src/HOL/Probability/Probability_Mass_Function.thy
 author Andreas Lochbihler Wed Nov 11 10:28:22 2015 +0100 (2015-11-11) changeset 61634 48e2de1b1df5 parent 61610 4f54d2759a0b child 61808 fc1556774cfe permissions -rw-r--r--
```     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)` 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_UNIV [simp]: "measure (measure_pmf p) UNIV = 1"
```
```   148 using measure_pmf.prob_space[of p] by simp
```
```   149
```
```   150 lemma measure_pmf_in_subprob_algebra[measurable (raw)]: "measure_pmf x \<in> space (subprob_algebra (count_space UNIV))"
```
```   151   by (simp add: space_subprob_algebra subprob_space_measure_pmf)
```
```   152
```
```   153 lemma measurable_pmf_measure1[simp]: "measurable (M :: 'a pmf) N = UNIV \<rightarrow> space N"
```
```   154   by (auto simp: measurable_def)
```
```   155
```
```   156 lemma measurable_pmf_measure2[simp]: "measurable N (M :: 'a pmf) = measurable N (count_space UNIV)"
```
```   157   by (intro measurable_cong_sets) simp_all
```
```   158
```
```   159 lemma measurable_pair_restrict_pmf2:
```
```   160   assumes "countable A"
```
```   161   assumes [measurable]: "\<And>y. y \<in> A \<Longrightarrow> (\<lambda>x. f (x, y)) \<in> measurable M L"
```
```   162   shows "f \<in> measurable (M \<Otimes>\<^sub>M restrict_space (measure_pmf N) A) L" (is "f \<in> measurable ?M _")
```
```   163 proof -
```
```   164   have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)"
```
```   165     by (simp add: restrict_count_space)
```
```   166
```
```   167   show ?thesis
```
```   168     by (intro measurable_compose_countable'[where f="\<lambda>a b. f (fst b, a)" and g=snd and I=A,
```
```   169                                             unfolded prod.collapse] assms)
```
```   170         measurable
```
```   171 qed
```
```   172
```
```   173 lemma measurable_pair_restrict_pmf1:
```
```   174   assumes "countable A"
```
```   175   assumes [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N L"
```
```   176   shows "f \<in> measurable (restrict_space (measure_pmf M) A \<Otimes>\<^sub>M N) L"
```
```   177 proof -
```
```   178   have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)"
```
```   179     by (simp add: restrict_count_space)
```
```   180
```
```   181   show ?thesis
```
```   182     by (intro measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A,
```
```   183                                             unfolded prod.collapse] assms)
```
```   184         measurable
```
```   185 qed
```
```   186
```
```   187 lift_definition pmf :: "'a pmf \<Rightarrow> 'a \<Rightarrow> real" is "\<lambda>M x. measure M {x}" .
```
```   188
```
```   189 lift_definition set_pmf :: "'a pmf \<Rightarrow> 'a set" is "\<lambda>M. {x. measure M {x} \<noteq> 0}" .
```
```   190 declare [[coercion set_pmf]]
```
```   191
```
```   192 lemma AE_measure_pmf: "AE x in (M::'a pmf). x \<in> M"
```
```   193   by transfer simp
```
```   194
```
```   195 lemma emeasure_pmf_single_eq_zero_iff:
```
```   196   fixes M :: "'a pmf"
```
```   197   shows "emeasure M {y} = 0 \<longleftrightarrow> y \<notin> M"
```
```   198   by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
```
```   199
```
```   200 lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \<longleftrightarrow> (\<forall>y\<in>M. P y)"
```
```   201   using AE_measure_singleton[of M] AE_measure_pmf[of M]
```
```   202   by (auto simp: set_pmf.rep_eq)
```
```   203
```
```   204 lemma AE_pmfI: "(\<And>y. y \<in> set_pmf M \<Longrightarrow> P y) \<Longrightarrow> almost_everywhere (measure_pmf M) P"
```
```   205 by(simp add: AE_measure_pmf_iff)
```
```   206
```
```   207 lemma countable_set_pmf [simp]: "countable (set_pmf p)"
```
```   208   by transfer (metis prob_space.finite_measure finite_measure.countable_support)
```
```   209
```
```   210 lemma pmf_positive: "x \<in> set_pmf p \<Longrightarrow> 0 < pmf p x"
```
```   211   by transfer (simp add: less_le measure_nonneg)
```
```   212
```
```   213 lemma pmf_nonneg: "0 \<le> pmf p x"
```
```   214   by transfer (simp add: measure_nonneg)
```
```   215
```
```   216 lemma pmf_le_1: "pmf p x \<le> 1"
```
```   217   by (simp add: pmf.rep_eq)
```
```   218
```
```   219 lemma set_pmf_not_empty: "set_pmf M \<noteq> {}"
```
```   220   using AE_measure_pmf[of M] by (intro notI) simp
```
```   221
```
```   222 lemma set_pmf_iff: "x \<in> set_pmf M \<longleftrightarrow> pmf M x \<noteq> 0"
```
```   223   by transfer simp
```
```   224
```
```   225 lemma set_pmf_eq: "set_pmf M = {x. pmf M x \<noteq> 0}"
```
```   226   by (auto simp: set_pmf_iff)
```
```   227
```
```   228 lemma emeasure_pmf_single:
```
```   229   fixes M :: "'a pmf"
```
```   230   shows "emeasure M {x} = pmf M x"
```
```   231   by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
```
```   232
```
```   233 lemma measure_pmf_single: "measure (measure_pmf M) {x} = pmf M x"
```
```   234 using emeasure_pmf_single[of M x] by(simp add: measure_pmf.emeasure_eq_measure)
```
```   235
```
```   236 lemma emeasure_measure_pmf_finite: "finite S \<Longrightarrow> emeasure (measure_pmf M) S = (\<Sum>s\<in>S. pmf M s)"
```
```   237   by (subst emeasure_eq_setsum_singleton) (auto simp: emeasure_pmf_single)
```
```   238
```
```   239 lemma measure_measure_pmf_finite: "finite S \<Longrightarrow> measure (measure_pmf M) S = setsum (pmf M) S"
```
```   240   using emeasure_measure_pmf_finite[of S M] by(simp add: measure_pmf.emeasure_eq_measure)
```
```   241
```
```   242 lemma nn_integral_measure_pmf_support:
```
```   243   fixes f :: "'a \<Rightarrow> ereal"
```
```   244   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"
```
```   245   shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>A. f x * pmf M x)"
```
```   246 proof -
```
```   247   have "(\<integral>\<^sup>+x. f x \<partial>M) = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)"
```
```   248     using nn by (intro nn_integral_cong_AE) (auto simp: AE_measure_pmf_iff split: split_indicator)
```
```   249   also have "\<dots> = (\<Sum>x\<in>A. f x * emeasure M {x})"
```
```   250     using assms by (intro nn_integral_indicator_finite) auto
```
```   251   finally show ?thesis
```
```   252     by (simp add: emeasure_measure_pmf_finite)
```
```   253 qed
```
```   254
```
```   255 lemma nn_integral_measure_pmf_finite:
```
```   256   fixes f :: "'a \<Rightarrow> ereal"
```
```   257   assumes f: "finite (set_pmf M)" and nn: "\<And>x. x \<in> set_pmf M \<Longrightarrow> 0 \<le> f x"
```
```   258   shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>set_pmf M. f x * pmf M x)"
```
```   259   using assms by (intro nn_integral_measure_pmf_support) auto
```
```   260 lemma integrable_measure_pmf_finite:
```
```   261   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
```
```   262   shows "finite (set_pmf M) \<Longrightarrow> integrable M f"
```
```   263   by (auto intro!: integrableI_bounded simp: nn_integral_measure_pmf_finite)
```
```   264
```
```   265 lemma integral_measure_pmf:
```
```   266   assumes [simp]: "finite A" and "\<And>a. a \<in> set_pmf M \<Longrightarrow> f a \<noteq> 0 \<Longrightarrow> a \<in> A"
```
```   267   shows "(\<integral>x. f x \<partial>measure_pmf M) = (\<Sum>a\<in>A. f a * pmf M a)"
```
```   268 proof -
```
```   269   have "(\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x * indicator A x \<partial>measure_pmf M)"
```
```   270     using assms(2) by (intro integral_cong_AE) (auto split: split_indicator simp: AE_measure_pmf_iff)
```
```   271   also have "\<dots> = (\<Sum>a\<in>A. f a * pmf M a)"
```
```   272     by (subst integral_indicator_finite_real) (auto simp: measure_def emeasure_measure_pmf_finite)
```
```   273   finally show ?thesis .
```
```   274 qed
```
```   275
```
```   276 lemma integrable_pmf: "integrable (count_space X) (pmf M)"
```
```   277 proof -
```
```   278   have " (\<integral>\<^sup>+ x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+ x. pmf M x \<partial>count_space (M \<inter> X))"
```
```   279     by (auto simp add: nn_integral_count_space_indicator set_pmf_iff intro!: nn_integral_cong split: split_indicator)
```
```   280   then have "integrable (count_space X) (pmf M) = integrable (count_space (M \<inter> X)) (pmf M)"
```
```   281     by (simp add: integrable_iff_bounded pmf_nonneg)
```
```   282   then show ?thesis
```
```   283     by (simp add: pmf.rep_eq measure_pmf.integrable_measure disjoint_family_on_def)
```
```   284 qed
```
```   285
```
```   286 lemma integral_pmf: "(\<integral>x. pmf M x \<partial>count_space X) = measure M X"
```
```   287 proof -
```
```   288   have "(\<integral>x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+x. pmf M x \<partial>count_space X)"
```
```   289     by (simp add: pmf_nonneg integrable_pmf nn_integral_eq_integral)
```
```   290   also have "\<dots> = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space (X \<inter> M))"
```
```   291     by (auto intro!: nn_integral_cong_AE split: split_indicator
```
```   292              simp: pmf.rep_eq measure_pmf.emeasure_eq_measure nn_integral_count_space_indicator
```
```   293                    AE_count_space set_pmf_iff)
```
```   294   also have "\<dots> = emeasure M (X \<inter> M)"
```
```   295     by (rule emeasure_countable_singleton[symmetric]) (auto intro: countable_set_pmf)
```
```   296   also have "\<dots> = emeasure M X"
```
```   297     by (auto intro!: emeasure_eq_AE simp: AE_measure_pmf_iff)
```
```   298   finally show ?thesis
```
```   299     by (simp add: measure_pmf.emeasure_eq_measure)
```
```   300 qed
```
```   301
```
```   302 lemma integral_pmf_restrict:
```
```   303   "(f::'a \<Rightarrow> 'b::{banach, second_countable_topology}) \<in> borel_measurable (count_space UNIV) \<Longrightarrow>
```
```   304     (\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x \<partial>restrict_space M M)"
```
```   305   by (auto intro!: integral_cong_AE simp add: integral_restrict_space AE_measure_pmf_iff)
```
```   306
```
```   307 lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1"
```
```   308 proof -
```
```   309   have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)"
```
```   310     by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf)
```
```   311   then show ?thesis
```
```   312     using measure_pmf.emeasure_space_1 by simp
```
```   313 qed
```
```   314
```
```   315 lemma emeasure_pmf_UNIV [simp]: "emeasure (measure_pmf M) UNIV = 1"
```
```   316 using measure_pmf.emeasure_space_1[of M] by simp
```
```   317
```
```   318 lemma in_null_sets_measure_pmfI:
```
```   319   "A \<inter> set_pmf p = {} \<Longrightarrow> A \<in> null_sets (measure_pmf p)"
```
```   320 using emeasure_eq_0_AE[where ?P="\<lambda>x. x \<in> A" and M="measure_pmf p"]
```
```   321 by(auto simp add: null_sets_def AE_measure_pmf_iff)
```
```   322
```
```   323 lemma measure_subprob: "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
```
```   324   by (simp add: space_subprob_algebra subprob_space_measure_pmf)
```
```   325
```
```   326 subsection \<open> Monad Interpretation \<close>
```
```   327
```
```   328 lemma measurable_measure_pmf[measurable]:
```
```   329   "(\<lambda>x. measure_pmf (M x)) \<in> measurable (count_space UNIV) (subprob_algebra (count_space UNIV))"
```
```   330   by (auto simp: space_subprob_algebra intro!: prob_space_imp_subprob_space) unfold_locales
```
```   331
```
```   332 lemma bind_measure_pmf_cong:
```
```   333   assumes "\<And>x. A x \<in> space (subprob_algebra N)" "\<And>x. B x \<in> space (subprob_algebra N)"
```
```   334   assumes "\<And>i. i \<in> set_pmf x \<Longrightarrow> A i = B i"
```
```   335   shows "bind (measure_pmf x) A = bind (measure_pmf x) B"
```
```   336 proof (rule measure_eqI)
```
```   337   show "sets (measure_pmf x \<guillemotright>= A) = sets (measure_pmf x \<guillemotright>= B)"
```
```   338     using assms by (subst (1 2) sets_bind) (auto simp: space_subprob_algebra)
```
```   339 next
```
```   340   fix X assume "X \<in> sets (measure_pmf x \<guillemotright>= A)"
```
```   341   then have X: "X \<in> sets N"
```
```   342     using assms by (subst (asm) sets_bind) (auto simp: space_subprob_algebra)
```
```   343   show "emeasure (measure_pmf x \<guillemotright>= A) X = emeasure (measure_pmf x \<guillemotright>= B) X"
```
```   344     using assms
```
```   345     by (subst (1 2) emeasure_bind[where N=N, OF _ _ X])
```
```   346        (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
```
```   347 qed
```
```   348
```
```   349 lift_definition bind_pmf :: "'a pmf \<Rightarrow> ('a \<Rightarrow> 'b pmf ) \<Rightarrow> 'b pmf" is bind
```
```   350 proof (clarify, intro conjI)
```
```   351   fix f :: "'a measure" and g :: "'a \<Rightarrow> 'b measure"
```
```   352   assume "prob_space f"
```
```   353   then interpret f: prob_space f .
