author | paulson <lp15@cam.ac.uk> |
Tue, 10 Nov 2015 14:18:41 +0000 | |
changeset 61609 | 77b453bd616f |
parent 61424 | c3658c18b7bc |
child 61610 | 4f54d2759a0b |
permissions | -rw-r--r-- |
58606 | 1 |
(* 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: emeasure_le_0_iff) } |
|
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then show "P x" by auto |
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qed |
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||
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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 ereal_divide': "b \<noteq> 0 \<Longrightarrow> ereal (a / b) = ereal a / ereal b" |
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using ereal_divide[of a b] by simp |
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||
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lemma (in finite_measure) countable_support: |
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"countable {x. measure M {x} \<noteq> 0}" |
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proof cases |
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assume "measure M (space M) = 0" |
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with bounded_measure measure_le_0_iff have "{x. measure M {x} \<noteq> 0} = {}" |
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by auto |
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then show ?thesis |
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by simp |
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next |
|
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let ?M = "measure M (space M)" and ?m = "\<lambda>x. measure M {x}" |
|
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assume "?M \<noteq> 0" |
|
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then have *: "{x. ?m x \<noteq> 0} = (\<Union>n. {x. ?M / Suc n < ?m x})" |
|
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using reals_Archimedean[of "?m x / ?M" for x] |
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by (auto simp: field_simps not_le[symmetric] measure_nonneg divide_le_0_iff measure_le_0_iff) |
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have **: "\<And>n. finite {x. ?M / Suc n < ?m x}" |
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proof (rule ccontr) |
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fix n assume "infinite {x. ?M / Suc n < ?m x}" (is "infinite ?X") |
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then obtain X where "finite X" "card X = Suc (Suc n)" "X \<subseteq> ?X" |
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by (metis infinite_arbitrarily_large) |
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parents:
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from this(3) have *: "\<And>x. x \<in> X \<Longrightarrow> ?M / Suc n \<le> ?m x" |
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by auto |
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{ fix x assume "x \<in> X" |
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from `?M \<noteq> 0` *[OF this] have "?m x \<noteq> 0" by (auto simp: field_simps measure_le_0_iff) |
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then have "{x} \<in> sets M" by (auto dest: measure_notin_sets) } |
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note singleton_sets = this |
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have "?M < (\<Sum>x\<in>X. ?M / Suc n)" |
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parents:
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using `?M \<noteq> 0` |
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by (simp add: `card X = Suc (Suc n)` of_nat_Suc field_simps less_le measure_nonneg) |
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also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)" |
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by (rule setsum_mono) fact |
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also have "\<dots> = measure M (\<Union>x\<in>X. {x})" |
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using singleton_sets `finite X` |
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by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def) |
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finally have "?M < measure M (\<Union>x\<in>X. {x})" . |
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moreover have "measure M (\<Union>x\<in>X. {x}) \<le> ?M" |
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using singleton_sets[THEN sets.sets_into_space] by (intro finite_measure_mono) auto |
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ultimately show False by simp |
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qed |
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show ?thesis |
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unfolding * by (intro countable_UN countableI_type countable_finite[OF **]) |
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qed |
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lemma (in finite_measure) AE_support_countable: |
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assumes [simp]: "sets M = UNIV" |
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shows "(AE x in M. measure M {x} \<noteq> 0) \<longleftrightarrow> (\<exists>S. countable S \<and> (AE x in M. x \<in> S))" |
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proof |
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assume "\<exists>S. countable S \<and> (AE x in M. x \<in> S)" |
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then obtain S where S[intro]: "countable S" and ae: "AE x in M. x \<in> S" |
|
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by auto |
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paulson <lp15@cam.ac.uk>
parents:
59665
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then have "emeasure M (\<Union>x\<in>{x\<in>S. emeasure M {x} \<noteq> 0}. {x}) = |
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(\<integral>\<^sup>+ x. emeasure M {x} * indicator {x\<in>S. emeasure M {x} \<noteq> 0} x \<partial>count_space UNIV)" |
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by (subst emeasure_UN_countable) |
|
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(auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space) |
|
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also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} * indicator S x \<partial>count_space UNIV)" |
|
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by (auto intro!: nn_integral_cong split: split_indicator) |
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also have "\<dots> = emeasure M (\<Union>x\<in>S. {x})" |
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by (subst emeasure_UN_countable) |
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(auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space) |
|
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also have "\<dots> = emeasure M (space M)" |
|
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using ae by (intro emeasure_eq_AE) auto |
|
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finally have "emeasure M {x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0} = emeasure M (space M)" |
|
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by (simp add: emeasure_single_in_space cong: rev_conj_cong) |
|
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with finite_measure_compl[of "{x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0}"] |
|
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have "AE x in M. x \<in> S \<and> emeasure M {x} \<noteq> 0" |
|
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by (intro AE_I[OF order_refl]) (auto simp: emeasure_eq_measure set_diff_eq cong: conj_cong) |
|
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then show "AE x in M. measure M {x} \<noteq> 0" |
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by (auto simp: emeasure_eq_measure) |
|
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qed (auto intro!: exI[of _ "{x. measure M {x} \<noteq> 0}"] countable_support) |
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||
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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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||
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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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|
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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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59000 | 142 |
by simp |
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||
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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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146 |
|
59048 | 147 |
lemma measure_pmf_in_subprob_algebra[measurable (raw)]: "measure_pmf x \<in> space (subprob_algebra (count_space UNIV))" |
148 |
by (simp add: space_subprob_algebra subprob_space_measure_pmf) |
|
149 |
||
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lemma measurable_pmf_measure1[simp]: "measurable (M :: 'a pmf) N = UNIV \<rightarrow> space N" |
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151 |
by (auto simp: measurable_def) |
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152 |
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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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155 |
|
59664 | 156 |
lemma measurable_pair_restrict_pmf2: |
157 |
assumes "countable A" |
|
158 |
assumes [measurable]: "\<And>y. y \<in> A \<Longrightarrow> (\<lambda>x. f (x, y)) \<in> measurable M L" |
|
159 |
shows "f \<in> measurable (M \<Otimes>\<^sub>M restrict_space (measure_pmf N) A) L" (is "f \<in> measurable ?M _") |
|
160 |
proof - |
|
161 |
have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)" |
|
162 |
by (simp add: restrict_count_space) |
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163 |
|
59664 | 164 |
show ?thesis |
165 |
by (intro measurable_compose_countable'[where f="\<lambda>a b. f (fst b, a)" and g=snd and I=A, |
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166 |
unfolded prod.collapse] assms) |
59664 | 167 |
measurable |
168 |
qed |
|
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169 |
|
59664 | 170 |
lemma measurable_pair_restrict_pmf1: |
171 |
assumes "countable A" |
|
172 |
assumes [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N L" |
|
173 |
shows "f \<in> measurable (restrict_space (measure_pmf M) A \<Otimes>\<^sub>M N) L" |
|
174 |
proof - |
|
175 |
have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)" |
|
176 |
by (simp add: restrict_count_space) |
|
59000 | 177 |
|
59664 | 178 |
show ?thesis |
179 |
by (intro measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A, |
|
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180 |
unfolded prod.collapse] assms) |
