src/HOL/Probability/Probability_Mass_Function.thy
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```     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:) }
```
```    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 (in finite_measure) AE_support_countable:
```
```    33   assumes [simp]: "sets M = UNIV"
```
```    34   shows "(AE x in M. measure M {x} \<noteq> 0) \<longleftrightarrow> (\<exists>S. countable S \<and> (AE x in M. x \<in> S))"
```
```    35 proof
```
```    36   assume "\<exists>S. countable S \<and> (AE x in M. x \<in> S)"
```
```    37   then obtain S where S[intro]: "countable S" and ae: "AE x in M. x \<in> S"
```
```    38     by auto
```
```    39   then have "emeasure M (\<Union>x\<in>{x\<in>S. emeasure M {x} \<noteq> 0}. {x}) =
```
```    40     (\<integral>\<^sup>+ x. emeasure M {x} * indicator {x\<in>S. emeasure M {x} \<noteq> 0} x \<partial>count_space UNIV)"
```
```    41     by (subst emeasure_UN_countable)
```
```    42        (auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space)
```
```    43   also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} * indicator S x \<partial>count_space UNIV)"
```
```    44     by (auto intro!: nn_integral_cong split: split_indicator)
```
```    45   also have "\<dots> = emeasure M (\<Union>x\<in>S. {x})"
```
```    46     by (subst emeasure_UN_countable)
```
```    47        (auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space)
```
```    48   also have "\<dots> = emeasure M (space M)"
```
```    49     using ae by (intro emeasure_eq_AE) auto
```
```    50   finally have "emeasure M {x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0} = emeasure M (space M)"
```
```    51     by (simp add: emeasure_single_in_space cong: rev_conj_cong)
```
```    52   with finite_measure_compl[of "{x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0}"]
```
```    53   have "AE x in M. x \<in> S \<and> emeasure M {x} \<noteq> 0"
```
```    54     by (intro AE_I[OF order_refl]) (auto simp: emeasure_eq_measure measure_nonneg set_diff_eq cong: conj_cong)
```
```    55   then show "AE x in M. measure M {x} \<noteq> 0"
```
```    56     by (auto simp: emeasure_eq_measure)
```
```    57 qed (auto intro!: exI[of _ "{x. measure M {x} \<noteq> 0}"] countable_support)
```
```    58
```
```    59 subsection \<open> PMF as measure \<close>
```
```    60
```
```    61 typedef 'a pmf = "{M :: 'a measure. prob_space M \<and> sets M = UNIV \<and> (AE x in M. measure M {x} \<noteq> 0)}"
```
```    62   morphisms measure_pmf Abs_pmf
```
```    63   by (intro exI[of _ "uniform_measure (count_space UNIV) {undefined}"])
```
```    64      (auto intro!: prob_space_uniform_measure AE_uniform_measureI)
```
```    65
```
```    66 declare [[coercion measure_pmf]]
```
```    67
```
```    68 lemma prob_space_measure_pmf: "prob_space (measure_pmf p)"
```
```    69   using pmf.measure_pmf[of p] by auto
```
```    70
```
```    71 interpretation measure_pmf: prob_space "measure_pmf M" for M
```
```    72   by (rule prob_space_measure_pmf)
```
```    73
```
```    74 interpretation measure_pmf: subprob_space "measure_pmf M" for M
```
```    75   by (rule prob_space_imp_subprob_space) unfold_locales
```
```    76
```
```    77 lemma subprob_space_measure_pmf: "subprob_space (measure_pmf x)"
```
```    78   by unfold_locales
```
```    79
```
```    80 locale pmf_as_measure
```
```    81 begin
```
```    82
```
```    83 setup_lifting type_definition_pmf
```
```    84
```
```    85 end
```
```    86
```
```    87 context
```
```    88 begin
```
```    89
```
```    90 interpretation pmf_as_measure .
