src/HOL/Probability/Probability_Mass_Function.thy
 author hoelzl Tue Mar 10 17:50:10 2015 +0100 (2015-03-10) changeset 59670 dee043d19729 parent 59667 651ea265d568 child 59681 f24ab09e4c37 permissions -rw-r--r--
generalized bind_cond_pmf_cancel
```     1 (*  Title:      HOL/Probability/Probability_Mass_Function.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3     Author:     Andreas Lochbihler, ETH Zurich
```
```     4 *)
```
```     5
```
```     6 section \<open> Probability mass function \<close>
```
```     7
```
```     8 theory Probability_Mass_Function
```
```     9 imports
```
```    10   Giry_Monad
```
```    11   "~~/src/HOL/Library/Multiset"
```
```    12 begin
```
```    13
```
```    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
```
```    28
```
```    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)
```
```    31
```
```    32 lemma ereal_divide': "b \<noteq> 0 \<Longrightarrow> ereal (a / b) = ereal a / ereal b"
```
```    33   using ereal_divide[of a b] by simp
```
```    34
```
```    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)` real_eq_of_nat[symmetric] real_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
```
```    76
```
```    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)
```
```   103
```
```   104 subsection \<open> PMF as measure \<close>
```
```   105
```
```   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)
```
```   110
```
```   111 declare [[coercion measure_pmf]]
```
```   112
```
```   113 lemma prob_space_measure_pmf: "prob_space (measure_pmf p)"
```
```   114   using pmf.measure_pmf[of p] by auto
```
```   115
```
```   116 interpretation measure_pmf!: prob_space "measure_pmf M" for M
```
```   117   by (rule prob_space_measure_pmf)
```
```   118
```
```   119 interpretation measure_pmf!: subprob_space "measure_pmf M" for M
```
```   120   by (rule prob_space_imp_subprob_space) unfold_locales
```
```   121
```
```   122 lemma subprob_space_measure_pmf: "subprob_space (measure_pmf x)"
```
```   123   by unfold_locales
```
```   124
```
```   125 locale pmf_as_measure
```
```   126 begin
```
```   127
```
```   128 setup_lifting type_definition_pmf
```
```   129
```
```   130 end
```
```   131
```
```   132 context
```
```   133 begin
```
```   134
```
```   135 interpretation pmf_as_measure .
```
```   136
```
```   137 lemma sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV"
```
```   138   by transfer blast
```
```   139
```
```   140 lemma sets_measure_pmf_count_space[measurable_cong]:
```
```   141   "sets (measure_pmf M) = sets (count_space UNIV)"
```
```   142   by simp
```
```   143
```
```   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
```
```   146
```
```   147 lemma measure_pmf_in_subprob_algebra[measurable (raw)]: "measure_pmf x \<in> space (subprob_algebra (count_space UNIV))"
```
```   148   by (simp add: space_subprob_algebra subprob_space_measure_pmf)
```
```   149
```
```   150 lemma measurable_pmf_measure1[simp]: "measurable (M :: 'a pmf) N = UNIV \<rightarrow> space N"
```
```   151   by (auto simp: measurable_def)
```
```   152
```
```   153 lemma measurable_pmf_measure2[simp]: "measurable N (M :: 'a pmf) = measurable N (count_space UNIV)"
```
```   154   by (intro measurable_cong_sets) simp_all
```
```   155
```
```   156 lemma measurable_pair_restrict_pmf2:
```
```   157   assumes "countable A"
```
```   158   assumes [measurable]: "\<And>y. y \<in> A \<Longrightarrow> (\<lambda>x. f (x, y)) \<in> measurable M L"
```
```   159   shows "f \<in> measurable (M \<Otimes>\<^sub>M restrict_space (measure_pmf N) A) L" (is "f \<in> measurable ?M _")
```
```   160 proof -
```
```   161   have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)"
```
```   162     by (simp add: restrict_count_space)
```
```   163
```
```   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 pair_collapse] assms)
```
```   167         measurable
```
```   168 qed
```
```   169
```
```   170 lemma measurable_pair_restrict_pmf1:
```
```   171   assumes "countable A"
```
```   172   assumes [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N L"
```
```   173   shows "f \<in> measurable (restrict_space (measure_pmf M) A \<Otimes>\<^sub>M N) L"
```
```   174 proof -
```
```   175   have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)"
```
```   176     by (simp add: restrict_count_space)
```
```   177
```
```   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 pair_collapse] assms)
```
```   181         measurable
```
```   182 qed
```
```   183
```
```   184 lift_definition pmf :: "'a pmf \<Rightarrow> 'a \<Rightarrow> real" is "\<lambda>M x. measure M {x}" .
```
```   185
```
```   186 lift_definition set_pmf :: "'a pmf \<Rightarrow> 'a set" is "\<lambda>M. {x. measure M {x} \<noteq> 0}" .
```
```   187 declare [[coercion set_pmf]]
```
```   188
```
```   189 lemma AE_measure_pmf: "AE x in (M::'a pmf). x \<in> M"
```
```   190   by transfer simp
```
```   191
```
```   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])
```
```   196
```
```   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)
```
```   200
```
```   201 lemma countable_set_pmf [simp]: "countable (set_pmf p)"
```
```   202   by transfer (metis prob_space.finite_measure finite_measure.countable_support)
```
```   203
```
```   204 lemma pmf_positive: "x \<in> set_pmf p \<Longrightarrow> 0 < pmf p x"
```
```   205   by transfer (simp add: less_le measure_nonneg)
```
```   206
```
```   207 lemma pmf_nonneg: "0 \<le> pmf p x"
```
```   208   by transfer (simp add: measure_nonneg)
```
```   209
```
```   210 lemma pmf_le_1: "pmf p x \<le> 1"
```
```   211   by (simp add: pmf.rep_eq)
```
```   212
```
```   213 lemma set_pmf_not_empty: "set_pmf M \<noteq> {}"
```
```   214   using AE_measure_pmf[of M] by (intro notI) simp
```
```   215
```
```   216 lemma set_pmf_iff: "x \<in> set_pmf M \<longleftrightarrow> pmf M x \<noteq> 0"
```
```   217   by transfer simp
```
```   218
```
```   219 lemma set_pmf_eq: "set_pmf M = {x. pmf M x \<noteq> 0}"
```
```   220   by (auto simp: set_pmf_iff)
```
```   221
```
```   222 lemma emeasure_pmf_single:
```
```   223   fixes M :: "'a pmf"
```
```   224   shows "emeasure M {x} = pmf M x"
```
```   225   by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
```
```   226
```
```   227 lemma emeasure_measure_pmf_finite: "finite S \<Longrightarrow> emeasure (measure_pmf M) S = (\<Sum>s\<in>S. pmf M s)"
```
```   228   by (subst emeasure_eq_setsum_singleton) (auto simp: emeasure_pmf_single)
```
```   229
```
```   230 lemma measure_measure_pmf_finite: "finite S \<Longrightarrow> measure (measure_pmf M) S = setsum (pmf M) S"
```
```   231   using emeasure_measure_pmf_finite[of S M] by(simp add: measure_pmf.emeasure_eq_measure)
```
```   232
```
```   233 lemma nn_integral_measure_pmf_support:
```
```   234   fixes f :: "'a \<Rightarrow> ereal"
```
```   235   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"
```
```   236   shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>A. f x * pmf M x)"
```
```   237 proof -
```
```   238   have "(\<integral>\<^sup>+x. f x \<partial>M) = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)"
```
```   239     using nn by (intro nn_integral_cong_AE) (auto simp: AE_measure_pmf_iff split: split_indicator)
```
```   240   also have "\<dots> = (\<Sum>x\<in>A. f x * emeasure M {x})"
```
```   241     using assms by (intro nn_integral_indicator_finite) auto
```
```   242   finally show ?thesis
```
```   243     by (simp add: emeasure_measure_pmf_finite)
```
```   244 qed
```
```   245
```
```   246 lemma nn_integral_measure_pmf_finite:
```
```   247   fixes f :: "'a \<Rightarrow> ereal"
```
```   248   assumes f: "finite (set_pmf M)" and nn: "\<And>x. x \<in> set_pmf M \<Longrightarrow> 0 \<le> f x"
```
```   249   shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>set_pmf M. f x * pmf M x)"
```
```   250   using assms by (intro nn_integral_measure_pmf_support) auto
```
```   251 lemma integrable_measure_pmf_finite:
```
```   252   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
```
```   253   shows "finite (set_pmf M) \<Longrightarrow> integrable M f"
```
```   254   by (auto intro!: integrableI_bounded simp: nn_integral_measure_pmf_finite)
```
```   255
```
```   256 lemma integral_measure_pmf:
```
```   257   assumes [simp]: "finite A" and "\<And>a. a \<in> set_pmf M \<Longrightarrow> f a \<noteq> 0 \<Longrightarrow> a \<in> A"
```
```   258   shows "(\<integral>x. f x \<partial>measure_pmf M) = (\<Sum>a\<in>A. f a * pmf M a)"
```
```   259 proof -
```
```   260   have "(\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x * indicator A x \<partial>measure_pmf M)"
```
```   261     using assms(2) by (intro integral_cong_AE) (auto split: split_indicator simp: AE_measure_pmf_iff)
```
```   262   also have "\<dots> = (\<Sum>a\<in>A. f a * pmf M a)"
```
```   263     by (subst integral_indicator_finite_real) (auto simp: measure_def emeasure_measure_pmf_finite)
```
```   264   finally show ?thesis .