59664 | 181 |
measurable |
182 |
qed |
|
183 |
||
184 |
lift_definition pmf :: "'a pmf \<Rightarrow> 'a \<Rightarrow> real" is "\<lambda>M x. measure M {x}" . |
|
185 |
||
186 |
lift_definition set_pmf :: "'a pmf \<Rightarrow> 'a set" is "\<lambda>M. {x. measure M {x} \<noteq> 0}" . |
|
187 |
declare [[coercion set_pmf]] |
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188 |
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189 |
lemma AE_measure_pmf: "AE x in (M::'a pmf). x \<in> M" |
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190 |
by transfer simp |
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191 |
|
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192 |
lemma emeasure_pmf_single_eq_zero_iff: |
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193 |
fixes M :: "'a pmf" |
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194 |
shows "emeasure M {y} = 0 \<longleftrightarrow> y \<notin> M" |
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195 |
by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure]) |
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|
196 |
|
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|
197 |
lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \<longleftrightarrow> (\<forall>y\<in>M. P y)" |
59664 | 198 |
using AE_measure_singleton[of M] AE_measure_pmf[of M] |
199 |
by (auto simp: set_pmf.rep_eq) |
|
200 |
||
201 |
lemma countable_set_pmf [simp]: "countable (set_pmf p)" |
|
202 |
by transfer (metis prob_space.finite_measure finite_measure.countable_support) |
|
203 |
||
204 |
lemma pmf_positive: "x \<in> set_pmf p \<Longrightarrow> 0 < pmf p x" |
|
205 |
by transfer (simp add: less_le measure_nonneg) |
|
206 |
||
207 |
lemma pmf_nonneg: "0 \<le> pmf p x" |
|
208 |
by transfer (simp add: measure_nonneg) |
|
209 |
||
210 |
lemma pmf_le_1: "pmf p x \<le> 1" |
|
211 |
by (simp add: pmf.rep_eq) |
|
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|
212 |
|
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|
213 |
lemma set_pmf_not_empty: "set_pmf M \<noteq> {}" |
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|
214 |
using AE_measure_pmf[of M] by (intro notI) simp |
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|
215 |
|
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|
216 |
lemma set_pmf_iff: "x \<in> set_pmf M \<longleftrightarrow> pmf M x \<noteq> 0" |
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|
217 |
by transfer simp |
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218 |
|
59664 | 219 |
lemma set_pmf_eq: "set_pmf M = {x. pmf M x \<noteq> 0}" |
220 |
by (auto simp: set_pmf_iff) |
|
221 |
||
222 |
lemma emeasure_pmf_single: |
|
223 |
fixes M :: "'a pmf" |
|
224 |
shows "emeasure M {x} = pmf M x" |
|
225 |
by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure]) |
|
226 |
||
60068 | 227 |
lemma measure_pmf_single: "measure (measure_pmf M) {x} = pmf M x" |
228 |
using emeasure_pmf_single[of M x] by(simp add: measure_pmf.emeasure_eq_measure) |
|
229 |
||
59000 | 230 |
lemma emeasure_measure_pmf_finite: "finite S \<Longrightarrow> emeasure (measure_pmf M) S = (\<Sum>s\<in>S. pmf M s)" |
231 |
by (subst emeasure_eq_setsum_singleton) (auto simp: emeasure_pmf_single) |
|
232 |
||
59023 | 233 |
lemma measure_measure_pmf_finite: "finite S \<Longrightarrow> measure (measure_pmf M) S = setsum (pmf M) S" |
59425 | 234 |
using emeasure_measure_pmf_finite[of S M] by(simp add: measure_pmf.emeasure_eq_measure) |
59023 | 235 |
|
59000 | 236 |
lemma nn_integral_measure_pmf_support: |
237 |
fixes f :: "'a \<Rightarrow> ereal" |
|
238 |
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" |
|
239 |
shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>A. f x * pmf M x)" |
|
240 |
proof - |
|
241 |
have "(\<integral>\<^sup>+x. f x \<partial>M) = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)" |
|
242 |
using nn by (intro nn_integral_cong_AE) (auto simp: AE_measure_pmf_iff split: split_indicator) |
|
243 |
also have "\<dots> = (\<Sum>x\<in>A. f x * emeasure M {x})" |
|
244 |
using assms by (intro nn_integral_indicator_finite) auto |
|
245 |
finally show ?thesis |
|
246 |
by (simp add: emeasure_measure_pmf_finite) |
|
247 |
qed |
|
248 |
||
249 |
lemma nn_integral_measure_pmf_finite: |
|
250 |
fixes f :: "'a \<Rightarrow> ereal" |
|
251 |
assumes f: "finite (set_pmf M)" and nn: "\<And>x. x \<in> set_pmf M \<Longrightarrow> 0 \<le> f x" |
|
252 |
shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>set_pmf M. f x * pmf M x)" |
|
253 |
using assms by (intro nn_integral_measure_pmf_support) auto |
|
254 |
lemma integrable_measure_pmf_finite: |
|
255 |
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" |
|
256 |
shows "finite (set_pmf M) \<Longrightarrow> integrable M f" |
|
257 |
by (auto intro!: integrableI_bounded simp: nn_integral_measure_pmf_finite) |
|
258 |
||
259 |
lemma integral_measure_pmf: |
|
260 |
assumes [simp]: "finite A" and "\<And>a. a \<in> set_pmf M \<Longrightarrow> f a \<noteq> 0 \<Longrightarrow> a \<in> A" |
|
261 |
shows "(\<integral>x. f x \<partial>measure_pmf M) = (\<Sum>a\<in>A. f a * pmf M a)" |
|
262 |
proof - |
|
263 |
have "(\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x * indicator A x \<partial>measure_pmf M)" |
|
264 |
using assms(2) by (intro integral_cong_AE) (auto split: split_indicator simp: AE_measure_pmf_iff) |
|
265 |
also have "\<dots> = (\<Sum>a\<in>A. f a * pmf M a)" |
|
266 |
by (subst integral_indicator_finite_real) (auto simp: measure_def emeasure_measure_pmf_finite) |
|
267 |
finally show ?thesis . |
|
268 |
qed |
|
269 |
||
270 |
lemma integrable_pmf: "integrable (count_space X) (pmf M)" |
|
271 |
proof - |
|
272 |
have " (\<integral>\<^sup>+ x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+ x. pmf M x \<partial>count_space (M \<inter> X))" |
|
273 |
by (auto simp add: nn_integral_count_space_indicator set_pmf_iff intro!: nn_integral_cong split: split_indicator) |
|
274 |
then have "integrable (count_space X) (pmf M) = integrable (count_space (M \<inter> X)) (pmf M)" |
|
275 |
by (simp add: integrable_iff_bounded pmf_nonneg) |
|
276 |
then show ?thesis |
|
59023 | 277 |
by (simp add: pmf.rep_eq measure_pmf.integrable_measure disjoint_family_on_def) |
59000 | 278 |
qed |
279 |
||
280 |
lemma integral_pmf: "(\<integral>x. pmf M x \<partial>count_space X) = measure M X" |
|
281 |
proof - |
|
282 |
have "(\<integral>x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+x. pmf M x \<partial>count_space X)" |
|
283 |
by (simp add: pmf_nonneg integrable_pmf nn_integral_eq_integral) |
|
284 |
also have "\<dots> = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space (X \<inter> M))" |
|
285 |
by (auto intro!: nn_integral_cong_AE split: split_indicator |
|
286 |
simp: pmf.rep_eq measure_pmf.emeasure_eq_measure nn_integral_count_space_indicator |
|
287 |
AE_count_space set_pmf_iff) |
|
288 |
also have "\<dots> = emeasure M (X \<inter> M)" |
|
289 |
by (rule emeasure_countable_singleton[symmetric]) (auto intro: countable_set_pmf) |
|
290 |
also have "\<dots> = emeasure M X" |
|
291 |
by (auto intro!: emeasure_eq_AE simp: AE_measure_pmf_iff) |
|
292 |
finally show ?thesis |
|
293 |
by (simp add: measure_pmf.emeasure_eq_measure) |
|
294 |
qed |
|
295 |
||
296 |
lemma integral_pmf_restrict: |
|
297 |
"(f::'a \<Rightarrow> 'b::{banach, second_countable_topology}) \<in> borel_measurable (count_space UNIV) \<Longrightarrow> |
|
298 |
(\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x \<partial>restrict_space M M)" |
|
299 |
by (auto intro!: integral_cong_AE simp add: integral_restrict_space AE_measure_pmf_iff) |
|
300 |
||
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|
301 |
lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1" |
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changeset
|
302 |
proof - |
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|
303 |
have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)" |
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changeset
|
304 |
by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf) |
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changeset
|
305 |
then show ?thesis |
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changeset
|
306 |
using measure_pmf.emeasure_space_1 by simp |
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changeset
|
307 |
qed |
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diff
changeset
|
308 |
|
59490 | 309 |
lemma emeasure_pmf_UNIV [simp]: "emeasure (measure_pmf M) UNIV = 1" |
310 |
using measure_pmf.emeasure_space_1[of M] by simp |
|
311 |
||
59023 | 312 |
lemma in_null_sets_measure_pmfI: |
313 |
"A \<inter> set_pmf p = {} \<Longrightarrow> A \<in> null_sets (measure_pmf p)" |
|
314 |
using emeasure_eq_0_AE[where ?P="\<lambda>x. x \<in> A" and M="measure_pmf p"] |
|
315 |
by(auto simp add: null_sets_def AE_measure_pmf_iff) |
|
316 |
||
59664 | 317 |
lemma measure_subprob: "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))" |
318 |
by (simp add: space_subprob_algebra subprob_space_measure_pmf) |
|
319 |
||
320 |
subsection \<open> Monad Interpretation \<close> |
|
321 |
||
322 |
lemma measurable_measure_pmf[measurable]: |
|
323 |
"(\<lambda>x. measure_pmf (M x)) \<in> measurable (count_space UNIV) (subprob_algebra (count_space UNIV))" |
|
324 |
by (auto simp: space_subprob_algebra intro!: prob_space_imp_subprob_space) unfold_locales |
|
325 |
||
326 |
lemma bind_measure_pmf_cong: |
|
327 |
assumes "\<And>x. A x \<in> space (subprob_algebra N)" "\<And>x. B x \<in> space (subprob_algebra N)" |
|
328 |
assumes "\<And>i. i \<in> set_pmf x \<Longrightarrow> A i = B i" |
|
329 |
shows "bind (measure_pmf x) A = bind (measure_pmf x) B" |
|
330 |
proof (rule measure_eqI) |
|
331 |
show "sets (measure_pmf x \<guillemotright>= A) = sets (measure_pmf x \<guillemotright>= B)" |
|
332 |
using assms by (subst (1 2) sets_bind) (auto simp: space_subprob_algebra) |
|
333 |
next |
|
334 |
fix X assume "X \<in> sets (measure_pmf x \<guillemotright>= A)" |
|
335 |
then have X: "X \<in> sets N" |
|
336 |
using assms by (subst (asm) sets_bind) (auto simp: space_subprob_algebra) |
|
337 |
show "emeasure (measure_pmf x \<guillemotright>= A) X = emeasure (measure_pmf x \<guillemotright>= B) X" |
|
338 |
using assms |
|
339 |
by (subst (1 2) emeasure_bind[where N=N, OF _ _ X]) |
|
340 |
(auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff) |
|
341 |
qed |
|
342 |
||
343 |
lift_definition bind_pmf :: "'a pmf \<Rightarrow> ('a \<Rightarrow> 'b pmf ) \<Rightarrow> 'b pmf" is bind |
|
344 |
proof (clarify, intro conjI) |
|
345 |
fix f :: "'a measure" and g :: "'a \<Rightarrow> 'b measure" |
|
346 |
assume "prob_space f" |
|
347 |
then interpret f: prob_space f . |
|
348 |
assume "sets f = UNIV" and ae_f: "AE x in f. measure f {x} \<noteq> 0" |
|
349 |
then have s_f[simp]: "sets f = sets (count_space UNIV)" |
|
350 |
by simp |
|
351 |
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)" |
|
352 |
then have g: "\<And>x. prob_space (g x)" and s_g[simp]: "\<And>x. sets (g x) = sets (count_space UNIV)" |
|
353 |