```
```    91
```
```    92 lemma sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV"
```
```    93   by transfer blast
```
```    94
```
```    95 lemma sets_measure_pmf_count_space[measurable_cong]:
```
```    96   "sets (measure_pmf M) = sets (count_space UNIV)"
```
```    97   by simp
```
```    98
```
```    99 lemma space_measure_pmf[simp]: "space (measure_pmf p) = UNIV"
```
```   100   using sets_eq_imp_space_eq[of "measure_pmf p" "count_space UNIV"] by simp
```
```   101
```
```   102 lemma measure_pmf_UNIV [simp]: "measure (measure_pmf p) UNIV = 1"
```
```   103 using measure_pmf.prob_space[of p] by simp
```
```   104
```
```   105 lemma measure_pmf_in_subprob_algebra[measurable (raw)]: "measure_pmf x \<in> space (subprob_algebra (count_space UNIV))"
```
```   106   by (simp add: space_subprob_algebra subprob_space_measure_pmf)
```
```   107
```
```   108 lemma measurable_pmf_measure1[simp]: "measurable (M :: 'a pmf) N = UNIV \<rightarrow> space N"
```
```   109   by (auto simp: measurable_def)
```
```   110
```
```   111 lemma measurable_pmf_measure2[simp]: "measurable N (M :: 'a pmf) = measurable N (count_space UNIV)"
```
```   112   by (intro measurable_cong_sets) simp_all
```
```   113
```
```   114 lemma measurable_pair_restrict_pmf2:
```
```   115   assumes "countable A"
```
```   116   assumes [measurable]: "\<And>y. y \<in> A \<Longrightarrow> (\<lambda>x. f (x, y)) \<in> measurable M L"
```
```   117   shows "f \<in> measurable (M \<Otimes>\<^sub>M restrict_space (measure_pmf N) A) L" (is "f \<in> measurable ?M _")
```
```   118 proof -
```
```   119   have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)"
```
```   120     by (simp add: restrict_count_space)
```
```   121
```
```   122   show ?thesis
```
```   123     by (intro measurable_compose_countable'[where f="\<lambda>a b. f (fst b, a)" and g=snd and I=A,
```
```   124                                             unfolded prod.collapse] assms)
```
```   125         measurable
```
```   126 qed
```
```   127
```
```   128 lemma measurable_pair_restrict_pmf1:
```
```   129   assumes "countable A"
```
```   130   assumes [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N L"
```
```   131   shows "f \<in> measurable (restrict_space (measure_pmf M) A \<Otimes>\<^sub>M N) L"
```
```   132 proof -
```
```   133   have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)"
```
```   134     by (simp add: restrict_count_space)
```
```   135
```
```   136   show ?thesis
```
```   137     by (intro measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A,
```
```   138                                             unfolded prod.collapse] assms)
```
```   139         measurable
```
```   140 qed
```
```   141
```
```   142 lift_definition pmf :: "'a pmf \<Rightarrow> 'a \<Rightarrow> real" is "\<lambda>M x. measure M {x}" .
```
```   143
```
```   144 lift_definition set_pmf :: "'a pmf \<Rightarrow> 'a set" is "\<lambda>M. {x. measure M {x} \<noteq> 0}" .
```
```   145 declare [[coercion set_pmf]]
```
```   146
```
```   147 lemma AE_measure_pmf: "AE x in (M::'a pmf). x \<in> M"
```
```   148   by transfer simp
```
```   149
```
```   150 lemma emeasure_pmf_single_eq_zero_iff:
```
```   151   fixes M :: "'a pmf"
```
```   152   shows "emeasure M {y} = 0 \<longleftrightarrow> y \<notin> M"
```
```   153   unfolding set_pmf.rep_eq by (simp add: measure_pmf.emeasure_eq_measure)
```
```   154
```
```   155 lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \<longleftrightarrow> (\<forall>y\<in>M. P y)"
```
```   156   using AE_measure_singleton[of M] AE_measure_pmf[of M]
```
```   157   by (auto simp: set_pmf.rep_eq)
```
```   158
```
```   159 lemma AE_pmfI: "(\<And>y. y \<in> set_pmf M \<Longrightarrow> P y) \<Longrightarrow> almost_everywhere (measure_pmf M) P"
```
```   160 by(simp add: AE_measure_pmf_iff)
```
```   161
```
```   162 lemma countable_set_pmf [simp]: "countable (set_pmf p)"
```
```   163   by transfer (metis prob_space.finite_measure finite_measure.countable_support)
```
```   164
```
```   165 lemma pmf_positive: "x \<in> set_pmf p \<Longrightarrow> 0 < pmf p x"
```
```   166   by transfer (simp add: less_le)
```
```   167
```
```   168 lemma pmf_nonneg[simp]: "0 \<le> pmf p x"
```
```   169   by transfer simp
```
```   170
```
```   171 lemma pmf_not_neg [simp]: "\<not>pmf p x < 0"
```
```   172   by (simp add: not_less pmf_nonneg)
```
```   173
```
```   174 lemma pmf_pos [simp]: "pmf p x \<noteq> 0 \<Longrightarrow> pmf p x > 0"
```
```   175   using pmf_nonneg[of p x] by linarith
```
```   176
```
```   177 lemma pmf_le_1: "pmf p x \<le> 1"
```
```   178   by (simp add: pmf.rep_eq)
```
```   179
```
