src/HOL/Probability/Probability_Mass_Function.thy
 author hoelzl Tue Feb 10 14:06:57 2015 +0100 (2015-02-10) changeset 59496 6faf024a1893 parent 59495 03944a830c4a parent 59492 ef195926dd98 child 59525 dfe6449aecd8 permissions -rw-r--r--
merged
1 (*  Title:      HOL/Probability/Probability_Mass_Function.thy
2     Author:     Johannes Hölzl, TU München
3     Author:     Andreas Lochbihler, ETH Zurich
4 *)
6 section \<open> Probability mass function \<close>
8 theory Probability_Mass_Function
9 imports
11   "~~/src/HOL/Number_Theory/Binomial"
12   "~~/src/HOL/Library/Multiset"
13 begin
15 lemma ereal_divide': "b \<noteq> 0 \<Longrightarrow> ereal (a / b) = ereal a / ereal b"
16   using ereal_divide[of a b] by simp
18 lemma (in finite_measure) countable_support:
19   "countable {x. measure M {x} \<noteq> 0}"
20 proof cases
21   assume "measure M (space M) = 0"
22   with bounded_measure measure_le_0_iff have "{x. measure M {x} \<noteq> 0} = {}"
23     by auto
24   then show ?thesis
25     by simp
26 next
27   let ?M = "measure M (space M)" and ?m = "\<lambda>x. measure M {x}"
28   assume "?M \<noteq> 0"
29   then have *: "{x. ?m x \<noteq> 0} = (\<Union>n. {x. ?M / Suc n < ?m x})"
30     using reals_Archimedean[of "?m x / ?M" for x]
31     by (auto simp: field_simps not_le[symmetric] measure_nonneg divide_le_0_iff measure_le_0_iff)
32   have **: "\<And>n. finite {x. ?M / Suc n < ?m x}"
33   proof (rule ccontr)
34     fix n assume "infinite {x. ?M / Suc n < ?m x}" (is "infinite ?X")
35     then obtain X where "finite X" "card X = Suc (Suc n)" "X \<subseteq> ?X"
36       by (metis infinite_arbitrarily_large)
37     from this(3) have *: "\<And>x. x \<in> X \<Longrightarrow> ?M / Suc n \<le> ?m x"
38       by auto
39     { fix x assume "x \<in> X"
40       from `?M \<noteq> 0` *[OF this] have "?m x \<noteq> 0" by (auto simp: field_simps measure_le_0_iff)
41       then have "{x} \<in> sets M" by (auto dest: measure_notin_sets) }
42     note singleton_sets = this
43     have "?M < (\<Sum>x\<in>X. ?M / Suc n)"
44       using `?M \<noteq> 0`
45       by (simp add: `card X = Suc (Suc n)` real_eq_of_nat[symmetric] real_of_nat_Suc field_simps less_le measure_nonneg)
46     also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"
47       by (rule setsum_mono) fact
48     also have "\<dots> = measure M (\<Union>x\<in>X. {x})"
49       using singleton_sets `finite X`
50       by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def)
51     finally have "?M < measure M (\<Union>x\<in>X. {x})" .
52     moreover have "measure M (\<Union>x\<in>X. {x}) \<le> ?M"
53       using singleton_sets[THEN sets.sets_into_space] by (intro finite_measure_mono) auto
54     ultimately show False by simp
55   qed
56   show ?thesis
57     unfolding * by (intro countable_UN countableI_type countable_finite[OF **])
58 qed
60 lemma (in finite_measure) AE_support_countable:
61   assumes [simp]: "sets M = UNIV"
62   shows "(AE x in M. measure M {x} \<noteq> 0) \<longleftrightarrow> (\<exists>S. countable S \<and> (AE x in M. x \<in> S))"
63 proof
64   assume "\<exists>S. countable S \<and> (AE x in M. x \<in> S)"
65   then obtain S where S[intro]: "countable S" and ae: "AE x in M. x \<in> S"
66     by auto
67   then have "emeasure M (\<Union>x\<in>{x\<in>S. emeasure M {x} \<noteq> 0}. {x}) =
68     (\<integral>\<^sup>+ x. emeasure M {x} * indicator {x\<in>S. emeasure M {x} \<noteq> 0} x \<partial>count_space UNIV)"
69     by (subst emeasure_UN_countable)
70        (auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space)
71   also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} * indicator S x \<partial>count_space UNIV)"
72     by (auto intro!: nn_integral_cong split: split_indicator)
73   also have "\<dots> = emeasure M (\<Union>x\<in>S. {x})"
74     by (subst emeasure_UN_countable)
75        (auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space)
76   also have "\<dots> = emeasure M (space M)"
77     using ae by (intro emeasure_eq_AE) auto
78   finally have "emeasure M {x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0} = emeasure M (space M)"
79     by (simp add: emeasure_single_in_space cong: rev_conj_cong)
80   with finite_measure_compl[of "{x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0}"]
81   have "AE x in M. x \<in> S \<and> emeasure M {x} \<noteq> 0"
82     by (intro AE_I[OF order_refl]) (auto simp: emeasure_eq_measure set_diff_eq cong: conj_cong)
83   then show "AE x in M. measure M {x} \<noteq> 0"
84     by (auto simp: emeasure_eq_measure)
85 qed (auto intro!: exI[of _ "{x. measure M {x} \<noteq> 0}"] countable_support)
87 subsection {* PMF as measure *}
89 typedef 'a pmf = "{M :: 'a measure. prob_space M \<and> sets M = UNIV \<and> (AE x in M. measure M {x} \<noteq> 0)}"
90   morphisms measure_pmf Abs_pmf
91   by (intro exI[of _ "uniform_measure (count_space UNIV) {undefined}"])
92      (auto intro!: prob_space_uniform_measure AE_uniform_measureI)
94 declare [[coercion measure_pmf]]
96 lemma prob_space_measure_pmf: "prob_space (measure_pmf p)"
97   using pmf.measure_pmf[of p] by auto
99 interpretation measure_pmf!: prob_space "measure_pmf M" for M
100   by (rule prob_space_measure_pmf)
102 interpretation measure_pmf!: subprob_space "measure_pmf M" for M
103   by (rule prob_space_imp_subprob_space) unfold_locales
105 lemma subprob_space_measure_pmf: "subprob_space (measure_pmf x)"
106   by unfold_locales
108 locale pmf_as_measure
109 begin
111 setup_lifting type_definition_pmf
113 end
115 context
116 begin
118 interpretation pmf_as_measure .
120 lift_definition pmf :: "'a pmf \<Rightarrow> 'a \<Rightarrow> real" is "\<lambda>M x. measure M {x}" .
122 lift_definition set_pmf :: "'a pmf \<Rightarrow> 'a set" is "\<lambda>M. {x. measure M {x} \<noteq> 0}" .
