (f -` {x})"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   505
proof (transfer fixing: f x)
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   506
  fix p :: "'b measure"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   507
  presume "prob_space p"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   508
  then interpret prob_space p .
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   509
  presume "sets p = UNIV"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   510
  then show "measure (distr p (count_space UNIV) f) {x} = measure p (f -` {x})"
59023
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   511
    by(simp add: measure_distr measurable_def emeasure_eq_measure)
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   512
qed simp_all
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   513
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   514
lemma nn_integral_pmf: "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = emeasure (measure_pmf p) A"
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   515
proof -
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   516
  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))"
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   517
    by(auto simp add: nn_integral_count_space_indicator indicator_def set_pmf_iff intro: nn_integral_cong)
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   518
  also have "\<dots> = emeasure (measure_pmf p) (\<Union>x\<in>A \<inter> set_pmf p. {x})"
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   519
    by(subst emeasure_UN_countable)(auto simp add: emeasure_pmf_single disjoint_family_on_def)
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   520
  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})"
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   521
    by(rule emeasure_Un_null_set[symmetric])(auto intro: in_null_sets_measure_pmfI)
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   522
  also have "\<dots> = emeasure (measure_pmf p) A"
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   523
    by(auto intro: arg_cong2[where f=emeasure])
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   524
  finally show ?thesis .
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   525
qed
4999a616336c register pmf as BNF
Andreas Lochbihler
parents: 59002
diff changeset
   526
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   527
lemma integral_map_pmf[simp]:
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   528
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   529
  shows "integral\<^sup>L (map_pmf g p) f = integral\<^sup>L p (\<lambda>x. f (g x))"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   530
  by (simp add: integral_distr map_pmf_rep_eq)
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   531
60068
ef2123db900c add various lemmas about pmfs
Andreas Lochbihler
parents: 59731
diff changeset
   532
lemma map_return_pmf [simp]: "map_pmf f (return_pmf x) = return_pmf (f x)"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   533
  by transfer (simp add: distr_return)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   534
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   535
lemma map_pmf_const[simp]: "map_pmf (\<lambda>_. c) M = return_pmf c"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   536
  by transfer (auto simp: prob_space.distr_const)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   537
60068
ef2123db900c add various lemmas about pmfs
Andreas Lochbihler
parents: 59731
diff changeset
   538
lemma pmf_return [simp]: "pmf (return_pmf x) y = indicator {y} x"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   539
  by transfer (simp add: measure_return)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   540
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   541
lemma nn_integral_return_pmf[simp]: "0 \<le> f x \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>return_pmf x) = f x"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   542
  unfolding return_pmf.rep_eq by (intro nn_integral_return) auto
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   543
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   544
lemma emeasure_return_pmf[simp]: "emeasure (return_pmf x) X = indicator X x"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   545
  unfolding return_pmf.rep_eq by (intro emeasure_return) auto
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   546
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   547
lemma measure_return_pmf [simp]: "measure_pmf.prob (return_pmf x) A = indicator A x"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   548
proof -
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   549
  have "ennreal (measure_pmf.prob (return_pmf x) A) = 
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   550
          emeasure (measure_pmf (return_pmf x)) A"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   551
    by (simp add: measure_pmf.emeasure_eq_measure)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   552
  also have "\<dots> = ennreal (indicator A x)" by (simp add: ennreal_indicator)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   553
  finally show ?thesis by simp
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   554
qed
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   555
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   556
lemma return_pmf_inj[simp]: "return_pmf x = return_pmf y \<longleftrightarrow> x = y"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   557
  by (metis insertI1 set_return_pmf singletonD)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   558
59665
37adca7fd48f add set_pmf lemmas to simpset
hoelzl
parents: 59664
diff changeset
   559
lemma map_pmf_eq_return_pmf_iff:
37adca7fd48f add set_pmf lemmas to simpset
hoelzl
parents: 59664
diff changeset
   560
  "map_pmf f p = return_pmf x \<longleftrightarrow> (\<forall>y \<in> set_pmf p. f y = x)"
37adca7fd48f add set_pmf lemmas to simpset
hoelzl
parents: 59664
diff changeset
   561
proof
37adca7fd48f add set_pmf lemmas to simpset
hoelzl
parents: 59664
diff changeset
   562
  assume "map_pmf f p = return_pmf x"
37adca7fd48f add set_pmf lemmas to simpset
hoelzl
parents: 59664
diff changeset
   563
  then have "set_pmf (map_pmf f p) = set_pmf (return_pmf x)" by simp
37adca7fd48f add set_pmf lemmas to simpset
hoelzl
parents: 59664
diff changeset
   564
  then show "\<forall>y \<in> set_pmf p. f y = x" by auto
37adca7fd48f add set_pmf lemmas to simpset
hoelzl
parents: 59664
diff changeset
   565
next
37adca7fd48f add set_pmf lemmas to simpset
hoelzl
parents: 59664
diff changeset
   566
  assume "\<forall>y \<in> set_pmf p. f y = x"
37adca7fd48f add set_pmf lemmas to simpset
hoelzl
parents: 59664
diff changeset
   567
  then show "map_pmf f p = return_pmf x"
37adca7fd48f add set_pmf lemmas to simpset
hoelzl
parents: 59664
diff changeset
   568
    unfolding map_pmf_const[symmetric, of _ p] by (intro map_pmf_cong) auto
37adca7fd48f add set_pmf lemmas to simpset
hoelzl
parents: 59664
diff changeset
   569
qed
37adca7fd48f add set_pmf lemmas to simpset
hoelzl
parents: 59664
diff changeset
   570
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   571
definition "pair_pmf A B = bind_pmf A (\<lambda>x. bind_pmf B (\<lambda>y. return_pmf (x, y)))"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   572
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   573
lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   574
  unfolding pair_pmf_def pmf_bind pmf_return
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   575
  apply (subst integral_measure_pmf[where A="{b}"])
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   576
  apply (auto simp: indicator_eq_0_iff)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   577
  apply (subst integral_measure_pmf[where A="{a}"])
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   578
  apply (auto simp: indicator_eq_0_iff setsum_nonneg_eq_0_iff pmf_nonneg)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   579
  done
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   580
59665
37adca7fd48f add set_pmf lemmas to simpset
hoelzl
parents: 59664
diff changeset
   581
lemma set_pair_pmf[simp]: "set_pmf (pair_pmf A B) = set_pmf A \<times> set_pmf B"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   582
  unfolding pair_pmf_def set_bind_pmf set_return_pmf by auto
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   583
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   584
lemma measure_pmf_in_subprob_space[measurable (raw)]:
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   585
  "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   586
  by (simp add: space_subprob_algebra) intro_locales
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   587
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   588
lemma nn_integral_pair_pmf': "(\<integral>\<^sup>+x. f x \<partial>pair_pmf A B) = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. f (a, b) \<partial>B \<partial>A)"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   589
proof -
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   590
  have "(\<integral>\<^sup>+x. f x \<partial>pair_pmf A B) = (\<integral>\<^sup>+x. f x * indicator (A \<times> B) x \<partial>pair_pmf A B)"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   591
    by (auto simp: AE_measure_pmf_iff intro!: nn_integral_cong_AE)
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   592
  also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. f (a, b) * indicator (A \<times> B) (a, b) \<partial>B \<partial>A)"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   593
    by (simp add: pair_pmf_def)
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   594
  also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. f (a, b) \<partial>B \<partial>A)"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   595
    by (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   596
  finally show ?thesis .
