1(* Title: HOL/Probability/Probability_Mass_Function.thy 2 Author: Johannes H��lzl, TU M��nchen 3 Author: Andreas Lochbihler, ETH Zurich 4*) 5 6section \<open> Probability mass function \<close> 7 8theory Probability_Mass_Function 9imports 10 Giry_Monad 11 "HOL-Library.Multiset" 12begin 13 14lemma AE_emeasure_singleton: 15 assumes x: "emeasure M {x} \<noteq> 0" and ae: "AE x in M. P x" shows "P x" 16proof - 17 from x have x_M: "{x} \<in> sets M" 18 by (auto intro: emeasure_notin_sets) 19 from ae obtain N where N: "{x\<in>space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M" 20 by (auto elim: AE_E) 21 { assume "\<not> P x" 22 with x_M[THEN sets.sets_into_space] N have "emeasure M {x} \<le> emeasure M N" 23 by (intro emeasure_mono) auto 24 with x N have False 25 by (auto simp:) } 26 then show "P x" by auto 27qed 28 29lemma AE_measure_singleton: "measure M {x} \<noteq> 0 \<Longrightarrow> AE x in M. P x \<Longrightarrow> P x" 30 by (metis AE_emeasure_singleton measure_def emeasure_empty measure_empty) 31 32lemma (in finite_measure) AE_support_countable: 33 assumes [simp]: "sets M = UNIV" 34 shows "(AE x in M. measure M {x} \<noteq> 0) \<longleftrightarrow> (\<exists>S. countable S \<and> (AE x in M. x \<in> S))" 35proof 36 assume "\<exists>S. countable S \<and> (AE x in M. x \<in> S)" 37 then obtain S where S[intro]: "countable S" and ae: "AE x in M. x \<in> S" 38 by auto 39 then have "emeasure M (\<Union>x\<in>{x\<in>S. emeasure M {x} \<noteq> 0}. {x}) = 40 (\<integral>\<^sup>+ x. emeasure M {x} * indicator {x\<in>S. emeasure M {x} \<noteq> 0} x \<partial>count_space UNIV)" 41 by (subst emeasure_UN_countable) 42 (auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space) 43 also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} * indicator S x \<partial>count_space UNIV)" 44 by (auto intro!: nn_integral_cong split: split_indicator) 45 also have "\<dots> = emeasure M (\<Union>x\<in>S. {x})" 46 by (subst emeasure_UN_countable) 47 (auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space) 48 also have "\<dots> = emeasure M (space M)" 49 using ae by (intro emeasure_eq_AE) auto 50 finally have "emeasure M {x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0} = emeasure M (space M)" 51 by (simp add: emeasure_single_in_space cong: rev_conj_cong) 52 with finite_measure_compl[of "{x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0}"] 53 have "AE x in M. x \<in> S \<and> emeasure M {x} \<noteq> 0" 54 by (intro AE_I[OF order_refl]) (auto simp: emeasure_eq_measure measure_nonneg set_diff_eq cong: conj_cong) 55 then show "AE x in M. measure M {x} \<noteq> 0" 56 by (auto simp: emeasure_eq_measure) 57qed (auto intro!: exI[of _ "{x. measure M {x} \<noteq> 0}"] countable_support) 58 59subsection \<open> PMF as measure \<close> 60 61typedef 'a pmf = "{M :: 'a measure. prob_space M \<and> sets M = UNIV \<and> (AE x in M. measure M {x} \<noteq> 0)}" 62 morphisms measure_pmf Abs_pmf 63 by (intro exI[of _ "uniform_measure (count_space UNIV) {undefined}"]) 64 (auto intro!: prob_space_uniform_measure AE_uniform_measureI) 65 66declare [[coercion measure_pmf]] 67 68lemma prob_space_measure_pmf: "prob_space (measure_pmf p)" 69 using pmf.measure_pmf[of p] by auto 70 71interpretation measure_pmf: prob_space "measure_pmf M" for M 72 by (rule prob_space_measure_pmf) 73 74interpretation measure_pmf: subprob_space "measure_pmf M" for M 75 by (rule prob_space_imp_subprob_space) unfold_locales 76 77lemma subprob_space_measure_pmf: "subprob_space (measure_pmf x)" 78 by unfold_locales 79 80locale pmf_as_measure 81begin 82 83setup_lifting type_definition_pmf 84 85end 86 87context 88begin 89 90interpretation pmf_as_measure . 91 92lemma sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV" 93 by transfer blast 94 95lemma sets_measure_pmf_count_space[measurable_cong]: 96 "sets (measure_pmf M) = sets (count_space UNIV)" 97 by simp 98 99lemma space_measure_pmf[simp]: "space (measure_pmf p) = UNIV" 100 using sets_eq_imp_space_eq[of "measure_pmf p" "count_space UNIV"] by simp 101 102lemma measure_pmf_UNIV [simp]: "measure (measure_pmf p) UNIV = 1" 103using measure_pmf.prob_space[of p] by simp 104 105lemma measure_pmf_in_subprob_algebra[measurable (raw)]: "measure_pmf x \<in> space (subprob_algebra (count_space UNIV))" 106 by (simp add: space_subprob_algebra subprob_space_measure_pmf) 107 108lemma measurable_pmf_measure1[simp]: "measurable (M :: 'a pmf) N = UNIV \<rightarrow> space N" 109 by (auto simp: measurable_def) 110 111lemma measurable_pmf_measure2[simp]: "measurable N (M :: 'a pmf) = measurable N (count_space UNIV)" 112 by (intro measurable_cong_sets) simp_all 113 114lemma measurable_pair_restrict_pmf2: 115 assumes "countable A" 116 assumes [measurable]: "\<And>y. y \<in> A \<Longrightarrow> (\<lambda>x. f (x, y)) \<in> measurable M L" 117 shows "f \<in> measurable (M \<Otimes>\<^sub>M restrict_space (measure_pmf N) A) L" (is "f \<in> measurable ?M _") 118proof - 119 have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)" 120 by (simp add: restrict_count_space) 121 122 show ?thesis 123 by (intro measurable_compose_countable'[where f="\<lambda>a b. f (fst b, a)" and g=snd and I=A, 124 unfolded prod.collapse] assms) 125 measurable 126qed 127 128lemma measurable_pair_restrict_pmf1: 129 assumes "countable A" 130 assumes [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N L" 131 shows "f \<in> measurable (restrict_space (measure_pmf M) A \<Otimes>\<^sub>M N) L" 132proof - 133 have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)" 134 by (simp add: restrict_count_space) 135 136 show ?thesis 137 by (intro measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A, 138 unfolded prod.collapse] assms) 139 measurable 140qed 141 142lift_definition pmf :: "'a pmf \<Rightarrow> 'a \<Rightarrow> real" is "\<lambda>M x. measure M {x}" . 143 144lift_definition set_pmf :: "'a pmf \<Rightarrow> 'a set" is "\<lambda>M. {x. measure M {x} \<noteq> 0}" . 145declare [[coercion set_pmf]] 146 147lemma AE_measure_pmf: "AE x in (M::'a pmf). x \<in> M" 148 by transfer simp 149 150lemma emeasure_pmf_single_eq_zero_iff: 151 fixes M :: "'a pmf" 152 shows "emeasure M {y} = 0 \<longleftrightarrow> y \<notin> M" 153 unfolding set_pmf.rep_eq by (simp add: measure_pmf.emeasure_eq_measure) 154 155lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \<longleftrightarrow> (\<forall>y\<in>M. P y)" 156 using AE_measure_singleton[of M] AE_measure_pmf[of M] 157 by (auto simp: set_pmf.rep_eq) 158 159lemma AE_pmfI: "(\<And>y. y \<in> set_pmf M \<Longrightarrow> P y) \<Longrightarrow> almost_everywhere (measure_pmf M) P" 160by(simp add: AE_measure_pmf_iff) 161 162lemma countable_set_pmf [simp]: "countable (set_pmf p)" 163 by transfer (metis prob_space.finite_measure finite_measure.countable_support) 164 165lemma pmf_positive: "x \<in> set_pmf p \<Longrightarrow> 0 < pmf p x" 166 by transfer (simp add: less_le) 167 168lemma pmf_nonneg[simp]: "0 \<le> pmf p x" 169 by transfer simp 170 171lemma pmf_not_neg [simp]: "\<not>pmf p x < 0" 172 by (simp add: not_less pmf_nonneg) 173 174lemma pmf_pos [simp]: "pmf p x \<noteq> 0 \<Longrightarrow> pmf p x > 0" 175 using pmf_nonneg[of p x] by linarith 176 177lemma pmf_le_1: "pmf p x \<le> 1" 178 by (simp add: pmf.rep_eq) 179 180lemma set_pmf_not_empty: "set_pmf M \<noteq> {}" 181 using AE_measure_pmf[of M] by (intro notI) simp 182 183lemma set_pmf_iff: "x \<in> set_pmf M \<longleftrightarrow> pmf M x \<noteq> 0" 184 by transfer simp 185 186lemma pmf_positive_iff: "0 < pmf p x \<longleftrightarrow> x \<in> set_pmf p" 187 unfolding less_le by (simp add: set_pmf_iff) 188 189lemma set_pmf_eq: "set_pmf M = {x. pmf M x \<noteq> 0}" 190 by (auto simp: set_pmf_iff) 191 192lemma set_pmf_eq': "set_pmf p = {x. pmf p x > 0}" 193proof safe 194 fix x assume "x \<in> set_pmf p" 195 hence "pmf p x \<noteq> 0" by (auto simp: set_pmf_eq) 196 with pmf_nonneg[of p x] show "pmf p x > 0" by simp 197qed (auto simp: set_pmf_eq) 198 199lemma emeasure_pmf_single: 200 fixes M :: "'a pmf" 201 shows "emeasure M {x} = pmf M x" 202 by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure]) 203 204lemma measure_pmf_single: "measure (measure_pmf M) {x} = pmf M x" 205 using emeasure_pmf_single[of M x] by(simp add: measure_pmf.emeasure_eq_measure pmf_nonneg measure_nonneg) 206 207lemma emeasure_measure_pmf_finite: "finite S \<Longrightarrow> emeasure (measure_pmf M) S = (\<Sum>s\<in>S. pmf M s)" 208 by (subst emeasure_eq_sum_singleton) (auto simp: emeasure_pmf_single pmf_nonneg) 209 210lemma measure_measure_pmf_finite: "finite S \<Longrightarrow> measure (measure_pmf M) S = sum (pmf M) S" 211 using emeasure_measure_pmf_finite[of S M] 212 by (simp add: measure_pmf.emeasure_eq_measure measure_nonneg sum_nonneg pmf_nonneg) 213 214lemma sum_pmf_eq_1: 215 assumes "finite A" "set_pmf p \<subseteq> A" 216 shows "(\<Sum>x\<in>A. pmf p x) = 1" 217proof - 218 have "(\<Sum>x\<in>A. pmf p x) = measure_pmf.prob p A" 219 by (simp add: measure_measure_pmf_finite assms) 220 also from assms have "\<dots> = 1" 221 by (subst measure_pmf.prob_eq_1) (auto simp: AE_measure_pmf_iff) 222 finally show ?thesis . 223qed 224 225lemma nn_integral_measure_pmf_support: 226 fixes f :: "'a \<Rightarrow> ennreal" 227 assumes f: "finite A" and nn: "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x" "\<And>x. x \<in> set_pmf M \<Longrightarrow> x \<notin> A \<Longrightarrow> f x = 0" 228 shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>A. f x * pmf M x)" 229proof - 230 have "(\<integral>\<^sup>+x. f x \<partial>M) = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)" 231 using nn by (intro nn_integral_cong_AE) (auto simp: AE_measure_pmf_iff split: split_indicator) 232 also have "\<dots> = (\<Sum>x\<in>A. f x * emeasure M {x})" 233 using assms by (intro nn_integral_indicator_finite) auto 234 finally show ?thesis 235 by (simp add: emeasure_measure_pmf_finite) 236qed 237 238lemma nn_integral_measure_pmf_finite: 239 fixes f :: "'a \<Rightarrow> ennreal" 240 assumes f: "finite (set_pmf M)" and nn: "\<And>x. x \<in> set_pmf M \<Longrightarrow> 0 \<le> f x" 241 shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>set_pmf M. f x * pmf M x)" 242 using assms by (intro nn_integral_measure_pmf_support) auto 243 244lemma integrable_measure_pmf_finite: 245 fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" 246 shows "finite (set_pmf M) \<Longrightarrow> integrable M f" 247 by (auto intro!: integrableI_bounded simp: nn_integral_measure_pmf_finite ennreal_mult_less_top) 248 249lemma integral_measure_pmf_real: 250 assumes [simp]: "finite A" and "\<And>a. a \<in> set_pmf M \<Longrightarrow> f a \<noteq> 0 \<Longrightarrow> a \<in> A" 251 shows "(\<integral>x. f x \<partial>measure_pmf M) = (\<Sum>a\<in>A. f a * pmf M a)" 252proof - 253 have "(\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x * indicator A x \<partial>measure_pmf M)" 254 using assms(2) by (intro integral_cong_AE) (auto split: split_indicator simp: AE_measure_pmf_iff) 255 also have "\<dots> = (\<Sum>a\<in>A. f a * pmf M a)" 256 by (subst integral_indicator_finite_real) 257 (auto simp: measure_def emeasure_measure_pmf_finite pmf_nonneg) 258 finally show ?thesis . 