124 lift_definition map_pmf :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf" is
125   "\<lambda>f M. distr M (count_space UNIV) f"
126 proof safe
127   fix M and f :: "'a \<Rightarrow> 'b"
128   let ?D = "distr M (count_space UNIV) f"
129   assume "prob_space M" and [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
130   interpret prob_space M by fact
131   from ae have "AE x in M. measure M (f -` {f x}) \<noteq> 0"
132   proof eventually_elim
133     fix x
134     have "measure M {x} \<le> measure M (f -` {f x})"
135       by (intro finite_measure_mono) auto
136     then show "measure M {x} \<noteq> 0 \<Longrightarrow> measure M (f -` {f x}) \<noteq> 0"
137       using measure_nonneg[of M "{x}"] by auto
138   qed
139   then show "AE x in ?D. measure ?D {x} \<noteq> 0"
140     by (simp add: AE_distr_iff measure_distr measurable_def)
141 qed (auto simp: measurable_def prob_space.prob_space_distr)
143 declare [[coercion set_pmf]]
145 lemma countable_set_pmf [simp]: "countable (set_pmf p)"
146   by transfer (metis prob_space.finite_measure finite_measure.countable_support)
148 lemma sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV"
149   by transfer metis
151 lemma sets_measure_pmf_count_space[measurable_cong]:
152   "sets (measure_pmf M) = sets (count_space UNIV)"
153   by simp
155 lemma space_measure_pmf[simp]: "space (measure_pmf p) = UNIV"
156   using sets_eq_imp_space_eq[of "measure_pmf p" "count_space UNIV"] by simp
158 lemma measure_pmf_in_subprob_algebra[measurable (raw)]: "measure_pmf x \<in> space (subprob_algebra (count_space UNIV))"
159   by (simp add: space_subprob_algebra subprob_space_measure_pmf)
161 lemma measurable_pmf_measure1[simp]: "measurable (M :: 'a pmf) N = UNIV \<rightarrow> space N"
162   by (auto simp: measurable_def)
164 lemma measurable_pmf_measure2[simp]: "measurable N (M :: 'a pmf) = measurable N (count_space UNIV)"
165   by (intro measurable_cong_sets) simp_all
167 lemma pmf_positive: "x \<in> set_pmf p \<Longrightarrow> 0 < pmf p x"
168   by transfer (simp add: less_le measure_nonneg)
170 lemma pmf_nonneg: "0 \<le> pmf p x"
171   by transfer (simp add: measure_nonneg)
173 lemma pmf_le_1: "pmf p x \<le> 1"
176 lemma emeasure_pmf_single:
177   fixes M :: "'a pmf"
178   shows "emeasure M {x} = pmf M x"
179   by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
181 lemma AE_measure_pmf: "AE x in (M::'a pmf). x \<in> M"
182   by transfer simp
184 lemma emeasure_pmf_single_eq_zero_iff:
185   fixes M :: "'a pmf"
186   shows "emeasure M {y} = 0 \<longleftrightarrow> y \<notin> M"
187   by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
189 lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \<longleftrightarrow> (\<forall>y\<in>M. P y)"
190 proof -
191   { fix y assume y: "y \<in> M" and P: "AE x in M. P x" "\<not> P y"
192     with P have "AE x in M. x \<noteq> y"
193       by auto
194     with y have False
195       by (simp add: emeasure_pmf_single_eq_zero_iff AE_iff_measurable[OF _ refl]) }
196   then show ?thesis
197     using AE_measure_pmf[of M] by auto
198 qed
200 lemma set_pmf_not_empty: "set_pmf M \<noteq> {}"
201   using AE_measure_pmf[of M] by (intro notI) simp
203 lemma set_pmf_iff: "x \<in> set_pmf M \<longleftrightarrow> pmf M x \<noteq> 0"
204   by transfer simp
206 lemma emeasure_measure_pmf_finite: "finite S \<Longrightarrow> emeasure (measure_pmf M) S = (\<Sum>s\<in>S. pmf M s)"
207   by (subst emeasure_eq_setsum_singleton) (auto simp: emeasure_pmf_single)
209 lemma measure_measure_pmf_finite: "finite S \<Longrightarrow> measure (measure_pmf M) S = setsum (pmf M) S"
210   using emeasure_measure_pmf_finite[of S M] by(simp add: measure_pmf.emeasure_eq_measure)
212 lemma nn_integral_measure_pmf_support:
213   fixes f :: "'a \<Rightarrow> ereal"
214   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"
215   shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>A. f x * pmf M x)"
216 proof -
217   have "(\<integral>\<^sup>+x. f x \<partial>M) = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)"
218     using nn by (intro nn_integral_cong_AE) (auto simp: AE_measure_pmf_iff split: split_indicator)
219   also have "\<dots> = (\<Sum>x\<in>A. f x * emeasure M {x})"
220     using assms by (intro nn_integral_indicator_finite) auto
221   finally show ?thesis
223 qed
225 lemma nn_integral_measure_pmf_finite:
226   fixes f :: "'a \<Rightarrow> ereal"
227   assumes f: "finite (set_pmf M)" and nn: "\<And>x. x \<in> set_pmf M \<Longrightarrow> 0 \<le> f x"
228   shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>set_pmf M. f x * pmf M x)"
229   using assms by (intro nn_integral_measure_pmf_support) auto
230 lemma integrable_measure_pmf_finite:
231   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
232   shows "finite (set_pmf M) \<Longrightarrow> integrable M f"
233   by (auto intro!: integrableI_bounded simp: nn_integral_measure_pmf_finite)
235 lemma integral_measure_pmf:
236   assumes [simp]: "finite A" and "\<And>a. a \<in> set_pmf M \<Longrightarrow> f a \<noteq> 0 \<Longrightarrow> a \<in> A"
237   shows "(\<integral>x. f x \<partial>measure_pmf M) = (\<Sum>a\<in>A. f a * pmf M a)"
238 proof -
239   have "(\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x * indicator A x \<partial>measure_pmf M)"
240     using assms(2) by (intro integral_cong_AE) (auto split: split_indicator simp: AE_measure_pmf_iff)
241   also have "\<dots> = (\<Sum>a\<in>A. f a * pmf M a)"
242     by (subst integral_indicator_finite_real) (auto simp: measure_def emeasure_measure_pmf_finite)
243   finally show ?thesis .
244 qed
246 lemma integrable_pmf: "integrable (count_space X) (pmf M)"
247 proof -
248   have " (\<integral>\<^sup>+ x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+ x. pmf M x \<partial>count_space (M \<inter> X))"
249     by (auto simp add: nn_integral_count_space_indicator set_pmf_iff intro!: nn_integral_cong split: split_indicator)
250   then have "integrable (count_space X) (pmf M) = integrable (count_space (M \<inter> X)) (pmf M)"
251     by (simp add: integrable_iff_bounded pmf_nonneg)
252   then show ?thesis
253     by (simp add: pmf.rep_eq measure_pmf.integrable_measure disjoint_family_on_def)
254 qed
256 lemma integral_pmf: "(\<integral>x. pmf M x \<partial>count_space X) = measure M X"
257 proof -
258   have "(\<integral>x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+x. pmf M x \<partial>count_space X)"
259     by (simp add: pmf_nonneg integrable_pmf nn_integral_eq_integral)
260   also have "\<dots> = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space (X \<inter> M))"
261     by (auto intro!: nn_integral_cong_AE split: split_indicator
262              simp: pmf.rep_eq measure_pmf.emeasure_eq_measure nn_integral_count_space_indicator
263                    AE_count_space set_pmf_iff)
264   also have "\<dots> = emeasure M (X \<inter> M)"
265     by (rule emeasure_countable_singleton[symmetric]) (auto intro: countable_set_pmf)
266   also have "\<dots> = emeasure M X"
267     by (auto intro!: emeasure_eq_AE simp: AE_measure_pmf_iff)
268   finally show ?thesis
270 qed
272 lemma integral_pmf_restrict:
273   "(f::'a \<Rightarrow> 'b::{banach, second_countable_topology}) \<in> borel_measurable (count_space UNIV) \<Longrightarrow>
274     (\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x \<partial>restrict_space M M)"
275   by (auto intro!: integral_cong_AE simp add: integral_restrict_space AE_measure_pmf_iff)
277 lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1"
278 proof -
279   have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)"
280     by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf)
281   then show ?thesis
282     using measure_pmf.emeasure_space_1 by simp
283 qed
285 lemma emeasure_pmf_UNIV [simp]: "emeasure (measure_pmf M) UNIV = 1"
286 using measure_pmf.emeasure_space_1[of M] by simp
288 lemma in_null_sets_measure_pmfI:
289   "A \<inter> set_pmf p = {} \<Longrightarrow> A \<in> null_sets (measure_pmf p)"
290 using emeasure_eq_0_AE[where ?P="\<lambda>x. x \<in> A" and M="measure_pmf p"]
291 by(auto simp add: null_sets_def AE_measure_pmf_iff)
293 lemma map_pmf_id[simp]: "map_pmf id = id"
294   by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI)
296 lemma map_pmf_ident[simp]: "map_pmf (\<lambda>x. x) = (\<lambda>x. x)"
297   using map_pmf_id unfolding id_def .
299 lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g"
300   by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def)
302 lemma map_pmf_comp: "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M"
303   using map_pmf_compose[of f g] by (simp add: comp_def)
305 lemma map_pmf_cong:
306   assumes "p = q"
307   shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> map_pmf f p = map_pmf g q"
308   unfolding `p = q`[symmetric] measure_pmf_inject[symmetric] map_pmf.rep_eq
309   by (auto simp add: emeasure_distr AE_measure_pmf_iff intro!: emeasure_eq_AE measure_eqI)
311 lemma emeasure_map_pmf[simp]: "emeasure (map_pmf f M) X = emeasure M (f -` X)"
312   unfolding map_pmf.rep_eq by (subst emeasure_distr) auto
314 lemma nn_integral_map_pmf[simp]: "(\<integral>\<^sup>+x. f x \<partial>map_pmf g M) = (\<integral>\<^sup>+x. f (g x) \<partial>M)"
315   unfolding map_pmf.rep_eq by (intro nn_integral_distr) auto
317 lemma ereal_pmf_map: "pmf (map_pmf f p) x = (\<integral>\<^sup>+ y. indicator (f -` {x}) y \<partial>measure_pmf p)"
318 proof(transfer fixing: f x)
319   fix p :: "'b measure"
320   presume "prob_space p"
321   then interpret prob_space p .