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   597
qed
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   598
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   599
lemma bind_pair_pmf:
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   600
  assumes M[measurable]: "M \<in> measurable (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) (subprob_algebra N)"
62026
ea3b1b0413b4 more symbols;
wenzelm
parents: 61808
diff changeset
   601
  shows "measure_pmf (pair_pmf A B) \<bind> M = (measure_pmf A \<bind> (\<lambda>x. measure_pmf B \<bind> (\<lambda>y. M (x, y))))"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   602
    (is "?L = ?R")
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   603
proof (rule measure_eqI)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   604
  have M'[measurable]: "M \<in> measurable (pair_pmf A B) (subprob_algebra N)"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   605
    using M[THEN measurable_space] by (simp_all add: space_pair_measure)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   606
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   607
  note measurable_bind[where N="count_space UNIV", measurable]
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   608
  note measure_pmf_in_subprob_space[simp]
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   609
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   610
  have sets_eq_N: "sets ?L = N"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   611
    by (subst sets_bind[OF sets_kernel[OF M']]) auto
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   612
  show "sets ?L = sets ?R"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   613
    using measurable_space[OF M]
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   614
    by (simp add: sets_eq_N space_pair_measure space_subprob_algebra)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   615
  fix X assume "X \<in> sets ?L"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   616
  then have X[measurable]: "X \<in> sets N"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   617
    unfolding sets_eq_N .
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   618
  then show "emeasure ?L X = emeasure ?R X"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   619
    apply (simp add: emeasure_bind[OF _ M' X])
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   620
    apply (simp add: nn_integral_bind[where B="count_space UNIV"] pair_pmf_def measure_pmf_bind[of A]
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   621
                     nn_integral_measure_pmf_finite)
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   622
    apply (subst emeasure_bind[OF _ _ X])
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   623
    apply measurable
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   624
    apply (subst emeasure_bind[OF _ _ X])
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   625
    apply measurable
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   626
    done
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   627
qed
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   628
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   629
lemma map_fst_pair_pmf: "map_pmf fst (pair_pmf A B) = A"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   630
  by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   631
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   632
lemma map_snd_pair_pmf: "map_pmf snd (pair_pmf A B) = B"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   633
  by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   634
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   635
lemma nn_integral_pmf':
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   636
  "inj_on f A \<Longrightarrow> (\<integral>\<^sup>+x. pmf p (f x) \<partial>count_space A) = emeasure p (f ` A)"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   637
  by (subst nn_integral_bij_count_space[where g=f and B="f`A"])
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   638
     (auto simp: bij_betw_def nn_integral_pmf)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   639
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   640
lemma pmf_le_0_iff[simp]: "pmf M p \<le> 0 \<longleftrightarrow> pmf M p = 0"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   641
  using pmf_nonneg[of M p] by arith
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   642
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   643
lemma min_pmf_0[simp]: "min (pmf M p) 0 = 0" "min 0 (pmf M p) = 0"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   644
  using pmf_nonneg[of M p] by arith+
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   645
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   646
lemma pmf_eq_0_set_pmf: "pmf M p = 0 \<longleftrightarrow> p \<notin> set_pmf M"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   647
  unfolding set_pmf_iff by simp
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   648
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   649
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"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   650
  by (auto simp: pmf.rep_eq map_pmf_rep_eq measure_distr AE_measure_pmf_iff inj_onD
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   651
           intro!: measure_pmf.finite_measure_eq_AE)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   652
60068
ef2123db900c add various lemmas about pmfs
Andreas Lochbihler
parents: 59731
diff changeset
   653
lemma pmf_map_inj': "inj f \<Longrightarrow> pmf (map_pmf f M) (f x) = pmf M x"
ef2123db900c add various lemmas about pmfs
Andreas Lochbihler
parents: 59731
diff changeset
   654
apply(cases "x \<in> set_pmf M")
ef2123db900c add various lemmas about pmfs
Andreas Lochbihler
parents: 59731
diff changeset
   655
 apply(simp add: pmf_map_inj[OF subset_inj_on])
ef2123db900c add various lemmas about pmfs
Andreas Lochbihler
parents: 59731
diff changeset
   656
apply(simp add: pmf_eq_0_set_pmf[symmetric])
ef2123db900c add various lemmas about pmfs
Andreas Lochbihler
parents: 59731
diff changeset
   657
apply(auto simp add: pmf_eq_0_set_pmf dest: injD)
ef2123db900c add various lemmas about pmfs
Andreas Lochbihler
parents: 59731
diff changeset
   658
done
ef2123db900c add various lemmas about pmfs
Andreas Lochbihler
parents: 59731
diff changeset
   659
ef2123db900c add various lemmas about pmfs
Andreas Lochbihler
parents: 59731
diff changeset
   660
lemma pmf_map_outside: "x \<notin> f ` set_pmf M \<Longrightarrow> pmf (map_pmf f M) x = 0"
ef2123db900c add various lemmas about pmfs
Andreas Lochbihler
parents: 59731
diff changeset
   661
unfolding pmf_eq_0_set_pmf by simp
ef2123db900c add various lemmas about pmfs
Andreas Lochbihler
parents: 59731
diff changeset
   662
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   663
subsection \<open> PMFs as function \<close>
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   664
58587
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   665
context
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   666
  fixes f :: "'a \<Rightarrow> real"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   667
  assumes nonneg: "\<And>x. 0 \<le> f x"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   668
  assumes prob: "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   669
begin
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   670
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   671
lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ennreal \<circ> f)"
58587
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   672
proof (intro conjI)
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   673
  have *[simp]: "\<And>x y. ennreal (f y) * indicator {x} y = ennreal (f x) * indicator {x} y"
58587
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   674
    by (simp split: split_indicator)
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   675
  show "AE x in density (count_space UNIV) (ennreal \<circ> f).