```   180 lemma set_pmf_not_empty: "set_pmf M \<noteq> {}"
```
```   181   using AE_measure_pmf[of M] by (intro notI) simp
```
```   182
```
```   183 lemma set_pmf_iff: "x \<in> set_pmf M \<longleftrightarrow> pmf M x \<noteq> 0"
```
```   184   by transfer simp
```
```   185
```
```   186 lemma pmf_positive_iff: "0 < pmf p x \<longleftrightarrow> x \<in> set_pmf p"
```
```   187   unfolding less_le by (simp add: set_pmf_iff)
```
```   188
```
```   189 lemma set_pmf_eq: "set_pmf M = {x. pmf M x \<noteq> 0}"
```
```   190   by (auto simp: set_pmf_iff)
```
```   191
```
```   192 lemma set_pmf_eq': "set_pmf p = {x. pmf p x > 0}"
```
```   193 proof safe
```
```   194   fix x assume "x \<in> set_pmf p"
```
```   195   hence "pmf p x \<noteq> 0" by (auto simp: set_pmf_eq)
```
```   196   with pmf_nonneg[of p x] show "pmf p x > 0" by simp
```
```   197 qed (auto simp: set_pmf_eq)
```
```   198
```
```   199 lemma emeasure_pmf_single:
```
```   200   fixes M :: "'a pmf"
```
```   201   shows "emeasure M {x} = pmf M x"
```
```   202   by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
```
```   203
```
```   204 lemma measure_pmf_single: "measure (measure_pmf M) {x} = pmf M x"
```
```   205   using emeasure_pmf_single[of M x] by(simp add: measure_pmf.emeasure_eq_measure pmf_nonneg measure_nonneg)
```
```   206
```
```   207 lemma emeasure_measure_pmf_finite: "finite S \<Longrightarrow> emeasure (measure_pmf M) S = (\<Sum>s\<in>S. pmf M s)"
```
```   208   by (subst emeasure_eq_sum_singleton) (auto simp: emeasure_pmf_single pmf_nonneg)
```
```   209
```
```   210 lemma measure_measure_pmf_finite: "finite S \<Longrightarrow> measure (measure_pmf M) S = sum (pmf M) S"
```
```   211   using emeasure_measure_pmf_finite[of S M]
```
```   212   by (simp add: measure_pmf.emeasure_eq_measure measure_nonneg sum_nonneg pmf_nonneg)
```
```   213
```
```   214 lemma sum_pmf_eq_1:
```
```   215   assumes "finite A" "set_pmf p \<subseteq> A"
```
```   216   shows   "(\<Sum>x\<in>A. pmf p x) = 1"
```
```   217 proof -
```
```   218   have "(\<Sum>x\<in>A. pmf p x) = measure_pmf.prob p A"
```
```   219     by (simp add: measure_measure_pmf_finite assms)
```
```   220   also from assms have "\<dots> = 1"
```
```   221     by (subst measure_pmf.prob_eq_1) (auto simp: AE_measure_pmf_iff)
```
```   222   finally show ?thesis .
```
```   223 qed
```
```   224
```
```   225 lemma nn_integral_measure_pmf_support:
```
```   226   fixes f :: "'a \<Rightarrow> ennreal"
```
```   227   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"
```
```   228   shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>A. f x * pmf M x)"
```
```   229 proof -
```
```   230   have "(\<integral>\<^sup>+x. f x \<partial>M) = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)"
```
```   231     using nn by (intro nn_integral_cong_AE) (auto simp: AE_measure_pmf_iff split: split_indicator)
```
```   232   also have "\<dots> = (\<Sum>x\<in>A. f x * emeasure M {x})"
```
```   233     using assms by (intro nn_integral_indicator_finite) auto
```
```   234   finally show ?thesis
```
```   235     by (simp add: emeasure_measure_pmf_finite)
```
```   236 qed
```
```   237
```
```   238 lemma nn_integral_measure_pmf_finite:
```
```   239   fixes f :: "'a \<Rightarrow> ennreal"
```
```   240   assumes f: "finite (set_pmf M)" and nn: "\<And>x. x \<in> set_pmf M \<Longrightarrow> 0 \<le> f x"
```
```   241   shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>set_pmf M. f x * pmf M x)"
```
```   242   using assms by (intro nn_integral_measure_pmf_support) auto
```
```   243
```
```   244 lemma integrable_measure_pmf_finite:
```
```   245   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
```
```   246   shows "finite (set_pmf M) \<Longrightarrow> integrable M f"
```
```   247   by (auto intro!: integrableI_bounded simp: nn_integral_measure_pmf_finite ennreal_mult_less_top)
```
```   248
```
```   249 lemma integral_measure_pmf_real:
```
```   250   assumes [simp]: "finite A" and "\<And>a. a \<in> set_pmf M \<Longrightarrow> f a \<noteq> 0 \<Longrightarrow> a \<in> A"
```
```   251   shows "(\<integral>x. f x \<partial>measure_pmf M) = (\<Sum>a\<in>A. f a * pmf M a)"
```
```   252 proof -
```
```   253   have "(\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x * indicator A x \<partial>measure_pmf M)"
```
```   254     using assms(2) by (intro integral_cong_AE) (auto split: split_indicator simp: AE_measure_pmf_iff)
```
```   255   also have "\<dots> = (\<Sum>a\<in>A. f a * pmf M a)"
```
```   256     by (subst integral_indicator_finite_real)
```
```   257        (auto simp: measure_def emeasure_measure_pmf_finite pmf_nonneg)
```
```   258   finally show ?thesis .