and ae_g: "\<And>x. AE y in g x. measure (g x) {y} \<noteq> 0" |
|
354 |
by auto |
|
355 |
||
356 |
have [measurable]: "g \<in> measurable f (subprob_algebra (count_space UNIV))" |
|
357 |
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
|
358 |
|
59664 | 359 |
show "prob_space (f \<guillemotright>= g)" |
360 |
using g by (intro f.prob_space_bind[where S="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
|
361 |
then interpret fg: prob_space "f \<guillemotright>= g" . |
59664 | 362 |
show [simp]: "sets (f \<guillemotright>= g) = UNIV" |
363 |
using sets_eq_imp_space_eq[OF s_f] |
|
364 |
by (subst sets_bind[where N="count_space UNIV"]) auto |
|
365 |
show "AE x in f \<guillemotright>= g. measure (f \<guillemotright>= g) {x} \<noteq> 0" |
|
366 |
apply (simp add: fg.prob_eq_0 AE_bind[where B="count_space UNIV"]) |
|
367 |
using ae_f |
|
368 |
apply eventually_elim |
|
369 |
using ae_g |
|
370 |
apply eventually_elim |
|
371 |
apply (auto dest: AE_measure_singleton) |
|
372 |
done |
|
373 |
qed |
|
374 |
||
375 |
lemma ereal_pmf_bind: "pmf (bind_pmf N f) i = (\<integral>\<^sup>+x. pmf (f x) i \<partial>measure_pmf N)" |
|
376 |
unfolding pmf.rep_eq bind_pmf.rep_eq |
|
377 |
by (auto simp: measure_pmf.measure_bind[where N="count_space UNIV"] measure_subprob measure_nonneg |
|
378 |
intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1]) |
|
379 |
||
380 |
lemma pmf_bind: "pmf (bind_pmf N f) i = (\<integral>x. pmf (f x) i \<partial>measure_pmf N)" |
|
381 |
using ereal_pmf_bind[of N f i] |
|
382 |
by (subst (asm) nn_integral_eq_integral) |
|
383 |
(auto simp: pmf_nonneg pmf_le_1 |
|
384 |
intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1]) |
|
385 |
||
386 |
lemma bind_pmf_const[simp]: "bind_pmf M (\<lambda>x. c) = c" |
|
387 |
by transfer (simp add: bind_const' prob_space_imp_subprob_space) |
|
388 |
||
59665 | 389 |
lemma set_bind_pmf[simp]: "set_pmf (bind_pmf M N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))" |
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
|
390 |
unfolding set_pmf_eq ereal_eq_0(1)[symmetric] ereal_pmf_bind |
59664 | 391 |
by (auto simp add: nn_integral_0_iff_AE AE_measure_pmf_iff set_pmf_eq not_le less_le pmf_nonneg) |
392 |
||
393 |
lemma bind_pmf_cong: |
|
394 |
assumes "p = q" |
|
395 |
shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g" |
|
396 |
unfolding `p = q`[symmetric] measure_pmf_inject[symmetric] bind_pmf.rep_eq |
|
397 |
by (auto simp: AE_measure_pmf_iff Pi_iff space_subprob_algebra subprob_space_measure_pmf |
|
398 |
sets_bind[where N="count_space UNIV"] emeasure_bind[where N="count_space UNIV"] |
|
399 |
intro!: nn_integral_cong_AE measure_eqI) |
|
400 |
||
401 |
lemma bind_pmf_cong_simp: |
|
402 |
"p = q \<Longrightarrow> (\<And>x. x \<in> set_pmf q =simp=> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g" |
|
403 |
by (simp add: simp_implies_def cong: bind_pmf_cong) |
|
404 |
||
405 |
lemma measure_pmf_bind: "measure_pmf (bind_pmf M f) = (measure_pmf M \<guillemotright>= (\<lambda>x. measure_pmf (f x)))" |
|
406 |
by transfer simp |
|
407 |
||
408 |
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)" |
|
409 |
using measurable_measure_pmf[of N] |
|
410 |
unfolding measure_pmf_bind |
|
411 |
apply (subst (1 3) nn_integral_max_0[symmetric]) |
|
412 |
apply (intro nn_integral_bind[where B="count_space UNIV"]) |
|
413 |
apply auto |
|
414 |
done |
|
415 |
||
416 |
lemma emeasure_bind_pmf[simp]: "emeasure (bind_pmf M N) X = (\<integral>\<^sup>+x. emeasure (N x) X \<partial>M)" |
|
417 |
using measurable_measure_pmf[of N] |
|
418 |
unfolding measure_pmf_bind |
|
419 |
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
|
420 |
|
59664 | 421 |
lift_definition return_pmf :: "'a \<Rightarrow> 'a pmf" is "return (count_space UNIV)" |
422 |
by (auto intro!: prob_space_return simp: AE_return measure_return) |
|
423 |
||
424 |
lemma bind_return_pmf: "bind_pmf (return_pmf x) f = f x" |
|
425 |
by transfer |
|
426 |
(auto intro!: prob_space_imp_subprob_space bind_return[where N="count_space UNIV"] |
|
427 |
simp: space_subprob_algebra) |
|
428 |
||
59665 | 429 |
lemma set_return_pmf[simp]: "set_pmf (return_pmf x) = {x}" |
59664 | 430 |
by transfer (auto simp add: measure_return split: split_indicator) |
431 |
||
432 |
lemma bind_return_pmf': "bind_pmf N return_pmf = N" |
|
433 |
proof (transfer, clarify) |
|
434 |
fix N :: "'a measure" assume "sets N = UNIV" then show "N \<guillemotright>= return (count_space UNIV) = N" |
|
435 |
by (subst return_sets_cong[where N=N]) (simp_all add: bind_return') |
|
436 |
qed |
|
437 |
||
438 |
lemma bind_assoc_pmf: "bind_pmf (bind_pmf A B) C = bind_pmf A (\<lambda>x. bind_pmf (B x) C)" |
|
439 |
by transfer |
|
440 |
(auto intro!: bind_assoc[where N="count_space UNIV" and R="count_space UNIV"] |
|
441 |
simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space) |
|
442 |
||
443 |
definition "map_pmf f M = bind_pmf M (\<lambda>x. return_pmf (f x))" |
|
444 |
||
445 |
lemma map_bind_pmf: "map_pmf f (bind_pmf M g) = bind_pmf M (\<lambda>x. map_pmf f (g x))" |
|
446 |
by (simp add: map_pmf_def bind_assoc_pmf) |
|
447 |
||
448 |
lemma bind_map_pmf: "bind_pmf (map_pmf f M) g = bind_pmf M (\<lambda>x. g (f x))" |
|
449 |
by (simp add: map_pmf_def bind_assoc_pmf bind_return_pmf) |
|
450 |
||
451 |
lemma map_pmf_transfer[transfer_rule]: |
|
452 |
"rel_fun op = (rel_fun cr_pmf cr_pmf) (\<lambda>f M. distr M (count_space UNIV) f) map_pmf" |
|
453 |
proof - |
|
454 |
have "rel_fun op = (rel_fun pmf_as_measure.cr_pmf pmf_as_measure.cr_pmf) |
|
455 |
(\<lambda>f M. M \<guillemotright>= (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
|
456 |
unfolding map_pmf_def[abs_def] comp_def by transfer_prover |
59664 | 457 |
then show ?thesis |
458 |
by (force simp: rel_fun_def cr_pmf_def bind_return_distr) |
|
459 |
qed |
|
460 |
||
461 |
lemma map_pmf_rep_eq: |
|
462 |
"measure_pmf (map_pmf f M) = distr (measure_pmf M) (count_space UNIV) f" |
|
463 |
unfolding map_pmf_def bind_pmf.rep_eq comp_def return_pmf.rep_eq |
|
464 |
using bind_return_distr[of M f "count_space UNIV"] by (simp add: comp_def) |
|
465 |
||
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
466 |
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
|
467 |
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
|
468 |
|
59053 | 469 |
lemma map_pmf_ident[simp]: "map_pmf (\<lambda>x. x) = (\<lambda>x. x)" |
470 |
using map_pmf_id unfolding id_def . |
|
471 |
||
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
472 |
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
|
473 |
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
|
474 |
|
59000 | 475 |
lemma map_pmf_comp: "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M" |
476 |
using map_pmf_compose[of f g] by (simp add: comp_def) |
|
477 |
||
59664 | 478 |
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" |
479 |
unfolding map_pmf_def by (rule bind_pmf_cong) auto |
|
480 |
||
481 |
lemma pmf_set_map: "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf" |
|
59665 | 482 |
by (auto simp add: comp_def fun_eq_iff map_pmf_def) |
59664 | 483 |
|
59665 | 484 |
lemma set_map_pmf[simp]: "set_pmf (map_pmf f M) = f`set_pmf M" |
59664 | 485 |
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
|
486 |
|
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
487 |
lemma emeasure_map_pmf[simp]: "emeasure (map_pmf f M) X = emeasure M (f -` X)" |
59664 | 488 |
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
|
489 |
|
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
490 |
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 | 491 |
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
|
492 |
|
59023 | 493 |
lemma ereal_pmf_map: "pmf (map_pmf f p) x = (\<integral>\<^sup>+ y. indicator (f -` {x}) y \<partial>measure_pmf p)" |
59664 | 494 |
proof (transfer fixing: f x) |
59023 | 495 |
fix p :: "'b measure" |
496 |
presume "prob_space p" |
|
497 |
then interpret prob_space p . |
|
498 |
presume "sets p = UNIV" |
|
499 |
then show "ereal (measure (distr p (count_space UNIV) f) {x}) = integral\<^sup>N p (indicator (f -` {x}))" |
|
500 |
by(simp add: measure_distr measurable_def emeasure_eq_measure) |
|
501 |
qed simp_all |
|
502 |
||
503 |
lemma nn_integral_pmf: "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = emeasure (measure_pmf p) A" |
|
504 |
proof - |
|
505 |
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))" |
|
506 |
by(auto simp add: nn_integral_count_space_indicator indicator_def set_pmf_iff intro: nn_integral_cong) |
|
507 |
also have "\<dots> = emeasure (measure_pmf p) (\<Union>x\<in>A \<inter> set_pmf p. {x})" |
|
508 |
by(subst emeasure_UN_countable)(auto simp add: emeasure_pmf_single disjoint_family_on_def) |
|
509 |
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})" |
|
510 |
by(rule emeasure_Un_null_set[symmetric])(auto intro: in_null_sets_measure_pmfI) |
|
511 |
also have "\<dots> = emeasure (measure_pmf p) A" |
|
512 |
by(auto intro: arg_cong2[where f=emeasure]) |
|
513 |
finally show ?thesis . |
|
514 |
qed |
|
515 |
||
60068 | 516 |
lemma map_return_pmf [simp]: "map_pmf f (return_pmf x) = return_pmf (f x)" |
59664 | 517 |
by transfer (simp add: distr_return) |
518 |
||
519 |
lemma map_pmf_const[simp]: "map_pmf (\<lambda>_. c) M = return_pmf c" |
|
520 |
by transfer (auto simp: prob_space.distr_const) |
|
521 |
||
60068 | 522 |
lemma pmf_return [simp]: "pmf (return_pmf x) y = indicator {y} x" |
59664 | 523 |
by transfer (simp add: measure_return) |
524 |
||
525 |
lemma nn_integral_return_pmf[simp]: "0 \<le> f x \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>return_pmf x) = f x" |
|
526 |
unfolding return_pmf.rep_eq by (intro nn_integral_return) auto |
|
527 |
||
528 |
lemma emeasure_return_pmf[simp]: "emeasure (return_pmf x) X = indicator X x" |
|
529 |
unfolding return_pmf.rep_eq by (intro emeasure_return) auto |
|
530 |
||
531 |
lemma return_pmf_inj[simp]: "return_pmf x = return_pmf y \<longleftrightarrow> x = y" |
|
532 |
by (metis insertI1 set_return_pmf singletonD) |
|
533 |
||
59665 | 534 |
lemma map_pmf_eq_return_pmf_iff: |
535 |
"map_pmf f p = return_pmf x \<longleftrightarrow> (\<forall>y \<in> set_pmf p. f y = x)" |
|
536 |
proof |
|
537 |
assume "map_pmf f p = return_pmf x" |
|
538 |
then have "set_pmf (map_pmf f p) = set_pmf (return_pmf x)" by simp |
|
539 |
then show "\<forall>y \<in> set_pmf p. f y = x" by auto |
|
540 |
next |
|
541 |
assume "\<forall>y \<in> set_pmf p. f y = x" |
|
542 |
then show "map_pmf f p = return_pmf x" |
|
543 |
unfolding map_pmf_const[symmetric, of _ p] by (intro map_pmf_cong) auto |
|
544 |
qed |
|
545 |
||
59664 | 546 |
definition "pair_pmf A B = bind_pmf A (\<lambda>x. bind_pmf B (\<lambda>y. return_pmf (x, y)))" |
547 |
||
548 |
lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b" |
|
549 |
unfolding pair_pmf_def pmf_bind pmf_return |
|
550 |
apply (subst integral_measure_pmf[where A="{b}"]) |
|
551 |
apply (auto simp: indicator_eq_0_iff) |
|
552 |
apply (subst integral_measure_pmf[where A="{a}"]) |
|
553 |
apply (auto simp: indicator_eq_0_iff setsum_nonneg_eq_0_iff pmf_nonneg) |
|
554 |
done |
|
555 |
||
