src/HOL/Probability/Probability_Mass_Function.thy
 author paulson Tue Nov 10 14:18:41 2015 +0000 (2015-11-10) changeset 61609 77b453bd616f parent 61424 c3658c18b7bc child 61610 4f54d2759a0b permissions -rw-r--r--
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
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/Library/Multiset"
12 begin
14 lemma AE_emeasure_singleton:
15   assumes x: "emeasure M {x} \<noteq> 0" and ae: "AE x in M. P x" shows "P x"
16 proof -
17   from x have x_M: "{x} \<in> sets M"
18     by (auto intro: emeasure_notin_sets)
19   from ae obtain N where N: "{x\<in>space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"
20     by (auto elim: AE_E)
21   { assume "\<not> P x"
22     with x_M[THEN sets.sets_into_space] N have "emeasure M {x} \<le> emeasure M N"
23       by (intro emeasure_mono) auto
24     with x N have False
25       by (auto simp: emeasure_le_0_iff) }
26   then show "P x" by auto
27 qed
29 lemma AE_measure_singleton: "measure M {x} \<noteq> 0 \<Longrightarrow> AE x in M. P x \<Longrightarrow> P x"
30   by (metis AE_emeasure_singleton measure_def emeasure_empty measure_empty)
32 lemma ereal_divide': "b \<noteq> 0 \<Longrightarrow> ereal (a / b) = ereal a / ereal b"
33   using ereal_divide[of a b] by simp
35 lemma (in finite_measure) countable_support:
36   "countable {x. measure M {x} \<noteq> 0}"
37 proof cases
38   assume "measure M (space M) = 0"
39   with bounded_measure measure_le_0_iff have "{x. measure M {x} \<noteq> 0} = {}"
40     by auto
41   then show ?thesis
42     by simp
43 next
44   let ?M = "measure M (space M)" and ?m = "\<lambda>x. measure M {x}"
45   assume "?M \<noteq> 0"
46   then have *: "{x. ?m x \<noteq> 0} = (\<Union>n. {x. ?M / Suc n < ?m x})"
47     using reals_Archimedean[of "?m x / ?M" for x]
48     by (auto simp: field_simps not_le[symmetric] measure_nonneg divide_le_0_iff measure_le_0_iff)
49   have **: "\<And>n. finite {x. ?M / Suc n < ?m x}"
50   proof (rule ccontr)
51     fix n assume "infinite {x. ?M / Suc n < ?m x}" (is "infinite ?X")
52     then obtain X where "finite X" "card X = Suc (Suc n)" "X \<subseteq> ?X"
53       by (metis infinite_arbitrarily_large)
54     from this(3) have *: "\<And>x. x \<in> X \<Longrightarrow> ?M / Suc n \<le> ?m x"
55       by auto
56     { fix x assume "x \<in> X"
57       from `?M \<noteq> 0` *[OF this] have "?m x \<noteq> 0" by (auto simp: field_simps measure_le_0_iff)
58       then have "{x} \<in> sets M" by (auto dest: measure_notin_sets) }
59     note singleton_sets = this
60     have "?M < (\<Sum>x\<in>X. ?M / Suc n)"
61       using `?M \<noteq> 0`
62       by (simp add: `card X = Suc (Suc n)` of_nat_Suc field_simps less_le measure_nonneg)
63     also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"
64       by (rule setsum_mono) fact
65     also have "\<dots> = measure M (\<Union>x\<in>X. {x})"
66       using singleton_sets `finite X`
67       by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def)
68     finally have "?M < measure M (\<Union>x\<in>X. {x})" .
69     moreover have "measure M (\<Union>x\<in>X. {x}) \<le> ?M"
70       using singleton_sets[THEN sets.sets_into_space] by (intro finite_measure_mono) auto
71     ultimately show False by simp
72   qed
73   show ?thesis
74     unfolding * by (intro countable_UN countableI_type countable_finite[OF **])
75 qed
77 lemma (in finite_measure) AE_support_countable:
78   assumes [simp]: "sets M = UNIV"
79   shows "(AE x in M. measure M {x} \<noteq> 0) \<longleftrightarrow> (\<exists>S. countable S \<and> (AE x in M. x \<in> S))"
80 proof
81   assume "\<exists>S. countable S \<and> (AE x in M. x \<in> S)"
82   then obtain S where S[intro]: "countable S" and ae: "AE x in M. x \<in> S"
83     by auto
84   then have "emeasure M (\<Union>x\<in>{x\<in>S. emeasure M {x} \<noteq> 0}. {x}) =
85     (\<integral>\<^sup>+ x. emeasure M {x} * indicator {x\<in>S. emeasure M {x} \<noteq> 0} x \<partial>count_space UNIV)"
86     by (subst emeasure_UN_countable)
87        (auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space)
88   also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} * indicator S x \<partial>count_space UNIV)"
89     by (auto intro!: nn_integral_cong split: split_indicator)
90   also have "\<dots> = emeasure M (\<Union>x\<in>S. {x})"
91     by (subst emeasure_UN_countable)
92        (auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space)
93   also have "\<dots> = emeasure M (space M)"
94     using ae by (intro emeasure_eq_AE) auto
95   finally have "emeasure M {x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0} = emeasure M (space M)"
96     by (simp add: emeasure_single_in_space cong: rev_conj_cong)
97   with finite_measure_compl[of "{x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0}"]
98   have "AE x in M. x \<in> S \<and> emeasure M {x} \<noteq> 0"
99     by (intro AE_I[OF order_refl]) (auto simp: emeasure_eq_measure set_diff_eq cong: conj_cong)
100   then show "AE x in M. measure M {x} \<noteq> 0"
101     by (auto simp: emeasure_eq_measure)
102 qed (auto intro!: exI[of _ "{x. measure M {x} \<noteq> 0}"] countable_support)
104 subsection \<open> PMF as measure \<close>
106 typedef 'a pmf = "{M :: 'a measure. prob_space M \<and> sets M = UNIV \<and> (AE x in M. measure M {x} \<noteq> 0)}"
107   morphisms measure_pmf Abs_pmf
108   by (intro exI[of _ "uniform_measure (count_space UNIV) {undefined}"])
109      (auto intro!: prob_space_uniform_measure AE_uniform_measureI)
111 declare [[coercion measure_pmf]]
113 lemma prob_space_measure_pmf: "prob_space (measure_pmf p)"
114   using pmf.measure_pmf[of p] by auto
116 interpretation measure_pmf!: prob_space "measure_pmf M" for M
117   by (rule prob_space_measure_pmf)
119 interpretation measure_pmf!: subprob_space "measure_pmf M" for M
120   by (rule prob_space_imp_subprob_space) unfold_locales
122 lemma subprob_space_measure_pmf: "subprob_space (measure_pmf x)"
123   by unfold_locales
125 locale pmf_as_measure
126 begin
128 setup_lifting type_definition_pmf
130 end
132 context
133 begin
135 interpretation pmf_as_measure .
137 lemma sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV"
138   by transfer blast
140 lemma sets_measure_pmf_count_space[measurable_cong]:
141   "sets (measure_pmf M) = sets (count_space UNIV)"
142   by simp
144 lemma space_measure_pmf[simp]: "space (measure_pmf p) = UNIV"
145   using sets_eq_imp_space_eq[of "measure_pmf p" "count_space UNIV"] by simp
147 lemma measure_pmf_in_subprob_algebra[measurable (raw)]: "measure_pmf x \<in> space (subprob_algebra (count_space UNIV))"
148   by (simp add: space_subprob_algebra subprob_space_measure_pmf)
150 lemma measurable_pmf_measure1[simp]: "measurable (M :: 'a pmf) N = UNIV \<rightarrow> space N"
151   by (auto simp: measurable_def)
153 lemma measurable_pmf_measure2[simp]: "measurable N (M :: 'a pmf) = measurable N (count_space UNIV)"
154   by (intro measurable_cong_sets) simp_all
156 lemma measurable_pair_restrict_pmf2:
157   assumes "countable A"
158   assumes [measurable]: "\<And>y. y \<in> A \<Longrightarrow> (\<lambda>x. f (x, y)) \<in> measurable M L"
159   shows "f \<in> measurable (M \<Otimes>\<^sub>M restrict_space (measure_pmf N) A) L" (is "f \<in> measurable ?M _")
160 proof -
161   have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)"
164   show ?thesis
165     by (intro measurable_compose_countable'[where f="\<lambda>a b. f (fst b, a)" and g=snd and I=A,
166                                             unfolded prod.collapse] assms)
167         measurable
168 qed
170 lemma measurable_pair_restrict_pmf1:
171   assumes "countable A"
172   assumes [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N L"
173   shows "f \<in> measurable (restrict_space (measure_pmf M) A \<Otimes>\<^sub>M N) L"
174 proof -
175   have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)"
178   show ?thesis
179     by (intro measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A,
180                                             unfolded prod.collapse] assms)
181         measurable
182 qed
184 lift_definition pmf :: "'a pmf \<Rightarrow> 'a \<Rightarrow> real" is "\<lambda>M x. measure M {x}" .
186 lift_definition set_pmf :: "'a pmf \<Rightarrow> 'a set" is "\<lambda>M. {x. measure M {x} \<noteq> 0}" .