```
```   265 qed
```
```   266
```
```   267 lemma integrable_pmf: "integrable (count_space X) (pmf M)"
```
```   268 proof -
```
```   269   have " (\<integral>\<^sup>+ x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+ x. pmf M x \<partial>count_space (M \<inter> X))"
```
```   270     by (auto simp add: nn_integral_count_space_indicator set_pmf_iff intro!: nn_integral_cong split: split_indicator)
```
```   271   then have "integrable (count_space X) (pmf M) = integrable (count_space (M \<inter> X)) (pmf M)"
```
```   272     by (simp add: integrable_iff_bounded pmf_nonneg)
```
```   273   then show ?thesis
```
```   274     by (simp add: pmf.rep_eq measure_pmf.integrable_measure disjoint_family_on_def)
```
```   275 qed
```
```   276
```
```   277 lemma integral_pmf: "(\<integral>x. pmf M x \<partial>count_space X) = measure M X"
```
```   278 proof -
```
```   279   have "(\<integral>x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+x. pmf M x \<partial>count_space X)"
```
```   280     by (simp add: pmf_nonneg integrable_pmf nn_integral_eq_integral)
```
```   281   also have "\<dots> = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space (X \<inter> M))"
```
```   282     by (auto intro!: nn_integral_cong_AE split: split_indicator
```
```   283              simp: pmf.rep_eq measure_pmf.emeasure_eq_measure nn_integral_count_space_indicator
```
```   284                    AE_count_space set_pmf_iff)
```
```   285   also have "\<dots> = emeasure M (X \<inter> M)"
```
```   286     by (rule emeasure_countable_singleton[symmetric]) (auto intro: countable_set_pmf)
```
```   287   also have "\<dots> = emeasure M X"
```
```   288     by (auto intro!: emeasure_eq_AE simp: AE_measure_pmf_iff)
```
```   289   finally show ?thesis
```
```   290     by (simp add: measure_pmf.emeasure_eq_measure)
```
```   291 qed
```
```   292
```
```   293 lemma integral_pmf_restrict:
```
```   294   "(f::'a \<Rightarrow> 'b::{banach, second_countable_topology}) \<in> borel_measurable (count_space UNIV) \<Longrightarrow>
```
```   295     (\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x \<partial>restrict_space M M)"
```
```   296   by (auto intro!: integral_cong_AE simp add: integral_restrict_space AE_measure_pmf_iff)
```
```   297
```
```   298 lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1"
```
```   299 proof -
```
```   300   have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)"
```
```   301     by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf)
```
```   302   then show ?thesis
```
```   303     using measure_pmf.emeasure_space_1 by simp
```
```   304 qed
```
```   305
```
```   306 lemma emeasure_pmf_UNIV [simp]: "emeasure (measure_pmf M) UNIV = 1"
```
```   307 using measure_pmf.emeasure_space_1[of M] by simp
```
```   308
```
```   309 lemma in_null_sets_measure_pmfI:
```
```   310   "A \<inter> set_pmf p = {} \<Longrightarrow> A \<in> null_sets (measure_pmf p)"
```
```   311 using emeasure_eq_0_AE[where ?P="\<lambda>x. x \<in> A" and M="measure_pmf p"]
```
```   312 by(auto simp add: null_sets_def AE_measure_pmf_iff)
```
```   313
```
```   314 lemma measure_subprob: "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
```
```   315   by (simp add: space_subprob_algebra subprob_space_measure_pmf)
```
```   316
```
```   317 subsection \<open> Monad Interpretation \<close>
```
```   318
```
```   319 lemma measurable_measure_pmf[measurable]:
```
```   320   "(\<lambda>x. measure_pmf (M x)) \<in> measurable (count_space UNIV) (subprob_algebra (count_space UNIV))"
```
```   321   by (auto simp: space_subprob_algebra intro!: prob_space_imp_subprob_space) unfold_locales
```
```   322
```
```   323 lemma bind_measure_pmf_cong:
```
```   324   assumes "\<And>x. A x \<in> space (subprob_algebra N)" "\<And>x. B x \<in> space (subprob_algebra N)"
```
```   325   assumes "\<And>i. i \<in> set_pmf x \<Longrightarrow> A i = B i"
```
```   326   shows "bind (measure_pmf x) A = bind (measure_pmf x) B"
```
```   327 proof (rule measure_eqI)
```
```   328   show "sets (measure_pmf x \<guillemotright>= A) = sets (measure_pmf x \<guillemotright>= B)"
```
```   329     using assms by (subst (1 2) sets_bind) (auto simp: space_subprob_algebra)
```
```   330 next
```
```   331   fix X assume "X \<in> sets (measure_pmf x \<guillemotright>= A)"
```
```   332   then have X: "X \<in> sets N"
```
```   333     using assms by (subst (asm) sets_bind) (auto simp: space_subprob_algebra)
```
```   334   show "emeasure (measure_pmf x \<guillemotright>= A) X = emeasure (measure_pmf x \<guillemotright>= B) X"
```
```   335     using assms
```
```   336     by (subst (1 2) emeasure_bind[where N=N, OF _ _ X])
```
```   337        (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
```
```   338 qed
```
```   339
```
```   340 lift_definition bind_pmf :: "'a pmf \<Rightarrow> ('a \<Rightarrow> 'b pmf ) \<Rightarrow> 'b pmf" is bind
```
```   341 proof (clarify, intro conjI)
```
```   342   fix f :: "'a measure" and g :: "'a \<Rightarrow> 'b measure"
```
```   343   assume "prob_space f"
```
```   344   then interpret f: prob_space f .
```
```   345   assume "sets f = UNIV" and ae_f: "AE x in f. measure f {x} \<noteq> 0"
```
```   346   then have s_f[simp]: "sets f = sets (count_space UNIV)"
```
```   347     by simp
```
```   348   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)"
```
```   349   then have g: "\<And>x. prob_space (g x)" and s_g[simp]: "\<And>x. sets (g x) = sets (count_space UNIV)"
```
```   350     and ae_g: "\<And>x. AE y in g x. measure (g x) {y} \<noteq> 0"
```
```   351     by auto
```
```   352
```
```   353   have [measurable]: "g \<in> measurable f (subprob_algebra (count_space UNIV))"
```
```   354     by (auto simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space g)
```
```   355
```
```   356   show "prob_space (f \<guillemotright>= g)"
```
```   357     using g by (intro f.prob_space_bind[where S="count_space UNIV"]) auto
```
```   358   then interpret fg: prob_space "f \<guillemotright>= g" .
```
```   359   show [simp]: "sets (f \<guillemotright>= g) = UNIV"
```
```   360     using sets_eq_imp_space_eq[OF s_f]
```
```   361     by (subst sets_bind[where N="count_space UNIV"]) auto
```
```   362   show "AE x in f \<guillemotright>= g. measure (f \<guillemotright>= g) {x} \<noteq> 0"
```
```   363     apply (simp add: fg.prob_eq_0 AE_bind[where B="count_space UNIV"])
```
```   364     using ae_f
```
```   365     apply eventually_elim
```
```   366     using ae_g
```
```   367     apply eventually_elim
```
```   368     apply (auto dest: AE_measure_singleton)
```
```   369     done
```
```   370 qed
```
```   371
```
```   372 lemma ereal_pmf_bind: "pmf (bind_pmf N f) i = (\<integral>\<^sup>+x. pmf (f x) i \<partial>measure_pmf N)"
```
```   373   unfolding pmf.rep_eq bind_pmf.rep_eq
```
```   374   by (auto simp: measure_pmf.measure_bind[where N="count_space UNIV"] measure_subprob measure_nonneg
```
```   375            intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])
```
```   376
```
```   377 lemma pmf_bind: "pmf (bind_pmf N f) i = (\<integral>x. pmf (f x) i \<partial>measure_pmf N)"
```
```   378   using ereal_pmf_bind[of N f i]
```
```   379   by (subst (asm) nn_integral_eq_integral)
```
```   380      (auto simp: pmf_nonneg pmf_le_1
```
```   381            intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])
```
```   382
```
```   383 lemma bind_pmf_const[simp]: "bind_pmf M (\<lambda>x. c) = c"
```
```   384   by transfer (simp add: bind_const' prob_space_imp_subprob_space)
```
```   385
```
```   386 lemma set_bind_pmf[simp]: "set_pmf (bind_pmf M N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))"
```
```   387   unfolding set_pmf_eq ereal_eq_0(1)[symmetric] ereal_pmf_bind
```
```   388   by (auto simp add: nn_integral_0_iff_AE AE_measure_pmf_iff set_pmf_eq not_le less_le pmf_nonneg)
```
```   389
```
```   390 lemma bind_pmf_cong:
```
```   391   assumes "p = q"
```
```   392   shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g"
```
```   393   unfolding `p = q`[symmetric] measure_pmf_inject[symmetric] bind_pmf.rep_eq
```
```   394   by (auto simp: AE_measure_pmf_iff Pi_iff space_subprob_algebra subprob_space_measure_pmf
```
```   395                  sets_bind[where N="count_space UNIV"] emeasure_bind[where N="count_space UNIV"]
```
```   396            intro!: nn_integral_cong_AE measure_eqI)
```
```   397
```
```   398 lemma bind_pmf_cong_simp:
```
```   399   "p = q \<Longrightarrow> (\<And>x. x \<in> set_pmf q =simp=> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g"
```
```   400   by (simp add: simp_implies_def cong: bind_pmf_cong)
```
```   401
```
```   402 lemma measure_pmf_bind: "measure_pmf (bind_pmf M f) = (measure_pmf M \<guillemotright>= (\<lambda>x. measure_pmf (f x)))"
```
```   403   by transfer simp
```
```   404
```
```   405 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)"
```
```   406   using measurable_measure_pmf[of N]
```
```   407   unfolding measure_pmf_bind
```
```   408   apply (subst (1 3) nn_integral_max_0[symmetric])
```
```   409   apply (intro nn_integral_bind[where B="count_space UNIV"])
```
```   410   apply auto
```
```   411   done
```
```   412
```
```   413 lemma emeasure_bind_pmf[simp]: "emeasure (bind_pmf M N) X = (\<integral>\<^sup>+x. emeasure (N x) X \<partial>M)"
