src/HOL/Probability/Probability_Mass_Function.thy
 author Andreas Lochbihler Fri Jan 09 09:16:51 2015 +0100 (2015-01-09) changeset 59325 922d31f5c3f5 parent 59134 a71f2e256ee2 child 59327 8a779359df67 permissions -rw-r--r--
simplify construction for distribution of rel_pmf over op OO
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 bind_return'': "sets M = sets N \<Longrightarrow> M \<guillemotright>= return N = M"
16    by (cases "space M = {}")
17       (simp_all add: bind_empty space_empty[symmetric] bind_nonempty join_return'
18                 cong: subprob_algebra_cong)
21 lemma (in prob_space) distr_const[simp]:
22   "c \<in> space N \<Longrightarrow> distr M N (\<lambda>x. c) = return N c"
23   by (rule measure_eqI) (auto simp: emeasure_distr emeasure_space_1)
25 lemma (in finite_measure) countable_support:
26   "countable {x. measure M {x} \<noteq> 0}"
27 proof cases
28   assume "measure M (space M) = 0"
29   with bounded_measure measure_le_0_iff have "{x. measure M {x} \<noteq> 0} = {}"
30     by auto
31   then show ?thesis
32     by simp
33 next
34   let ?M = "measure M (space M)" and ?m = "\<lambda>x. measure M {x}"
35   assume "?M \<noteq> 0"
36   then have *: "{x. ?m x \<noteq> 0} = (\<Union>n. {x. ?M / Suc n < ?m x})"
37     using reals_Archimedean[of "?m x / ?M" for x]
38     by (auto simp: field_simps not_le[symmetric] measure_nonneg divide_le_0_iff measure_le_0_iff)
39   have **: "\<And>n. finite {x. ?M / Suc n < ?m x}"
40   proof (rule ccontr)
41     fix n assume "infinite {x. ?M / Suc n < ?m x}" (is "infinite ?X")
42     then obtain X where "finite X" "card X = Suc (Suc n)" "X \<subseteq> ?X"
43       by (metis infinite_arbitrarily_large)
44     from this(3) have *: "\<And>x. x \<in> X \<Longrightarrow> ?M / Suc n \<le> ?m x"
45       by auto
46     { fix x assume "x \<in> X"
47       from `?M \<noteq> 0` *[OF this] have "?m x \<noteq> 0" by (auto simp: field_simps measure_le_0_iff)
48       then have "{x} \<in> sets M" by (auto dest: measure_notin_sets) }
49     note singleton_sets = this
50     have "?M < (\<Sum>x\<in>X. ?M / Suc n)"
51       using `?M \<noteq> 0`
52       by (simp add: `card X = Suc (Suc n)` real_eq_of_nat[symmetric] real_of_nat_Suc field_simps less_le measure_nonneg)
53     also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"
54       by (rule setsum_mono) fact
55     also have "\<dots> = measure M (\<Union>x\<in>X. {x})"
56       using singleton_sets `finite X`
57       by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def)
58     finally have "?M < measure M (\<Union>x\<in>X. {x})" .
59     moreover have "measure M (\<Union>x\<in>X. {x}) \<le> ?M"
60       using singleton_sets[THEN sets.sets_into_space] by (intro finite_measure_mono) auto
61     ultimately show False by simp
62   qed
63   show ?thesis
64     unfolding * by (intro countable_UN countableI_type countable_finite[OF **])
65 qed
67 lemma (in finite_measure) AE_support_countable:
68   assumes [simp]: "sets M = UNIV"
69   shows "(AE x in M. measure M {x} \<noteq> 0) \<longleftrightarrow> (\<exists>S. countable S \<and> (AE x in M. x \<in> S))"
70 proof
71   assume "\<exists>S. countable S \<and> (AE x in M. x \<in> S)"
72   then obtain S where S[intro]: "countable S" and ae: "AE x in M. x \<in> S"
73     by auto
74   then have "emeasure M (\<Union>x\<in>{x\<in>S. emeasure M {x} \<noteq> 0}. {x}) =
75     (\<integral>\<^sup>+ x. emeasure M {x} * indicator {x\<in>S. emeasure M {x} \<noteq> 0} x \<partial>count_space UNIV)"
76     by (subst emeasure_UN_countable)
77        (auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space)
78   also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} * indicator S x \<partial>count_space UNIV)"
79     by (auto intro!: nn_integral_cong split: split_indicator)
80   also have "\<dots> = emeasure M (\<Union>x\<in>S. {x})"
81     by (subst emeasure_UN_countable)
82        (auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space)
83   also have "\<dots> = emeasure M (space M)"
84     using ae by (intro emeasure_eq_AE) auto
85   finally have "emeasure M {x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0} = emeasure M (space M)"
86     by (simp add: emeasure_single_in_space cong: rev_conj_cong)
87   with finite_measure_compl[of "{x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0}"]
88   have "AE x in M. x \<in> S \<and> emeasure M {x} \<noteq> 0"
89     by (intro AE_I[OF order_refl]) (auto simp: emeasure_eq_measure set_diff_eq cong: conj_cong)
90   then show "AE x in M. measure M {x} \<noteq> 0"
91     by (auto simp: emeasure_eq_measure)
92 qed (auto intro!: exI[of _ "{x. measure M {x} \<noteq> 0}"] countable_support)
94 subsection {* PMF as measure *}
96 typedef 'a pmf = "{M :: 'a measure. prob_space M \<and> sets M = UNIV \<and> (AE x in M. measure M {x} \<noteq> 0)}"
97   morphisms measure_pmf Abs_pmf
98   by (intro exI[of _ "uniform_measure (count_space UNIV) {undefined}"])
99      (auto intro!: prob_space_uniform_measure AE_uniform_measureI)
101 declare [[coercion measure_pmf]]
103 lemma prob_space_measure_pmf: "prob_space (measure_pmf p)"
104   using pmf.measure_pmf[of p] by auto
106 interpretation measure_pmf!: prob_space "measure_pmf M" for M
107   by (rule prob_space_measure_pmf)
109 interpretation measure_pmf!: subprob_space "measure_pmf M" for M
110   by (rule prob_space_imp_subprob_space) unfold_locales
112 lemma subprob_space_measure_pmf: "subprob_space (measure_pmf x)"
113   by unfold_locales
115 locale pmf_as_measure
116 begin
118 setup_lifting type_definition_pmf
120 end
122 context
123 begin
125 interpretation pmf_as_measure .
127 lift_definition pmf :: "'a pmf \<Rightarrow> 'a \<Rightarrow> real" is "\<lambda>M x. measure M {x}" .
129 lift_definition set_pmf :: "'a pmf \<Rightarrow> 'a set" is "\<lambda>M. {x. measure M {x} \<noteq> 0}" .