259qed 260 261lemma integrable_pmf: "integrable (count_space X) (pmf M)" 262proof - 263 have " (\<integral>\<^sup>+ x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+ x. pmf M x \<partial>count_space (M \<inter> X))" 264 by (auto simp add: nn_integral_count_space_indicator set_pmf_iff intro!: nn_integral_cong split: split_indicator) 265 then have "integrable (count_space X) (pmf M) = integrable (count_space (M \<inter> X)) (pmf M)" 266 by (simp add: integrable_iff_bounded pmf_nonneg) 267 then show ?thesis 268 by (simp add: pmf.rep_eq measure_pmf.integrable_measure disjoint_family_on_def) 269qed 270 271lemma integral_pmf: "(\<integral>x. pmf M x \<partial>count_space X) = measure M X" 272proof - 273 have "(\<integral>x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+x. pmf M x \<partial>count_space X)" 274 by (simp add: pmf_nonneg integrable_pmf nn_integral_eq_integral) 275 also have "\<dots> = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space (X \<inter> M))" 276 by (auto intro!: nn_integral_cong_AE split: split_indicator 277 simp: pmf.rep_eq measure_pmf.emeasure_eq_measure nn_integral_count_space_indicator 278 AE_count_space set_pmf_iff) 279 also have "\<dots> = emeasure M (X \<inter> M)" 280 by (rule emeasure_countable_singleton[symmetric]) (auto intro: countable_set_pmf) 281 also have "\<dots> = emeasure M X" 282 by (auto intro!: emeasure_eq_AE simp: AE_measure_pmf_iff) 283 finally show ?thesis 284 by (simp add: measure_pmf.emeasure_eq_measure measure_nonneg integral_nonneg pmf_nonneg) 285qed 286 287lemma integral_pmf_restrict: 288 "(f::'a \<Rightarrow> 'b::{banach, second_countable_topology}) \<in> borel_measurable (count_space UNIV) \<Longrightarrow> 289 (\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x \<partial>restrict_space M M)" 290 by (auto intro!: integral_cong_AE simp add: integral_restrict_space AE_measure_pmf_iff) 291 292lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1" 293proof - 294 have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)" 295 by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf) 296 then show ?thesis 297 using measure_pmf.emeasure_space_1 by simp 298qed 299 300lemma emeasure_pmf_UNIV [simp]: "emeasure (measure_pmf M) UNIV = 1" 301using measure_pmf.emeasure_space_1[of M] by simp 302 303lemma in_null_sets_measure_pmfI: 304 "A \<inter> set_pmf p = {} \<Longrightarrow> A \<in> null_sets (measure_pmf p)" 305using emeasure_eq_0_AE[where ?P="\<lambda>x. x \<in> A" and M="measure_pmf p"] 306by(auto simp add: null_sets_def AE_measure_pmf_iff) 307 308lemma measure_subprob: "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))" 309 by (simp add: space_subprob_algebra subprob_space_measure_pmf) 310 311subsection \<open> Monad Interpretation \<close> 312 313lemma measurable_measure_pmf[measurable]: 314 "(\<lambda>x. measure_pmf (M x)) \<in> measurable (count_space UNIV) (subprob_algebra (count_space UNIV))" 315 by (auto simp: space_subprob_algebra intro!: prob_space_imp_subprob_space) unfold_locales 316 317lemma bind_measure_pmf_cong: 318 assumes "\<And>x. A x \<in> space (subprob_algebra N)" "\<And>x. B x \<in> space (subprob_algebra N)" 319 assumes "\<And>i. i \<in> set_pmf x \<Longrightarrow> A i = B i" 320 shows "bind (measure_pmf x) A = bind (measure_pmf x) B" 321proof (rule measure_eqI) 322 show "sets (measure_pmf x \<bind> A) = sets (measure_pmf x \<bind> B)" 323 using assms by (subst (1 2) sets_bind) (auto simp: space_subprob_algebra) 324next 325 fix X assume "X \<in> sets (measure_pmf x \<bind> A)" 326 then have X: "X \<in> sets N" 327 using assms by (subst (asm) sets_bind) (auto simp: space_subprob_algebra) 328 show "emeasure (measure_pmf x \<bind> A) X = emeasure (measure_pmf x \<bind> B) X" 329 using assms 330 by (subst (1 2) emeasure_bind[where N=N, OF _ _ X]) 331 (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff) 332qed 333 334lift_definition bind_pmf :: "'a pmf \<Rightarrow> ('a \<Rightarrow> 'b pmf ) \<Rightarrow> 'b pmf" is bind 335proof (clarify, intro conjI) 336 fix f :: "'a measure" and g :: "'a \<Rightarrow> 'b measure" 337 assume "prob_space f" 338 then interpret f: prob_space f . 339 assume "sets f = UNIV" and ae_f: "AE x in f. measure f {x} \<noteq> 0" 340 then have s_f[simp]: "sets f = sets (count_space UNIV)" 341 by simp 342 assume g: "\<And>x. prob_space (g x) \<and> sets (g x) = UNIV \<and> (AE y in g x. measure (g x) {y} \<noteq> 0)" 343 then have g: "\<And>x. prob_space (g x)" and s_g[simp]: "\<And>x. sets (g x) = sets (count_space UNIV)" 344 and ae_g: "\<And>x. AE y in g x. measure (g x) {y} \<noteq> 0" 345 by auto 346 347 have [measurable]: "g \<in> measurable f (subprob_algebra (count_space UNIV))" 348 by (auto simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space g) 349 350 show "prob_space (f \<bind> g)" 351 using g by (intro f.prob_space_bind[where S="count_space UNIV"]) auto 352 then interpret fg: prob_space "f \<bind> g" . 353 show [simp]: "sets (f \<bind> g) = UNIV" 354 using sets_eq_imp_space_eq[OF s_f] 355 by (subst sets_bind[where N="count_space UNIV"]) auto 356 show "AE x in f \<bind> g. measure (f \<bind> g) {x} \<noteq> 0" 357 apply (simp add: fg.prob_eq_0 AE_bind[where B="count_space UNIV"]) 358 using ae_f 359 apply eventually_elim 360 using ae_g 361 apply eventually_elim 362 apply (auto dest: AE_measure_singleton) 363 done 364qed 365 366adhoc_overloading Monad_Syntax.bind bind_pmf 367 368lemma ennreal_pmf_bind: "pmf (bind_pmf N f) i = (\<integral>\<^sup>+x. pmf (f x) i \<partial>measure_pmf N)" 369 unfolding pmf.rep_eq bind_pmf.rep_eq 370 by (auto simp: measure_pmf.measure_bind[where N="count_space UNIV"] measure_subprob measure_nonneg 371 intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1]) 372 373lemma pmf_bind: "pmf (bind_pmf N f) i = (\<integral>x. pmf (f x) i \<partial>measure_pmf N)" 374 using ennreal_pmf_bind[of N f i] 375 by (subst (asm) nn_integral_eq_integral) 376 (auto simp: pmf_nonneg pmf_le_1 pmf_nonneg integral_nonneg 377 intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1]) 378 379lemma bind_pmf_const[simp]: "bind_pmf M (\<lambda>x. c) = c" 380 by transfer (simp add: bind_const' prob_space_imp_subprob_space) 381 382lemma set_bind_pmf[simp]: "set_pmf (bind_pmf M N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))" 383proof - 384 have "set_pmf (bind_pmf M N) = {x. ennreal (pmf (bind_pmf M N) x) \<noteq> 0}" 385 by (simp add: set_pmf_eq pmf_nonneg) 386 also have "\<dots> = (\<Union>M\<in>set_pmf M. set_pmf (N M))" 387 unfolding ennreal_pmf_bind 388 by (subst nn_integral_0_iff_AE) (auto simp: AE_measure_pmf_iff pmf_nonneg set_pmf_eq) 389 finally show ?thesis . 390qed 391 392lemma bind_pmf_cong [fundef_cong]: 393 assumes "p = q" 394 shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g" 395 unfolding \<open>p = q\<close>[symmetric] measure_pmf_inject[symmetric] bind_pmf.rep_eq 396 by (auto simp: AE_measure_pmf_iff Pi_iff space_subprob_algebra subprob_space_measure_pmf 397 sets_bind[where N="count_space UNIV"] emeasure_bind[where N="count_space UNIV"] 398 intro!: nn_integral_cong_AE measure_eqI) 399 400lemma bind_pmf_cong_simp: 401 "p = q \<Longrightarrow> (\<And>x. x \<in> set_pmf q =simp=> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g" 402 by (simp add: simp_implies_def cong: bind_pmf_cong) 403 404lemma measure_pmf_bind: "measure_pmf (bind_pmf M f) = (measure_pmf M \<bind> (\<lambda>x. measure_pmf (f x)))" 405 by transfer simp 406 407lemma 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)" 408 using measurable_measure_pmf[of N] 409 unfolding measure_pmf_bind 410 apply (intro nn_integral_bind[where B="count_space UNIV"]) 411 apply auto 412 done 413 414lemma emeasure_bind_pmf[simp]: "emeasure (bind_pmf M N) X = (\<integral>\<^sup>+x. emeasure (N x) X \<partial>M)" 415 using measurable_measure_pmf[of N] 416 unfolding measure_pmf_bind 417 by (subst emeasure_bind[where N="count_space UNIV"]) auto 418 419lift_definition return_pmf :: "'a \<Rightarrow> 'a pmf" is "return (count_space UNIV)" 420 by (auto intro!: prob_space_return simp: AE_return measure_return) 421 422lemma bind_return_pmf: "bind_pmf (return_pmf x) f = f x" 423 by transfer 424 (auto intro!: prob_space_imp_subprob_space bind_return[where N="count_space UNIV"] 425 simp: space_subprob_algebra) 426 427lemma set_return_pmf[simp]: "set_pmf (return_pmf x) = {x}" 428 by transfer (auto simp add: measure_return split: split_indicator) 429 430lemma bind_return_pmf': "bind_pmf N return_pmf = N" 431proof (transfer, clarify) 432 fix N :: "'a measure" assume "sets N = UNIV" then show "N \<bind> return (count_space UNIV) = N" 433 by (subst return_sets_cong[where N=N]) (simp_all add: bind_return') 434qed 435 436lemma bind_assoc_pmf: "bind_pmf (bind_pmf A B) C = bind_pmf A (\<lambda>x. bind_pmf (B x) C)" 437 by transfer 438 (auto intro!: bind_assoc[where N="count_space UNIV" and R="count_space UNIV"] 439 simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space) 440 441definition "map_pmf f M = bind_pmf M (\<lambda>x. return_pmf (f x))" 442 443lemma map_bind_pmf: "map_pmf f (bind_pmf M g) = bind_pmf M (\<lambda>x. map_pmf f (g x))" 444 by (simp add: map_pmf_def bind_assoc_pmf) 445 446lemma bind_map_pmf: "bind_pmf (map_pmf f M) g = bind_pmf M (\<lambda>x. g (f x))" 447 by (simp add: map_pmf_def bind_assoc_pmf bind_return_pmf) 448 449lemma map_pmf_transfer[transfer_rule]: 450 "rel_fun (=) (rel_fun cr_pmf cr_pmf) (\<lambda>f M. distr M (count_space UNIV) f) map_pmf" 451proof - 452 have "rel_fun (=) (rel_fun pmf_as_measure.cr_pmf pmf_as_measure.cr_pmf) 453 (\<lambda>f M. M \<bind> (return (count_space UNIV) o f)) map_pmf" 454 unfolding map_pmf_def[abs_def] comp_def by transfer_prover 455 then show ?thesis 456 by (force simp: rel_fun_def cr_pmf_def bind_return_distr) 457qed 458 459lemma map_pmf_rep_eq: 460 "measure_pmf (map_pmf f M) = distr (measure_pmf M) (count_space UNIV) f" 461 unfolding map_pmf_def bind_pmf.rep_eq comp_def return_pmf.rep_eq 462 using bind_return_distr[of M f "count_space UNIV"] by (simp add: comp_def) 463 464lemma map_pmf_id[simp]: "map_pmf id = id" 465 by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI) 466 467lemma map_pmf_ident[simp]: "map_pmf (\<lambda>x. x) = (\<lambda>x. x)" 468 using map_pmf_id unfolding id_def . 469 470lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" 471 by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def) 472 473lemma map_pmf_comp: "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M" 474 using map_pmf_compose[of f g] by (simp add: comp_def) 475 476lemma 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" 477 unfolding map_pmf_def by (rule bind_pmf_cong) auto 478 479lemma pmf_set_map: "set_pmf \<circ> map_pmf f = (`) f \<circ> set_pmf" 480 by (auto simp add: comp_def fun_eq_iff map_pmf_def) 481 482lemma set_map_pmf[simp]: "set_pmf (map_pmf f M) = f`set_pmf M" 483 using pmf_set_map[of f] by (auto simp: comp_def fun_eq_iff) 484 485lemma emeasure_map_pmf[simp]: "emeasure (map_pmf f M) X = emeasure M (f -` X)" 486 unfolding map_pmf_rep_eq by (subst emeasure_distr) auto 487 488lemma measure_map_pmf[simp]: "measure (map_pmf f M) X = measure M (f -` X)" 489using emeasure_map_pmf[of f M X] by(simp add: measure_pmf.emeasure_eq_measure measure_nonneg) 490 491lemma nn_integral_map_pmf[simp]: "(\<integral>\<^sup>+x. f x \<partial>map_pmf g M) = (\<integral>\<^sup>+x. f (g x) \<partial>M)" 492 unfolding map_pmf_rep_eq by (intro nn_integral_distr) auto 493 494lemma ennreal_pmf_map: "pmf (map_pmf f p) x = (\<integral>\<^sup>+ y. indicator (f -` {x}) y \<partial>measure_pmf p)" 495proof (transfer fixing: f x) 496 fix p :: "'b measure" 497 presume "prob_space p" 498 then interpret prob_space p . 499 presume "sets p = UNIV" 500 then show "ennreal (measure (distr p (count_space UNIV) f) {x}) = integral\<^sup>N p (indicator (f -` {x}))" 501 by(simp add: measure_distr measurable_def emeasure_eq_measure) 502qed simp_all 503 504lemma pmf_map: "pmf (map_pmf f p) x = measure p (f -` {x})" 505proof (transfer fixing: f x) 506 fix p :: "'b measure" 507 presume "prob_space p" 508 then interpret prob_space p . 509 presume "sets p = UNIV" 510 then show "measure (distr p (count_space UNIV) f) {x} = measure p (f -` {x})" 511 by(simp add: measure_distr measurable_def emeasure_eq_measure) 512qed simp_all 513 514lemma nn_integral_pmf: "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = emeasure (measure_pmf p) A" 515proof - 516 have "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = (\<integral>\<^sup>+ x. pmf p x \<partial>count_space (A \<inter> set_pmf p))" 517 by(auto simp add: nn_integral_count_space_indicator indicator_def set_pmf_iff intro: nn_integral_cong) 518 also have "\<dots> = emeasure (measure_pmf p) (\<Union>x\<in>A \<inter> set_pmf p. {x})" 519 by(subst emeasure_UN_countable)(auto simp add: emeasure_pmf_single disjoint_family_on_def) 520 also have "\<dots> = emeasure (measure_pmf p) ((\<Union>x\<in>A \<inter> set_pmf p. {x}) \<union> {x. x \<in> A \<and> x \<notin> set_pmf p})" 521 by(rule emeasure_Un_null_set[symmetric])(auto intro: in_null_sets_measure_pmfI) 522 also have "\<dots> = emeasure (measure_pmf p) A" 523 by(auto intro: arg_cong2[where f=emeasure]) 524 finally show ?thesis . 525qed 526 527lemma integral_map_pmf[simp]: 528 fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" 529 shows "integral\<^sup>L (map_pmf g p) f = integral\<^sup>L p (\<lambda>x. f (g x))" 530 by (simp add: integral_distr map_pmf_rep_eq) 531 532lemma pmf_abs_summable [intro]: "pmf p abs_summable_on A" 533 by (rule abs_summable_on_subset[OF _ subset_UNIV]) 534 (auto simp: abs_summable_on_def integrable_iff_bounded nn_integral_pmf) 535 536lemma measure_pmf_conv_infsetsum: "measure (measure_pmf p) A = infsetsum (pmf p) A" 537 unfolding infsetsum_def by (simp add: integral_eq_nn_integral nn_integral_pmf measure_def) 538 539lemma infsetsum_pmf_eq_1: 540 assumes "set_pmf p \<subseteq> A" 541 shows "infsetsum (pmf p) A = 1" 542proof - 543 have "infsetsum (pmf p) A = lebesgue_integral (count_space UNIV) (pmf p)" 544 using assms unfolding infsetsum_altdef set_lebesgue_integral_def 545 by (intro Bochner_Integration.integral_cong) (auto simp: indicator_def set_pmf_eq) 546 also have "\<dots> = 1" 547 by (subst integral_eq_nn_integral) (auto simp: nn_integral_pmf) 548 finally show ?thesis . 