```
```   354   assume "sets f = UNIV" and ae_f: "AE x in f. measure f {x} \<noteq> 0"
```
```   355   then have s_f[simp]: "sets f = sets (count_space UNIV)"
```
```   356     by simp
```
```   357   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)"
```
```   358   then have g: "\<And>x. prob_space (g x)" and s_g[simp]: "\<And>x. sets (g x) = sets (count_space UNIV)"
```
```   359     and ae_g: "\<And>x. AE y in g x. measure (g x) {y} \<noteq> 0"
```
```   360     by auto
```
```   361
```
```   362   have [measurable]: "g \<in> measurable f (subprob_algebra (count_space UNIV))"
```
```   363     by (auto simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space g)
```
```   364
```
```   365   show "prob_space (f \<guillemotright>= g)"
```
```   366     using g by (intro f.prob_space_bind[where S="count_space UNIV"]) auto
```
```   367   then interpret fg: prob_space "f \<guillemotright>= g" .
```
```   368   show [simp]: "sets (f \<guillemotright>= g) = UNIV"
```
```   369     using sets_eq_imp_space_eq[OF s_f]
```
```   370     by (subst sets_bind[where N="count_space UNIV"]) auto
```
```   371   show "AE x in f \<guillemotright>= g. measure (f \<guillemotright>= g) {x} \<noteq> 0"
```
```   372     apply (simp add: fg.prob_eq_0 AE_bind[where B="count_space UNIV"])
```
```   373     using ae_f
```
```   374     apply eventually_elim
```
```   375     using ae_g
```
```   376     apply eventually_elim
```
```   377     apply (auto dest: AE_measure_singleton)
```
```   378     done
```
```   379 qed
```
```   380
```
```   381 lemma ereal_pmf_bind: "pmf (bind_pmf N f) i = (\<integral>\<^sup>+x. pmf (f x) i \<partial>measure_pmf N)"
```
```   382   unfolding pmf.rep_eq bind_pmf.rep_eq
```
```   383   by (auto simp: measure_pmf.measure_bind[where N="count_space UNIV"] measure_subprob measure_nonneg
```
```   384            intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])
```
```   385
```
```   386 lemma pmf_bind: "pmf (bind_pmf N f) i = (\<integral>x. pmf (f x) i \<partial>measure_pmf N)"
```
```   387   using ereal_pmf_bind[of N f i]
```
```   388   by (subst (asm) nn_integral_eq_integral)
```
```   389      (auto simp: pmf_nonneg pmf_le_1
```
```   390            intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])
```
```   391
```
```   392 lemma bind_pmf_const[simp]: "bind_pmf M (\<lambda>x. c) = c"
```
```   393   by transfer (simp add: bind_const' prob_space_imp_subprob_space)
```
```   394
```
```   395 lemma set_bind_pmf[simp]: "set_pmf (bind_pmf M N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))"
```
```   396   unfolding set_pmf_eq ereal_eq_0(1)[symmetric] ereal_pmf_bind
```
```   397   by (auto simp add: nn_integral_0_iff_AE AE_measure_pmf_iff set_pmf_eq not_le less_le pmf_nonneg)
```
```   398
```
```   399 lemma bind_pmf_cong:
```
```   400   assumes "p = q"
```
```   401   shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g"
```
```   402   unfolding `p = q`[symmetric] measure_pmf_inject[symmetric] bind_pmf.rep_eq
```
```   403   by (auto simp: AE_measure_pmf_iff Pi_iff space_subprob_algebra subprob_space_measure_pmf
```
```   404                  sets_bind[where N="count_space UNIV"] emeasure_bind[where N="count_space UNIV"]
```
```   405            intro!: nn_integral_cong_AE measure_eqI)
```
```   406
```
```   407 lemma bind_pmf_cong_simp:
```
```   408   "p = q \<Longrightarrow> (\<And>x. x \<in> set_pmf q =simp=> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g"
```
```   409   by (simp add: simp_implies_def cong: bind_pmf_cong)
```
```   410
```
```   411 lemma measure_pmf_bind: "measure_pmf (bind_pmf M f) = (measure_pmf M \<guillemotright>= (\<lambda>x. measure_pmf (f x)))"
```
```   412   by transfer simp
```
```   413
```
```   414 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)"
```
```   415   using measurable_measure_pmf[of N]
```
```   416   unfolding measure_pmf_bind
```
```   417   apply (subst (1 3) nn_integral_max_0[symmetric])
```
```   418   apply (intro nn_integral_bind[where B="count_space UNIV"])
```
```   419   apply auto
```
```   420   done
```
```   421
```
```   422 lemma emeasure_bind_pmf[simp]: "emeasure (bind_pmf M N) X = (\<integral>\<^sup>+x. emeasure (N x) X \<partial>M)"
```
```   423   using measurable_measure_pmf[of N]
```
```   424   unfolding measure_pmf_bind
```
```   425   by (subst emeasure_bind[where N="count_space UNIV"]) auto
```
```   426
```
```   427 lift_definition return_pmf :: "'a \<Rightarrow> 'a pmf" is "return (count_space UNIV)"
```
```   428   by (auto intro!: prob_space_return simp: AE_return measure_return)
```
```   429
```
```   430 lemma bind_return_pmf: "bind_pmf (return_pmf x) f = f x"
```
```   431   by transfer
```
```   432      (auto intro!: prob_space_imp_subprob_space bind_return[where N="count_space UNIV"]
```
```   433            simp: space_subprob_algebra)
```
```   434
```
```   435 lemma set_return_pmf[simp]: "set_pmf (return_pmf x) = {x}"
```
```   436   by transfer (auto simp add: measure_return split: split_indicator)
```
```   437
```
```   438 lemma bind_return_pmf': "bind_pmf N return_pmf = N"
```
```   439 proof (transfer, clarify)
```
```   440   fix N :: "'a measure" assume "sets N = UNIV" then show "N \<guillemotright>= return (count_space UNIV) = N"
```
```   441     by (subst return_sets_cong[where N=N]) (simp_all add: bind_return')
```
```   442 qed
```
```   443
```
```   444 lemma bind_assoc_pmf: "bind_pmf (bind_pmf A B) C = bind_pmf A (\<lambda>x. bind_pmf (B x) C)"
```
```   445   by transfer
```
```   446      (auto intro!: bind_assoc[where N="count_space UNIV" and R="count_space UNIV"]
```
```   447            simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space)
```
```   448
```
```   449 definition "map_pmf f M = bind_pmf M (\<lambda>x. return_pmf (f x))"
```
```   450
```
```   451 lemma map_bind_pmf: "map_pmf f (bind_pmf M g) = bind_pmf M (\<lambda>x. map_pmf f (g x))"
```
```   452   by (simp add: map_pmf_def bind_assoc_pmf)
```
```   453
```
```   454 lemma bind_map_pmf: "bind_pmf (map_pmf f M) g = bind_pmf M (\<lambda>x. g (f x))"
```
```   455   by (simp add: map_pmf_def bind_assoc_pmf bind_return_pmf)
```
```   456
```
```   457 lemma map_pmf_transfer[transfer_rule]:
```
```   458   "rel_fun op = (rel_fun cr_pmf cr_pmf) (\<lambda>f M. distr M (count_space UNIV) f) map_pmf"
```
```   459 proof -
```
```   460   have "rel_fun op = (rel_fun pmf_as_measure.cr_pmf pmf_as_measure.cr_pmf)
```
```   461      (\<lambda>f M. M \<guillemotright>= (return (count_space UNIV) o f)) map_pmf"
```
```   462     unfolding map_pmf_def[abs_def] comp_def by transfer_prover
```
```   463   then show ?thesis
```
```   464     by (force simp: rel_fun_def cr_pmf_def bind_return_distr)
```
```   465 qed
```
```   466
```
```   467 lemma map_pmf_rep_eq:
```
```   468   "measure_pmf (map_pmf f M) = distr (measure_pmf M) (count_space UNIV) f"
```
```   469   unfolding map_pmf_def bind_pmf.rep_eq comp_def return_pmf.rep_eq
```
```   470   using bind_return_distr[of M f "count_space UNIV"] by (simp add: comp_def)
```
```   471
```
```   472 lemma map_pmf_id[simp]: "map_pmf id = id"
```
```   473   by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI)
```
```   474
```
```   475 lemma map_pmf_ident[simp]: "map_pmf (\<lambda>x. x) = (\<lambda>x. x)"
```
```   476   using map_pmf_id unfolding id_def .
```
```   477
```
```   478 lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g"
```
```   479   by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def)
```
```   480
```
```   481 lemma map_pmf_comp: "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M"
```
```   482   using map_pmf_compose[of f g] by (simp add: comp_def)
```
```   483
```
```   484 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"
```
```   485   unfolding map_pmf_def by (rule bind_pmf_cong) auto
```
```   486
```
```   487 lemma pmf_set_map: "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
```
```   488   by (auto simp add: comp_def fun_eq_iff map_pmf_def)
```
```   489
```
```   490 lemma set_map_pmf[simp]: "set_pmf (map_pmf f M) = f`set_pmf M"
```
```   491   using pmf_set_map[of f] by (auto simp: comp_def fun_eq_iff)
```
```   492
```
```   493 lemma emeasure_map_pmf[simp]: "emeasure (map_pmf f M) X = emeasure M (f -` X)"
```
```   494   unfolding map_pmf_rep_eq by (subst emeasure_distr) auto
```
```   495
```
```   496 lemma measure_map_pmf[simp]: "measure (map_pmf f M) X = measure M (f -` X)"
```
```   497 using emeasure_map_pmf[of f M X] by(simp add: measure_pmf.emeasure_eq_measure)
```
```   498
```
```   499 lemma nn_integral_map_pmf[simp]: "(\<integral>\<^sup>+x. f x \<partial>map_pmf g M) = (\<integral>\<^sup>+x. f (g x) \<partial>M)"
```
```   500   unfolding map_pmf_rep_eq by (intro nn_integral_distr) auto
```
```   501
```
```   502 lemma ereal_pmf_map: "pmf (map_pmf f p) x = (\<integral>\<^sup>+ y. indicator (f -` {x}) y \<partial>measure_pmf p)"
```
```   503 proof (transfer fixing: f x)
```
```   504   fix p :: "'b measure"
```
```   505   presume "prob_space p"
```
```   506   then interpret prob_space p .
```
```   507   presume "sets p = UNIV"
```
```   508   then show "ereal (measure (distr p (count_space UNIV) f) {x}) = integral\<^sup>N p (indicator (f -` {x}))"
```
```   509     by(simp add: measure_distr measurable_def emeasure_eq_measure)
```
```   510 qed simp_all
```
```   511
```
```   512 lemma nn_integral_pmf: "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = emeasure (measure_pmf p) A"
```
```   513 proof -
```
```   514   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))"
```
```   515     by(auto simp add: nn_integral_count_space_indicator indicator_def set_pmf_iff intro: nn_integral_cong)
```
```   516   also have "\<dots> = emeasure (measure_pmf p) (\<Union>x\<in>A \<inter> set_pmf p. {x})"
```
```   517     by(subst emeasure_UN_countable)(auto simp add: emeasure_pmf_single disjoint_family_on_def)
```
```   518   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})"
```
```   519     by(rule emeasure_Un_null_set[symmetric])(auto intro: in_null_sets_measure_pmfI)
```
```   520   also have "\<dots> = emeasure (measure_pmf p) A"
```
```   521     by(auto intro: arg_cong2[where f=emeasure])
```
```   522   finally show ?thesis .