131 lift_definition map_pmf :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf" is
132   "\<lambda>f M. distr M (count_space UNIV) f"
133 proof safe
134   fix M and f :: "'a \<Rightarrow> 'b"
135   let ?D = "distr M (count_space UNIV) f"
136   assume "prob_space M" and [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
137   interpret prob_space M by fact
138   from ae have "AE x in M. measure M (f -` {f x}) \<noteq> 0"
139   proof eventually_elim
140     fix x
141     have "measure M {x} \<le> measure M (f -` {f x})"
142       by (intro finite_measure_mono) auto
143     then show "measure M {x} \<noteq> 0 \<Longrightarrow> measure M (f -` {f x}) \<noteq> 0"
144       using measure_nonneg[of M "{x}"] by auto
145   qed
146   then show "AE x in ?D. measure ?D {x} \<noteq> 0"
147     by (simp add: AE_distr_iff measure_distr measurable_def)
148 qed (auto simp: measurable_def prob_space.prob_space_distr)
150 declare [[coercion set_pmf]]
152 lemma countable_set_pmf [simp]: "countable (set_pmf p)"
153   by transfer (metis prob_space.finite_measure finite_measure.countable_support)
155 lemma sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV"
156   by transfer metis
158 lemma sets_measure_pmf_count_space[measurable_cong]:
159   "sets (measure_pmf M) = sets (count_space UNIV)"
160   by simp
162 lemma space_measure_pmf[simp]: "space (measure_pmf p) = UNIV"
163   using sets_eq_imp_space_eq[of "measure_pmf p" "count_space UNIV"] by simp
165 lemma measure_pmf_in_subprob_algebra[measurable (raw)]: "measure_pmf x \<in> space (subprob_algebra (count_space UNIV))"
166   by (simp add: space_subprob_algebra subprob_space_measure_pmf)
168 lemma measurable_pmf_measure1[simp]: "measurable (M :: 'a pmf) N = UNIV \<rightarrow> space N"
169   by (auto simp: measurable_def)
171 lemma measurable_pmf_measure2[simp]: "measurable N (M :: 'a pmf) = measurable N (count_space UNIV)"
172   by (intro measurable_cong_sets) simp_all
174 lemma pmf_positive: "x \<in> set_pmf p \<Longrightarrow> 0 < pmf p x"
175   by transfer (simp add: less_le measure_nonneg)
177 lemma pmf_nonneg: "0 \<le> pmf p x"
178   by transfer (simp add: measure_nonneg)
180 lemma pmf_le_1: "pmf p x \<le> 1"
183 lemma emeasure_pmf_single:
184   fixes M :: "'a pmf"
185   shows "emeasure M {x} = pmf M x"
186   by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
188 lemma AE_measure_pmf: "AE x in (M::'a pmf). x \<in> M"
189   by transfer simp
191 lemma emeasure_pmf_single_eq_zero_iff:
192   fixes M :: "'a pmf"
193   shows "emeasure M {y} = 0 \<longleftrightarrow> y \<notin> M"
194   by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
196 lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \<longleftrightarrow> (\<forall>y\<in>M. P y)"
197 proof -
198   { fix y assume y: "y \<in> M" and P: "AE x in M. P x" "\<not> P y"
199     with P have "AE x in M. x \<noteq> y"
200       by auto
201     with y have False
202       by (simp add: emeasure_pmf_single_eq_zero_iff AE_iff_measurable[OF _ refl]) }
203   then show ?thesis
204     using AE_measure_pmf[of M] by auto
205 qed
207 lemma set_pmf_not_empty: "set_pmf M \<noteq> {}"
208   using AE_measure_pmf[of M] by (intro notI) simp
210 lemma set_pmf_iff: "x \<in> set_pmf M \<longleftrightarrow> pmf M x \<noteq> 0"
211   by transfer simp
213 lemma emeasure_measure_pmf_finite: "finite S \<Longrightarrow> emeasure (measure_pmf M) S = (\<Sum>s\<in>S. pmf M s)"
214   by (subst emeasure_eq_setsum_singleton) (auto simp: emeasure_pmf_single)
216 lemma measure_measure_pmf_finite: "finite S \<Longrightarrow> measure (measure_pmf M) S = setsum (pmf M) S"
217 using emeasure_measure_pmf_finite[of S M]
220 lemma nn_integral_measure_pmf_support:
221   fixes f :: "'a \<Rightarrow> ereal"
222   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"
223   shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>A. f x * pmf M x)"
224 proof -
225   have "(\<integral>\<^sup>+x. f x \<partial>M) = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)"
226     using nn by (intro nn_integral_cong_AE) (auto simp: AE_measure_pmf_iff split: split_indicator)
227   also have "\<dots> = (\<Sum>x\<in>A. f x * emeasure M {x})"
228     using assms by (intro nn_integral_indicator_finite) auto
229   finally show ?thesis
231 qed
233 lemma nn_integral_measure_pmf_finite:
234   fixes f :: "'a \<Rightarrow> ereal"
235   assumes f: "finite (set_pmf M)" and nn: "\<And>x. x \<in> set_pmf M \<Longrightarrow> 0 \<le> f x"
236   shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>set_pmf M. f x * pmf M x)"
237   using assms by (intro nn_integral_measure_pmf_support) auto
238 lemma integrable_measure_pmf_finite:
239   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
240   shows "finite (set_pmf M) \<Longrightarrow> integrable M f"
241   by (auto intro!: integrableI_bounded simp: nn_integral_measure_pmf_finite)
243 lemma integral_measure_pmf:
244   assumes [simp]: "finite A" and "\<And>a. a \<in> set_pmf M \<Longrightarrow> f a \<noteq> 0 \<Longrightarrow> a \<in> A"
245   shows "(\<integral>x. f x \<partial>measure_pmf M) = (\<Sum>a\<in>A. f a * pmf M a)"
246 proof -
247   have "(\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x * indicator A x \<partial>measure_pmf M)"
248     using assms(2) by (intro integral_cong_AE) (auto split: split_indicator simp: AE_measure_pmf_iff)
249   also have "\<dots> = (\<Sum>a\<in>A. f a * pmf M a)"
250     by (subst integral_indicator_finite_real) (auto simp: measure_def emeasure_measure_pmf_finite)
251   finally show ?thesis .
252 qed
254 lemma integrable_pmf: "integrable (count_space X) (pmf M)"
255 proof -
256   have " (\<integral>\<^sup>+ x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+ x. pmf M x \<partial>count_space (M \<inter> X))"
257     by (auto simp add: nn_integral_count_space_indicator set_pmf_iff intro!: nn_integral_cong split: split_indicator)
258   then have "integrable (count_space X) (pmf M) = integrable (count_space (M \<inter> X)) (pmf M)"
259     by (simp add: integrable_iff_bounded pmf_nonneg)
260   then show ?thesis
261     by (simp add: pmf.rep_eq measure_pmf.integrable_measure disjoint_family_on_def)
262 qed
264 lemma integral_pmf: "(\<integral>x. pmf M x \<partial>count_space X) = measure M X"
265 proof -
266   have "(\<integral>x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+x. pmf M x \<partial>count_space X)"
267     by (simp add: pmf_nonneg integrable_pmf nn_integral_eq_integral)
268   also have "\<dots> = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space (X \<inter> M))"
269     by (auto intro!: nn_integral_cong_AE split: split_indicator
270              simp: pmf.rep_eq measure_pmf.emeasure_eq_measure nn_integral_count_space_indicator
271                    AE_count_space set_pmf_iff)
272   also have "\<dots> = emeasure M (X \<inter> M)"
273     by (rule emeasure_countable_singleton[symmetric]) (auto intro: countable_set_pmf)
274   also have "\<dots> = emeasure M X"
275     by (auto intro!: emeasure_eq_AE simp: AE_measure_pmf_iff)
276   finally show ?thesis
278 qed
280 lemma integral_pmf_restrict:
281   "(f::'a \<Rightarrow> 'b::{banach, second_countable_topology}) \<in> borel_measurable (count_space UNIV) \<Longrightarrow>
282     (\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x \<partial>restrict_space M M)"
283   by (auto intro!: integral_cong_AE simp add: integral_restrict_space AE_measure_pmf_iff)
285 lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1"
286 proof -
287   have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)"
288     by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf)
289   then show ?thesis
290     using measure_pmf.emeasure_space_1 by simp
291 qed
293 lemma in_null_sets_measure_pmfI:
294   "A \<inter> set_pmf p = {} \<Longrightarrow> A \<in> null_sets (measure_pmf p)"
295 using emeasure_eq_0_AE[where ?P="\<lambda>x. x \<in> A" and M="measure_pmf p"]
296 by(auto simp add: null_sets_def AE_measure_pmf_iff)
298 lemma map_pmf_id[simp]: "map_pmf id = id"
299   by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI)
301 lemma map_pmf_ident[simp]: "map_pmf (\<lambda>x. x) = (\<lambda>x. x)"
302   using map_pmf_id unfolding id_def .