187 declare [[coercion set_pmf]]
189 lemma AE_measure_pmf: "AE x in (M::'a pmf). x \<in> M"
190   by transfer simp
192 lemma emeasure_pmf_single_eq_zero_iff:
193   fixes M :: "'a pmf"
194   shows "emeasure M {y} = 0 \<longleftrightarrow> y \<notin> M"
195   by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
197 lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \<longleftrightarrow> (\<forall>y\<in>M. P y)"
198   using AE_measure_singleton[of M] AE_measure_pmf[of M]
199   by (auto simp: set_pmf.rep_eq)
201 lemma countable_set_pmf [simp]: "countable (set_pmf p)"
202   by transfer (metis prob_space.finite_measure finite_measure.countable_support)
204 lemma pmf_positive: "x \<in> set_pmf p \<Longrightarrow> 0 < pmf p x"
205   by transfer (simp add: less_le measure_nonneg)
207 lemma pmf_nonneg: "0 \<le> pmf p x"
208   by transfer (simp add: measure_nonneg)
210 lemma pmf_le_1: "pmf p x \<le> 1"
213 lemma set_pmf_not_empty: "set_pmf M \<noteq> {}"
214   using AE_measure_pmf[of M] by (intro notI) simp
216 lemma set_pmf_iff: "x \<in> set_pmf M \<longleftrightarrow> pmf M x \<noteq> 0"
217   by transfer simp
219 lemma set_pmf_eq: "set_pmf M = {x. pmf M x \<noteq> 0}"
220   by (auto simp: set_pmf_iff)
222 lemma emeasure_pmf_single:
223   fixes M :: "'a pmf"
224   shows "emeasure M {x} = pmf M x"
225   by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
227 lemma measure_pmf_single: "measure (measure_pmf M) {x} = pmf M x"
228 using emeasure_pmf_single[of M x] by(simp add: measure_pmf.emeasure_eq_measure)
230 lemma emeasure_measure_pmf_finite: "finite S \<Longrightarrow> emeasure (measure_pmf M) S = (\<Sum>s\<in>S. pmf M s)"
231   by (subst emeasure_eq_setsum_singleton) (auto simp: emeasure_pmf_single)
233 lemma measure_measure_pmf_finite: "finite S \<Longrightarrow> measure (measure_pmf M) S = setsum (pmf M) S"
234   using emeasure_measure_pmf_finite[of S M] by(simp add: measure_pmf.emeasure_eq_measure)
236 lemma nn_integral_measure_pmf_support:
237   fixes f :: "'a \<Rightarrow> ereal"
238   assumes f: "finite A" and nn: "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x" "\<And>x. x \<in> set_pmf M \<Longrightarrow> x \<notin> A \<Longrightarrow> f x = 0"
239   shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>A. f x * pmf M x)"
240 proof -
241   have "(\<integral>\<^sup>+x. f x \<partial>M) = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)"
242     using nn by (intro nn_integral_cong_AE) (auto simp: AE_measure_pmf_iff split: split_indicator)
243   also have "\<dots> = (\<Sum>x\<in>A. f x * emeasure M {x})"
244     using assms by (intro nn_integral_indicator_finite) auto
245   finally show ?thesis
247 qed
249 lemma nn_integral_measure_pmf_finite:
250   fixes f :: "'a \<Rightarrow> ereal"
251   assumes f: "finite (set_pmf M)" and nn: "\<And>x. x \<in> set_pmf M \<Longrightarrow> 0 \<le> f x"
252   shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>set_pmf M. f x * pmf M x)"
253   using assms by (intro nn_integral_measure_pmf_support) auto
254 lemma integrable_measure_pmf_finite:
255   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
256   shows "finite (set_pmf M) \<Longrightarrow> integrable M f"
257   by (auto intro!: integrableI_bounded simp: nn_integral_measure_pmf_finite)
259 lemma integral_measure_pmf:
260   assumes [simp]: "finite A" and "\<And>a. a \<in> set_pmf M \<Longrightarrow> f a \<noteq> 0 \<Longrightarrow> a \<in> A"
261   shows "(\<integral>x. f x \<partial>measure_pmf M) = (\<Sum>a\<in>A. f a * pmf M a)"
262 proof -
263   have "(\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x * indicator A x \<partial>measure_pmf M)"
264     using assms(2) by (intro integral_cong_AE) (auto split: split_indicator simp: AE_measure_pmf_iff)
265   also have "\<dots> = (\<Sum>a\<in>A. f a * pmf M a)"
266     by (subst integral_indicator_finite_real) (auto simp: measure_def emeasure_measure_pmf_finite)
267   finally show ?thesis .
268 qed
270 lemma integrable_pmf: "integrable (count_space X) (pmf M)"
271 proof -
272   have " (\<integral>\<^sup>+ x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+ x. pmf M x \<partial>count_space (M \<inter> X))"
273     by (auto simp add: nn_integral_count_space_indicator set_pmf_iff intro!: nn_integral_cong split: split_indicator)
274   then have "integrable (count_space X) (pmf M) = integrable (count_space (M \<inter> X)) (pmf M)"
275     by (simp add: integrable_iff_bounded pmf_nonneg)
276   then show ?thesis
277     by (simp add: pmf.rep_eq measure_pmf.integrable_measure disjoint_family_on_def)
278 qed
280 lemma integral_pmf: "(\<integral>x. pmf M x \<partial>count_space X) = measure M X"
281 proof -
282   have "(\<integral>x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+x. pmf M x \<partial>count_space X)"
283     by (simp add: pmf_nonneg integrable_pmf nn_integral_eq_integral)
284   also have "\<dots> = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space (X \<inter> M))"
285     by (auto intro!: nn_integral_cong_AE split: split_indicator
286              simp: pmf.rep_eq measure_pmf.emeasure_eq_measure nn_integral_count_space_indicator
287                    AE_count_space set_pmf_iff)
288   also have "\<dots> = emeasure M (X \<inter> M)"
289     by (rule emeasure_countable_singleton[symmetric]) (auto intro: countable_set_pmf)
290   also have "\<dots> = emeasure M X"
291     by (auto intro!: emeasure_eq_AE simp: AE_measure_pmf_iff)
292   finally show ?thesis
294 qed
296 lemma integral_pmf_restrict:
297   "(f::'a \<Rightarrow> 'b::{banach, second_countable_topology}) \<in> borel_measurable (count_space UNIV) \<Longrightarrow>
298     (\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x \<partial>restrict_space M M)"
299   by (auto intro!: integral_cong_AE simp add: integral_restrict_space AE_measure_pmf_iff)
301 lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1"
302 proof -
303   have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)"
304     by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf)
305   then show ?thesis
306     using measure_pmf.emeasure_space_1 by simp
307 qed
309 lemma emeasure_pmf_UNIV [simp]: "emeasure (measure_pmf M) UNIV = 1"
310 using measure_pmf.emeasure_space_1[of M] by simp
312 lemma in_null_sets_measure_pmfI:
313   "A \<inter> set_pmf p = {} \<Longrightarrow> A \<in> null_sets (measure_pmf p)"
314 using emeasure_eq_0_AE[where ?P="\<lambda>x. x \<in> A" and M="measure_pmf p"]
315 by(auto simp add: null_sets_def AE_measure_pmf_iff)
317 lemma measure_subprob: "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
318   by (simp add: space_subprob_algebra subprob_space_measure_pmf)
320 subsection \<open> Monad Interpretation \<close>
322 lemma measurable_measure_pmf[measurable]:
323   "(\<lambda>x. measure_pmf (M x)) \<in> measurable (count_space UNIV) (subprob_algebra (count_space UNIV))"
324   by (auto simp: space_subprob_algebra intro!: prob_space_imp_subprob_space) unfold_locales
326 lemma bind_measure_pmf_cong:
327   assumes "\<And>x. A x \<in> space (subprob_algebra N)" "\<And>x. B x \<in> space (subprob_algebra N)"
328   assumes "\<And>i. i \<in> set_pmf x \<Longrightarrow> A i = B i"
329   shows "bind (measure_pmf x) A = bind (measure_pmf x) B"
330 proof (rule measure_eqI)
331   show "sets (measure_pmf x \<guillemotright>= A) = sets (measure_pmf x \<guillemotright>= B)"
332     using assms by (subst (1 2) sets_bind) (auto simp: space_subprob_algebra)
333 next
334   fix X assume "X \<in> sets (measure_pmf x \<guillemotright>= A)"
335   then have X: "X \<in> sets N"
336     using assms by (subst (asm) sets_bind) (auto simp: space_subprob_algebra)
337   show "emeasure (measure_pmf x \<guillemotright>= A) X = emeasure (measure_pmf x \<guillemotright>= B) X"
338     using assms
339     by (subst (1 2) emeasure_bind[where N=N, OF _ _ X])
340        (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
341 qed
343 lift_definition bind_pmf :: "'a pmf \<Rightarrow> ('a \<Rightarrow> 'b pmf ) \<Rightarrow> 'b pmf" is bind
344 proof (clarify, intro conjI)
345   fix f :: "'a measure" and g :: "'a \<Rightarrow> 'b measure"
346   assume "prob_space f"
347   then interpret f: prob_space f .
348   assume "sets f = UNIV" and ae_f: "AE x in f. measure f {x} \<noteq> 0"
349   then have s_f[simp]: "sets f = sets (count_space UNIV)"
350     by simp
351   assume g: "\<And>x. prob_space (g x) \<and> sets (g x) = UNIV \<and> (AE y in g x. measure (g x) {y} \<noteq> 0)"
352   then have g: "\<And>x. prob_space (g x)" and s_g[simp]: "\<And>x. sets (g x) = sets (count_space UNIV)"
353     and ae_g: "\<And>x. AE y in g x. measure (g x) {y} \<noteq> 0"
354     by auto
356   have [measurable]: "g \<in> measurable f (subprob_algebra (count_space UNIV))"
357     by (auto simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space g)
359   show "prob_space (f \<guillemotright>= g)"
360     using g by (intro f.prob_space_bind[where S="count_space UNIV"]) auto
361   then interpret fg: prob_space "f \<guillemotright>= g" .