322   presume "sets p = UNIV"
323   then show "ereal (measure (distr p (count_space UNIV) f) {x}) = integral\<^sup>N p (indicator (f -` {x}))"
324     by(simp add: measure_distr measurable_def emeasure_eq_measure)
325 qed simp_all
327 lemma pmf_set_map:
328   fixes f :: "'a \<Rightarrow> 'b"
329   shows "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
330 proof (rule, transfer, clarsimp simp add: measure_distr measurable_def)
331   fix f :: "'a \<Rightarrow> 'b" and M :: "'a measure"
332   assume "prob_space M" and ae: "AE x in M. measure M {x} \<noteq> 0" and [simp]: "sets M = UNIV"
333   interpret prob_space M by fact
334   show "{x. measure M (f -` {x}) \<noteq> 0} = f ` {x. measure M {x} \<noteq> 0}"
335   proof safe
336     fix x assume "measure M (f -` {x}) \<noteq> 0"
337     moreover have "measure M (f -` {x}) = measure M {y. f y = x \<and> measure M {y} \<noteq> 0}"
338       using ae by (intro finite_measure_eq_AE) auto
339     ultimately have "{y. f y = x \<and> measure M {y} \<noteq> 0} \<noteq> {}"
340       by (metis measure_empty)
341     then show "x \<in> f ` {x. measure M {x} \<noteq> 0}"
342       by auto
343   next
344     fix x assume "measure M {x} \<noteq> 0"
345     then have "0 < measure M {x}"
346       using measure_nonneg[of M "{x}"] by auto
347     also have "measure M {x} \<le> measure M (f -` {f x})"
348       by (intro finite_measure_mono) auto
349     finally show "measure M (f -` {f x}) = 0 \<Longrightarrow> False"
350       by simp
351   qed
352 qed
354 lemma set_map_pmf: "set_pmf (map_pmf f M) = f`set_pmf M"
355   using pmf_set_map[of f] by (auto simp: comp_def fun_eq_iff)
357 lemma nn_integral_pmf: "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = emeasure (measure_pmf p) A"
358 proof -
359   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))"
360     by(auto simp add: nn_integral_count_space_indicator indicator_def set_pmf_iff intro: nn_integral_cong)
361   also have "\<dots> = emeasure (measure_pmf p) (\<Union>x\<in>A \<inter> set_pmf p. {x})"
362     by(subst emeasure_UN_countable)(auto simp add: emeasure_pmf_single disjoint_family_on_def)
363   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})"
364     by(rule emeasure_Un_null_set[symmetric])(auto intro: in_null_sets_measure_pmfI)
365   also have "\<dots> = emeasure (measure_pmf p) A"
366     by(auto intro: arg_cong2[where f=emeasure])
367   finally show ?thesis .
368 qed
370 subsection {* PMFs as function *}
372 context
373   fixes f :: "'a \<Rightarrow> real"
374   assumes nonneg: "\<And>x. 0 \<le> f x"
375   assumes prob: "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
376 begin
378 lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ereal \<circ> f)"
379 proof (intro conjI)
380   have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
381     by (simp split: split_indicator)
382   show "AE x in density (count_space UNIV) (ereal \<circ> f).
383     measure (density (count_space UNIV) (ereal \<circ> f)) {x} \<noteq> 0"
384     by (simp add: AE_density nonneg measure_def emeasure_density max_def)
385   show "prob_space (density (count_space UNIV) (ereal \<circ> f))"
386     by default (simp add: emeasure_density prob)
387 qed simp
389 lemma pmf_embed_pmf: "pmf embed_pmf x = f x"
390 proof transfer
391   have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
392     by (simp split: split_indicator)
393   fix x show "measure (density (count_space UNIV) (ereal \<circ> f)) {x} = f x"
394     by transfer (simp add: measure_def emeasure_density nonneg max_def)
395 qed
397 end
399 lemma embed_pmf_transfer:
400   "rel_fun (eq_onp (\<lambda>f. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ereal (f x) \<partial>count_space UNIV) = 1)) pmf_as_measure.cr_pmf (\<lambda>f. density (count_space UNIV) (ereal \<circ> f)) embed_pmf"
401   by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer)
403 lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)"
404 proof (transfer, elim conjE)
405   fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
406   assume "prob_space M" then interpret prob_space M .
407   show "M = density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))"
408   proof (rule measure_eqI)
409     fix A :: "'a set"
410     have "(\<integral>\<^sup>+ x. ereal (measure M {x}) * indicator A x \<partial>count_space UNIV) =
411       (\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)"
412       by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator)
413     also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))"
414       by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space)
415     also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})"
416       by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support)
417          (auto simp: disjoint_family_on_def)
418     also have "\<dots> = emeasure M A"
419       using ae by (intro emeasure_eq_AE) auto
420     finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))) A"
421       using emeasure_space_1 by (simp add: emeasure_density)
422   qed simp
423 qed
425 lemma td_pmf_embed_pmf:
426   "type_definition pmf embed_pmf {f::'a \<Rightarrow> real. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ereal (f x) \<partial>count_space UNIV) = 1}"
427   unfolding type_definition_def
428 proof safe
429   fix p :: "'a pmf"
430   have "(\<integral>\<^sup>+ x. 1 \<partial>measure_pmf p) = 1"
431     using measure_pmf.emeasure_space_1[of p] by simp
432   then show *: "(\<integral>\<^sup>+ x. ereal (pmf p x) \<partial>count_space UNIV) = 1"
433     by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const)
435   show "embed_pmf (pmf p) = p"
436     by (intro measure_pmf_inject[THEN iffD1])
437        (simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def)
438 next
439   fix f :: "'a \<Rightarrow> real" assume "\<forall>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
440   then show "pmf (embed_pmf f) = f"
441     by (auto intro!: pmf_embed_pmf)
442 qed (rule pmf_nonneg)
444 end
446 locale pmf_as_function
447 begin
449 setup_lifting td_pmf_embed_pmf
451 lemma set_pmf_transfer[transfer_rule]:
452   assumes "bi_total A"
453   shows "rel_fun (pcr_pmf A) (rel_set A) (\<lambda>f. {x. f x \<noteq> 0}) set_pmf"
454   using `bi_total A`
455   by (auto simp: pcr_pmf_def cr_pmf_def rel_fun_def rel_set_def bi_total_def Bex_def set_pmf_iff)
456      metis+
458 end
460 context
461 begin
463 interpretation pmf_as_function .
465 lemma pmf_eqI: "(\<And>i. pmf M i = pmf N i) \<Longrightarrow> M = N"
466   by transfer auto
468 lemma pmf_eq_iff: "M = N \<longleftrightarrow> (\<forall>i. pmf M i = pmf N i)"
469   by (auto intro: pmf_eqI)
471 end
473 context
474 begin
476 interpretation pmf_as_function .