304 lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g"
305   by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def)
307 lemma map_pmf_comp: "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M"
308   using map_pmf_compose[of f g] by (simp add: comp_def)
310 lemma map_pmf_cong:
311   assumes "p = q"
312   shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> map_pmf f p = map_pmf g q"
313   unfolding `p = q`[symmetric] measure_pmf_inject[symmetric] map_pmf.rep_eq
314   by (auto simp add: emeasure_distr AE_measure_pmf_iff intro!: emeasure_eq_AE measure_eqI)
316 lemma emeasure_map_pmf[simp]: "emeasure (map_pmf f M) X = emeasure M (f -` X)"
317   unfolding map_pmf.rep_eq by (subst emeasure_distr) auto
319 lemma nn_integral_map_pmf[simp]: "(\<integral>\<^sup>+x. f x \<partial>map_pmf g M) = (\<integral>\<^sup>+x. f (g x) \<partial>M)"
320   unfolding map_pmf.rep_eq by (intro nn_integral_distr) auto
322 lemma ereal_pmf_map: "pmf (map_pmf f p) x = (\<integral>\<^sup>+ y. indicator (f -` {x}) y \<partial>measure_pmf p)"
323 proof(transfer fixing: f x)
324   fix p :: "'b measure"
325   presume "prob_space p"
326   then interpret prob_space p .
327   presume "sets p = UNIV"
328   then show "ereal (measure (distr p (count_space UNIV) f) {x}) = integral\<^sup>N p (indicator (f -` {x}))"
329     by(simp add: measure_distr measurable_def emeasure_eq_measure)
330 qed simp_all
332 lemma pmf_set_map:
333   fixes f :: "'a \<Rightarrow> 'b"
334   shows "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
335 proof (rule, transfer, clarsimp simp add: measure_distr measurable_def)
336   fix f :: "'a \<Rightarrow> 'b" and M :: "'a measure"
337   assume "prob_space M" and ae: "AE x in M. measure M {x} \<noteq> 0" and [simp]: "sets M = UNIV"
338   interpret prob_space M by fact
339   show "{x. measure M (f -` {x}) \<noteq> 0} = f ` {x. measure M {x} \<noteq> 0}"
340   proof safe
341     fix x assume "measure M (f -` {x}) \<noteq> 0"
342     moreover have "measure M (f -` {x}) = measure M {y. f y = x \<and> measure M {y} \<noteq> 0}"
343       using ae by (intro finite_measure_eq_AE) auto
344     ultimately have "{y. f y = x \<and> measure M {y} \<noteq> 0} \<noteq> {}"
345       by (metis measure_empty)
346     then show "x \<in> f ` {x. measure M {x} \<noteq> 0}"
347       by auto
348   next
349     fix x assume "measure M {x} \<noteq> 0"
350     then have "0 < measure M {x}"
351       using measure_nonneg[of M "{x}"] by auto
352     also have "measure M {x} \<le> measure M (f -` {f x})"
353       by (intro finite_measure_mono) auto
354     finally show "measure M (f -` {f x}) = 0 \<Longrightarrow> False"
355       by simp
356   qed
357 qed
359 lemma set_map_pmf: "set_pmf (map_pmf f M) = f`set_pmf M"
360   using pmf_set_map[of f] by (auto simp: comp_def fun_eq_iff)
362 lemma nn_integral_pmf: "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = emeasure (measure_pmf p) A"
363 proof -
364   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))"
365     by(auto simp add: nn_integral_count_space_indicator indicator_def set_pmf_iff intro: nn_integral_cong)
366   also have "\<dots> = emeasure (measure_pmf p) (\<Union>x\<in>A \<inter> set_pmf p. {x})"
367     by(subst emeasure_UN_countable)(auto simp add: emeasure_pmf_single disjoint_family_on_def)
368   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})"
369     by(rule emeasure_Un_null_set[symmetric])(auto intro: in_null_sets_measure_pmfI)
370   also have "\<dots> = emeasure (measure_pmf p) A"
371     by(auto intro: arg_cong2[where f=emeasure])
372   finally show ?thesis .
373 qed
375 subsection {* PMFs as function *}
377 context
378   fixes f :: "'a \<Rightarrow> real"
379   assumes nonneg: "\<And>x. 0 \<le> f x"
380   assumes prob: "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
381 begin
383 lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ereal \<circ> f)"
384 proof (intro conjI)
385   have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
386     by (simp split: split_indicator)
387   show "AE x in density (count_space UNIV) (ereal \<circ> f).
388     measure (density (count_space UNIV) (ereal \<circ> f)) {x} \<noteq> 0"
389     by (simp add: AE_density nonneg measure_def emeasure_density max_def)
390   show "prob_space (density (count_space UNIV) (ereal \<circ> f))"
391     by default (simp add: emeasure_density prob)
392 qed simp
394 lemma pmf_embed_pmf: "pmf embed_pmf x = f x"
395 proof transfer
396   have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
397     by (simp split: split_indicator)
398   fix x show "measure (density (count_space UNIV) (ereal \<circ> f)) {x} = f x"
399     by transfer (simp add: measure_def emeasure_density nonneg max_def)
400 qed
402 end
404 lemma embed_pmf_transfer:
405   "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"
406   by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer)
408 lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)"
409 proof (transfer, elim conjE)
410   fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
411   assume "prob_space M" then interpret prob_space M .
412   show "M = density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))"
413   proof (rule measure_eqI)
414     fix A :: "'a set"
415     have "(\<integral>\<^sup>+ x. ereal (measure M {x}) * indicator A x \<partial>count_space UNIV) =
416       (\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)"
417       by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator)
418     also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))"
419       by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space)
420     also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})"
421       by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support)
422          (auto simp: disjoint_family_on_def)
423     also have "\<dots> = emeasure M A"
424       using ae by (intro emeasure_eq_AE) auto
425     finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))) A"
426       using emeasure_space_1 by (simp add: emeasure_density)
427   qed simp
428 qed
430 lemma td_pmf_embed_pmf:
431   "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}"
432   unfolding type_definition_def
433 proof safe
434   fix p :: "'a pmf"
435   have "(\<integral>\<^sup>+ x. 1 \<partial>measure_pmf p) = 1"
436     using measure_pmf.emeasure_space_1[of p] by simp
437   then show *: "(\<integral>\<^sup>+ x. ereal (pmf p x) \<partial>count_space UNIV) = 1"
438     by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const)
440   show "embed_pmf (pmf p) = p"
441     by (intro measure_pmf_inject[THEN iffD1])
442        (simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def)
443 next
444   fix f :: "'a \<Rightarrow> real" assume "\<forall>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
445   then show "pmf (embed_pmf f) = f"
446     by (auto intro!: pmf_embed_pmf)
447 qed (rule pmf_nonneg)
449 end
451 locale pmf_as_function
452 begin
454 setup_lifting td_pmf_embed_pmf
456 lemma set_pmf_transfer[transfer_rule]:
457   assumes "bi_total A"
458   shows "rel_fun (pcr_pmf A) (rel_set A) (\<lambda>f. {x. f x \<noteq> 0}) set_pmf"
459   using `bi_total A`
460   by (auto simp: pcr_pmf_def cr_pmf_def rel_fun_def rel_set_def bi_total_def Bex_def set_pmf_iff)
461      metis+
463 end
465 context
466 begin
468 interpretation pmf_as_function .
470 lemma pmf_eqI: "(\<And>i. pmf M i = pmf N i) \<Longrightarrow> M = N"
471   by transfer auto
473 lemma pmf_eq_iff: "M = N \<longleftrightarrow> (\<forall>i. pmf M i = pmf N i)"
474   by (auto intro: pmf_eqI)
476 end
478 context
479 begin
481 interpretation pmf_as_function .