549qed 550 551lemma map_return_pmf [simp]: "map_pmf f (return_pmf x) = return_pmf (f x)" 552 by transfer (simp add: distr_return) 553 554lemma map_pmf_const[simp]: "map_pmf (\<lambda>_. c) M = return_pmf c" 555 by transfer (auto simp: prob_space.distr_const) 556 557lemma pmf_return [simp]: "pmf (return_pmf x) y = indicator {y} x" 558 by transfer (simp add: measure_return) 559 560lemma nn_integral_return_pmf[simp]: "0 \<le> f x \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>return_pmf x) = f x" 561 unfolding return_pmf.rep_eq by (intro nn_integral_return) auto 562 563lemma emeasure_return_pmf[simp]: "emeasure (return_pmf x) X = indicator X x" 564 unfolding return_pmf.rep_eq by (intro emeasure_return) auto 565 566lemma measure_return_pmf [simp]: "measure_pmf.prob (return_pmf x) A = indicator A x" 567proof - 568 have "ennreal (measure_pmf.prob (return_pmf x) A) = 569 emeasure (measure_pmf (return_pmf x)) A" 570 by (simp add: measure_pmf.emeasure_eq_measure) 571 also have "\<dots> = ennreal (indicator A x)" by (simp add: ennreal_indicator) 572 finally show ?thesis by simp 573qed 574 575lemma return_pmf_inj[simp]: "return_pmf x = return_pmf y \<longleftrightarrow> x = y" 576 by (metis insertI1 set_return_pmf singletonD) 577 578lemma map_pmf_eq_return_pmf_iff: 579 "map_pmf f p = return_pmf x \<longleftrightarrow> (\<forall>y \<in> set_pmf p. f y = x)" 580proof 581 assume "map_pmf f p = return_pmf x" 582 then have "set_pmf (map_pmf f p) = set_pmf (return_pmf x)" by simp 583 then show "\<forall>y \<in> set_pmf p. f y = x" by auto 584next 585 assume "\<forall>y \<in> set_pmf p. f y = x" 586 then show "map_pmf f p = return_pmf x" 587 unfolding map_pmf_const[symmetric, of _ p] by (intro map_pmf_cong) auto 588qed 589 590definition "pair_pmf A B = bind_pmf A (\<lambda>x. bind_pmf B (\<lambda>y. return_pmf (x, y)))" 591 592lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b" 593 unfolding pair_pmf_def pmf_bind pmf_return 594 apply (subst integral_measure_pmf_real[where A="{b}"]) 595 apply (auto simp: indicator_eq_0_iff) 596 apply (subst integral_measure_pmf_real[where A="{a}"]) 597 apply (auto simp: indicator_eq_0_iff sum_nonneg_eq_0_iff pmf_nonneg) 598 done 599 600lemma set_pair_pmf[simp]: "set_pmf (pair_pmf A B) = set_pmf A \<times> set_pmf B" 601 unfolding pair_pmf_def set_bind_pmf set_return_pmf by auto 602 603lemma measure_pmf_in_subprob_space[measurable (raw)]: 604 "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))" 605 by (simp add: space_subprob_algebra) intro_locales 606 607lemma 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)" 608proof - 609 have "(\<integral>\<^sup>+x. f x \<partial>pair_pmf A B) = (\<integral>\<^sup>+x. f x * indicator (A \<times> B) x \<partial>pair_pmf A B)" 610 by (auto simp: AE_measure_pmf_iff intro!: nn_integral_cong_AE) 611 also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. f (a, b) * indicator (A \<times> B) (a, b) \<partial>B \<partial>A)" 612 by (simp add: pair_pmf_def) 613 also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. f (a, b) \<partial>B \<partial>A)" 614 by (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff) 615 finally show ?thesis . 616qed 617 618lemma bind_pair_pmf: 619 assumes M[measurable]: "M \<in> measurable (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) (subprob_algebra N)" 620 shows "measure_pmf (pair_pmf A B) \<bind> M = (measure_pmf A \<bind> (\<lambda>x. measure_pmf B \<bind> (\<lambda>y. M (x, y))))" 621 (is "?L = ?R") 622proof (rule measure_eqI) 623 have M'[measurable]: "M \<in> measurable (pair_pmf A B) (subprob_algebra N)" 624 using M[THEN measurable_space] by (simp_all add: space_pair_measure) 625 626 note measurable_bind[where N="count_space UNIV", measurable] 627 note measure_pmf_in_subprob_space[simp] 628 629 have sets_eq_N: "sets ?L = N" 630 by (subst sets_bind[OF sets_kernel[OF M']]) auto 631 show "sets ?L = sets ?R" 632 using measurable_space[OF M] 633 by (simp add: sets_eq_N space_pair_measure space_subprob_algebra) 634 fix X assume "X \<in> sets ?L" 635 then have X[measurable]: "X \<in> sets N" 636 unfolding sets_eq_N . 637 then show "emeasure ?L X = emeasure ?R X" 638 apply (simp add: emeasure_bind[OF _ M' X]) 639 apply (simp add: nn_integral_bind[where B="count_space UNIV"] pair_pmf_def measure_pmf_bind[of A] 640 nn_integral_measure_pmf_finite) 641 apply (subst emeasure_bind[OF _ _ X]) 642 apply measurable 643 apply (subst emeasure_bind[OF _ _ X]) 644 apply measurable 645 done 646qed 647 648lemma map_fst_pair_pmf: "map_pmf fst (pair_pmf A B) = A" 649 by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf') 650 651lemma map_snd_pair_pmf: "map_pmf snd (pair_pmf A B) = B" 652 by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf') 653 654lemma nn_integral_pmf': 655 "inj_on f A \<Longrightarrow> (\<integral>\<^sup>+x. pmf p (f x) \<partial>count_space A) = emeasure p (f ` A)" 656 by (subst nn_integral_bij_count_space[where g=f and B="f`A"]) 657 (auto simp: bij_betw_def nn_integral_pmf) 658 659lemma pmf_le_0_iff[simp]: "pmf M p \<le> 0 \<longleftrightarrow> pmf M p = 0" 660 using pmf_nonneg[of M p] by arith 661 662lemma min_pmf_0[simp]: "min (pmf M p) 0 = 0" "min 0 (pmf M p) = 0" 663 using pmf_nonneg[of M p] by arith+ 664 665lemma pmf_eq_0_set_pmf: "pmf M p = 0 \<longleftrightarrow> p \<notin> set_pmf M" 666 unfolding set_pmf_iff by simp 667 668lemma 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" 669 by (auto simp: pmf.rep_eq map_pmf_rep_eq measure_distr AE_measure_pmf_iff inj_onD 670 intro!: measure_pmf.finite_measure_eq_AE) 671 672lemma pmf_map_inj': "inj f \<Longrightarrow> pmf (map_pmf f M) (f x) = pmf M x" 673apply(cases "x \<in> set_pmf M") 674 apply(simp add: pmf_map_inj[OF subset_inj_on]) 675apply(simp add: pmf_eq_0_set_pmf[symmetric]) 676apply(auto simp add: pmf_eq_0_set_pmf dest: injD) 677done 678 679lemma pmf_map_outside: "x \<notin> f ` set_pmf M \<Longrightarrow> pmf (map_pmf f M) x = 0" 680 unfolding pmf_eq_0_set_pmf by simp 681 682lemma measurable_set_pmf[measurable]: "Measurable.pred (count_space UNIV) (\<lambda>x. x \<in> set_pmf M)" 683 by simp 684 685 686subsection \<open> PMFs as function \<close> 687 688context 689 fixes f :: "'a \<Rightarrow> real" 690 assumes nonneg: "\<And>x. 0 \<le> f x" 691 assumes prob: "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1" 692begin 693 694lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ennreal \<circ> f)" 695proof (intro conjI) 696 have *[simp]: "\<And>x y. ennreal (f y) * indicator {x} y = ennreal (f x) * indicator {x} y" 697 by (simp split: split_indicator) 698 show "AE x in density (count_space UNIV) (ennreal \<circ> f). 699 measure (density (count_space UNIV) (ennreal \<circ> f)) {x} \<noteq> 0" 700 by (simp add: AE_density nonneg measure_def emeasure_density max_def) 701 show "prob_space (density (count_space UNIV) (ennreal \<circ> f))" 702 by standard (simp add: emeasure_density prob) 703qed simp 704 705lemma pmf_embed_pmf: "pmf embed_pmf x = f x" 706proof transfer 707 have *[simp]: "\<And>x y. ennreal (f y) * indicator {x} y = ennreal (f x) * indicator {x} y" 708 by (simp split: split_indicator) 709 fix x show "measure (density (count_space UNIV) (ennreal \<circ> f)) {x} = f x" 710 by transfer (simp add: measure_def emeasure_density nonneg max_def) 711qed 712 713lemma set_embed_pmf: "set_pmf embed_pmf = {x. f x \<noteq> 0}" 714by(auto simp add: set_pmf_eq pmf_embed_pmf) 715 716end 717 718lemma embed_pmf_transfer: 719 "rel_fun (eq_onp (\<lambda>f. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ennreal (f x) \<partial>count_space UNIV) = 1)) pmf_as_measure.cr_pmf (\<lambda>f. density (count_space UNIV) (ennreal \<circ> f)) embed_pmf" 720 by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer) 721 722lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)" 723proof (transfer, elim conjE) 724 fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0" 725 assume "prob_space M" then interpret prob_space M . 726 show "M = density (count_space UNIV) (\<lambda>x. ennreal (measure M {x}))" 727 proof (rule measure_eqI) 728 fix A :: "'a set" 729 have "(\<integral>\<^sup>+ x. ennreal (measure M {x}) * indicator A x \<partial>count_space UNIV) = 730 (\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)" 731 by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator) 732 also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))" 733 by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space) 734 also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})" 735 by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support) 736 (auto simp: disjoint_family_on_def) 737 also have "\<dots> = emeasure M A" 738 using ae by (intro emeasure_eq_AE) auto 739 finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ennreal (measure M {x}))) A" 740 using emeasure_space_1 by (simp add: emeasure_density) 741 qed simp 742qed 743 744lemma td_pmf_embed_pmf: 745 "type_definition pmf embed_pmf {f::'a \<Rightarrow> real. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ennreal (f x) \<partial>count_space UNIV) = 1}" 746 unfolding type_definition_def 747proof safe 748 fix p :: "'a pmf" 749 have "(\<integral>\<^sup>+ x. 1 \<partial>measure_pmf p) = 1" 750 using measure_pmf.emeasure_space_1[of p] by simp 751 then show *: "(\<integral>\<^sup>+ x. ennreal (pmf p x) \<partial>count_space UNIV) = 1" 752 by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const) 753 754 show "embed_pmf (pmf p) = p" 755 by (intro measure_pmf_inject[THEN iffD1]) 756 (simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def) 757next 758 fix f :: "'a \<Rightarrow> real" assume "\<forall>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1" 759 then show "pmf (embed_pmf f) = f" 760 by (auto intro!: pmf_embed_pmf) 761qed (rule pmf_nonneg) 762 763end 764 765lemma nn_integral_measure_pmf: "(\<integral>\<^sup>+ x. f x \<partial>measure_pmf p) = \<integral>\<^sup>+ x. ennreal (pmf p x) * f x \<partial>count_space UNIV" 766by(simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg) 767 768lemma integral_measure_pmf: 769 fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" 770 assumes A: "finite A" 771 shows "(\<And>a. a \<in> set_pmf M \<Longrightarrow> f a \<noteq> 0 \<Longrightarrow> a \<in> A) \<Longrightarrow> (LINT x|M. f x) = (\<Sum>a\<in>A. pmf M a *\<^sub>R f a)" 772 unfolding measure_pmf_eq_density 773 apply (simp add: integral_density) 774 apply (subst lebesgue_integral_count_space_finite_support) 775 apply (auto intro!: finite_subset[OF _ \<open>finite A\<close>] sum.mono_neutral_left simp: pmf_eq_0_set_pmf) 776 done 777 778lemma expectation_return_pmf [simp]: 779 fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" 780 shows "measure_pmf.expectation (return_pmf x) f = f x" 781 by (subst integral_measure_pmf[of "{x}"]) simp_all 782 783lemma pmf_expectation_bind: 784 fixes p :: "'a pmf" and f :: "'a \<Rightarrow> 'b pmf" 785 and h :: "'b \<Rightarrow> 'c::{banach, second_countable_topology}" 786 assumes "finite A" "\<And>x. x \<in> A \<Longrightarrow> finite (set_pmf (f x))" "set_pmf p \<subseteq> A" 787 shows "measure_pmf.expectation (p \<bind> f) h = 788 (\<Sum>a\<in>A. pmf p a *\<^sub>R measure_pmf.expectation (f a) h)" 789proof - 790 have "measure_pmf.expectation (p \<bind> f) h = (\<Sum>a\<in>(\<Union>x\<in>A. set_pmf (f x)). pmf (p \<bind> f) a *\<^sub>R h a)" 791 using assms by (intro integral_measure_pmf) auto 792 also have "\<dots> = (\<Sum>x\<in>(\<Union>x\<in>A. set_pmf (f x)). (\<Sum>a\<in>A. (pmf p a * pmf (f a) x) *\<^sub>R h x))" 793 proof (intro sum.cong refl, goal_cases) 794 case (1 x) 795 thus ?case 796 by (subst pmf_bind, subst integral_measure_pmf[of A]) 797 (insert assms, auto simp: scaleR_sum_left) 798 qed 799 also have "\<dots> = (\<Sum>j\<in>A. pmf p j *\<^sub>R (\<Sum>i\<in>(\<Union>x\<in>A. set_pmf (f x)). pmf (f j) i *\<^sub>R h i))" 800 by (subst sum.swap) (simp add: scaleR_sum_right) 801 also have "\<dots> = (\<Sum>j\<in>A. pmf p j *\<^sub>R measure_pmf.expectation (f j) h)" 802 proof (intro sum.cong refl, goal_cases) 803 case (1 x) 804 thus ?case 805 by (subst integral_measure_pmf[of "(\<Union>x\<in>A. set_pmf (f x))"]) 806 (insert assms, auto simp: scaleR_sum_left) 807 qed 808 finally show ?thesis . 809qed 810 811lemma continuous_on_LINT_pmf: \<comment> \<open>This is dominated convergence!?