478 subsubsection \<open> Bernoulli Distribution \<close>
480 lift_definition bernoulli_pmf :: "real \<Rightarrow> bool pmf" is
481   "\<lambda>p b. ((\<lambda>p. if b then p else 1 - p) \<circ> min 1 \<circ> max 0) p"
482   by (auto simp: nn_integral_count_space_finite[where A="{False, True}"] UNIV_bool
483            split: split_max split_min)
485 lemma pmf_bernoulli_True[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) True = p"
486   by transfer simp
488 lemma pmf_bernoulli_False[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) False = 1 - p"
489   by transfer simp
491 lemma set_pmf_bernoulli: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (bernoulli_pmf p) = UNIV"
492   by (auto simp add: set_pmf_iff UNIV_bool)
494 lemma nn_integral_bernoulli_pmf[simp]:
495   assumes [simp]: "0 \<le> p" "p \<le> 1" "\<And>x. 0 \<le> f x"
496   shows "(\<integral>\<^sup>+x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
497   by (subst nn_integral_measure_pmf_support[of UNIV])
498      (auto simp: UNIV_bool field_simps)
500 lemma integral_bernoulli_pmf[simp]:
501   assumes [simp]: "0 \<le> p" "p \<le> 1"
502   shows "(\<integral>x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
503   by (subst integral_measure_pmf[of UNIV]) (auto simp: UNIV_bool)
505 subsubsection \<open> Geometric Distribution \<close>
507 lift_definition geometric_pmf :: "nat pmf" is "\<lambda>n. 1 / 2^Suc n"
508 proof
509   note geometric_sums[of "1 / 2"]
510   note sums_mult[OF this, of "1 / 2"]
511   from sums_suminf_ereal[OF this]
512   show "(\<integral>\<^sup>+ x. ereal (1 / 2 ^ Suc x) \<partial>count_space UNIV) = 1"
513     by (simp add: nn_integral_count_space_nat field_simps)
514 qed simp
516 lemma pmf_geometric[simp]: "pmf geometric_pmf n = 1 / 2^Suc n"
517   by transfer rule
519 lemma set_pmf_geometric[simp]: "set_pmf geometric_pmf = UNIV"
520   by (auto simp: set_pmf_iff)
522 subsubsection \<open> Uniform Multiset Distribution \<close>
524 context
525   fixes M :: "'a multiset" assumes M_not_empty: "M \<noteq> {#}"
526 begin
528 lift_definition pmf_of_multiset :: "'a pmf" is "\<lambda>x. count M x / size M"
529 proof
530   show "(\<integral>\<^sup>+ x. ereal (real (count M x) / real (size M)) \<partial>count_space UNIV) = 1"
531     using M_not_empty
532     by (simp add: zero_less_divide_iff nn_integral_count_space nonempty_has_size
533                   setsum_divide_distrib[symmetric])
534        (auto simp: size_multiset_overloaded_eq intro!: setsum.cong)
535 qed simp
537 lemma pmf_of_multiset[simp]: "pmf pmf_of_multiset x = count M x / size M"
538   by transfer rule
540 lemma set_pmf_of_multiset[simp]: "set_pmf pmf_of_multiset = set_of M"
541   by (auto simp: set_pmf_iff)
543 end
545 subsubsection \<open> Uniform Distribution \<close>
547 context
548   fixes S :: "'a set" assumes S_not_empty: "S \<noteq> {}" and S_finite: "finite S"
549 begin
551 lift_definition pmf_of_set :: "'a pmf" is "\<lambda>x. indicator S x / card S"
552 proof
553   show "(\<integral>\<^sup>+ x. ereal (indicator S x / real (card S)) \<partial>count_space UNIV) = 1"
554     using S_not_empty S_finite by (subst nn_integral_count_space'[of S]) auto
555 qed simp
557 lemma pmf_of_set[simp]: "pmf pmf_of_set x = indicator S x / card S"
558   by transfer rule
560 lemma set_pmf_of_set[simp]: "set_pmf pmf_of_set = S"
561   using S_finite S_not_empty by (auto simp: set_pmf_iff)
563 lemma emeasure_pmf_of_set[simp]: "emeasure pmf_of_set S = 1"
564   by (rule measure_pmf.emeasure_eq_1_AE) (auto simp: AE_measure_pmf_iff)
566 end
568 subsubsection \<open> Poisson Distribution \<close>
570 context
571   fixes rate :: real assumes rate_pos: "0 < rate"
572 begin
574 lift_definition poisson_pmf :: "nat pmf" is "\<lambda>k. rate ^ k / fact k * exp (-rate)"
575 proof
576   (* Proof by Manuel Eberl *)
578   have summable: "summable (\<lambda>x::nat. rate ^ x / fact x)" using summable_exp
579     by (simp add: field_simps field_divide_inverse[symmetric])
580   have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x * exp (-rate) \<partial>count_space UNIV) =
581           exp (-rate) * (\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV)"
582     by (simp add: field_simps nn_integral_cmult[symmetric])
583   also from rate_pos have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV) = (\<Sum>x. rate ^ x / fact x)"
584     by (simp_all add: nn_integral_count_space_nat suminf_ereal summable suminf_ereal_finite)
585   also have "... = exp rate" unfolding exp_def
586     by (simp add: field_simps field_divide_inverse[symmetric] transfer_int_nat_factorial)
587   also have "ereal (exp (-rate)) * ereal (exp rate) = 1"
589   finally show "(\<integral>\<^sup>+ x. ereal (rate ^ x / real (fact x) * exp (- rate)) \<partial>count_space UNIV) = 1" .
590 qed (simp add: rate_pos[THEN less_imp_le])
592 lemma pmf_poisson[simp]: "pmf poisson_pmf k = rate ^ k / fact k * exp (-rate)"
593   by transfer rule
595 lemma set_pmf_poisson[simp]: "set_pmf poisson_pmf = UNIV"
596   using rate_pos by (auto simp: set_pmf_iff)
598 end
600 subsubsection \<open> Binomial Distribution \<close>
602 context
603   fixes n :: nat and p :: real assumes p_nonneg: "0 \<le> p" and p_le_1: "p \<le> 1"
604 begin
606 lift_definition binomial_pmf :: "nat pmf" is "\<lambda>k. (n choose k) * p^k * (1 - p)^(n - k)"
607 proof
608   have "(\<integral>\<^sup>+k. ereal (real (n choose k) * p ^ k * (1 - p) ^ (n - k)) \<partial>count_space UNIV) =
609     ereal (\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k))"
610     using p_le_1 p_nonneg by (subst nn_integral_count_space') auto
611   also have "(\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k)) = (p + (1 - p)) ^ n"
612     by (subst binomial_ring) (simp add: atLeast0AtMost real_of_nat_def)
613   finally show "(\<integral>\<^sup>+ x. ereal (real (n choose x) * p ^ x * (1 - p) ^ (n - x)) \<partial>count_space UNIV) = 1"
614     by simp
615 qed (insert p_nonneg p_le_1, simp)
617 lemma pmf_binomial[simp]: "pmf binomial_pmf k = (n choose k) * p^k * (1 - p)^(n - k)"
618   by transfer rule
620 lemma set_pmf_binomial_eq: "set_pmf binomial_pmf = (if p = 0 then {0} else if p = 1 then {n} else {.. n})"
621   using p_nonneg p_le_1 unfolding set_eq_iff set_pmf_iff pmf_binomial by (auto simp: set_pmf_iff)
623 end
625 end
627 lemma set_pmf_binomial_0[simp]: "set_pmf (binomial_pmf n 0) = {0}"
630 lemma set_pmf_binomial_1[simp]: "set_pmf (binomial_pmf n 1) = {n}"
633 lemma set_pmf_binomial[simp]: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (binomial_pmf n p) = {..n}"
636 subsection \<open> Monad Interpretation \<close>
638 lemma measurable_measure_pmf[measurable]:
639   "(\<lambda>x. measure_pmf (M x)) \<in> measurable (count_space UNIV) (subprob_algebra (count_space UNIV))"
640   by (auto simp: space_subprob_algebra intro!: prob_space_imp_subprob_space) unfold_locales
642 lemma bind_pmf_cong:
643   assumes "\<And>x. A x \<in> space (subprob_algebra N)" "\<And>x. B x \<in> space (subprob_algebra N)"
644   assumes "\<And>i. i \<in> set_pmf x \<Longrightarrow> A i = B i"
645   shows "bind (measure_pmf x) A = bind (measure_pmf x) B"
646 proof (rule measure_eqI)
647   show "sets (measure_pmf x \<guillemotright>= A) = sets (measure_pmf x \<guillemotright>= B)"
648     using assms by (subst (1 2) sets_bind) (auto simp: space_subprob_algebra)
649 next
650   fix X assume "X \<in> sets (measure_pmf x \<guillemotright>= A)"
651   then have X: "X \<in> sets N"
652     using assms by (subst (asm) sets_bind) (auto simp: space_subprob_algebra)
653   show "emeasure (measure_pmf x \<guillemotright>= A) X = emeasure (measure_pmf x \<guillemotright>= B) X"
654     using assms
655     by (subst (1 2) emeasure_bind[where N=N, OF _ _ X])
656        (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
657 qed
659 context
660 begin
662 interpretation pmf_as_measure .