```
```   523 qed
```
```   524
```
```   525 lemma map_return_pmf [simp]: "map_pmf f (return_pmf x) = return_pmf (f x)"
```
```   526   by transfer (simp add: distr_return)
```
```   527
```
```   528 lemma map_pmf_const[simp]: "map_pmf (\<lambda>_. c) M = return_pmf c"
```
```   529   by transfer (auto simp: prob_space.distr_const)
```
```   530
```
```   531 lemma pmf_return [simp]: "pmf (return_pmf x) y = indicator {y} x"
```
```   532   by transfer (simp add: measure_return)
```
```   533
```
```   534 lemma nn_integral_return_pmf[simp]: "0 \<le> f x \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>return_pmf x) = f x"
```
```   535   unfolding return_pmf.rep_eq by (intro nn_integral_return) auto
```
```   536
```
```   537 lemma emeasure_return_pmf[simp]: "emeasure (return_pmf x) X = indicator X x"
```
```   538   unfolding return_pmf.rep_eq by (intro emeasure_return) auto
```
```   539
```
```   540 lemma return_pmf_inj[simp]: "return_pmf x = return_pmf y \<longleftrightarrow> x = y"
```
```   541   by (metis insertI1 set_return_pmf singletonD)
```
```   542
```
```   543 lemma map_pmf_eq_return_pmf_iff:
```
```   544   "map_pmf f p = return_pmf x \<longleftrightarrow> (\<forall>y \<in> set_pmf p. f y = x)"
```
```   545 proof
```
```   546   assume "map_pmf f p = return_pmf x"
```
```   547   then have "set_pmf (map_pmf f p) = set_pmf (return_pmf x)" by simp
```
```   548   then show "\<forall>y \<in> set_pmf p. f y = x" by auto
```
```   549 next
```
```   550   assume "\<forall>y \<in> set_pmf p. f y = x"
```
```   551   then show "map_pmf f p = return_pmf x"
```
```   552     unfolding map_pmf_const[symmetric, of _ p] by (intro map_pmf_cong) auto
```
```   553 qed
```
```   554
```
```   555 definition "pair_pmf A B = bind_pmf A (\<lambda>x. bind_pmf B (\<lambda>y. return_pmf (x, y)))"
```
```   556
```
```   557 lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b"
```
```   558   unfolding pair_pmf_def pmf_bind pmf_return
```
```   559   apply (subst integral_measure_pmf[where A="{b}"])
```
```   560   apply (auto simp: indicator_eq_0_iff)
```
```   561   apply (subst integral_measure_pmf[where A="{a}"])
```
```   562   apply (auto simp: indicator_eq_0_iff setsum_nonneg_eq_0_iff pmf_nonneg)
```
```   563   done
```
```   564
```
```   565 lemma set_pair_pmf[simp]: "set_pmf (pair_pmf A B) = set_pmf A \<times> set_pmf B"
```
```   566   unfolding pair_pmf_def set_bind_pmf set_return_pmf by auto
```
```   567
```
```   568 lemma measure_pmf_in_subprob_space[measurable (raw)]:
```
```   569   "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
```
```   570   by (simp add: space_subprob_algebra) intro_locales
```
```   571
```
```   572 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)"
```
```   573 proof -
```
```   574   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)"
```
```   575     by (subst nn_integral_max_0[symmetric])
```
```   576        (auto simp: AE_measure_pmf_iff intro!: nn_integral_cong_AE)
```
```   577   also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) * indicator (A \<times> B) (a, b) \<partial>B \<partial>A)"
```
```   578     by (simp add: pair_pmf_def)
```
```   579   also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) \<partial>B \<partial>A)"
```
```   580     by (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
```
```   581   finally show ?thesis
```
```   582     unfolding nn_integral_max_0 .
```
```   583 qed
```
```   584
```
```   585 lemma bind_pair_pmf:
```
```   586   assumes M[measurable]: "M \<in> measurable (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) (subprob_algebra N)"
```
```   587   shows "measure_pmf (pair_pmf A B) \<guillemotright>= M = (measure_pmf A \<guillemotright>= (\<lambda>x. measure_pmf B \<guillemotright>= (\<lambda>y. M (x, y))))"
```
```   588     (is "?L = ?R")
```
```   589 proof (rule measure_eqI)
```
```   590   have M'[measurable]: "M \<in> measurable (pair_pmf A B) (subprob_algebra N)"
```
```   591     using M[THEN measurable_space] by (simp_all add: space_pair_measure)
```
```   592
```
```   593   note measurable_bind[where N="count_space UNIV", measurable]
```
```   594   note measure_pmf_in_subprob_space[simp]
```
```   595
```
```   596   have sets_eq_N: "sets ?L = N"
```
```   597     by (subst sets_bind[OF sets_kernel[OF M']]) auto
```
```   598   show "sets ?L = sets ?R"
```
```   599     using measurable_space[OF M]
```
```   600     by (simp add: sets_eq_N space_pair_measure space_subprob_algebra)
```
```   601   fix X assume "X \<in> sets ?L"
```
```   602   then have X[measurable]: "X \<in> sets N"
```
```   603     unfolding sets_eq_N .
```
```   604   then show "emeasure ?L X = emeasure ?R X"
```
```   605     apply (simp add: emeasure_bind[OF _ M' X])
```
```   606     apply (simp add: nn_integral_bind[where B="count_space UNIV"] pair_pmf_def measure_pmf_bind[of A]
```
```   607                      nn_integral_measure_pmf_finite emeasure_nonneg one_ereal_def[symmetric])
```
```   608     apply (subst emeasure_bind[OF _ _ X])
```
```   609     apply measurable
```
```   610     apply (subst emeasure_bind[OF _ _ X])
```
```   611     apply measurable
```
```   612     done
```
```   613 qed
```
```   614
```
```   615 lemma map_fst_pair_pmf: "map_pmf fst (pair_pmf A B) = A"
```
```   616   by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
```
```   617
```
```   618 lemma map_snd_pair_pmf: "map_pmf snd (pair_pmf A B) = B"
```
```   619   by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
```
```   620
```
```   621 lemma nn_integral_pmf':
```
```   622   "inj_on f A \<Longrightarrow> (\<integral>\<^sup>+x. pmf p (f x) \<partial>count_space A) = emeasure p (f ` A)"
```
```   623   by (subst nn_integral_bij_count_space[where g=f and B="f`A"])
```
```   624      (auto simp: bij_betw_def nn_integral_pmf)
```
```   625
```
```   626 lemma pmf_le_0_iff[simp]: "pmf M p \<le> 0 \<longleftrightarrow> pmf M p = 0"
```
```   627   using pmf_nonneg[of M p] by simp
```
```   628
```
```   629 lemma min_pmf_0[simp]: "min (pmf M p) 0 = 0" "min 0 (pmf M p) = 0"
```
```   630   using pmf_nonneg[of M p] by simp_all
```
```   631
```
```   632 lemma pmf_eq_0_set_pmf: "pmf M p = 0 \<longleftrightarrow> p \<notin> set_pmf M"
```
```   633   unfolding set_pmf_iff by simp
```
```   634
```
```   635 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"
```
```   636   by (auto simp: pmf.rep_eq map_pmf_rep_eq measure_distr AE_measure_pmf_iff inj_onD
```
```   637            intro!: measure_pmf.finite_measure_eq_AE)
```
```   638
```
```   639 lemma pmf_map_inj': "inj f \<Longrightarrow> pmf (map_pmf f M) (f x) = pmf M x"
```
```   640 apply(cases "x \<in> set_pmf M")
```
```   641  apply(simp add: pmf_map_inj[OF subset_inj_on])
```
```   642 apply(simp add: pmf_eq_0_set_pmf[symmetric])
```
```   643 apply(auto simp add: pmf_eq_0_set_pmf dest: injD)
```
```   644 done
```
```   645
```
```   646 lemma pmf_map_outside: "x \<notin> f ` set_pmf M \<Longrightarrow> pmf (map_pmf f M) x = 0"
```
```   647 unfolding pmf_eq_0_set_pmf by simp
```
```   648
```
```   649 subsection \<open> PMFs as function \<close>
```
```   650
```
```   651 context
```
```   652   fixes f :: "'a \<Rightarrow> real"
```
```   653   assumes nonneg: "\<And>x. 0 \<le> f x"
```
```   654   assumes prob: "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
```
```   655 begin
```
```   656
```
```   657 lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ereal \<circ> f)"
```
```   658 proof (intro conjI)
```
```   659   have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
```
```   660     by (simp split: split_indicator)
```
```   661   show "AE x in density (count_space UNIV) (ereal \<circ> f).
```
```   662     measure (density (count_space UNIV) (ereal \<circ> f)) {x} \<noteq> 0"
```
```   663     by (simp add: AE_density nonneg measure_def emeasure_density max_def)
```
```   664   show "prob_space (density (count_space UNIV) (ereal \<circ> f))"
```
```   665     by standard (simp add: emeasure_density prob)
```
```   666 qed simp
```
```   667
```
```   668 lemma pmf_embed_pmf: "pmf embed_pmf x = f x"
```
```   669 proof transfer
```
```   670   have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
```
```   671     by (simp split: split_indicator)
```
```   672   fix x show "measure (density (count_space UNIV) (ereal \<circ> f)) {x} = f x"
```
```   673     by transfer (simp add: measure_def emeasure_density nonneg max_def)
```
```   674 qed
```
```   675
```
```   676 lemma set_embed_pmf: "set_pmf embed_pmf = {x. f x \<noteq> 0}"
```
```   677 by(auto simp add: set_pmf_eq assms pmf_embed_pmf)
```
```   678
```
```   679 end
```
```   680
```
```   681 lemma embed_pmf_transfer:
```
```   682   "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"
```
```   683   by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer)
```
```   684
```
```   685 lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)"
```
```   686 proof (transfer, elim conjE)
```
```   687   fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
```
```   688   assume "prob_space M" then interpret prob_space M .
```
```   689   show "M = density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))"
```
```   690   proof (rule measure_eqI)
```
```   691     fix A :: "'a set"
```
```   692     have "(\<integral>\<^sup>+ x. ereal (measure M {x}) * indicator A x \<partial>count_space UNIV) =
```
```   693       (\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)"
```
```   694       by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator)
```
```   695     also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))"
```
```   696       by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space)
```
```   697     also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})"
```
```   698       by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support)
```
```   699          (auto simp: disjoint_family_on_def)
```
```   700     also have "\<dots> = emeasure M A"
```
```   701       using ae by (intro emeasure_eq_AE) auto
```
```   702     finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))) A"
```
```   703       using emeasure_space_1 by (simp add: emeasure_density)
```
```   704   qed simp
```
```   705 qed
```
```   706
```
```   707 lemma td_pmf_embed_pmf:
```
```   708   "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}"
```
```   709   unfolding type_definition_def
```
```   710 proof safe
```
```   711   fix p :: "'a pmf"
```
```   712   have "(\<integral>\<^sup>+ x. 1 \<partial>measure_pmf p) = 1"
```
```   713     using measure_pmf.emeasure_space_1[of p] by simp
```
```   714   then show *: "(\<integral>\<^sup>+ x. ereal (pmf p x) \<partial>count_space UNIV) = 1"
```
```   715     by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const)
```
```   716
```
```   717   show "embed_pmf (pmf p) = p"
```
```   718     by (intro measure_pmf_inject[THEN iffD1])
```
```   719        (simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def)
```
```   720 next
```
```   721   fix f :: "'a \<Rightarrow> real" assume "\<forall>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
```
```   722   then show "pmf (embed_pmf f) = f"
```
```   723     by (auto intro!: pmf_embed_pmf)
```
```   724 qed (rule pmf_nonneg)
```
```   725
```
```   726 end
```
```   727
```
```   728 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"
```
```   729 by(simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg)
```
```   730
```
```   731 locale pmf_as_function
```
```   732 begin
```
```   733
```
```   734 setup_lifting td_pmf_embed_pmf
```
```   735
```
```   736 lemma set_pmf_transfer[transfer_rule]:
```
```   737   assumes "bi_total A"
```
```   738   shows "rel_fun (pcr_pmf A) (rel_set A) (\<lambda>f. {x. f x \<noteq> 0}) set_pmf"
```
```   739   using `bi_total A`
```
```   740   by (auto simp: pcr_pmf_def cr_pmf_def rel_fun_def rel_set_def bi_total_def Bex_def set_pmf_iff)
```
```   741      metis+
```
```   742
```
```   743 end
```
```   744
```
```   745 context
```
```   746 begin
```
```   747
```
```   748 interpretation pmf_as_function .