\<close> 812 fixes f :: "'i \<Rightarrow> 'a::topological_space \<Rightarrow> 'b::{banach, second_countable_topology}" 813 assumes f: "\<And>i. i \<in> set_pmf M \<Longrightarrow> continuous_on A (f i)" 814 and bnd: "\<And>a i. a \<in> A \<Longrightarrow> i \<in> set_pmf M \<Longrightarrow> norm (f i a) \<le> B" 815 shows "continuous_on A (\<lambda>a. LINT i|M. f i a)" 816proof cases 817 assume "finite M" with f show ?thesis 818 using integral_measure_pmf[OF \<open>finite M\<close>] 819 by (subst integral_measure_pmf[OF \<open>finite M\<close>]) 820 (auto intro!: continuous_on_sum continuous_on_scaleR continuous_on_const) 821next 822 assume "infinite M" 823 let ?f = "\<lambda>i x. pmf (map_pmf (to_nat_on M) M) i *\<^sub>R f (from_nat_into M i) x" 824 825 show ?thesis 826 proof (rule uniform_limit_theorem) 827 show "\<forall>\<^sub>F n in sequentially. continuous_on A (\<lambda>a. \<Sum>i<n. ?f i a)" 828 by (intro always_eventually allI continuous_on_sum continuous_on_scaleR continuous_on_const f 829 from_nat_into set_pmf_not_empty) 830 show "uniform_limit A (\<lambda>n a. \<Sum>i<n. ?f i a) (\<lambda>a. LINT i|M. f i a) sequentially" 831 proof (subst uniform_limit_cong[where g="\<lambda>n a. \<Sum>i<n. ?f i a"]) 832 fix a assume "a \<in> A" 833 have 1: "(LINT i|M. f i a) = (LINT i|map_pmf (to_nat_on M) M. f (from_nat_into M i) a)" 834 by (auto intro!: integral_cong_AE AE_pmfI) 835 have 2: "\<dots> = (LINT i|count_space UNIV. pmf (map_pmf (to_nat_on M) M) i *\<^sub>R f (from_nat_into M i) a)" 836 by (simp add: measure_pmf_eq_density integral_density) 837 have "(\<lambda>n. ?f n a) sums (LINT i|M. f i a)" 838 unfolding 1 2 839 proof (intro sums_integral_count_space_nat) 840 have A: "integrable M (\<lambda>i. f i a)" 841 using \<open>a\<in>A\<close> by (auto intro!: measure_pmf.integrable_const_bound AE_pmfI bnd) 842 have "integrable (map_pmf (to_nat_on M) M) (\<lambda>i. f (from_nat_into M i) a)" 843 by (auto simp add: map_pmf_rep_eq integrable_distr_eq intro!: AE_pmfI integrable_cong_AE_imp[OF A]) 844 then show "integrable (count_space UNIV) (\<lambda>n. ?f n a)" 845 by (simp add: measure_pmf_eq_density integrable_density) 846 qed 847 then show "(LINT i|M. f i a) = (\<Sum> n. ?f n a)" 848 by (simp add: sums_unique) 849 next 850 show "uniform_limit A (\<lambda>n a. \<Sum>i<n. ?f i a) (\<lambda>a. (\<Sum> n. ?f n a)) sequentially" 851 proof (rule Weierstrass_m_test) 852 fix n a assume "a\<in>A" 853 then show "norm (?f n a) \<le> pmf (map_pmf (to_nat_on M) M) n * B" 854 using bnd by (auto intro!: mult_mono simp: from_nat_into set_pmf_not_empty) 855 next 856 have "integrable (map_pmf (to_nat_on M) M) (\<lambda>n. B)" 857 by auto 858 then show "summable (\<lambda>n. pmf (map_pmf (to_nat_on (set_pmf M)) M) n * B)" 859 by (fastforce simp add: measure_pmf_eq_density integrable_density integrable_count_space_nat_iff summable_mult2) 860 qed 861 qed simp 862 qed simp 863qed 864 865lemma continuous_on_LBINT: 866 fixes f :: "real \<Rightarrow> real" 867 assumes f: "\<And>b. a \<le> b \<Longrightarrow> set_integrable lborel {a..b} f" 868 shows "continuous_on UNIV (\<lambda>b. LBINT x:{a..b}. f x)" 869proof (subst set_borel_integral_eq_integral) 870 { fix b :: real assume "a \<le> b" 871 from f[OF this] have "continuous_on {a..b} (\<lambda>b. integral {a..b} f)" 872 by (intro indefinite_integral_continuous_1 set_borel_integral_eq_integral) } 873 note * = this 874 875 have "continuous_on (\<Union>b\<in>{a..}. {a <..< b}) (\<lambda>b. integral {a..b} f)" 876 proof (intro continuous_on_open_UN) 877 show "b \<in> {a..} \<Longrightarrow> continuous_on {a<..<b} (\<lambda>b. integral {a..b} f)" for b 878 using *[of b] by (rule continuous_on_subset) auto 879 qed simp 880 also have "(\<Union>b\<in>{a..}. {a <..< b}) = {a <..}" 881 by (auto simp: lt_ex gt_ex less_imp_le) (simp add: Bex_def less_imp_le gt_ex cong: rev_conj_cong) 882 finally have "continuous_on {a+1 ..} (\<lambda>b. integral {a..b} f)" 883 by (rule continuous_on_subset) auto 884 moreover have "continuous_on {a..a+1} (\<lambda>b. integral {a..b} f)" 885 by (rule *) simp 886 moreover 887 have "x \<le> a \<Longrightarrow> {a..x} = (if a = x then {a} else {})" for x 888 by auto 889 then have "continuous_on {..a} (\<lambda>b. integral {a..b} f)" 890 by (subst continuous_on_cong[OF refl, where g="\<lambda>x. 0"]) (auto intro!: continuous_on_const) 891 ultimately have "continuous_on ({..a} \<union> {a..a+1} \<union> {a+1 ..}) (\<lambda>b. integral {a..b} f)" 892 by (intro continuous_on_closed_Un) auto 893 also have "{..a} \<union> {a..a+1} \<union> {a+1 ..} = UNIV" 894 by auto 895 finally show "continuous_on UNIV (\<lambda>b. integral {a..b} f)" 896 by auto 897next 898 show "set_integrable lborel {a..b} f" for b 899 using f by (cases "a \<le> b") auto 900qed 901 902locale pmf_as_function 903begin 904 905setup_lifting td_pmf_embed_pmf 906 907lemma set_pmf_transfer[transfer_rule]: 908 assumes "bi_total A" 909 shows "rel_fun (pcr_pmf A) (rel_set A) (\<lambda>f. {x. f x \<noteq> 0}) set_pmf" 910 using \<open>bi_total A\<close> 911 by (auto simp: pcr_pmf_def cr_pmf_def rel_fun_def rel_set_def bi_total_def Bex_def set_pmf_iff) 912 metis+ 913 914end 915 916context 917begin 918 919interpretation pmf_as_function . 920 921lemma pmf_eqI: "(\<And>i. pmf M i = pmf N i) \<Longrightarrow> M = N" 922 by transfer auto 923 924lemma pmf_eq_iff: "M = N \<longleftrightarrow> (\<forall>i. pmf M i = pmf N i)" 925 by (auto intro: pmf_eqI) 926 927lemma pmf_neq_exists_less: 928 assumes "M \<noteq> N" 929 shows "\<exists>x. pmf M x < pmf N x" 930proof (rule ccontr) 931 assume "\<not>(\<exists>x. pmf M x < pmf N x)" 932 hence ge: "pmf M x \<ge> pmf N x" for x by (auto simp: not_less) 933 from assms obtain x where "pmf M x \<noteq> pmf N x" by (auto simp: pmf_eq_iff) 934 with ge[of x] have gt: "pmf M x > pmf N x" by simp 935 have "1 = measure (measure_pmf M) UNIV" by simp 936 also have "\<dots> = measure (measure_pmf N) {x} + measure (measure_pmf N) (UNIV - {x})" 937 by (subst measure_pmf.finite_measure_Union [symmetric]) simp_all 938 also from gt have "measure (measure_pmf N) {x} < measure (measure_pmf M) {x}" 939 by (simp add: measure_pmf_single) 940 also have "measure (measure_pmf N) (UNIV - {x}) \<le> measure (measure_pmf M) (UNIV - {x})" 941 by (subst (1 2) integral_pmf [symmetric]) 942 (intro integral_mono integrable_pmf, simp_all add: ge) 943 also have "measure (measure_pmf M) {x} + \<dots> = 1" 944 by (subst measure_pmf.finite_measure_Union [symmetric]) simp_all 945 finally show False by simp_all 946qed 947 948lemma 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))" 949 unfolding pmf_eq_iff pmf_bind 950proof 951 fix i 952 interpret B: prob_space "restrict_space B B" 953 by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE) 954 (auto simp: AE_measure_pmf_iff) 955 interpret A: prob_space "restrict_space A A" 956 by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE) 957 (auto simp: AE_measure_pmf_iff) 958 959 interpret AB: pair_prob_space "restrict_space A A" "restrict_space B B" 960 by unfold_locales 961 962 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)" 963 by (rule Bochner_Integration.integral_cong) (auto intro!: integral_pmf_restrict) 964 also have "\<dots> = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>restrict_space A A)" 965 by (intro integral_pmf_restrict B.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2 966 countable_set_pmf borel_measurable_count_space) 967 also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>restrict_space B B)" 968 by (rule AB.Fubini_integral[symmetric]) 969 (auto intro!: AB.integrable_const_bound[where B=1] measurable_pair_restrict_pmf2 970 simp: pmf_nonneg pmf_le_1 measurable_restrict_space1) 971 also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>B)" 972 by (intro integral_pmf_restrict[symmetric] A.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2 973 countable_set_pmf borel_measurable_count_space) 974 also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>A \<partial>B)" 975 by (rule Bochner_Integration.integral_cong) (auto intro!: integral_pmf_restrict[symmetric]) 976 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)" . 977qed 978 979lemma pair_map_pmf1: "pair_pmf (map_pmf f A) B = map_pmf (apfst f) (pair_pmf A B)" 980proof (safe intro!: pmf_eqI) 981 fix a :: "'a" and b :: "'b" 982 have [simp]: "\<And>c d. indicator (apfst f -` {(a, b)}) (c, d) = indicator (f -` {a}) c * (indicator {b} d::ennreal)" 983 by (auto split: split_indicator) 984 985 have "ennreal (pmf (pair_pmf (map_pmf f A) B) (a, b)) = 986 ennreal (pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b))" 987 unfolding pmf_pair ennreal_pmf_map 988 by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_multc pmf_nonneg 989 emeasure_map_pmf[symmetric] ennreal_mult del: emeasure_map_pmf) 990 then show "pmf (pair_pmf (map_pmf f A) B) (a, b) = pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b)" 991 by (simp add: pmf_nonneg) 992qed 993 994lemma pair_map_pmf2: "pair_pmf A (map_pmf f B) = map_pmf (apsnd f) (pair_pmf A B)" 995proof (safe intro!: pmf_eqI) 996 fix a :: "'a" and b :: "'b" 997 have [simp]: "\<And>c d. indicator (apsnd f -` {(a, b)}) (c, d) = indicator {a} c * (indicator (f -` {b}) d::ennreal)" 998 by (auto split: split_indicator) 999 1000 have "ennreal (pmf (pair_pmf A (map_pmf f B)) (a, b)) = 1001 ennreal (pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b))" 1002 unfolding pmf_pair ennreal_pmf_map 1003 by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_cmult nn_integral_multc pmf_nonneg 1004 emeasure_map_pmf[symmetric] ennreal_mult del: emeasure_map_pmf) 1005 then show "pmf (pair_pmf A (map_pmf f B)) (a, b) = pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b)" 1006 by (simp add: pmf_nonneg) 1007qed 1008 1009lemma 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)" 1010 by (simp add: pair_map_pmf2 pair_map_pmf1 map_pmf_comp split_beta') 1011 1012end 1013 1014lemma pair_return_pmf1: "pair_pmf (return_pmf x) y = map_pmf (Pair x) y" 1015by(simp add: pair_pmf_def bind_return_pmf map_pmf_def) 1016 1017lemma pair_return_pmf2: "pair_pmf x (return_pmf y) = map_pmf (\<lambda>x. (x, y)) x" 1018by(simp add: pair_pmf_def bind_return_pmf map_pmf_def) 1019 1020lemma pair_pair_pmf: "pair_pmf (pair_pmf u v) w = map_pmf (\<lambda>(x, (y, z)). ((x, y), z)) (pair_pmf u (pair_pmf v w))" 1021by(simp add: pair_pmf_def bind_return_pmf map_pmf_def bind_assoc_pmf) 1022 1023lemma pair_commute_pmf: "pair_pmf x y = map_pmf (\<lambda>(x, y). (y, x)) (pair_pmf y x)" 1024unfolding pair_pmf_def by(subst bind_commute_pmf)(simp add: map_pmf_def bind_assoc_pmf bind_return_pmf) 1025 1026lemma set_pmf_subset_singleton: "set_pmf p \<subseteq> {x} \<longleftrightarrow> p = return_pmf x" 1027proof(intro iffI pmf_eqI) 1028 fix i 1029 assume x: "set_pmf p \<subseteq> {x}" 1030 hence *: "set_pmf p = {x}" using set_pmf_not_empty[of p] by auto 1031 have "ennreal (pmf p x) = \<integral>\<^sup>+ i. indicator {x} i \<partial>p" by(simp add: emeasure_pmf_single) 1032 also have "\<dots> = \<integral>\<^sup>+ i. 1 \<partial>p" by(rule nn_integral_cong_AE)(simp add: AE_measure_pmf_iff * ) 1033 also have "\<dots> = 1" by simp 1034 finally show "pmf p i = pmf (return_pmf x) i" using x 1035 by(auto split: split_indicator simp add: pmf_eq_0_set_pmf) 1036qed auto 1037 1038lemma bind_eq_return_pmf: 1039 "bind_pmf p f = return_pmf x \<longleftrightarrow> (\<forall>y\<in>set_pmf p. f y = return_pmf x)" 1040 (is "?lhs \<longleftrightarrow> ?rhs") 1041proof(intro iffI strip) 1042 fix y 1043 assume y: "y \<in> set_pmf p" 1044 assume "?lhs" 1045 hence "set_pmf (bind_pmf p f) = {x}" by simp 1046 hence "(\<Union>y\<in>set_pmf p. set_pmf (f y)) = {x}" by simp 1047 hence "set_pmf (f y) \<subseteq> {x}" using y by auto 1048 thus "f y = return_pmf x" by(simp add: set_pmf_subset_singleton) 1049next 1050 assume *: ?rhs 1051 show ?lhs 1052 proof(rule pmf_eqI) 1053 fix i 1054 have "ennreal (pmf (bind_pmf p f) i) = \<integral>\<^sup>+ y. ennreal (pmf (f y) i) \<partial>p" 1055 by (simp add: ennreal_pmf_bind) 1056 also have "\<dots> = \<integral>\<^sup>+ y. ennreal (pmf (return_pmf x) i) \<partial>p" 1057 by(rule nn_integral_cong_AE)(simp add: AE_measure_pmf_iff * ) 1058 also have "\<dots> = ennreal (pmf (return_pmf x) i)" 1059 by simp 1060 finally show "pmf (bind_pmf p f) i = pmf (return_pmf x) i" 1061 by (simp add: pmf_nonneg) 1062 qed 1063qed 1064 1065lemma pmf_False_conv_True: "pmf p False = 1 - pmf p True" 1066proof - 1067 have "pmf p False + pmf p True = measure p {False} + measure p {True}" 1068 by(simp add: measure_pmf_single) 1069 also have "\<dots> = measure p ({False} \<union> {True})" 1070 by(subst measure_pmf.finite_measure_Union) simp_all 1071 also have "{False} \<union> {True} = space p" by auto 1072 finally show ?thesis by simp 1073qed 1074 1075lemma pmf_True_conv_False: "pmf p True = 1 - pmf p False" 1076by(simp add: pmf_False_conv_True) 1077 1078subsection \<open> Conditional Probabilities \<close> 1079 1080lemma measure_pmf_zero_iff: "measure (measure_pmf p) s = 0 \<longleftrightarrow> set_pmf p \<inter> s = {}" 1081 by (subst measure_pmf.prob_eq_0) (auto simp: AE_measure_pmf_iff) 1082 1083context 1084 fixes p :: "'a pmf" and s :: "'a set" 1085 assumes not_empty: "set_pmf p \<inter> s \<noteq> {}" 1086begin 1087 1088interpretation pmf_as_measure . 