```
```   414   using measurable_measure_pmf[of N]
```
```   415   unfolding measure_pmf_bind
```
```   416   by (subst emeasure_bind[where N="count_space UNIV"]) auto
```
```   417
```
```   418 lift_definition return_pmf :: "'a \<Rightarrow> 'a pmf" is "return (count_space UNIV)"
```
```   419   by (auto intro!: prob_space_return simp: AE_return measure_return)
```
```   420
```
```   421 lemma bind_return_pmf: "bind_pmf (return_pmf x) f = f x"
```
```   422   by transfer
```
```   423      (auto intro!: prob_space_imp_subprob_space bind_return[where N="count_space UNIV"]
```
```   424            simp: space_subprob_algebra)
```
```   425
```
```   426 lemma set_return_pmf[simp]: "set_pmf (return_pmf x) = {x}"
```
```   427   by transfer (auto simp add: measure_return split: split_indicator)
```
```   428
```
```   429 lemma bind_return_pmf': "bind_pmf N return_pmf = N"
```
```   430 proof (transfer, clarify)
```
```   431   fix N :: "'a measure" assume "sets N = UNIV" then show "N \<guillemotright>= return (count_space UNIV) = N"
```
```   432     by (subst return_sets_cong[where N=N]) (simp_all add: bind_return')
```
```   433 qed
```
```   434
```
```   435 lemma bind_assoc_pmf: "bind_pmf (bind_pmf A B) C = bind_pmf A (\<lambda>x. bind_pmf (B x) C)"
```
```   436   by transfer
```
```   437      (auto intro!: bind_assoc[where N="count_space UNIV" and R="count_space UNIV"]
```
```   438            simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space)
```
```   439
```
```   440 definition "map_pmf f M = bind_pmf M (\<lambda>x. return_pmf (f x))"
```
```   441
```
```   442 lemma map_bind_pmf: "map_pmf f (bind_pmf M g) = bind_pmf M (\<lambda>x. map_pmf f (g x))"
```
```   443   by (simp add: map_pmf_def bind_assoc_pmf)
```
```   444
```
```   445 lemma bind_map_pmf: "bind_pmf (map_pmf f M) g = bind_pmf M (\<lambda>x. g (f x))"
```
```   446   by (simp add: map_pmf_def bind_assoc_pmf bind_return_pmf)
```
```   447
```
```   448 lemma map_pmf_transfer[transfer_rule]:
```
```   449   "rel_fun op = (rel_fun cr_pmf cr_pmf) (\<lambda>f M. distr M (count_space UNIV) f) map_pmf"
```
```   450 proof -
```
```   451   have "rel_fun op = (rel_fun pmf_as_measure.cr_pmf pmf_as_measure.cr_pmf)
```
```   452      (\<lambda>f M. M \<guillemotright>= (return (count_space UNIV) o f)) map_pmf"
```
```   453     unfolding map_pmf_def[abs_def] comp_def by transfer_prover
```
```   454   then show ?thesis
```
```   455     by (force simp: rel_fun_def cr_pmf_def bind_return_distr)
```
```   456 qed
```
```   457
```
```   458 lemma map_pmf_rep_eq:
```
```   459   "measure_pmf (map_pmf f M) = distr (measure_pmf M) (count_space UNIV) f"
```
```   460   unfolding map_pmf_def bind_pmf.rep_eq comp_def return_pmf.rep_eq
```
```   461   using bind_return_distr[of M f "count_space UNIV"] by (simp add: comp_def)
```
```   462
```
```   463 lemma map_pmf_id[simp]: "map_pmf id = id"
```
```   464   by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI)
```
```   465
```
```   466 lemma map_pmf_ident[simp]: "map_pmf (\<lambda>x. x) = (\<lambda>x. x)"
```
```   467   using map_pmf_id unfolding id_def .
```
```   468
```
```   469 lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g"
```
```   470   by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def)
```
```   471
```
```   472 lemma map_pmf_comp: "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M"
```
```   473   using map_pmf_compose[of f g] by (simp add: comp_def)
```
```   474
```
```   475 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"
```
```   476   unfolding map_pmf_def by (rule bind_pmf_cong) auto
```
```   477
```
```   478 lemma pmf_set_map: "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
```
```   479   by (auto simp add: comp_def fun_eq_iff map_pmf_def)
```
```   480
```
```   481 lemma set_map_pmf[simp]: "set_pmf (map_pmf f M) = f`set_pmf M"
```
```   482   using pmf_set_map[of f] by (auto simp: comp_def fun_eq_iff)
```
```   483
```
```   484 lemma emeasure_map_pmf[simp]: "emeasure (map_pmf f M) X = emeasure M (f -` X)"
```
```   485   unfolding map_pmf_rep_eq by (subst emeasure_distr) auto
```
```   486
```
```   487 lemma nn_integral_map_pmf[simp]: "(\<integral>\<^sup>+x. f x \<partial>map_pmf g M) = (\<integral>\<^sup>+x. f (g x) \<partial>M)"
```
```   488   unfolding map_pmf_rep_eq by (intro nn_integral_distr) auto
```
```   489
```
```   490 lemma ereal_pmf_map: "pmf (map_pmf f p) x = (\<integral>\<^sup>+ y. indicator (f -` {x}) y \<partial>measure_pmf p)"
```
```   491 proof (transfer fixing: f x)
```
```   492   fix p :: "'b measure"
```
```   493   presume "prob_space p"
```
```   494   then interpret prob_space p .
```
```   495   presume "sets p = UNIV"
```
```   496   then show "ereal (measure (distr p (count_space UNIV) f) {x}) = integral\<^sup>N p (indicator (f -` {x}))"
```
```   497     by(simp add: measure_distr measurable_def emeasure_eq_measure)
```
```   498 qed simp_all
```
```   499
```
```   500 lemma nn_integral_pmf: "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = emeasure (measure_pmf p) A"
```
```   501 proof -
```
```   502   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))"
```
```   503     by(auto simp add: nn_integral_count_space_indicator indicator_def set_pmf_iff intro: nn_integral_cong)
```
```   504   also have "\<dots> = emeasure (measure_pmf p) (\<Union>x\<in>A \<inter> set_pmf p. {x})"
```
```   505     by(subst emeasure_UN_countable)(auto simp add: emeasure_pmf_single disjoint_family_on_def)
```
```   506   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})"
```
```   507     by(rule emeasure_Un_null_set[symmetric])(auto intro: in_null_sets_measure_pmfI)
```
```   508   also have "\<dots> = emeasure (measure_pmf p) A"
```
```   509     by(auto intro: arg_cong2[where f=emeasure])
```
```   510   finally show ?thesis .
```
```   511 qed
```
```   512
```
```   513 lemma map_return_pmf: "map_pmf f (return_pmf x) = return_pmf (f x)"
```
```   514   by transfer (simp add: distr_return)
```
```   515
```
```   516 lemma map_pmf_const[simp]: "map_pmf (\<lambda>_. c) M = return_pmf c"
```
```   517   by transfer (auto simp: prob_space.distr_const)
```
```   518
```
```   519 lemma pmf_return: "pmf (return_pmf x) y = indicator {y} x"
```
```   520   by transfer (simp add: measure_return)
```
```   521
```
```   522 lemma nn_integral_return_pmf[simp]: "0 \<le> f x \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>return_pmf x) = f x"
```
```   523   unfolding return_pmf.rep_eq by (intro nn_integral_return) auto
```
```   524
```
```   525 lemma emeasure_return_pmf[simp]: "emeasure (return_pmf x) X = indicator X x"
```
```   526   unfolding return_pmf.rep_eq by (intro emeasure_return) auto
```
```   527
```
```   528 lemma return_pmf_inj[simp]: "return_pmf x = return_pmf y \<longleftrightarrow> x = y"
```
```   529   by (metis insertI1 set_return_pmf singletonD)
```
```   530
```
```   531 lemma map_pmf_eq_return_pmf_iff:
```
```   532   "map_pmf f p = return_pmf x \<longleftrightarrow> (\<forall>y \<in> set_pmf p. f y = x)"
```
```   533 proof
```
```   534   assume "map_pmf f p = return_pmf x"
```
```   535   then have "set_pmf (map_pmf f p) = set_pmf (return_pmf x)" by simp
```
```   536   then show "\<forall>y \<in> set_pmf p. f y = x" by auto
```
```   537 next
```
```   538   assume "\<forall>y \<in> set_pmf p. f y = x"
```
```   539   then show "map_pmf f p = return_pmf x"
```
```   540     unfolding map_pmf_const[symmetric, of _ p] by (intro map_pmf_cong) auto
```
```   541 qed
```
```   542
```
```   543 definition "pair_pmf A B = bind_pmf A (\<lambda>x. bind_pmf B (\<lambda>y. return_pmf (x, y)))"
```
```   544
```
```   545 lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b"
```
```   546   unfolding pair_pmf_def pmf_bind pmf_return
```
```   547   apply (subst integral_measure_pmf[where A="{b}"])
```
```   548   apply (auto simp: indicator_eq_0_iff)
```
```   549   apply (subst integral_measure_pmf[where A="{a}"])
```
```   550   apply (auto simp: indicator_eq_0_iff setsum_nonneg_eq_0_iff pmf_nonneg)
```
```   551   done
```
```   552
```
```   553 lemma set_pair_pmf[simp]: "set_pmf (pair_pmf A B) = set_pmf A \<times> set_pmf B"
```
```   554   unfolding pair_pmf_def set_bind_pmf set_return_pmf by auto
```
```   555
```
```   556 lemma measure_pmf_in_subprob_space[measurable (raw)]:
```
```   557   "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
```
```   558   by (simp add: space_subprob_algebra) intro_locales
```
```   559
```
```   560 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)"
```
```   561 proof -
```
```   562   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)"
```
```   563     by (subst nn_integral_max_0[symmetric])
```
```   564        (auto simp: AE_measure_pmf_iff intro!: nn_integral_cong_AE)
```
```   565   also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) * indicator (A \<times> B) (a, b) \<partial>B \<partial>A)"
```
```   566     by (simp add: pair_pmf_def)
```
```   567   also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) \<partial>B \<partial>A)"
```
```   568     by (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
```
```   569   finally show ?thesis
```
```   570     unfolding nn_integral_max_0 .