59665 | 556 |
lemma set_pair_pmf[simp]: "set_pmf (pair_pmf A B) = set_pmf A \<times> set_pmf B" |
59664 | 557 |
unfolding pair_pmf_def set_bind_pmf set_return_pmf by auto |
558 |
||
559 |
lemma measure_pmf_in_subprob_space[measurable (raw)]: |
|
560 |
"measure_pmf M \<in> space (subprob_algebra (count_space UNIV))" |
|
561 |
by (simp add: space_subprob_algebra) intro_locales |
|
562 |
||
563 |
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)" |
|
564 |
proof - |
|
565 |
have "(\<integral>\<^sup>+x. f x \<partial>pair_pmf A B) = (\<integral>\<^sup>+x. max 0 (f x) * indicator (A \<times> B) x \<partial>pair_pmf A B)" |
|
566 |
by (subst nn_integral_max_0[symmetric]) |
|
59665 | 567 |
(auto simp: AE_measure_pmf_iff intro!: nn_integral_cong_AE) |
59664 | 568 |
also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) * indicator (A \<times> B) (a, b) \<partial>B \<partial>A)" |
569 |
by (simp add: pair_pmf_def) |
|
570 |
also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) \<partial>B \<partial>A)" |
|
571 |
by (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff) |
|
572 |
finally show ?thesis |
|
573 |
unfolding nn_integral_max_0 . |
|
574 |
qed |
|
575 |
||
576 |
lemma bind_pair_pmf: |
|
577 |
assumes M[measurable]: "M \<in> measurable (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) (subprob_algebra N)" |
|
578 |
shows "measure_pmf (pair_pmf A B) \<guillemotright>= M = (measure_pmf A \<guillemotright>= (\<lambda>x. measure_pmf B \<guillemotright>= (\<lambda>y. M (x, y))))" |
|
579 |
(is "?L = ?R") |
|
580 |
proof (rule measure_eqI) |
|
581 |
have M'[measurable]: "M \<in> measurable (pair_pmf A B) (subprob_algebra N)" |
|
582 |
using M[THEN measurable_space] by (simp_all add: space_pair_measure) |
|
583 |
||
584 |
note measurable_bind[where N="count_space UNIV", measurable] |
|
585 |
note measure_pmf_in_subprob_space[simp] |
|
586 |
||
587 |
have sets_eq_N: "sets ?L = N" |
|
588 |
by (subst sets_bind[OF sets_kernel[OF M']]) auto |
|
589 |
show "sets ?L = sets ?R" |
|
590 |
using measurable_space[OF M] |
|
591 |
by (simp add: sets_eq_N space_pair_measure space_subprob_algebra) |
|
592 |
fix X assume "X \<in> sets ?L" |
|
593 |
then have X[measurable]: "X \<in> sets N" |
|
594 |
unfolding sets_eq_N . |
|
595 |
then show "emeasure ?L X = emeasure ?R X" |
|
596 |
apply (simp add: emeasure_bind[OF _ M' X]) |
|
597 |
apply (simp add: nn_integral_bind[where B="count_space UNIV"] pair_pmf_def measure_pmf_bind[of A] |
|
60068 | 598 |
nn_integral_measure_pmf_finite emeasure_nonneg one_ereal_def[symmetric]) |
59664 | 599 |
apply (subst emeasure_bind[OF _ _ X]) |
600 |
apply measurable |
|
601 |
apply (subst emeasure_bind[OF _ _ X]) |
|
602 |
apply measurable |
|
603 |
done |
|
604 |
qed |
|
605 |
||
606 |
lemma map_fst_pair_pmf: "map_pmf fst (pair_pmf A B) = A" |
|
607 |
by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf') |
|
608 |
||
609 |
lemma map_snd_pair_pmf: "map_pmf snd (pair_pmf A B) = B" |
|
610 |
by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf') |
|
611 |
||
612 |
lemma nn_integral_pmf': |
|
613 |
"inj_on f A \<Longrightarrow> (\<integral>\<^sup>+x. pmf p (f x) \<partial>count_space A) = emeasure p (f ` A)" |
|
614 |
by (subst nn_integral_bij_count_space[where g=f and B="f`A"]) |
|
615 |
(auto simp: bij_betw_def nn_integral_pmf) |
|
616 |
||
617 |
lemma pmf_le_0_iff[simp]: "pmf M p \<le> 0 \<longleftrightarrow> pmf M p = 0" |
|
618 |
using pmf_nonneg[of M p] by simp |
|
619 |
||
620 |
lemma min_pmf_0[simp]: "min (pmf M p) 0 = 0" "min 0 (pmf M p) = 0" |
|
621 |
using pmf_nonneg[of M p] by simp_all |
|
622 |
||
623 |
lemma pmf_eq_0_set_pmf: "pmf M p = 0 \<longleftrightarrow> p \<notin> set_pmf M" |
|
624 |
unfolding set_pmf_iff by simp |
|
625 |
||
626 |
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" |
|
627 |
by (auto simp: pmf.rep_eq map_pmf_rep_eq measure_distr AE_measure_pmf_iff inj_onD |
|
628 |
intro!: measure_pmf.finite_measure_eq_AE) |
|
629 |
||
60068 | 630 |
lemma pmf_map_inj': "inj f \<Longrightarrow> pmf (map_pmf f M) (f x) = pmf M x" |
631 |
apply(cases "x \<in> set_pmf M") |
|
632 |
apply(simp add: pmf_map_inj[OF subset_inj_on]) |
|
633 |
apply(simp add: pmf_eq_0_set_pmf[symmetric]) |
|
634 |
apply(auto simp add: pmf_eq_0_set_pmf dest: injD) |
|
635 |
done |
|
636 |
||
637 |
lemma pmf_map_outside: "x \<notin> f ` set_pmf M \<Longrightarrow> pmf (map_pmf f M) x = 0" |
|
638 |
unfolding pmf_eq_0_set_pmf by simp |
|
639 |
||
59664 | 640 |
subsection \<open> PMFs as function \<close> |
59000 | 641 |
|
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
642 |
context |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
643 |
fixes f :: "'a \<Rightarrow> real" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
644 |
assumes nonneg: "\<And>x. 0 \<le> f x" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
645 |
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
|
646 |
begin |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
647 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
648 |
lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ereal \<circ> f)" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
649 |
proof (intro conjI) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
650 |
have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
651 |
by (simp split: split_indicator) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
652 |
show "AE x in density (count_space UNIV) (ereal \<circ> f). |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
653 |
measure (density (count_space UNIV) (ereal \<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
|
654 |
by (simp add: AE_density nonneg measure_def emeasure_density max_def) |
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
655 |
show "prob_space (density (count_space UNIV) (ereal \<circ> f))" |
61169 | 656 |
by standard (simp add: emeasure_density prob) |
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
657 |
qed simp |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
658 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
659 |
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
|
660 |
proof transfer |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
661 |
have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
662 |
by (simp split: split_indicator) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
663 |
fix x show "measure (density (count_space UNIV) (ereal \<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
|
664 |
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
|
665 |
qed |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
666 |
|
60068 | 667 |
lemma set_embed_pmf: "set_pmf embed_pmf = {x. f x \<noteq> 0}" |
668 |
by(auto simp add: set_pmf_eq assms pmf_embed_pmf) |
|
669 |
||
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
670 |
end |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
671 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
672 |
lemma embed_pmf_transfer: |
58730 | 673 |
"rel_fun (eq_onp (\<lambda>f. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ereal (f x) \<partial>count_space UNIV) = 1)) pmf_as_measure.cr_pmf (\<lambda>f. density (count_space UNIV) (ereal \<circ> f)) embed_pmf" |
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
674 |
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
|
675 |
|
59000 | 676 |
lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)" |
677 |
proof (transfer, elim conjE) |
|
678 |
fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0" |
|
679 |
assume "prob_space M" then interpret prob_space M . |
|
680 |
show "M = density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))" |
|
681 |
proof (rule measure_eqI) |
|
682 |
fix A :: "'a set" |
|
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
|
683 |
have "(\<integral>\<^sup>+ x. ereal (measure M {x}) * indicator A x \<partial>count_space UNIV) = |
59000 | 684 |
(\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)" |
685 |
by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator) |
|
686 |
also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))" |
|
687 |
by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space) |
|
688 |
also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})" |
|
689 |
by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support) |
|
690 |
(auto simp: disjoint_family_on_def) |
|
691 |
also have "\<dots> = emeasure M A" |
|
692 |
using ae by (intro emeasure_eq_AE) auto |
|
693 |
finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))) A" |
|
694 |
using emeasure_space_1 by (simp add: emeasure_density) |
|
695 |
qed simp |
|
696 |
qed |
|
697 |
||
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
698 |
lemma td_pmf_embed_pmf: |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
699 |
"type_definition pmf embed_pmf {f::'a \<Rightarrow> real. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ereal (f x) \<partial>count_space UNIV) = 1}" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
700 |
unfolding type_definition_def |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
701 |
proof safe |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
702 |
fix p :: "'a pmf" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
703 |
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
|
704 |
using measure_pmf.emeasure_space_1[of p] by simp |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
705 |
then show *: "(\<integral>\<^sup>+ x. ereal (pmf p x) \<partial>count_space UNIV) = 1" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
706 |
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
|
707 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
708 |
show "embed_pmf (pmf p) = p" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
709 |
by (intro measure_pmf_inject[THEN iffD1]) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
710 |
(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
|
711 |
next |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
712 |
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
|
713 |
then show "pmf (embed_pmf f) = f" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
714 |
by (auto intro!: pmf_embed_pmf) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
715 |
qed (rule pmf_nonneg) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
716 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
717 |
end |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
718 |
|
60068 | 719 |
lemma nn_integral_measure_pmf: "(\<integral>\<^sup>+ x. f x \<partial>measure_pmf p) = \<integral>\<^sup>+ x. ereal (pmf p x) * f x \<partial>count_space UNIV" |
720 |
by(simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg) |
|
721 |
||
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
722 |
locale pmf_as_function |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
723 |
begin |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
724 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
725 |
setup_lifting td_pmf_embed_pmf |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
726 |
|
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