483 subsubsection \<open> Bernoulli Distribution \<close>
485 lift_definition bernoulli_pmf :: "real \<Rightarrow> bool pmf" is
486   "\<lambda>p b. ((\<lambda>p. if b then p else 1 - p) \<circ> min 1 \<circ> max 0) p"
487   by (auto simp: nn_integral_count_space_finite[where A="{False, True}"] UNIV_bool
488            split: split_max split_min)
490 lemma pmf_bernoulli_True[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) True = p"
491   by transfer simp
493 lemma pmf_bernoulli_False[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) False = 1 - p"
494   by transfer simp
496 lemma set_pmf_bernoulli: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (bernoulli_pmf p) = UNIV"
497   by (auto simp add: set_pmf_iff UNIV_bool)
499 lemma nn_integral_bernoulli_pmf[simp]:
500   assumes [simp]: "0 \<le> p" "p \<le> 1" "\<And>x. 0 \<le> f x"
501   shows "(\<integral>\<^sup>+x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
502   by (subst nn_integral_measure_pmf_support[of UNIV])
503      (auto simp: UNIV_bool field_simps)
505 lemma integral_bernoulli_pmf[simp]:
506   assumes [simp]: "0 \<le> p" "p \<le> 1"
507   shows "(\<integral>x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
508   by (subst integral_measure_pmf[of UNIV]) (auto simp: UNIV_bool)
510 subsubsection \<open> Geometric Distribution \<close>
512 lift_definition geometric_pmf :: "nat pmf" is "\<lambda>n. 1 / 2^Suc n"
513 proof
514   note geometric_sums[of "1 / 2"]
515   note sums_mult[OF this, of "1 / 2"]
516   from sums_suminf_ereal[OF this]
517   show "(\<integral>\<^sup>+ x. ereal (1 / 2 ^ Suc x) \<partial>count_space UNIV) = 1"
518     by (simp add: nn_integral_count_space_nat field_simps)
519 qed simp
521 lemma pmf_geometric[simp]: "pmf geometric_pmf n = 1 / 2^Suc n"
522   by transfer rule
524 lemma set_pmf_geometric[simp]: "set_pmf geometric_pmf = UNIV"
525   by (auto simp: set_pmf_iff)
527 subsubsection \<open> Uniform Multiset Distribution \<close>
529 context
530   fixes M :: "'a multiset" assumes M_not_empty: "M \<noteq> {#}"
531 begin
533 lift_definition pmf_of_multiset :: "'a pmf" is "\<lambda>x. count M x / size M"
534 proof
535   show "(\<integral>\<^sup>+ x. ereal (real (count M x) / real (size M)) \<partial>count_space UNIV) = 1"
536     using M_not_empty
537     by (simp add: zero_less_divide_iff nn_integral_count_space nonempty_has_size
538                   setsum_divide_distrib[symmetric])
539        (auto simp: size_multiset_overloaded_eq intro!: setsum.cong)
540 qed simp
542 lemma pmf_of_multiset[simp]: "pmf pmf_of_multiset x = count M x / size M"
543   by transfer rule
545 lemma set_pmf_of_multiset[simp]: "set_pmf pmf_of_multiset = set_of M"
546   by (auto simp: set_pmf_iff)
548 end
550 subsubsection \<open> Uniform Distribution \<close>
552 context
553   fixes S :: "'a set" assumes S_not_empty: "S \<noteq> {}" and S_finite: "finite S"
554 begin
556 lift_definition pmf_of_set :: "'a pmf" is "\<lambda>x. indicator S x / card S"
557 proof
558   show "(\<integral>\<^sup>+ x. ereal (indicator S x / real (card S)) \<partial>count_space UNIV) = 1"
559     using S_not_empty S_finite by (subst nn_integral_count_space'[of S]) auto
560 qed simp
562 lemma pmf_of_set[simp]: "pmf pmf_of_set x = indicator S x / card S"
563   by transfer rule
565 lemma set_pmf_of_set[simp]: "set_pmf pmf_of_set = S"
566   using S_finite S_not_empty by (auto simp: set_pmf_iff)
568 lemma emeasure_pmf_of_set[simp]: "emeasure pmf_of_set S = 1"
569   by (rule measure_pmf.emeasure_eq_1_AE) (auto simp: AE_measure_pmf_iff)
571 end
573 subsubsection \<open> Poisson Distribution \<close>
575 context
576   fixes rate :: real assumes rate_pos: "0 < rate"
577 begin
579 lift_definition poisson_pmf :: "nat pmf" is "\<lambda>k. rate ^ k / fact k * exp (-rate)"
580 proof
581   (* Proof by Manuel Eberl *)
583   have summable: "summable (\<lambda>x::nat. rate ^ x / fact x)" using summable_exp
584     by (simp add: field_simps field_divide_inverse[symmetric])
585   have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x * exp (-rate) \<partial>count_space UNIV) =
586           exp (-rate) * (\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV)"
587     by (simp add: field_simps nn_integral_cmult[symmetric])
588   also from rate_pos have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV) = (\<Sum>x. rate ^ x / fact x)"
589     by (simp_all add: nn_integral_count_space_nat suminf_ereal summable suminf_ereal_finite)
590   also have "... = exp rate" unfolding exp_def
591     by (simp add: field_simps field_divide_inverse[symmetric] transfer_int_nat_factorial)
592   also have "ereal (exp (-rate)) * ereal (exp rate) = 1"
594   finally show "(\<integral>\<^sup>+ x. ereal (rate ^ x / real (fact x) * exp (- rate)) \<partial>count_space UNIV) = 1" .
595 qed (simp add: rate_pos[THEN less_imp_le])
597 lemma pmf_poisson[simp]: "pmf poisson_pmf k = rate ^ k / fact k * exp (-rate)"
598   by transfer rule
600 lemma set_pmf_poisson[simp]: "set_pmf poisson_pmf = UNIV"
601   using rate_pos by (auto simp: set_pmf_iff)
603 end
605 subsubsection \<open> Binomial Distribution \<close>
607 context
608   fixes n :: nat and p :: real assumes p_nonneg: "0 \<le> p" and p_le_1: "p \<le> 1"
609 begin
611 lift_definition binomial_pmf :: "nat pmf" is "\<lambda>k. (n choose k) * p^k * (1 - p)^(n - k)"
612 proof
613   have "(\<integral>\<^sup>+k. ereal (real (n choose k) * p ^ k * (1 - p) ^ (n - k)) \<partial>count_space UNIV) =
614     ereal (\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k))"
615     using p_le_1 p_nonneg by (subst nn_integral_count_space') auto
616   also have "(\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k)) = (p + (1 - p)) ^ n"
617     by (subst binomial_ring) (simp add: atLeast0AtMost real_of_nat_def)
618   finally show "(\<integral>\<^sup>+ x. ereal (real (n choose x) * p ^ x * (1 - p) ^ (n - x)) \<partial>count_space UNIV) = 1"
619     by simp
620 qed (insert p_nonneg p_le_1, simp)
622 lemma pmf_binomial[simp]: "pmf binomial_pmf k = (n choose k) * p^k * (1 - p)^(n - k)"
623   by transfer rule
625 lemma set_pmf_binomial_eq: "set_pmf binomial_pmf = (if p = 0 then {0} else if p = 1 then {n} else {.. n})"
626   using p_nonneg p_le_1 unfolding set_eq_iff set_pmf_iff pmf_binomial by (auto simp: set_pmf_iff)
628 end
630 end
632 lemma set_pmf_binomial_0[simp]: "set_pmf (binomial_pmf n 0) = {0}"
635 lemma set_pmf_binomial_1[simp]: "set_pmf (binomial_pmf n 1) = {n}"
638 lemma set_pmf_binomial[simp]: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (binomial_pmf n p) = {..n}"
641 subsection \<open> Monad Interpretation \<close>
643 lemma measurable_measure_pmf[measurable]:
644   "(\<lambda>x. measure_pmf (M x)) \<in> measurable (count_space UNIV) (subprob_algebra (count_space UNIV))"
645   by (auto simp: space_subprob_algebra intro!: prob_space_imp_subprob_space) unfold_locales
647 lemma bind_pmf_cong:
648   assumes "\<And>x. A x \<in> space (subprob_algebra N)" "\<And>x. B x \<in> space (subprob_algebra N)"
649   assumes "\<And>i. i \<in> set_pmf x \<Longrightarrow> A i = B i"
650   shows "bind (measure_pmf x) A = bind (measure_pmf x) B"
651 proof (rule measure_eqI)
652   show "sets (measure_pmf x \<guillemotright>= A) = sets (measure_pmf x \<guillemotright>= B)"
653     using assms by (subst (1 2) sets_bind) (auto simp: space_subprob_algebra)
654 next
655   fix X assume "X \<in> sets (measure_pmf x \<guillemotright>= A)"
656   then have X: "X \<in> sets N"
657     using assms by (subst (asm) sets_bind) (auto simp: space_subprob_algebra)
658   show "emeasure (measure_pmf x \<guillemotright>= A) X = emeasure (measure_pmf x \<guillemotright>= B) X"
659     using assms
660     by (subst (1 2) emeasure_bind[where N=N, OF _ _ X])
661        (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
662 qed
664 context
665 begin
667 interpretation pmf_as_measure .