```
```   571 qed
```
```   572
```
```   573 lemma bind_pair_pmf:
```
```   574   assumes M[measurable]: "M \<in> measurable (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) (subprob_algebra N)"
```
```   575   shows "measure_pmf (pair_pmf A B) \<guillemotright>= M = (measure_pmf A \<guillemotright>= (\<lambda>x. measure_pmf B \<guillemotright>= (\<lambda>y. M (x, y))))"
```
```   576     (is "?L = ?R")
```
```   577 proof (rule measure_eqI)
```
```   578   have M'[measurable]: "M \<in> measurable (pair_pmf A B) (subprob_algebra N)"
```
```   579     using M[THEN measurable_space] by (simp_all add: space_pair_measure)
```
```   580
```
```   581   note measurable_bind[where N="count_space UNIV", measurable]
```
```   582   note measure_pmf_in_subprob_space[simp]
```
```   583
```
```   584   have sets_eq_N: "sets ?L = N"
```
```   585     by (subst sets_bind[OF sets_kernel[OF M']]) auto
```
```   586   show "sets ?L = sets ?R"
```
```   587     using measurable_space[OF M]
```
```   588     by (simp add: sets_eq_N space_pair_measure space_subprob_algebra)
```
```   589   fix X assume "X \<in> sets ?L"
```
```   590   then have X[measurable]: "X \<in> sets N"
```
```   591     unfolding sets_eq_N .
```
```   592   then show "emeasure ?L X = emeasure ?R X"
```
```   593     apply (simp add: emeasure_bind[OF _ M' X])
```
```   594     apply (simp add: nn_integral_bind[where B="count_space UNIV"] pair_pmf_def measure_pmf_bind[of A]
```
```   595                      nn_integral_measure_pmf_finite emeasure_nonneg pmf_return one_ereal_def[symmetric])
```
```   596     apply (subst emeasure_bind[OF _ _ X])
```
```   597     apply measurable
```
```   598     apply (subst emeasure_bind[OF _ _ X])
```
```   599     apply measurable
```
```   600     done
```
```   601 qed
```
```   602
```
```   603 lemma map_fst_pair_pmf: "map_pmf fst (pair_pmf A B) = A"
```
```   604   by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
```
```   605
```
```   606 lemma map_snd_pair_pmf: "map_pmf snd (pair_pmf A B) = B"
```
```   607   by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
```
```   608
```
```   609 lemma nn_integral_pmf':
```
```   610   "inj_on f A \<Longrightarrow> (\<integral>\<^sup>+x. pmf p (f x) \<partial>count_space A) = emeasure p (f ` A)"
```
```   611   by (subst nn_integral_bij_count_space[where g=f and B="f`A"])
```
```   612      (auto simp: bij_betw_def nn_integral_pmf)
```
```   613
```
```   614 lemma pmf_le_0_iff[simp]: "pmf M p \<le> 0 \<longleftrightarrow> pmf M p = 0"
```
```   615   using pmf_nonneg[of M p] by simp
```
```   616
```
```   617 lemma min_pmf_0[simp]: "min (pmf M p) 0 = 0" "min 0 (pmf M p) = 0"
```
```   618   using pmf_nonneg[of M p] by simp_all
```
```   619
```
```   620 lemma pmf_eq_0_set_pmf: "pmf M p = 0 \<longleftrightarrow> p \<notin> set_pmf M"
```
```   621   unfolding set_pmf_iff by simp
```
```   622
```
```   623 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"
```
```   624   by (auto simp: pmf.rep_eq map_pmf_rep_eq measure_distr AE_measure_pmf_iff inj_onD
```
```   625            intro!: measure_pmf.finite_measure_eq_AE)
```
```   626
```
```   627 subsection \<open> PMFs as function \<close>
```
```   628
```
```   629 context
```
```   630   fixes f :: "'a \<Rightarrow> real"
```
```   631   assumes nonneg: "\<And>x. 0 \<le> f x"
```
```   632   assumes prob: "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
```
```   633 begin
```
```   634
```
```   635 lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ereal \<circ> f)"
```
```   636 proof (intro conjI)
```
```   637   have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
```
```   638     by (simp split: split_indicator)
```
```   639   show "AE x in density (count_space UNIV) (ereal \<circ> f).
```
```   640     measure (density (count_space UNIV) (ereal \<circ> f)) {x} \<noteq> 0"
```
```   641     by (simp add: AE_density nonneg measure_def emeasure_density max_def)
```
```   642   show "prob_space (density (count_space UNIV) (ereal \<circ> f))"
```
```   643     by default (simp add: emeasure_density prob)
```
```   644 qed simp
```
```   645
```
```   646 lemma pmf_embed_pmf: "pmf embed_pmf x = f x"
```
```   647 proof transfer
```
```   648   have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
```
```   649     by (simp split: split_indicator)
```
```   650   fix x show "measure (density (count_space UNIV) (ereal \<circ> f)) {x} = f x"
```
```   651     by transfer (simp add: measure_def emeasure_density nonneg max_def)
```
```   652 qed
```
```   653
```
```   654 end
```
```   655
```
```   656 lemma embed_pmf_transfer:
```
```   657   "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"
```
```   658   by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer)
```
```   659
```
```   660 lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)"
```
```   661 proof (transfer, elim conjE)
```
```   662   fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
```
```   663   assume "prob_space M" then interpret prob_space M .
```
```   664   show "M = density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))"
```
```   665   proof (rule measure_eqI)
```
```   666     fix A :: "'a set"
```
```   667     have "(\<integral>\<^sup>+ x. ereal (measure M {x}) * indicator A x \<partial>count_space UNIV) =
```
```   668       (\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)"
```
```   669       by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator)
```
```   670     also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))"
```
```   671       by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space)
```
```   672     also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})"
```
```   673       by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support)
```
```   674          (auto simp: disjoint_family_on_def)
```
```   675     also have "\<dots> = emeasure M A"
```
```   676       using ae by (intro emeasure_eq_AE) auto
```
```   677     finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))) A"
```
```   678       using emeasure_space_1 by (simp add: emeasure_density)
```
```   679   qed simp
```
```   680 qed
```
```   681
```
```   682 lemma td_pmf_embed_pmf:
```
```   683   "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}"
```
```   684   unfolding type_definition_def
```
```   685 proof safe
```
```   686   fix p :: "'a pmf"
```
```   687   have "(\<integral>\<^sup>+ x. 1 \<partial>measure_pmf p) = 1"
```
```   688     using measure_pmf.emeasure_space_1[of p] by simp
```
```   689   then show *: "(\<integral>\<^sup>+ x. ereal (pmf p x) \<partial>count_space UNIV) = 1"
```
```   690     by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const)
```
```   691
```
```   692   show "embed_pmf (pmf p) = p"
```
```   693     by (intro measure_pmf_inject[THEN iffD1])
```
```   694        (simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def)
```
```   695 next
```
```   696   fix f :: "'a \<Rightarrow> real" assume "\<forall>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
```
```   697   then show "pmf (embed_pmf f) = f"
```
```   698     by (auto intro!: pmf_embed_pmf)
```
```   699 qed (rule pmf_nonneg)
```
```   700
```
```   701 end
```
```   702
```
```   703 locale pmf_as_function
```
```   704 begin
```
```   705
```
```   706 setup_lifting td_pmf_embed_pmf
```
```   707
```
```   708 lemma set_pmf_transfer[transfer_rule]:
```
```   709   assumes "bi_total A"
```
```   710   shows "rel_fun (pcr_pmf A) (rel_set A) (\<lambda>f. {x. f x \<noteq> 0}) set_pmf"
```
```   711   using `bi_total A`
```
```   712   by (auto simp: pcr_pmf_def cr_pmf_def rel_fun_def rel_set_def bi_total_def Bex_def set_pmf_iff)
```
```   713      metis+
```
```   714
```
```   715 end
```
```   716
```
```   717 context
```
```   718 begin
```
```   719
```
```   720 interpretation pmf_as_function .
```
```   721
```
```   722 lemma pmf_eqI: "(\<And>i. pmf M i = pmf N i) \<Longrightarrow> M = N"
```
```   723   by transfer auto
```
```   724
```
```   725 lemma pmf_eq_iff: "M = N \<longleftrightarrow> (\<forall>i. pmf M i = pmf N i)"
```
```   726   by (auto intro: pmf_eqI)
```
```   727
```
```   728 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))"
```
```   729   unfolding pmf_eq_iff pmf_bind
```
```   730 proof
```
```   731   fix i
```
```   732   interpret B: prob_space "restrict_space B B"
```
```   733     by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
```
```   734        (auto simp: AE_measure_pmf_iff)
```
```   735   interpret A: prob_space "restrict_space A A"
```
```   736     by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
```
```   737        (auto simp: AE_measure_pmf_iff)
```
```   738
```
```   739   interpret AB: pair_prob_space "restrict_space A A" "restrict_space B B"
```
```   740     by unfold_locales
```
```   741
```
```   742   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)"
```
```   743     by (rule integral_cong) (auto intro!: integral_pmf_restrict)
```
```   744   also have "\<dots> = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>restrict_space A A)"
```
```   745     by (intro integral_pmf_restrict B.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
```
```   746               countable_set_pmf borel_measurable_count_space)
```
```   747   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>restrict_space B B)"
```
```   748     by (rule AB.Fubini_integral[symmetric])
```
```   749        (auto intro!: AB.integrable_const_bound[where B=1] measurable_pair_restrict_pmf2
```
```   750              simp: pmf_nonneg pmf_le_1 measurable_restrict_space1)
```
```   751   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>B)"
```
```   752     by (intro integral_pmf_restrict[symmetric] A.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
```
```   753               countable_set_pmf borel_measurable_count_space)
```
```   754   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>A \<partial>B)"
```
```   755     by (rule integral_cong) (auto intro!: integral_pmf_restrict[symmetric])
```
```   756   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)" .