|
727 |
lemma set_pmf_transfer[transfer_rule]: |
58730 | 728 |
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
|
729 |
shows "rel_fun (pcr_pmf A) (rel_set A) (\<lambda>f. {x. f x \<noteq> 0}) set_pmf" |
58730 | 730 |
using `bi_total A` |
731 |
by (auto simp: pcr_pmf_def cr_pmf_def rel_fun_def rel_set_def bi_total_def Bex_def set_pmf_iff) |
|
732 |
metis+ |
|
733 |
||
59000 | 734 |
end |
735 |
||
736 |
context |
|
737 |
begin |
|
738 |
||
739 |
interpretation pmf_as_function . |
|
740 |
||
741 |
lemma pmf_eqI: "(\<And>i. pmf M i = pmf N i) \<Longrightarrow> M = N" |
|
742 |
by transfer auto |
|
743 |
||
744 |
lemma pmf_eq_iff: "M = N \<longleftrightarrow> (\<forall>i. pmf M i = pmf N i)" |
|
745 |
by (auto intro: pmf_eqI) |
|
746 |
||
59664 | 747 |
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))" |
748 |
unfolding pmf_eq_iff pmf_bind |
|
749 |
proof |
|
750 |
fix i |
|
751 |
interpret B: prob_space "restrict_space B B" |
|
752 |
by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE) |
|
753 |
(auto simp: AE_measure_pmf_iff) |
|
754 |
interpret A: prob_space "restrict_space A A" |
|
755 |
by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE) |
|
756 |
(auto simp: AE_measure_pmf_iff) |
|
757 |
||
758 |
interpret AB: pair_prob_space "restrict_space A A" "restrict_space B B" |
|
759 |
by unfold_locales |
|
760 |
||
761 |
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)" |
|
762 |
by (rule integral_cong) (auto intro!: integral_pmf_restrict) |
|
763 |
also have "\<dots> = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>restrict_space A A)" |
|
764 |
by (intro integral_pmf_restrict B.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2 |
|
765 |
countable_set_pmf borel_measurable_count_space) |
|
766 |
also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>restrict_space B B)" |
|
767 |
by (rule AB.Fubini_integral[symmetric]) |
|
768 |
(auto intro!: AB.integrable_const_bound[where B=1] measurable_pair_restrict_pmf2 |
|
769 |
simp: pmf_nonneg pmf_le_1 measurable_restrict_space1) |
|
770 |
also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>B)" |
|
771 |
by (intro integral_pmf_restrict[symmetric] A.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2 |
|
772 |
countable_set_pmf borel_measurable_count_space) |
|
773 |
also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>A \<partial>B)" |
|
774 |
by (rule integral_cong) (auto intro!: integral_pmf_restrict[symmetric]) |
|
775 |
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)" . |
|
776 |
qed |
|
777 |
||
778 |
lemma pair_map_pmf1: "pair_pmf (map_pmf f A) B = map_pmf (apfst f) (pair_pmf A B)" |
|
779 |
proof (safe intro!: pmf_eqI) |
|
780 |
fix a :: "'a" and b :: "'b" |
|
781 |
have [simp]: "\<And>c d. indicator (apfst f -` {(a, b)}) (c, d) = indicator (f -` {a}) c * (indicator {b} d::ereal)" |
|
782 |
by (auto split: split_indicator) |
|
783 |
||
784 |
have "ereal (pmf (pair_pmf (map_pmf f A) B) (a, b)) = |
|
785 |
ereal (pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b))" |
|
786 |
unfolding pmf_pair ereal_pmf_map |
|
787 |
by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_multc pmf_nonneg |
|
788 |
emeasure_map_pmf[symmetric] del: emeasure_map_pmf) |
|
789 |
then show "pmf (pair_pmf (map_pmf f A) B) (a, b) = pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b)" |
|
790 |
by simp |
|
791 |
qed |
|
792 |
||
793 |
lemma pair_map_pmf2: "pair_pmf A (map_pmf f B) = map_pmf (apsnd f) (pair_pmf A B)" |
|
794 |
proof (safe intro!: pmf_eqI) |
|
795 |
fix a :: "'a" and b :: "'b" |
|
796 |
have [simp]: "\<And>c d. indicator (apsnd f -` {(a, b)}) (c, d) = indicator {a} c * (indicator (f -` {b}) d::ereal)" |
|
797 |
by (auto split: split_indicator) |
|
798 |
||
799 |
have "ereal (pmf (pair_pmf A (map_pmf f B)) (a, b)) = |
|
800 |
ereal (pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b))" |
|
801 |
unfolding pmf_pair ereal_pmf_map |
|
802 |
by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_cmult nn_integral_multc pmf_nonneg |
|
803 |
emeasure_map_pmf[symmetric] del: emeasure_map_pmf) |
|
804 |
then show "pmf (pair_pmf A (map_pmf f B)) (a, b) = pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b)" |
|
805 |
by simp |
|
806 |
qed |
|
807 |
||
808 |
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)" |
|
809 |
by (simp add: pair_map_pmf2 pair_map_pmf1 map_pmf_comp split_beta') |
|
810 |
||
59000 | 811 |
end |
812 |
||
59664 | 813 |
subsection \<open> Conditional Probabilities \<close> |
814 |
||
59670 | 815 |
lemma measure_pmf_zero_iff: "measure (measure_pmf p) s = 0 \<longleftrightarrow> set_pmf p \<inter> s = {}" |
816 |
by (subst measure_pmf.prob_eq_0) (auto simp: AE_measure_pmf_iff) |
|
817 |
||
59664 | 818 |
context |
819 |
fixes p :: "'a pmf" and s :: "'a set" |
|
820 |
assumes not_empty: "set_pmf p \<inter> s \<noteq> {}" |
|
821 |
begin |
|
822 |
||
823 |
interpretation pmf_as_measure . |
|
824 |
||
825 |
lemma emeasure_measure_pmf_not_zero: "emeasure (measure_pmf p) s \<noteq> 0" |
|
826 |
proof |
|
827 |
assume "emeasure (measure_pmf p) s = 0" |
|
828 |
then have "AE x in measure_pmf p. x \<notin> s" |
|
829 |
by (rule AE_I[rotated]) auto |
|
830 |
with not_empty show False |
|
831 |
by (auto simp: AE_measure_pmf_iff) |
|
832 |
qed |
|
833 |
||
834 |
lemma measure_measure_pmf_not_zero: "measure (measure_pmf p) s \<noteq> 0" |
|
835 |
using emeasure_measure_pmf_not_zero unfolding measure_pmf.emeasure_eq_measure by simp |
|
836 |
||
837 |
lift_definition cond_pmf :: "'a pmf" is |
|
838 |
"uniform_measure (measure_pmf p) s" |
|
839 |
proof (intro conjI) |
|
840 |
show "prob_space (uniform_measure (measure_pmf p) s)" |
|
841 |
by (intro prob_space_uniform_measure) (auto simp: emeasure_measure_pmf_not_zero) |
|
842 |
show "AE x in uniform_measure (measure_pmf p) s. measure (uniform_measure (measure_pmf p) s) {x} \<noteq> 0" |
|
843 |
by (simp add: emeasure_measure_pmf_not_zero measure_measure_pmf_not_zero AE_uniform_measure |
|
844 |
AE_measure_pmf_iff set_pmf.rep_eq) |
|
845 |
qed simp |
|
846 |
||
847 |
lemma pmf_cond: "pmf cond_pmf x = (if x \<in> s then pmf p x / measure p s else 0)" |
|
848 |
by transfer (simp add: emeasure_measure_pmf_not_zero pmf.rep_eq) |
|
849 |
||
59665 | 850 |
lemma set_cond_pmf[simp]: "set_pmf cond_pmf = set_pmf p \<inter> s" |
59664 | 851 |
by (auto simp add: set_pmf_iff pmf_cond measure_measure_pmf_not_zero split: split_if_asm) |
852 |
||
853 |
end |
|
854 |
||
855 |
lemma cond_map_pmf: |
|
856 |
assumes "set_pmf p \<inter> f -` s \<noteq> {}" |
|
857 |
shows "cond_pmf (map_pmf f p) s = map_pmf f (cond_pmf p (f -` s))" |
|
858 |
proof - |
|
859 |
have *: "set_pmf (map_pmf f p) \<inter> s \<noteq> {}" |
|
59665 | 860 |
using assms by auto |
59664 | 861 |
{ fix x |
862 |
have "ereal (pmf (map_pmf f (cond_pmf p (f -` s))) x) = |
|
863 |
emeasure p (f -` s \<inter> f -` {x}) / emeasure p (f -` s)" |
|
864 |
unfolding ereal_pmf_map cond_pmf.rep_eq[OF assms] by (simp add: nn_integral_uniform_measure) |
|
865 |
also have "f -` s \<inter> f -` {x} = (if x \<in> s then f -` {x} else {})" |
|
866 |
by auto |
|
867 |
also have "emeasure p (if x \<in> s then f -` {x} else {}) / emeasure p (f -` s) = |
|
868 |
ereal (pmf (cond_pmf (map_pmf f p) s) x)" |
|
869 |
using measure_measure_pmf_not_zero[OF *] |
|
870 |
by (simp add: pmf_cond[OF *] ereal_divide' ereal_pmf_map measure_pmf.emeasure_eq_measure[symmetric] |
|
871 |
del: ereal_divide) |
|
872 |
finally have "ereal (pmf (cond_pmf (map_pmf f p) s) x) = ereal (pmf (map_pmf f (cond_pmf p (f -` s))) x)" |
|
873 |
by simp } |
|
874 |
then show ?thesis |
|
875 |
by (intro pmf_eqI) simp |
|
876 |
qed |
|
877 |
||
878 |
lemma bind_cond_pmf_cancel: |
|
59670 | 879 |
assumes [simp]: "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}" |
880 |
assumes [simp]: "\<And>y. y \<in> set_pmf q \<Longrightarrow> set_pmf p \<inter> {x. R x y} \<noteq> {}" |
|
881 |
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}" |
|
882 |
shows "bind_pmf p (\<lambda>x. cond_pmf q {y. R x y}) = q" |
|
59664 | 883 |
proof (rule pmf_eqI) |
59670 | 884 |
fix i |
885 |
have "ereal (pmf (bind_pmf p (\<lambda>x. cond_pmf q {y. R x y})) i) = |
|
886 |
(\<integral>\<^sup>+x. ereal (pmf q i / measure p {x. R x i}) * ereal (indicator {x. R x i} x) \<partial>p)" |
|
887 |
by (auto simp add: ereal_pmf_bind AE_measure_pmf_iff pmf_cond pmf_eq_0_set_pmf intro!: nn_integral_cong_AE) |
|
888 |
also have "\<dots> = (pmf q i * measure p {x. R x i}) / measure p {x. R x i}" |
|
889 |
by (simp add: pmf_nonneg measure_nonneg zero_ereal_def[symmetric] ereal_indicator |
|
890 |
nn_integral_cmult measure_pmf.emeasure_eq_measure) |
|
891 |
also have "\<dots> = pmf q i" |
|
892 |
by (cases "pmf q i = 0") (simp_all add: pmf_eq_0_set_pmf measure_measure_pmf_not_zero) |
|
893 |
finally show "pmf (bind_pmf p (\<lambda>x. cond_pmf q {y. R x y})) i = pmf q i" |
|
894 |
by simp |
|
59664 | 895 |
qed |
896 |
||
897 |
subsection \<open> Relator \<close> |
|
898 |
||
899 |
inductive rel_pmf :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf \<Rightarrow> bool" |
|
900 |
for R p q |
|
901 |
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
|
902 |
"\<lbrakk> \<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y; |
59664 | 903 |
map_pmf fst pq = p; map_pmf snd pq = q \<rbrakk> |
904 |
\<Longrightarrow> rel_pmf R p q" |
|
905 |
||
59681 | 906 |
lemma rel_pmfI: |
907 |
assumes R: "rel_set R (set_pmf p) (set_pmf q)" |
|
908 |
assumes eq: "\<And>x y. x \<in> set_pmf p \<Longrightarrow> y \<in> set_pmf q \<Longrightarrow> R x y \<Longrightarrow> |
|
909 |
measure p {x. R x y} = measure q {y. R x y}" |
|
910 |
shows "rel_pmf R p q" |
|
911 |
proof |
|
912 |
let ?pq = "bind_pmf p (\<lambda>x. bind_pmf (cond_pmf q {y. R x y}) (\<lambda>y. return_pmf (x, y)))" |
|
913 |
have "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}" |
|
914 |
using R by (auto simp: rel_set_def) |
|
915 |
then show "\<And>x y. (x, y) \<in> set_pmf ?pq \<Longrightarrow> R x y" |
|
916 |
by auto |
|
917 |
show "map_pmf fst ?pq = p" |
|
60068 | 918 |
by (simp add: map_bind_pmf bind_return_pmf') |
59681 | 919 |
|
920 |
show "map_pmf snd ?pq = q" |
|
921 |
using R eq |
|
60068 | 922 |
apply (simp add: bind_cond_pmf_cancel map_bind_pmf bind_return_pmf') |
59681 | 923 |
apply (rule bind_cond_pmf_cancel) |
924 |
apply (auto simp: rel_set_def) |
|
925 |
done |
|
926 |
qed |
|
927 |
||
928 |
lemma rel_pmf_imp_rel_set: "rel_pmf R p q \<Longrightarrow> rel_set R (set_pmf p) (set_pmf q)" |
|
929 |
by (force simp add: rel_pmf.simps rel_set_def) |
|
930 |
||
931 |
lemma rel_pmfD_measure: |
|
932 |
assumes rel_R: "rel_pmf R p q" and R: "\<And>a b. R a b \<Longrightarrow> R a y \<longleftrightarrow> R x b" |
|
933 |
assumes "x \<in> set_pmf p" "y \<in> set_pmf q" |
|
934 |
shows "measure p {x. R x y} = measure q {y. R x y}" |
|
935 |
proof - |
|
936 |
from rel_R obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y" |
|
937 |
and eq: "p = map_pmf fst pq" "q = map_pmf snd pq" |
|
938 |
by (auto elim: rel_pmf.cases) |
|
939 |
have "measure p {x. R x y} = measure pq {x. R (fst x) y}" |
|
940 |