```
```   259 qed
```
```   260
```
```   261 lemma integrable_pmf: "integrable (count_space X) (pmf M)"
```
```   262 proof -
```
```   263   have " (\<integral>\<^sup>+ x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+ x. pmf M x \<partial>count_space (M \<inter> X))"
```
```   264     by (auto simp add: nn_integral_count_space_indicator set_pmf_iff intro!: nn_integral_cong split: split_indicator)
```
```   265   then have "integrable (count_space X) (pmf M) = integrable (count_space (M \<inter> X)) (pmf M)"
```
```   266     by (simp add: integrable_iff_bounded pmf_nonneg)
```
```   267   then show ?thesis
```
```   268     by (simp add: pmf.rep_eq measure_pmf.integrable_measure disjoint_family_on_def)
```
```   269 qed
```
```   270
```
```   271 lemma integral_pmf: "(\<integral>x. pmf M x \<partial>count_space X) = measure M X"
```
```   272 proof -
```
```   273   have "(\<integral>x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+x. pmf M x \<partial>count_space X)"
```
```   274     by (simp add: pmf_nonneg integrable_pmf nn_integral_eq_integral)
```
```   275   also have "\<dots> = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space (X \<inter> M))"
```
```   276     by (auto intro!: nn_integral_cong_AE split: split_indicator
```
```   277              simp: pmf.rep_eq measure_pmf.emeasure_eq_measure nn_integral_count_space_indicator
```
```   278                    AE_count_space set_pmf_iff)
```
```   279   also have "\<dots> = emeasure M (X \<inter> M)"
```
```   280     by (rule emeasure_countable_singleton[symmetric]) (auto intro: countable_set_pmf)
```
```   281   also have "\<dots> = emeasure M X"
```
```   282     by (auto intro!: emeasure_eq_AE simp: AE_measure_pmf_iff)
```
```   283   finally show ?thesis
```
```   284     by (simp add: measure_pmf.emeasure_eq_measure measure_nonneg integral_nonneg pmf_nonneg)
```
```   285 qed
```
```   286
```
```   287 lemma integral_pmf_restrict:
```
```   288   "(f::'a \<Rightarrow> 'b::{banach, second_countable_topology}) \<in> borel_measurable (count_space UNIV) \<Longrightarrow>
```
```   289     (\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x \<partial>restrict_space M M)"
```
```   290   by (auto intro!: integral_cong_AE simp add: integral_restrict_space AE_measure_pmf_iff)
```
```   291
```
```   292 lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1"
```
```   293 proof -
```
```   294   have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)"
```
```   295     by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf)
```
```   296   then show ?thesis
```
```   297     using measure_pmf.emeasure_space_1 by simp
```
```   298 qed
```
```   299
```
```   300 lemma emeasure_pmf_UNIV [simp]: "emeasure (measure_pmf M) UNIV = 1"
```
```   301 using measure_pmf.emeasure_space_1[of M] by simp
```
```   302
```
```   303 lemma in_null_sets_measure_pmfI:
```
```   304   "A \<inter> set_pmf p = {} \<Longrightarrow> A \<in> null_sets (measure_pmf p)"
```
```   305 using emeasure_eq_0_AE[where ?P="\<lambda>x. x \<in> A" and M="measure_pmf p"]
```
```   306 by(auto simp add: null_sets_def AE_measure_pmf_iff)
```
```   307
```
```   308 lemma measure_subprob: "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
```
```   309   by (simp add: space_subprob_algebra subprob_space_measure_pmf)
```
```   310
```
```   311 subsection \<open> Monad Interpretation \<close>
```
```   312
```
```   313 lemma measurable_measure_pmf[measurable]:
```
```   314   "(\<lambda>x. measure_pmf (M x)) \<in> measurable (count_space UNIV) (subprob_algebra (count_space UNIV))"
```
```   315   by (auto simp: space_subprob_algebra intro!: prob_space_imp_subprob_space) unfold_locales
```
```   316
```
```   317 lemma bind_measure_pmf_cong:
```
```   318   assumes "\<And>x. A x \<in> space (subprob_algebra N)" "\<And>x. B x \<in> space (subprob_algebra N)"
```
```   319   assumes "\<And>i. i \<in> set_pmf x \<Longrightarrow> A i = B i"
```
```   320   shows "bind (measure_pmf x) A = bind (measure_pmf x) B"
```
```   321 proof (rule measure_eqI)
```
```   322   show "sets (measure_pmf x \<bind> A) = sets (measure_pmf x \<bind> B)"
```
```   323     using assms by (subst (1 2) sets_bind) (auto simp: space_subprob_algebra)
```
```   324 next
```
```   325   fix X assume "X \<in> sets (measure_pmf x \<bind> A)"
```
```   326   then have X: "X \<in> sets N"
```
```   327     using assms by (subst (asm) sets_bind) (auto simp: space_subprob_algebra)
```
```   328   show "emeasure (measure_pmf x \<bind> A) X = emeasure (measure_pmf x \<bind> B) X"
```
```   329     using assms
```
```   330     by (subst (1 2) emeasure_bind[where N=N, OF _ _ X])
```
```   331        (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
```
```   332 qed
```
```   333
```
```   334 lift_definition bind_pmf :: "'a pmf \<Rightarrow> ('a \<Rightarrow> 'b pmf ) \<Rightarrow> 'b pmf" is bind
```
```   335 proof (clarify, intro conjI)
```
```   336   fix f :: "'a measure" and g :: "'a \<Rightarrow> 'b measure"
```
```   337   assume "prob_space f"
```
```   338   then interpret f: prob_space f .