```
```   749
```
```   750 lemma pmf_eqI: "(\<And>i. pmf M i = pmf N i) \<Longrightarrow> M = N"
```
```   751   by transfer auto
```
```   752
```
```   753 lemma pmf_eq_iff: "M = N \<longleftrightarrow> (\<forall>i. pmf M i = pmf N i)"
```
```   754   by (auto intro: pmf_eqI)
```
```   755
```
```   756 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))"
```
```   757   unfolding pmf_eq_iff pmf_bind
```
```   758 proof
```
```   759   fix i
```
```   760   interpret B: prob_space "restrict_space B B"
```
```   761     by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
```
```   762        (auto simp: AE_measure_pmf_iff)
```
```   763   interpret A: prob_space "restrict_space A A"
```
```   764     by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
```
```   765        (auto simp: AE_measure_pmf_iff)
```
```   766
```
```   767   interpret AB: pair_prob_space "restrict_space A A" "restrict_space B B"
```
```   768     by unfold_locales
```
```   769
```
```   770   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)"
```
```   771     by (rule integral_cong) (auto intro!: integral_pmf_restrict)
```
```   772   also have "\<dots> = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>restrict_space A A)"
```
```   773     by (intro integral_pmf_restrict B.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
```
```   774               countable_set_pmf borel_measurable_count_space)
```
```   775   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>restrict_space B B)"
```
```   776     by (rule AB.Fubini_integral[symmetric])
```
```   777        (auto intro!: AB.integrable_const_bound[where B=1] measurable_pair_restrict_pmf2
```
```   778              simp: pmf_nonneg pmf_le_1 measurable_restrict_space1)
```
```   779   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>B)"
```
```   780     by (intro integral_pmf_restrict[symmetric] A.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
```
```   781               countable_set_pmf borel_measurable_count_space)
```
```   782   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>A \<partial>B)"
```
```   783     by (rule integral_cong) (auto intro!: integral_pmf_restrict[symmetric])
```
```   784   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)" .
```
```   785 qed
```
```   786
```
```   787 lemma pair_map_pmf1: "pair_pmf (map_pmf f A) B = map_pmf (apfst f) (pair_pmf A B)"
```
```   788 proof (safe intro!: pmf_eqI)
```
```   789   fix a :: "'a" and b :: "'b"
```
```   790   have [simp]: "\<And>c d. indicator (apfst f -` {(a, b)}) (c, d) = indicator (f -` {a}) c * (indicator {b} d::ereal)"
```
```   791     by (auto split: split_indicator)
```
```   792
```
```   793   have "ereal (pmf (pair_pmf (map_pmf f A) B) (a, b)) =
```
```   794          ereal (pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b))"
```
```   795     unfolding pmf_pair ereal_pmf_map
```
```   796     by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_multc pmf_nonneg
```
```   797                   emeasure_map_pmf[symmetric] del: emeasure_map_pmf)
```
```   798   then show "pmf (pair_pmf (map_pmf f A) B) (a, b) = pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b)"
```
```   799     by simp
```
```   800 qed
```
```   801
```
```   802 lemma pair_map_pmf2: "pair_pmf A (map_pmf f B) = map_pmf (apsnd f) (pair_pmf A B)"
```
```   803 proof (safe intro!: pmf_eqI)
```
```   804   fix a :: "'a" and b :: "'b"
```
```   805   have [simp]: "\<And>c d. indicator (apsnd f -` {(a, b)}) (c, d) = indicator {a} c * (indicator (f -` {b}) d::ereal)"
```
```   806     by (auto split: split_indicator)
```
```   807
```
```   808   have "ereal (pmf (pair_pmf A (map_pmf f B)) (a, b)) =
```
```   809          ereal (pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b))"
```
```   810     unfolding pmf_pair ereal_pmf_map
```
```   811     by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_cmult nn_integral_multc pmf_nonneg
```
```   812                   emeasure_map_pmf[symmetric] del: emeasure_map_pmf)
```
```   813   then show "pmf (pair_pmf A (map_pmf f B)) (a, b) = pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b)"
```
```   814     by simp
```
```   815 qed
```
```   816
```
```   817 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)"
```
```   818   by (simp add: pair_map_pmf2 pair_map_pmf1 map_pmf_comp split_beta')
```
```   819
```
```   820 end
```
```   821
```
```   822 lemma pair_return_pmf1: "pair_pmf (return_pmf x) y = map_pmf (Pair x) y"
```
```   823 by(simp add: pair_pmf_def bind_return_pmf map_pmf_def)
```
```   824
```
```   825 lemma pair_return_pmf2: "pair_pmf x (return_pmf y) = map_pmf (\<lambda>x. (x, y)) x"
```
```   826 by(simp add: pair_pmf_def bind_return_pmf map_pmf_def)
```
```   827
```
```   828 lemma pair_pair_pmf: "pair_pmf (pair_pmf u v) w = map_pmf (\<lambda>(x, (y, z)). ((x, y), z)) (pair_pmf u (pair_pmf v w))"
```
```   829 by(simp add: pair_pmf_def bind_return_pmf map_pmf_def bind_assoc_pmf)
```
```   830
```
```   831 lemma pair_commute_pmf: "pair_pmf x y = map_pmf (\<lambda>(x, y). (y, x)) (pair_pmf y x)"
```
```   832 unfolding pair_pmf_def by(subst bind_commute_pmf)(simp add: map_pmf_def bind_assoc_pmf bind_return_pmf)
```
```   833
```
```   834 lemma set_pmf_subset_singleton: "set_pmf p \<subseteq> {x} \<longleftrightarrow> p = return_pmf x"
```
```   835 proof(intro iffI pmf_eqI)
```
```   836   fix i
```
```   837   assume x: "set_pmf p \<subseteq> {x}"
```
```   838   hence *: "set_pmf p = {x}" using set_pmf_not_empty[of p] by auto
```
```   839   have "ereal (pmf p x) = \<integral>\<^sup>+ i. indicator {x} i \<partial>p" by(simp add: emeasure_pmf_single)
```
```   840   also have "\<dots> = \<integral>\<^sup>+ i. 1 \<partial>p" by(rule nn_integral_cong_AE)(simp add: AE_measure_pmf_iff * )
```
```   841   also have "\<dots> = 1" by simp
```
```   842   finally show "pmf p i = pmf (return_pmf x) i" using x
```
```   843     by(auto split: split_indicator simp add: pmf_eq_0_set_pmf)
```
```   844 qed auto
```
```   845
```
```   846 lemma bind_eq_return_pmf:
```
```   847   "bind_pmf p f = return_pmf x \<longleftrightarrow> (\<forall>y\<in>set_pmf p. f y = return_pmf x)"
```
```   848   (is "?lhs \<longleftrightarrow> ?rhs")
```
```   849 proof(intro iffI strip)
```
```   850   fix y
```
```   851   assume y: "y \<in> set_pmf p"
```
```   852   assume "?lhs"
```
```   853   hence "set_pmf (bind_pmf p f) = {x}" by simp
```
```   854   hence "(\<Union>y\<in>set_pmf p. set_pmf (f y)) = {x}" by simp
```
```   855   hence "set_pmf (f y) \<subseteq> {x}" using y by auto
```
```   856   thus "f y = return_pmf x" by(simp add: set_pmf_subset_singleton)
```
```   857 next
```
```   858   assume *: ?rhs
```
```   859   show ?lhs
```
```   860   proof(rule pmf_eqI)
```
```   861     fix i
```
```   862     have "ereal (pmf (bind_pmf p f) i) = \<integral>\<^sup>+ y. ereal (pmf (f y) i) \<partial>p" by(simp add: ereal_pmf_bind)
```
```   863     also have "\<dots> = \<integral>\<^sup>+ y. ereal (pmf (return_pmf x) i) \<partial>p"
```
```   864       by(rule nn_integral_cong_AE)(simp add: AE_measure_pmf_iff * )
```
```   865     also have "\<dots> = ereal (pmf (return_pmf x) i)" by simp
```
```   866     finally show "pmf (bind_pmf p f) i = pmf (return_pmf x) i" by simp
```
```   867   qed
```
```   868 qed
```
```   869
```
```   870 lemma pmf_False_conv_True: "pmf p False = 1 - pmf p True"
```
```   871 proof -
```
```   872   have "pmf p False + pmf p True = measure p {False} + measure p {True}"
```
```   873     by(simp add: measure_pmf_single)
```
```   874   also have "\<dots> = measure p ({False} \<union> {True})"
```
```   875     by(subst measure_pmf.finite_measure_Union) simp_all
```
```   876   also have "{False} \<union> {True} = space p" by auto
```
```   877   finally show ?thesis by simp
```
```   878 qed
```
```   879
```
```   880 lemma pmf_True_conv_False: "pmf p True = 1 - pmf p False"
```
```   881 by(simp add: pmf_False_conv_True)
```
```   882
```
```   883 subsection \<open> Conditional Probabilities \<close>
```
```   884
```
```   885 lemma measure_pmf_zero_iff: "measure (measure_pmf p) s = 0 \<longleftrightarrow> set_pmf p \<inter> s = {}"
```
```   886   by (subst measure_pmf.prob_eq_0) (auto simp: AE_measure_pmf_iff)
```
```   887
```
```   888 context
```
```   889   fixes p :: "'a pmf" and s :: "'a set"
```
```   890   assumes not_empty: "set_pmf p \<inter> s \<noteq> {}"
```
```   891 begin
```
```   892
```
```   893 interpretation pmf_as_measure .