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   676
    measure (density (count_space UNIV) (ennreal \<circ> f)) {x} \<noteq> 0"
59092
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents: 59053
diff changeset
   677
    by (simp add: AE_density nonneg measure_def emeasure_density max_def)
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   678
  show "prob_space (density (count_space UNIV) (ennreal \<circ> f))"
61169
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 60602
diff changeset
   679
    by standard (simp add: emeasure_density prob)
58587
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   680
qed simp
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   681
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   682
lemma pmf_embed_pmf: "pmf embed_pmf x = f x"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   683
proof transfer
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   684
  have *[simp]: "\<And>x y. ennreal (f y) * indicator {x} y = ennreal (f x) * indicator {x} y"
58587
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   685
    by (simp split: split_indicator)
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   686
  fix x show "measure (density (count_space UNIV) (ennreal \<circ> f)) {x} = f x"
59092
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents: 59053
diff changeset
   687
    by transfer (simp add: measure_def emeasure_density nonneg max_def)
58587
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   688
qed
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   689
60068
ef2123db900c add various lemmas about pmfs
Andreas Lochbihler
parents: 59731
diff changeset
   690
lemma set_embed_pmf: "set_pmf embed_pmf = {x. f x \<noteq> 0}"
63092
a949b2a5f51d eliminated use of empty "assms";
wenzelm
parents: 63040
diff changeset
   691
by(auto simp add: set_pmf_eq pmf_embed_pmf)
60068
ef2123db900c add various lemmas about pmfs
Andreas Lochbihler
parents: 59731
diff changeset
   692
58587
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   693
end
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   694
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   695
lemma embed_pmf_transfer:
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   696
  "rel_fun (eq_onp (\<lambda>f. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ennreal (f x) \<partial>count_space UNIV) = 1)) pmf_as_measure.cr_pmf (\<lambda>f. density (count_space UNIV) (ennreal \<circ> f)) embed_pmf"
58587
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   697
  by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   698
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   699
lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   700
proof (transfer, elim conjE)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   701
  fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   702
  assume "prob_space M" then interpret prob_space M .
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   703
  show "M = density (count_space UNIV) (\<lambda>x. ennreal (measure M {x}))"
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   704
  proof (rule measure_eqI)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   705
    fix A :: "'a set"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   706
    have "(\<integral>\<^sup>+ x. ennreal (measure M {x}) * indicator A x \<partial>count_space UNIV) =
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   707
      (\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   708
      by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   709
    also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   710
      by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   711
    also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   712
      by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   713
         (auto simp: disjoint_family_on_def)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   714
    also have "\<dots> = emeasure M A"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   715
      using ae by (intro emeasure_eq_AE) auto
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   716
    finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ennreal (measure M {x}))) A"
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   717
      using emeasure_space_1 by (simp add: emeasure_density)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   718
  qed simp
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   719
qed
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   720
58587
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   721
lemma td_pmf_embed_pmf:
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   722
  "type_definition pmf embed_pmf {f::'a \<Rightarrow> real. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ennreal (f x) \<partial>count_space UNIV) = 1}"
58587
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   723
  unfolding type_definition_def
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   724
proof safe
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   725
  fix p :: "'a pmf"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   726
  have "(\<integral>\<^sup>+ x. 1 \<partial>measure_pmf p) = 1"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   727
    using measure_pmf.emeasure_space_1[of p] by simp
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   728
  then show *: "(\<integral>\<^sup>+ x. ennreal (pmf p x) \<partial>count_space UNIV) = 1"
58587
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   729
    by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   730
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   731
  show "embed_pmf (pmf p) = p"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   732
    by (intro measure_pmf_inject[THEN iffD1])
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   733
       (simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   734
next
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   735
  fix f :: "'a \<Rightarrow> real" assume "\<forall>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   736
  then show "pmf (embed_pmf f) = f"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   737
    by (auto intro!: pmf_embed_pmf)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   738
qed (rule pmf_nonneg)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   739
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   740
end
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   741
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   742
lemma nn_integral_measure_pmf: "(\<integral>\<^sup>+ x. f x \<partial>measure_pmf p) = \<integral>\<^sup>+ x. ennreal (pmf p x) * f x \<partial>count_space UNIV"
60068
ef2123db900c add various lemmas about pmfs
Andreas Lochbihler
parents: 59731
diff changeset
   743
by(simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg)
ef2123db900c add various lemmas about pmfs
Andreas Lochbihler
parents: 59731
diff changeset
   744
58587
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   745
locale pmf_as_function
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   746
begin
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   747
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   748
setup_lifting td_pmf_embed_pmf
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   749
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59665
diff changeset
   750
lemma set_pmf_transfer[transfer_rule]:
58730
b3fd0628f849 add transfer rule for set_pmf
hoelzl
parents: 58606
diff changeset
   751
  assumes "bi_total A"
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59665
diff changeset
   752
  shows "rel_fun (pcr_pmf A) (rel_set A) (\<lambda>f. {x. f x \<noteq> 0}) set_pmf"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61634
diff changeset
   753
  using \<open>bi_total A\<close>
58730
b3fd0628f849 add transfer rule for set_pmf
hoelzl
parents: 58606
diff changeset
   754
  by (auto simp: pcr_pmf_def cr_pmf_def rel_fun_def rel_set_def bi_total_def Bex_def set_pmf_iff)
b3fd0628f849 add transfer rule for set_pmf
hoelzl
parents: 58606
diff changeset
   755
     metis+
b3fd0628f849 add transfer rule for set_pmf
hoelzl
parents: 58606
diff changeset
   756
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   757
end
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   758
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   759
context
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   760
begin
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   761
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   762
interpretation pmf_as_function .