```
```   339   assume "sets f = UNIV" and ae_f: "AE x in f. measure f {x} \<noteq> 0"
```
```   340   then have s_f[simp]: "sets f = sets (count_space UNIV)"
```
```   341     by simp
```
```   342   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)"
```
```   343   then have g: "\<And>x. prob_space (g x)" and s_g[simp]: "\<And>x. sets (g x) = sets (count_space UNIV)"
```
```   344     and ae_g: "\<And>x. AE y in g x. measure (g x) {y} \<noteq> 0"
```
```   345     by auto
```
```   346
```
```   347   have [measurable]: "g \<in> measurable f (subprob_algebra (count_space UNIV))"
```
```   348     by (auto simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space g)
```
```   349
```
```   350   show "prob_space (f \<bind> g)"
```
```   351     using g by (intro f.prob_space_bind[where S="count_space UNIV"]) auto
```
```   352   then interpret fg: prob_space "f \<bind> g" .
```
```   353   show [simp]: "sets (f \<bind> g) = UNIV"
```
```   354     using sets_eq_imp_space_eq[OF s_f]
```
```   355     by (subst sets_bind[where N="count_space UNIV"]) auto
```
```   356   show "AE x in f \<bind> g. measure (f \<bind> g) {x} \<noteq> 0"
```
```   357     apply (simp add: fg.prob_eq_0 AE_bind[where B="count_space UNIV"])
```
```   358     using ae_f
```
```   359     apply eventually_elim
```
```   360     using ae_g
```
```   361     apply eventually_elim
```
```   362     apply (auto dest: AE_measure_singleton)
```
```   363     done
```
```   364 qed
```
```   365
```
```   366 adhoc_overloading Monad_Syntax.bind bind_pmf
```
```   367
```
```   368 lemma ennreal_pmf_bind: "pmf (bind_pmf N f) i = (\<integral>\<^sup>+x. pmf (f x) i \<partial>measure_pmf N)"
```
```   369   unfolding pmf.rep_eq bind_pmf.rep_eq
```
```   370   by (auto simp: measure_pmf.measure_bind[where N="count_space UNIV"] measure_subprob measure_nonneg
```
```   371            intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])
```
```   372
```
```   373 lemma pmf_bind: "pmf (bind_pmf N f) i = (\<integral>x. pmf (f x) i \<partial>measure_pmf N)"
```
```   374   using ennreal_pmf_bind[of N f i]
```
```   375   by (subst (asm) nn_integral_eq_integral)
```
```   376      (auto simp: pmf_nonneg pmf_le_1 pmf_nonneg integral_nonneg
```
```   377            intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])
```
```   378
```
```   379 lemma bind_pmf_const[simp]: "bind_pmf M (\<lambda>x. c) = c"
```
```   380   by transfer (simp add: bind_const' prob_space_imp_subprob_space)
```
```   381
```
```   382 lemma set_bind_pmf[simp]: "set_pmf (bind_pmf M N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))"
```
```   383 proof -
```
```   384   have "set_pmf (bind_pmf M N) = {x. ennreal (pmf (bind_pmf M N) x) \<noteq> 0}"
```
```   385     by (simp add: set_pmf_eq pmf_nonneg)
```
```   386   also have "\<dots> = (\<Union>M\<in>set_pmf M. set_pmf (N M))"
```
```   387     unfolding ennreal_pmf_bind
```
```   388     by (subst nn_integral_0_iff_AE) (auto simp: AE_measure_pmf_iff pmf_nonneg set_pmf_eq)
```
```   389   finally show ?thesis .