```
```   757 qed
```
```   758
```
```   759 lemma pair_map_pmf1: "pair_pmf (map_pmf f A) B = map_pmf (apfst f) (pair_pmf A B)"
```
```   760 proof (safe intro!: pmf_eqI)
```
```   761   fix a :: "'a" and b :: "'b"
```
```   762   have [simp]: "\<And>c d. indicator (apfst f -` {(a, b)}) (c, d) = indicator (f -` {a}) c * (indicator {b} d::ereal)"
```
```   763     by (auto split: split_indicator)
```
```   764
```
```   765   have "ereal (pmf (pair_pmf (map_pmf f A) B) (a, b)) =
```
```   766          ereal (pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b))"
```
```   767     unfolding pmf_pair ereal_pmf_map
```
```   768     by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_multc pmf_nonneg
```
```   769                   emeasure_map_pmf[symmetric] del: emeasure_map_pmf)
```
```   770   then show "pmf (pair_pmf (map_pmf f A) B) (a, b) = pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b)"
```
```   771     by simp
```
```   772 qed
```
```   773
```
```   774 lemma pair_map_pmf2: "pair_pmf A (map_pmf f B) = map_pmf (apsnd f) (pair_pmf A B)"
```
```   775 proof (safe intro!: pmf_eqI)
```
```   776   fix a :: "'a" and b :: "'b"
```
```   777   have [simp]: "\<And>c d. indicator (apsnd f -` {(a, b)}) (c, d) = indicator {a} c * (indicator (f -` {b}) d::ereal)"
```
```   778     by (auto split: split_indicator)
```
```   779
```
```   780   have "ereal (pmf (pair_pmf A (map_pmf f B)) (a, b)) =
```
```   781          ereal (pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b))"
```
```   782     unfolding pmf_pair ereal_pmf_map
```
```   783     by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_cmult nn_integral_multc pmf_nonneg
```
```   784                   emeasure_map_pmf[symmetric] del: emeasure_map_pmf)
```
```   785   then show "pmf (pair_pmf A (map_pmf f B)) (a, b) = pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b)"
```
```   786     by simp
```
```   787 qed
```
```   788
```
```   789 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)"
```
```   790   by (simp add: pair_map_pmf2 pair_map_pmf1 map_pmf_comp split_beta')
```
```   791
```
```   792 end
```
```   793
```
```   794 subsection \<open> Conditional Probabilities \<close>
```
```   795
```
```   796 lemma measure_pmf_zero_iff: "measure (measure_pmf p) s = 0 \<longleftrightarrow> set_pmf p \<inter> s = {}"
```
```   797   by (subst measure_pmf.prob_eq_0) (auto simp: AE_measure_pmf_iff)
```
```   798
```
```   799 context
```
```   800   fixes p :: "'a pmf" and s :: "'a set"
```
```   801   assumes not_empty: "set_pmf p \<inter> s \<noteq> {}"
```
```   802 begin
```
```   803
```
```   804 interpretation pmf_as_measure .
```
```   805
```
```   806 lemma emeasure_measure_pmf_not_zero: "emeasure (measure_pmf p) s \<noteq> 0"
```
```   807 proof
```
```   808   assume "emeasure (measure_pmf p) s = 0"
```
```   809   then have "AE x in measure_pmf p. x \<notin> s"
```
```   810     by (rule AE_I[rotated]) auto
```
```   811   with not_empty show False
```
```   812     by (auto simp: AE_measure_pmf_iff)
```
```   813 qed
```
```   814
```
```   815 lemma measure_measure_pmf_not_zero: "measure (measure_pmf p) s \<noteq> 0"
```
```   816   using emeasure_measure_pmf_not_zero unfolding measure_pmf.emeasure_eq_measure by simp
```
```   817
```
```   818 lift_definition cond_pmf :: "'a pmf" is
```
```   819   "uniform_measure (measure_pmf p) s"
```
```   820 proof (intro conjI)
```
```   821   show "prob_space (uniform_measure (measure_pmf p) s)"
```
```   822     by (intro prob_space_uniform_measure) (auto simp: emeasure_measure_pmf_not_zero)
```
```   823   show "AE x in uniform_measure (measure_pmf p) s. measure (uniform_measure (measure_pmf p) s) {x} \<noteq> 0"
```
```   824     by (simp add: emeasure_measure_pmf_not_zero measure_measure_pmf_not_zero AE_uniform_measure
```
```   825                   AE_measure_pmf_iff set_pmf.rep_eq)
```
```   826 qed simp
```
```   827
```
```   828 lemma pmf_cond: "pmf cond_pmf x = (if x \<in> s then pmf p x / measure p s else 0)"
```
```   829   by transfer (simp add: emeasure_measure_pmf_not_zero pmf.rep_eq)
```
```   830
```
```   831 lemma set_cond_pmf[simp]: "set_pmf cond_pmf = set_pmf p \<inter> s"
```
```   832   by (auto simp add: set_pmf_iff pmf_cond measure_measure_pmf_not_zero split: split_if_asm)
```
```   833
```
```   834 end
```
```   835
```
```   836 lemma cond_map_pmf:
```
```   837   assumes "set_pmf p \<inter> f -` s \<noteq> {}"
```
```   838   shows "cond_pmf (map_pmf f p) s = map_pmf f (cond_pmf p (f -` s))"
```
```   839 proof -
```
```   840   have *: "set_pmf (map_pmf f p) \<inter> s \<noteq> {}"
```
```   841     using assms by auto
```
```   842   { fix x
```
```   843     have "ereal (pmf (map_pmf f (cond_pmf p (f -` s))) x) =
```
```   844       emeasure p (f -` s \<inter> f -` {x}) / emeasure p (f -` s)"
```
```   845       unfolding ereal_pmf_map cond_pmf.rep_eq[OF assms] by (simp add: nn_integral_uniform_measure)
```
```   846     also have "f -` s \<inter> f -` {x} = (if x \<in> s then f -` {x} else {})"
```
```   847       by auto
```
```   848     also have "emeasure p (if x \<in> s then f -` {x} else {}) / emeasure p (f -` s) =
```
```   849       ereal (pmf (cond_pmf (map_pmf f p) s) x)"
```
```   850       using measure_measure_pmf_not_zero[OF *]
```
```   851       by (simp add: pmf_cond[OF *] ereal_divide' ereal_pmf_map measure_pmf.emeasure_eq_measure[symmetric]
```
```   852                del: ereal_divide)
```
```   853     finally have "ereal (pmf (cond_pmf (map_pmf f p) s) x) = ereal (pmf (map_pmf f (cond_pmf p (f -` s))) x)"
```
```   854       by simp }
```
```   855   then show ?thesis
```
```   856     by (intro pmf_eqI) simp
```
```   857 qed
```
```   858
```
```   859 lemma bind_cond_pmf_cancel:
```
```   860   assumes [simp]: "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}"
```
```   861   assumes [simp]: "\<And>y. y \<in> set_pmf q \<Longrightarrow> set_pmf p \<inter> {x. R x y} \<noteq> {}"
```
```   862   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}"
```
```   863   shows "bind_pmf p (\<lambda>x. cond_pmf q {y. R x y}) = q"
```
```   864 proof (rule pmf_eqI)
```
```   865   fix i
```
```   866   have "ereal (pmf (bind_pmf p (\<lambda>x. cond_pmf q {y. R x y})) i) =
```
```   867     (\<integral>\<^sup>+x. ereal (pmf q i / measure p {x. R x i}) * ereal (indicator {x. R x i} x) \<partial>p)"
```
```   868     by (auto simp add: ereal_pmf_bind AE_measure_pmf_iff pmf_cond pmf_eq_0_set_pmf intro!: nn_integral_cong_AE)
```
```   869   also have "\<dots> = (pmf q i * measure p {x. R x i}) / measure p {x. R x i}"
```
```   870     by (simp add: pmf_nonneg measure_nonneg zero_ereal_def[symmetric] ereal_indicator
```
```   871                   nn_integral_cmult measure_pmf.emeasure_eq_measure)
```
```   872   also have "\<dots> = pmf q i"
```
```   873     by (cases "pmf q i = 0") (simp_all add: pmf_eq_0_set_pmf measure_measure_pmf_not_zero)
```
```   874   finally show "pmf (bind_pmf p (\<lambda>x. cond_pmf q {y. R x y})) i = pmf q i"
```
```   875     by simp
```
```   876 qed
```
```   877
```
```   878 subsection \<open> Relator \<close>
```
```   879
```
```   880 inductive rel_pmf :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf \<Rightarrow> bool"
```
```   881 for R p q
```
```   882 where
```
```   883   "\<lbrakk> \<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y;
```
```   884      map_pmf fst pq = p; map_pmf snd pq = q \<rbrakk>
```
```   885   \<Longrightarrow> rel_pmf R p q"
```
```   886
```
```   887 bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: rel_pmf
```
```   888 proof -
```
```   889   show "map_pmf id = id" by (rule map_pmf_id)
```
```   890   show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose)
```
```   891   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"
```
```   892     by (intro map_pmf_cong refl)
```
```   893
```
```   894   show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
```
```   895     by (rule pmf_set_map)
```
```   896
```
```   897   { fix p :: "'s pmf"
```
```   898     have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq"
```
```   899       by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"])
```
```   900          (auto intro: countable_set_pmf)
```
```   901     also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq"
```
```   902       by (metis Field_natLeq card_of_least natLeq_Well_order)
```
```   903     finally show "(card_of (set_pmf p), natLeq) \<in> ordLeq" . }
```
```   904
```
```   905   show "\<And>R. rel_pmf R =
```
```   906          (BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf fst))\<inverse>\<inverse> OO