1089 1090lemma emeasure_measure_pmf_not_zero: "emeasure (measure_pmf p) s \<noteq> 0" 1091proof 1092 assume "emeasure (measure_pmf p) s = 0" 1093 then have "AE x in measure_pmf p. x \<notin> s" 1094 by (rule AE_I[rotated]) auto 1095 with not_empty show False 1096 by (auto simp: AE_measure_pmf_iff) 1097qed 1098 1099lemma measure_measure_pmf_not_zero: "measure (measure_pmf p) s \<noteq> 0" 1100 using emeasure_measure_pmf_not_zero by (simp add: measure_pmf.emeasure_eq_measure measure_nonneg) 1101 1102lift_definition cond_pmf :: "'a pmf" is 1103 "uniform_measure (measure_pmf p) s" 1104proof (intro conjI) 1105 show "prob_space (uniform_measure (measure_pmf p) s)" 1106 by (intro prob_space_uniform_measure) (auto simp: emeasure_measure_pmf_not_zero) 1107 show "AE x in uniform_measure (measure_pmf p) s. measure (uniform_measure (measure_pmf p) s) {x} \<noteq> 0" 1108 by (simp add: emeasure_measure_pmf_not_zero measure_measure_pmf_not_zero AE_uniform_measure 1109 AE_measure_pmf_iff set_pmf.rep_eq less_top[symmetric]) 1110qed simp 1111 1112lemma pmf_cond: "pmf cond_pmf x = (if x \<in> s then pmf p x / measure p s else 0)" 1113 by transfer (simp add: emeasure_measure_pmf_not_zero pmf.rep_eq) 1114 1115lemma set_cond_pmf[simp]: "set_pmf cond_pmf = set_pmf p \<inter> s" 1116 by (auto simp add: set_pmf_iff pmf_cond measure_measure_pmf_not_zero split: if_split_asm) 1117 1118end 1119 1120lemma measure_pmf_posI: "x \<in> set_pmf p \<Longrightarrow> x \<in> A \<Longrightarrow> measure_pmf.prob p A > 0" 1121 using measure_measure_pmf_not_zero[of p A] by (subst zero_less_measure_iff) blast 1122 1123lemma cond_map_pmf: 1124 assumes "set_pmf p \<inter> f -` s \<noteq> {}" 1125 shows "cond_pmf (map_pmf f p) s = map_pmf f (cond_pmf p (f -` s))" 1126proof - 1127 have *: "set_pmf (map_pmf f p) \<inter> s \<noteq> {}" 1128 using assms by auto 1129 { fix x 1130 have "ennreal (pmf (map_pmf f (cond_pmf p (f -` s))) x) = 1131 emeasure p (f -` s \<inter> f -` {x}) / emeasure p (f -` s)" 1132 unfolding ennreal_pmf_map cond_pmf.rep_eq[OF assms] by (simp add: nn_integral_uniform_measure) 1133 also have "f -` s \<inter> f -` {x} = (if x \<in> s then f -` {x} else {})" 1134 by auto 1135 also have "emeasure p (if x \<in> s then f -` {x} else {}) / emeasure p (f -` s) = 1136 ennreal (pmf (cond_pmf (map_pmf f p) s) x)" 1137 using measure_measure_pmf_not_zero[OF *] 1138 by (simp add: pmf_cond[OF *] ennreal_pmf_map measure_pmf.emeasure_eq_measure 1139 divide_ennreal pmf_nonneg measure_nonneg zero_less_measure_iff pmf_map) 1140 finally have "ennreal (pmf (cond_pmf (map_pmf f p) s) x) = ennreal (pmf (map_pmf f (cond_pmf p (f -` s))) x)" 1141 by simp } 1142 then show ?thesis 1143 by (intro pmf_eqI) (simp add: pmf_nonneg) 1144qed 1145 1146lemma bind_cond_pmf_cancel: 1147 assumes [simp]: "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}" 1148 assumes [simp]: "\<And>y. y \<in> set_pmf q \<Longrightarrow> set_pmf p \<inter> {x. R x y} \<noteq> {}" 1149 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}" 1150 shows "bind_pmf p (\<lambda>x. cond_pmf q {y. R x y}) = q" 1151proof (rule pmf_eqI) 1152 fix i 1153 have "ennreal (pmf (bind_pmf p (\<lambda>x. cond_pmf q {y. R x y})) i) = 1154 (\<integral>\<^sup>+x. ennreal (pmf q i / measure p {x. R x i}) * ennreal (indicator {x. R x i} x) \<partial>p)" 1155 by (auto simp add: ennreal_pmf_bind AE_measure_pmf_iff pmf_cond pmf_eq_0_set_pmf pmf_nonneg measure_nonneg 1156 intro!: nn_integral_cong_AE) 1157 also have "\<dots> = (pmf q i * measure p {x. R x i}) / measure p {x. R x i}" 1158 by (simp add: pmf_nonneg measure_nonneg zero_ennreal_def[symmetric] ennreal_indicator 1159 nn_integral_cmult measure_pmf.emeasure_eq_measure ennreal_mult[symmetric]) 1160 also have "\<dots> = pmf q i" 1161 by (cases "pmf q i = 0") 1162 (simp_all add: pmf_eq_0_set_pmf measure_measure_pmf_not_zero pmf_nonneg) 1163 finally show "pmf (bind_pmf p (\<lambda>x. cond_pmf q {y. R x y})) i = pmf q i" 1164 by (simp add: pmf_nonneg) 1165qed 1166 1167subsection \<open> Relator \<close> 1168 1169inductive rel_pmf :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf \<Rightarrow> bool" 1170for R p q 1171where 1172 "\<lbrakk> \<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y; 1173 map_pmf fst pq = p; map_pmf snd pq = q \<rbrakk> 1174 \<Longrightarrow> rel_pmf R p q" 1175 1176lemma rel_pmfI: 1177 assumes R: "rel_set R (set_pmf p) (set_pmf q)" 1178 assumes eq: "\<And>x y. x \<in> set_pmf p \<Longrightarrow> y \<in> set_pmf q \<Longrightarrow> R x y \<Longrightarrow> 1179 measure p {x. R x y} = measure q {y. R x y}" 1180 shows "rel_pmf R p q" 1181proof 1182 let ?pq = "bind_pmf p (\<lambda>x. bind_pmf (cond_pmf q {y. R x y}) (\<lambda>y. return_pmf (x, y)))" 1183 have "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}" 1184 using R by (auto simp: rel_set_def) 1185 then show "\<And>x y. (x, y) \<in> set_pmf ?pq \<Longrightarrow> R x y" 1186 by auto 1187 show "map_pmf fst ?pq = p" 1188 by (simp add: map_bind_pmf bind_return_pmf') 1189 1190 show "map_pmf snd ?pq = q" 1191 using R eq 1192 apply (simp add: bind_cond_pmf_cancel map_bind_pmf bind_return_pmf') 1193 apply (rule bind_cond_pmf_cancel) 1194 apply (auto simp: rel_set_def) 1195 done 1196qed 1197 1198lemma rel_pmf_imp_rel_set: "rel_pmf R p q \<Longrightarrow> rel_set R (set_pmf p) (set_pmf q)" 1199 by (force simp add: rel_pmf.simps rel_set_def) 1200 1201lemma rel_pmfD_measure: 1202 assumes rel_R: "rel_pmf R p q" and R: "\<And>a b. R a b \<Longrightarrow> R a y \<longleftrightarrow> R x b" 1203 assumes "x \<in> set_pmf p" "y \<in> set_pmf q" 1204 shows "measure p {x. R x y} = measure q {y. R x y}" 1205proof - 1206 from rel_R obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y" 1207 and eq: "p = map_pmf fst pq" "q = map_pmf snd pq" 1208 by (auto elim: rel_pmf.cases) 1209 have "measure p {x. R x y} = measure pq {x. R (fst x) y}" 1210 by (simp add: eq map_pmf_rep_eq measure_distr) 1211 also have "\<dots> = measure pq {y. R x (snd y)}" 1212 by (intro measure_pmf.finite_measure_eq_AE) 1213 (auto simp: AE_measure_pmf_iff R dest!: pq) 1214 also have "\<dots> = measure q {y. R x y}" 1215 by (simp add: eq map_pmf_rep_eq measure_distr) 1216 finally show "measure p {x. R x y} = measure q {y. R x y}" . 1217qed 1218 1219lemma rel_pmf_measureD: 1220 assumes "rel_pmf R p q" 1221 shows "measure (measure_pmf p) A \<le> measure (measure_pmf q) {y. \<exists>x\<in>A. R x y}" (is "?lhs \<le> ?rhs") 1222using assms 1223proof cases 1224 fix pq 1225 assume R: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y" 1226 and p[symmetric]: "map_pmf fst pq = p" 1227 and q[symmetric]: "map_pmf snd pq = q" 1228 have "?lhs = measure (measure_pmf pq) (fst -` A)" by(simp add: p) 1229 also have "\<dots> \<le> measure (measure_pmf pq) {y. \<exists>x\<in>A. R x (snd y)}" 1230 by(rule measure_pmf.finite_measure_mono_AE)(auto 4 3 simp add: AE_measure_pmf_iff dest: R) 1231 also have "\<dots> = ?rhs" by(simp add: q) 1232 finally show ?thesis . 1233qed 1234 1235lemma rel_pmf_iff_measure: 1236 assumes "symp R" "transp R" 1237 shows "rel_pmf R p q \<longleftrightarrow> 1238 rel_set R (set_pmf p) (set_pmf q) \<and> 1239 (\<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})" 1240 by (safe intro!: rel_pmf_imp_rel_set rel_pmfI) 1241 (auto intro!: rel_pmfD_measure dest: sympD[OF \<open>symp R\<close>] transpD[OF \<open>transp R\<close>]) 1242 1243lemma quotient_rel_set_disjoint: 1244 "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 = {}" 1245 using in_quotient_imp_closed[of UNIV "{(x, y). R x y}" C] 1246 by (auto 0 0 simp: equivp_equiv rel_set_def set_eq_iff elim: equivpE) 1247 (blast dest: equivp_symp)+ 1248 1249lemma quotientD: "equiv X R \<Longrightarrow> A \<in> X // R \<Longrightarrow> x \<in> A \<Longrightarrow> A = R `` {x}" 1250 by (metis Image_singleton_iff equiv_class_eq_iff quotientE) 1251 1252lemma rel_pmf_iff_equivp: 1253 assumes "equivp R" 1254 shows "rel_pmf R p q \<longleftrightarrow> (\<forall>C\<in>UNIV // {(x, y). R x y}. measure p C = measure q C)" 1255 (is "_ \<longleftrightarrow> (\<forall>C\<in>_//?R. _)") 1256proof (subst rel_pmf_iff_measure, safe) 1257 show "symp R" "transp R" 1258 using assms by (auto simp: equivp_reflp_symp_transp) 1259next 1260 fix C assume C: "C \<in> UNIV // ?R" and R: "rel_set R (set_pmf p) (set_pmf q)" 1261 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}" 1262 1263 show "measure p C = measure q C" 1264 proof (cases "p \<inter> C = {}") 1265 case True 1266 then have "q \<inter> C = {}" 1267 using quotient_rel_set_disjoint[OF assms C R] by simp 1268 with True show ?thesis 1269 unfolding measure_pmf_zero_iff[symmetric] by simp 1270 next 1271 case False 1272 then have "q \<inter> C \<noteq> {}" 1273 using quotient_rel_set_disjoint[OF assms C R] by simp 1274 with False obtain x y where in_set: "x \<in> set_pmf p" "y \<in> set_pmf q" and in_C: "x \<in> C" "y \<in> C" 1275 by auto 1276 then have "R x y" 1277 using in_quotient_imp_in_rel[of UNIV ?R C x y] C assms 1278 by (simp add: equivp_equiv) 1279 with in_set eq have "measure p {x. R x y} = measure q {y. R x y}" 1280 by auto 1281 moreover have "{y. R x y} = C" 1282 using assms \<open>x \<in> C\<close> C quotientD[of UNIV ?R C x] by (simp add: equivp_equiv) 1283 moreover have "{x. R x y} = C" 1284 using assms \<open>y \<in> C\<close> C quotientD[of UNIV "?R" C y] sympD[of R] 1285 by (auto simp add: equivp_equiv elim: equivpE) 1286 ultimately show ?thesis 1287 by auto 1288 qed 1289next 1290 assume eq: "\<forall>C\<in>UNIV // ?R. measure p C = measure q C" 1291 show "rel_set R (set_pmf p) (set_pmf q)" 1292 unfolding rel_set_def 1293 proof safe 1294 fix x assume x: "x \<in> set_pmf p" 1295 have "{y. R x y} \<in> UNIV // ?R" 1296 by (auto simp: quotient_def) 1297 with eq have *: "measure q {y. R x y} = measure p {y. R x y}" 1298 by auto 1299 have "measure q {y. R x y} \<noteq> 0" 1300 using x assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp) 1301 then show "\<exists>y\<in>set_pmf q. R x y" 1302 unfolding measure_pmf_zero_iff by auto 1303 next 1304 fix y assume y: "y \<in> set_pmf q" 1305 have "{x. R x y} \<in> UNIV // ?R" 1306 using assms by (auto simp: quotient_def dest: equivp_symp) 1307 with eq have *: "measure p {x. R x y} = measure q {x. R x y}" 1308 by auto 1309 have "measure p {x. R x y} \<noteq> 0" 1310 using y assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp) 1311 then show "\<exists>x\<in>set_pmf p. R x y" 1312 unfolding measure_pmf_zero_iff by auto 1313 qed 1314 1315 fix x y assume "x \<in> set_pmf p" "y \<in> set_pmf q" "R x y" 1316 have "{y. R x y} \<in> UNIV // ?R" "{x. R x y} = {y. R x y}" 1317 using assms \<open>R x y\<close> by (auto simp: quotient_def dest: equivp_symp equivp_transp) 1318 with eq show "measure p {x. R x y} = measure q {y. R x y}" 1319 by auto 1320qed 1321 1322bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: rel_pmf 1323proof - 1324 show "map_pmf id = id" by (rule map_pmf_id) 1325 show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose) 1326 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" 1327 by (intro map_pmf_cong refl) 1328 1329 show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = (`) f \<circ> set_pmf" 1330 by (rule pmf_set_map) 1331 1332 show "(card_of (set_pmf p), natLeq) \<in> ordLeq" for p :: "'s pmf" 1333 proof - 1334 have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq" 1335 by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"]) 1336 (auto intro: countable_set_pmf) 1337 also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq" 1338 by (metis Field_natLeq card_of_least natLeq_Well_order) 1339 finally show ?thesis . 