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   763
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   764
lemma pmf_eqI: "(\<And>i. pmf M i = pmf N i) \<Longrightarrow> M = N"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   765
  by transfer auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   766
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   767
lemma pmf_eq_iff: "M = N \<longleftrightarrow> (\<forall>i. pmf M i = pmf N i)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   768
  by (auto intro: pmf_eqI)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   769
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   770
lemma pmf_neq_exists_less:
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   771
  assumes "M \<noteq> N"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   772
  shows   "\<exists>x. pmf M x < pmf N x"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   773
proof (rule ccontr)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   774
  assume "\<not>(\<exists>x. pmf M x < pmf N x)"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   775
  hence ge: "pmf M x \<ge> pmf N x" for x by (auto simp: not_less)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   776
  from assms obtain x where "pmf M x \<noteq> pmf N x" by (auto simp: pmf_eq_iff)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   777
  with ge[of x] have gt: "pmf M x > pmf N x" by simp
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   778
  have "1 = measure (measure_pmf M) UNIV" by simp
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   779
  also have "\<dots> = measure (measure_pmf N) {x} + measure (measure_pmf N) (UNIV - {x})"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   780
    by (subst measure_pmf.finite_measure_Union [symmetric]) simp_all
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   781
  also from gt have "measure (measure_pmf N) {x} < measure (measure_pmf M) {x}" 
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   782
    by (simp add: measure_pmf_single)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   783
  also have "measure (measure_pmf N) (UNIV - {x}) \<le> measure (measure_pmf M) (UNIV - {x})"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   784
    by (subst (1 2) integral_pmf [symmetric]) 
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   785
       (intro integral_mono integrable_pmf, simp_all add: ge)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   786
  also have "measure (measure_pmf M) {x} + \<dots> = 1"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   787
    by (subst measure_pmf.finite_measure_Union [symmetric]) simp_all
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   788
  finally show False by simp_all
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   789
qed
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   790
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   791
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))"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   792
  unfolding pmf_eq_iff pmf_bind
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   793
proof
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   794
  fix i
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   795
  interpret B: prob_space "restrict_space B B"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   796
    by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   797
       (auto simp: AE_measure_pmf_iff)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   798
  interpret A: prob_space "restrict_space A A"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   799
    by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   800
       (auto simp: AE_measure_pmf_iff)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   801
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   802
  interpret AB: pair_prob_space "restrict_space A A" "restrict_space B B"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   803
    by unfold_locales
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   804
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   805
  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)"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   806
    by (rule integral_cong) (auto intro!: integral_pmf_restrict)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   807
  also have "\<dots> = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>restrict_space A A)"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   808
    by (intro integral_pmf_restrict B.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   809
              countable_set_pmf borel_measurable_count_space)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   810
  also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>restrict_space B B)"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   811
    by (rule AB.Fubini_integral[symmetric])
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   812
       (auto intro!: AB.integrable_const_bound[where B=1] measurable_pair_restrict_pmf2
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   813
             simp: pmf_nonneg pmf_le_1 measurable_restrict_space1)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   814
  also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>B)"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   815
    by (intro integral_pmf_restrict[symmetric] A.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   816
              countable_set_pmf borel_measurable_count_space)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   817
  also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>A \<partial>B)"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   818
    by (rule integral_cong) (auto intro!: integral_pmf_restrict[symmetric])
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   819
  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)" .
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   820
qed
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   821
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   822
lemma pair_map_pmf1: "pair_pmf (map_pmf f A) B = map_pmf (apfst f) (pair_pmf A B)"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   823
proof (safe intro!: pmf_eqI)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   824
  fix a :: "'a" and b :: "'b"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   825
  have [simp]: "\<And>c d. indicator (apfst f -` {(a, b)}) (c, d) = indicator (f -` {a}) c * (indicator {b} d::ennreal)"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   826
    by (auto split: split_indicator)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   827
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   828
  have "ennreal (pmf (pair_pmf (map_pmf f A) B) (a, b)) =
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   829
         ennreal (pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b))"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   830
    unfolding pmf_pair ennreal_pmf_map
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   831
    by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_multc pmf_nonneg
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   832
                  emeasure_map_pmf[symmetric] ennreal_mult del: emeasure_map_pmf)
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   833
  then show "pmf (pair_pmf (map_pmf f A) B) (a, b) = pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b)"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   834
    by (simp add: pmf_nonneg)
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   835
qed
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   836
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   837
lemma pair_map_pmf2: "pair_pmf A (map_pmf f B) = map_pmf (apsnd f) (pair_pmf A B)"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   838
proof (safe intro!: pmf_eqI)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   839
  fix a :: "'a" and b :: "'b"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   840
  have [simp]: "\<And>c d. indicator (apsnd f -` {(a, b)}) (c, d) = indicator {a} c * (indicator (f -` {b}) d::ennreal)"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   841
    by (auto split: split_indicator)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   842
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   843
  have "ennreal (pmf (pair_pmf A (map_pmf f B)) (a, b)) =
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   844
         ennreal (pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b))"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   845
    unfolding pmf_pair ennreal_pmf_map
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   846