669 lift_definition join_pmf :: "'a pmf pmf \<Rightarrow> 'a pmf" is "\<lambda>M. measure_pmf M \<guillemotright>= measure_pmf"
670 proof (intro conjI)
671   fix M :: "'a pmf pmf"
673   interpret bind: prob_space "measure_pmf M \<guillemotright>= measure_pmf"
674     apply (intro measure_pmf.prob_space_bind[where S="count_space UNIV"] AE_I2)
675     apply (auto intro!: subprob_space_measure_pmf simp: space_subprob_algebra)
676     apply unfold_locales
677     done
678   show "prob_space (measure_pmf M \<guillemotright>= measure_pmf)"
679     by intro_locales
680   show "sets (measure_pmf M \<guillemotright>= measure_pmf) = UNIV"
681     by (subst sets_bind) auto
682   have "AE x in measure_pmf M \<guillemotright>= measure_pmf. emeasure (measure_pmf M \<guillemotright>= measure_pmf) {x} \<noteq> 0"
683     by (auto simp: AE_bind[where B="count_space UNIV"] measure_pmf_in_subprob_algebra
684                    emeasure_bind[where N="count_space UNIV"] AE_measure_pmf_iff nn_integral_0_iff_AE
685                    measure_pmf.emeasure_eq_measure measure_le_0_iff set_pmf_iff pmf.rep_eq)
686   then show "AE x in measure_pmf M \<guillemotright>= measure_pmf. measure (measure_pmf M \<guillemotright>= measure_pmf) {x} \<noteq> 0"
687     unfolding bind.emeasure_eq_measure by simp
688 qed
690 lemma pmf_join: "pmf (join_pmf N) i = (\<integral>M. pmf M i \<partial>measure_pmf N)"
691 proof (transfer fixing: N i)
692   have N: "subprob_space (measure_pmf N)"
693     by (rule prob_space_imp_subprob_space) intro_locales
694   show "measure (measure_pmf N \<guillemotright>= measure_pmf) {i} = integral\<^sup>L (measure_pmf N) (\<lambda>M. measure M {i})"
695     using measurable_measure_pmf[of "\<lambda>x. x"]
696     by (intro subprob_space.measure_bind[where N="count_space UNIV", OF N]) auto
697 qed (auto simp: Transfer.Rel_def rel_fun_def cr_pmf_def)
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 definition "bind_pmf M f = join_pmf (map_pmf f M)"
733 lemma (in pmf_as_measure) bind_transfer[transfer_rule]:
734   "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"
735 proof (auto simp: pmf_as_measure.cr_pmf_def rel_fun_def bind_pmf_def join_pmf.rep_eq map_pmf.rep_eq)
736   fix M f and g :: "'a \<Rightarrow> 'b pmf" assume "\<forall>x. f x = measure_pmf (g x)"
737   then have f: "f = (\<lambda>x. measure_pmf (g x))"
738     by auto
739   show "measure_pmf M \<guillemotright>= f = distr (measure_pmf M) (count_space UNIV) g \<guillemotright>= measure_pmf"
740     unfolding f by (subst bind_distr[OF _ measurable_measure_pmf]) auto
741 qed
743 lemma pmf_bind: "pmf (bind_pmf N f) i = (\<integral>x. pmf (f x) i \<partial>measure_pmf N)"
744   by (auto intro!: integral_distr simp: bind_pmf_def pmf_join map_pmf.rep_eq)
746 lemma bind_return_pmf: "bind_pmf (return_pmf x) f = f x"
747   unfolding bind_pmf_def map_return_pmf join_return_pmf ..
749 lemma join_eq_bind_pmf: "join_pmf M = bind_pmf M id"
752 lemma bind_pmf_const[simp]: "bind_pmf M (\<lambda>x. c) = c"
753   unfolding bind_pmf_def map_pmf_const join_return_pmf ..
755 lemma set_bind_pmf: "set_pmf (bind_pmf M N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))"
756   apply (simp add: set_eq_iff set_pmf_iff pmf_bind)
757   apply (subst integral_nonneg_eq_0_iff_AE)
758   apply (auto simp: pmf_nonneg pmf_le_1 AE_measure_pmf_iff
759               intro!: measure_pmf.integrable_const_bound[where B=1])
760   done
762 lemma measurable_pair_restrict_pmf2:
763   assumes "countable A"
764   assumes "\<And>y. y \<in> A \<Longrightarrow> (\<lambda>x. f (x, y)) \<in> measurable M L"
765   shows "f \<in> measurable (M \<Otimes>\<^sub>M restrict_space (measure_pmf N) A) L"
766   apply (subst measurable_cong_sets)
767   apply (rule sets_pair_measure_cong sets_restrict_space_cong sets_measure_pmf_count_space refl)+
769   apply (subst split_eta[symmetric])
770   unfolding measurable_split_conv
771   apply (rule measurable_compose_countable'[OF _ measurable_snd `countable A`])
772   apply (rule measurable_compose[OF measurable_fst])
773   apply fact
774   done
776 lemma measurable_pair_restrict_pmf1:
777   assumes "countable A"
778   assumes "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N L"
779   shows "f \<in> measurable (restrict_space (measure_pmf M) A \<Otimes>\<^sub>M N) L"
780   apply (subst measurable_cong_sets)
781   apply (rule sets_pair_measure_cong sets_restrict_space_cong sets_measure_pmf_count_space refl)+
783   apply (subst split_eta[symmetric])
784   unfolding measurable_split_conv
785   apply (rule measurable_compose_countable'[OF _ measurable_fst `countable A`])
786   apply (rule measurable_compose[OF measurable_snd])
787   apply fact
788   done
790 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))"
791   unfolding pmf_eq_iff pmf_bind
792 proof
793   fix i
794   interpret B: prob_space "restrict_space B B"
795     by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
796        (auto simp: AE_measure_pmf_iff)
797   interpret A: prob_space "restrict_space A A"
798     by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
799        (auto simp: AE_measure_pmf_iff)
801   interpret AB: pair_prob_space "restrict_space A A" "restrict_space B B"
802     by unfold_locales
804   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)"
805     by (rule integral_cong) (auto intro!: integral_pmf_restrict)
806   also have "\<dots> = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>restrict_space A A)"
807     by (intro integral_pmf_restrict B.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
808               countable_set_pmf borel_measurable_count_space)
809   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>restrict_space B B)"
810     by (rule AB.Fubini_integral[symmetric])
811        (auto intro!: AB.integrable_const_bound[where B=1] measurable_pair_restrict_pmf2
812              simp: pmf_nonneg pmf_le_1 measurable_restrict_space1)
813   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>B)"
814     by (intro integral_pmf_restrict[symmetric] A.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>A \<partial>B)"
817     by (rule integral_cong) (auto intro!: integral_pmf_restrict[symmetric])
818   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)" .
819 qed
822 context
823 begin
825 interpretation pmf_as_measure .