```
```   390 qed
```
```   391
```
```   392 lemma bind_pmf_cong [fundef_cong]:
```
```   393   assumes "p = q"
```
```   394   shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g"
```
```   395   unfolding \<open>p = q\<close>[symmetric] measure_pmf_inject[symmetric] bind_pmf.rep_eq
```
```   396   by (auto simp: AE_measure_pmf_iff Pi_iff space_subprob_algebra subprob_space_measure_pmf
```
```   397                  sets_bind[where N="count_space UNIV"] emeasure_bind[where N="count_space UNIV"]
```
```   398            intro!: nn_integral_cong_AE measure_eqI)
```
```   399
```
```   400 lemma bind_pmf_cong_simp:
```
```   401   "p = q \<Longrightarrow> (\<And>x. x \<in> set_pmf q =simp=> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g"
```
```   402   by (simp add: simp_implies_def cong: bind_pmf_cong)
```
```   403
```
```   404 lemma measure_pmf_bind: "measure_pmf (bind_pmf M f) = (measure_pmf M \<bind> (\<lambda>x. measure_pmf (f x)))"
```
```   405   by transfer simp
```
```   406
```
```   407 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)"
```
```   408   using measurable_measure_pmf[of N]
```
```   409   unfolding measure_pmf_bind
```
```   410   apply (intro nn_integral_bind[where B="count_space UNIV"])
```
```   411   apply auto
```
```   412   done
```
```   413
```
```   414 lemma emeasure_bind_pmf[simp]: "emeasure (bind_pmf M N) X = (\<integral>\<^sup>+x. emeasure (N x) X \<partial>M)"
```
```   415   using measurable_measure_pmf[of N]
```
```   416   unfolding measure_pmf_bind
```
```   417   by (subst emeasure_bind[where N="count_space UNIV"]) auto
```
```   418
```
```   419 lift_definition return_pmf :: "'a \<Rightarrow> 'a pmf" is "return (count_space UNIV)"
```
```   420   by (auto intro!: prob_space_return simp: AE_return measure_return)
```
```   421
```
```   422 lemma bind_return_pmf: "bind_pmf (return_pmf x) f = f x"
```
```   423   by transfer
```
```   424      (auto intro!: prob_space_imp_subprob_space bind_return[where N="count_space UNIV"]
```
```   425            simp: space_subprob_algebra)
```
```   426
```
```   427 lemma set_return_pmf[simp]: "set_pmf (return_pmf x) = {x}"
```
```   428   by transfer (auto simp add: measure_return split: split_indicator)
```
```   429
```
```   430 lemma bind_return_pmf': "bind_pmf N return_pmf = N"
```
```   431 proof (transfer, clarify)
```
```   432   fix N :: "'a measure" assume "sets N = UNIV" then show "N \<bind> return (count_space UNIV) = N"
```
```   433     by (subst return_sets_cong[where N=N]) (simp_all add: bind_return')
```
```   434 qed
```
```   435
```
```   436 lemma bind_assoc_pmf: "bind_pmf (bind_pmf A B) C = bind_pmf A (\<lambda>x. bind_pmf (B x) C)"
```
```   437   by transfer
```
```   438      (auto intro!: bind_assoc[where N="count_space UNIV" and R="count_space UNIV"]
```
```   439            simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space)
```
```   440
```
```   441 definition "map_pmf f M = bind_pmf M (\<lambda>x. return_pmf (f x))"
```
```   442
```
```   443 lemma map_bind_pmf: "map_pmf f (bind_pmf M g) = bind_pmf M (\<lambda>x. map_pmf f (g x))"
```
```   444   by (simp add: map_pmf_def bind_assoc_pmf)
```
```   445
```
```   446 lemma bind_map_pmf: "bind_pmf (map_pmf f M) g = bind_pmf M (\<lambda>x. g (f x))"
```
```   447   by (simp add: map_pmf_def bind_assoc_pmf bind_return_pmf)
```
```   448
```
```   449 lemma map_pmf_transfer[transfer_rule]:
```
```   450   "rel_fun op = (rel_fun cr_pmf cr_pmf) (\<lambda>f M. distr M (count_space UNIV) f) map_pmf"
```
```   451 proof -
```
```   452   have "rel_fun op = (rel_fun pmf_as_measure.cr_pmf pmf_as_measure.cr_pmf)
```
```   453      (\<lambda>f M. M \<bind> (return (count_space UNIV) o f)) map_pmf"
```
```   454     unfolding map_pmf_def[abs_def] comp_def by transfer_prover
```
```   455   then show ?thesis
```
```   456     by (force simp: rel_fun_def cr_pmf_def bind_return_distr)
```
```   457 qed
```
```   458
```
```   459 lemma map_pmf_rep_eq:
```
```   460   "measure_pmf (map_pmf f M) = distr (measure_pmf M) (count_space UNIV) f"
```
```   461   unfolding map_pmf_def bind_pmf.rep_eq comp_def return_pmf.rep_eq
```
```   462   using bind_return_distr[of M f "count_space UNIV"] by (simp add: comp_def)
```
```   463
```
```   464 lemma map_pmf_id[simp]: "map_pmf id = id"
```
```   465   by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI)
```
```   466
```
```   467 lemma map_pmf_ident[simp]: "map_pmf (\<lambda>x. x) = (\<lambda>x. x)"
```
```   468   using map_pmf_id unfolding id_def .