    by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_cmult nn_integral_multc pmf_nonneg
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   847
                  emeasure_map_pmf[symmetric] ennreal_mult del: emeasure_map_pmf)
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   848
  then show "pmf (pair_pmf A (map_pmf f B)) (a, b) = pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b)"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   849
    by (simp add: pmf_nonneg)
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   850
qed
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   851
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   852
lemma map_pair: "map_pmf (\<lambda>(a, b). (f a, g b)) (pair_pmf A B) = pair_pmf (map_pmf f A) (map_pmf g B)"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   853
  by (simp add: pair_map_pmf2 pair_map_pmf1 map_pmf_comp split_beta')
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   854
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   855
end
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58730
diff changeset
   856
61634
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   857
lemma pair_return_pmf1: "pair_pmf (return_pmf x) y = map_pmf (Pair x) y"
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   858
by(simp add: pair_pmf_def bind_return_pmf map_pmf_def)
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   859
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   860
lemma pair_return_pmf2: "pair_pmf x (return_pmf y) = map_pmf (\<lambda>x. (x, y)) x"
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   861
by(simp add: pair_pmf_def bind_return_pmf map_pmf_def)
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   862
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   863
lemma pair_pair_pmf: "pair_pmf (pair_pmf u v) w = map_pmf (\<lambda>(x, (y, z)). ((x, y), z)) (pair_pmf u (pair_pmf v w))"
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   864
by(simp add: pair_pmf_def bind_return_pmf map_pmf_def bind_assoc_pmf)
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   865
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   866
lemma pair_commute_pmf: "pair_pmf x y = map_pmf (\<lambda>(x, y). (y, x)) (pair_pmf y x)"
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   867
unfolding pair_pmf_def by(subst bind_commute_pmf)(simp add: map_pmf_def bind_assoc_pmf bind_return_pmf)
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   868
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   869
lemma set_pmf_subset_singleton: "set_pmf p \<subseteq> {x} \<longleftrightarrow> p = return_pmf x"
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   870
proof(intro iffI pmf_eqI)
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   871
  fix i
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   872
  assume x: "set_pmf p \<subseteq> {x}"
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   873
  hence *: "set_pmf p = {x}" using set_pmf_not_empty[of p] by auto
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   874
  have "ennreal (pmf p x) = \<integral>\<^sup>+ i. indicator {x} i \<partial>p" by(simp add: emeasure_pmf_single)
61634
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   875
  also have "\<dots> = \<integral>\<^sup>+ i. 1 \<partial>p" by(rule nn_integral_cong_AE)(simp add: AE_measure_pmf_iff * )
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   876
  also have "\<dots> = 1" by simp
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   877
  finally show "pmf p i = pmf (return_pmf x) i" using x
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   878
    by(auto split: split_indicator simp add: pmf_eq_0_set_pmf)
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   879
qed auto
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   880
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   881
lemma bind_eq_return_pmf:
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   882
  "bind_pmf p f = return_pmf x \<longleftrightarrow> (\<forall>y\<in>set_pmf p. f y = return_pmf x)"
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   883
  (is "?lhs \<longleftrightarrow> ?rhs")
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   884
proof(intro iffI strip)
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   885
  fix y
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   886
  assume y: "y \<in> set_pmf p"
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   887
  assume "?lhs"
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   888
  hence "set_pmf (bind_pmf p f) = {x}" by simp
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   889
  hence "(\<Union>y\<in>set_pmf p. set_pmf (f y)) = {x}" by simp
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   890
  hence "set_pmf (f y) \<subseteq> {x}" using y by auto
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   891
  thus "f y = return_pmf x" by(simp add: set_pmf_subset_singleton)
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   892
next
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   893
  assume *: ?rhs
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   894
  show ?lhs
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   895
  proof(rule pmf_eqI)
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   896
    fix i
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   897
    have "ennreal (pmf (bind_pmf p f) i) = \<integral>\<^sup>+ y. ennreal (pmf (f y) i) \<partial>p"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   898
      by (simp add: ennreal_pmf_bind)
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   899
    also have "\<dots> = \<integral>\<^sup>+ y. ennreal (pmf (return_pmf x) i) \<partial>p"
61634
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   900
      by(rule nn_integral_cong_AE)(simp add: AE_measure_pmf_iff * )
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   901
    also have "\<dots> = ennreal (pmf (return_pmf x) i)"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   902
      by simp
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   903
    finally show "pmf (bind_pmf p f) i = pmf (return_pmf x) i"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   904
      by (simp add: pmf_nonneg)
61634
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   905
  qed
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   906
qed
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   907
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   908
lemma pmf_False_conv_True: "pmf p False = 1 - pmf p True"
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   909
proof -
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   910
  have "pmf p False + pmf p True = measure p {False} + measure p {True}"
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   911
    by(simp add: measure_pmf_single)
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   912
  also have "\<dots> = measure p ({False} \<union> {True})"
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   913
    by(subst measure_pmf.finite_measure_Union) simp_all
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   914
  also have "{False} \<union> {True} = space p" by auto
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   915
  finally show ?thesis by simp
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   916
qed
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   917
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   918
lemma pmf_True_conv_False: "pmf p True = 1 - pmf p False"
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   919
by(simp add: pmf_False_conv_True)
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
   920
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   921
subsection \<open> Conditional Probabilities \<close>
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   922
59670
dee043d19729 generalized bind_cond_pmf_cancel
hoelzl
parents: 59667
diff changeset
   923
lemma measure_pmf_zero_iff: "measure (measure_pmf p) s = 0 \<longleftrightarrow> set_pmf p \<inter> s = {}"
dee043d19729 generalized bind_cond_pmf_cancel
hoelzl
parents: 59667
diff changeset
   924
  by (subst measure_pmf.prob_eq_0) (auto simp: AE_measure_pmf_iff)
dee043d19729 generalized bind_cond_pmf_cancel
hoelzl
parents: 59667
diff changeset
   925
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   926
context
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   927
  fixes p :: "'a pmf" and s :: "'a set"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   928
  assumes not_empty: "set_pmf p \<inter> s \<noteq> {}"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   929
begin
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   930
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   931
interpretation pmf_as_measure .