```
```   907          BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf snd)"
```
```   908      by (auto simp add: fun_eq_iff BNF_Def.Grp_def OO_def rel_pmf.simps)
```
```   909
```
```   910   { fix p :: "'a pmf" and f :: "'a \<Rightarrow> 'b" and g x
```
```   911     assume p: "\<And>z. z \<in> set_pmf p \<Longrightarrow> f z = g z"
```
```   912       and x: "x \<in> set_pmf p"
```
```   913     thus "f x = g x" by simp }
```
```   914
```
```   915   fix R :: "'a => 'b \<Rightarrow> bool" and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
```
```   916   { fix p q r
```
```   917     assume pq: "rel_pmf R p q"
```
```   918       and qr:"rel_pmf S q r"
```
```   919     from pq obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
```
```   920       and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto
```
```   921     from qr obtain qr where qr: "\<And>y z. (y, z) \<in> set_pmf qr \<Longrightarrow> S y z"
```
```   922       and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto
```
```   923
```
```   924     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)))"
```
```   925     have pr_welldefined: "\<And>y. y \<in> q \<Longrightarrow> qr \<inter> {yz. fst yz = y} \<noteq> {}"
```
```   926       by (force simp: q')
```
```   927
```
```   928     have "rel_pmf (R OO S) p r"
```
```   929     proof (rule rel_pmf.intros)
```
```   930       fix x z assume "(x, z) \<in> pr"
```
```   931       then have "\<exists>y. (x, y) \<in> pq \<and> (y, z) \<in> qr"
```
```   932         by (auto simp: q pr_welldefined pr_def split_beta)
```
```   933       with pq qr show "(R OO S) x z"
```
```   934         by blast
```
```   935     next
```
```   936       have "map_pmf snd pr = map_pmf snd (bind_pmf q (\<lambda>y. cond_pmf qr {yz. fst yz = y}))"
```
```   937         by (simp add: pr_def q split_beta bind_map_pmf map_pmf_def[symmetric] map_bind_pmf map_return_pmf map_pmf_comp)
```
```   938       then show "map_pmf snd pr = r"
```
```   939         unfolding r q' bind_map_pmf by (subst (asm) bind_cond_pmf_cancel) (auto simp: eq_commute)
```
```   940     qed (simp add: pr_def map_bind_pmf split_beta map_return_pmf map_pmf_def[symmetric] p map_pmf_comp) }
```
```   941   then show "rel_pmf R OO rel_pmf S \<le> rel_pmf (R OO S)"
```
```   942     by(auto simp add: le_fun_def)
```
```   943 qed (fact natLeq_card_order natLeq_cinfinite)+
```
```   944
```
```   945 lemma rel_pmf_conj[simp]:
```
```   946   "rel_pmf (\<lambda>x y. P \<and> Q x y) x y \<longleftrightarrow> P \<and> rel_pmf Q x y"
```
```   947   "rel_pmf (\<lambda>x y. Q x y \<and> P) x y \<longleftrightarrow> P \<and> rel_pmf Q x y"
```
```   948   using set_pmf_not_empty by (fastforce simp: pmf.in_rel subset_eq)+
```
```   949
```
```   950 lemma rel_pmf_top[simp]: "rel_pmf top = top"
```
```   951   by (auto simp: pmf.in_rel[abs_def] fun_eq_iff map_fst_pair_pmf map_snd_pair_pmf
```
```   952            intro: exI[of _ "pair_pmf x y" for x y])
```
```   953
```
```   954 lemma rel_pmf_return_pmf1: "rel_pmf R (return_pmf x) M \<longleftrightarrow> (\<forall>a\<in>M. R x a)"
```
```   955 proof safe
```
```   956   fix a assume "a \<in> M" "rel_pmf R (return_pmf x) M"
```
```   957   then obtain pq where *: "\<And>a b. (a, b) \<in> set_pmf pq \<Longrightarrow> R a b"
```
```   958     and eq: "return_pmf x = map_pmf fst pq" "M = map_pmf snd pq"
```
```   959     by (force elim: rel_pmf.cases)
```
```   960   moreover have "set_pmf (return_pmf x) = {x}"
```
```   961     by simp
```
```   962   with `a \<in> M` have "(x, a) \<in> pq"
```
```   963     by (force simp: eq)
```
```   964   with * show "R x a"
```
```   965     by auto
```
```   966 qed (auto intro!: rel_pmf.intros[where pq="pair_pmf (return_pmf x) M"]
```
```   967           simp: map_fst_pair_pmf map_snd_pair_pmf)
```
```   968
```
```   969 lemma rel_pmf_return_pmf2: "rel_pmf R M (return_pmf x) \<longleftrightarrow> (\<forall>a\<in>M. R a x)"
```
```   970   by (subst pmf.rel_flip[symmetric]) (simp add: rel_pmf_return_pmf1)
```
```   971
```
```   972 lemma rel_return_pmf[simp]: "rel_pmf R (return_pmf x1) (return_pmf x2) = R x1 x2"
```
```   973   unfolding rel_pmf_return_pmf2 set_return_pmf by simp
```
```   974
```
```   975 lemma rel_pmf_False[simp]: "rel_pmf (\<lambda>x y. False) x y = False"
```
```   976   unfolding pmf.in_rel fun_eq_iff using set_pmf_not_empty by fastforce
```
```   977
```
```   978 lemma rel_pmf_rel_prod:
```
```   979   "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'"
```
```   980 proof safe
```
```   981   assume "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
```
```   982   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"
```
```   983     and eq: "map_pmf fst pq = pair_pmf A A'" "map_pmf snd pq = pair_pmf B B'"
```
```   984     by (force elim: rel_pmf.cases)
```
```   985   show "rel_pmf R A B"
```
```   986   proof (rule rel_pmf.intros)
```
```   987     let ?f = "\<lambda>(a, b). (fst a, fst b)"
```
```   988     have [simp]: "(\<lambda>x. fst (?f x)) = fst o fst" "(\<lambda>x. snd (?f x)) = fst o snd"
```
```   989       by auto
```
```   990
```
```   991     show "map_pmf fst (map_pmf ?f pq) = A"
```
```   992       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
```
```   993     show "map_pmf snd (map_pmf ?f pq) = B"
```
```   994       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
```
```   995
```
```   996     fix a b assume "(a, b) \<in> set_pmf (map_pmf ?f pq)"
```
```   997     then obtain c d where "((a, c), (b, d)) \<in> set_pmf pq"
```
```   998       by auto
```
```   999     from pq[OF this] show "R a b" ..
```
```  1000   qed
```
```  1001   show "rel_pmf S A' B'"
```
```  1002   proof (rule rel_pmf.intros)
```
```  1003     let ?f = "\<lambda>(a, b). (snd a, snd b)"
```
```  1004     have [simp]: "(\<lambda>x. fst (?f x)) = snd o fst" "(\<lambda>x. snd (?f x)) = snd o snd"
```
```  1005       by auto
```
```  1006
```
```  1007     show "map_pmf fst (map_pmf ?f pq) = A'"
```
```  1008       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
```
```  1009     show "map_pmf snd (map_pmf ?f pq) = B'"
```
```  1010       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
```
```  1011
```
```  1012     fix c d assume "(c, d) \<in> set_pmf (map_pmf ?f pq)"
```
```  1013     then obtain a b where "((a, c), (b, d)) \<in> set_pmf pq"
```
```  1014       by auto
```
```  1015     from pq[OF this] show "S c d" ..
```
```  1016   qed
```
```  1017 next
```
```  1018   assume "rel_pmf R A B" "rel_pmf S A' B'"
```
```  1019   then obtain Rpq Spq
```
```  1020     where Rpq: "\<And>a b. (a, b) \<in> set_pmf Rpq \<Longrightarrow> R a b"
```
```  1021         "map_pmf fst Rpq = A" "map_pmf snd Rpq = B"
```
```  1022       and Spq: "\<And>a b. (a, b) \<in> set_pmf Spq \<Longrightarrow> S a b"
```
```  1023         "map_pmf fst Spq = A'" "map_pmf snd Spq = B'"
```
```  1024     by (force elim: rel_pmf.cases)
```
```  1025
```
```  1026   let ?f = "(\<lambda>((a, c), (b, d)). ((a, b), (c, d)))"
```
```  1027   let ?pq = "map_pmf ?f (pair_pmf Rpq Spq)"
```
```  1028   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))"
```
```  1029     by auto
```
```  1030
```
```  1031   show "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
```
```  1032     by (rule rel_pmf.intros[where pq="?pq"])
```
```  1033        (auto simp: map_snd_pair_pmf map_fst_pair_pmf map_pmf_comp Rpq Spq
```
```  1034                    map_pair)
```
```  1035 qed
```
```  1036
```
```  1037 lemma rel_pmf_reflI:
```
```  1038   assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> P x x"
```
```  1039   shows "rel_pmf P p p"
```
```  1040   by (rule rel_pmf.intros[where pq="map_pmf (\<lambda>x. (x, x)) p"])
```
```  1041      (auto simp add: pmf.map_comp o_def assms)
```
```  1042
```
```  1043 context
```
```  1044 begin
```
```  1045
```
```  1046 interpretation pmf_as_measure .
```
```  1047
```
```  1048 definition "join_pmf M = bind_pmf M (\<lambda>x. x)"
```
```  1049
```
```  1050 lemma bind_eq_join_pmf: "bind_pmf M f = join_pmf (map_pmf f M)"
```
```  1051   unfolding join_pmf_def bind_map_pmf ..
```
```  1052
```
```  1053 lemma join_eq_bind_pmf: "join_pmf M = bind_pmf M id"
```
```  1054   by (simp add: join_pmf_def id_def)
```
```  1055
```
```  1056 lemma pmf_join: "pmf (join_pmf N) i = (\<integral>M. pmf M i \<partial>measure_pmf N)"
```
```  1057   unfolding join_pmf_def pmf_bind ..
```
```  1058
```
```  1059 lemma ereal_pmf_join: "ereal (pmf (join_pmf N) i) = (\<integral>\<^sup>+M. pmf M i \<partial>measure_pmf N)"
```
```  1060   unfolding join_pmf_def ereal_pmf_bind ..