362   show [simp]: "sets (f \<guillemotright>= g) = UNIV"
363     using sets_eq_imp_space_eq[OF s_f]
364     by (subst sets_bind[where N="count_space UNIV"]) auto
365   show "AE x in f \<guillemotright>= g. measure (f \<guillemotright>= g) {x} \<noteq> 0"
366     apply (simp add: fg.prob_eq_0 AE_bind[where B="count_space UNIV"])
367     using ae_f
368     apply eventually_elim
369     using ae_g
370     apply eventually_elim
371     apply (auto dest: AE_measure_singleton)
372     done
373 qed
375 lemma ereal_pmf_bind: "pmf (bind_pmf N f) i = (\<integral>\<^sup>+x. pmf (f x) i \<partial>measure_pmf N)"
376   unfolding pmf.rep_eq bind_pmf.rep_eq
377   by (auto simp: measure_pmf.measure_bind[where N="count_space UNIV"] measure_subprob measure_nonneg
378            intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])
380 lemma pmf_bind: "pmf (bind_pmf N f) i = (\<integral>x. pmf (f x) i \<partial>measure_pmf N)"
381   using ereal_pmf_bind[of N f i]
382   by (subst (asm) nn_integral_eq_integral)
383      (auto simp: pmf_nonneg pmf_le_1
384            intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])
386 lemma bind_pmf_const[simp]: "bind_pmf M (\<lambda>x. c) = c"
387   by transfer (simp add: bind_const' prob_space_imp_subprob_space)
389 lemma set_bind_pmf[simp]: "set_pmf (bind_pmf M N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))"
390   unfolding set_pmf_eq ereal_eq_0(1)[symmetric] ereal_pmf_bind
391   by (auto simp add: nn_integral_0_iff_AE AE_measure_pmf_iff set_pmf_eq not_le less_le pmf_nonneg)
393 lemma bind_pmf_cong:
394   assumes "p = q"
395   shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g"
396   unfolding `p = q`[symmetric] measure_pmf_inject[symmetric] bind_pmf.rep_eq
397   by (auto simp: AE_measure_pmf_iff Pi_iff space_subprob_algebra subprob_space_measure_pmf
398                  sets_bind[where N="count_space UNIV"] emeasure_bind[where N="count_space UNIV"]
399            intro!: nn_integral_cong_AE measure_eqI)
401 lemma bind_pmf_cong_simp:
402   "p = q \<Longrightarrow> (\<And>x. x \<in> set_pmf q =simp=> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g"
403   by (simp add: simp_implies_def cong: bind_pmf_cong)
405 lemma measure_pmf_bind: "measure_pmf (bind_pmf M f) = (measure_pmf M \<guillemotright>= (\<lambda>x. measure_pmf (f x)))"
406   by transfer simp
408 lemma nn_integral_bind_pmf[simp]: "(\<integral>\<^sup>+x. f x \<partial>bind_pmf M N) = (\<integral>\<^sup>+x. \<integral>\<^sup>+y. f y \<partial>N x \<partial>M)"
409   using measurable_measure_pmf[of N]
410   unfolding measure_pmf_bind
411   apply (subst (1 3) nn_integral_max_0[symmetric])
412   apply (intro nn_integral_bind[where B="count_space UNIV"])
413   apply auto
414   done
416 lemma emeasure_bind_pmf[simp]: "emeasure (bind_pmf M N) X = (\<integral>\<^sup>+x. emeasure (N x) X \<partial>M)"
417   using measurable_measure_pmf[of N]
418   unfolding measure_pmf_bind
419   by (subst emeasure_bind[where N="count_space UNIV"]) auto
421 lift_definition return_pmf :: "'a \<Rightarrow> 'a pmf" is "return (count_space UNIV)"
422   by (auto intro!: prob_space_return simp: AE_return measure_return)
424 lemma bind_return_pmf: "bind_pmf (return_pmf x) f = f x"
425   by transfer
426      (auto intro!: prob_space_imp_subprob_space bind_return[where N="count_space UNIV"]
427            simp: space_subprob_algebra)
429 lemma set_return_pmf[simp]: "set_pmf (return_pmf x) = {x}"
430   by transfer (auto simp add: measure_return split: split_indicator)
432 lemma bind_return_pmf': "bind_pmf N return_pmf = N"
433 proof (transfer, clarify)
434   fix N :: "'a measure" assume "sets N = UNIV" then show "N \<guillemotright>= return (count_space UNIV) = N"
435     by (subst return_sets_cong[where N=N]) (simp_all add: bind_return')
436 qed
438 lemma bind_assoc_pmf: "bind_pmf (bind_pmf A B) C = bind_pmf A (\<lambda>x. bind_pmf (B x) C)"
439   by transfer
440      (auto intro!: bind_assoc[where N="count_space UNIV" and R="count_space UNIV"]
441            simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space)
443 definition "map_pmf f M = bind_pmf M (\<lambda>x. return_pmf (f x))"
445 lemma map_bind_pmf: "map_pmf f (bind_pmf M g) = bind_pmf M (\<lambda>x. map_pmf f (g x))"
446   by (simp add: map_pmf_def bind_assoc_pmf)
448 lemma bind_map_pmf: "bind_pmf (map_pmf f M) g = bind_pmf M (\<lambda>x. g (f x))"
449   by (simp add: map_pmf_def bind_assoc_pmf bind_return_pmf)
451 lemma map_pmf_transfer[transfer_rule]:
452   "rel_fun op = (rel_fun cr_pmf cr_pmf) (\<lambda>f M. distr M (count_space UNIV) f) map_pmf"
453 proof -
454   have "rel_fun op = (rel_fun pmf_as_measure.cr_pmf pmf_as_measure.cr_pmf)
455      (\<lambda>f M. M \<guillemotright>= (return (count_space UNIV) o f)) map_pmf"
456     unfolding map_pmf_def[abs_def] comp_def by transfer_prover
457   then show ?thesis
458     by (force simp: rel_fun_def cr_pmf_def bind_return_distr)
459 qed
461 lemma map_pmf_rep_eq:
462   "measure_pmf (map_pmf f M) = distr (measure_pmf M) (count_space UNIV) f"
463   unfolding map_pmf_def bind_pmf.rep_eq comp_def return_pmf.rep_eq
464   using bind_return_distr[of M f "count_space UNIV"] by (simp add: comp_def)
466 lemma map_pmf_id[simp]: "map_pmf id = id"
467   by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI)
469 lemma map_pmf_ident[simp]: "map_pmf (\<lambda>x. x) = (\<lambda>x. x)"
470   using map_pmf_id unfolding id_def .
472 lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g"
473   by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def)
475 lemma map_pmf_comp: "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M"
476   using map_pmf_compose[of f g] by (simp add: comp_def)
478 lemma map_pmf_cong: "p = q \<Longrightarrow> (\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> map_pmf f p = map_pmf g q"
479   unfolding map_pmf_def by (rule bind_pmf_cong) auto
481 lemma pmf_set_map: "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
482   by (auto simp add: comp_def fun_eq_iff map_pmf_def)
484 lemma set_map_pmf[simp]: "set_pmf (map_pmf f M) = f`set_pmf M"
485   using pmf_set_map[of f] by (auto simp: comp_def fun_eq_iff)
487 lemma emeasure_map_pmf[simp]: "emeasure (map_pmf f M) X = emeasure M (f -` X)"
488   unfolding map_pmf_rep_eq by (subst emeasure_distr) auto
490 lemma nn_integral_map_pmf[simp]: "(\<integral>\<^sup>+x. f x \<partial>map_pmf g M) = (\<integral>\<^sup>+x. f (g x) \<partial>M)"
491   unfolding map_pmf_rep_eq by (intro nn_integral_distr) auto
493 lemma ereal_pmf_map: "pmf (map_pmf f p) x = (\<integral>\<^sup>+ y. indicator (f -` {x}) y \<partial>measure_pmf p)"
494 proof (transfer fixing: f x)
495   fix p :: "'b measure"
496   presume "prob_space p"
497   then interpret prob_space p .
498   presume "sets p = UNIV"
499   then show "ereal (measure (distr p (count_space UNIV) f) {x}) = integral\<^sup>N p (indicator (f -` {x}))"
500     by(simp add: measure_distr measurable_def emeasure_eq_measure)
501 qed simp_all
503 lemma nn_integral_pmf: "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = emeasure (measure_pmf p) A"
504 proof -
505   have "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = (\<integral>\<^sup>+ x. pmf p x \<partial>count_space (A \<inter> set_pmf p))"
506     by(auto simp add: nn_integral_count_space_indicator indicator_def set_pmf_iff intro: nn_integral_cong)
507   also have "\<dots> = emeasure (measure_pmf p) (\<Union>x\<in>A \<inter> set_pmf p. {x})"
508     by(subst emeasure_UN_countable)(auto simp add: emeasure_pmf_single disjoint_family_on_def)
509   also have "\<dots> = emeasure (measure_pmf p) ((\<Union>x\<in>A \<inter> set_pmf p. {x}) \<union> {x. x \<in> A \<and> x \<notin> set_pmf p})"
510     by(rule emeasure_Un_null_set[symmetric])(auto intro: in_null_sets_measure_pmfI)
511   also have "\<dots> = emeasure (measure_pmf p) A"
512     by(auto intro: arg_cong2[where f=emeasure])
513   finally show ?thesis .
514 qed
516 lemma map_return_pmf [simp]: "map_pmf f (return_pmf x) = return_pmf (f x)"
517   by transfer (simp add: distr_return)
519 lemma map_pmf_const[simp]: "map_pmf (\<lambda>_. c) M = return_pmf c"
520   by transfer (auto simp: prob_space.distr_const)
522 lemma pmf_return [simp]: "pmf (return_pmf x) y = indicator {y} x"
523   by transfer (simp add: measure_return)
525 lemma nn_integral_return_pmf[simp]: "0 \<le> f x \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>return_pmf x) = f x"
526   unfolding return_pmf.rep_eq by (intro nn_integral_return) auto
528 lemma emeasure_return_pmf[simp]: "emeasure (return_pmf x) X = indicator X x"
529   unfolding return_pmf.rep_eq by (intro emeasure_return) auto
531 lemma return_pmf_inj[simp]: "return_pmf x = return_pmf y \<longleftrightarrow> x = y"
532   by (metis insertI1 set_return_pmf singletonD)
534 lemma map_pmf_eq_return_pmf_iff:
535   "map_pmf f p = return_pmf x \<longleftrightarrow> (\<forall>y \<in> set_pmf p. f y = x)"
536 proof
537   assume "map_pmf f p = return_pmf x"
538   then have "set_pmf (map_pmf f p) = set_pmf (return_pmf x)" by simp
539   then show "\<forall>y \<in> set_pmf p. f y = x" by auto
540 next
541   assume "\<forall>y \<in> set_pmf p. f y = x"
542   then show "map_pmf f p = return_pmf x"
543     unfolding map_pmf_const[symmetric, of _ p] by (intro map_pmf_cong) auto
544 qed
546 definition "pair_pmf A B = bind_pmf A (\<lambda>x. bind_pmf B (\<lambda>y. return_pmf (x, y)))"
548 lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b"
549   unfolding pair_pmf_def pmf_bind pmf_return
550   apply (subst integral_measure_pmf[where A="{b}"])
551   apply (auto simp: indicator_eq_0_iff)
552   apply (subst integral_measure_pmf[where A="{a}"])
553   apply (auto simp: indicator_eq_0_iff setsum_nonneg_eq_0_iff pmf_nonneg)
554   done
556 lemma set_pair_pmf[simp]: "set_pmf (pair_pmf A B) = set_pmf A \<times> set_pmf B"
557   unfolding pair_pmf_def set_bind_pmf set_return_pmf by auto
559 lemma measure_pmf_in_subprob_space[measurable (raw)]:
560   "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
561   by (simp add: space_subprob_algebra) intro_locales
563 lemma nn_integral_pair_pmf': "(\<integral>\<^sup>+x. f x \<partial>pair_pmf A B) = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. f (a, b) \<partial>B \<partial>A)"
564 proof -
565   have "(\<integral>\<^sup>+x. f x \<partial>pair_pmf A B) = (\<integral>\<^sup>+x. max 0 (f x) * indicator (A \<times> B) x \<partial>pair_pmf A B)"
566     by (subst nn_integral_max_0[symmetric])
567        (auto simp: AE_measure_pmf_iff intro!: nn_integral_cong_AE)
568   also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) * indicator (A \<times> B) (a, b) \<partial>B \<partial>A)"
570   also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) \<partial>B \<partial>A)"
571     by (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
572   finally show ?thesis
573     unfolding nn_integral_max_0 .