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   932
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   933
lemma emeasure_measure_pmf_not_zero: "emeasure (measure_pmf p) s \<noteq> 0"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   934
proof
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   935
  assume "emeasure (measure_pmf p) s = 0"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   936
  then have "AE x in measure_pmf p. x \<notin> s"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   937
    by (rule AE_I[rotated]) auto
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   938
  with not_empty show False
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   939
    by (auto simp: AE_measure_pmf_iff)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   940
qed
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   941
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   942
lemma measure_measure_pmf_not_zero: "measure (measure_pmf p) s \<noteq> 0"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   943
  using emeasure_measure_pmf_not_zero by (simp add: measure_pmf.emeasure_eq_measure measure_nonneg)
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   944
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   945
lift_definition cond_pmf :: "'a pmf" is
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   946
  "uniform_measure (measure_pmf p) s"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   947
proof (intro conjI)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   948
  show "prob_space (uniform_measure (measure_pmf p) s)"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   949
    by (intro prob_space_uniform_measure) (auto simp: emeasure_measure_pmf_not_zero)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   950
  show "AE x in uniform_measure (measure_pmf p) s. measure (uniform_measure (measure_pmf p) s) {x} \<noteq> 0"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   951
    by (simp add: emeasure_measure_pmf_not_zero measure_measure_pmf_not_zero AE_uniform_measure
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   952
                  AE_measure_pmf_iff set_pmf.rep_eq less_top[symmetric])
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   953
qed simp
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   954
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   955
lemma pmf_cond: "pmf cond_pmf x = (if x \<in> s then pmf p x / measure p s else 0)"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   956
  by transfer (simp add: emeasure_measure_pmf_not_zero pmf.rep_eq)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   957
59665
37adca7fd48f add set_pmf lemmas to simpset
hoelzl
parents: 59664
diff changeset
   958
lemma set_cond_pmf[simp]: "set_pmf cond_pmf = set_pmf p \<inter> s"
62390
842917225d56 more canonical names
nipkow
parents: 62324
diff changeset
   959
  by (auto simp add: set_pmf_iff pmf_cond measure_measure_pmf_not_zero split: if_split_asm)
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   960
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   961
end
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   962
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   963
lemma measure_pmf_posI: "x \<in> set_pmf p \<Longrightarrow> x \<in> A \<Longrightarrow> measure_pmf.prob p A > 0"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   964
  using measure_measure_pmf_not_zero[of p A] by (subst zero_less_measure_iff) blast
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 63092
diff changeset
   965
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   966
lemma cond_map_pmf:
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   967
  assumes "set_pmf p \<inter> f -` s \<noteq> {}"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   968
  shows "cond_pmf (map_pmf f p) s = map_pmf f (cond_pmf p (f -` s))"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   969
proof -
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   970
  have *: "set_pmf (map_pmf f p) \<inter> s \<noteq> {}"
59665
37adca7fd48f add set_pmf lemmas to simpset
hoelzl
parents: 59664
diff changeset
   971
    using assms by auto
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   972
  { fix x
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   973
    have "ennreal (pmf (map_pmf f (cond_pmf p (f -` s))) x) =
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   974
      emeasure p (f -` s \<inter> f -` {x}) / emeasure p (f -` s)"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   975
      unfolding ennreal_pmf_map cond_pmf.rep_eq[OF assms] by (simp add: nn_integral_uniform_measure)
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   976
    also have "f -` s \<inter> f -` {x} = (if x \<in> s then f -` {x} else {})"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   977
      by auto
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   978
    also have "emeasure p (if x \<in> s then f -` {x} else {}) / emeasure p (f -` s) =
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   979
      ennreal (pmf (cond_pmf (map_pmf f p) s) x)"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   980
      using measure_measure_pmf_not_zero[OF *]
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   981
      by (simp add: pmf_cond[OF *] ennreal_pmf_map measure_pmf.emeasure_eq_measure
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   982
                    divide_ennreal pmf_nonneg measure_nonneg zero_less_measure_iff pmf_map)
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   983
    finally have "ennreal (pmf (cond_pmf (map_pmf f p) s) x) = ennreal (pmf (map_pmf f (cond_pmf p (f -` s))) x)"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   984
      by simp }
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   985
  then show ?thesis
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   986
    by (intro pmf_eqI) (simp add: pmf_nonneg)
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   987
qed
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   988
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   989
lemma bind_cond_pmf_cancel:
59670
dee043d19729 generalized bind_cond_pmf_cancel
hoelzl
parents: 59667
diff changeset
   990
  assumes [simp]: "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}"
dee043d19729 generalized bind_cond_pmf_cancel
hoelzl
parents: 59667
diff changeset
   991
  assumes [simp]: "\<And>y. y \<in> set_pmf q \<Longrightarrow> set_pmf p \<inter> {x. R x y} \<noteq> {}"
dee043d19729 generalized bind_cond_pmf_cancel
hoelzl
parents: 59667
diff changeset
   992
  assumes [simp]: "\<And>x y. x \<in> set_pmf p \<Longrightarrow> y \<in> set_pmf q \<Longrightarrow> R x y \<Longrightarrow> measure q {y. R x y} = measure p {x. R x y}"
dee043d19729 generalized bind_cond_pmf_cancel
hoelzl
parents: 59667
diff changeset
   993
  shows "bind_pmf p (\<lambda>x. cond_pmf q {y. R x y}) = q"
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
   994
proof (rule pmf_eqI)
59670
dee043d19729 generalized bind_cond_pmf_cancel
hoelzl
parents: 59667
diff changeset
   995
  fix i
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   996
  have "ennreal (pmf (bind_pmf p (\<lambda>x. cond_pmf q {y. R x y})) i) =
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   997
    (\<integral>\<^sup>+x. ennreal (pmf q i / measure p {x. R x i}) * ennreal (indicator {x. R x i} x) \<partial>p)"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   998
    by (auto simp add: ennreal_pmf_bind AE_measure_pmf_iff pmf_cond pmf_eq_0_set_pmf pmf_nonneg measure_nonneg
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   999
             intro!: nn_integral_cong_AE)
59670
dee043d19729 generalized bind_cond_pmf_cancel
hoelzl