1340 qed 1341 1342 show "\<And>R. rel_pmf R = (\<lambda>x y. \<exists>z. set_pmf z \<subseteq> {(x, y). R x y} \<and> 1343 map_pmf fst z = x \<and> map_pmf snd z = y)" 1344 by (auto simp add: fun_eq_iff rel_pmf.simps) 1345 1346 show "rel_pmf R OO rel_pmf S \<le> rel_pmf (R OO S)" 1347 for R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool" 1348 proof - 1349 { fix p q r 1350 assume pq: "rel_pmf R p q" 1351 and qr:"rel_pmf S q r" 1352 from pq obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y" 1353 and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto 1354 from qr obtain qr where qr: "\<And>y z. (y, z) \<in> set_pmf qr \<Longrightarrow> S y z" 1355 and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto 1356 1357 define pr where "pr = 1358 bind_pmf pq (\<lambda>xy. bind_pmf (cond_pmf qr {yz. fst yz = snd xy}) 1359 (\<lambda>yz. return_pmf (fst xy, snd yz)))" 1360 have pr_welldefined: "\<And>y. y \<in> q \<Longrightarrow> qr \<inter> {yz. fst yz = y} \<noteq> {}" 1361 by (force simp: q') 1362 1363 have "rel_pmf (R OO S) p r" 1364 proof (rule rel_pmf.intros) 1365 fix x z assume "(x, z) \<in> pr" 1366 then have "\<exists>y. (x, y) \<in> pq \<and> (y, z) \<in> qr" 1367 by (auto simp: q pr_welldefined pr_def split_beta) 1368 with pq qr show "(R OO S) x z" 1369 by blast 1370 next 1371 have "map_pmf snd pr = map_pmf snd (bind_pmf q (\<lambda>y. cond_pmf qr {yz. fst yz = y}))" 1372 by (simp add: pr_def q split_beta bind_map_pmf map_pmf_def[symmetric] map_bind_pmf map_pmf_comp) 1373 then show "map_pmf snd pr = r" 1374 unfolding r q' bind_map_pmf by (subst (asm) bind_cond_pmf_cancel) (auto simp: eq_commute) 1375 qed (simp add: pr_def map_bind_pmf split_beta map_pmf_def[symmetric] p map_pmf_comp) 1376 } 1377 then show ?thesis 1378 by(auto simp add: le_fun_def) 1379 qed 1380qed (fact natLeq_card_order natLeq_cinfinite)+ 1381 1382lemma map_pmf_idI: "(\<And>x. x \<in> set_pmf p \<Longrightarrow> f x = x) \<Longrightarrow> map_pmf f p = p" 1383by(simp cong: pmf.map_cong) 1384 1385lemma rel_pmf_conj[simp]: 1386 "rel_pmf (\<lambda>x y. P \<and> Q x y) x y \<longleftrightarrow> P \<and> rel_pmf Q x y" 1387 "rel_pmf (\<lambda>x y. Q x y \<and> P) x y \<longleftrightarrow> P \<and> rel_pmf Q x y" 1388 using set_pmf_not_empty by (fastforce simp: pmf.in_rel subset_eq)+ 1389 1390lemma rel_pmf_top[simp]: "rel_pmf top = top" 1391 by (auto simp: pmf.in_rel[abs_def] fun_eq_iff map_fst_pair_pmf map_snd_pair_pmf 1392 intro: exI[of _ "pair_pmf x y" for x y]) 1393 1394lemma rel_pmf_return_pmf1: "rel_pmf R (return_pmf x) M \<longleftrightarrow> (\<forall>a\<in>M. R x a)" 1395proof safe 1396 fix a assume "a \<in> M" "rel_pmf R (return_pmf x) M" 1397 then obtain pq where *: "\<And>a b. (a, b) \<in> set_pmf pq \<Longrightarrow> R a b" 1398 and eq: "return_pmf x = map_pmf fst pq" "M = map_pmf snd pq" 1399 by (force elim: rel_pmf.cases) 1400 moreover have "set_pmf (return_pmf x) = {x}" 1401 by simp 1402 with \<open>a \<in> M\<close> have "(x, a) \<in> pq" 1403 by (force simp: eq) 1404 with * show "R x a" 1405 by auto 1406qed (auto intro!: rel_pmf.intros[where pq="pair_pmf (return_pmf x) M"] 1407 simp: map_fst_pair_pmf map_snd_pair_pmf) 1408 1409lemma rel_pmf_return_pmf2: "rel_pmf R M (return_pmf x) \<longleftrightarrow> (\<forall>a\<in>M. R a x)" 1410 by (subst pmf.rel_flip[symmetric]) (simp add: rel_pmf_return_pmf1) 1411 1412lemma rel_return_pmf[simp]: "rel_pmf R (return_pmf x1) (return_pmf x2) = R x1 x2" 1413 unfolding rel_pmf_return_pmf2 set_return_pmf by simp 1414 1415lemma rel_pmf_False[simp]: "rel_pmf (\<lambda>x y. False) x y = False" 1416 unfolding pmf.in_rel fun_eq_iff using set_pmf_not_empty by fastforce 1417 1418lemma rel_pmf_rel_prod: 1419 "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'" 1420proof safe 1421 assume "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')" 1422 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" 1423 and eq: "map_pmf fst pq = pair_pmf A A'" "map_pmf snd pq = pair_pmf B B'" 1424 by (force elim: rel_pmf.cases) 1425 show "rel_pmf R A B" 1426 proof (rule rel_pmf.intros) 1427 let ?f = "\<lambda>(a, b). (fst a, fst b)" 1428 have [simp]: "(\<lambda>x. fst (?f x)) = fst o fst" "(\<lambda>x. snd (?f x)) = fst o snd" 1429 by auto 1430 1431 show "map_pmf fst (map_pmf ?f pq) = A" 1432 by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf) 1433 show "map_pmf snd (map_pmf ?f pq) = B" 1434 by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf) 1435 1436 fix a b assume "(a, b) \<in> set_pmf (map_pmf ?f pq)" 1437 then obtain c d where "((a, c), (b, d)) \<in> set_pmf pq" 1438 by auto 1439 from pq[OF this] show "R a b" .. 1440 qed 1441 show "rel_pmf S A' B'" 1442 proof (rule rel_pmf.intros) 1443 let ?f = "\<lambda>(a, b). (snd a, snd b)" 1444 have [simp]: "(\<lambda>x. fst (?f x)) = snd o fst" "(\<lambda>x. snd (?f x)) = snd o snd" 1445 by auto 1446 1447 show "map_pmf fst (map_pmf ?f pq) = A'" 1448 by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf) 1449 show "map_pmf snd (map_pmf ?f pq) = B'" 1450 by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf) 1451 1452 fix c d assume "(c, d) \<in> set_pmf (map_pmf ?f pq)" 1453 then obtain a b where "((a, c), (b, d)) \<in> set_pmf pq" 1454 by auto 1455 from pq[OF this] show "S c d" .. 1456 qed 1457next 1458 assume "rel_pmf R A B" "rel_pmf S A' B'" 1459 then obtain Rpq Spq 1460 where Rpq: "\<And>a b. (a, b) \<in> set_pmf Rpq \<Longrightarrow> R a b" 1461 "map_pmf fst Rpq = A" "map_pmf snd Rpq = B" 1462 and Spq: "\<And>a b. (a, b) \<in> set_pmf Spq \<Longrightarrow> S a b" 1463 "map_pmf fst Spq = A'" "map_pmf snd Spq = B'" 1464 by (force elim: rel_pmf.cases) 1465 1466 let ?f = "(\<lambda>((a, c), (b, d)). ((a, b), (c, d)))" 1467 let ?pq = "map_pmf ?f (pair_pmf Rpq Spq)" 1468 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))" 1469 by auto 1470 1471 show "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')" 1472 by (rule rel_pmf.intros[where pq="?pq"]) 1473 (auto simp: map_snd_pair_pmf map_fst_pair_pmf map_pmf_comp Rpq Spq 1474 map_pair) 1475qed 1476 1477lemma rel_pmf_reflI: 1478 assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> P x x" 1479 shows "rel_pmf P p p" 1480 by (rule rel_pmf.intros[where pq="map_pmf (\<lambda>x. (x, x)) p"]) 1481 (auto simp add: pmf.map_comp o_def assms) 1482 1483lemma rel_pmf_bij_betw: 1484 assumes f: "bij_betw f (set_pmf p) (set_pmf q)" 1485 and eq: "\<And>x. x \<in> set_pmf p \<Longrightarrow> pmf p x = pmf q (f x)" 1486 shows "rel_pmf (\<lambda>x y. f x = y) p q" 1487proof(rule rel_pmf.intros) 1488 let ?pq = "map_pmf (\<lambda>x. (x, f x)) p" 1489 show "map_pmf fst ?pq = p" by(simp add: pmf.map_comp o_def) 1490 1491 have "map_pmf f p = q" 1492 proof(rule pmf_eqI) 1493 fix i 1494 show "pmf (map_pmf f p) i = pmf q i" 1495 proof(cases "i \<in> set_pmf q") 1496 case True 1497 with f obtain j where "i = f j" "j \<in> set_pmf p" 1498 by(auto simp add: bij_betw_def image_iff) 1499 thus ?thesis using f by(simp add: bij_betw_def pmf_map_inj eq) 1500 next 1501 case False thus ?thesis 1502 by(subst pmf_map_outside)(auto simp add: set_pmf_iff eq[symmetric]) 1503 qed 1504 qed 1505 then show "map_pmf snd ?pq = q" by(simp add: pmf.map_comp o_def) 1506qed auto 1507 1508context 1509begin 1510 1511interpretation pmf_as_measure . 1512 1513definition "join_pmf M = bind_pmf M (\<lambda>x. x)" 1514 1515lemma bind_eq_join_pmf: "bind_pmf M f = join_pmf (map_pmf f M)" 1516 unfolding join_pmf_def bind_map_pmf .. 1517 1518lemma join_eq_bind_pmf: "join_pmf M = bind_pmf M id" 1519 by (simp add: join_pmf_def id_def) 1520 1521lemma pmf_join: "pmf (join_pmf N) i = (\<integral>M. pmf M i \<partial>measure_pmf N)" 1522 unfolding join_pmf_def pmf_bind .. 1523 1524lemma ennreal_pmf_join: "ennreal (pmf (join_pmf N) i) = (\<integral>\<^sup>+M. pmf M i \<partial>measure_pmf N)" 1525 unfolding join_pmf_def ennreal_pmf_bind .. 1526 1527lemma set_pmf_join_pmf[simp]: "set_pmf (join_pmf f) = (\<Union>p\<in>set_pmf f. set_pmf p)" 1528 by (simp add: join_pmf_def) 1529 1530lemma join_return_pmf: "join_pmf (return_pmf M) = M" 1531 by (simp add: integral_return pmf_eq_iff pmf_join return_pmf.rep_eq) 1532 1533lemma map_join_pmf: "map_pmf f (join_pmf AA) = join_pmf (map_pmf (map_pmf f) AA)" 1534 by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf) 1535 1536lemma join_map_return_pmf: "join_pmf (map_pmf return_pmf A) = A" 1537 by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf') 1538 1539end 1540 1541lemma rel_pmf_joinI: 1542 assumes "rel_pmf (rel_pmf P) p q" 1543 shows "rel_pmf P (join_pmf p) (join_pmf q)" 1544proof - 1545 from assms obtain pq where p: "p = map_pmf fst pq" 1546 and q: "q = map_pmf snd pq" 1547 and P: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> rel_pmf P x y" 1548 by cases auto 1549 from P obtain PQ 1550 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" 1551 and x: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf fst (PQ x y) = x" 1552 and y: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf snd (PQ x y) = y" 1553 by(metis rel_pmf.simps) 1554 1555 let ?r = "bind_pmf pq (\<lambda>(x, y). PQ x y)" 1556 have "\<And>a b. (a, b) \<in> set_pmf ?r \<Longrightarrow> P a b" by (auto intro: PQ) 1557 moreover have "map_pmf fst ?r = join_pmf p" "map_pmf snd ?r = join_pmf q" 1558 by (simp_all add: p q x y join_pmf_def map_bind_pmf bind_map_pmf split_def cong: bind_pmf_cong) 1559 ultimately show ?thesis .. 1560qed 1561 1562lemma rel_pmf_bindI: 1563 assumes pq: "rel_pmf R p q" 1564 and fg: "\<And>x y. R x y \<Longrightarrow> rel_pmf P (f x) (g y)" 1565 shows "rel_pmf P (bind_pmf p f) (bind_pmf q g)" 1566 unfolding bind_eq_join_pmf 1567 by (rule rel_pmf_joinI) 1568 (auto simp add: pmf.rel_map intro: pmf.rel_mono[THEN le_funD, THEN le_funD, THEN le_boolD, THEN mp, OF _ pq] fg) 1569 1570text \<open> 1571 Proof that \<^const>\<open>rel_pmf\<close> preserves orders. 1572 Antisymmetry proof follows Thm. 1 in N. Saheb-Djahromi, Cpo's of measures for nondeterminism, 1573 Theoretical Computer Science 12(1):19--37, 1980, 1574 \<^url>\<open>https://doi.org/10.1016/0304-3975(80)90003-1\<close> 1575\<close> 1576 1577lemma 1578 assumes *: "rel_pmf R p q" 1579 and refl: "reflp R" and trans: "transp R" 1580 shows measure_Ici: "measure p {y. R x y} \<le> measure q {y. R x y}" (is ?thesis1) 1581 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) 1582proof - 1583 from * obtain pq 1584 where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y" 1585 and p: "p = map_pmf fst pq" 1586 and q: "q = map_pmf snd pq" 1587 by cases auto 1588 show ?thesis1 ?thesis2 unfolding p q map_pmf_rep_eq using refl trans 1589 by(auto 4 3 simp add: measure_distr reflpD AE_measure_pmf_iff intro!: measure_pmf.finite_measure_mono_AE dest!: pq elim: transpE) 1590qed 1591 1592lemma rel_pmf_inf: 1593 fixes p q :: "'a pmf" 1594 assumes 1: "rel_pmf R p q" 1595 assumes 2: "rel_pmf R q p" 1596 and refl: "reflp R" and trans: "transp R" 1597 shows "rel_pmf (inf R R\<inverse>\<inverse>) p q" 1598proof (subst rel_pmf_iff_equivp, safe) 1599 show "equivp (inf R R\<inverse>\<inverse>)" 1600 using trans refl by (auto simp: equivp_reflp_symp_transp intro: sympI transpI reflpI dest: transpD reflpD) 1601 1602 fix C assume "C \<in> UNIV // {(x, y). inf R R\<inverse>\<inverse> x y}" 1603 then obtain x where C: "C = {y. R x y \<and> R y x}" 1604 by (auto elim: quotientE) 1605 1606 let ?R = "\<lambda>x y. R x y \<and> R y x" 1607 let ?\<mu>R = "\<lambda>y. measure q {x. ?R x y}" 1608 have "measure p {y. ?R x y} = measure p ({y. R x y} - {y. R x y \<and> \<not> R y x})" 1609 by(auto intro!: arg_cong[where f="measure p"]) 1610 also have "\<dots> = measure p {y. R x y} - measure p {y. R x y \<and> \<not> R y x}" 1611 by (rule measure_pmf.finite_measure_Diff) auto 1612 also have "measure p {y. R x y \<and> \<not> R y x} = measure q {y. R x y \<and> \<not> R y x}" 1613 using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ioi) 1614 also have "measure p {y. R x y} = measure q {y. R x y}" 1615 using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ici) 1616 also have "measure q {y. R x y} - measure q {y. R x y \<and> \<not> R y x} = 1617 measure q ({y. R x y} - {y. R x y \<and> \<not> R y x})" 1618 by(rule measure_pmf.finite_measure_Diff[symmetric]) auto 1619 also have "\<dots> = ?\<mu>R x" 1620 by(auto intro!: arg_cong[where f="measure q"]) 1621 finally show "measure p C = measure q C" 1622 by (simp add: C conj_commute) 1623qed 1624 1625lemma rel_pmf_antisym: 1626 fixes p q :: "'a pmf" 1627 assumes 1: "rel_pmf R p q" 1628 assumes 2: "rel_pmf R q p" 1629 and refl: "reflp R" and trans: "transp R" and antisym: "antisymp R" 1630 shows "p = q" 1631proof - 1632 from 1 2 refl trans have "rel_pmf (inf R R\<inverse>\<inverse>) p q" by(rule rel_pmf_inf) 1633 also have "inf R R\<inverse>\<inverse> = (=)" 1634 using refl antisym by (auto intro!: ext simp add: reflpD dest: antisympD) 1635 finally show ?thesis unfolding pmf.rel_eq . 1636qed 1637 1638lemma reflp_rel_pmf: "reflp R \<Longrightarrow> reflp (rel_pmf R)" 1639 by (fact pmf.rel_reflp) 1640 1641lemma antisymp_rel_pmf: 1642 "\<lbrakk> reflp R; transp R; antisymp R \<rbrakk> 1643 \<Longrightarrow> antisymp (rel_pmf R)" 1644by(rule antisympI)(blast intro: rel_pmf_antisym) 1645 1646lemma transp_rel_pmf: 1647 assumes "transp R" 1648 shows "transp (rel_pmf R)" 1649 using assms by (fact pmf.rel_transp) 1650 1651 1652subsection \<open> Distributions \<close> 1653 1654context 1655begin 1656 1657interpretation pmf_as_function . 