574 qed
576 lemma bind_pair_pmf:
577   assumes M[measurable]: "M \<in> measurable (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) (subprob_algebra N)"
578   shows "measure_pmf (pair_pmf A B) \<guillemotright>= M = (measure_pmf A \<guillemotright>= (\<lambda>x. measure_pmf B \<guillemotright>= (\<lambda>y. M (x, y))))"
579     (is "?L = ?R")
580 proof (rule measure_eqI)
581   have M'[measurable]: "M \<in> measurable (pair_pmf A B) (subprob_algebra N)"
582     using M[THEN measurable_space] by (simp_all add: space_pair_measure)
584   note measurable_bind[where N="count_space UNIV", measurable]
585   note measure_pmf_in_subprob_space[simp]
587   have sets_eq_N: "sets ?L = N"
588     by (subst sets_bind[OF sets_kernel[OF M']]) auto
589   show "sets ?L = sets ?R"
590     using measurable_space[OF M]
591     by (simp add: sets_eq_N space_pair_measure space_subprob_algebra)
592   fix X assume "X \<in> sets ?L"
593   then have X[measurable]: "X \<in> sets N"
594     unfolding sets_eq_N .
595   then show "emeasure ?L X = emeasure ?R X"
596     apply (simp add: emeasure_bind[OF _ M' X])
597     apply (simp add: nn_integral_bind[where B="count_space UNIV"] pair_pmf_def measure_pmf_bind[of A]
598                      nn_integral_measure_pmf_finite emeasure_nonneg one_ereal_def[symmetric])
599     apply (subst emeasure_bind[OF _ _ X])
600     apply measurable
601     apply (subst emeasure_bind[OF _ _ X])
602     apply measurable
603     done
604 qed
606 lemma map_fst_pair_pmf: "map_pmf fst (pair_pmf A B) = A"
607   by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
609 lemma map_snd_pair_pmf: "map_pmf snd (pair_pmf A B) = B"
610   by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
612 lemma nn_integral_pmf':
613   "inj_on f A \<Longrightarrow> (\<integral>\<^sup>+x. pmf p (f x) \<partial>count_space A) = emeasure p (f ` A)"
614   by (subst nn_integral_bij_count_space[where g=f and B="f`A"])
615      (auto simp: bij_betw_def nn_integral_pmf)
617 lemma pmf_le_0_iff[simp]: "pmf M p \<le> 0 \<longleftrightarrow> pmf M p = 0"
618   using pmf_nonneg[of M p] by simp
620 lemma min_pmf_0[simp]: "min (pmf M p) 0 = 0" "min 0 (pmf M p) = 0"
621   using pmf_nonneg[of M p] by simp_all
623 lemma pmf_eq_0_set_pmf: "pmf M p = 0 \<longleftrightarrow> p \<notin> set_pmf M"
624   unfolding set_pmf_iff by simp
626 lemma pmf_map_inj: "inj_on f (set_pmf M) \<Longrightarrow> x \<in> set_pmf M \<Longrightarrow> pmf (map_pmf f M) (f x) = pmf M x"
627   by (auto simp: pmf.rep_eq map_pmf_rep_eq measure_distr AE_measure_pmf_iff inj_onD
628            intro!: measure_pmf.finite_measure_eq_AE)
630 lemma pmf_map_inj': "inj f \<Longrightarrow> pmf (map_pmf f M) (f x) = pmf M x"
631 apply(cases "x \<in> set_pmf M")
634 apply(auto simp add: pmf_eq_0_set_pmf dest: injD)
635 done
637 lemma pmf_map_outside: "x \<notin> f ` set_pmf M \<Longrightarrow> pmf (map_pmf f M) x = 0"
638 unfolding pmf_eq_0_set_pmf by simp
640 subsection \<open> PMFs as function \<close>
642 context
643   fixes f :: "'a \<Rightarrow> real"
644   assumes nonneg: "\<And>x. 0 \<le> f x"
645   assumes prob: "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
646 begin
648 lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ereal \<circ> f)"
649 proof (intro conjI)
650   have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
651     by (simp split: split_indicator)
652   show "AE x in density (count_space UNIV) (ereal \<circ> f).
653     measure (density (count_space UNIV) (ereal \<circ> f)) {x} \<noteq> 0"
654     by (simp add: AE_density nonneg measure_def emeasure_density max_def)
655   show "prob_space (density (count_space UNIV) (ereal \<circ> f))"
656     by standard (simp add: emeasure_density prob)
657 qed simp
659 lemma pmf_embed_pmf: "pmf embed_pmf x = f x"
660 proof transfer
661   have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
662     by (simp split: split_indicator)
663   fix x show "measure (density (count_space UNIV) (ereal \<circ> f)) {x} = f x"
664     by transfer (simp add: measure_def emeasure_density nonneg max_def)
665 qed
667 lemma set_embed_pmf: "set_pmf embed_pmf = {x. f x \<noteq> 0}"
668 by(auto simp add: set_pmf_eq assms pmf_embed_pmf)
670 end
672 lemma embed_pmf_transfer:
673   "rel_fun (eq_onp (\<lambda>f. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ereal (f x) \<partial>count_space UNIV) = 1)) pmf_as_measure.cr_pmf (\<lambda>f. density (count_space UNIV) (ereal \<circ> f)) embed_pmf"
674   by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer)
676 lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)"
677 proof (transfer, elim conjE)
678   fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
679   assume "prob_space M" then interpret prob_space M .
680   show "M = density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))"
681   proof (rule measure_eqI)
682     fix A :: "'a set"
683     have "(\<integral>\<^sup>+ x. ereal (measure M {x}) * indicator A x \<partial>count_space UNIV) =
684       (\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)"
685       by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator)
686     also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))"
687       by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space)
688     also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})"
689       by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support)
690          (auto simp: disjoint_family_on_def)
691     also have "\<dots> = emeasure M A"
692       using ae by (intro emeasure_eq_AE) auto
693     finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))) A"
694       using emeasure_space_1 by (simp add: emeasure_density)
695   qed simp
696 qed
698 lemma td_pmf_embed_pmf:
699   "type_definition pmf embed_pmf {f::'a \<Rightarrow> real. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ereal (f x) \<partial>count_space UNIV) = 1}"
700   unfolding type_definition_def
701 proof safe
702   fix p :: "'a pmf"
703   have "(\<integral>\<^sup>+ x. 1 \<partial>measure_pmf p) = 1"
704     using measure_pmf.emeasure_space_1[of p] by simp
705   then show *: "(\<integral>\<^sup>+ x. ereal (pmf p x) \<partial>count_space UNIV) = 1"
706     by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const)
708   show "embed_pmf (pmf p) = p"
709     by (intro measure_pmf_inject[THEN iffD1])
710        (simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def)
711 next
712   fix f :: "'a \<Rightarrow> real" assume "\<forall>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
713   then show "pmf (embed_pmf f) = f"
714     by (auto intro!: pmf_embed_pmf)
715 qed (rule pmf_nonneg)
717 end
719 lemma nn_integral_measure_pmf: "(\<integral>\<^sup>+ x. f x \<partial>measure_pmf p) = \<integral>\<^sup>+ x. ereal (pmf p x) * f x \<partial>count_space UNIV"
720 by(simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg)
722 locale pmf_as_function
723 begin
725 setup_lifting td_pmf_embed_pmf
727 lemma set_pmf_transfer[transfer_rule]:
728   assumes "bi_total A"
729   shows "rel_fun (pcr_pmf A) (rel_set A) (\<lambda>f. {x. f x \<noteq> 0}) set_pmf"
730   using `bi_total A`
731   by (auto simp: pcr_pmf_def cr_pmf_def rel_fun_def rel_set_def bi_total_def Bex_def set_pmf_iff)
732      metis+
734 end
736 context
737 begin
739 interpretation pmf_as_function .
741 lemma pmf_eqI: "(\<And>i. pmf M i = pmf N i) \<Longrightarrow> M = N"
742   by transfer auto
744 lemma pmf_eq_iff: "M = N \<longleftrightarrow> (\<forall>i. pmf M i = pmf N i)"
745   by (auto intro: pmf_eqI)
747 lemma bind_commute_pmf: "bind_pmf A (\<lambda>x. bind_pmf B (C x)) = bind_pmf B (\<lambda>y. bind_pmf A (\<lambda>x. C x y))"
748   unfolding pmf_eq_iff pmf_bind
749 proof
750   fix i
751   interpret B: prob_space "restrict_space B B"
752     by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
753        (auto simp: AE_measure_pmf_iff)
754   interpret A: prob_space "restrict_space A A"
755     by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
756        (auto simp: AE_measure_pmf_iff)
758   interpret AB: pair_prob_space "restrict_space A A" "restrict_space B B"
759     by unfold_locales
761   have "(\<integral> x. \<integral> y. pmf (C x y) i \<partial>B \<partial>A) = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>A)"
762     by (rule integral_cong) (auto intro!: integral_pmf_restrict)
763   also have "\<dots> = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>restrict_space A A)"
764     by (intro integral_pmf_restrict B.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
765               countable_set_pmf borel_measurable_count_space)
766   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>restrict_space B B)"
767     by (rule AB.Fubini_integral[symmetric])
768        (auto intro!: AB.integrable_const_bound[where B=1] measurable_pair_restrict_pmf2
769              simp: pmf_nonneg pmf_le_1 measurable_restrict_space1)
770   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>B)"
771     by (intro integral_pmf_restrict[symmetric] A.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
772               countable_set_pmf borel_measurable_count_space)
773   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>A \<partial>B)"
774     by (rule integral_cong) (auto intro!: integral_pmf_restrict[symmetric])
775   finally show "(\<integral> x. \<integral> y. pmf (C x y) i \<partial>B \<partial>A) = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>A \<partial>B)" .