by (simp add: eq map_pmf_rep_eq measure_distr) |
|
941 |
also have "\<dots> = measure pq {y. R x (snd y)}" |
|
942 |
by (intro measure_pmf.finite_measure_eq_AE) |
|
943 |
(auto simp: AE_measure_pmf_iff R dest!: pq) |
|
944 |
also have "\<dots> = measure q {y. R x y}" |
|
945 |
by (simp add: eq map_pmf_rep_eq measure_distr) |
|
946 |
finally show "measure p {x. R x y} = measure q {y. R x y}" . |
|
947 |
qed |
|
948 |
||
949 |
lemma rel_pmf_iff_measure: |
|
950 |
assumes "symp R" "transp R" |
|
951 |
shows "rel_pmf R p q \<longleftrightarrow> |
|
952 |
rel_set R (set_pmf p) (set_pmf q) \<and> |
|
953 |
(\<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})" |
|
954 |
by (safe intro!: rel_pmf_imp_rel_set rel_pmfI) |
|
955 |
(auto intro!: rel_pmfD_measure dest: sympD[OF \<open>symp R\<close>] transpD[OF \<open>transp R\<close>]) |
|
956 |
||
957 |
lemma quotient_rel_set_disjoint: |
|
958 |
"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
|
959 |
using in_quotient_imp_closed[of UNIV "{(x, y). R x y}" C] |
59681 | 960 |
by (auto 0 0 simp: equivp_equiv rel_set_def set_eq_iff elim: equivpE) |
961 |
(blast dest: equivp_symp)+ |
|
962 |
||
963 |
lemma quotientD: "equiv X R \<Longrightarrow> A \<in> X // R \<Longrightarrow> x \<in> A \<Longrightarrow> A = R `` {x}" |
|
964 |
by (metis Image_singleton_iff equiv_class_eq_iff quotientE) |
|
965 |
||
966 |
lemma rel_pmf_iff_equivp: |
|
967 |
assumes "equivp R" |
|
968 |
shows "rel_pmf R p q \<longleftrightarrow> (\<forall>C\<in>UNIV // {(x, y). R x y}. measure p C = measure q C)" |
|
969 |
(is "_ \<longleftrightarrow> (\<forall>C\<in>_//?R. _)") |
|
970 |
proof (subst rel_pmf_iff_measure, safe) |
|
971 |
show "symp R" "transp R" |
|
972 |
using assms by (auto simp: equivp_reflp_symp_transp) |
|
973 |
next |
|
974 |
fix C assume C: "C \<in> UNIV // ?R" and R: "rel_set R (set_pmf p) (set_pmf q)" |
|
975 |
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
|
976 |
|
59681 | 977 |
show "measure p C = measure q C" |
978 |
proof cases |
|
979 |
assume "p \<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
|
980 |
moreover then have "q \<inter> C = {}" |
59681 | 981 |
using quotient_rel_set_disjoint[OF assms C R] by simp |
982 |
ultimately show ?thesis |
|
983 |
unfolding measure_pmf_zero_iff[symmetric] by simp |
|
984 |
next |
|
985 |
assume "p \<inter> C \<noteq> {}" |
|
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
|
986 |
moreover then have "q \<inter> C \<noteq> {}" |
59681 | 987 |
using quotient_rel_set_disjoint[OF assms C R] by simp |
988 |
ultimately 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" |
|
989 |
by auto |
|
990 |
then have "R x y" |
|
991 |
using in_quotient_imp_in_rel[of UNIV ?R C x y] C assms |
|
992 |
by (simp add: equivp_equiv) |
|
993 |
with in_set eq have "measure p {x. R x y} = measure q {y. R x y}" |
|
994 |
by auto |
|
995 |
moreover have "{y. R x y} = C" |
|
996 |
using assms `x \<in> C` C quotientD[of UNIV ?R C x] by (simp add: equivp_equiv) |
|
997 |
moreover have "{x. R x y} = C" |
|
998 |
using assms `y \<in> C` C quotientD[of UNIV "?R" C y] sympD[of R] |
|
999 |
by (auto simp add: equivp_equiv elim: equivpE) |
|
1000 |
ultimately show ?thesis |
|
1001 |
by auto |
|
1002 |
qed |
|
1003 |
next |
|
1004 |
assume eq: "\<forall>C\<in>UNIV // ?R. measure p C = measure q C" |
|
1005 |
show "rel_set R (set_pmf p) (set_pmf q)" |
|
1006 |
unfolding rel_set_def |
|
1007 |
proof safe |
|
1008 |
fix x assume x: "x \<in> set_pmf p" |
|
1009 |
have "{y. R x y} \<in> UNIV // ?R" |
|
1010 |
by (auto simp: quotient_def) |
|
1011 |
with eq have *: "measure q {y. R x y} = measure p {y. R x y}" |
|
1012 |
by auto |
|
1013 |
have "measure q {y. R x y} \<noteq> 0" |
|
1014 |
using x assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp) |
|
1015 |
then show "\<exists>y\<in>set_pmf q. R x y" |
|
1016 |
unfolding measure_pmf_zero_iff by auto |
|
1017 |
next |
|
1018 |
fix y assume y: "y \<in> set_pmf q" |
|
1019 |
have "{x. R x y} \<in> UNIV // ?R" |
|
1020 |
using assms by (auto simp: quotient_def dest: equivp_symp) |
|
1021 |
with eq have *: "measure p {x. R x y} = measure q {x. R x y}" |
|
1022 |
by auto |
|
1023 |
have "measure p {x. R x y} \<noteq> 0" |
|
1024 |
using y assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp) |
|
1025 |
then show "\<exists>x\<in>set_pmf p. R x y" |
|
1026 |
unfolding measure_pmf_zero_iff by auto |
|
1027 |
qed |
|
1028 |
||
1029 |
fix x y assume "x \<in> set_pmf p" "y \<in> set_pmf q" "R x y" |
|
1030 |
have "{y. R x y} \<in> UNIV // ?R" "{x. R x y} = {y. R x y}" |
|
1031 |
using assms `R x y` by (auto simp: quotient_def dest: equivp_symp equivp_transp) |
|
1032 |
with eq show "measure p {x. R x y} = measure q {y. R x y}" |
|
1033 |
by auto |
|
1034 |
qed |
|
1035 |
||
59664 | 1036 |
bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: rel_pmf |
1037 |
proof - |
|
1038 |
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
|
1039 |
show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose) |
59664 | 1040 |
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" |
1041 |
by (intro map_pmf_cong refl) |
|
1042 |
||
1043 |
show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf" |
|
1044 |
by (rule pmf_set_map) |
|
1045 |
||
60595 | 1046 |
show "(card_of (set_pmf p), natLeq) \<in> ordLeq" for p :: "'s pmf" |
1047 |
proof - |
|
59664 | 1048 |
have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq" |
1049 |
by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"]) |
|
1050 |
(auto intro: countable_set_pmf) |
|
1051 |
also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq" |
|
1052 |
by (metis Field_natLeq card_of_least natLeq_Well_order) |
|
60595 | 1053 |
finally show ?thesis . |
1054 |
qed |
|
59664 | 1055 |
|
1056 |
show "\<And>R. rel_pmf R = |
|
1057 |
(BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf fst))\<inverse>\<inverse> OO |
|
1058 |
BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf snd)" |
|
1059 |
by (auto simp add: fun_eq_iff BNF_Def.Grp_def OO_def rel_pmf.simps) |
|
1060 |
||
60595 | 1061 |
show "rel_pmf R OO rel_pmf S \<le> rel_pmf (R OO S)" |
1062 |
for R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool" |
|
1063 |
proof - |
|
1064 |
{ fix p q r |
|
1065 |
assume pq: "rel_pmf R p q" |
|
1066 |
and qr:"rel_pmf S q r" |
|
1067 |
from pq obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y" |
|
1068 |
and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto |
|
1069 |
from qr obtain qr where qr: "\<And>y z. (y, z) \<in> set_pmf qr \<Longrightarrow> S y z" |
|
1070 |
and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto |
|
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
|
1071 |
|
60595 | 1072 |
def pr \<equiv> "bind_pmf pq (\<lambda>xy. bind_pmf (cond_pmf qr {yz. fst yz = snd xy}) (\<lambda>yz. return_pmf (fst xy, snd yz)))" |
1073 |
have pr_welldefined: "\<And>y. y \<in> q \<Longrightarrow> qr \<inter> {yz. fst yz = y} \<noteq> {}" |
|
1074 |
by (force simp: q') |
|
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
|
1075 |
|
60595 | 1076 |
have "rel_pmf (R OO S) p r" |
1077 |
proof (rule rel_pmf.intros) |
|
1078 |
fix x z assume "(x, z) \<in> pr" |
|
1079 |
then have "\<exists>y. (x, y) \<in> pq \<and> (y, z) \<in> qr" |
|
1080 |
by (auto simp: q pr_welldefined pr_def split_beta) |
|
1081 |
with pq qr show "(R OO S) x z" |
|
1082 |
by blast |
|
1083 |
next |
|
1084 |
have "map_pmf snd pr = map_pmf snd (bind_pmf q (\<lambda>y. cond_pmf qr {yz. fst yz = y}))" |
|
1085 |
by (simp add: pr_def q split_beta bind_map_pmf map_pmf_def[symmetric] map_bind_pmf map_pmf_comp) |
|
1086 |
then show "map_pmf snd pr = r" |
|
1087 |
unfolding r q' bind_map_pmf by (subst (asm) bind_cond_pmf_cancel) (auto simp: eq_commute) |
|
1088 |
qed (simp add: pr_def map_bind_pmf split_beta map_pmf_def[symmetric] p map_pmf_comp) |
|
1089 |
} |
|
1090 |
then show ?thesis |
|
1091 |
by(auto simp add: le_fun_def) |
|
1092 |
qed |
|
59664 | 1093 |
qed (fact natLeq_card_order natLeq_cinfinite)+ |
1094 |
||
59665 | 1095 |
lemma rel_pmf_conj[simp]: |
1096 |
"rel_pmf (\<lambda>x y. P \<and> Q x y) x y \<longleftrightarrow> P \<and> rel_pmf Q x y" |
|
1097 |
"rel_pmf (\<lambda>x y. Q x y \<and> P) x y \<longleftrightarrow> P \<and> rel_pmf Q x y" |
|
1098 |
using set_pmf_not_empty by (fastforce simp: pmf.in_rel subset_eq)+ |
|
1099 |
||
1100 |
lemma rel_pmf_top[simp]: "rel_pmf top = top" |
|
1101 |
by (auto simp: pmf.in_rel[abs_def] fun_eq_iff map_fst_pair_pmf map_snd_pair_pmf |
|
1102 |
intro: exI[of _ "pair_pmf x y" for x y]) |
|
1103 |
||
59664 | 1104 |
lemma rel_pmf_return_pmf1: "rel_pmf R (return_pmf x) M \<longleftrightarrow> (\<forall>a\<in>M. R x a)" |
1105 |
proof safe |
|
1106 |
fix a assume "a \<in> M" "rel_pmf R (return_pmf x) M" |
|
1107 |
then obtain pq where *: "\<And>a b. (a, b) \<in> set_pmf pq \<Longrightarrow> R a b" |
|
1108 |
and eq: "return_pmf x = map_pmf fst pq" "M = map_pmf snd pq" |
|
1109 |
by (force elim: rel_pmf.cases) |
|
1110 |
moreover have "set_pmf (return_pmf x) = {x}" |
|
59665 | 1111 |
by simp |
59664 | 1112 |
with `a \<in> M` have "(x, a) \<in> pq" |
59665 | 1113 |
by (force simp: eq) |
59664 | 1114 |
with * show "R x a" |
1115 |
by auto |
|
1116 |
qed (auto intro!: rel_pmf.intros[where pq="pair_pmf (return_pmf x) M"] |
|
59665 | 1117 |
simp: map_fst_pair_pmf map_snd_pair_pmf) |
59664 | 1118 |
|
1119 |
lemma rel_pmf_return_pmf2: "rel_pmf R M (return_pmf x) \<longleftrightarrow> (\<forall>a\<in>M. R a x)" |
|
1120 |
by (subst pmf.rel_flip[symmetric]) (simp add: rel_pmf_return_pmf1) |
|
1121 |
||
1122 |
lemma rel_return_pmf[simp]: "rel_pmf R (return_pmf x1) (return_pmf x2) = R x1 x2" |
|
1123 |
unfolding rel_pmf_return_pmf2 set_return_pmf by simp |
|
1124 |
||
1125 |
lemma rel_pmf_False[simp]: "rel_pmf (\<lambda>x y. False) x y = False" |
|
1126 |
unfolding pmf.in_rel fun_eq_iff using set_pmf_not_empty by fastforce |
|
1127 |
||
1128 |
lemma rel_pmf_rel_prod: |
|
1129 |
"rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B') \<longleftrightarrow> rel_pmf R A B \<and> rel_pmf S A' B'" |
|
1130 |
proof safe |
|
1131 |
assume "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')" |
|
1132 |
then obtain pq where pq: "\<And>a b c d. ((a, c), (b, d)) \<in> set_pmf pq \<Longrightarrow> R a b \<and> S c d" |
|
1133 |
and eq: "map_pmf fst pq = pair_pmf A A'" "map_pmf snd pq = pair_pmf B B'" |
|
1134 |
by (force elim: rel_pmf.cases) |
|
1135 |
show "rel_pmf R A B" |
|
1136 |
proof (rule rel_pmf.intros) |
|
1137 |
let ?f = "\<lambda>(a, b). (fst a, fst b)" |
|
1138 |
have [simp]: "(\<lambda>x. fst (?f x)) = fst o fst" "(\<lambda>x. snd (?f x)) = fst o snd" |
|
1139 |
by auto |
|
1140 |
||
1141 |
show "map_pmf fst (map_pmf ?f pq) = A" |
|
1142 |
by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf) |
|
1143 |
show "map_pmf snd (map_pmf ?f pq) = B" |
|
1144 |
by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf) |
|
1145 |
||
1146 |
fix a b assume "(a, b) \<in> set_pmf (map_pmf ?f pq)" |
|
1147 |
then obtain c d where "((a, c), (b, d)) \<in> set_pmf pq" |
|
59665 | 1148 |
by auto |
59664 | 1149 |
from pq[OF this] show "R a b" .. |
1150 |
qed |