```
```   894
```
```   895 lemma emeasure_measure_pmf_not_zero: "emeasure (measure_pmf p) s \<noteq> 0"
```
```   896 proof
```
```   897   assume "emeasure (measure_pmf p) s = 0"
```
```   898   then have "AE x in measure_pmf p. x \<notin> s"
```
```   899     by (rule AE_I[rotated]) auto
```
```   900   with not_empty show False
```
```   901     by (auto simp: AE_measure_pmf_iff)
```
```   902 qed
```
```   903
```
```   904 lemma measure_measure_pmf_not_zero: "measure (measure_pmf p) s \<noteq> 0"
```
```   905   using emeasure_measure_pmf_not_zero unfolding measure_pmf.emeasure_eq_measure by simp
```
```   906
```
```   907 lift_definition cond_pmf :: "'a pmf" is
```
```   908   "uniform_measure (measure_pmf p) s"
```
```   909 proof (intro conjI)
```
```   910   show "prob_space (uniform_measure (measure_pmf p) s)"
```
```   911     by (intro prob_space_uniform_measure) (auto simp: emeasure_measure_pmf_not_zero)
```
```   912   show "AE x in uniform_measure (measure_pmf p) s. measure (uniform_measure (measure_pmf p) s) {x} \<noteq> 0"
```
```   913     by (simp add: emeasure_measure_pmf_not_zero measure_measure_pmf_not_zero AE_uniform_measure
```
```   914                   AE_measure_pmf_iff set_pmf.rep_eq)
```
```   915 qed simp
```
```   916
```
```   917 lemma pmf_cond: "pmf cond_pmf x = (if x \<in> s then pmf p x / measure p s else 0)"
```
```   918   by transfer (simp add: emeasure_measure_pmf_not_zero pmf.rep_eq)
```
```   919
```
```   920 lemma set_cond_pmf[simp]: "set_pmf cond_pmf = set_pmf p \<inter> s"
```
```   921   by (auto simp add: set_pmf_iff pmf_cond measure_measure_pmf_not_zero split: split_if_asm)
```
```   922
```
```   923 end
```
```   924
```
```   925 lemma cond_map_pmf:
```
```   926   assumes "set_pmf p \<inter> f -` s \<noteq> {}"
```
```   927   shows "cond_pmf (map_pmf f p) s = map_pmf f (cond_pmf p (f -` s))"
```
```   928 proof -
```
```   929   have *: "set_pmf (map_pmf f p) \<inter> s \<noteq> {}"
```
```   930     using assms by auto
```
```   931   { fix x
```
```   932     have "ereal (pmf (map_pmf f (cond_pmf p (f -` s))) x) =
```
```   933       emeasure p (f -` s \<inter> f -` {x}) / emeasure p (f -` s)"
```
```   934       unfolding ereal_pmf_map cond_pmf.rep_eq[OF assms] by (simp add: nn_integral_uniform_measure)
```
```   935     also have "f -` s \<inter> f -` {x} = (if x \<in> s then f -` {x} else {})"
```
```   936       by auto
```
```   937     also have "emeasure p (if x \<in> s then f -` {x} else {}) / emeasure p (f -` s) =
```
```   938       ereal (pmf (cond_pmf (map_pmf f p) s) x)"
```
```   939       using measure_measure_pmf_not_zero[OF *]
```
```   940       by (simp add: pmf_cond[OF *] ereal_divide' ereal_pmf_map measure_pmf.emeasure_eq_measure[symmetric]
```
```   941                del: ereal_divide)
```
```   942     finally have "ereal (pmf (cond_pmf (map_pmf f p) s) x) = ereal (pmf (map_pmf f (cond_pmf p (f -` s))) x)"
```
```   943       by simp }
```
```   944   then show ?thesis
```
```   945     by (intro pmf_eqI) simp
```
```   946 qed
```
```   947
```
```   948 lemma bind_cond_pmf_cancel:
```
```   949   assumes [simp]: "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}"
```
```   950   assumes [simp]: "\<And>y. y \<in> set_pmf q \<Longrightarrow> set_pmf p \<inter> {x. R x y} \<noteq> {}"
```
```   951   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}"
```
```   952   shows "bind_pmf p (\<lambda>x. cond_pmf q {y. R x y}) = q"
```
```   953 proof (rule pmf_eqI)
```
```   954   fix i
```
```   955   have "ereal (pmf (bind_pmf p (\<lambda>x. cond_pmf q {y. R x y})) i) =
```
```   956     (\<integral>\<^sup>+x. ereal (pmf q i / measure p {x. R x i}) * ereal (indicator {x. R x i} x) \<partial>p)"
```
```   957     by (auto simp add: ereal_pmf_bind AE_measure_pmf_iff pmf_cond pmf_eq_0_set_pmf intro!: nn_integral_cong_AE)
```
```   958   also have "\<dots> = (pmf q i * measure p {x. R x i}) / measure p {x. R x i}"
```
```   959     by (simp add: pmf_nonneg measure_nonneg zero_ereal_def[symmetric] ereal_indicator
```
```   960                   nn_integral_cmult measure_pmf.emeasure_eq_measure)
```
```   961   also have "\<dots> = pmf q i"
```
```   962     by (cases "pmf q i = 0") (simp_all add: pmf_eq_0_set_pmf measure_measure_pmf_not_zero)
```
```   963   finally show "pmf (bind_pmf p (\<lambda>x. cond_pmf q {y. R x y})) i = pmf q i"
```
```   964     by simp
```
```   965 qed
```
```   966
```
```   967 subsection \<open> Relator \<close>
```
```   968
```
```   969 inductive rel_pmf :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf \<Rightarrow> bool"
```
```   970 for R p q
```
```   971 where
```
```   972   "\<lbrakk> \<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y;
```
```   973      map_pmf fst pq = p; map_pmf snd pq = q \<rbrakk>
```
```   974   \<Longrightarrow> rel_pmf R p q"
```
```   975
```
```   976 lemma rel_pmfI:
```
```   977   assumes R: "rel_set R (set_pmf p) (set_pmf q)"
```
```   978   assumes eq: "\<And>x y. x \<in> set_pmf p \<Longrightarrow> y \<in> set_pmf q \<Longrightarrow> R x y \<Longrightarrow>
```
```   979     measure p {x. R x y} = measure q {y. R x y}"
```
```   980   shows "rel_pmf R p q"
```
```   981 proof
```
```   982   let ?pq = "bind_pmf p (\<lambda>x. bind_pmf (cond_pmf q {y. R x y}) (\<lambda>y. return_pmf (x, y)))"
```
```   983   have "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}"
```
```   984     using R by (auto simp: rel_set_def)
```
```   985   then show "\<And>x y. (x, y) \<in> set_pmf ?pq \<Longrightarrow> R x y"
```
```   986     by auto
```
```   987   show "map_pmf fst ?pq = p"
```
```   988     by (simp add: map_bind_pmf bind_return_pmf')
```
```   989
```
```   990   show "map_pmf snd ?pq = q"
```
```   991     using R eq
```
```   992     apply (simp add: bind_cond_pmf_cancel map_bind_pmf bind_return_pmf')
```
```   993     apply (rule bind_cond_pmf_cancel)
```
```   994     apply (auto simp: rel_set_def)
```
```   995     done
```
```   996 qed
```
```   997
```
```   998 lemma rel_pmf_imp_rel_set: "rel_pmf R p q \<Longrightarrow> rel_set R (set_pmf p) (set_pmf q)"
```
```   999   by (force simp add: rel_pmf.simps rel_set_def)
```
```  1000
```
```  1001 lemma rel_pmfD_measure:
```
```  1002   assumes rel_R: "rel_pmf R p q" and R: "\<And>a b. R a b \<Longrightarrow> R a y \<longleftrightarrow> R x b"
```
```  1003   assumes "x \<in> set_pmf p" "y \<in> set_pmf q"
```
```  1004   shows "measure p {x. R x y} = measure q {y. R x y}"
```
```  1005 proof -
```
```  1006   from rel_R obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
```
```  1007     and eq: "p = map_pmf fst pq" "q = map_pmf snd pq"
```
```  1008     by (auto elim: rel_pmf.cases)
```
```  1009   have "measure p {x. R x y} = measure pq {x. R (fst x) y}"
```
```  1010     by (simp add: eq map_pmf_rep_eq measure_distr)
```
```  1011   also have "\<dots> = measure pq {y. R x (snd y)}"
```
```  1012     by (intro measure_pmf.finite_measure_eq_AE)
```
```  1013        (auto simp: AE_measure_pmf_iff R dest!: pq)
```
```  1014   also have "\<dots> = measure q {y. R x y}"
```
```  1015     by (simp add: eq map_pmf_rep_eq measure_distr)
```
```  1016   finally show "measure p {x. R x y} = measure q {y. R x y}" .
```
```  1017 qed
```
```  1018
```
```  1019 lemma rel_pmf_measureD:
```
```  1020   assumes "rel_pmf R p q"
```
```  1021   shows "measure (measure_pmf p) A \<le> measure (measure_pmf q) {y. \<exists>x\<in>A. R x y}" (is "?lhs \<le> ?rhs")
```
```  1022 using assms
```
```  1023 proof cases
```
```  1024   fix pq
```
```  1025   assume R: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
```
```  1026     and p[symmetric]: "map_pmf fst pq = p"
```
```  1027     and q[symmetric]: "map_pmf snd pq = q"
```
```  1028   have "?lhs = measure (measure_pmf pq) (fst -` A)" by(simp add: p)
```
```  1029   also have "\<dots> \<le> measure (measure_pmf pq) {y. \<exists>x\<in>A. R x (snd y)}"
```
```  1030     by(rule measure_pmf.finite_measure_mono_AE)(auto 4 3 simp add: AE_measure_pmf_iff dest: R)
```
```  1031   also have "\<dots> = ?rhs" by(simp add: q)
```
```  1032   finally show ?thesis .
```
```  1033 qed
```
```  1034
```
```  1035 lemma rel_pmf_iff_measure:
```
```  1036   assumes "symp R" "transp R"
```
```  1037   shows "rel_pmf R p q \<longleftrightarrow>
```
```  1038     rel_set R (set_pmf p) (set_pmf q) \<and>
```
```  1039     (\<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})"
```
```  1040   by (safe intro!: rel_pmf_imp_rel_set rel_pmfI)
```
```  1041      (auto intro!: rel_pmfD_measure dest: sympD[OF \<open>symp R\<close>] transpD[OF \<open>transp R\<close>])
```
```  1042
```
```  1043 lemma quotient_rel_set_disjoint:
```
```  1044   "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 = {}"
```
```  1045   using in_quotient_imp_closed[of UNIV "{(x, y). R x y}" C]
```
```  1046   by (auto 0 0 simp: equivp_equiv rel_set_def set_eq_iff elim: equivpE)
```
```  1047      (blast dest: equivp_symp)+
```
```  1048
```
```  1049 lemma quotientD: "equiv X R \<Longrightarrow> A \<in> X // R \<Longrightarrow> x \<in> A \<Longrightarrow> A = R `` {x}"
```
```  1050   by (metis Image_singleton_iff equiv_class_eq_iff quotientE)
```
```  1051
```
```  1052 lemma rel_pmf_iff_equivp:
```
```  1053   assumes "equivp R"
```
```  1054   shows "rel_pmf R p q \<longleftrightarrow> (\<forall>C\<in>UNIV // {(x, y). R x y}. measure p C = measure q C)"
```
```  1055     (is "_ \<longleftrightarrow>   (\<forall>C\<in>_//?R. _)")
```
```  1056 proof (subst rel_pmf_iff_measure, safe)
```
```  1057   show "symp R" "transp R"
```
```  1058     using assms by (auto simp: equivp_reflp_symp_transp)
```
```  1059 next
```
```  1060   fix C assume C: "C \<in> UNIV // ?R" and R: "rel_set R (set_pmf p) (set_pmf q)"
```
```  1061   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}"
```
```  1062
```
```  1063   show "measure p C = measure q C"
```
```  1064   proof cases
```
```  1065     assume "p \<inter> C = {}"
```
```  1066     moreover then have "q \<inter> C = {}"
```
```  1067       using quotient_rel_set_disjoint[OF assms C R] by simp
```
```  1068     ultimately show ?thesis
```
```  1069       unfolding measure_pmf_zero_iff[symmetric] by simp
```
```  1070   next
```
```  1071     assume "p \<inter> C \<noteq> {}"
```
```  1072     moreover then have "q \<inter> C \<noteq> {}"
```
```  1073       using quotient_rel_set_disjoint[OF assms C R] by simp
```
```  1074     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"
```
```  1075       by auto
```
```  1076     then have "R x y"
```
```  1077       using in_quotient_imp_in_rel[of UNIV ?R C x y] C assms
```
```  1078       by (simp add: equivp_equiv)
```
```  1079     with in_set eq have "measure p {x. R x y} = measure q {y. R x y}"
```
```  1080       by auto
```
```  1081     moreover have "{y. R x y} = C"
```
```  1082       using assms `x \<in> C` C quotientD[of UNIV ?R C x] by (simp add: equivp_equiv)
```
```  1083     moreover have "{x. R x y} = C"
```
```  1084       using assms `y \<in> C` C quotientD[of UNIV "?R" C y] sympD[of R]
```
```  1085       by (auto simp add: equivp_equiv elim: equivpE)
```
```  1086     ultimately show ?thesis
```
```  1087       by auto
```
```  1088   qed
```
```  1089 next
```
```  1090   assume eq: "\<forall>C\<in>UNIV // ?R. measure p C = measure q C"
```
```  1091   show "rel_set R (set_pmf p) (set_pmf q)"
```
```  1092     unfolding rel_set_def
```
```  1093   proof safe
```
```  1094     fix x assume x: "x \<in> set_pmf p"
```
```  1095     have "{y. R x y} \<in> UNIV // ?R"
```
```  1096       by (auto simp: quotient_def)
```
```  1097     with eq have *: "measure q {y. R x y} = measure p {y. R x y}"
```
```  1098       by auto
```
```  1099     have "measure q {y. R x y} \<noteq> 0"
```
```  1100       using x assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp)
```
```  1101     then show "\<exists>y\<in>set_pmf q. R x y"
```
```  1102       unfolding measure_pmf_zero_iff by auto
```
```  1103   next
```
```  1104     fix y assume y: "y \<in> set_pmf q"
```
```  1105     have "{x. R x y} \<in> UNIV // ?R"
```
```  1106       using assms by (auto simp: quotient_def dest: equivp_symp)
```
```  1107     with eq have *: "measure p {x. R x y} = measure q {x. R x y}"
```
```  1108       by auto
```
```  1109     have "measure p {x. R x y} \<noteq> 0"
```
```  1110       using y assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp)
```
```  1111     then show "\<exists>x\<in>set_pmf p. R x y"
```
```  1112       unfolding measure_pmf_zero_iff by auto
```
```  1113   qed
```
```  1114
```
```  1115   fix x y assume "x \<in> set_pmf p" "y \<in> set_pmf q" "R x y"
```
```  1116   have "{y. R x y} \<in> UNIV // ?R" "{x. R x y} = {y. R x y}"
```
```  1117     using assms `R x y` by (auto simp: quotient_def dest: equivp_symp equivp_transp)
```
```  1118   with eq show "measure p {x. R x y} = measure q {y. R x y}"
```
```  1119     by auto
```
```  1120 qed
```
```  1121
```
```  1122 bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: rel_pmf
```
```  1123 proof -
```
```  1124   show "map_pmf id = id" by (rule map_pmf_id)
```
```  1125   show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose)
```
```  1126   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"
```
```  1127     by (intro map_pmf_cong refl)
```
```  1128
```
```  1129   show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
```
```  1130     by (rule pmf_set_map)
```
```  1131
```
```  1132   show "(card_of (set_pmf p), natLeq) \<in> ordLeq" for p :: "'s pmf"
```
```  1133   proof -
```
```  1134     have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq"
```
```  1135       by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"])
```
```  1136          (auto intro: countable_set_pmf)
```
```  1137     also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq"
```
```  1138       by (metis Field_natLeq card_of_least natLeq_Well_order)
```
```  1139     finally show ?thesis .