827 lemma measure_pmf_bind: "measure_pmf (bind_pmf M f) = (measure_pmf M \<guillemotright>= (\<lambda>x. measure_pmf (f x)))"
828   by transfer simp
830 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)"
831   using measurable_measure_pmf[of N]
832   unfolding measure_pmf_bind
833   apply (subst (1 3) nn_integral_max_0[symmetric])
834   apply (intro nn_integral_bind[where B="count_space UNIV"])
835   apply auto
836   done
838 lemma emeasure_bind_pmf[simp]: "emeasure (bind_pmf M N) X = (\<integral>\<^sup>+x. emeasure (N x) X \<partial>M)"
839   using measurable_measure_pmf[of N]
840   unfolding measure_pmf_bind
841   by (subst emeasure_bind[where N="count_space UNIV"]) auto
843 lemma bind_return_pmf': "bind_pmf N return_pmf = N"
844 proof (transfer, clarify)
845   fix N :: "'a measure" assume "sets N = UNIV" then show "N \<guillemotright>= return (count_space UNIV) = N"
846     by (subst return_sets_cong[where N=N]) (simp_all add: bind_return')
847 qed
849 lemma bind_return_pmf'': "bind_pmf N (\<lambda>x. return_pmf (f x)) = map_pmf f N"
850 proof (transfer, clarify)
851   fix N :: "'b measure" and f :: "'b \<Rightarrow> 'a" assume "prob_space N" "sets N = UNIV"
852   then show "N \<guillemotright>= (\<lambda>x. return (count_space UNIV) (f x)) = distr N (count_space UNIV) f"
853     by (subst bind_return_distr[symmetric])
854        (auto simp: prob_space.not_empty measurable_def comp_def)
855 qed
857 lemma bind_assoc_pmf: "bind_pmf (bind_pmf A B) C = bind_pmf A (\<lambda>x. bind_pmf (B x) C)"
858   by transfer
859      (auto intro!: bind_assoc[where N="count_space UNIV" and R="count_space UNIV"]
860            simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space)
862 end
864 lemma map_join_pmf: "map_pmf f (join_pmf AA) = join_pmf (map_pmf (map_pmf f) AA)"
865   unfolding bind_pmf_def[symmetric]
866   unfolding bind_return_pmf''[symmetric] join_eq_bind_pmf bind_assoc_pmf
869 definition "pair_pmf A B = bind_pmf A (\<lambda>x. bind_pmf B (\<lambda>y. return_pmf (x, y)))"
871 lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b"
872   unfolding pair_pmf_def pmf_bind pmf_return
873   apply (subst integral_measure_pmf[where A="{b}"])
874   apply (auto simp: indicator_eq_0_iff)
875   apply (subst integral_measure_pmf[where A="{a}"])
876   apply (auto simp: indicator_eq_0_iff setsum_nonneg_eq_0_iff pmf_nonneg)
877   done
879 lemma set_pair_pmf: "set_pmf (pair_pmf A B) = set_pmf A \<times> set_pmf B"
880   unfolding pair_pmf_def set_bind_pmf set_return_pmf by auto
882 lemma measure_pmf_in_subprob_space[measurable (raw)]:
883   "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
884   by (simp add: space_subprob_algebra) intro_locales
886 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)"
887 proof -
888   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)"
889     by (subst nn_integral_max_0[symmetric])
890        (auto simp: AE_measure_pmf_iff set_pair_pmf intro!: nn_integral_cong_AE)
891   also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) * indicator (A \<times> B) (a, b) \<partial>B \<partial>A)"
893   also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) \<partial>B \<partial>A)"
894     by (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
895   finally show ?thesis
896     unfolding nn_integral_max_0 .
897 qed
899 lemma pair_map_pmf1: "pair_pmf (map_pmf f A) B = map_pmf (apfst f) (pair_pmf A B)"
900 proof (safe intro!: pmf_eqI)
901   fix a :: "'a" and b :: "'b"
902   have [simp]: "\<And>c d. indicator (apfst f -` {(a, b)}) (c, d) = indicator (f -` {a}) c * (indicator {b} d::ereal)"
903     by (auto split: split_indicator)
905   have "ereal (pmf (pair_pmf (map_pmf f A) B) (a, b)) =
906          ereal (pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b))"
907     unfolding pmf_pair ereal_pmf_map
908     by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_multc pmf_nonneg
909                   emeasure_map_pmf[symmetric] del: emeasure_map_pmf)
910   then show "pmf (pair_pmf (map_pmf f A) B) (a, b) = pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b)"
911     by simp
912 qed
914 lemma pair_map_pmf2: "pair_pmf A (map_pmf f B) = map_pmf (apsnd f) (pair_pmf A B)"
915 proof (safe intro!: pmf_eqI)
916   fix a :: "'a" and b :: "'b"
917   have [simp]: "\<And>c d. indicator (apsnd f -` {(a, b)}) (c, d) = indicator {a} c * (indicator (f -` {b}) d::ereal)"
918     by (auto split: split_indicator)
920   have "ereal (pmf (pair_pmf A (map_pmf f B)) (a, b)) =
921          ereal (pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b))"
922     unfolding pmf_pair ereal_pmf_map
923     by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_cmult nn_integral_multc pmf_nonneg
924                   emeasure_map_pmf[symmetric] del: emeasure_map_pmf)
925   then show "pmf (pair_pmf A (map_pmf f B)) (a, b) = pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b)"
926     by simp
927 qed
929 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)"
930   by (simp add: pair_map_pmf2 pair_map_pmf1 map_pmf_comp split_beta')
932 lemma bind_pair_pmf:
933   assumes M[measurable]: "M \<in> measurable (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) (subprob_algebra N)"
934   shows "measure_pmf (pair_pmf A B) \<guillemotright>= M = (measure_pmf A \<guillemotright>= (\<lambda>x. measure_pmf B \<guillemotright>= (\<lambda>y. M (x, y))))"
935     (is "?L = ?R")
936 proof (rule measure_eqI)
937   have M'[measurable]: "M \<in> measurable (pair_pmf A B) (subprob_algebra N)"
938     using M[THEN measurable_space] by (simp_all add: space_pair_measure)
940   note measurable_bind[where N="count_space UNIV", measurable]
941   note measure_pmf_in_subprob_space[simp]
943   have sets_eq_N: "sets ?L = N"
944     by (subst sets_bind[OF sets_kernel[OF M']]) auto
945   show "sets ?L = sets ?R"
946     using measurable_space[OF M]
947     by (simp add: sets_eq_N space_pair_measure space_subprob_algebra)
948   fix X assume "X \<in> sets ?L"
949   then have X[measurable]: "X \<in> sets N"
950     unfolding sets_eq_N .
951   then show "emeasure ?L X = emeasure ?R X"
952     apply (simp add: emeasure_bind[OF _ M' X])
953     apply (simp add: nn_integral_bind[where B="count_space UNIV"] pair_pmf_def measure_pmf_bind[of A]
954       nn_integral_measure_pmf_finite set_return_pmf emeasure_nonneg pmf_return one_ereal_def[symmetric])
955     apply (subst emeasure_bind[OF _ _ X])
956     apply measurable
957     apply (subst emeasure_bind[OF _ _ X])
958     apply measurable
959     done
960 qed
962 lemma join_map_return_pmf: "join_pmf (map_pmf return_pmf A) = A"
963   unfolding bind_pmf_def[symmetric] bind_return_pmf' ..