776 qed
778 lemma pair_map_pmf1: "pair_pmf (map_pmf f A) B = map_pmf (apfst f) (pair_pmf A B)"
779 proof (safe intro!: pmf_eqI)
780   fix a :: "'a" and b :: "'b"
781   have [simp]: "\<And>c d. indicator (apfst f -` {(a, b)}) (c, d) = indicator (f -` {a}) c * (indicator {b} d::ereal)"
782     by (auto split: split_indicator)
784   have "ereal (pmf (pair_pmf (map_pmf f A) B) (a, b)) =
785          ereal (pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b))"
786     unfolding pmf_pair ereal_pmf_map
787     by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_multc pmf_nonneg
788                   emeasure_map_pmf[symmetric] del: emeasure_map_pmf)
789   then show "pmf (pair_pmf (map_pmf f A) B) (a, b) = pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b)"
790     by simp
791 qed
793 lemma pair_map_pmf2: "pair_pmf A (map_pmf f B) = map_pmf (apsnd f) (pair_pmf A B)"
794 proof (safe intro!: pmf_eqI)
795   fix a :: "'a" and b :: "'b"
796   have [simp]: "\<And>c d. indicator (apsnd f -` {(a, b)}) (c, d) = indicator {a} c * (indicator (f -` {b}) d::ereal)"
797     by (auto split: split_indicator)
799   have "ereal (pmf (pair_pmf A (map_pmf f B)) (a, b)) =
800          ereal (pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b))"
801     unfolding pmf_pair ereal_pmf_map
802     by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_cmult nn_integral_multc pmf_nonneg
803                   emeasure_map_pmf[symmetric] del: emeasure_map_pmf)
804   then show "pmf (pair_pmf A (map_pmf f B)) (a, b) = pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b)"
805     by simp
806 qed
808 lemma map_pair: "map_pmf (\<lambda>(a, b). (f a, g b)) (pair_pmf A B) = pair_pmf (map_pmf f A) (map_pmf g B)"
809   by (simp add: pair_map_pmf2 pair_map_pmf1 map_pmf_comp split_beta')
811 end
813 subsection \<open> Conditional Probabilities \<close>
815 lemma measure_pmf_zero_iff: "measure (measure_pmf p) s = 0 \<longleftrightarrow> set_pmf p \<inter> s = {}"
816   by (subst measure_pmf.prob_eq_0) (auto simp: AE_measure_pmf_iff)
818 context
819   fixes p :: "'a pmf" and s :: "'a set"
820   assumes not_empty: "set_pmf p \<inter> s \<noteq> {}"
821 begin
823 interpretation pmf_as_measure .
825 lemma emeasure_measure_pmf_not_zero: "emeasure (measure_pmf p) s \<noteq> 0"
826 proof
827   assume "emeasure (measure_pmf p) s = 0"
828   then have "AE x in measure_pmf p. x \<notin> s"
829     by (rule AE_I[rotated]) auto
830   with not_empty show False
831     by (auto simp: AE_measure_pmf_iff)
832 qed
834 lemma measure_measure_pmf_not_zero: "measure (measure_pmf p) s \<noteq> 0"
835   using emeasure_measure_pmf_not_zero unfolding measure_pmf.emeasure_eq_measure by simp
837 lift_definition cond_pmf :: "'a pmf" is
838   "uniform_measure (measure_pmf p) s"
839 proof (intro conjI)
840   show "prob_space (uniform_measure (measure_pmf p) s)"
841     by (intro prob_space_uniform_measure) (auto simp: emeasure_measure_pmf_not_zero)
842   show "AE x in uniform_measure (measure_pmf p) s. measure (uniform_measure (measure_pmf p) s) {x} \<noteq> 0"
843     by (simp add: emeasure_measure_pmf_not_zero measure_measure_pmf_not_zero AE_uniform_measure
844                   AE_measure_pmf_iff set_pmf.rep_eq)
845 qed simp
847 lemma pmf_cond: "pmf cond_pmf x = (if x \<in> s then pmf p x / measure p s else 0)"
848   by transfer (simp add: emeasure_measure_pmf_not_zero pmf.rep_eq)
850 lemma set_cond_pmf[simp]: "set_pmf cond_pmf = set_pmf p \<inter> s"
851   by (auto simp add: set_pmf_iff pmf_cond measure_measure_pmf_not_zero split: split_if_asm)
853 end
855 lemma cond_map_pmf:
856   assumes "set_pmf p \<inter> f -` s \<noteq> {}"
857   shows "cond_pmf (map_pmf f p) s = map_pmf f (cond_pmf p (f -` s))"
858 proof -
859   have *: "set_pmf (map_pmf f p) \<inter> s \<noteq> {}"
860     using assms by auto
861   { fix x
862     have "ereal (pmf (map_pmf f (cond_pmf p (f -` s))) x) =
863       emeasure p (f -` s \<inter> f -` {x}) / emeasure p (f -` s)"
864       unfolding ereal_pmf_map cond_pmf.rep_eq[OF assms] by (simp add: nn_integral_uniform_measure)
865     also have "f -` s \<inter> f -` {x} = (if x \<in> s then f -` {x} else {})"
866       by auto
867     also have "emeasure p (if x \<in> s then f -` {x} else {}) / emeasure p (f -` s) =
868       ereal (pmf (cond_pmf (map_pmf f p) s) x)"
869       using measure_measure_pmf_not_zero[OF *]
870       by (simp add: pmf_cond[OF *] ereal_divide' ereal_pmf_map measure_pmf.emeasure_eq_measure[symmetric]
871                del: ereal_divide)
872     finally have "ereal (pmf (cond_pmf (map_pmf f p) s) x) = ereal (pmf (map_pmf f (cond_pmf p (f -` s))) x)"
873       by simp }
874   then show ?thesis
875     by (intro pmf_eqI) simp
876 qed
878 lemma bind_cond_pmf_cancel:
879   assumes [simp]: "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}"
880   assumes [simp]: "\<And>y. y \<in> set_pmf q \<Longrightarrow> set_pmf p \<inter> {x. R x y} \<noteq> {}"
881   assumes [simp]: "\<And>x y. x \<in> set_pmf p \<Longrightarrow> y \<in> set_pmf q \<Longrightarrow> R x y \<Longrightarrow> measure q {y. R x y} = measure p {x. R x y}"
882   shows "bind_pmf p (\<lambda>x. cond_pmf q {y. R x y}) = q"
883 proof (rule pmf_eqI)
884   fix i
885   have "ereal (pmf (bind_pmf p (\<lambda>x. cond_pmf q {y. R x y})) i) =
886     (\<integral>\<^sup>+x. ereal (pmf q i / measure p {x. R x i}) * ereal (indicator {x. R x i} x) \<partial>p)"
887     by (auto simp add: ereal_pmf_bind AE_measure_pmf_iff pmf_cond pmf_eq_0_set_pmf intro!: nn_integral_cong_AE)
888   also have "\<dots> = (pmf q i * measure p {x. R x i}) / measure p {x. R x i}"
889     by (simp add: pmf_nonneg measure_nonneg zero_ereal_def[symmetric] ereal_indicator
890                   nn_integral_cmult measure_pmf.emeasure_eq_measure)
891   also have "\<dots> = pmf q i"
892     by (cases "pmf q i = 0") (simp_all add: pmf_eq_0_set_pmf measure_measure_pmf_not_zero)
893   finally show "pmf (bind_pmf p (\<lambda>x. cond_pmf q {y. R x y})) i = pmf q i"
894     by simp
895 qed
897 subsection \<open> Relator \<close>
899 inductive rel_pmf :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf \<Rightarrow> bool"
900 for R p q
901 where
902   "\<lbrakk> \<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y;
903      map_pmf fst pq = p; map_pmf snd pq = q \<rbrakk>
904   \<Longrightarrow> rel_pmf R p q"
906 lemma rel_pmfI:
907   assumes R: "rel_set R (set_pmf p) (set_pmf q)"
908   assumes eq: "\<And>x y. x \<in> set_pmf p \<Longrightarrow> y \<in> set_pmf q \<Longrightarrow> R x y \<Longrightarrow>
909     measure p {x. R x y} = measure q {y. R x y}"
910   shows "rel_pmf R p q"
911 proof
912   let ?pq = "bind_pmf p (\<lambda>x. bind_pmf (cond_pmf q {y. R x y}) (\<lambda>y. return_pmf (x, y)))"
913   have "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}"
914     using R by (auto simp: rel_set_def)
915   then show "\<And>x y. (x, y) \<in> set_pmf ?pq \<Longrightarrow> R x y"
916     by auto
917   show "map_pmf fst ?pq = p"
918     by (simp add: map_bind_pmf bind_return_pmf')
920   show "map_pmf snd ?pq = q"
921     using R eq
922     apply (simp add: bind_cond_pmf_cancel map_bind_pmf bind_return_pmf')
923     apply (rule bind_cond_pmf_cancel)
924     apply (auto simp: rel_set_def)
925     done
926 qed
928 lemma rel_pmf_imp_rel_set: "rel_pmf R p q \<Longrightarrow> rel_set R (set_pmf p) (set_pmf q)"
929   by (force simp add: rel_pmf.simps rel_set_def)
931 lemma rel_pmfD_measure:
932   assumes rel_R: "rel_pmf R p q" and R: "\<And>a b. R a b \<Longrightarrow> R a y \<longleftrightarrow> R x b"
933   assumes "x \<in> set_pmf p" "y \<in> set_pmf q"
934   shows "measure p {x. R x y} = measure q {y. R x y}"
935 proof -
936   from rel_R obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
937     and eq: "p = map_pmf fst pq" "q = map_pmf snd pq"
938     by (auto elim: rel_pmf.cases)
939   have "measure p {x. R x y} = measure pq {x. R (fst x) y}"
940     by (simp add: eq map_pmf_rep_eq measure_distr)
941   also have "\<dots> = measure pq {y. R x (snd y)}"
942     by (intro measure_pmf.finite_measure_eq_AE)
943        (auto simp: AE_measure_pmf_iff R dest!: pq)
944   also have "\<dots> = measure q {y. R x y}"
945     by (simp add: eq map_pmf_rep_eq measure_distr)
946   finally show "measure p {x. R x y} = measure q {y. R x y}" .