```
```  1140   qed
```
```  1141
```
```  1142   show "\<And>R. rel_pmf R =
```
```  1143          (BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf fst))\<inverse>\<inverse> OO
```
```  1144          BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf snd)"
```
```  1145      by (auto simp add: fun_eq_iff BNF_Def.Grp_def OO_def rel_pmf.simps)
```
```  1146
```
```  1147   show "rel_pmf R OO rel_pmf S \<le> rel_pmf (R OO S)"
```
```  1148     for R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
```
```  1149   proof -
```
```  1150     { fix p q r
```
```  1151       assume pq: "rel_pmf R p q"
```
```  1152         and qr:"rel_pmf S q r"
```
```  1153       from pq obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
```
```  1154         and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto
```
```  1155       from qr obtain qr where qr: "\<And>y z. (y, z) \<in> set_pmf qr \<Longrightarrow> S y z"
```
```  1156         and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto
```
```  1157
```
```  1158       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)))"
```
```  1159       have pr_welldefined: "\<And>y. y \<in> q \<Longrightarrow> qr \<inter> {yz. fst yz = y} \<noteq> {}"
```
```  1160         by (force simp: q')
```
```  1161
```
```  1162       have "rel_pmf (R OO S) p r"
```
```  1163       proof (rule rel_pmf.intros)
```
```  1164         fix x z assume "(x, z) \<in> pr"
```
```  1165         then have "\<exists>y. (x, y) \<in> pq \<and> (y, z) \<in> qr"
```
```  1166           by (auto simp: q pr_welldefined pr_def split_beta)
```
```  1167         with pq qr show "(R OO S) x z"
```
```  1168           by blast
```
```  1169       next
```
```  1170         have "map_pmf snd pr = map_pmf snd (bind_pmf q (\<lambda>y. cond_pmf qr {yz. fst yz = y}))"
```
```  1171           by (simp add: pr_def q split_beta bind_map_pmf map_pmf_def[symmetric] map_bind_pmf map_pmf_comp)
```
```  1172         then show "map_pmf snd pr = r"
```
```  1173           unfolding r q' bind_map_pmf by (subst (asm) bind_cond_pmf_cancel) (auto simp: eq_commute)
```
```  1174       qed (simp add: pr_def map_bind_pmf split_beta map_pmf_def[symmetric] p map_pmf_comp)
```
```  1175     }
```
```  1176     then show ?thesis
```
```  1177       by(auto simp add: le_fun_def)
```
```  1178   qed
```
```  1179 qed (fact natLeq_card_order natLeq_cinfinite)+
```
```  1180
```
```  1181 lemma map_pmf_idI: "(\<And>x. x \<in> set_pmf p \<Longrightarrow> f x = x) \<Longrightarrow> map_pmf f p = p"
```
```  1182 by(simp cong: pmf.map_cong)
```
```  1183
```
```  1184 lemma rel_pmf_conj[simp]:
```
```  1185   "rel_pmf (\<lambda>x y. P \<and> Q x y) x y \<longleftrightarrow> P \<and> rel_pmf Q x y"
```
```  1186   "rel_pmf (\<lambda>x y. Q x y \<and> P) x y \<longleftrightarrow> P \<and> rel_pmf Q x y"
```
```  1187   using set_pmf_not_empty by (fastforce simp: pmf.in_rel subset_eq)+
```
```  1188
```
```  1189 lemma rel_pmf_top[simp]: "rel_pmf top = top"
```
```  1190   by (auto simp: pmf.in_rel[abs_def] fun_eq_iff map_fst_pair_pmf map_snd_pair_pmf
```
```  1191            intro: exI[of _ "pair_pmf x y" for x y])
```
```  1192
```
```  1193 lemma rel_pmf_return_pmf1: "rel_pmf R (return_pmf x) M \<longleftrightarrow> (\<forall>a\<in>M. R x a)"
```
```  1194 proof safe
```
```  1195   fix a assume "a \<in> M" "rel_pmf R (return_pmf x) M"
```
```  1196   then obtain pq where *: "\<And>a b. (a, b) \<in> set_pmf pq \<Longrightarrow> R a b"
```
```  1197     and eq: "return_pmf x = map_pmf fst pq" "M = map_pmf snd pq"
```
```  1198     by (force elim: rel_pmf.cases)
```
```  1199   moreover have "set_pmf (return_pmf x) = {x}"
```
```  1200     by simp
```
```  1201   with `a \<in> M` have "(x, a) \<in> pq"
```
```  1202     by (force simp: eq)
```
```  1203   with * show "R x a"
```
```  1204     by auto
```
```  1205 qed (auto intro!: rel_pmf.intros[where pq="pair_pmf (return_pmf x) M"]
```
```  1206           simp: map_fst_pair_pmf map_snd_pair_pmf)
```
```  1207
```
```  1208 lemma rel_pmf_return_pmf2: "rel_pmf R M (return_pmf x) \<longleftrightarrow> (\<forall>a\<in>M. R a x)"
```
```  1209   by (subst pmf.rel_flip[symmetric]) (simp add: rel_pmf_return_pmf1)
```
```  1210
```
```  1211 lemma rel_return_pmf[simp]: "rel_pmf R (return_pmf x1) (return_pmf x2) = R x1 x2"
```
```  1212   unfolding rel_pmf_return_pmf2 set_return_pmf by simp
```
```  1213
```
```  1214 lemma rel_pmf_False[simp]: "rel_pmf (\<lambda>x y. False) x y = False"
```
```  1215   unfolding pmf.in_rel fun_eq_iff using set_pmf_not_empty by fastforce
```
```  1216
```
```  1217 lemma rel_pmf_rel_prod:
```
```  1218   "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'"
```
```  1219 proof safe
```
```  1220   assume "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
```
```  1221   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"
```
```  1222     and eq: "map_pmf fst pq = pair_pmf A A'" "map_pmf snd pq = pair_pmf B B'"
```
```  1223     by (force elim: rel_pmf.cases)
```
```  1224   show "rel_pmf R A B"
```
```  1225   proof (rule rel_pmf.intros)
```
```  1226     let ?f = "\<lambda>(a, b). (fst a, fst b)"
```
```  1227     have [simp]: "(\<lambda>x. fst (?f x)) = fst o fst" "(\<lambda>x. snd (?f x)) = fst o snd"
```
```  1228       by auto
```
```  1229
```
```  1230     show "map_pmf fst (map_pmf ?f pq) = A"
```
```  1231       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
```
```  1232     show "map_pmf snd (map_pmf ?f pq) = B"
```
```  1233       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
```
```  1234
```
```  1235     fix a b assume "(a, b) \<in> set_pmf (map_pmf ?f pq)"
```
```  1236     then obtain c d where "((a, c), (b, d)) \<in> set_pmf pq"
```
```  1237       by auto
```
```  1238     from pq[OF this] show "R a b" ..
```
```  1239   qed
```
```  1240   show "rel_pmf S A' B'"
```
```  1241   proof (rule rel_pmf.intros)
```
```  1242     let ?f = "\<lambda>(a, b). (snd a, snd b)"
```
```  1243     have [simp]: "(\<lambda>x. fst (?f x)) = snd o fst" "(\<lambda>x. snd (?f x)) = snd o snd"
```
```  1244       by auto
```
```  1245
```
```  1246     show "map_pmf fst (map_pmf ?f pq) = A'"
```
```  1247       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
```
```  1248     show "map_pmf snd (map_pmf ?f pq) = B'"
```
```  1249       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
```
```  1250
```
```  1251     fix c d assume "(c, d) \<in> set_pmf (map_pmf ?f pq)"
```
```  1252     then obtain a b where "((a, c), (b, d)) \<in> set_pmf pq"
```
```  1253       by auto
```
```  1254     from pq[OF this] show "S c d" ..
```
```  1255   qed
```
```  1256 next
```
```  1257   assume "rel_pmf R A B" "rel_pmf S A' B'"
```
```  1258   then obtain Rpq Spq
```
```  1259     where Rpq: "\<And>a b. (a, b) \<in> set_pmf Rpq \<Longrightarrow> R a b"
```
```  1260         "map_pmf fst Rpq = A" "map_pmf snd Rpq = B"
```
```  1261       and Spq: "\<And>a b. (a, b) \<in> set_pmf Spq \<Longrightarrow> S a b"
```
```  1262         "map_pmf fst Spq = A'" "map_pmf snd Spq = B'"
```
```  1263     by (force elim: rel_pmf.cases)
```
```  1264
```
```  1265   let ?f = "(\<lambda>((a, c), (b, d)). ((a, b), (c, d)))"
```
```  1266   let ?pq = "map_pmf ?f (pair_pmf Rpq Spq)"
```
```  1267   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))"
```
```  1268     by auto
```
```  1269
```
```  1270   show "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
```
```  1271     by (rule rel_pmf.intros[where pq="?pq"])
```
```  1272        (auto simp: map_snd_pair_pmf map_fst_pair_pmf map_pmf_comp Rpq Spq
```
```  1273                    map_pair)
```
```  1274 qed
```
```  1275
```
```  1276 lemma rel_pmf_reflI:
```
```  1277   assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> P x x"
```
```  1278   shows "rel_pmf P p p"
```
```  1279   by (rule rel_pmf.intros[where pq="map_pmf (\<lambda>x. (x, x)) p"])
```
```  1280      (auto simp add: pmf.map_comp o_def assms)
```
```  1281
```
```  1282 lemma rel_pmf_bij_betw:
```
```  1283   assumes f: "bij_betw f (set_pmf p) (set_pmf q)"
```
```  1284   and eq: "\<And>x. x \<in> set_pmf p \<Longrightarrow> pmf p x = pmf q (f x)"
```
```  1285   shows "rel_pmf (\<lambda>x y. f x = y) p q"
```
```  1286 proof(rule rel_pmf.intros)
```
```  1287   let ?pq = "map_pmf (\<lambda>x. (x, f x)) p"
```
```  1288   show "map_pmf fst ?pq = p" by(simp add: pmf.map_comp o_def)
```
```  1289
```
```  1290   have "map_pmf f p = q"
```
```  1291   proof(rule pmf_eqI)
```
```  1292     fix i
```
```  1293     show "pmf (map_pmf f p) i = pmf q i"
```
```  1294     proof(cases "i \<in> set_pmf q")
```
```  1295       case True
```
```  1296       with f obtain j where "i = f j" "j \<in> set_pmf p"
```
```  1297         by(auto simp add: bij_betw_def image_iff)
```
```  1298       thus ?thesis using f by(simp add: bij_betw_def pmf_map_inj eq)
```
```  1299     next
```
```  1300       case False thus ?thesis
```
```  1301         by(subst pmf_map_outside)(auto simp add: set_pmf_iff eq[symmetric])
```
```  1302     qed
```
```  1303   qed
```
```  1304   then show "map_pmf snd ?pq = q" by(simp add: pmf.map_comp o_def)
```
```  1305 qed auto
```
```  1306
```
```  1307 context
```
```  1308 begin
```
```  1309
```
```  1310 interpretation pmf_as_measure .
```
```  1311
```
```  1312 definition "join_pmf M = bind_pmf M (\<lambda>x. x)"
```
```  1313
```
```  1314 lemma bind_eq_join_pmf: "bind_pmf M f = join_pmf (map_pmf f M)"
```
```  1315   unfolding join_pmf_def bind_map_pmf ..
```
```  1316
```
```  1317 lemma join_eq_bind_pmf: "join_pmf M = bind_pmf M id"
```
```  1318   by (simp add: join_pmf_def id_def)
```
```  1319
```
```  1320 lemma pmf_join: "pmf (join_pmf N) i = (\<integral>M. pmf M i \<partial>measure_pmf N)"
```
```  1321   unfolding join_pmf_def pmf_bind ..