965 lemma map_fst_pair_pmf: "map_pmf fst (pair_pmf A B) = A"
966   by (simp add: pair_pmf_def bind_return_pmf''[symmetric] bind_assoc_pmf bind_return_pmf bind_return_pmf')
968 lemma map_snd_pair_pmf: "map_pmf snd (pair_pmf A B) = B"
969   by (simp add: pair_pmf_def bind_return_pmf''[symmetric] bind_assoc_pmf bind_return_pmf bind_return_pmf')
971 lemma nn_integral_pmf':
972   "inj_on f A \<Longrightarrow> (\<integral>\<^sup>+x. pmf p (f x) \<partial>count_space A) = emeasure p (f ` A)"
973   by (subst nn_integral_bij_count_space[where g=f and B="f`A"])
974      (auto simp: bij_betw_def nn_integral_pmf)
976 lemma pmf_le_0_iff[simp]: "pmf M p \<le> 0 \<longleftrightarrow> pmf M p = 0"
977   using pmf_nonneg[of M p] by simp
979 lemma min_pmf_0[simp]: "min (pmf M p) 0 = 0" "min 0 (pmf M p) = 0"
980   using pmf_nonneg[of M p] by simp_all
982 lemma pmf_eq_0_set_pmf: "pmf M p = 0 \<longleftrightarrow> p \<notin> set_pmf M"
983   unfolding set_pmf_iff by simp
985 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"
986   by (auto simp: pmf.rep_eq map_pmf.rep_eq measure_distr AE_measure_pmf_iff inj_onD
987            intro!: measure_pmf.finite_measure_eq_AE)
989 inductive rel_pmf :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf \<Rightarrow> bool"
990 for R p q
991 where
992   "\<lbrakk> \<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y;
993      map_pmf fst pq = p; map_pmf snd pq = q \<rbrakk>
994   \<Longrightarrow> rel_pmf R p q"
996 bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: rel_pmf
997 proof -
998   show "map_pmf id = id" by (rule map_pmf_id)
999   show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose)
1000   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"
1001     by (intro map_pmf_cong refl)
1003   show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
1004     by (rule pmf_set_map)
1006   { fix p :: "'s pmf"
1007     have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq"
1008       by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"])
1009          (auto intro: countable_set_pmf)
1010     also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq"
1011       by (metis Field_natLeq card_of_least natLeq_Well_order)
1012     finally show "(card_of (set_pmf p), natLeq) \<in> ordLeq" . }
1014   show "\<And>R. rel_pmf R =
1015          (BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf fst))\<inverse>\<inverse> OO
1016          BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf snd)"
1017      by (auto simp add: fun_eq_iff BNF_Def.Grp_def OO_def rel_pmf.simps)
1019   { fix p :: "'a pmf" and f :: "'a \<Rightarrow> 'b" and g x
1020     assume p: "\<And>z. z \<in> set_pmf p \<Longrightarrow> f z = g z"
1021       and x: "x \<in> set_pmf p"
1022     thus "f x = g x" by simp }
1024   fix R :: "'a => 'b \<Rightarrow> bool" and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
1025   { fix p q r
1026     assume pq: "rel_pmf R p q"
1027       and qr:"rel_pmf S q r"
1028     from pq obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
1029       and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto
1030     from qr obtain qr where qr: "\<And>y z. (y, z) \<in> set_pmf qr \<Longrightarrow> S y z"
1031       and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto
1033     note pmf_nonneg[intro, simp]
1035     def A \<equiv> "\<lambda>y. {x. (x, y) \<in> set_pmf pq}"
1036     then have "\<And>y. A y \<subseteq> set_pmf p" by (auto simp add: p set_map_pmf intro: rev_image_eqI)
1037     then have [simp]: "\<And>y. countable (A y)" by (rule countable_subset) simp
1038     have A: "\<And>x y. (x, y) \<in> set_pmf pq \<longleftrightarrow> x \<in> A y"
1041     let ?P = "\<lambda>y. to_nat_on (A y)"
1042     def pp \<equiv> "map_pmf (\<lambda>(x, y). (y, ?P y x)) pq"
1043     let ?pp = "\<lambda>y x. pmf pp (y, x)"
1044     { fix x y have "x \<in> A y \<Longrightarrow> ?pp y (?P y x) = pmf pq (x, y)"
1045         unfolding pp_def
1046         by (intro pmf_map_inj[of "\<lambda>(x, y). (y, ?P y x)" pq "(x, y)", simplified])
1047            (auto simp: inj_on_def A) }
1048     note pmf_pp = this
1049     have pp_0: "\<And>y x. pmf q y = 0 \<Longrightarrow> ?pp y x = 0"
1050     proof(erule contrapos_pp)
1051       fix y x
1052       assume "?pp y x \<noteq> 0"
1053       hence "(y, x) \<in> set_pmf pp" by(simp add: set_pmf_iff)
1054       hence "y \<in> set_pmf q" by(auto simp add: pp_def q set_map_pmf intro: rev_image_eqI)
1055       thus "pmf q y \<noteq> 0" by(simp add: set_pmf_iff)
1056     qed
1058     def B \<equiv> "\<lambda>y. {z. (y, z) \<in> set_pmf qr}"
1059     then have "\<And>y. B y \<subseteq> set_pmf r" by (auto simp add: r set_map_pmf intro: rev_image_eqI)
1060     then have [simp]: "\<And>y. countable (B y)" by (rule countable_subset) simp
1061     have B: "\<And>y z. (y, z) \<in> set_pmf qr \<longleftrightarrow> z \<in> B y"
1064     let ?R = "\<lambda>y. to_nat_on (B y)"
1065     def rr \<equiv> "map_pmf (\<lambda>(y, z). (y, ?R y z)) qr"
1066     let ?rr = "\<lambda>y z. pmf rr (y, z)"
1067     { fix y z have "z \<in> B y \<Longrightarrow> ?rr y (?R y z) = pmf qr (y, z)"
1068         unfolding rr_def
1069         by (intro pmf_map_inj[of "\<lambda>(y, z). (y, ?R y z)" qr "(y, z)", simplified])
1070            (auto simp: inj_on_def B) }
1071     note pmf_rr = this
1072     have rr_0: "\<And>y z. pmf q y = 0 \<Longrightarrow> ?rr y z = 0"
1073     proof(erule contrapos_pp)
1074       fix y z
1075       assume "?rr y z \<noteq> 0"
1076       hence "(y, z) \<in> set_pmf rr" by(simp add: set_pmf_iff)
1077       hence "y \<in> set_pmf q" by(auto simp add: rr_def q' set_map_pmf intro: rev_image_eqI)
1078       thus "pmf q y \<noteq> 0" by(simp add: set_pmf_iff)
1079     qed
1081     have nn_integral_pp2: "\<And>y. (\<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV) = pmf q y"
1082       by (simp add: nn_integral_pmf' inj_on_def pp_def q)
1083          (auto simp add: ereal_pmf_map intro!: arg_cong2[where f=emeasure])
1084     have nn_integral_rr1: "\<And>y. (\<integral>\<^sup>+ x. ?rr y x \<partial>count_space UNIV) = pmf q y"
1085       by (simp add: nn_integral_pmf' inj_on_def rr_def q')
1086          (auto simp add: ereal_pmf_map intro!: arg_cong2[where f=emeasure])
1087     have eq: "\<And>y. (\<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV) = (\<integral>\<^sup>+ z. ?rr y z \<partial>count_space UNIV)"
1090     def assign \<equiv> "\<lambda>y x z. ?pp y x * ?rr y z / pmf q y"
1091     have assign_nonneg [simp]: "\<And>y x z. 0 \<le> assign y x z" by(simp add: assign_def)
1092     have assign_eq_0_outside: "\<And>y x z. \<lbrakk> ?pp y x = 0 \<or> ?rr y z = 0 \<rbrakk> \<Longrightarrow> assign y x z = 0"
1094     have nn_integral_assign1: "\<And>y z. (\<integral>\<^sup>+ x. assign y x z \<partial>count_space UNIV) = ?rr y z"
1095     proof -
1096       fix y z
1097       have "(\<integral>\<^sup>+ x. assign y x z \<partial>count_space UNIV) =
1098             (\<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV) * (?rr y z / pmf q y)"
1099         by(simp add: assign_def nn_integral_multc times_ereal.simps(1)[symmetric] divide_real_def mult.assoc del: times_ereal.simps(1))
1100       also have "\<dots> = ?rr y z" by(simp add: rr_0 nn_integral_pp2)
1101       finally show "?thesis y z" .
1102     qed
1103     have nn_integral_assign2: "\<And>y x. (\<integral>\<^sup>+ z. assign y x z \<partial>count_space UNIV) = ?pp y x"
1104     proof -
1105       fix x y
1106       have "(\<integral>\<^sup>+ z. assign y x z \<partial>count_space UNIV) = (\<integral>\<^sup>+ z. ?rr y z \<partial>count_space UNIV) * (?pp y x / pmf q y)"
1107         by(simp add: assign_def divide_real_def mult.commute[where a="?pp y x"] mult.assoc nn_integral_multc times_ereal.simps(1)[symmetric] del: times_ereal.simps(1))
1108       also have "\<dots> = ?pp y x" by(simp add: nn_integral_rr1 pp_0)
1109       finally show "?thesis y x" .