947 qed
949 lemma rel_pmf_iff_measure:
950   assumes "symp R" "transp R"
951   shows "rel_pmf R p q \<longleftrightarrow>
952     rel_set R (set_pmf p) (set_pmf q) \<and>
953     (\<forall>x\<in>set_pmf p. \<forall>y\<in>set_pmf q. R x y \<longrightarrow> measure p {x. R x y} = measure q {y. R x y})"
954   by (safe intro!: rel_pmf_imp_rel_set rel_pmfI)
955      (auto intro!: rel_pmfD_measure dest: sympD[OF \<open>symp R\<close>] transpD[OF \<open>transp R\<close>])
957 lemma quotient_rel_set_disjoint:
958   "equivp R \<Longrightarrow> C \<in> UNIV // {(x, y). R x y} \<Longrightarrow> rel_set R A B \<Longrightarrow> A \<inter> C = {} \<longleftrightarrow> B \<inter> C = {}"
959   using in_quotient_imp_closed[of UNIV "{(x, y). R x y}" C]
960   by (auto 0 0 simp: equivp_equiv rel_set_def set_eq_iff elim: equivpE)
961      (blast dest: equivp_symp)+
963 lemma quotientD: "equiv X R \<Longrightarrow> A \<in> X // R \<Longrightarrow> x \<in> A \<Longrightarrow> A = R `` {x}"
964   by (metis Image_singleton_iff equiv_class_eq_iff quotientE)
966 lemma rel_pmf_iff_equivp:
967   assumes "equivp R"
968   shows "rel_pmf R p q \<longleftrightarrow> (\<forall>C\<in>UNIV // {(x, y). R x y}. measure p C = measure q C)"
969     (is "_ \<longleftrightarrow>   (\<forall>C\<in>_//?R. _)")
970 proof (subst rel_pmf_iff_measure, safe)
971   show "symp R" "transp R"
972     using assms by (auto simp: equivp_reflp_symp_transp)
973 next
974   fix C assume C: "C \<in> UNIV // ?R" and R: "rel_set R (set_pmf p) (set_pmf q)"
975   assume eq: "\<forall>x\<in>set_pmf p. \<forall>y\<in>set_pmf q. R x y \<longrightarrow> measure p {x. R x y} = measure q {y. R x y}"
977   show "measure p C = measure q C"
978   proof cases
979     assume "p \<inter> C = {}"
980     moreover then have "q \<inter> C = {}"
981       using quotient_rel_set_disjoint[OF assms C R] by simp
982     ultimately show ?thesis
983       unfolding measure_pmf_zero_iff[symmetric] by simp
984   next
985     assume "p \<inter> C \<noteq> {}"
986     moreover then have "q \<inter> C \<noteq> {}"
987       using quotient_rel_set_disjoint[OF assms C R] by simp
988     ultimately obtain x y where in_set: "x \<in> set_pmf p" "y \<in> set_pmf q" and in_C: "x \<in> C" "y \<in> C"
989       by auto
990     then have "R x y"
991       using in_quotient_imp_in_rel[of UNIV ?R C x y] C assms
993     with in_set eq have "measure p {x. R x y} = measure q {y. R x y}"
994       by auto
995     moreover have "{y. R x y} = C"
996       using assms `x \<in> C` C quotientD[of UNIV ?R C x] by (simp add: equivp_equiv)
997     moreover have "{x. R x y} = C"
998       using assms `y \<in> C` C quotientD[of UNIV "?R" C y] sympD[of R]
999       by (auto simp add: equivp_equiv elim: equivpE)
1000     ultimately show ?thesis
1001       by auto
1002   qed
1003 next
1004   assume eq: "\<forall>C\<in>UNIV // ?R. measure p C = measure q C"
1005   show "rel_set R (set_pmf p) (set_pmf q)"
1006     unfolding rel_set_def
1007   proof safe
1008     fix x assume x: "x \<in> set_pmf p"
1009     have "{y. R x y} \<in> UNIV // ?R"
1010       by (auto simp: quotient_def)
1011     with eq have *: "measure q {y. R x y} = measure p {y. R x y}"
1012       by auto
1013     have "measure q {y. R x y} \<noteq> 0"
1014       using x assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp)
1015     then show "\<exists>y\<in>set_pmf q. R x y"
1016       unfolding measure_pmf_zero_iff by auto
1017   next
1018     fix y assume y: "y \<in> set_pmf q"
1019     have "{x. R x y} \<in> UNIV // ?R"
1020       using assms by (auto simp: quotient_def dest: equivp_symp)
1021     with eq have *: "measure p {x. R x y} = measure q {x. R x y}"
1022       by auto
1023     have "measure p {x. R x y} \<noteq> 0"
1024       using y assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp)
1025     then show "\<exists>x\<in>set_pmf p. R x y"
1026       unfolding measure_pmf_zero_iff by auto
1027   qed
1029   fix x y assume "x \<in> set_pmf p" "y \<in> set_pmf q" "R x y"
1030   have "{y. R x y} \<in> UNIV // ?R" "{x. R x y} = {y. R x y}"
1031     using assms `R x y` by (auto simp: quotient_def dest: equivp_symp equivp_transp)
1032   with eq show "measure p {x. R x y} = measure q {y. R x y}"
1033     by auto
1034 qed
1036 bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: rel_pmf
1037 proof -
1038   show "map_pmf id = id" by (rule map_pmf_id)
1039   show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose)
1040   show "\<And>f g::'a \<Rightarrow> 'b. \<And>p. (\<And>x. x \<in> set_pmf p \<Longrightarrow> f x = g x) \<Longrightarrow> map_pmf f p = map_pmf g p"
1041     by (intro map_pmf_cong refl)
1043   show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
1044     by (rule pmf_set_map)
1046   show "(card_of (set_pmf p), natLeq) \<in> ordLeq" for p :: "'s pmf"
1047   proof -
1048     have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq"
1049       by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"])
1050          (auto intro: countable_set_pmf)
1051     also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq"
1052       by (metis Field_natLeq card_of_least natLeq_Well_order)
1053     finally show ?thesis .
1054   qed
1056   show "\<And>R. rel_pmf R =
1057          (BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf fst))\<inverse>\<inverse> OO
1058          BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf snd)"
1059      by (auto simp add: fun_eq_iff BNF_Def.Grp_def OO_def rel_pmf.simps)
1061   show "rel_pmf R OO rel_pmf S \<le> rel_pmf (R OO S)"
1062     for R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
1063   proof -
1064     { fix p q r
1065       assume pq: "rel_pmf R p q"
1066         and qr:"rel_pmf S q r"
1067       from pq obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
1068         and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto
1069       from qr obtain qr where qr: "\<And>y z. (y, z) \<in> set_pmf qr \<Longrightarrow> S y z"
1070         and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto
1072       def pr \<equiv> "bind_pmf pq (\<lambda>xy. bind_pmf (cond_pmf qr {yz. fst yz = snd xy}) (\<lambda>yz. return_pmf (fst xy, snd yz)))"
1073       have pr_welldefined: "\<And>y. y \<in> q \<Longrightarrow> qr \<inter> {yz. fst yz = y} \<noteq> {}"
1074         by (force simp: q')
1076       have "rel_pmf (R OO S) p r"
1077       proof (rule rel_pmf.intros)
1078         fix x z assume "(x, z) \<in> pr"
1079         then have "\<exists>y. (x, y) \<in> pq \<and> (y, z) \<in> qr"
1080           by (auto simp: q pr_welldefined pr_def split_beta)
1081         with pq qr show "(R OO S) x z"
1082           by blast
1083       next
1084         have "map_pmf snd pr = map_pmf snd (bind_pmf q (\<lambda>y. cond_pmf qr {yz. fst yz = y}))"
1085           by (simp add: pr_def q split_beta bind_map_pmf map_pmf_def[symmetric] map_bind_pmf map_pmf_comp)
1086         then show "map_pmf snd pr = r"
1087           unfolding r q' bind_map_pmf by (subst (asm) bind_cond_pmf_cancel) (auto simp: eq_commute)
1088       qed (simp add: pr_def map_bind_pmf split_beta map_pmf_def[symmetric] p map_pmf_comp)
1089     }
1090     then show ?thesis
1092   qed
1093 qed (fact natLeq_card_order natLeq_cinfinite)+
1095 lemma rel_pmf_conj[simp]:
1096   "rel_pmf (\<lambda>x y. P \<and> Q x y) x y \<longleftrightarrow> P \<and> rel_pmf Q x y"
1097   "rel_pmf (\<lambda>x y. Q x y \<and> P) x y \<longleftrightarrow> P \<and> rel_pmf Q x y"
1098   using set_pmf_not_empty by (fastforce simp: pmf.in_rel subset_eq)+
1100 lemma rel_pmf_top[simp]: "rel_pmf top = top"
1101   by (auto simp: pmf.in_rel[abs_def] fun_eq_iff map_fst_pair_pmf map_snd_pair_pmf
1102            intro: exI[of _ "pair_pmf x y" for x y])
1104 lemma rel_pmf_return_pmf1: "rel_pmf R (return_pmf x) M \<longleftrightarrow> (\<forall>a\<in>M. R x a)"
1105 proof safe
1106   fix a assume "a \<in> M" "rel_pmf R (return_pmf x) M"
1107   then obtain pq where *: "\<And>a b. (a, b) \<in> set_pmf pq \<Longrightarrow> R a b"
1108     and eq: "return_pmf x = map_pmf fst pq" "M = map_pmf snd pq"
1109     by (force elim: rel_pmf.cases)
1110   moreover have "set_pmf (return_pmf x) = {x}"
1111     by simp
1112   with `a \<in> M` have "(x, a) \<in> pq"
1113     by (force simp: eq)
1114   with * show "R x a"
1115     by auto
1116 qed (auto intro!: rel_pmf.intros[where pq="pair_pmf (return_pmf x) M"]
1117           simp: map_fst_pair_pmf map_snd_pair_pmf)
1119 lemma rel_pmf_return_pmf2: "rel_pmf R M (return_pmf x) \<longleftrightarrow> (\<forall>a\<in>M. R a x)"
1120   by (subst pmf.rel_flip[symmetric]) (simp add: rel_pmf_return_pmf1)
1122 lemma rel_return_pmf[simp]: "rel_pmf R (return_pmf x1) (return_pmf x2) = R x1 x2"
1123   unfolding rel_pmf_return_pmf2 set_return_pmf by simp
1125 lemma rel_pmf_False[simp]: "rel_pmf (\<lambda>x y. False) x y = False"
1126   unfolding pmf.in_rel fun_eq_iff using set_pmf_not_empty by fastforce
1128 lemma rel_pmf_rel_prod:
1129   "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B') \<longleftrightarrow> rel_pmf R A B \<and> rel_pmf S A' B'"
1130 proof safe
1131   assume "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
1132   then obtain pq where pq: "\<And>a b c d. ((a, c), (b, d)) \<in> set_pmf pq \<Longrightarrow> R a b \<and> S c d"
1133     and eq: "map_pmf fst pq = pair_pmf A A'" "map_pmf snd pq = pair_pmf B B'"
1134     by (force elim: rel_pmf.cases)
1135   show "rel_pmf R A B"
1136   proof (rule rel_pmf.intros)
1137     let ?f = "\<lambda>(a, b). (fst a, fst b)"
1138     have [simp]: "(\<lambda>x. fst (?f x)) = fst o fst" "(\<lambda>x. snd (?f x)) = fst o snd"
1139       by auto
1141     show "map_pmf fst (map_pmf ?f pq) = A"
1142       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
1143     show "map_pmf snd (map_pmf ?f pq) = B"
1144       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
1146     fix a b assume "(a, b) \<in> set_pmf (map_pmf ?f pq)"
1147     then obtain c d where "((a, c), (b, d)) \<in> set_pmf pq"
1148       by auto
1149     from pq[OF this] show "R a b" ..
1150   qed
1151   show "rel_pmf S A' B'"
1152   proof (rule rel_pmf.intros)
1153     let ?f = "\<lambda>(a, b). (snd a, snd b)"
1154     have [simp]: "(\<lambda>x. fst (?f x)) = snd o fst" "(\<lambda>x. snd (?f x)) = snd o snd"
1155       by auto
1157     show "map_pmf fst (map_pmf ?f pq) = A'"
1158       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
1159     show "map_pmf snd (map_pmf ?f pq) = B'"
1160       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
1162     fix c d assume "(c, d) \<in> set_pmf (map_pmf ?f pq)"
1163     then obtain a b where "((a, c), (b, d)) \<in> set_pmf pq"
1164       by auto
1165     from pq[OF this] show "S c d" ..