```
```  1322
```
```  1323 lemma ereal_pmf_join: "ereal (pmf (join_pmf N) i) = (\<integral>\<^sup>+M. pmf M i \<partial>measure_pmf N)"
```
```  1324   unfolding join_pmf_def ereal_pmf_bind ..
```
```  1325
```
```  1326 lemma set_pmf_join_pmf[simp]: "set_pmf (join_pmf f) = (\<Union>p\<in>set_pmf f. set_pmf p)"
```
```  1327   by (simp add: join_pmf_def)
```
```  1328
```
```  1329 lemma join_return_pmf: "join_pmf (return_pmf M) = M"
```
```  1330   by (simp add: integral_return pmf_eq_iff pmf_join return_pmf.rep_eq)
```
```  1331
```
```  1332 lemma map_join_pmf: "map_pmf f (join_pmf AA) = join_pmf (map_pmf (map_pmf f) AA)"
```
```  1333   by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf)
```
```  1334
```
```  1335 lemma join_map_return_pmf: "join_pmf (map_pmf return_pmf A) = A"
```
```  1336   by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
```
```  1337
```
```  1338 end
```
```  1339
```
```  1340 lemma rel_pmf_joinI:
```
```  1341   assumes "rel_pmf (rel_pmf P) p q"
```
```  1342   shows "rel_pmf P (join_pmf p) (join_pmf q)"
```
```  1343 proof -
```
```  1344   from assms obtain pq where p: "p = map_pmf fst pq"
```
```  1345     and q: "q = map_pmf snd pq"
```
```  1346     and P: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> rel_pmf P x y"
```
```  1347     by cases auto
```
```  1348   from P obtain PQ
```
```  1349     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"
```
```  1350     and x: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf fst (PQ x y) = x"
```
```  1351     and y: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf snd (PQ x y) = y"
```
```  1352     by(metis rel_pmf.simps)
```
```  1353
```
```  1354   let ?r = "bind_pmf pq (\<lambda>(x, y). PQ x y)"
```
```  1355   have "\<And>a b. (a, b) \<in> set_pmf ?r \<Longrightarrow> P a b" by (auto intro: PQ)
```
```  1356   moreover have "map_pmf fst ?r = join_pmf p" "map_pmf snd ?r = join_pmf q"
```
```  1357     by (simp_all add: p q x y join_pmf_def map_bind_pmf bind_map_pmf split_def cong: bind_pmf_cong)
```
```  1358   ultimately show ?thesis ..
```
```  1359 qed
```
```  1360
```
```  1361 lemma rel_pmf_bindI:
```
```  1362   assumes pq: "rel_pmf R p q"
```
```  1363   and fg: "\<And>x y. R x y \<Longrightarrow> rel_pmf P (f x) (g y)"
```
```  1364   shows "rel_pmf P (bind_pmf p f) (bind_pmf q g)"
```
```  1365   unfolding bind_eq_join_pmf
```
```  1366   by (rule rel_pmf_joinI)
```
```  1367      (auto simp add: pmf.rel_map intro: pmf.rel_mono[THEN le_funD, THEN le_funD, THEN le_boolD, THEN mp, OF _ pq] fg)
```
```  1368
```
```  1369 text {*
```
```  1370   Proof that @{const rel_pmf} preserves orders.
```
```  1371   Antisymmetry proof follows Thm. 1 in N. Saheb-Djahromi, Cpo's of measures for nondeterminism,
```
```  1372   Theoretical Computer Science 12(1):19--37, 1980,
```
```  1373   @{url "http://dx.doi.org/10.1016/0304-3975(80)90003-1"}
```
```  1374 *}
```
```  1375
```
```  1376 lemma
```
```  1377   assumes *: "rel_pmf R p q"
```
```  1378   and refl: "reflp R" and trans: "transp R"
```
```  1379   shows measure_Ici: "measure p {y. R x y} \<le> measure q {y. R x y}" (is ?thesis1)
```
```  1380   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)
```
```  1381 proof -
```
```  1382   from * obtain pq
```
```  1383     where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
```
```  1384     and p: "p = map_pmf fst pq"
```
```  1385     and q: "q = map_pmf snd pq"
```
```  1386     by cases auto
```
```  1387   show ?thesis1 ?thesis2 unfolding p q map_pmf_rep_eq using refl trans
```
```  1388     by(auto 4 3 simp add: measure_distr reflpD AE_measure_pmf_iff intro!: measure_pmf.finite_measure_mono_AE dest!: pq elim: transpE)
```
```  1389 qed
```
```  1390
```
```  1391 lemma rel_pmf_inf:
```
```  1392   fixes p q :: "'a pmf"
```
```  1393   assumes 1: "rel_pmf R p q"
```
```  1394   assumes 2: "rel_pmf R q p"
```
```  1395   and refl: "reflp R" and trans: "transp R"
```
```  1396   shows "rel_pmf (inf R R\<inverse>\<inverse>) p q"
```
```  1397 proof (subst rel_pmf_iff_equivp, safe)
```
```  1398   show "equivp (inf R R\<inverse>\<inverse>)"
```
```  1399     using trans refl by (auto simp: equivp_reflp_symp_transp intro: sympI transpI reflpI dest: transpD reflpD)
```
```  1400
```
```  1401   fix C assume "C \<in> UNIV // {(x, y). inf R R\<inverse>\<inverse> x y}"
```
```  1402   then obtain x where C: "C = {y. R x y \<and> R y x}"
```
```  1403     by (auto elim: quotientE)
```
```  1404
```
```  1405   let ?R = "\<lambda>x y. R x y \<and> R y x"
```
```  1406   let ?\<mu>R = "\<lambda>y. measure q {x. ?R x y}"
```
```  1407   have "measure p {y. ?R x y} = measure p ({y. R x y} - {y. R x y \<and> \<not> R y x})"
```
```  1408     by(auto intro!: arg_cong[where f="measure p"])
```
```  1409   also have "\<dots> = measure p {y. R x y} - measure p {y. R x y \<and> \<not> R y x}"
```
```  1410     by (rule measure_pmf.finite_measure_Diff) auto
```
```  1411   also have "measure p {y. R x y \<and> \<not> R y x} = measure q {y. R x y \<and> \<not> R y x}"
```
```  1412     using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ioi)
```
```  1413   also have "measure p {y. R x y} = measure q {y. R x y}"
```
```  1414     using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ici)
```
```  1415   also have "measure q {y. R x y} - measure q {y. R x y \<and> \<not> R y x} =
```
```  1416     measure q ({y. R x y} - {y. R x y \<and> \<not> R y x})"
```
```  1417     by(rule measure_pmf.finite_measure_Diff[symmetric]) auto
```
```  1418   also have "\<dots> = ?\<mu>R x"
```
```  1419     by(auto intro!: arg_cong[where f="measure q"])
```
```  1420   finally show "measure p C = measure q C"
```
```  1421     by (simp add: C conj_commute)
```
```  1422 qed
```
```  1423
```
```  1424 lemma rel_pmf_antisym:
```
```  1425   fixes p q :: "'a pmf"
```
```  1426   assumes 1: "rel_pmf R p q"
```
```  1427   assumes 2: "rel_pmf R q p"
```
```  1428   and refl: "reflp R" and trans: "transp R" and antisym: "antisymP R"
```
```  1429   shows "p = q"
```
```  1430 proof -
```
```  1431   from 1 2 refl trans have "rel_pmf (inf R R\<inverse>\<inverse>) p q" by(rule rel_pmf_inf)
```
```  1432   also have "inf R R\<inverse>\<inverse> = op ="
```
```  1433     using refl antisym by (auto intro!: ext simp add: reflpD dest: antisymD)
```
```  1434   finally show ?thesis unfolding pmf.rel_eq .
```
```  1435 qed
```
```  1436
```
```  1437 lemma reflp_rel_pmf: "reflp R \<Longrightarrow> reflp (rel_pmf R)"
```
```  1438 by(blast intro: reflpI rel_pmf_reflI reflpD)
```
```  1439
```
```  1440 lemma antisymP_rel_pmf:
```
```  1441   "\<lbrakk> reflp R; transp R; antisymP R \<rbrakk>
```
```  1442   \<Longrightarrow> antisymP (rel_pmf R)"
```
```  1443 by(rule antisymI)(blast intro: rel_pmf_antisym)
```
```  1444
```
```  1445 lemma transp_rel_pmf:
```
```  1446   assumes "transp R"
```
```  1447   shows "transp (rel_pmf R)"
```
```  1448 proof (rule transpI)
```
```  1449   fix x y z
```
```  1450   assume "rel_pmf R x y" and "rel_pmf R y z"
```
```  1451   hence "rel_pmf (R OO R) x z" by (simp add: pmf.rel_compp relcompp.relcompI)
```
```  1452   thus "rel_pmf R x z"
```
```  1453     using assms by (metis (no_types) pmf.rel_mono rev_predicate2D transp_relcompp_less_eq)
```
```  1454 qed
```
```  1455
```
```  1456 subsection \<open> Distributions \<close>
```
```  1457
```
```  1458 context
```
```  1459 begin
```
```  1460
```
```  1461 interpretation pmf_as_function .