1110     qed
1112     def a \<equiv> "embed_pmf (\<lambda>(y, x, z). assign y x z)"
1113     { fix y x z
1114       have "assign y x z = pmf a (y, x, z)"
1115         unfolding a_def
1116       proof (subst pmf_embed_pmf)
1117         have "(\<integral>\<^sup>+ x. ereal ((\<lambda>(y, x, z). assign y x z) x) \<partial>count_space UNIV) =
1118           (\<integral>\<^sup>+ x. ereal ((\<lambda>(y, x, z). assign y x z) x) \<partial>(count_space ((\<lambda>((y, x), z). (y, x, z)) ` (pp \<times> UNIV))))"
1119           by (force simp add: nn_integral_count_space_indicator pmf_eq_0_set_pmf split: split_indicator
1120                     intro!: nn_integral_cong assign_eq_0_outside)
1121         also have "\<dots> = (\<integral>\<^sup>+ x. ereal ((\<lambda>((y, x), z). assign y x z) x) \<partial>(count_space (pp \<times> UNIV)))"
1122           by (subst nn_integral_bij_count_space[OF inj_on_imp_bij_betw, symmetric])
1123              (auto simp: inj_on_def intro!: nn_integral_cong)
1124         also have "\<dots> = (\<integral>\<^sup>+ y. \<integral>\<^sup>+z. ereal ((\<lambda>((y, x), z). assign y x z) (y, z)) \<partial>count_space UNIV \<partial>count_space pp)"
1125           by (subst sigma_finite_measure.nn_integral_fst)
1126              (auto simp: pair_measure_countable sigma_finite_measure_count_space_countable)
1127         also have "\<dots> = (\<integral>\<^sup>+ z. ?pp (fst z) (snd z) \<partial>count_space pp)"
1128           by (subst nn_integral_assign2[symmetric]) (auto intro!: nn_integral_cong)
1129         finally show "(\<integral>\<^sup>+ x. ereal ((\<lambda>(y, x, z). assign y x z) x) \<partial>count_space UNIV) = 1"
1130           by (simp add: nn_integral_pmf emeasure_pmf)
1131       qed auto }
1132     note a = this
1134     def pr \<equiv> "map_pmf (\<lambda>(y, x, z). (from_nat_into (A y) x, from_nat_into (B y) z)) a"
1136     have "rel_pmf (R OO S) p r"
1137     proof
1138       have pp_eq: "pp = map_pmf (\<lambda>(y, x, z). (y, x)) a"
1139       proof (rule pmf_eqI)
1140         fix i
1141         show "pmf pp i = pmf (map_pmf (\<lambda>(y, x, z). (y, x)) a) i"
1142           using nn_integral_assign2[of "fst i" "snd i", symmetric]
1143           by (auto simp add: a nn_integral_pmf' inj_on_def ereal.inject[symmetric] ereal_pmf_map
1144                    simp del: ereal.inject intro!: arg_cong2[where f=emeasure])
1145       qed
1146       moreover have pq_eq: "pq = map_pmf (\<lambda>(y, x). (from_nat_into (A y) x, y)) pp"
1147         by (simp add: pp_def map_pmf_comp split_beta A[symmetric] cong: map_pmf_cong)
1148       ultimately show "map_pmf fst pr = p"
1149         unfolding p pr_def by (simp add: map_pmf_comp split_beta)
1151       have rr_eq: "rr = map_pmf (\<lambda>(y, x, z). (y, z)) a"
1152       proof (rule pmf_eqI)
1153         fix i show "pmf rr i = pmf (map_pmf (\<lambda>(y, x, z). (y, z)) a) i"
1154           using nn_integral_assign1[of "fst i" "snd i", symmetric]
1155           by (auto simp add: a nn_integral_pmf' inj_on_def ereal.inject[symmetric] ereal_pmf_map
1156                    simp del: ereal.inject intro!: arg_cong2[where f=emeasure])
1157       qed
1158       moreover have qr_eq: "qr = map_pmf (\<lambda>(y, z). (y, from_nat_into (B y) z)) rr"
1159         by (simp add: rr_def map_pmf_comp split_beta B[symmetric] cong: map_pmf_cong)
1160       ultimately show "map_pmf snd pr = r"
1161         unfolding r pr_def by (simp add: map_pmf_comp split_beta)
1163       fix x z assume "(x, z) \<in> set_pmf pr"
1164       then have "\<exists>y. (x, y) \<in> set_pmf pq \<and> (y, z) \<in> set_pmf qr"
1165         by (force simp add: pp_eq pq_eq rr_eq qr_eq set_map_pmf pr_def image_image)
1166       with pq qr show "(R OO S) x z"
1167         by blast
1168     qed }
1169   then show "rel_pmf R OO rel_pmf S \<le> rel_pmf (R OO S)"
1171 qed (fact natLeq_card_order natLeq_cinfinite)+
1173 lemma rel_pmf_return_pmf1: "rel_pmf R (return_pmf x) M \<longleftrightarrow> (\<forall>a\<in>M. R x a)"
1174 proof safe
1175   fix a assume "a \<in> M" "rel_pmf R (return_pmf x) M"
1176   then obtain pq where *: "\<And>a b. (a, b) \<in> set_pmf pq \<Longrightarrow> R a b"
1177     and eq: "return_pmf x = map_pmf fst pq" "M = map_pmf snd pq"
1178     by (force elim: rel_pmf.cases)
1179   moreover have "set_pmf (return_pmf x) = {x}"
1181   with `a \<in> M` have "(x, a) \<in> pq"
1182     by (force simp: eq set_map_pmf)
1183   with * show "R x a"
1184     by auto
1185 qed (auto intro!: rel_pmf.intros[where pq="pair_pmf (return_pmf x) M"]
1186           simp: map_fst_pair_pmf map_snd_pair_pmf set_pair_pmf set_return_pmf)
1188 lemma rel_pmf_return_pmf2: "rel_pmf R M (return_pmf x) \<longleftrightarrow> (\<forall>a\<in>M. R a x)"
1189   by (subst pmf.rel_flip[symmetric]) (simp add: rel_pmf_return_pmf1)
1191 lemma rel_pmf_rel_prod:
1192   "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'"
1193 proof safe
1194   assume "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
1195   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"
1196     and eq: "map_pmf fst pq = pair_pmf A A'" "map_pmf snd pq = pair_pmf B B'"
1197     by (force elim: rel_pmf.cases)
1198   show "rel_pmf R A B"
1199   proof (rule rel_pmf.intros)
1200     let ?f = "\<lambda>(a, b). (fst a, fst b)"
1201     have [simp]: "(\<lambda>x. fst (?f x)) = fst o fst" "(\<lambda>x. snd (?f x)) = fst o snd"
1202       by auto
1204     show "map_pmf fst (map_pmf ?f pq) = A"
1205       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
1206     show "map_pmf snd (map_pmf ?f pq) = B"
1207       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
1209     fix a b assume "(a, b) \<in> set_pmf (map_pmf ?f pq)"
1210     then obtain c d where "((a, c), (b, d)) \<in> set_pmf pq"
1211       by (auto simp: set_map_pmf)
1212     from pq[OF this] show "R a b" ..
1213   qed
1214   show "rel_pmf S A' B'"
1215   proof (rule rel_pmf.intros)
1216     let ?f = "\<lambda>(a, b). (snd a, snd b)"
1217     have [simp]: "(\<lambda>x. fst (?f x)) = snd o fst" "(\<lambda>x. snd (?f x)) = snd o snd"
1218       by auto
1220     show "map_pmf fst (map_pmf ?f pq) = A'"
1221       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
1222     show "map_pmf snd (map_pmf ?f pq) = B'"
1223       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
1225     fix c d assume "(c, d) \<in> set_pmf (map_pmf ?f pq)"
1226     then obtain a b where "((a, c), (b, d)) \<in> set_pmf pq"
1227       by (auto simp: set_map_pmf)
1228     from pq[OF this] show "S c d" ..
1229   qed
1230 next
1231   assume "rel_pmf R A B" "rel_pmf S A' B'"
1232   then obtain Rpq Spq
1233     where Rpq: "\<And>a b. (a, b) \<in> set_pmf Rpq \<Longrightarrow> R a b"
1234         "map_pmf fst Rpq = A" "map_pmf snd Rpq = B"
1235       and Spq: "\<And>a b. (a, b) \<in> set_pmf Spq \<Longrightarrow> S a b"
1236         "map_pmf fst Spq = A'" "map_pmf snd Spq = B'"
1237     by (force elim: rel_pmf.cases)
1239   let ?f = "(\<lambda>((a, c), (b, d)). ((a, b), (c, d)))"
1240   let ?pq = "map_pmf ?f (pair_pmf Rpq Spq)"
1241   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))"
1242     by auto
1244   show "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
1245     by (rule rel_pmf.intros[where pq="?pq"])
1246        (auto simp: map_snd_pair_pmf map_fst_pair_pmf set_pair_pmf set_map_pmf map_pmf_comp Rpq Spq
1247                    map_pair)
1248 qed
1250 end