1166   qed
1167 next
1168   assume "rel_pmf R A B" "rel_pmf S A' B'"
1169   then obtain Rpq Spq
1170     where Rpq: "\<And>a b. (a, b) \<in> set_pmf Rpq \<Longrightarrow> R a b"
1171         "map_pmf fst Rpq = A" "map_pmf snd Rpq = B"
1172       and Spq: "\<And>a b. (a, b) \<in> set_pmf Spq \<Longrightarrow> S a b"
1173         "map_pmf fst Spq = A'" "map_pmf snd Spq = B'"
1174     by (force elim: rel_pmf.cases)
1176   let ?f = "(\<lambda>((a, c), (b, d)). ((a, b), (c, d)))"
1177   let ?pq = "map_pmf ?f (pair_pmf Rpq Spq)"
1178   have [simp]: "(\<lambda>x. fst (?f x)) = (\<lambda>(a, b). (fst a, fst b))" "(\<lambda>x. snd (?f x)) = (\<lambda>(a, b). (snd a, snd b))"
1179     by auto
1181   show "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
1182     by (rule rel_pmf.intros[where pq="?pq"])
1183        (auto simp: map_snd_pair_pmf map_fst_pair_pmf map_pmf_comp Rpq Spq
1184                    map_pair)
1185 qed
1187 lemma rel_pmf_reflI:
1188   assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> P x x"
1189   shows "rel_pmf P p p"
1190   by (rule rel_pmf.intros[where pq="map_pmf (\<lambda>x. (x, x)) p"])
1191      (auto simp add: pmf.map_comp o_def assms)
1193 context
1194 begin
1196 interpretation pmf_as_measure .
1198 definition "join_pmf M = bind_pmf M (\<lambda>x. x)"
1200 lemma bind_eq_join_pmf: "bind_pmf M f = join_pmf (map_pmf f M)"
1201   unfolding join_pmf_def bind_map_pmf ..
1203 lemma join_eq_bind_pmf: "join_pmf M = bind_pmf M id"
1204   by (simp add: join_pmf_def id_def)
1206 lemma pmf_join: "pmf (join_pmf N) i = (\<integral>M. pmf M i \<partial>measure_pmf N)"
1207   unfolding join_pmf_def pmf_bind ..
1209 lemma ereal_pmf_join: "ereal (pmf (join_pmf N) i) = (\<integral>\<^sup>+M. pmf M i \<partial>measure_pmf N)"
1210   unfolding join_pmf_def ereal_pmf_bind ..
1212 lemma set_pmf_join_pmf[simp]: "set_pmf (join_pmf f) = (\<Union>p\<in>set_pmf f. set_pmf p)"
1215 lemma join_return_pmf: "join_pmf (return_pmf M) = M"
1216   by (simp add: integral_return pmf_eq_iff pmf_join return_pmf.rep_eq)
1218 lemma map_join_pmf: "map_pmf f (join_pmf AA) = join_pmf (map_pmf (map_pmf f) AA)"
1219   by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf)
1221 lemma join_map_return_pmf: "join_pmf (map_pmf return_pmf A) = A"
1222   by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
1224 end
1226 lemma rel_pmf_joinI:
1227   assumes "rel_pmf (rel_pmf P) p q"
1228   shows "rel_pmf P (join_pmf p) (join_pmf q)"
1229 proof -
1230   from assms obtain pq where p: "p = map_pmf fst pq"
1231     and q: "q = map_pmf snd pq"
1232     and P: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> rel_pmf P x y"
1233     by cases auto
1234   from P obtain PQ
1235     where PQ: "\<And>x y a b. \<lbrakk> (x, y) \<in> set_pmf pq; (a, b) \<in> set_pmf (PQ x y) \<rbrakk> \<Longrightarrow> P a b"
1236     and x: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf fst (PQ x y) = x"
1237     and y: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf snd (PQ x y) = y"
1238     by(metis rel_pmf.simps)
1240   let ?r = "bind_pmf pq (\<lambda>(x, y). PQ x y)"
1241   have "\<And>a b. (a, b) \<in> set_pmf ?r \<Longrightarrow> P a b" by (auto intro: PQ)
1242   moreover have "map_pmf fst ?r = join_pmf p" "map_pmf snd ?r = join_pmf q"
1243     by (simp_all add: p q x y join_pmf_def map_bind_pmf bind_map_pmf split_def cong: bind_pmf_cong)
1244   ultimately show ?thesis ..
1245 qed
1247 lemma rel_pmf_bindI:
1248   assumes pq: "rel_pmf R p q"
1249   and fg: "\<And>x y. R x y \<Longrightarrow> rel_pmf P (f x) (g y)"
1250   shows "rel_pmf P (bind_pmf p f) (bind_pmf q g)"
1251   unfolding bind_eq_join_pmf
1252   by (rule rel_pmf_joinI)
1253      (auto simp add: pmf.rel_map intro: pmf.rel_mono[THEN le_funD, THEN le_funD, THEN le_boolD, THEN mp, OF _ pq] fg)
1255 text {*
1256   Proof that @{const rel_pmf} preserves orders.
1257   Antisymmetry proof follows Thm. 1 in N. Saheb-Djahromi, Cpo's of measures for nondeterminism,
1258   Theoretical Computer Science 12(1):19--37, 1980,
1259   @{url "http://dx.doi.org/10.1016/0304-3975(80)90003-1"}
1260 *}
1262 lemma
1263   assumes *: "rel_pmf R p q"
1264   and refl: "reflp R" and trans: "transp R"
1265   shows measure_Ici: "measure p {y. R x y} \<le> measure q {y. R x y}" (is ?thesis1)
1266   and measure_Ioi: "measure p {y. R x y \<and> \<not> R y x} \<le> measure q {y. R x y \<and> \<not> R y x}" (is ?thesis2)
1267 proof -
1268   from * obtain pq
1269     where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
1270     and p: "p = map_pmf fst pq"
1271     and q: "q = map_pmf snd pq"
1272     by cases auto
1273   show ?thesis1 ?thesis2 unfolding p q map_pmf_rep_eq using refl trans
1274     by(auto 4 3 simp add: measure_distr reflpD AE_measure_pmf_iff intro!: measure_pmf.finite_measure_mono_AE dest!: pq elim: transpE)
1275 qed
1277 lemma rel_pmf_inf:
1278   fixes p q :: "'a pmf"
1279   assumes 1: "rel_pmf R p q"
1280   assumes 2: "rel_pmf R q p"
1281   and refl: "reflp R" and trans: "transp R"
1282   shows "rel_pmf (inf R R\<inverse>\<inverse>) p q"
1283 proof (subst rel_pmf_iff_equivp, safe)
1284   show "equivp (inf R R\<inverse>\<inverse>)"
1285     using trans refl by (auto simp: equivp_reflp_symp_transp intro: sympI transpI reflpI dest: transpD reflpD)
1287   fix C assume "C \<in> UNIV // {(x, y). inf R R\<inverse>\<inverse> x y}"
1288   then obtain x where C: "C = {y. R x y \<and> R y x}"
1289     by (auto elim: quotientE)
1291   let ?R = "\<lambda>x y. R x y \<and> R y x"
1292   let ?\<mu>R = "\<lambda>y. measure q {x. ?R x y}"
1293   have "measure p {y. ?R x y} = measure p ({y. R x y} - {y. R x y \<and> \<not> R y x})"
1294     by(auto intro!: arg_cong[where f="measure p"])
1295   also have "\<dots> = measure p {y. R x y} - measure p {y. R x y \<and> \<not> R y x}"
1296     by (rule measure_pmf.finite_measure_Diff) auto
1297   also have "measure p {y. R x y \<and> \<not> R y x} = measure q {y. R x y \<and> \<not> R y x}"
1298     using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ioi)
1299   also have "measure p {y. R x y} = measure q {y. R x y}"
1300     using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ici)
1301   also have "measure q {y. R x y} - measure q {y. R x y \<and> \<not> R y x} =
1302     measure q ({y. R x y} - {y. R x y \<and> \<not> R y x})"
1303     by(rule measure_pmf.finite_measure_Diff[symmetric]) auto
1304   also have "\<dots> = ?\<mu>R x"
1305     by(auto intro!: arg_cong[where f="measure q"])
1306   finally show "measure p C = measure q C"
1307     by (simp add: C conj_commute)
1308 qed
1310 lemma rel_pmf_antisym:
1311   fixes p q :: "'a pmf"
1312   assumes 1: "rel_pmf R p q"
1313   assumes 2: "rel_pmf R q p"
1314   and refl: "reflp R" and trans: "transp R" and antisym: "antisymP R"
1315   shows "p = q"
1316 proof -
1317   from 1 2 refl trans have "rel_pmf (inf R R\<inverse>\<inverse>) p q" by(rule rel_pmf_inf)
1318   also have "inf R R\<inverse>\<inverse> = op ="
1319     using refl antisym by (auto intro!: ext simp add: reflpD dest: antisymD)
1320   finally show ?thesis unfolding pmf.rel_eq .
1321 qed
1323 lemma reflp_rel_pmf: "reflp R \<Longrightarrow> reflp (rel_pmf R)"
1324 by(blast intro: reflpI rel_pmf_reflI reflpD)
1326 lemma antisymP_rel_pmf:
1327   "\<lbrakk> reflp R; transp R; antisymP R \<rbrakk>
1328   \<Longrightarrow> antisymP (rel_pmf R)"
1329 by(rule antisymI)(blast intro: rel_pmf_antisym)
1331 lemma transp_rel_pmf:
1332   assumes "transp R"
1333   shows "transp (rel_pmf R)"
1334 proof (rule transpI)
1335   fix x y z
1336   assume "rel_pmf R x y" and "rel_pmf R y z"
1337   hence "rel_pmf (R OO R) x z" by (simp add: pmf.rel_compp relcompp.relcompI)
1338   thus "rel_pmf R x z"
1339     using assms by (metis (no_types) pmf.rel_mono rev_predicate2D transp_relcompp_less_eq)
1340 qed
1342 subsection \<open> Distributions \<close>
1344 context
1345 begin
1347 interpretation pmf_as_function .