```
```  1462
```
```  1463 subsubsection \<open> Bernoulli Distribution \<close>
```
```  1464
```
```  1465 lift_definition bernoulli_pmf :: "real \<Rightarrow> bool pmf" is
```
```  1466   "\<lambda>p b. ((\<lambda>p. if b then p else 1 - p) \<circ> min 1 \<circ> max 0) p"
```
```  1467   by (auto simp: nn_integral_count_space_finite[where A="{False, True}"] UNIV_bool
```
```  1468            split: split_max split_min)
```
```  1469
```
```  1470 lemma pmf_bernoulli_True[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) True = p"
```
```  1471   by transfer simp
```
```  1472
```
```  1473 lemma pmf_bernoulli_False[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) False = 1 - p"
```
```  1474   by transfer simp
```
```  1475
```
```  1476 lemma set_pmf_bernoulli: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (bernoulli_pmf p) = UNIV"
```
```  1477   by (auto simp add: set_pmf_iff UNIV_bool)
```
```  1478
```
```  1479 lemma nn_integral_bernoulli_pmf[simp]:
```
```  1480   assumes [simp]: "0 \<le> p" "p \<le> 1" "\<And>x. 0 \<le> f x"
```
```  1481   shows "(\<integral>\<^sup>+x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
```
```  1482   by (subst nn_integral_measure_pmf_support[of UNIV])
```
```  1483      (auto simp: UNIV_bool field_simps)
```
```  1484
```
```  1485 lemma integral_bernoulli_pmf[simp]:
```
```  1486   assumes [simp]: "0 \<le> p" "p \<le> 1"
```
```  1487   shows "(\<integral>x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
```
```  1488   by (subst integral_measure_pmf[of UNIV]) (auto simp: UNIV_bool)
```
```  1489
```
```  1490 lemma pmf_bernoulli_half [simp]: "pmf (bernoulli_pmf (1 / 2)) x = 1 / 2"
```
```  1491 by(cases x) simp_all
```
```  1492
```
```  1493 lemma measure_pmf_bernoulli_half: "measure_pmf (bernoulli_pmf (1 / 2)) = uniform_count_measure UNIV"
```
```  1494 by(rule measure_eqI)(simp_all add: nn_integral_pmf[symmetric] emeasure_uniform_count_measure nn_integral_count_space_finite sets_uniform_count_measure)
```
```  1495
```
```  1496 subsubsection \<open> Geometric Distribution \<close>
```
```  1497
```
```  1498 context
```
```  1499   fixes p :: real assumes p[arith]: "0 < p" "p \<le> 1"
```
```  1500 begin
```
```  1501
```
```  1502 lift_definition geometric_pmf :: "nat pmf" is "\<lambda>n. (1 - p)^n * p"
```
```  1503 proof
```
```  1504   have "(\<Sum>i. ereal (p * (1 - p) ^ i)) = ereal (p * (1 / (1 - (1 - p))))"
```
```  1505     by (intro sums_suminf_ereal sums_mult geometric_sums) auto
```
```  1506   then show "(\<integral>\<^sup>+ x. ereal ((1 - p)^x * p) \<partial>count_space UNIV) = 1"
```
```  1507     by (simp add: nn_integral_count_space_nat field_simps)
```
```  1508 qed simp
```
```  1509
```
```  1510 lemma pmf_geometric[simp]: "pmf geometric_pmf n = (1 - p)^n * p"
```
```  1511   by transfer rule
```
```  1512
```
```  1513 end
```
```  1514
```
```  1515 lemma set_pmf_geometric: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (geometric_pmf p) = UNIV"
```
```  1516   by (auto simp: set_pmf_iff)
```
```  1517
```
```  1518 subsubsection \<open> Uniform Multiset Distribution \<close>
```
```  1519
```
```  1520 context
```
```  1521   fixes M :: "'a multiset" assumes M_not_empty: "M \<noteq> {#}"
```
```  1522 begin
```
```  1523
```
```  1524 lift_definition pmf_of_multiset :: "'a pmf" is "\<lambda>x. count M x / size M"
```
```  1525 proof
```
```  1526   show "(\<integral>\<^sup>+ x. ereal (real (count M x) / real (size M)) \<partial>count_space UNIV) = 1"
```
```  1527     using M_not_empty
```
```  1528     by (simp add: zero_less_divide_iff nn_integral_count_space nonempty_has_size
```
```  1529                   setsum_divide_distrib[symmetric])
```
```  1530        (auto simp: size_multiset_overloaded_eq intro!: setsum.cong)
```
```  1531 qed simp
```
```  1532
```
```  1533 lemma pmf_of_multiset[simp]: "pmf pmf_of_multiset x = count M x / size M"
```
```  1534   by transfer rule
```
```  1535
```
```  1536 lemma set_pmf_of_multiset[simp]: "set_pmf pmf_of_multiset = set_mset M"
```
```  1537   by (auto simp: set_pmf_iff)
```
```  1538
```
```  1539 end
```
```  1540
```
```  1541 subsubsection \<open> Uniform Distribution \<close>
```
```  1542
```
```  1543 context
```
```  1544   fixes S :: "'a set" assumes S_not_empty: "S \<noteq> {}" and S_finite: "finite S"
```
```  1545 begin
```
```  1546
```
```  1547 lift_definition pmf_of_set :: "'a pmf" is "\<lambda>x. indicator S x / card S"
```
```  1548 proof
```
```  1549   show "(\<integral>\<^sup>+ x. ereal (indicator S x / real (card S)) \<partial>count_space UNIV) = 1"
```
```  1550     using S_not_empty S_finite by (subst nn_integral_count_space'[of S]) auto
```
```  1551 qed simp
```
```  1552
```
```  1553 lemma pmf_of_set[simp]: "pmf pmf_of_set x = indicator S x / card S"
```
```  1554   by transfer rule
```
```  1555
```
```  1556 lemma set_pmf_of_set[simp]: "set_pmf pmf_of_set = S"
```
```  1557   using S_finite S_not_empty by (auto simp: set_pmf_iff)
```
```  1558
```
```  1559 lemma emeasure_pmf_of_set_space[simp]: "emeasure pmf_of_set S = 1"
```
```  1560   by (rule measure_pmf.emeasure_eq_1_AE) (auto simp: AE_measure_pmf_iff)
```
```  1561
```
```  1562 lemma nn_integral_pmf_of_set':
```
```  1563   "(\<And>x. x \<in> S \<Longrightarrow> f x \<ge> 0) \<Longrightarrow> nn_integral (measure_pmf pmf_of_set) f = setsum f S / card S"
```
```  1564 apply(subst nn_integral_measure_pmf_finite, simp_all add: S_finite)
```
```  1565 apply(simp add: setsum_ereal_left_distrib[symmetric])
```
```  1566 apply(subst ereal_divide', simp add: S_not_empty S_finite)
```
```  1567 apply(simp add: ereal_times_divide_eq one_ereal_def[symmetric])
```
```  1568 done
```
```  1569
```
```  1570 lemma nn_integral_pmf_of_set:
```
```  1571   "nn_integral (measure_pmf pmf_of_set) f = setsum (max 0 \<circ> f) S / card S"
```
```  1572 apply(subst nn_integral_max_0[symmetric])
```
```  1573 apply(subst nn_integral_pmf_of_set')
```
```  1574 apply simp_all
```
```  1575 done
```
```  1576
```
```  1577 lemma integral_pmf_of_set:
```
```  1578   "integral\<^sup>L (measure_pmf pmf_of_set) f = setsum f S / card S"
```
```  1579 apply(simp add: real_lebesgue_integral_def integrable_measure_pmf_finite nn_integral_pmf_of_set S_finite)
```
```  1580 apply(subst real_of_ereal_minus')
```
```  1581  apply(simp add: ereal_max_0 S_finite del: ereal_max)
```
```  1582 apply(simp add: ereal_max_0 S_finite S_not_empty del: ereal_max)
```
```  1583 apply(simp add: field_simps S_finite S_not_empty)
```
```  1584 apply(subst setsum.distrib[symmetric])
```
```  1585 apply(rule setsum.cong; simp_all)
```
```  1586 done
```
```  1587
```
```  1588 lemma emeasure_pmf_of_set:
```
```  1589   "emeasure (measure_pmf pmf_of_set) A = card (S \<inter> A) / card S"
```
```  1590 apply(subst nn_integral_indicator[symmetric], simp)
```
```  1591 apply(subst nn_integral_pmf_of_set)
```
```  1592 apply(simp add: o_def max_def ereal_indicator[symmetric] S_not_empty S_finite real_of_nat_indicator[symmetric] of_nat_setsum[symmetric] setsum_indicator_eq_card del: of_nat_setsum)
```
```  1593 done
```
```  1594
```
```  1595 end
```
```  1596
```
```  1597 lemma pmf_of_set_singleton: "pmf_of_set {x} = return_pmf x"
```
```  1598 by(rule pmf_eqI)(simp add: indicator_def)
```
```  1599
```
```  1600 lemma map_pmf_of_set_inj:
```
```  1601   assumes f: "inj_on f A"
```
```  1602   and [simp]: "A \<noteq> {}" "finite A"
```
```  1603   shows "map_pmf f (pmf_of_set A) = pmf_of_set (f ` A)" (is "?lhs = ?rhs")
```
```  1604 proof(rule pmf_eqI)
```
```  1605   fix i
```
```  1606   show "pmf ?lhs i = pmf ?rhs i"
```
```  1607   proof(cases "i \<in> f ` A")
```
```  1608     case True
```
```  1609     then obtain i' where "i = f i'" "i' \<in> A" by auto
```
```  1610     thus ?thesis using f by(simp add: card_image pmf_map_inj)
```
```  1611   next
```
```  1612     case False
```
```  1613     hence "pmf ?lhs i = 0" by(simp add: pmf_eq_0_set_pmf set_map_pmf)
```
```  1614     moreover have "pmf ?rhs i = 0" using False by simp
```
```  1615     ultimately show ?thesis by simp
```
```  1616   qed
```
```  1617 qed
```
```  1618
```
```  1619 lemma bernoulli_pmf_half_conv_pmf_of_set: "bernoulli_pmf (1 / 2) = pmf_of_set UNIV"
```
```  1620 by(rule pmf_eqI) simp_all
```
```  1621
```
```  1622
```
```  1623
```
```  1624 lemma measure_pmf_of_set:
```
```  1625   assumes "S \<noteq> {}" "finite S"
```
```  1626   shows "measure (measure_pmf (pmf_of_set S)) A = card (S \<inter> A) / card S"
```
```  1627 using emeasure_pmf_of_set[OF assms, of A]
```
```  1628 unfolding measure_pmf.emeasure_eq_measure by simp
```
```  1629
```
```  1630 subsubsection \<open> Poisson Distribution \<close>
```
```  1631
```
```  1632 context
```
```  1633   fixes rate :: real assumes rate_pos: "0 < rate"
```
```  1634 begin
```
```  1635
```
```  1636 lift_definition poisson_pmf :: "nat pmf" is "\<lambda>k. rate ^ k / fact k * exp (-rate)"
```
```  1637 proof  (* by Manuel Eberl *)
```
```  1638   have summable: "summable (\<lambda>x::nat. rate ^ x / fact x)" using summable_exp
```
```  1639     by (simp add: field_simps divide_inverse [symmetric])
```
```  1640   have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x * exp (-rate) \<partial>count_space UNIV) =
```
```  1641           exp (-rate) * (\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV)"
```
```  1642     by (simp add: field_simps nn_integral_cmult[symmetric])
```
```  1643   also from rate_pos have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV) = (\<Sum>x. rate ^ x / fact x)"
```
```  1644     by (simp_all add: nn_integral_count_space_nat suminf_ereal summable suminf_ereal_finite)
```
```  1645   also have "... = exp rate" unfolding exp_def
```
```  1646     by (simp add: field_simps divide_inverse [symmetric])
```
```  1647   also have "ereal (exp (-rate)) * ereal (exp rate) = 1"
```
```  1648     by (simp add: mult_exp_exp)
```
```  1649   finally show "(\<integral>\<^sup>+ x. ereal (rate ^ x / (fact x) * exp (- rate)) \<partial>count_space UNIV) = 1" .
```
```  1650 qed (simp add: rate_pos[THEN less_imp_le])
```
```  1651
```
```  1652 lemma pmf_poisson[simp]: "pmf poisson_pmf k = rate ^ k / fact k * exp (-rate)"
```
```  1653   by transfer rule
```
```  1654
```
```  1655 lemma set_pmf_poisson[simp]: "set_pmf poisson_pmf = UNIV"
```
```  1656   using rate_pos by (auto simp: set_pmf_iff)
```
```  1657
```
```  1658 end
```
```  1659
```
```  1660 subsubsection \<open> Binomial Distribution \<close>
```
```  1661
```
```  1662 context
```
```  1663   fixes n :: nat and p :: real assumes p_nonneg: "0 \<le> p" and p_le_1: "p \<le> 1"
```
```  1664 begin
```
```  1665
```
```  1666 lift_definition binomial_pmf :: "nat pmf" is "\<lambda>k. (n choose k) * p^k * (1 - p)^(n - k)"
```
```  1667 proof
```
```  1668   have "(\<integral>\<^sup>+k. ereal (real (n choose k) * p ^ k * (1 - p) ^ (n - k)) \<partial>count_space UNIV) =
```
```  1669     ereal (\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k))"
```
```  1670     using p_le_1 p_nonneg by (subst nn_integral_count_space') auto
```
```  1671   also have "(\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k)) = (p + (1 - p)) ^ n"
```
```  1672     by (subst binomial_ring) (simp add: atLeast0AtMost)
```
```  1673   finally show "(\<integral>\<^sup>+ x. ereal (real (n choose x) * p ^ x * (1 - p) ^ (n - x)) \<partial>count_space UNIV) = 1"
```
```  1674     by simp
```
```  1675 qed (insert p_nonneg p_le_1, simp)
```
```  1676
```
```  1677 lemma pmf_binomial[simp]: "pmf binomial_pmf k = (n choose k) * p^k * (1 - p)^(n - k)"
```
```  1678   by transfer rule
```
```  1679
```
```  1680 lemma set_pmf_binomial_eq: "set_pmf binomial_pmf = (if p = 0 then {0} else if p = 1 then {n} else {.. n})"
```
```  1681   using p_nonneg p_le_1 unfolding set_eq_iff set_pmf_iff pmf_binomial by (auto simp: set_pmf_iff)
```
```  1682
```
```  1683 end
```
```  1684
```
```  1685 end
```
```  1686
```
```  1687 lemma set_pmf_binomial_0[simp]: "set_pmf (binomial_pmf n 0) = {0}"
```
```  1688   by (simp add: set_pmf_binomial_eq)
```
```  1689
```
```  1690 lemma set_pmf_binomial_1[simp]: "set_pmf (binomial_pmf n 1) = {n}"
```
```  1691   by (simp add: set_pmf_binomial_eq)
```
```  1692
```
```  1693 lemma set_pmf_binomial[simp]: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (binomial_pmf n p) = {..n}"
```
```  1694   by (simp add: set_pmf_binomial_eq)
```
```  1695
```
```  1696 context begin interpretation lifting_syntax .
```
```  1697
```
```  1698 lemma bind_pmf_parametric [transfer_rule]:
```
```  1699   "(rel_pmf A ===> (A ===> rel_pmf B) ===> rel_pmf B) bind_pmf bind_pmf"
```
```  1700 by(blast intro: rel_pmf_bindI dest: rel_funD)
```
```  1701
```
```  1702 lemma return_pmf_parametric [transfer_rule]: "(A ===> rel_pmf A) return_pmf return_pmf"
```
```  1703 by(rule rel_funI) simp
```
```  1704
```
```  1705 end
```
```  1706
```
```  1707 end
```