parents: 59667
diff changeset
  1000
  also have "\<dots> = (pmf q i * measure p {x. R x i}) / measure p {x. R x i}"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
  1001
    by (simp add: pmf_nonneg measure_nonneg zero_ennreal_def[symmetric] ennreal_indicator
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
  1002
                  nn_integral_cmult measure_pmf.emeasure_eq_measure ennreal_mult[symmetric])
59670
dee043d19729 generalized bind_cond_pmf_cancel
hoelzl
parents: 59667
diff changeset
  1003
  also have "\<dots> = pmf q i"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
  1004
    by (cases "pmf q i = 0")
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
  1005
       (simp_all add: pmf_eq_0_set_pmf measure_measure_pmf_not_zero pmf_nonneg)
59670
dee043d19729 generalized bind_cond_pmf_cancel
hoelzl
parents: 59667
diff changeset
  1006
  finally show "pmf (bind_pmf p (\<lambda>x. cond_pmf q {y. R x y})) i = pmf q i"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
  1007
    by (simp add: pmf_nonneg)
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
  1008
qed
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
  1009
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
  1010
subsection \<open> Relator \<close>
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
  1011
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
  1012
inductive rel_pmf :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf \<Rightarrow> bool"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
  1013
for R p q
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
  1014
where
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59665
diff changeset
  1015
  "\<lbrakk> \<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y;
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
  1016
     map_pmf fst pq = p; map_pmf snd pq = q \<rbrakk>
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
  1017
  \<Longrightarrow> rel_pmf R p q"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
  1018
59681
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1019
lemma rel_pmfI:
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1020
  assumes R: "rel_set R (set_pmf p) (set_pmf q)"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1021
  assumes eq: "\<And>x y. x \<in> set_pmf p \<Longrightarrow> y \<in> set_pmf q \<Longrightarrow> R x y \<Longrightarrow>
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1022
    measure p {x. R x y} = measure q {y. R x y}"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1023
  shows "rel_pmf R p q"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1024
proof
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1025
  let ?pq = "bind_pmf p (\<lambda>x. bind_pmf (cond_pmf q {y. R x y}) (\<lambda>y. return_pmf (x, y)))"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1026
  have "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1027
    using R by (auto simp: rel_set_def)
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1028
  then show "\<And>x y. (x, y) \<in> set_pmf ?pq \<Longrightarrow> R x y"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1029
    by auto
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1030
  show "map_pmf fst ?pq = p"
60068
ef2123db900c add various lemmas about pmfs
Andreas Lochbihler
parents: 59731
diff changeset
  1031
    by (simp add: map_bind_pmf bind_return_pmf')
59681
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1032
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1033
  show "map_pmf snd ?pq = q"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1034
    using R eq
60068
ef2123db900c add various lemmas about pmfs
Andreas Lochbihler
parents: 59731
diff changeset
  1035
    apply (simp add: bind_cond_pmf_cancel map_bind_pmf bind_return_pmf')
59681
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1036
    apply (rule bind_cond_pmf_cancel)
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1037
    apply (auto simp: rel_set_def)
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1038
    done
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1039
qed
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1040
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1041
lemma rel_pmf_imp_rel_set: "rel_pmf R p q \<Longrightarrow> rel_set R (set_pmf p) (set_pmf q)"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1042
  by (force simp add: rel_pmf.simps rel_set_def)
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1043
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1044
lemma rel_pmfD_measure:
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1045
  assumes rel_R: "rel_pmf R p q" and R: "\<And>a b. R a b \<Longrightarrow> R a y \<longleftrightarrow> R x b"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1046
  assumes "x \<in> set_pmf p" "y \<in> set_pmf q"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1047
  shows "measure p {x. R x y} = measure q {y. R x y}"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1048
proof -
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1049
  from rel_R obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1050
    and eq: "p = map_pmf fst pq" "q = map_pmf snd pq"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1051
    by (auto elim: rel_pmf.cases)
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1052
  have "measure p {x. R x y} = measure pq {x. R (fst x) y}"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1053
    by (simp add: eq map_pmf_rep_eq measure_distr)
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1054
  also have "\<dots> = measure pq {y. R x (snd y)}"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1055
    by (intro measure_pmf.finite_measure_eq_AE)
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1056
       (auto simp: AE_measure_pmf_iff R dest!: pq)
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1057
  also have "\<dots> = measure q {y. R x y}"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1058
    by (simp add: eq map_pmf_rep_eq measure_distr)
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1059
  finally show "measure p {x. R x y} = measure q {y. R x y}" .
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1060
qed
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1061
61634
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
  1062
lemma rel_pmf_measureD:
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
  1063
  assumes "rel_pmf R p q"
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
  1064
  shows "measure (measure_pmf p) A \<le> measure (measure_pmf q) {y. \<exists>x\<in>A. R x y}" (is "?lhs \<le> ?rhs")
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
  1065
using assms
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
  1066
proof cases
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
  1067
  fix pq
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
  1068
  assume R: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
  1069
    and p[symmetric]: "map_pmf fst pq = p"
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
  1070
    and q[symmetric]: "map_pmf snd pq = q"
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
  1071
  have "?lhs = measure (measure_pmf pq) (fst -` A)" by(simp add: p)
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
  1072
  also have "\<dots> \<le> measure (measure_pmf pq) {y. \<exists>x\<in>A. R x (snd y)}"
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
  1073
    by(rule measure_pmf.finite_measure_mono_AE)(auto 4 3 simp add: AE_measure_pmf_iff dest: R)
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
  1074
  also have "\<dots> = ?rhs" by(simp add: q)
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
  1075
  finally show ?thesis .