1658 1659subsubsection \<open> Bernoulli Distribution \<close> 1660 1661lift_definition bernoulli_pmf :: "real \<Rightarrow> bool pmf" is 1662 "\<lambda>p b. ((\<lambda>p. if b then p else 1 - p) \<circ> min 1 \<circ> max 0) p" 1663 by (auto simp: nn_integral_count_space_finite[where A="{False, True}"] UNIV_bool 1664 split: split_max split_min) 1665 1666lemma pmf_bernoulli_True[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) True = p" 1667 by transfer simp 1668 1669lemma pmf_bernoulli_False[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) False = 1 - p" 1670 by transfer simp 1671 1672lemma set_pmf_bernoulli[simp]: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (bernoulli_pmf p) = UNIV" 1673 by (auto simp add: set_pmf_iff UNIV_bool) 1674 1675lemma nn_integral_bernoulli_pmf[simp]: 1676 assumes [simp]: "0 \<le> p" "p \<le> 1" "\<And>x. 0 \<le> f x" 1677 shows "(\<integral>\<^sup>+x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)" 1678 by (subst nn_integral_measure_pmf_support[of UNIV]) 1679 (auto simp: UNIV_bool field_simps) 1680 1681lemma integral_bernoulli_pmf[simp]: 1682 assumes [simp]: "0 \<le> p" "p \<le> 1" 1683 shows "(\<integral>x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)" 1684 by (subst integral_measure_pmf[of UNIV]) (auto simp: UNIV_bool) 1685 1686lemma pmf_bernoulli_half [simp]: "pmf (bernoulli_pmf (1 / 2)) x = 1 / 2" 1687by(cases x) simp_all 1688 1689lemma measure_pmf_bernoulli_half: "measure_pmf (bernoulli_pmf (1 / 2)) = uniform_count_measure UNIV" 1690 by (rule measure_eqI) 1691 (simp_all add: nn_integral_pmf[symmetric] emeasure_uniform_count_measure ennreal_divide_numeral[symmetric] 1692 nn_integral_count_space_finite sets_uniform_count_measure divide_ennreal_def mult_ac 1693 ennreal_of_nat_eq_real_of_nat) 1694 1695subsubsection \<open> Geometric Distribution \<close> 1696 1697context 1698 fixes p :: real assumes p[arith]: "0 < p" "p \<le> 1" 1699begin 1700 1701lift_definition geometric_pmf :: "nat pmf" is "\<lambda>n. (1 - p)^n * p" 1702proof 1703 have "(\<Sum>i. ennreal (p * (1 - p) ^ i)) = ennreal (p * (1 / (1 - (1 - p))))" 1704 by (intro suminf_ennreal_eq sums_mult geometric_sums) auto 1705 then show "(\<integral>\<^sup>+ x. ennreal ((1 - p)^x * p) \<partial>count_space UNIV) = 1" 1706 by (simp add: nn_integral_count_space_nat field_simps) 1707qed simp 1708 1709lemma pmf_geometric[simp]: "pmf geometric_pmf n = (1 - p)^n * p" 1710 by transfer rule 1711 1712end 1713 1714lemma set_pmf_geometric: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (geometric_pmf p) = UNIV" 1715 by (auto simp: set_pmf_iff) 1716 1717subsubsection \<open> Uniform Multiset Distribution \<close> 1718 1719context 1720 fixes M :: "'a multiset" assumes M_not_empty: "M \<noteq> {#}" 1721begin 1722 1723lift_definition pmf_of_multiset :: "'a pmf" is "\<lambda>x. count M x / size M" 1724proof 1725 show "(\<integral>\<^sup>+ x. ennreal (real (count M x) / real (size M)) \<partial>count_space UNIV) = 1" 1726 using M_not_empty 1727 by (simp add: zero_less_divide_iff nn_integral_count_space nonempty_has_size 1728 sum_divide_distrib[symmetric]) 1729 (auto simp: size_multiset_overloaded_eq intro!: sum.cong) 1730qed simp 1731 1732lemma pmf_of_multiset[simp]: "pmf pmf_of_multiset x = count M x / size M" 1733 by transfer rule 1734 1735lemma set_pmf_of_multiset[simp]: "set_pmf pmf_of_multiset = set_mset M" 1736 by (auto simp: set_pmf_iff) 1737 1738end 1739 1740subsubsection \<open> Uniform Distribution \<close> 1741 1742context 1743 fixes S :: "'a set" assumes S_not_empty: "S \<noteq> {}" and S_finite: "finite S" 1744begin 1745 1746lift_definition pmf_of_set :: "'a pmf" is "\<lambda>x. indicator S x / card S" 1747proof 1748 show "(\<integral>\<^sup>+ x. ennreal (indicator S x / real (card S)) \<partial>count_space UNIV) = 1" 1749 using S_not_empty S_finite 1750 by (subst nn_integral_count_space'[of S]) 1751 (auto simp: ennreal_of_nat_eq_real_of_nat ennreal_mult[symmetric]) 1752qed simp 1753 1754lemma pmf_of_set[simp]: "pmf pmf_of_set x = indicator S x / card S" 1755 by transfer rule 1756 1757lemma set_pmf_of_set[simp]: "set_pmf pmf_of_set = S" 1758 using S_finite S_not_empty by (auto simp: set_pmf_iff) 1759 1760lemma emeasure_pmf_of_set_space[simp]: "emeasure pmf_of_set S = 1" 1761 by (rule measure_pmf.emeasure_eq_1_AE) (auto simp: AE_measure_pmf_iff) 1762 1763lemma nn_integral_pmf_of_set: "nn_integral (measure_pmf pmf_of_set) f = sum f S / card S" 1764 by (subst nn_integral_measure_pmf_finite) 1765 (simp_all add: sum_distrib_right[symmetric] card_gt_0_iff S_not_empty S_finite divide_ennreal_def 1766 divide_ennreal[symmetric] ennreal_of_nat_eq_real_of_nat[symmetric] ennreal_times_divide) 1767 1768lemma integral_pmf_of_set: "integral\<^sup>L (measure_pmf pmf_of_set) f = sum f S / card S" 1769 by (subst integral_measure_pmf[of S]) (auto simp: S_finite sum_divide_distrib) 1770 1771lemma emeasure_pmf_of_set: "emeasure (measure_pmf pmf_of_set) A = card (S \<inter> A) / card S" 1772 by (subst nn_integral_indicator[symmetric], simp) 1773 (simp add: S_finite S_not_empty card_gt_0_iff indicator_def sum.If_cases divide_ennreal 1774 ennreal_of_nat_eq_real_of_nat nn_integral_pmf_of_set) 1775 1776lemma measure_pmf_of_set: "measure (measure_pmf pmf_of_set) A = card (S \<inter> A) / card S" 1777 using emeasure_pmf_of_set[of A] 1778 by (simp add: measure_nonneg measure_pmf.emeasure_eq_measure) 1779 1780end 1781 1782lemma pmf_expectation_bind_pmf_of_set: 1783 fixes A :: "'a set" and f :: "'a \<Rightarrow> 'b pmf" 1784 and h :: "'b \<Rightarrow> 'c::{banach, second_countable_topology}" 1785 assumes "A \<noteq> {}" "finite A" "\<And>x. x \<in> A \<Longrightarrow> finite (set_pmf (f x))" 1786 shows "measure_pmf.expectation (pmf_of_set A \<bind> f) h = 1787 (\<Sum>a\<in>A. measure_pmf.expectation (f a) h /\<^sub>R real (card A))" 1788 using assms by (subst pmf_expectation_bind[of A]) (auto simp: field_split_simps) 1789 1790lemma map_pmf_of_set: 1791 assumes "finite A" "A \<noteq> {}" 1792 shows "map_pmf f (pmf_of_set A) = pmf_of_multiset (image_mset f (mset_set A))" 1793 (is "?lhs = ?rhs") 1794proof (intro pmf_eqI) 1795 fix x 1796 from assms have "ennreal (pmf ?lhs x) = ennreal (pmf ?rhs x)" 1797 by (subst ennreal_pmf_map) 1798 (simp_all add: emeasure_pmf_of_set mset_set_empty_iff count_image_mset Int_commute) 1799 thus "pmf ?lhs x = pmf ?rhs x" by simp 1800qed 1801 1802lemma pmf_bind_pmf_of_set: 1803 assumes "A \<noteq> {}" "finite A" 1804 shows "pmf (bind_pmf (pmf_of_set A) f) x = 1805 (\<Sum>xa\<in>A. pmf (f xa) x) / real_of_nat (card A)" (is "?lhs = ?rhs") 1806proof - 1807 from assms have "card A > 0" by auto 1808 with assms have "ennreal ?lhs = ennreal ?rhs" 1809 by (subst ennreal_pmf_bind) 1810 (simp_all add: nn_integral_pmf_of_set max_def pmf_nonneg divide_ennreal [symmetric] 1811 sum_nonneg ennreal_of_nat_eq_real_of_nat) 1812 thus ?thesis by (subst (asm) ennreal_inj) (auto intro!: sum_nonneg divide_nonneg_nonneg) 1813qed 1814 1815lemma pmf_of_set_singleton: "pmf_of_set {x} = return_pmf x" 1816by(rule pmf_eqI)(simp add: indicator_def) 1817 1818lemma map_pmf_of_set_inj: 1819 assumes f: "inj_on f A" 1820 and [simp]: "A \<noteq> {}" "finite A" 1821 shows "map_pmf f (pmf_of_set A) = pmf_of_set (f ` A)" (is "?lhs = ?rhs") 1822proof(rule pmf_eqI) 1823 fix i 1824 show "pmf ?lhs i = pmf ?rhs i" 1825 proof(cases "i \<in> f ` A") 1826 case True 1827 then obtain i' where "i = f i'" "i' \<in> A" by auto 1828 thus ?thesis using f by(simp add: card_image pmf_map_inj) 1829 next 1830 case False 1831 hence "pmf ?lhs i = 0" by(simp add: pmf_eq_0_set_pmf set_map_pmf) 1832 moreover have "pmf ?rhs i = 0" using False by simp 1833 ultimately show ?thesis by simp 1834 qed 1835qed 1836 1837lemma map_pmf_of_set_bij_betw: 1838 assumes "bij_betw f A B" "A \<noteq> {}" "finite A" 1839 shows "map_pmf f (pmf_of_set A) = pmf_of_set B" 1840proof - 1841 have "map_pmf f (pmf_of_set A) = pmf_of_set (f ` A)" 1842 by (intro map_pmf_of_set_inj assms bij_betw_imp_inj_on[OF assms(1)]) 1843 also from assms have "f ` A = B" by (simp add: bij_betw_def) 1844 finally show ?thesis . 1845qed 1846 1847text \<open> 1848 Choosing an element uniformly at random from the union of a disjoint family 1849 of finite non-empty sets with the same size is the same as first choosing a set 1850 from the family uniformly at random and then choosing an element from the chosen set 1851 uniformly at random. 1852\<close> 1853lemma pmf_of_set_UN: 1854 assumes "finite (\<Union>(f ` A))" "A \<noteq> {}" "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> {}" 1855 "\<And>x. x \<in> A \<Longrightarrow> card (f x) = n" "disjoint_family_on f A" 1856 shows "pmf_of_set (\<Union>(f ` A)) = do {x \<leftarrow> pmf_of_set A; pmf_of_set (f x)}" 1857 (is "?lhs = ?rhs") 1858proof (intro pmf_eqI) 1859 fix x 1860 from assms have [simp]: "finite A" 1861 using infinite_disjoint_family_imp_infinite_UNION[of A f] by blast 1862 from assms have "ereal (pmf (pmf_of_set (\<Union>(f ` A))) x) = 1863 ereal (indicator (\<Union>x\<in>A. f x) x / real (card (\<Union>x\<in>A. f x)))" 1864 by (subst pmf_of_set) auto 1865 also from assms have "card (\<Union>x\<in>A. f x) = card A * n" 1866 by (subst card_UN_disjoint) (auto simp: disjoint_family_on_def) 1867 also from assms 1868 have "indicator (\<Union>x\<in>A. f x) x / real \<dots> = 1869 indicator (\<Union>x\<in>A. f x) x / (n * real (card A))" 1870 by (simp add: sum_divide_distrib [symmetric] mult_ac) 1871 also from assms have "indicator (\<Union>x\<in>A. f x) x = (\<Sum>y\<in>A. indicator (f y) x)" 1872 by (intro indicator_UN_disjoint) simp_all 1873 also from assms have "ereal ((\<Sum>y\<in>A. indicator (f y) x) / (real n * real (card A))) = 1874 ereal (pmf ?rhs x)" 1875 by (subst pmf_bind_pmf_of_set) (simp_all add: sum_divide_distrib) 1876 finally show "pmf ?lhs x = pmf ?rhs x" by simp 1877qed 1878 1879lemma bernoulli_pmf_half_conv_pmf_of_set: "bernoulli_pmf (1 / 2) = pmf_of_set UNIV" 1880 by (rule pmf_eqI) simp_all 1881 1882subsubsection \<open> Poisson Distribution \<close> 1883 1884context 1885 fixes rate :: real assumes rate_pos: "0 < rate" 1886begin 1887 1888lift_definition poisson_pmf :: "nat pmf" is "\<lambda>k. rate ^ k / fact k * exp (-rate)" 1889proof (* by Manuel Eberl *) 1890 have summable: "summable (\<lambda>x::nat. rate ^ x / fact x)" using summable_exp 1891 by (simp add: field_simps divide_inverse [symmetric]) 1892 have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x * exp (-rate) \<partial>count_space UNIV) = 1893 exp (-rate) * (\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV)" 1894 by (simp add: field_simps nn_integral_cmult[symmetric] ennreal_mult'[symmetric]) 1895 also from rate_pos have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV) = (\<Sum>x. rate ^ x / fact x)" 1896 by (simp_all add: nn_integral_count_space_nat suminf_ennreal summable ennreal_suminf_neq_top) 1897 also have "... = exp rate" unfolding exp_def 1898 by (simp add: field_simps divide_inverse [symmetric]) 1899 also have "ennreal (exp (-rate)) * ennreal (exp rate) = 1" 1900 by (simp add: mult_exp_exp ennreal_mult[symmetric]) 1901 finally show "(\<integral>\<^sup>+ x. ennreal (rate ^ x / (fact x) * exp (- rate)) \<partial>count_space UNIV) = 1" . 1902qed (simp add: rate_pos[THEN less_imp_le]) 1903 1904lemma pmf_poisson[simp]: "pmf poisson_pmf k = rate ^ k / fact k * exp (-rate)" 1905 by transfer rule 1906 1907lemma set_pmf_poisson[simp]: "set_pmf poisson_pmf = UNIV" 1908 using rate_pos by (auto simp: set_pmf_iff) 1909 1910end 1911 1912subsubsection \<open> Binomial Distribution \<close> 1913 1914context 1915 fixes n :: nat and p :: real assumes p_nonneg: "0 \<le> p" and p_le_1: "p \<le> 1" 1916begin 1917 1918lift_definition binomial_pmf :: "nat pmf" is "\<lambda>k. (n choose k) * p^k * (1 - p)^(n - k)" 1919proof 1920 have "(\<integral>\<^sup>+k. ennreal (real (n choose k) * p ^ k * (1 - p) ^ (n - k)) \<partial>count_space UNIV) = 1921 ennreal (\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k))" 1922 using p_le_1 p_nonneg by (subst nn_integral_count_space') auto 1923 also have "(\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k)) = (p + (1 - p)) ^ n" 1924 by (subst binomial_ring) (simp add: atLeast0AtMost) 1925 finally show "(\<integral>\<^sup>+ x. ennreal (real (n choose x) * p ^ x * (1 - p) ^ (n - x)) \<partial>count_space UNIV) = 1" 1926 by simp 1927qed (insert p_nonneg p_le_1, simp) 1928 1929lemma pmf_binomial[simp]: "pmf binomial_pmf k = (n choose k) * p^k * (1 - p)^(n - k)" 1930 by transfer rule 1931 1932lemma set_pmf_binomial_eq: "set_pmf binomial_pmf = (if p = 0 then {0} else if p = 1 then {n} else {.. n})" 1933 using p_nonneg p_le_1 unfolding set_eq_iff set_pmf_iff pmf_binomial by (auto simp: set_pmf_iff) 1934 1935end 1936 1937end 1938 1939lemma set_pmf_binomial_0[simp]: "set_pmf (binomial_pmf n 0) = {0}" 1940 by (simp add: set_pmf_binomial_eq) 1941 1942lemma set_pmf_binomial_1[simp]: "set_pmf (binomial_pmf n 1) = {n}" 1943 by (simp add: set_pmf_binomial_eq) 1944 1945lemma set_pmf_binomial[simp]: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (binomial_pmf n p) = {..n}" 1946 by (simp add: set_pmf_binomial_eq) 1947 1948context includes lifting_syntax 1949begin 1950 1951lemma bind_pmf_parametric [transfer_rule]: 1952 "(rel_pmf A ===> (A ===> rel_pmf B) ===> rel_pmf B) bind_pmf bind_pmf" 1953by(blast intro: rel_pmf_bindI dest: rel_funD) 1954 1955lemma return_pmf_parametric [transfer_rule]: "(A ===> rel_pmf A) return_pmf return_pmf" 1956by(rule rel_funI) simp 1957 1958end 1959 1960 1961primrec replicate_pmf :: "nat \<Rightarrow> 'a pmf \<Rightarrow> 'a list pmf" where 1962 "replicate_pmf 0 _ = return_pmf []" 1963| "replicate_pmf (Suc n) p = do {x \<leftarrow> p; xs \<leftarrow> replicate_pmf n p; return_pmf (x#xs)}" 1964 1965lemma replicate_pmf_1: "replicate_pmf 1 p = map_pmf (\<lambda>x. [x]) p" 