664 lift_definition join_pmf :: "'a pmf pmf \<Rightarrow> 'a pmf" is "\<lambda>M. measure_pmf M \<guillemotright>= measure_pmf"
665 proof (intro conjI)
666   fix M :: "'a pmf pmf"
668   interpret bind: prob_space "measure_pmf M \<guillemotright>= measure_pmf"
669     apply (intro measure_pmf.prob_space_bind[where S="count_space UNIV"] AE_I2)
670     apply (auto intro!: subprob_space_measure_pmf simp: space_subprob_algebra)
671     apply unfold_locales
672     done
673   show "prob_space (measure_pmf M \<guillemotright>= measure_pmf)"
674     by intro_locales
675   show "sets (measure_pmf M \<guillemotright>= measure_pmf) = UNIV"
676     by (subst sets_bind) auto
677   have "AE x in measure_pmf M \<guillemotright>= measure_pmf. emeasure (measure_pmf M \<guillemotright>= measure_pmf) {x} \<noteq> 0"
678     by (auto simp: AE_bind[where B="count_space UNIV"] measure_pmf_in_subprob_algebra
679                    emeasure_bind[where N="count_space UNIV"] AE_measure_pmf_iff nn_integral_0_iff_AE
680                    measure_pmf.emeasure_eq_measure measure_le_0_iff set_pmf_iff pmf.rep_eq)
681   then show "AE x in measure_pmf M \<guillemotright>= measure_pmf. measure (measure_pmf M \<guillemotright>= measure_pmf) {x} \<noteq> 0"
682     unfolding bind.emeasure_eq_measure by simp
683 qed
685 lemma pmf_join: "pmf (join_pmf N) i = (\<integral>M. pmf M i \<partial>measure_pmf N)"
686 proof (transfer fixing: N i)
687   have N: "subprob_space (measure_pmf N)"
688     by (rule prob_space_imp_subprob_space) intro_locales
689   show "measure (measure_pmf N \<guillemotright>= measure_pmf) {i} = integral\<^sup>L (measure_pmf N) (\<lambda>M. measure M {i})"
690     using measurable_measure_pmf[of "\<lambda>x. x"]
691     by (intro subprob_space.measure_bind[where N="count_space UNIV", OF N]) auto
692 qed (auto simp: Transfer.Rel_def rel_fun_def cr_pmf_def)
694 lemma ereal_pmf_join: "ereal (pmf (join_pmf N) i) = (\<integral>\<^sup>+M. pmf M i \<partial>measure_pmf N)"
695   unfolding pmf_join
696   by (intro nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])
697      (auto simp: pmf_le_1 pmf_nonneg)
699 lemma set_pmf_join_pmf: "set_pmf (join_pmf f) = (\<Union>p\<in>set_pmf f. set_pmf p)"
700 apply(simp add: set_eq_iff set_pmf_iff pmf_join)
701 apply(subst integral_nonneg_eq_0_iff_AE)
702 apply(auto simp add: pmf_le_1 pmf_nonneg AE_measure_pmf_iff intro!: measure_pmf.integrable_const_bound[where B=1])
703 done
705 lift_definition return_pmf :: "'a \<Rightarrow> 'a pmf" is "return (count_space UNIV)"
706   by (auto intro!: prob_space_return simp: AE_return measure_return)
708 lemma join_return_pmf: "join_pmf (return_pmf M) = M"
709   by (simp add: integral_return pmf_eq_iff pmf_join return_pmf.rep_eq)
711 lemma map_return_pmf: "map_pmf f (return_pmf x) = return_pmf (f x)"
712   by transfer (simp add: distr_return)
714 lemma map_pmf_const[simp]: "map_pmf (\<lambda>_. c) M = return_pmf c"
715   by transfer (auto simp: prob_space.distr_const)
717 lemma set_return_pmf: "set_pmf (return_pmf x) = {x}"
718   by transfer (auto simp add: measure_return split: split_indicator)
720 lemma pmf_return: "pmf (return_pmf x) y = indicator {y} x"
721   by transfer (simp add: measure_return)
723 lemma nn_integral_return_pmf[simp]: "0 \<le> f x \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>return_pmf x) = f x"
724   unfolding return_pmf.rep_eq by (intro nn_integral_return) auto
726 lemma emeasure_return_pmf[simp]: "emeasure (return_pmf x) X = indicator X x"
727   unfolding return_pmf.rep_eq by (intro emeasure_return) auto
729 end
731 lemma return_pmf_inj[simp]: "return_pmf x = return_pmf y \<longleftrightarrow> x = y"
732   by (metis insertI1 set_return_pmf singletonD)
734 definition "bind_pmf M f = join_pmf (map_pmf f M)"
736 lemma (in pmf_as_measure) bind_transfer[transfer_rule]:
737   "rel_fun pmf_as_measure.cr_pmf (rel_fun (rel_fun op = pmf_as_measure.cr_pmf) pmf_as_measure.cr_pmf) op \<guillemotright>= bind_pmf"
738 proof (auto simp: pmf_as_measure.cr_pmf_def rel_fun_def bind_pmf_def join_pmf.rep_eq map_pmf.rep_eq)
739   fix M f and g :: "'a \<Rightarrow> 'b pmf" assume "\<forall>x. f x = measure_pmf (g x)"
740   then have f: "f = (\<lambda>x. measure_pmf (g x))"
741     by auto
742   show "measure_pmf M \<guillemotright>= f = distr (measure_pmf M) (count_space UNIV) g \<guillemotright>= measure_pmf"
743     unfolding f by (subst bind_distr[OF _ measurable_measure_pmf]) auto
744 qed
746 lemma ereal_pmf_bind: "pmf (bind_pmf N f) i = (\<integral>\<^sup>+x. pmf (f x) i \<partial>measure_pmf N)"
747   by (auto intro!: nn_integral_distr simp: bind_pmf_def ereal_pmf_join map_pmf.rep_eq)
749 lemma pmf_bind: "pmf (bind_pmf N f) i = (\<integral>x. pmf (f x) i \<partial>measure_pmf N)"
750   by (auto intro!: integral_distr simp: bind_pmf_def pmf_join map_pmf.rep_eq)
752 lemma bind_return_pmf: "bind_pmf (return_pmf x) f = f x"
753   unfolding bind_pmf_def map_return_pmf join_return_pmf ..
755 lemma join_eq_bind_pmf: "join_pmf M = bind_pmf M id"
758 lemma bind_pmf_const[simp]: "bind_pmf M (\<lambda>x. c) = c"
759   unfolding bind_pmf_def map_pmf_const join_return_pmf ..
761 lemma set_bind_pmf: "set_pmf (bind_pmf M N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))"
762   apply (simp add: set_eq_iff set_pmf_iff pmf_bind)
763   apply (subst integral_nonneg_eq_0_iff_AE)
764   apply (auto simp: pmf_nonneg pmf_le_1 AE_measure_pmf_iff
765               intro!: measure_pmf.integrable_const_bound[where B=1])
766   done
769 lemma measurable_pair_restrict_pmf2:
770   assumes "countable A"
771   assumes [measurable]: "\<And>y. y \<in> A \<Longrightarrow> (\<lambda>x. f (x, y)) \<in> measurable M L"
772   shows "f \<in> measurable (M \<Otimes>\<^sub>M restrict_space (measure_pmf N) A) L" (is "f \<in> measurable ?M _")
773 proof -
774   have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)"
777   show ?thesis
778     by (intro measurable_compose_countable'[where f="\<lambda>a b. f (fst b, a)" and g=snd and I=A,
779                                             unfolded pair_collapse] assms)
780         measurable
781 qed
783 lemma measurable_pair_restrict_pmf1:
784   assumes "countable A"
785   assumes [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N L"
786   shows "f \<in> measurable (restrict_space (measure_pmf M) A \<Otimes>\<^sub>M N) L"
787 proof -
788   have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)"
791   show ?thesis
792     by (intro measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A,
793                                             unfolded pair_collapse] assms)
794         measurable
795 qed
797 lemma bind_commute_pmf: "bind_pmf A (\<lambda>x. bind_pmf B (C x)) = bind_pmf B (\<lambda>y. bind_pmf A (\<lambda>x. C x y))"
798   unfolding pmf_eq_iff pmf_bind
799 proof
800   fix i
801   interpret B: prob_space "restrict_space B B"
802     by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
803        (auto simp: AE_measure_pmf_iff)
804   interpret A: prob_space "restrict_space A A"
805     by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
806        (auto simp: AE_measure_pmf_iff)
808   interpret AB: pair_prob_space "restrict_space A A" "restrict_space B B"
809     by unfold_locales
811   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)"
812     by (rule integral_cong) (auto intro!: integral_pmf_restrict)
813   also have "\<dots> = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>restrict_space A A)"
814     by (intro integral_pmf_restrict B.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
815               countable_set_pmf borel_measurable_count_space)
816   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>restrict_space B B)"
817     by (rule AB.Fubini_integral[symmetric])
818        (auto intro!: AB.integrable_const_bound[where B=1] measurable_pair_restrict_pmf2
819              simp: pmf_nonneg pmf_le_1 measurable_restrict_space1)
820   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>B)"
821     by (intro integral_pmf_restrict[symmetric] A.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
822               countable_set_pmf borel_measurable_count_space)
823   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>A \<partial>B)"
824     by (rule integral_cong) (auto intro!: integral_pmf_restrict[symmetric])
825   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)" .