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
  1076
qed
48e2de1b1df5 add various lemmas
Andreas Lochbihler
parents: 61610
diff changeset
  1077
59681
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1078
lemma rel_pmf_iff_measure:
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1079
  assumes "symp R" "transp R"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1080
  shows "rel_pmf R p q \<longleftrightarrow>
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1081
    rel_set R (set_pmf p) (set_pmf q) \<and>
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1082
    (\<forall>x\<in>set_pmf p. \<forall>y\<in>set_pmf q. R x y \<longrightarrow> measure p {x. R x y} = measure q {y. R x y})"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1083
  by (safe intro!: rel_pmf_imp_rel_set rel_pmfI)
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1084
     (auto intro!: rel_pmfD_measure dest: sympD[OF \<open>symp R\<close>] transpD[OF \<open>transp R\<close>])
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1085
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1086
lemma quotient_rel_set_disjoint:
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1087
  "equivp R \<Longrightarrow> C \<in> UNIV // {(x, y). R x y} \<Longrightarrow> rel_set R A B \<Longrightarrow> A \<inter> C = {} \<longleftrightarrow> B \<inter> C = {}"
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
  1088
  using in_quotient_imp_closed[of UNIV "{(x, y). R x y}" C]
59681
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1089
  by (auto 0 0 simp: equivp_equiv rel_set_def set_eq_iff elim: equivpE)
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1090
     (blast dest: equivp_symp)+
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1091
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1092
lemma quotientD: "equiv X R \<Longrightarrow> A \<in> X // R \<Longrightarrow> x \<in> A \<Longrightarrow> A = R `` {x}"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1093
  by (metis Image_singleton_iff equiv_class_eq_iff quotientE)
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1094
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1095
lemma rel_pmf_iff_equivp:
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1096
  assumes "equivp R"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1097
  shows "rel_pmf R p q \<longleftrightarrow> (\<forall>C\<in>UNIV // {(x, y). R x y}. measure p C = measure q C)"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1098
    (is "_ \<longleftrightarrow>   (\<forall>C\<in>_//?R. _)")
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1099
proof (subst rel_pmf_iff_measure, safe)
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1100
  show "symp R" "transp R"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1101
    using assms by (auto simp: equivp_reflp_symp_transp)
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1102
next
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1103
  fix C assume C: "C \<in> UNIV // ?R" and R: "rel_set R (set_pmf p) (set_pmf q)"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1104
  assume eq: "\<forall>x\<in>set_pmf p. \<forall>y\<in>set_pmf q. R x y \<longrightarrow> measure p {x. R x y} = measure q {y. R x y}"
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
  1105
59681
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1106
  show "measure p C = measure q C"
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1107
  proof (cases "p \<inter> C = {}")
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1108
    case True
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1109
    then have "q \<inter> C = {}"
59681
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1110
      using quotient_rel_set_disjoint[OF assms C R] by simp
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1111
    with True show ?thesis
59681
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1112
      unfolding measure_pmf_zero_iff[symmetric] by simp
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1113
  next
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1114
    case False
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1115
    then have "q \<inter> C \<noteq> {}"
59681
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1116
      using quotient_rel_set_disjoint[OF assms C R] by simp
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63343
diff changeset
  1117
    with False obtain x y where in_set: "x \<in> set_pmf p" "y \<in> set_pmf q" and in_C: "x \<in> C" "y \<in> C"
59681
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1118
      by auto
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1119
    then have "R x y"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1120
      using in_quotient_imp_in_rel[of UNIV ?R C x y] C assms
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1121
      by (simp add: equivp_equiv)
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1122
    with in_set eq have "measure p {x. R x y} = measure q {y. R x y}"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1123
      by auto
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1124
    moreover have "{y. R x y} = C"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61634
diff changeset
  1125
      using assms \<open>x \<in> C\<close> C quotientD[of UNIV ?R C x] by (simp add: equivp_equiv)
59681
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1126
    moreover have "{x. R x y} = C"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61634
diff changeset
  1127
      using assms \<open>y \<in> C\<close> C quotientD[of UNIV "?R" C y] sympD[of R]
59681
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1128
      by (auto simp add: equivp_equiv elim: equivpE)
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1129
    ultimately show ?thesis
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1130
      by auto
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1131
  qed
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1132
next
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1133
  assume eq: "\<forall>C\<in>UNIV // ?R. measure p C = measure q C"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1134
  show "rel_set R (set_pmf p) (set_pmf q)"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1135
    unfolding rel_set_def
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1136
  proof safe
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1137
    fix x assume x: "x \<in> set_pmf p"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1138
    have "{y. R x y} \<in> UNIV // ?R"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1139
      by (auto simp: quotient_def)
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1140
    with eq have *: "measure q {y. R x y} = measure p {y. R x y}"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1141
      by auto
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1142
    have "measure q {y. R x y} \<noteq> 0"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1143
      using x assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp)
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1144
    then show "\<exists>y\<in>set_pmf q. R x y"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1145
      unfolding measure_pmf_zero_iff by auto
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1146
  next
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1147
    fix y assume y: "y \<in> set_pmf q"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1148
    have "{x. R x y} \<in> UNIV // ?R"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1149
      using assms by (auto simp: quotient_def dest: equivp_symp)
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1150
    with eq have *: "measure p {x. R x y} = measure q {x. R x y}"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1151
      by auto
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1152
    have "measure p {x. R x y} \<noteq> 0"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1153
      using y assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp)
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1154
    then show "\<exists>x\<in>set_pmf p. R x y"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1155
      unfolding measure_pmf_zero_iff by auto
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1156
  qed
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1157
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1158
  fix x y assume "x \<in> set_pmf p" "y \<in> set_pmf q" "R x y"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1159
  have "{y. R x y} \<in> UNIV // ?R" "{x. R x y} = {y. R x y}"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61634
diff changeset
  1160
    using assms \<open>R x y\<close> by (auto simp: quotient_def dest: equivp_symp equivp_transp)
59681
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1161
  with eq show "measure p {x. R x y} = measure q {y. R x y}"
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1162
    by auto
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1163
qed
f24ab09e4c37 rel_pmf on equivalence relation
hoelzl
parents: 59670
diff changeset
  1164
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
  1165
bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: rel_pmf
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
  1166
proof -
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
  1167
  show "map_pmf id = id" by (rule map_pmf_id)
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59665
diff changeset
  1168
  show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose)
59664
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
  1169
  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"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
  1170
    by (intro map_pmf_cong refl)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
  1171
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
  1172
  show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
  1173
    by (rule pmf_set_map)
224741ede5ae build pmf's on bind
hoelzl
parents: 59557
diff changeset
  1174
</