```
```  1061
```
```  1062 lemma set_pmf_join_pmf[simp]: "set_pmf (join_pmf f) = (\<Union>p\<in>set_pmf f. set_pmf p)"
```
```  1063   by (simp add: join_pmf_def)
```
```  1064
```
```  1065 lemma join_return_pmf: "join_pmf (return_pmf M) = M"
```
```  1066   by (simp add: integral_return pmf_eq_iff pmf_join return_pmf.rep_eq)
```
```  1067
```
```  1068 lemma map_join_pmf: "map_pmf f (join_pmf AA) = join_pmf (map_pmf (map_pmf f) AA)"
```
```  1069   by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf)
```
```  1070
```
```  1071 lemma join_map_return_pmf: "join_pmf (map_pmf return_pmf A) = A"
```
```  1072   by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
```
```  1073
```
```  1074 end
```
```  1075
```
```  1076 lemma rel_pmf_joinI:
```
```  1077   assumes "rel_pmf (rel_pmf P) p q"
```
```  1078   shows "rel_pmf P (join_pmf p) (join_pmf q)"
```
```  1079 proof -
```
```  1080   from assms obtain pq where p: "p = map_pmf fst pq"
```
```  1081     and q: "q = map_pmf snd pq"
```
```  1082     and P: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> rel_pmf P x y"
```
```  1083     by cases auto
```
```  1084   from P obtain PQ
```
```  1085     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"
```
```  1086     and x: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf fst (PQ x y) = x"
```
```  1087     and y: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf snd (PQ x y) = y"
```
```  1088     by(metis rel_pmf.simps)
```
```  1089
```
```  1090   let ?r = "bind_pmf pq (\<lambda>(x, y). PQ x y)"
```
```  1091   have "\<And>a b. (a, b) \<in> set_pmf ?r \<Longrightarrow> P a b" by (auto intro: PQ)
```
```  1092   moreover have "map_pmf fst ?r = join_pmf p" "map_pmf snd ?r = join_pmf q"
```
```  1093     by (simp_all add: p q x y join_pmf_def map_bind_pmf bind_map_pmf split_def cong: bind_pmf_cong)
```
```  1094   ultimately show ?thesis ..
```
```  1095 qed
```
```  1096
```
```  1097 lemma rel_pmf_bindI:
```
```  1098   assumes pq: "rel_pmf R p q"
```
```  1099   and fg: "\<And>x y. R x y \<Longrightarrow> rel_pmf P (f x) (g y)"
```
```  1100   shows "rel_pmf P (bind_pmf p f) (bind_pmf q g)"
```
```  1101   unfolding bind_eq_join_pmf
```
```  1102   by (rule rel_pmf_joinI)
```
```  1103      (auto simp add: pmf.rel_map intro: pmf.rel_mono[THEN le_funD, THEN le_funD, THEN le_boolD, THEN mp, OF _ pq] fg)
```
```  1104
```
```  1105 text {*
```
```  1106   Proof that @{const rel_pmf} preserves orders.
```
```  1107   Antisymmetry proof follows Thm. 1 in N. Saheb-Djahromi, Cpo's of measures for nondeterminism,
```
```  1108   Theoretical Computer Science 12(1):19--37, 1980,
```
```  1109   @{url "http://dx.doi.org/10.1016/0304-3975(80)90003-1"}
```
```  1110 *}
```
```  1111
```
```  1112 lemma
```
```  1113   assumes *: "rel_pmf R p q"
```
```  1114   and refl: "reflp R" and trans: "transp R"
```
```  1115   shows measure_Ici: "measure p {y. R x y} \<le> measure q {y. R x y}" (is ?thesis1)
```
```  1116   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)
```
```  1117 proof -
```
```  1118   from * obtain pq
```
```  1119     where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
```
```  1120     and p: "p = map_pmf fst pq"
```
```  1121     and q: "q = map_pmf snd pq"
```
```  1122     by cases auto
```
```  1123   show ?thesis1 ?thesis2 unfolding p q map_pmf_rep_eq using refl trans
```
```  1124     by(auto 4 3 simp add: measure_distr reflpD AE_measure_pmf_iff intro!: measure_pmf.finite_measure_mono_AE dest!: pq elim: transpE)
```
```  1125 qed
```
```  1126
```
```  1127 lemma rel_pmf_inf:
```
```  1128   fixes p q :: "'a pmf"
```
```  1129   assumes 1: "rel_pmf R p q"
```
```  1130   assumes 2: "rel_pmf R q p"
```
```  1131   and refl: "reflp R" and trans: "transp R"
```
```  1132   shows "rel_pmf (inf R R\<inverse>\<inverse>) p q"
```
```  1133 proof
```
```  1134   let ?R = "\<lambda>x y. R x y \<and> R y x"
```
```  1135   let ?\<mu>R = "\<lambda>y. measure q {x. ?R x y}"
```
```  1136   { fix x
```
```  1137     have "measure p {y. ?R x y} = measure p ({y. R x y} - {y. R x y \<and> \<not> R y x})"
```
```  1138       by(auto intro!: arg_cong[where f="measure p"])
```
```  1139     also have "\<dots> = measure p {y. R x y} - measure p {y. R x y \<and> \<not> R y x}"
```
```  1140       by (rule measure_pmf.finite_measure_Diff) auto
```
```  1141     also have "measure p {y. R x y \<and> \<not> R y x} = measure q {y. R x y \<and> \<not> R y x}"
```
```  1142       using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ioi)
```
```  1143     also have "measure p {y. R x y} = measure q {y. R x y}"
```
```  1144       using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ici)
```
```  1145     also have "measure q {y. R x y} - measure q {y. R x y \<and> ~ R y x} =
```
```  1146       measure q ({y. R x y} - {y. R x y \<and> \<not> R y x})"
```
```  1147       by(rule measure_pmf.finite_measure_Diff[symmetric]) auto
```
```  1148     also have "\<dots> = ?\<mu>R x"
```
```  1149       by(auto intro!: arg_cong[where f="measure q"])
```
```  1150     also note calculation }
```
```  1151   note eq = this
```
```  1152
```
```  1153   def pq \<equiv> "bind_pmf p (\<lambda>x. bind_pmf (cond_pmf q {y. ?R y x}) (\<lambda>y. return_pmf (x, y)))"
```
```  1154
```
```  1155   show "map_pmf fst pq = p"
```
```  1156     by(simp add: pq_def map_bind_pmf map_return_pmf bind_return_pmf')
```
```  1157
```
```  1158   { fix y assume "y \<in> set_pmf p" then have "set_pmf q \<inter> {x. ?R x y} \<noteq> {}"
```
```  1159       unfolding measure_pmf_zero_iff[symmetric] eq[symmetric] by (auto simp: measure_pmf_zero_iff intro: reflpD[OF refl]) }
```
```  1160   note set_p = this
```
```  1161   moreover
```
```  1162   { fix x assume "x \<in> set_pmf q" then have "set_pmf p \<inter> {y. R x y \<and> R y x} \<noteq> {}"
```
```  1163       unfolding measure_pmf_zero_iff[symmetric] eq by (auto simp: measure_pmf_zero_iff intro: reflpD[OF refl]) }
```
```  1164   ultimately
```
```  1165   show "map_pmf snd pq = q"
```
```  1166     unfolding pq_def map_bind_pmf map_return_pmf bind_return_pmf' snd_conv
```
```  1167     by (subst bind_cond_pmf_cancel)
```
```  1168        (auto simp add: eq AE_measure_pmf_iff dest: transpD[OF trans]
```
```  1169              intro!: measure_pmf.finite_measure_eq_AE)
```
```  1170
```
```  1171   fix x y assume "(x, y) \<in> set_pmf pq" with set_p show "inf R R\<inverse>\<inverse> x y"
```
```  1172     by (auto simp add: pq_def)
```
```  1173 qed
```
```  1174
```
```  1175 lemma rel_pmf_antisym:
```
```  1176   fixes p q :: "'a pmf"
```
```  1177   assumes 1: "rel_pmf R p q"
```
```  1178   assumes 2: "rel_pmf R q p"
```
```  1179   and refl: "reflp R" and trans: "transp R" and antisym: "antisymP R"
```
```  1180   shows "p = q"
```
```  1181 proof -
```
```  1182   from 1 2 refl trans have "rel_pmf (inf R R\<inverse>\<inverse>) p q" by(rule rel_pmf_inf)
```
```  1183   also have "inf R R\<inverse>\<inverse> = op ="
```
```  1184     using refl antisym by (auto intro!: ext simp add: reflpD dest: antisymD)
```
```  1185   finally show ?thesis unfolding pmf.rel_eq .
```
```  1186 qed
```
```  1187
```
```  1188 lemma reflp_rel_pmf: "reflp R \<Longrightarrow> reflp (rel_pmf R)"
```
```  1189 by(blast intro: reflpI rel_pmf_reflI reflpD)
```
```  1190
```
```  1191 lemma antisymP_rel_pmf:
```
```  1192   "\<lbrakk> reflp R; transp R; antisymP R \<rbrakk>
```
```  1193   \<Longrightarrow> antisymP (rel_pmf R)"
```
```  1194 by(rule antisymI)(blast intro: rel_pmf_antisym)
```
```  1195
```
```  1196 lemma transp_rel_pmf:
```
```  1197   assumes "transp R"
```
```  1198   shows "transp (rel_pmf R)"
```
```  1199 proof (rule transpI)
```
```  1200   fix x y z
```
```  1201   assume "rel_pmf R x y" and "rel_pmf R y z"
```
```  1202   hence "rel_pmf (R OO R) x z" by (simp add: pmf.rel_compp relcompp.relcompI)
```
```  1203   thus "rel_pmf R x z"
```
```  1204     using assms by (metis (no_types) pmf.rel_mono rev_predicate2D transp_relcompp_less_eq)
```
```  1205 qed
```
```  1206
```
```  1207 subsection \<open> Distributions \<close>
```
```  1208
```
```  1209 context
```
```  1210 begin
```
```  1211
```
```  1212 interpretation pmf_as_function .