```
```   469
```
```   470 lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g"
```
```   471   by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def)
```
```   472
```
```   473 lemma map_pmf_comp: "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M"
```
```   474   using map_pmf_compose[of f g] by (simp add: comp_def)
```
```   475
```
```   476 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"
```
```   477   unfolding map_pmf_def by (rule bind_pmf_cong) auto
```
```   478
```
```   479 lemma pmf_set_map: "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
```
```   480   by (auto simp add: comp_def fun_eq_iff map_pmf_def)
```
```   481
```
```   482 lemma set_map_pmf[simp]: "set_pmf (map_pmf f M) = f`set_pmf M"
```
```   483   using pmf_set_map[of f] by (auto simp: comp_def fun_eq_iff)
```
```   484
```
```   485 lemma emeasure_map_pmf[simp]: "emeasure (map_pmf f M) X = emeasure M (f -` X)"
```
```   486   unfolding map_pmf_rep_eq by (subst emeasure_distr) auto
```
```   487
```
```   488 lemma measure_map_pmf[simp]: "measure (map_pmf f M) X = measure M (f -` X)"
```
```   489 using emeasure_map_pmf[of f M X] by(simp add: measure_pmf.emeasure_eq_measure measure_nonneg)
```
```   490
```
```   491 lemma nn_integral_map_pmf[simp]: "(\<integral>\<^sup>+x. f x \<partial>map_pmf g M) = (\<integral>\<^sup>+x. f (g x) \<partial>M)"
```
```   492   unfolding map_pmf_rep_eq by (intro nn_integral_distr) auto
```
```   493
```
```   494 lemma ennreal_pmf_map: "pmf (map_pmf f p) x = (\<integral>\<^sup>+ y. indicator (f -` {x}) y \<partial>measure_pmf p)"
```
```   495 proof (transfer fixing: f x)
```
```   496   fix p :: "'b measure"
```
```   497   presume "prob_space p"
```
```   498   then interpret prob_space p .
```
```   499   presume "sets p = UNIV"
```
```   500   then show "ennreal (measure (distr p (count_space UNIV) f) {x}) = integral\<^sup>N p (indicator (f -` {x}))"
```
```   501     by(simp add: measure_distr measurable_def emeasure_eq_measure)
```
```   502 qed simp_all
```
```   503
```
```   504 lemma pmf_map: "pmf (map_pmf f p) x = measure p (f -` {x})"
```
```   505 proof (transfer fixing: f x)
```
```   506   fix p :: "'b measure"
```
```   507   presume "prob_space p"
```
```   508   then interpret prob_space p .
```
```   509   presume "sets p = UNIV"
```
```   510   then show "measure (distr p (count_space UNIV) f) {x} = measure p (f -` {x})"
```
```   511     by(simp add: measure_distr measurable_def emeasure_eq_measure)
```
```   512 qed simp_all
```
```   513
```
```   514 lemma nn_integral_pmf: "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = emeasure (measure_pmf p) A"
```
```   515 proof -
```
```   516   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))"
```
```   517     by(auto simp add: nn_integral_count_space_indicator indicator_def set_pmf_iff intro: nn_integral_cong)
```
```   518   also have "\<dots> = emeasure (measure_pmf p) (\<Union>x\<in>A \<inter> set_pmf p. {x})"
```
```   519     by(subst emeasure_UN_countable)(auto simp add: emeasure_pmf_single disjoint_family_on_def)
```
```   520   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})"
```
```   521     by(rule emeasure_Un_null_set[symmetric])(auto intro: in_null_sets_measure_pmfI)
```
```   522   also have "\<dots> = emeasure (measure_pmf p) A"
```
```   523     by(auto intro: arg_cong2[where f=emeasure])
```
```   524   finally show ?thesis .