826 qed
829 context
830 begin
832 interpretation pmf_as_measure .
834 lemma measure_pmf_bind: "measure_pmf (bind_pmf M f) = (measure_pmf M \<guillemotright>= (\<lambda>x. measure_pmf (f x)))"
835   by transfer simp
837 lemma nn_integral_bind_pmf[simp]: "(\<integral>\<^sup>+x. f x \<partial>bind_pmf M N) = (\<integral>\<^sup>+x. \<integral>\<^sup>+y. f y \<partial>N x \<partial>M)"
838   using measurable_measure_pmf[of N]
839   unfolding measure_pmf_bind
840   apply (subst (1 3) nn_integral_max_0[symmetric])
841   apply (intro nn_integral_bind[where B="count_space UNIV"])
842   apply auto
843   done
845 lemma emeasure_bind_pmf[simp]: "emeasure (bind_pmf M N) X = (\<integral>\<^sup>+x. emeasure (N x) X \<partial>M)"
846   using measurable_measure_pmf[of N]
847   unfolding measure_pmf_bind
848   by (subst emeasure_bind[where N="count_space UNIV"]) auto
850 lemma bind_return_pmf': "bind_pmf N return_pmf = N"
851 proof (transfer, clarify)
852   fix N :: "'a measure" assume "sets N = UNIV" then show "N \<guillemotright>= return (count_space UNIV) = N"
853     by (subst return_sets_cong[where N=N]) (simp_all add: bind_return')
854 qed
856 lemma bind_return_pmf'': "bind_pmf N (\<lambda>x. return_pmf (f x)) = map_pmf f N"
857 proof (transfer, clarify)
858   fix N :: "'b measure" and f :: "'b \<Rightarrow> 'a" assume "prob_space N" "sets N = UNIV"
859   then show "N \<guillemotright>= (\<lambda>x. return (count_space UNIV) (f x)) = distr N (count_space UNIV) f"
860     by (subst bind_return_distr[symmetric])
861        (auto simp: prob_space.not_empty measurable_def comp_def)
862 qed
864 lemma bind_assoc_pmf: "bind_pmf (bind_pmf A B) C = bind_pmf A (\<lambda>x. bind_pmf (B x) C)"
865   by transfer
866      (auto intro!: bind_assoc[where N="count_space UNIV" and R="count_space UNIV"]
867            simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space)
869 end
871 lemma map_bind_pmf: "map_pmf f (bind_pmf M g) = bind_pmf M (\<lambda>x. map_pmf f (g x))"
872   unfolding bind_return_pmf''[symmetric] bind_assoc_pmf[of M] ..
874 lemma bind_map_pmf: "bind_pmf (map_pmf f M) g = bind_pmf M (\<lambda>x. g (f x))"
875   unfolding bind_return_pmf''[symmetric] bind_assoc_pmf bind_return_pmf ..
877 lemma map_join_pmf: "map_pmf f (join_pmf AA) = join_pmf (map_pmf (map_pmf f) AA)"
878   unfolding bind_pmf_def[symmetric]
879   unfolding bind_return_pmf''[symmetric] join_eq_bind_pmf bind_assoc_pmf
882 definition "pair_pmf A B = bind_pmf A (\<lambda>x. bind_pmf B (\<lambda>y. return_pmf (x, y)))"
884 lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b"
885   unfolding pair_pmf_def pmf_bind pmf_return
886   apply (subst integral_measure_pmf[where A="{b}"])
887   apply (auto simp: indicator_eq_0_iff)
888   apply (subst integral_measure_pmf[where A="{a}"])
889   apply (auto simp: indicator_eq_0_iff setsum_nonneg_eq_0_iff pmf_nonneg)
890   done
892 lemma set_pair_pmf: "set_pmf (pair_pmf A B) = set_pmf A \<times> set_pmf B"
893   unfolding pair_pmf_def set_bind_pmf set_return_pmf by auto
895 lemma measure_pmf_in_subprob_space[measurable (raw)]:
896   "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
897   by (simp add: space_subprob_algebra) intro_locales
899 lemma nn_integral_pair_pmf': "(\<integral>\<^sup>+x. f x \<partial>pair_pmf A B) = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. f (a, b) \<partial>B \<partial>A)"
900 proof -
901   have "(\<integral>\<^sup>+x. f x \<partial>pair_pmf A B) = (\<integral>\<^sup>+x. max 0 (f x) * indicator (A \<times> B) x \<partial>pair_pmf A B)"
902     by (subst nn_integral_max_0[symmetric])
903        (auto simp: AE_measure_pmf_iff set_pair_pmf intro!: nn_integral_cong_AE)
904   also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) * indicator (A \<times> B) (a, b) \<partial>B \<partial>A)"
906   also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) \<partial>B \<partial>A)"
907     by (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
908   finally show ?thesis
909     unfolding nn_integral_max_0 .
910 qed
912 lemma pair_map_pmf1: "pair_pmf (map_pmf f A) B = map_pmf (apfst f) (pair_pmf A B)"
913 proof (safe intro!: pmf_eqI)
914   fix a :: "'a" and b :: "'b"
915   have [simp]: "\<And>c d. indicator (apfst f -` {(a, b)}) (c, d) = indicator (f -` {a}) c * (indicator {b} d::ereal)"
916     by (auto split: split_indicator)
918   have "ereal (pmf (pair_pmf (map_pmf f A) B) (a, b)) =
919          ereal (pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b))"
920     unfolding pmf_pair ereal_pmf_map
921     by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_multc pmf_nonneg
922                   emeasure_map_pmf[symmetric] del: emeasure_map_pmf)
923   then show "pmf (pair_pmf (map_pmf f A) B) (a, b) = pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b)"
924     by simp
925 qed
927 lemma pair_map_pmf2: "pair_pmf A (map_pmf f B) = map_pmf (apsnd f) (pair_pmf A B)"
928 proof (safe intro!: pmf_eqI)
929   fix a :: "'a" and b :: "'b"
930   have [simp]: "\<And>c d. indicator (apsnd f -` {(a, b)}) (c, d) = indicator {a} c * (indicator (f -` {b}) d::ereal)"
931     by (auto split: split_indicator)
933   have "ereal (pmf (pair_pmf A (map_pmf f B)) (a, b)) =
934          ereal (pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b))"
935     unfolding pmf_pair ereal_pmf_map
936     by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_cmult nn_integral_multc pmf_nonneg
937                   emeasure_map_pmf[symmetric] del: emeasure_map_pmf)
938   then show "pmf (pair_pmf A (map_pmf f B)) (a, b) = pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b)"
939     by simp
940 qed
942 lemma map_pair: "map_pmf (\<lambda>(a, b). (f a, g b)) (pair_pmf A B) = pair_pmf (map_pmf f A) (map_pmf g B)"
943   by (simp add: pair_map_pmf2 pair_map_pmf1 map_pmf_comp split_beta')
945 lemma bind_pair_pmf:
946   assumes M[measurable]: "M \<in> measurable (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) (subprob_algebra N)"
947   shows "measure_pmf (pair_pmf A B) \<guillemotright>= M = (measure_pmf A \<guillemotright>= (\<lambda>x. measure_pmf B \<guillemotright>= (\<lambda>y. M (x, y))))"
948     (is "?L = ?R")
949 proof (rule measure_eqI)
950   have M'[measurable]: "M \<in> measurable (pair_pmf A B) (subprob_algebra N)"
951     using M[THEN measurable_space] by (simp_all add: space_pair_measure)
953   note measurable_bind[where N="count_space UNIV", measurable]
954   note measure_pmf_in_subprob_space[simp]
956   have sets_eq_N: "sets ?L = N"
957     by (subst sets_bind[OF sets_kernel[OF M']]) auto
958   show "sets ?L = sets ?R"
959     using measurable_space[OF M]
960     by (simp add: sets_eq_N space_pair_measure space_subprob_algebra)
961   fix X assume "X \<in> sets ?L"
962   then have X[measurable]: "X \<in> sets N"
963     unfolding sets_eq_N .