1966 by (simp add: map_pmf_def bind_return_pmf) 1967 1968lemma set_replicate_pmf: 1969 "set_pmf (replicate_pmf n p) = {xs\<in>lists (set_pmf p). length xs = n}" 1970 by (induction n) (auto simp: length_Suc_conv) 1971 1972lemma replicate_pmf_distrib: 1973 "replicate_pmf (m + n) p = 1974 do {xs \<leftarrow> replicate_pmf m p; ys \<leftarrow> replicate_pmf n p; return_pmf (xs @ ys)}" 1975 by (induction m) (simp_all add: bind_return_pmf bind_return_pmf' bind_assoc_pmf) 1976 1977lemma power_diff': 1978 assumes "b \<le> a" 1979 shows "x ^ (a - b) = (if x = 0 \<and> a = b then 1 else x ^ a / (x::'a::field) ^ b)" 1980proof (cases "x = 0") 1981 case True 1982 with assms show ?thesis by (cases "a - b") simp_all 1983qed (insert assms, simp_all add: power_diff) 1984 1985 1986lemma binomial_pmf_Suc: 1987 assumes "p \<in> {0..1}" 1988 shows "binomial_pmf (Suc n) p = 1989 do {b \<leftarrow> bernoulli_pmf p; 1990 k \<leftarrow> binomial_pmf n p; 1991 return_pmf ((if b then 1 else 0) + k)}" (is "_ = ?rhs") 1992proof (intro pmf_eqI) 1993 fix k 1994 have A: "indicator {Suc a} (Suc b) = indicator {a} b" for a b 1995 by (simp add: indicator_def) 1996 show "pmf (binomial_pmf (Suc n) p) k = pmf ?rhs k" 1997 by (cases k; cases "k > n") 1998 (insert assms, auto simp: pmf_bind measure_pmf_single A field_split_simps algebra_simps 1999 not_less less_eq_Suc_le [symmetric] power_diff') 2000qed 2001 2002lemma binomial_pmf_0: "p \<in> {0..1} \<Longrightarrow> binomial_pmf 0 p = return_pmf 0" 2003 by (rule pmf_eqI) (simp_all add: indicator_def) 2004 2005lemma binomial_pmf_altdef: 2006 assumes "p \<in> {0..1}" 2007 shows "binomial_pmf n p = map_pmf (length \<circ> filter id) (replicate_pmf n (bernoulli_pmf p))" 2008 by (induction n) 2009 (insert assms, auto simp: binomial_pmf_Suc map_pmf_def bind_return_pmf bind_assoc_pmf 2010 bind_return_pmf' binomial_pmf_0 intro!: bind_pmf_cong) 2011 2012 2013subsection \<open>PMFs from association lists\<close> 2014 2015definition pmf_of_list ::" ('a \<times> real) list \<Rightarrow> 'a pmf" where 2016 "pmf_of_list xs = embed_pmf (\<lambda>x. sum_list (map snd (filter (\<lambda>z. fst z = x) xs)))" 2017 2018definition pmf_of_list_wf where 2019 "pmf_of_list_wf xs \<longleftrightarrow> (\<forall>x\<in>set (map snd xs) . x \<ge> 0) \<and> sum_list (map snd xs) = 1" 2020 2021lemma pmf_of_list_wfI: 2022 "(\<And>x. x \<in> set (map snd xs) \<Longrightarrow> x \<ge> 0) \<Longrightarrow> sum_list (map snd xs) = 1 \<Longrightarrow> pmf_of_list_wf xs" 2023 unfolding pmf_of_list_wf_def by simp 2024 2025context 2026begin 2027 2028private lemma pmf_of_list_aux: 2029 assumes "\<And>x. x \<in> set (map snd xs) \<Longrightarrow> x \<ge> 0" 2030 assumes "sum_list (map snd xs) = 1" 2031 shows "(\<integral>\<^sup>+ x. ennreal (sum_list (map snd [z\<leftarrow>xs . fst z = x])) \<partial>count_space UNIV) = 1" 2032proof - 2033 have "(\<integral>\<^sup>+ x. ennreal (sum_list (map snd (filter (\<lambda>z. fst z = x) xs))) \<partial>count_space UNIV) = 2034 (\<integral>\<^sup>+ x. ennreal (sum_list (map (\<lambda>(x',p). indicator {x'} x * p) xs)) \<partial>count_space UNIV)" 2035 apply (intro nn_integral_cong ennreal_cong, subst sum_list_map_filter') 2036 apply (rule arg_cong[where f = sum_list]) 2037 apply (auto cong: map_cong) 2038 done 2039 also have "\<dots> = (\<Sum>(x',p)\<leftarrow>xs. (\<integral>\<^sup>+ x. ennreal (indicator {x'} x * p) \<partial>count_space UNIV))" 2040 using assms(1) 2041 proof (induction xs) 2042 case (Cons x xs) 2043 from Cons.prems have "snd x \<ge> 0" by simp 2044 moreover have "b \<ge> 0" if "(a,b) \<in> set xs" for a b 2045 using Cons.prems[of b] that by force 2046 ultimately have "(\<integral>\<^sup>+ y. ennreal (\<Sum>(x', p)\<leftarrow>x # xs. indicator {x'} y * p) \<partial>count_space UNIV) = 2047 (\<integral>\<^sup>+ y. ennreal (indicator {fst x} y * snd x) + 2048 ennreal (\<Sum>(x', p)\<leftarrow>xs. indicator {x'} y * p) \<partial>count_space UNIV)" 2049 by (intro nn_integral_cong, subst ennreal_plus [symmetric]) 2050 (auto simp: case_prod_unfold indicator_def intro!: sum_list_nonneg) 2051 also have "\<dots> = (\<integral>\<^sup>+ y. ennreal (indicator {fst x} y * snd x) \<partial>count_space UNIV) + 2052 (\<integral>\<^sup>+ y. ennreal (\<Sum>(x', p)\<leftarrow>xs. indicator {x'} y * p) \<partial>count_space UNIV)" 2053 by (intro nn_integral_add) 2054 (force intro!: sum_list_nonneg AE_I2 intro: Cons simp: indicator_def)+ 2055 also have "(\<integral>\<^sup>+ y. ennreal (\<Sum>(x', p)\<leftarrow>xs. indicator {x'} y * p) \<partial>count_space UNIV) = 2056 (\<Sum>(x', p)\<leftarrow>xs. (\<integral>\<^sup>+ y. ennreal (indicator {x'} y * p) \<partial>count_space UNIV))" 2057 using Cons(1) by (intro Cons) simp_all 2058 finally show ?case by (simp add: case_prod_unfold) 2059 qed simp 2060 also have "\<dots> = (\<Sum>(x',p)\<leftarrow>xs. ennreal p * (\<integral>\<^sup>+ x. indicator {x'} x \<partial>count_space UNIV))" 2061 using assms(1) 2062 by (simp cong: map_cong only: case_prod_unfold, subst nn_integral_cmult [symmetric]) 2063 (auto intro!: assms(1) simp: max_def times_ereal.simps [symmetric] mult_ac ereal_indicator 2064 simp del: times_ereal.simps)+ 2065 also from assms have "\<dots> = sum_list (map snd xs)" by (simp add: case_prod_unfold sum_list_ennreal) 2066 also have "\<dots> = 1" using assms(2) by simp 2067 finally show ?thesis . 2068qed 2069 2070lemma pmf_pmf_of_list: 2071 assumes "pmf_of_list_wf xs" 2072 shows "pmf (pmf_of_list xs) x = sum_list (map snd (filter (\<lambda>z. fst z = x) xs))" 2073 using assms pmf_of_list_aux[of xs] unfolding pmf_of_list_def pmf_of_list_wf_def 2074 by (subst pmf_embed_pmf) (auto intro!: sum_list_nonneg) 2075 2076end 2077 2078lemma set_pmf_of_list: 2079 assumes "pmf_of_list_wf xs" 2080 shows "set_pmf (pmf_of_list xs) \<subseteq> set (map fst xs)" 2081proof clarify 2082 fix x assume A: "x \<in> set_pmf (pmf_of_list xs)" 2083 show "x \<in> set (map fst xs)" 2084 proof (rule ccontr) 2085 assume "x \<notin> set (map fst xs)" 2086 hence "[z\<leftarrow>xs . fst z = x] = []" by (auto simp: filter_empty_conv) 2087 with A assms show False by (simp add: pmf_pmf_of_list set_pmf_eq) 2088 qed 2089qed 2090 2091lemma finite_set_pmf_of_list: 2092 assumes "pmf_of_list_wf xs" 2093 shows "finite (set_pmf (pmf_of_list xs))" 2094 using assms by (rule finite_subset[OF set_pmf_of_list]) simp_all 2095 2096lemma emeasure_Int_set_pmf: 2097 "emeasure (measure_pmf p) (A \<inter> set_pmf p) = emeasure (measure_pmf p) A" 2098 by (rule emeasure_eq_AE) (auto simp: AE_measure_pmf_iff) 2099 2100lemma measure_Int_set_pmf: 2101 "measure (measure_pmf p) (A \<inter> set_pmf p) = measure (measure_pmf p) A" 2102 using emeasure_Int_set_pmf[of p A] by (simp add: Sigma_Algebra.measure_def) 2103 2104lemma measure_prob_cong_0: 2105 assumes "\<And>x. x \<in> A - B \<Longrightarrow> pmf p x = 0" 2106 assumes "\<And>x. x \<in> B - A \<Longrightarrow> pmf p x = 0" 2107 shows "measure (measure_pmf p) A = measure (measure_pmf p) B" 2108proof - 2109 have "measure_pmf.prob p A = measure_pmf.prob p (A \<inter> set_pmf p)" 2110 by (simp add: measure_Int_set_pmf) 2111 also have "A \<inter> set_pmf p = B \<inter> set_pmf p" 2112 using assms by (auto simp: set_pmf_eq) 2113 also have "measure_pmf.prob p \<dots> = measure_pmf.prob p B" 2114 by (simp add: measure_Int_set_pmf) 2115 finally show ?thesis . 2116qed 2117 2118lemma emeasure_pmf_of_list: 2119 assumes "pmf_of_list_wf xs" 2120 shows "emeasure (pmf_of_list xs) A = ennreal (sum_list (map snd (filter (\<lambda>x. fst x \<in> A) xs)))" 2121proof - 2122 have "emeasure (pmf_of_list xs) A = nn_integral (measure_pmf (pmf_of_list xs)) (indicator A)" 2123 by simp 2124 also from assms 2125 have "\<dots> = (\<Sum>x\<in>set_pmf (pmf_of_list xs) \<inter> A. ennreal (sum_list (map snd [z\<leftarrow>xs . fst z = x])))" 2126 by (subst nn_integral_measure_pmf_finite) (simp_all add: finite_set_pmf_of_list pmf_pmf_of_list Int_def) 2127 also from assms 2128 have "\<dots> = ennreal (\<Sum>x\<in>set_pmf (pmf_of_list xs) \<inter> A. sum_list (map snd [z\<leftarrow>xs . fst z = x]))" 2129 by (subst sum_ennreal) (auto simp: pmf_of_list_wf_def intro!: sum_list_nonneg) 2130 also have "\<dots> = ennreal (\<Sum>x\<in>set_pmf (pmf_of_list xs) \<inter> A. 2131 indicator A x * pmf (pmf_of_list xs) x)" (is "_ = ennreal ?S") 2132 using assms by (intro ennreal_cong sum.cong) (auto simp: pmf_pmf_of_list) 2133 also have "?S = (\<Sum>x\<in>set_pmf (pmf_of_list xs). indicator A x * pmf (pmf_of_list xs) x)" 2134 using assms by (intro sum.mono_neutral_left set_pmf_of_list finite_set_pmf_of_list) auto 2135 also have "\<dots> = (\<Sum>x\<in>set (map fst xs). indicator A x * pmf (pmf_of_list xs) x)" 2136 using assms by (intro sum.mono_neutral_left set_pmf_of_list) (auto simp: set_pmf_eq) 2137 also have "\<dots> = (\<Sum>x\<in>set (map fst xs). indicator A x * 2138 sum_list (map snd (filter (\<lambda>z. fst z = x) xs)))" 2139 using assms by (simp add: pmf_pmf_of_list) 2140 also have "\<dots> = (\<Sum>x\<in>set (map fst xs). sum_list (map snd (filter (\<lambda>z. fst z = x \<and> x \<in> A) xs)))" 2141 by (intro sum.cong) (auto simp: indicator_def) 2142 also have "\<dots> = (\<Sum>x\<in>set (map fst xs). (\<Sum>xa = 0..<length xs. 2143 if fst (xs ! xa) = x \<and> x \<in> A then snd (xs ! xa) else 0))" 2144 by (intro sum.cong refl, subst sum_list_map_filter', subst sum_list_sum_nth) simp 2145 also have "\<dots> = (\<Sum>xa = 0..<length xs. (\<Sum>x\<in>set (map fst xs). 2146 if fst (xs ! xa) = x \<and> x \<in> A then snd (xs ! xa) else 0))" 2147 by (rule sum.swap) 2148 also have "\<dots> = (\<Sum>xa = 0..<length xs. if fst (xs ! xa) \<in> A then 2149 (\<Sum>x\<in>set (map fst xs). if x = fst (xs ! xa) then snd (xs ! xa) else 0) else 0)" 2150 by (auto intro!: sum.cong sum.neutral simp del: sum.delta) 2151 also have "\<dots> = (\<Sum>xa = 0..<length xs. if fst (xs ! xa) \<in> A then snd (xs ! xa) else 0)" 2152 by (intro sum.cong refl) (simp_all add: sum.delta) 2153 also have "\<dots> = sum_list (map snd (filter (\<lambda>x. fst x \<in> A) xs))" 2154 by (subst sum_list_map_filter', subst sum_list_sum_nth) simp_all 2155 finally show ?thesis . 2156qed 2157 2158lemma measure_pmf_of_list: 2159 assumes "pmf_of_list_wf xs" 2160 shows "measure (pmf_of_list xs) A = sum_list (map snd (filter (\<lambda>x. fst x \<in> A) xs))" 2161 using assms unfolding pmf_of_list_wf_def Sigma_Algebra.measure_def 2162 by (subst emeasure_pmf_of_list [OF assms], subst enn2real_ennreal) (auto intro!: sum_list_nonneg) 2163 2164(* TODO Move? *) 2165lemma sum_list_nonneg_eq_zero_iff: 2166 fixes xs :: "'a :: linordered_ab_group_add list" 2167 shows "(\<And>x. x \<in> set xs \<Longrightarrow> x \<ge> 0) \<Longrightarrow> sum_list xs = 0 \<longleftrightarrow> set xs \<subseteq> {0}" 2168proof (induction xs) 2169 case (Cons x xs) 2170 from Cons.prems have "sum_list (x#xs) = 0 \<longleftrightarrow> x = 0 \<and> sum_list xs = 0" 2171 unfolding sum_list_simps by (subst add_nonneg_eq_0_iff) (auto intro: sum_list_nonneg) 2172 with Cons.IH Cons.prems show ?case by simp 2173qed simp_all 2174 2175lemma sum_list_filter_nonzero: 2176 "sum_list (filter (\<lambda>x. x \<noteq> 0) xs) = sum_list xs" 2177 by (induction xs) simp_all 2178(* END MOVE *) 2179 2180lemma set_pmf_of_list_eq: 2181 assumes "pmf_of_list_wf xs" "\<And>x. x \<in> snd ` set xs \<Longrightarrow> x > 0" 2182 shows "set_pmf (pmf_of_list xs) = fst ` set xs" 2183proof 2184 { 2185 fix x assume A: "x \<in> fst ` set xs" and B: "x \<notin> set_pmf (pmf_of_list xs)" 2186 then obtain y where y: "(x, y) \<in> set xs" by auto 2187 from B have "sum_list (map snd [z\<leftarrow>xs. fst z = x]) = 0" 2188 by (simp add: pmf_pmf_of_list[OF assms(1)] set_pmf_eq) 2189 moreover from y have "y \<in> snd ` {xa \<in> set xs. fst xa = x}" by force 2190 ultimately have "y = 0" using assms(1) 2191 by (subst (asm) sum_list_nonneg_eq_zero_iff) (auto simp: pmf_of_list_wf_def) 2192 with assms(2) y have False by force 2193 } 2194 thus "fst ` set xs \<subseteq> set_pmf (pmf_of_list xs)" by blast 2195qed (insert set_pmf_of_list[OF assms(1)], simp_all) 2196 2197lemma pmf_of_list_remove_zeros: 2198 assumes "pmf_of_list_wf xs" 2199 defines "xs' \<equiv> filter (\<lambda>z. snd z \<noteq> 0) xs" 2200 shows "pmf_of_list_wf xs'" "pmf_of_list xs' = pmf_of_list xs" 2201proof - 2202 have "map snd [z\<leftarrow>xs . snd z \<noteq> 0] = filter (\<lambda>x. x \<noteq> 0) (map snd xs)" 2203 by (induction xs) simp_all 2204 with assms(1) show wf: "pmf_of_list_wf xs'" 2205 by (auto simp: pmf_of_list_wf_def xs'_def sum_list_filter_nonzero) 2206 have "sum_list (map snd [z\<leftarrow>xs' . fst z = i]) = sum_list (map snd [z\<leftarrow>xs . fst z = i])" for i 2207 unfolding xs'_def by (induction xs) simp_all 2208 with assms(1) wf show "pmf_of_list xs' = pmf_of_list xs" 2209 by (intro pmf_eqI) (simp_all add: pmf_pmf_of_list) 2210qed 2211 2212end 2213