|
1151 |
show "rel_pmf S A' B'" |
|
1152 |
proof (rule rel_pmf.intros) |
|
1153 |
let ?f = "\<lambda>(a, b). (snd a, snd b)" |
|
1154 |
have [simp]: "(\<lambda>x. fst (?f x)) = snd o fst" "(\<lambda>x. snd (?f x)) = snd o snd" |
|
1155 |
by auto |
|
1156 |
||
1157 |
show "map_pmf fst (map_pmf ?f pq) = A'" |
|
1158 |
by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf) |
|
1159 |
show "map_pmf snd (map_pmf ?f pq) = B'" |
|
1160 |
by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf) |
|
1161 |
||
1162 |
fix c d assume "(c, d) \<in> set_pmf (map_pmf ?f pq)" |
|
1163 |
then obtain a b where "((a, c), (b, d)) \<in> set_pmf pq" |
|
59665 | 1164 |
by auto |
59664 | 1165 |
from pq[OF this] show "S c d" .. |
1166 |
qed |
|
1167 |
next |
|
1168 |
assume "rel_pmf R A B" "rel_pmf S A' B'" |
|
1169 |
then obtain Rpq Spq |
|
1170 |
where Rpq: "\<And>a b. (a, b) \<in> set_pmf Rpq \<Longrightarrow> R a b" |
|
1171 |
"map_pmf fst Rpq = A" "map_pmf snd Rpq = B" |
|
1172 |
and Spq: "\<And>a b. (a, b) \<in> set_pmf Spq \<Longrightarrow> S a b" |
|
1173 |
"map_pmf fst Spq = A'" "map_pmf snd Spq = B'" |
|
1174 |
by (force elim: rel_pmf.cases) |
|
1175 |
||
1176 |
let ?f = "(\<lambda>((a, c), (b, d)). ((a, b), (c, d)))" |
|
1177 |
let ?pq = "map_pmf ?f (pair_pmf Rpq Spq)" |
|
1178 |
have [simp]: "(\<lambda>x. fst (?f x)) = (\<lambda>(a, b). (fst a, fst b))" "(\<lambda>x. snd (?f x)) = (\<lambda>(a, b). (snd a, snd b))" |
|
1179 |
by auto |
|
1180 |
||
1181 |
show "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')" |
|
1182 |
by (rule rel_pmf.intros[where pq="?pq"]) |
|
59665 | 1183 |
(auto simp: map_snd_pair_pmf map_fst_pair_pmf map_pmf_comp Rpq Spq |
59664 | 1184 |
map_pair) |
1185 |
qed |
|
1186 |
||
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
|
1187 |
lemma rel_pmf_reflI: |
59664 | 1188 |
assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> P x x" |
1189 |
shows "rel_pmf P p p" |
|
59665 | 1190 |
by (rule rel_pmf.intros[where pq="map_pmf (\<lambda>x. (x, x)) p"]) |
1191 |
(auto simp add: pmf.map_comp o_def assms) |
|
59664 | 1192 |
|
1193 |
context |
|
1194 |
begin |
|
1195 |
||
1196 |
interpretation pmf_as_measure . |
|
1197 |
||
1198 |
definition "join_pmf M = bind_pmf M (\<lambda>x. x)" |
|
1199 |
||
1200 |
lemma bind_eq_join_pmf: "bind_pmf M f = join_pmf (map_pmf f M)" |
|
1201 |
unfolding join_pmf_def bind_map_pmf .. |
|
1202 |
||
1203 |
lemma join_eq_bind_pmf: "join_pmf M = bind_pmf M id" |
|
1204 |
by (simp add: join_pmf_def id_def) |
|
1205 |
||
1206 |
lemma pmf_join: "pmf (join_pmf N) i = (\<integral>M. pmf M i \<partial>measure_pmf N)" |
|
1207 |
unfolding join_pmf_def pmf_bind .. |
|
1208 |
||
1209 |
lemma ereal_pmf_join: "ereal (pmf (join_pmf N) i) = (\<integral>\<^sup>+M. pmf M i \<partial>measure_pmf N)" |
|
1210 |
unfolding join_pmf_def ereal_pmf_bind .. |
|
1211 |
||
59665 | 1212 |
lemma set_pmf_join_pmf[simp]: "set_pmf (join_pmf f) = (\<Union>p\<in>set_pmf f. set_pmf p)" |
1213 |
by (simp add: join_pmf_def) |
|
59664 | 1214 |
|
1215 |
lemma join_return_pmf: "join_pmf (return_pmf M) = M" |
|
1216 |
by (simp add: integral_return pmf_eq_iff pmf_join return_pmf.rep_eq) |
|
1217 |
||
1218 |
lemma map_join_pmf: "map_pmf f (join_pmf AA) = join_pmf (map_pmf (map_pmf f) AA)" |
|
1219 |
by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf) |
|
1220 |
||
1221 |
lemma join_map_return_pmf: "join_pmf (map_pmf return_pmf A) = A" |
|
1222 |
by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf') |
|
1223 |
||
1224 |
end |
|
1225 |
||
1226 |
lemma rel_pmf_joinI: |
|
1227 |
assumes "rel_pmf (rel_pmf P) p q" |
|
1228 |
shows "rel_pmf P (join_pmf p) (join_pmf q)" |
|
1229 |
proof - |
|
1230 |
from assms obtain pq where p: "p = map_pmf fst pq" |
|
1231 |
and q: "q = map_pmf snd pq" |
|
1232 |
and P: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> rel_pmf P x y" |
|
1233 |
by cases 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
|
1234 |
from P obtain PQ |
59664 | 1235 |
where PQ: "\<And>x y a b. \<lbrakk> (x, y) \<in> set_pmf pq; (a, b) \<in> set_pmf (PQ x y) \<rbrakk> \<Longrightarrow> P a b" |
1236 |
and x: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf fst (PQ x y) = x" |
|
1237 |
and y: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf snd (PQ x y) = y" |
|
1238 |
by(metis rel_pmf.simps) |
|
1239 |
||
1240 |
let ?r = "bind_pmf pq (\<lambda>(x, y). PQ x y)" |
|
59665 | 1241 |
have "\<And>a b. (a, b) \<in> set_pmf ?r \<Longrightarrow> P a b" by (auto intro: PQ) |
59664 | 1242 |
moreover have "map_pmf fst ?r = join_pmf p" "map_pmf snd ?r = join_pmf q" |
1243 |
by (simp_all add: p q x y join_pmf_def map_bind_pmf bind_map_pmf split_def cong: bind_pmf_cong) |
|
1244 |
ultimately show ?thesis .. |
|
1245 |
qed |
|
1246 |
||
1247 |
lemma rel_pmf_bindI: |
|
1248 |
assumes pq: "rel_pmf R p q" |
|
1249 |
and fg: "\<And>x y. R x y \<Longrightarrow> rel_pmf P (f x) (g y)" |
|
1250 |
shows "rel_pmf P (bind_pmf p f) (bind_pmf q g)" |
|
1251 |
unfolding bind_eq_join_pmf |
|
1252 |
by (rule rel_pmf_joinI) |
|
1253 |
(auto simp add: pmf.rel_map intro: pmf.rel_mono[THEN le_funD, THEN le_funD, THEN le_boolD, THEN mp, OF _ pq] fg) |
|
1254 |
||
1255 |
text {* |
|
1256 |
Proof that @{const rel_pmf} preserves orders. |
|
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
|
1257 |
Antisymmetry proof follows Thm. 1 in N. Saheb-Djahromi, Cpo's of measures for nondeterminism, |
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
|
1258 |
Theoretical Computer Science 12(1):19--37, 1980, |
59664 | 1259 |
@{url "http://dx.doi.org/10.1016/0304-3975(80)90003-1"} |
1260 |
*} |
|
1261 |
||
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
|
1262 |
lemma |
59664 | 1263 |
assumes *: "rel_pmf R p q" |
1264 |
and refl: "reflp R" and trans: "transp R" |
|
1265 |
shows measure_Ici: "measure p {y. R x y} \<le> measure q {y. R x y}" (is ?thesis1) |
|
1266 |
and measure_Ioi: "measure p {y. R x y \<and> \<not> R y x} \<le> measure q {y. R x y \<and> \<not> R y x}" (is ?thesis2) |
|
1267 |
proof - |
|
1268 |
from * obtain pq |
|
1269 |
where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y" |
|
1270 |
and p: "p = map_pmf fst pq" |
|
1271 |
and q: "q = map_pmf snd pq" |
|
1272 |
by cases auto |
|
1273 |
show ?thesis1 ?thesis2 unfolding p q map_pmf_rep_eq using refl trans |
|
1274 |
by(auto 4 3 simp add: measure_distr reflpD AE_measure_pmf_iff intro!: measure_pmf.finite_measure_mono_AE dest!: pq elim: transpE) |
|
1275 |
qed |
|
1276 |
||
1277 |
lemma rel_pmf_inf: |
|
1278 |
fixes p q :: "'a pmf" |
|
1279 |
assumes 1: "rel_pmf R p q" |
|
1280 |
assumes 2: "rel_pmf R q p" |
|
1281 |
and refl: "reflp R" and trans: "transp R" |
|
1282 |
shows "rel_pmf (inf R R\<inverse>\<inverse>) p q" |
|
59681 | 1283 |
proof (subst rel_pmf_iff_equivp, safe) |
1284 |
show "equivp (inf R R\<inverse>\<inverse>)" |
|
1285 |
using trans refl by (auto simp: equivp_reflp_symp_transp intro: sympI transpI reflpI dest: transpD reflpD) |
|
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
|
1286 |
|
59681 | 1287 |
fix C assume "C \<in> UNIV // {(x, y). inf R R\<inverse>\<inverse> x y}" |
1288 |
then obtain x where C: "C = {y. R x y \<and> R y x}" |
|
1289 |
by (auto elim: quotientE) |
|
1290 |
||
59670 | 1291 |
let ?R = "\<lambda>x y. R x y \<and> R y x" |
1292 |
let ?\<mu>R = "\<lambda>y. measure q {x. ?R x y}" |
|
59681 | 1293 |
have "measure p {y. ?R x y} = measure p ({y. R x y} - {y. R x y \<and> \<not> R y x})" |
1294 |
by(auto intro!: arg_cong[where f="measure p"]) |
|
1295 |
also have "\<dots> = measure p {y. R x y} - measure p {y. R x y \<and> \<not> R y x}" |
|
1296 |
by (rule measure_pmf.finite_measure_Diff) auto |
|
1297 |
also have "measure p {y. R x y \<and> \<not> R y x} = measure q {y. R x y \<and> \<not> R y x}" |
|
1298 |
using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ioi) |
|
1299 |
also have "measure p {y. R x y} = measure q {y. R x y}" |
|
1300 |
using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ici) |
|
1301 |
also have "measure q {y. R x y} - measure q {y. R x y \<and> \<not> R y x} = |
|
1302 |
measure q ({y. R x y} - {y. R x y \<and> \<not> R y x})" |
|
1303 |
by(rule measure_pmf.finite_measure_Diff[symmetric]) auto |
|
1304 |
also have "\<dots> = ?\<mu>R x" |
|
1305 |
by(auto intro!: arg_cong[where f="measure q"]) |
|
1306 |
finally show "measure p C = measure q C" |
|
1307 |
by (simp add: C conj_commute) |
|
59664 | 1308 |
qed |
1309 |
||
1310 |
lemma rel_pmf_antisym: |
|
1311 |
fixes p q :: "'a pmf" |
|
1312 |
assumes 1: "rel_pmf R p q" |
|
1313 |
assumes 2: "rel_pmf R q p" |
|
1314 |
and refl: "reflp R" and trans: "transp R" and antisym: "antisymP R" |
|
1315 |
shows "p = q" |
|
1316 |
proof - |
|
1317 |
from 1 2 refl trans have "rel_pmf (inf R R\<inverse>\<inverse>) p q" by(rule rel_pmf_inf) |
|
1318 |
also have "inf R R\<inverse>\<inverse> = op =" |
|
59665 | 1319 |
using refl antisym by (auto intro!: ext simp add: reflpD dest: antisymD) |
59664 | 1320 |
finally show ?thesis unfolding pmf.rel_eq . |
1321 |
qed |
|
1322 |
||
1323 |
lemma reflp_rel_pmf: "reflp R \<Longrightarrow> reflp (rel_pmf R)" |
|
1324 |
by(blast intro: reflpI rel_pmf_reflI reflpD) |
|
1325 |
||
1326 |
lemma antisymP_rel_pmf: |
|
1327 |
"\<lbrakk> reflp R; transp R; antisymP R \<rbrakk> |
|
1328 |
\<Longrightarrow> antisymP (rel_pmf R)" |
|
1329 |
by(rule antisymI)(blast intro: rel_pmf_antisym) |
|
1330 |
||
1331 |
lemma transp_rel_pmf: |
|
1332 |
assumes "transp R" |
|
1333 |
shows "transp (rel_pmf R)" |
|
1334 |
proof (rule transpI) |
|
1335 |
fix x y z |
|
1336 |
assume "rel_pmf R x y" and "rel_pmf R y z" |
|
1337 |
hence "rel_pmf (R OO R) x z" by (simp add: pmf.rel_compp relcompp.relcompI) |
|
1338 |
thus "rel_pmf R x z" |
|
1339 |
using assms by (metis (no_types) pmf.rel_mono rev_predicate2D transp_relcompp_less_eq) |
|
1340 |
qed |
|
1341 |
||
1342 |
subsection \<open> Distributions \<close> |
|
1343 |
||
59000 | 1344 |
context |
1345 |
begin |
|
1346 |
||
1347 |
interpretation pmf_as_function . |
|
1348 |
||
59093 | 1349 |
subsubsection \<open> Bernoulli Distribution \<close> |
1350 |
||
59000 | 1351 |
lift_definition bernoulli_pmf :: "real \<Rightarrow> bool pmf" is |
1352 |
"\<lambda>p b. ((\<lambda>p. if b then p else 1 - p) \<circ> min 1 \<circ> max 0) p" |
|
1353 |
by (auto simp: nn_integral_count_space_finite[where A="{False, True}"] UNIV_bool |
|
1354 |
split: split_max split_min) |
|
1355 |
||
1356 |
lemma pmf_bernoulli_True[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) True = p" |
|
1357 |
by transfer simp |
|
1358 |
||
1359 |
lemma pmf_bernoulli_False[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) False = 1 - p" |
|
1360 |
by transfer simp |
|
1361 |
||
1362 |
lemma set_pmf_bernoulli: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (bernoulli_pmf p) = UNIV" |
|
1363 |
by (auto simp add: set_pmf_iff UNIV_bool) |
|
1364 |
||
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
|
1365 |
lemma nn_integral_bernoulli_pmf[simp]: |
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1366 |
assumes [simp]: "0 \<le> p" "p \<le> 1" "\<And>x. 0 \<le> f x" |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1367 |