964   then show "emeasure ?L X = emeasure ?R X"
965     apply (simp add: emeasure_bind[OF _ M' X])
966     apply (simp add: nn_integral_bind[where B="count_space UNIV"] pair_pmf_def measure_pmf_bind[of A]
967       nn_integral_measure_pmf_finite set_return_pmf emeasure_nonneg pmf_return one_ereal_def[symmetric])
968     apply (subst emeasure_bind[OF _ _ X])
969     apply measurable
970     apply (subst emeasure_bind[OF _ _ X])
971     apply measurable
972     done
973 qed
975 lemma join_map_return_pmf: "join_pmf (map_pmf return_pmf A) = A"
976   unfolding bind_pmf_def[symmetric] bind_return_pmf' ..
978 lemma map_fst_pair_pmf: "map_pmf fst (pair_pmf A B) = A"
979   by (simp add: pair_pmf_def bind_return_pmf''[symmetric] bind_assoc_pmf bind_return_pmf bind_return_pmf')
981 lemma map_snd_pair_pmf: "map_pmf snd (pair_pmf A B) = B"
982   by (simp add: pair_pmf_def bind_return_pmf''[symmetric] bind_assoc_pmf bind_return_pmf bind_return_pmf')
984 lemma nn_integral_pmf':
985   "inj_on f A \<Longrightarrow> (\<integral>\<^sup>+x. pmf p (f x) \<partial>count_space A) = emeasure p (f ` A)"
986   by (subst nn_integral_bij_count_space[where g=f and B="f`A"])
987      (auto simp: bij_betw_def nn_integral_pmf)
989 lemma pmf_le_0_iff[simp]: "pmf M p \<le> 0 \<longleftrightarrow> pmf M p = 0"
990   using pmf_nonneg[of M p] by simp
992 lemma min_pmf_0[simp]: "min (pmf M p) 0 = 0" "min 0 (pmf M p) = 0"
993   using pmf_nonneg[of M p] by simp_all
995 lemma pmf_eq_0_set_pmf: "pmf M p = 0 \<longleftrightarrow> p \<notin> set_pmf M"
996   unfolding set_pmf_iff by simp
998 lemma pmf_map_inj: "inj_on f (set_pmf M) \<Longrightarrow> x \<in> set_pmf M \<Longrightarrow> pmf (map_pmf f M) (f x) = pmf M x"
999   by (auto simp: pmf.rep_eq map_pmf.rep_eq measure_distr AE_measure_pmf_iff inj_onD
1000            intro!: measure_pmf.finite_measure_eq_AE)
1002 subsection \<open> Conditional Probabilities \<close>
1004 context
1005   fixes p :: "'a pmf" and s :: "'a set"
1006   assumes not_empty: "set_pmf p \<inter> s \<noteq> {}"
1007 begin
1009 interpretation pmf_as_measure .
1011 lemma emeasure_measure_pmf_not_zero: "emeasure (measure_pmf p) s \<noteq> 0"
1012 proof
1013   assume "emeasure (measure_pmf p) s = 0"
1014   then have "AE x in measure_pmf p. x \<notin> s"
1015     by (rule AE_I[rotated]) auto
1016   with not_empty show False
1017     by (auto simp: AE_measure_pmf_iff)
1018 qed
1020 lemma measure_measure_pmf_not_zero: "measure (measure_pmf p) s \<noteq> 0"
1021   using emeasure_measure_pmf_not_zero unfolding measure_pmf.emeasure_eq_measure by simp
1023 lift_definition cond_pmf :: "'a pmf" is
1024   "uniform_measure (measure_pmf p) s"
1025 proof (intro conjI)
1026   show "prob_space (uniform_measure (measure_pmf p) s)"
1027     by (intro prob_space_uniform_measure) (auto simp: emeasure_measure_pmf_not_zero)
1028   show "AE x in uniform_measure (measure_pmf p) s. measure (uniform_measure (measure_pmf p) s) {x} \<noteq> 0"
1029     by (simp add: emeasure_measure_pmf_not_zero measure_measure_pmf_not_zero AE_uniform_measure
1030                   AE_measure_pmf_iff set_pmf.rep_eq)
1031 qed simp
1033 lemma pmf_cond: "pmf cond_pmf x = (if x \<in> s then pmf p x / measure p s else 0)"
1034   by transfer (simp add: emeasure_measure_pmf_not_zero pmf.rep_eq)
1036 lemma set_cond_pmf: "set_pmf cond_pmf = set_pmf p \<inter> s"
1037   by (auto simp add: set_pmf_iff pmf_cond measure_measure_pmf_not_zero split: split_if_asm)
1039 end
1041 lemma cond_map_pmf:
1042   assumes "set_pmf p \<inter> f -` s \<noteq> {}"
1043   shows "cond_pmf (map_pmf f p) s = map_pmf f (cond_pmf p (f -` s))"
1044 proof -
1045   have *: "set_pmf (map_pmf f p) \<inter> s \<noteq> {}"
1046     using assms by (simp add: set_map_pmf) auto
1047   { fix x
1048     have "ereal (pmf (map_pmf f (cond_pmf p (f -` s))) x) =
1049       emeasure p (f -` s \<inter> f -` {x}) / emeasure p (f -` s)"
1050       unfolding ereal_pmf_map cond_pmf.rep_eq[OF assms] by (simp add: nn_integral_uniform_measure)
1051     also have "f -` s \<inter> f -` {x} = (if x \<in> s then f -` {x} else {})"
1052       by auto
1053     also have "emeasure p (if x \<in> s then f -` {x} else {}) / emeasure p (f -` s) =
1054       ereal (pmf (cond_pmf (map_pmf f p) s) x)"
1055       using measure_measure_pmf_not_zero[OF *]
1056       by (simp add: pmf_cond[OF *] ereal_divide' ereal_pmf_map measure_pmf.emeasure_eq_measure[symmetric]
1057                del: ereal_divide)
1058     finally have "ereal (pmf (cond_pmf (map_pmf f p) s) x) = ereal (pmf (map_pmf f (cond_pmf p (f -` s))) x)"
1059       by simp }
1060   then show ?thesis
1061     by (intro pmf_eqI) simp
1062 qed
1064 lemma bind_cond_pmf_cancel:
1065   assumes in_S: "\<And>x. x \<in> set_pmf p \<Longrightarrow> x \<in> S x"
1066   assumes S_eq: "\<And>x y. x \<in> S y \<Longrightarrow> S x = S y"
1067   shows "bind_pmf p (\<lambda>x. cond_pmf p (S x)) = p"
1068 proof (rule pmf_eqI)
1069   have [simp]: "\<And>x. x \<in> p \<Longrightarrow> p \<inter> (S x) \<noteq> {}"
1070     using in_S by auto
1071   fix z
1072   have pmf_le: "pmf p z \<le> measure p (S z)"
1073   proof cases
1074     assume "z \<in> p" from in_S[OF this] show ?thesis
1075       by (auto intro!: measure_pmf.finite_measure_mono simp: pmf.rep_eq)
1076   qed (simp add: set_pmf_iff measure_nonneg)
1078   have "ereal (pmf (bind_pmf p (\<lambda>x. cond_pmf p (S x))) z) =
1079     (\<integral>\<^sup>+ x. ereal (pmf p z / measure p (S z)) * indicator (S z) x \<partial>p)"
1080     by (subst ereal_pmf_bind)
1081        (auto intro!: nn_integral_cong_AE dest!: S_eq split: split_indicator
1082              simp: AE_measure_pmf_iff pmf_cond pmf_eq_0_set_pmf in_S)
1083   also have "\<dots> = pmf p z"
1084     using pmf_le pmf_nonneg[of p z]
1085     by (subst nn_integral_cmult) (simp_all add: measure_nonneg measure_pmf.emeasure_eq_measure)
1086   finally show "pmf (bind_pmf p (\<lambda>x. cond_pmf p (S x))) z = pmf p z"
1087     by simp
1088 qed
1090 inductive rel_pmf :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf \<Rightarrow> bool"
1091 for R p q
1092 where
1093   "\<lbrakk> \<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y;
1094      map_pmf fst pq = p; map_pmf snd pq = q \<rbrakk>
1095   \<Longrightarrow> rel_pmf R p q"
1097 bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: rel_pmf
1098 proof -
1099   show "map_pmf id = id" by (rule map_pmf_id)
1100   show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose)
1101   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"
1102     by (intro map_pmf_cong refl)