1349 subsubsection \<open> Bernoulli Distribution \<close>
1351 lift_definition bernoulli_pmf :: "real \<Rightarrow> bool pmf" is
1352   "\<lambda>p b. ((\<lambda>p. if b then p else 1 - p) \<circ> min 1 \<circ> max 0) p"
1353   by (auto simp: nn_integral_count_space_finite[where A="{False, True}"] UNIV_bool
1354            split: split_max split_min)
1356 lemma pmf_bernoulli_True[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) True = p"
1357   by transfer simp
1359 lemma pmf_bernoulli_False[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) False = 1 - p"
1360   by transfer simp
1362 lemma set_pmf_bernoulli: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (bernoulli_pmf p) = UNIV"
1363   by (auto simp add: set_pmf_iff UNIV_bool)
1365 lemma nn_integral_bernoulli_pmf[simp]:
1366   assumes [simp]: "0 \<le> p" "p \<le> 1" "\<And>x. 0 \<le> f x"
1367   shows "(\<integral>\<^sup>+x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
1368   by (subst nn_integral_measure_pmf_support[of UNIV])
1369      (auto simp: UNIV_bool field_simps)
1371 lemma integral_bernoulli_pmf[simp]:
1372   assumes [simp]: "0 \<le> p" "p \<le> 1"
1373   shows "(\<integral>x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
1374   by (subst integral_measure_pmf[of UNIV]) (auto simp: UNIV_bool)
1376 lemma pmf_bernoulli_half [simp]: "pmf (bernoulli_pmf (1 / 2)) x = 1 / 2"
1377 by(cases x) simp_all
1379 lemma measure_pmf_bernoulli_half: "measure_pmf (bernoulli_pmf (1 / 2)) = uniform_count_measure UNIV"
1380 by(rule measure_eqI)(simp_all add: nn_integral_pmf[symmetric] emeasure_uniform_count_measure nn_integral_count_space_finite sets_uniform_count_measure)
1382 subsubsection \<open> Geometric Distribution \<close>
1384 context
1385   fixes p :: real assumes p[arith]: "0 < p" "p \<le> 1"
1386 begin
1388 lift_definition geometric_pmf :: "nat pmf" is "\<lambda>n. (1 - p)^n * p"
1389 proof
1390   have "(\<Sum>i. ereal (p * (1 - p) ^ i)) = ereal (p * (1 / (1 - (1 - p))))"
1391     by (intro sums_suminf_ereal sums_mult geometric_sums) auto
1392   then show "(\<integral>\<^sup>+ x. ereal ((1 - p)^x * p) \<partial>count_space UNIV) = 1"
1393     by (simp add: nn_integral_count_space_nat field_simps)
1394 qed simp
1396 lemma pmf_geometric[simp]: "pmf geometric_pmf n = (1 - p)^n * p"
1397   by transfer rule
1399 end
1401 lemma set_pmf_geometric: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (geometric_pmf p) = UNIV"
1402   by (auto simp: set_pmf_iff)
1404 subsubsection \<open> Uniform Multiset Distribution \<close>
1406 context
1407   fixes M :: "'a multiset" assumes M_not_empty: "M \<noteq> {#}"
1408 begin
1410 lift_definition pmf_of_multiset :: "'a pmf" is "\<lambda>x. count M x / size M"
1411 proof
1412   show "(\<integral>\<^sup>+ x. ereal (real (count M x) / real (size M)) \<partial>count_space UNIV) = 1"
1413     using M_not_empty
1414     by (simp add: zero_less_divide_iff nn_integral_count_space nonempty_has_size
1415                   setsum_divide_distrib[symmetric])
1416        (auto simp: size_multiset_overloaded_eq intro!: setsum.cong)
1417 qed simp
1419 lemma pmf_of_multiset[simp]: "pmf pmf_of_multiset x = count M x / size M"
1420   by transfer rule
1422 lemma set_pmf_of_multiset[simp]: "set_pmf pmf_of_multiset = set_mset M"
1423   by (auto simp: set_pmf_iff)
1425 end
1427 subsubsection \<open> Uniform Distribution \<close>
1429 context
1430   fixes S :: "'a set" assumes S_not_empty: "S \<noteq> {}" and S_finite: "finite S"
1431 begin
1433 lift_definition pmf_of_set :: "'a pmf" is "\<lambda>x. indicator S x / card S"
1434 proof
1435   show "(\<integral>\<^sup>+ x. ereal (indicator S x / real (card S)) \<partial>count_space UNIV) = 1"
1436     using S_not_empty S_finite by (subst nn_integral_count_space'[of S]) auto
1437 qed simp
1439 lemma pmf_of_set[simp]: "pmf pmf_of_set x = indicator S x / card S"
1440   by transfer rule
1442 lemma set_pmf_of_set[simp]: "set_pmf pmf_of_set = S"
1443   using S_finite S_not_empty by (auto simp: set_pmf_iff)
1445 lemma emeasure_pmf_of_set[simp]: "emeasure pmf_of_set S = 1"
1446   by (rule measure_pmf.emeasure_eq_1_AE) (auto simp: AE_measure_pmf_iff)
1448 lemma nn_integral_pmf_of_set':
1449   "(\<And>x. x \<in> S \<Longrightarrow> f x \<ge> 0) \<Longrightarrow> nn_integral (measure_pmf pmf_of_set) f = setsum f S / card S"
1450 apply(subst nn_integral_measure_pmf_finite, simp_all add: S_finite)
1452 apply(subst ereal_divide', simp add: S_not_empty S_finite)
1454 done
1456 lemma nn_integral_pmf_of_set:
1457   "nn_integral (measure_pmf pmf_of_set) f = setsum (max 0 \<circ> f) S / card S"
1458 apply(subst nn_integral_max_0[symmetric])
1459 apply(subst nn_integral_pmf_of_set')
1460 apply simp_all
1461 done
1463 lemma integral_pmf_of_set:
1464   "integral\<^sup>L (measure_pmf pmf_of_set) f = setsum f S / card S"
1465 apply(simp add: real_lebesgue_integral_def integrable_measure_pmf_finite nn_integral_pmf_of_set S_finite)
1466 apply(subst real_of_ereal_minus')
1467  apply(simp add: ereal_max_0 S_finite del: ereal_max)
1468 apply(simp add: ereal_max_0 S_finite S_not_empty del: ereal_max)
1469 apply(simp add: field_simps S_finite S_not_empty)
1470 apply(subst setsum.distrib[symmetric])
1471 apply(rule setsum.cong; simp_all)
1472 done
1474 end
1476 lemma pmf_of_set_singleton: "pmf_of_set {x} = return_pmf x"
1479 lemma map_pmf_of_set_inj:
1480   assumes f: "inj_on f A"
1481   and [simp]: "A \<noteq> {}" "finite A"
1482   shows "map_pmf f (pmf_of_set A) = pmf_of_set (f ` A)" (is "?lhs = ?rhs")
1483 proof(rule pmf_eqI)
1484   fix i
1485   show "pmf ?lhs i = pmf ?rhs i"
1486   proof(cases "i \<in> f ` A")
1487     case True
1488     then obtain i' where "i = f i'" "i' \<in> A" by auto
1489     thus ?thesis using f by(simp add: card_image pmf_map_inj)
1490   next
1491     case False
1492     hence "pmf ?lhs i = 0" by(simp add: pmf_eq_0_set_pmf set_map_pmf)
1493     moreover have "pmf ?rhs i = 0" using False by simp
1494     ultimately show ?thesis by simp
1495   qed
1496 qed
1498 lemma bernoulli_pmf_half_conv_pmf_of_set: "bernoulli_pmf (1 / 2) = pmf_of_set UNIV"
1499 by(rule pmf_eqI) simp_all
1501 subsubsection \<open> Poisson Distribution \<close>
1503 context
1504   fixes rate :: real assumes rate_pos: "0 < rate"
1505 begin
1507 lift_definition poisson_pmf :: "nat pmf" is "\<lambda>k. rate ^ k / fact k * exp (-rate)"
1508 proof  (* by Manuel Eberl *)
1509   have summable: "summable (\<lambda>x::nat. rate ^ x / fact x)" using summable_exp
1510     by (simp add: field_simps divide_inverse [symmetric])
1511   have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x * exp (-rate) \<partial>count_space UNIV) =
1512           exp (-rate) * (\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV)"
1513     by (simp add: field_simps nn_integral_cmult[symmetric])
1514   also from rate_pos have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV) = (\<Sum>x. rate ^ x / fact x)"
1515     by (simp_all add: nn_integral_count_space_nat suminf_ereal summable suminf_ereal_finite)
1516   also have "... = exp rate" unfolding exp_def
1517     by (simp add: field_simps divide_inverse [symmetric])
1518   also have "ereal (exp (-rate)) * ereal (exp rate) = 1"
1520   finally show "(\<integral>\<^sup>+ x. ereal (rate ^ x / (fact x) * exp (- rate)) \<partial>count_space UNIV) = 1" .
1521 qed (simp add: rate_pos[THEN less_imp_le])
1523 lemma pmf_poisson[simp]: "pmf poisson_pmf k = rate ^ k / fact k * exp (-rate)"
1524   by transfer rule
1526 lemma set_pmf_poisson[simp]: "set_pmf poisson_pmf = UNIV"
1527   using rate_pos by (auto simp: set_pmf_iff)
1529 end
1531 subsubsection \<open> Binomial Distribution \<close>
1533 context
1534   fixes n :: nat and p :: real assumes p_nonneg: "0 \<le> p" and p_le_1: "p \<le> 1"
1535 begin
1537 lift_definition binomial_pmf :: "nat pmf" is "\<lambda>k. (n choose k) * p^k * (1 - p)^(n - k)"
1538 proof
1539   have "(\<integral>\<^sup>+k. ereal (real (n choose k) * p ^ k * (1 - p) ^ (n - k)) \<partial>count_space UNIV) =
1540     ereal (\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k))"
1541     using p_le_1 p_nonneg by (subst nn_integral_count_space') auto
1542   also have "(\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k)) = (p + (1 - p)) ^ n"
1543     by (subst binomial_ring) (simp add: atLeast0AtMost)
1544   finally show "(\<integral>\<^sup>+ x. ereal (real (n choose x) * p ^ x * (1 - p) ^ (n - x)) \<partial>count_space UNIV) = 1"
1545     by simp
1546 qed (insert p_nonneg p_le_1, simp)
1548 lemma pmf_binomial[simp]: "pmf binomial_pmf k = (n choose k) * p^k * (1 - p)^(n - k)"
1549   by transfer rule
1551 lemma set_pmf_binomial_eq: "set_pmf binomial_pmf = (if p = 0 then {0} else if p = 1 then {n} else {.. n})"
1552   using p_nonneg p_le_1 unfolding set_eq_iff set_pmf_iff pmf_binomial by (auto simp: set_pmf_iff)
1554 end
1556 end
1558 lemma set_pmf_binomial_0[simp]: "set_pmf (binomial_pmf n 0) = {0}"