shows "(\<integral>\<^sup>+x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)" |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1368 |
by (subst nn_integral_measure_pmf_support[of UNIV]) |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1369 |
(auto simp: UNIV_bool field_simps) |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1370 |
|
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
|
1371 |
lemma integral_bernoulli_pmf[simp]: |
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1372 |
assumes [simp]: "0 \<le> p" "p \<le> 1" |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1373 |
shows "(\<integral>x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)" |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1374 |
by (subst integral_measure_pmf[of UNIV]) (auto simp: UNIV_bool) |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1375 |
|
59525 | 1376 |
lemma pmf_bernoulli_half [simp]: "pmf (bernoulli_pmf (1 / 2)) x = 1 / 2" |
1377 |
by(cases x) simp_all |
|
1378 |
||
1379 |
lemma measure_pmf_bernoulli_half: "measure_pmf (bernoulli_pmf (1 / 2)) = uniform_count_measure UNIV" |
|
1380 |
by(rule measure_eqI)(simp_all add: nn_integral_pmf[symmetric] emeasure_uniform_count_measure nn_integral_count_space_finite sets_uniform_count_measure) |
|
1381 |
||
59093 | 1382 |
subsubsection \<open> Geometric Distribution \<close> |
1383 |
||
60602 | 1384 |
context |
1385 |
fixes p :: real assumes p[arith]: "0 < p" "p \<le> 1" |
|
1386 |
begin |
|
1387 |
||
1388 |
lift_definition geometric_pmf :: "nat pmf" is "\<lambda>n. (1 - p)^n * p" |
|
59000 | 1389 |
proof |
60602 | 1390 |
have "(\<Sum>i. ereal (p * (1 - p) ^ i)) = ereal (p * (1 / (1 - (1 - p))))" |
1391 |
by (intro sums_suminf_ereal sums_mult geometric_sums) auto |
|
1392 |
then show "(\<integral>\<^sup>+ x. ereal ((1 - p)^x * p) \<partial>count_space UNIV) = 1" |
|
59000 | 1393 |
by (simp add: nn_integral_count_space_nat field_simps) |
1394 |
qed simp |
|
1395 |
||
60602 | 1396 |
lemma pmf_geometric[simp]: "pmf geometric_pmf n = (1 - p)^n * p" |
59000 | 1397 |
by transfer rule |
1398 |
||
60602 | 1399 |
end |
1400 |
||
1401 |
lemma set_pmf_geometric: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (geometric_pmf p) = UNIV" |
|
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
|
1402 |
by (auto simp: set_pmf_iff) |
59000 | 1403 |
|
59093 | 1404 |
subsubsection \<open> Uniform Multiset Distribution \<close> |
1405 |
||
59000 | 1406 |
context |
1407 |
fixes M :: "'a multiset" assumes M_not_empty: "M \<noteq> {#}" |
|
1408 |
begin |
|
1409 |
||
1410 |
lift_definition pmf_of_multiset :: "'a pmf" is "\<lambda>x. count M x / size M" |
|
1411 |
proof |
|
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
|
1412 |
show "(\<integral>\<^sup>+ x. ereal (real (count M x) / real (size M)) \<partial>count_space UNIV) = 1" |
59000 | 1413 |
using M_not_empty |
1414 |
by (simp add: zero_less_divide_iff nn_integral_count_space nonempty_has_size |
|
1415 |
setsum_divide_distrib[symmetric]) |
|
1416 |
(auto simp: size_multiset_overloaded_eq intro!: setsum.cong) |
|
1417 |
qed simp |
|
1418 |
||
1419 |
lemma pmf_of_multiset[simp]: "pmf pmf_of_multiset x = count M x / size M" |
|
1420 |
by transfer rule |
|
1421 |
||
60495 | 1422 |
lemma set_pmf_of_multiset[simp]: "set_pmf pmf_of_multiset = set_mset M" |
59000 | 1423 |
by (auto simp: set_pmf_iff) |
1424 |
||
1425 |
end |
|
1426 |
||
59093 | 1427 |
subsubsection \<open> Uniform Distribution \<close> |
1428 |
||
59000 | 1429 |
context |
1430 |
fixes S :: "'a set" assumes S_not_empty: "S \<noteq> {}" and S_finite: "finite S" |
|
1431 |
begin |
|
1432 |
||
1433 |
lift_definition pmf_of_set :: "'a pmf" is "\<lambda>x. indicator S x / card S" |
|
1434 |
proof |
|
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
|
1435 |
show "(\<integral>\<^sup>+ x. ereal (indicator S x / real (card S)) \<partial>count_space UNIV) = 1" |
59000 | 1436 |
using S_not_empty S_finite by (subst nn_integral_count_space'[of S]) auto |
1437 |
qed simp |
|
1438 |
||
1439 |
lemma pmf_of_set[simp]: "pmf pmf_of_set x = indicator S x / card S" |
|
1440 |
by transfer rule |
|
1441 |
||
1442 |
lemma set_pmf_of_set[simp]: "set_pmf pmf_of_set = S" |
|
1443 |
using S_finite S_not_empty by (auto simp: set_pmf_iff) |
|
1444 |
||
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1445 |
lemma emeasure_pmf_of_set[simp]: "emeasure pmf_of_set S = 1" |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1446 |
by (rule measure_pmf.emeasure_eq_1_AE) (auto simp: AE_measure_pmf_iff) |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1447 |
|
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
|
1448 |
lemma nn_integral_pmf_of_set': |
60068 | 1449 |
"(\<And>x. x \<in> S \<Longrightarrow> f x \<ge> 0) \<Longrightarrow> nn_integral (measure_pmf pmf_of_set) f = setsum f S / card S" |
1450 |
apply(subst nn_integral_measure_pmf_finite, simp_all add: S_finite) |
|
1451 |
apply(simp add: setsum_ereal_left_distrib[symmetric]) |
|
1452 |
apply(subst ereal_divide', simp add: S_not_empty S_finite) |
|
1453 |
apply(simp add: ereal_times_divide_eq one_ereal_def[symmetric]) |
|
1454 |
done |
|
1455 |
||
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
|
1456 |
lemma nn_integral_pmf_of_set: |
60068 | 1457 |
"nn_integral (measure_pmf pmf_of_set) f = setsum (max 0 \<circ> f) S / card S" |
1458 |
apply(subst nn_integral_max_0[symmetric]) |
|
1459 |
apply(subst nn_integral_pmf_of_set') |
|
1460 |
apply simp_all |
|
1461 |
done |
|
1462 |
||
1463 |
lemma integral_pmf_of_set: |
|
1464 |
"integral\<^sup>L (measure_pmf pmf_of_set) f = setsum f S / card S" |
|
1465 |
apply(simp add: real_lebesgue_integral_def integrable_measure_pmf_finite nn_integral_pmf_of_set S_finite) |
|
1466 |
apply(subst real_of_ereal_minus') |
|
1467 |
apply(simp add: ereal_max_0 S_finite del: ereal_max) |
|
1468 |
apply(simp add: ereal_max_0 S_finite S_not_empty del: ereal_max) |
|
1469 |
apply(simp add: field_simps S_finite S_not_empty) |
|
1470 |
apply(subst setsum.distrib[symmetric]) |
|
1471 |
apply(rule setsum.cong; simp_all) |
|
1472 |
done |
|
1473 |
||
59000 | 1474 |
end |
1475 |
||
60068 | 1476 |
lemma pmf_of_set_singleton: "pmf_of_set {x} = return_pmf x" |
1477 |
by(rule pmf_eqI)(simp add: indicator_def) |
|
1478 |
||
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
|
1479 |
lemma map_pmf_of_set_inj: |
60068 | 1480 |
assumes f: "inj_on f A" |
1481 |
and [simp]: "A \<noteq> {}" "finite A" |
|
1482 |
shows "map_pmf f (pmf_of_set A) = pmf_of_set (f ` A)" (is "?lhs = ?rhs") |
|
1483 |
proof(rule pmf_eqI) |
|
1484 |
fix i |
|
1485 |
show "pmf ?lhs i = pmf ?rhs i" |
|
1486 |
proof(cases "i \<in> f ` A") |
|
1487 |
case True |
|
1488 |
then obtain i' where "i = f i'" "i' \<in> A" by auto |
|
1489 |
thus ?thesis using f by(simp add: card_image pmf_map_inj) |
|
1490 |
next |
|
1491 |
case False |
|
1492 |
hence "pmf ?lhs i = 0" by(simp add: pmf_eq_0_set_pmf set_map_pmf) |
|
1493 |
moreover have "pmf ?rhs i = 0" using False by simp |
|
1494 |
ultimately show ?thesis by simp |
|
1495 |
qed |
|
1496 |
qed |
|
1497 |
||
1498 |
lemma bernoulli_pmf_half_conv_pmf_of_set: "bernoulli_pmf (1 / 2) = pmf_of_set UNIV" |
|
1499 |
by(rule pmf_eqI) simp_all |
|
1500 |
||
59093 | 1501 |
subsubsection \<open> Poisson Distribution \<close> |
1502 |
||
1503 |
context |
|
1504 |
fixes rate :: real assumes rate_pos: "0 < rate" |
|
1505 |
begin |
|
1506 |
||
1507 |
lift_definition poisson_pmf :: "nat pmf" is "\<lambda>k. rate ^ k / fact k * exp (-rate)" |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59667
diff
changeset
|
1508 |
proof (* by Manuel Eberl *) |
59093 | 1509 |
have summable: "summable (\<lambda>x::nat. rate ^ x / fact x)" using summable_exp |
59557 | 1510 |
by (simp add: field_simps divide_inverse [symmetric]) |
59093 | 1511 |
have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x * exp (-rate) \<partial>count_space UNIV) = |
1512 |
exp (-rate) * (\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV)" |
|
1513 |
by (simp add: field_simps nn_integral_cmult[symmetric]) |
|
1514 |
also from rate_pos have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV) = (\<Sum>x. rate ^ x / fact x)" |
|
1515 |
by (simp_all add: nn_integral_count_space_nat suminf_ereal summable suminf_ereal_finite) |
|
1516 |
also have "... = exp rate" unfolding exp_def |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59667
diff
changeset
|
1517 |
by (simp add: field_simps divide_inverse [symmetric]) |
59093 | 1518 |
also have "ereal (exp (-rate)) * ereal (exp rate) = 1" |
1519 |
by (simp add: mult_exp_exp) |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59667
diff
changeset
|
1520 |
finally show "(\<integral>\<^sup>+ x. ereal (rate ^ x / (fact x) * exp (- rate)) \<partial>count_space UNIV) = 1" . |
59093 | 1521 |
qed (simp add: rate_pos[THEN less_imp_le]) |
1522 |
||
1523 |
lemma pmf_poisson[simp]: "pmf poisson_pmf k = rate ^ k / fact k * exp (-rate)" |
|
1524 |
by transfer rule |
|
1525 |
||
1526 |
lemma set_pmf_poisson[simp]: "set_pmf poisson_pmf = UNIV" |
|
1527 |
using rate_pos by (auto simp: set_pmf_iff) |
|
1528 |
||
59000 | 1529 |
end |
1530 |
||
59093 | 1531 |
subsubsection \<open> Binomial Distribution \<close> |
1532 |
||
1533 |
context |
|
1534 |
fixes n :: nat and p :: real assumes p_nonneg: "0 \<le> p" and p_le_1: "p \<le> 1" |
|
1535 |
begin |
|
1536 |
||
1537 |
lift_definition binomial_pmf :: "nat pmf" is "\<lambda>k. (n choose k) * p^k * (1 - p)^(n - k)" |
|
1538 |
proof |
|
1539 |
have "(\<integral>\<^sup>+k. ereal (real (n choose k) * p ^ k * (1 - p) ^ (n - k)) \<partial>count_space UNIV) = |
|
1540 |
ereal (\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k))" |
|
1541 |
using p_le_1 p_nonneg by (subst nn_integral_count_space') auto |
|
1542 |
also have "(\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k)) = (p + (1 - p)) ^ n" |
|
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
|
1543 |
by (subst binomial_ring) (simp add: atLeast0AtMost) |
59093 | 1544 |
finally show "(\<integral>\<^sup>+ x. ereal (real (n choose x) * p ^ x * (1 - p) ^ (n - x)) \<partial>count_space UNIV) = 1" |
1545 |
by simp |
|
1546 |
qed (insert p_nonneg p_le_1, simp) |
|
1547 |
||
1548 |
lemma pmf_binomial[simp]: "pmf binomial_pmf k = (n choose k) * p^k * (1 - p)^(n - k)" |
|
1549 |
by transfer rule |
|
1550 |
||
1551 |
lemma set_pmf_binomial_eq: "set_pmf binomial_pmf = (if p = 0 then {0} else if p = 1 then {n} else {.. n})" |
|
1552 |
using p_nonneg p_le_1 unfolding set_eq_iff set_pmf_iff pmf_binomial by (auto simp: set_pmf_iff) |
|
1553 |
||
1554 |
end |
|
1555 |
||
1556 |
end |
|
1557 |
||
1558 |
lemma set_pmf_binomial_0[simp]: "set_pmf (binomial_pmf n 0) = {0}" |
|
1559 |
by (simp add: set_pmf_binomial_eq) |
|
1560 |
||
1561 |
lemma set_pmf_binomial_1[simp]: "set_pmf (binomial_pmf n 1) = {n}" |
|
1562 |
by (simp add: set_pmf_binomial_eq) |
|
1563 |
||
1564 |
lemma set_pmf_binomial[simp]: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (binomial_pmf n p) = {..n}" |
|
1565 |
by (simp add: set_pmf_binomial_eq) |
|
1566 |
||
59000 | 1567 |
end |