1104   show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
1105     by (rule pmf_set_map)
1107   { fix p :: "'s pmf"
1108     have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq"
1109       by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"])
1110          (auto intro: countable_set_pmf)
1111     also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq"
1112       by (metis Field_natLeq card_of_least natLeq_Well_order)
1113     finally show "(card_of (set_pmf p), natLeq) \<in> ordLeq" . }
1115   show "\<And>R. rel_pmf R =
1116          (BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf fst))\<inverse>\<inverse> OO
1117          BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf snd)"
1118      by (auto simp add: fun_eq_iff BNF_Def.Grp_def OO_def rel_pmf.simps)
1120   { fix p :: "'a pmf" and f :: "'a \<Rightarrow> 'b" and g x
1121     assume p: "\<And>z. z \<in> set_pmf p \<Longrightarrow> f z = g z"
1122       and x: "x \<in> set_pmf p"
1123     thus "f x = g x" by simp }
1125   fix R :: "'a => 'b \<Rightarrow> bool" and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
1126   { fix p q r
1127     assume pq: "rel_pmf R p q"
1128       and qr:"rel_pmf S q r"
1129     from pq obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
1130       and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto
1131     from qr obtain qr where qr: "\<And>y z. (y, z) \<in> set_pmf qr \<Longrightarrow> S y z"
1132       and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto
1134     def pr \<equiv> "bind_pmf pq (\<lambda>(x, y). bind_pmf (cond_pmf qr {(y', z). y' = y}) (\<lambda>(y', z). return_pmf (x, z)))"
1135     have pr_welldefined: "\<And>y. y \<in> q \<Longrightarrow> qr \<inter> {(y', z). y' = y} \<noteq> {}"
1136       by (force simp: q' set_map_pmf)
1138     have "rel_pmf (R OO S) p r"
1139     proof (rule rel_pmf.intros)
1140       fix x z assume "(x, z) \<in> pr"
1141       then have "\<exists>y. (x, y) \<in> pq \<and> (y, z) \<in> qr"
1142         by (auto simp: q pr_welldefined pr_def set_bind_pmf split_beta set_return_pmf set_cond_pmf set_map_pmf)
1143       with pq qr show "(R OO S) x z"
1144         by blast
1145     next
1146       have "map_pmf snd pr = map_pmf snd (bind_pmf q (\<lambda>y. cond_pmf qr {(y', z). y' = y}))"
1147         by (simp add: pr_def q split_beta bind_map_pmf bind_return_pmf'' map_bind_pmf map_return_pmf)
1148       then show "map_pmf snd pr = r"
1149         unfolding r q' bind_map_pmf by (subst (asm) bind_cond_pmf_cancel) auto
1150     qed (simp add: pr_def map_bind_pmf split_beta map_return_pmf bind_return_pmf'' p) }
1151   then show "rel_pmf R OO rel_pmf S \<le> rel_pmf (R OO S)"
1153 qed (fact natLeq_card_order natLeq_cinfinite)+
1155 lemma rel_pmf_return_pmf1: "rel_pmf R (return_pmf x) M \<longleftrightarrow> (\<forall>a\<in>M. R x a)"
1156 proof safe
1157   fix a assume "a \<in> M" "rel_pmf R (return_pmf x) M"
1158   then obtain pq where *: "\<And>a b. (a, b) \<in> set_pmf pq \<Longrightarrow> R a b"
1159     and eq: "return_pmf x = map_pmf fst pq" "M = map_pmf snd pq"
1160     by (force elim: rel_pmf.cases)
1161   moreover have "set_pmf (return_pmf x) = {x}"
1163   with `a \<in> M` have "(x, a) \<in> pq"
1164     by (force simp: eq set_map_pmf)
1165   with * show "R x a"
1166     by auto
1167 qed (auto intro!: rel_pmf.intros[where pq="pair_pmf (return_pmf x) M"]
1168           simp: map_fst_pair_pmf map_snd_pair_pmf set_pair_pmf set_return_pmf)
1170 lemma rel_pmf_return_pmf2: "rel_pmf R M (return_pmf x) \<longleftrightarrow> (\<forall>a\<in>M. R a x)"
1171   by (subst pmf.rel_flip[symmetric]) (simp add: rel_pmf_return_pmf1)
1173 lemma rel_return_pmf[simp]: "rel_pmf R (return_pmf x1) (return_pmf x2) = R x1 x2"
1174   unfolding rel_pmf_return_pmf2 set_return_pmf by simp
1176 lemma rel_pmf_False[simp]: "rel_pmf (\<lambda>x y. False) x y = False"
1177   unfolding pmf.in_rel fun_eq_iff using set_pmf_not_empty by fastforce
1179 lemma rel_pmf_rel_prod:
1180   "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'"
1181 proof safe
1182   assume "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
1183   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"
1184     and eq: "map_pmf fst pq = pair_pmf A A'" "map_pmf snd pq = pair_pmf B B'"
1185     by (force elim: rel_pmf.cases)
1186   show "rel_pmf R A B"
1187   proof (rule rel_pmf.intros)
1188     let ?f = "\<lambda>(a, b). (fst a, fst b)"
1189     have [simp]: "(\<lambda>x. fst (?f x)) = fst o fst" "(\<lambda>x. snd (?f x)) = fst o snd"
1190       by auto
1192     show "map_pmf fst (map_pmf ?f pq) = A"
1193       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
1194     show "map_pmf snd (map_pmf ?f pq) = B"
1195       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
1197     fix a b assume "(a, b) \<in> set_pmf (map_pmf ?f pq)"
1198     then obtain c d where "((a, c), (b, d)) \<in> set_pmf pq"
1199       by (auto simp: set_map_pmf)
1200     from pq[OF this] show "R a b" ..
1201   qed
1202   show "rel_pmf S A' B'"
1203   proof (rule rel_pmf.intros)
1204     let ?f = "\<lambda>(a, b). (snd a, snd b)"
1205     have [simp]: "(\<lambda>x. fst (?f x)) = snd o fst" "(\<lambda>x. snd (?f x)) = snd o snd"
1206       by auto
1208     show "map_pmf fst (map_pmf ?f pq) = A'"
1209       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
1210     show "map_pmf snd (map_pmf ?f pq) = B'"
1211       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
1213     fix c d assume "(c, d) \<in> set_pmf (map_pmf ?f pq)"
1214     then obtain a b where "((a, c), (b, d)) \<in> set_pmf pq"
1215       by (auto simp: set_map_pmf)
1216     from pq[OF this] show "S c d" ..
1217   qed
1218 next
1219   assume "rel_pmf R A B" "rel_pmf S A' B'"
1220   then obtain Rpq Spq
1221     where Rpq: "\<And>a b. (a, b) \<in> set_pmf Rpq \<Longrightarrow> R a b"
1222         "map_pmf fst Rpq = A" "map_pmf snd Rpq = B"
1223       and Spq: "\<And>a b. (a, b) \<in> set_pmf Spq \<Longrightarrow> S a b"
1224         "map_pmf fst Spq = A'" "map_pmf snd Spq = B'"
1225     by (force elim: rel_pmf.cases)
1227   let ?f = "(\<lambda>((a, c), (b, d)). ((a, b), (c, d)))"
1228   let ?pq = "map_pmf ?f (pair_pmf Rpq Spq)"
1229   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))"
1230     by auto
1232   show "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
1233     by (rule rel_pmf.intros[where pq="?pq"])
1234        (auto simp: map_snd_pair_pmf map_fst_pair_pmf set_pair_pmf set_map_pmf map_pmf_comp Rpq Spq
1235                    map_pair)
1236 qed
1238 end