```
```  1213
```
```  1214 subsubsection \<open> Bernoulli Distribution \<close>
```
```  1215
```
```  1216 lift_definition bernoulli_pmf :: "real \<Rightarrow> bool pmf" is
```
```  1217   "\<lambda>p b. ((\<lambda>p. if b then p else 1 - p) \<circ> min 1 \<circ> max 0) p"
```
```  1218   by (auto simp: nn_integral_count_space_finite[where A="{False, True}"] UNIV_bool
```
```  1219            split: split_max split_min)
```
```  1220
```
```  1221 lemma pmf_bernoulli_True[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) True = p"
```
```  1222   by transfer simp
```
```  1223
```
```  1224 lemma pmf_bernoulli_False[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) False = 1 - p"
```
```  1225   by transfer simp
```
```  1226
```
```  1227 lemma set_pmf_bernoulli: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (bernoulli_pmf p) = UNIV"
```
```  1228   by (auto simp add: set_pmf_iff UNIV_bool)
```
```  1229
```
```  1230 lemma nn_integral_bernoulli_pmf[simp]:
```
```  1231   assumes [simp]: "0 \<le> p" "p \<le> 1" "\<And>x. 0 \<le> f x"
```
```  1232   shows "(\<integral>\<^sup>+x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
```
```  1233   by (subst nn_integral_measure_pmf_support[of UNIV])
```
```  1234      (auto simp: UNIV_bool field_simps)
```
```  1235
```
```  1236 lemma integral_bernoulli_pmf[simp]:
```
```  1237   assumes [simp]: "0 \<le> p" "p \<le> 1"
```
```  1238   shows "(\<integral>x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
```
```  1239   by (subst integral_measure_pmf[of UNIV]) (auto simp: UNIV_bool)
```
```  1240
```
```  1241 lemma pmf_bernoulli_half [simp]: "pmf (bernoulli_pmf (1 / 2)) x = 1 / 2"
```
```  1242 by(cases x) simp_all
```
```  1243
```
```  1244 lemma measure_pmf_bernoulli_half: "measure_pmf (bernoulli_pmf (1 / 2)) = uniform_count_measure UNIV"
```
```  1245 by(rule measure_eqI)(simp_all add: nn_integral_pmf[symmetric] emeasure_uniform_count_measure nn_integral_count_space_finite sets_uniform_count_measure)
```
```  1246
```
```  1247 subsubsection \<open> Geometric Distribution \<close>
```
```  1248
```
```  1249 lift_definition geometric_pmf :: "nat pmf" is "\<lambda>n. 1 / 2^Suc n"
```
```  1250 proof
```
```  1251   note geometric_sums[of "1 / 2"]
```
```  1252   note sums_mult[OF this, of "1 / 2"]
```
```  1253   from sums_suminf_ereal[OF this]
```
```  1254   show "(\<integral>\<^sup>+ x. ereal (1 / 2 ^ Suc x) \<partial>count_space UNIV) = 1"
```
```  1255     by (simp add: nn_integral_count_space_nat field_simps)
```
```  1256 qed simp
```
```  1257
```
```  1258 lemma pmf_geometric[simp]: "pmf geometric_pmf n = 1 / 2^Suc n"
```
```  1259   by transfer rule
```
```  1260
```
```  1261 lemma set_pmf_geometric[simp]: "set_pmf geometric_pmf = UNIV"
```
```  1262   by (auto simp: set_pmf_iff)
```
```  1263
```
```  1264 subsubsection \<open> Uniform Multiset Distribution \<close>
```
```  1265
```
```  1266 context
```
```  1267   fixes M :: "'a multiset" assumes M_not_empty: "M \<noteq> {#}"
```
```  1268 begin
```
```  1269
```
```  1270 lift_definition pmf_of_multiset :: "'a pmf" is "\<lambda>x. count M x / size M"
```
```  1271 proof
```
```  1272   show "(\<integral>\<^sup>+ x. ereal (real (count M x) / real (size M)) \<partial>count_space UNIV) = 1"
```
```  1273     using M_not_empty
```
```  1274     by (simp add: zero_less_divide_iff nn_integral_count_space nonempty_has_size
```
```  1275                   setsum_divide_distrib[symmetric])
```
```  1276        (auto simp: size_multiset_overloaded_eq intro!: setsum.cong)
```
```  1277 qed simp
```
```  1278
```
```  1279 lemma pmf_of_multiset[simp]: "pmf pmf_of_multiset x = count M x / size M"
```
```  1280   by transfer rule
```
```  1281
```
```  1282 lemma set_pmf_of_multiset[simp]: "set_pmf pmf_of_multiset = set_of M"
```
```  1283   by (auto simp: set_pmf_iff)
```
```  1284
```
```  1285 end
```
```  1286
```
```  1287 subsubsection \<open> Uniform Distribution \<close>
```
```  1288
```
```  1289 context
```
```  1290   fixes S :: "'a set" assumes S_not_empty: "S \<noteq> {}" and S_finite: "finite S"
```
```  1291 begin
```
```  1292
```
```  1293 lift_definition pmf_of_set :: "'a pmf" is "\<lambda>x. indicator S x / card S"
```
```  1294 proof
```
```  1295   show "(\<integral>\<^sup>+ x. ereal (indicator S x / real (card S)) \<partial>count_space UNIV) = 1"
```
```  1296     using S_not_empty S_finite by (subst nn_integral_count_space'[of S]) auto
```
```  1297 qed simp
```
```  1298
```
```  1299 lemma pmf_of_set[simp]: "pmf pmf_of_set x = indicator S x / card S"
```
```  1300   by transfer rule
```
```  1301
```
```  1302 lemma set_pmf_of_set[simp]: "set_pmf pmf_of_set = S"
```
```  1303   using S_finite S_not_empty by (auto simp: set_pmf_iff)
```
```  1304
```
```  1305 lemma emeasure_pmf_of_set[simp]: "emeasure pmf_of_set S = 1"
```
```  1306   by (rule measure_pmf.emeasure_eq_1_AE) (auto simp: AE_measure_pmf_iff)
```
```  1307
```
```  1308 end
```
```  1309
```
```  1310 subsubsection \<open> Poisson Distribution \<close>
```
```  1311
```
```  1312 context
```
```  1313   fixes rate :: real assumes rate_pos: "0 < rate"
```
```  1314 begin
```
```  1315
```
```  1316 lift_definition poisson_pmf :: "nat pmf" is "\<lambda>k. rate ^ k / fact k * exp (-rate)"
```
```  1317 proof
```
```  1318   (* Proof by Manuel Eberl *)
```
```  1319
```
```  1320   have summable: "summable (\<lambda>x::nat. rate ^ x / fact x)" using summable_exp
```
```  1321     by (simp add: field_simps divide_inverse [symmetric])
```
```  1322   have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x * exp (-rate) \<partial>count_space UNIV) =
```
```  1323           exp (-rate) * (\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV)"
```
```  1324     by (simp add: field_simps nn_integral_cmult[symmetric])
```
```  1325   also from rate_pos have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV) = (\<Sum>x. rate ^ x / fact x)"
```
```  1326     by (simp_all add: nn_integral_count_space_nat suminf_ereal summable suminf_ereal_finite)
```
```  1327   also have "... = exp rate" unfolding exp_def
```
```  1328     by (simp add: field_simps divide_inverse [symmetric] transfer_int_nat_factorial)
```
```  1329   also have "ereal (exp (-rate)) * ereal (exp rate) = 1"
```
```  1330     by (simp add: mult_exp_exp)
```
```  1331   finally show "(\<integral>\<^sup>+ x. ereal (rate ^ x / real (fact x) * exp (- rate)) \<partial>count_space UNIV) = 1" .
```
```  1332 qed (simp add: rate_pos[THEN less_imp_le])
```
```  1333
```
```  1334 lemma pmf_poisson[simp]: "pmf poisson_pmf k = rate ^ k / fact k * exp (-rate)"
```
```  1335   by transfer rule
```
```  1336
```
```  1337 lemma set_pmf_poisson[simp]: "set_pmf poisson_pmf = UNIV"
```
```  1338   using rate_pos by (auto simp: set_pmf_iff)
```
```  1339
```
```  1340 end
```
```  1341
```
```  1342 subsubsection \<open> Binomial Distribution \<close>
```
```  1343
```
```  1344 context
```
```  1345   fixes n :: nat and p :: real assumes p_nonneg: "0 \<le> p" and p_le_1: "p \<le> 1"
```
```  1346 begin
```
```  1347
```
```  1348 lift_definition binomial_pmf :: "nat pmf" is "\<lambda>k. (n choose k) * p^k * (1 - p)^(n - k)"
```
```  1349 proof
```
```  1350   have "(\<integral>\<^sup>+k. ereal (real (n choose k) * p ^ k * (1 - p) ^ (n - k)) \<partial>count_space UNIV) =
```
```  1351     ereal (\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k))"
```
```  1352     using p_le_1 p_nonneg by (subst nn_integral_count_space') auto
```
```  1353   also have "(\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k)) = (p + (1 - p)) ^ n"
```
```  1354     by (subst binomial_ring) (simp add: atLeast0AtMost real_of_nat_def)
```
```  1355   finally show "(\<integral>\<^sup>+ x. ereal (real (n choose x) * p ^ x * (1 - p) ^ (n - x)) \<partial>count_space UNIV) = 1"
```
```  1356     by simp
```
```  1357 qed (insert p_nonneg p_le_1, simp)
```
```  1358
```
```  1359 lemma pmf_binomial[simp]: "pmf binomial_pmf k = (n choose k) * p^k * (1 - p)^(n - k)"
```
```  1360   by transfer rule
```
```  1361
```
```  1362 lemma set_pmf_binomial_eq: "set_pmf binomial_pmf = (if p = 0 then {0} else if p = 1 then {n} else {.. n})"
```
```  1363   using p_nonneg p_le_1 unfolding set_eq_iff set_pmf_iff pmf_binomial by (auto simp: set_pmf_iff)
```
```  1364
```
```  1365 end
```
```  1366
```
```  1367 end
```
```  1368
```
```  1369 lemma set_pmf_binomial_0[simp]: "set_pmf (binomial_pmf n 0) = {0}"
```
```  1370   by (simp add: set_pmf_binomial_eq)
```
```  1371
```
```  1372 lemma set_pmf_binomial_1[simp]: "set_pmf (binomial_pmf n 1) = {n}"
```
```  1373   by (simp add: set_pmf_binomial_eq)
```
```  1374
```
```  1375 lemma set_pmf_binomial[simp]: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (binomial_pmf n p) = {..n}"
```
```  1376   by (simp add: set_pmf_binomial_eq)
```
```  1377
```
```  1378 end
```