```
```   525 qed
```
```   526
```
```   527 lemma integral_map_pmf[simp]:
```
```   528   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
```
```   529   shows "integral\<^sup>L (map_pmf g p) f = integral\<^sup>L p (\<lambda>x. f (g x))"
```
```   530   by (simp add: integral_distr map_pmf_rep_eq)
```
```   531
```
```   532 lemma map_return_pmf [simp]: "map_pmf f (return_pmf x) = return_pmf (f x)"
```
```   533   by transfer (simp add: distr_return)
```
```   534
```
```   535 lemma map_pmf_const[simp]: "map_pmf (\<lambda>_. c) M = return_pmf c"
```
```   536   by transfer (auto simp: prob_space.distr_const)
```
```   537
```
```   538 lemma pmf_return [simp]: "pmf (return_pmf x) y = indicator {y} x"
```
```   539   by transfer (simp add: measure_return)
```
```   540
```
```   541 lemma nn_integral_return_pmf[simp]: "0 \<le> f x \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>return_pmf x) = f x"
```
```   542   unfolding return_pmf.rep_eq by (intro nn_integral_return) auto
```
```   543
```
```   544 lemma emeasure_return_pmf[simp]: "emeasure (return_pmf x) X = indicator X x"
```
```   545   unfolding return_pmf.rep_eq by (intro emeasure_return) auto
```
```   546
```
```   547 lemma measure_return_pmf [simp]: "measure_pmf.prob (return_pmf x) A = indicator A x"
```
```   548 proof -
```
```   549   have "ennreal (measure_pmf.prob (return_pmf x) A) =
```
```   550           emeasure (measure_pmf (return_pmf x)) A"
```
```   551     by (simp add: measure_pmf.emeasure_eq_measure)
```
```   552   also have "\<dots> = ennreal (indicator A x)" by (simp add: ennreal_indicator)
```
```   553   finally show ?thesis by simp
```
```   554 qed
```
```   555
```
```   556 lemma return_pmf_inj[simp]: "return_pmf x = return_pmf y \<longleftrightarrow> x = y"
```
```   557   by (metis insertI1 set_return_pmf singletonD)
```
```   558
```
```   559 lemma map_pmf_eq_return_pmf_iff:
```
```   560   "map_pmf f p = return_pmf x \<longleftrightarrow> (\<forall>y \<in> set_pmf p. f y = x)"
```
```   561 proof
```
```   562   assume "map_pmf f p = return_pmf x"
```
```   563   then have "set_pmf (map_pmf f p) = set_pmf (return_pmf x)" by simp
```
```   564   then show "\<forall>y \<in> set_pmf p. f y = x" by auto
```
```   565 next
```
```   566   assume "\<forall>y \<in> set_pmf p. f y = x"
```
```   567   then show "map_pmf f p = return_pmf x"
```
```   568     unfolding map_pmf_const[symmetric, of _ p] by (intro map_pmf_cong) auto
```
```   569 qed
```
```   570
```
```   571 definition "pair_pmf A B = bind_pmf A (\<lambda>x. bind_pmf B (\<lambda>y. return_pmf (x, y)))"
```
```   572
```
```   573 lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b"
```
```   574   unfolding pair_pmf_def pmf_bind pmf_return
```
```   575   apply (subst integral_measure_pmf_real[where A="{b}"])
```
```   576   apply (auto simp: indicator_eq_0_iff)
```
```   577   apply (subst integral_measure_pmf_real[where A="{a}"])
```
```   578   apply (auto simp: indicator_eq_0_iff sum_nonneg_eq_0_iff pmf_nonneg)
```
```   579   done
```
```   580
```
```   581 lemma set_pair_pmf[simp]: "set_pmf (pair_pmf A B) = set_pmf A \<times> set_pmf B"
```
```   582   unfolding pair_pmf_def set_bind_pmf set_return_pmf by auto
```
```   583
```
```   584 lemma measure_pmf_in_subprob_space[measurable (raw)]:
```
```   585   "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
```
```   586   by (simp add: space_subprob_algebra) intro_locales
```
```   587
```
```   588 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)"
```
```   589 proof -
```
```   590   have "(\<integral>\<^sup>+x. f x \<partial>pair_pmf A B) = (\<integral>\<^sup>+x. f x * indicator (A \<times> B) x \<partial>pair_pmf A B)"
```
```   591     by (auto simp: AE_measure_pmf_iff intro!: nn_integral_cong_AE)
```
```   592   also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. f (a, b) * indicator (A \<times> B) (a, b) \<partial>B \<partial>A)"
```
```   593     by (simp add: pair_pmf_def)
```
```   594   also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. f (a, b) \<partial>B \<partial>A)"
```
```   595     by (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
```
```   596   finally show ?thesis .
```
```   597 qed
```
```   598
```
```   599 lemma bind_pair_pmf:
```
```   600   assumes M[measurable]: "M \<in> measurable (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) (subprob_algebra N)"
```
```   601   shows "measure_pmf (pair_pmf A B) \<bind> M = (measure_pmf A \<bind> (\<lambda>x. measure_pmf B \<bind> (\<lambda>y. M (x, y))))"
```
```   602     (is "?L = ?R")
```
```   603 proof (rule measure_eqI)
```
```   604   have M'[measurable]: "M \<in> measurable (pair_pmf A B) (subprob_algebra N)"
```
```   605     using M[THEN measurable_space] by (simp_all add: space_pair_measure)
```
```   606
```
```   607   note measurable_bind[where N="count_space UNIV", measurable]
```
```   608   note measure_pmf_in_subprob_space[simp]
```
```   609
```
```   610   have sets_eq_N: "sets ?L = N"
```
```   611     by (subst sets_bind[OF sets_kernel[OF M']]) auto
```
```   612   show "sets ?L = sets ?R"
```
```   613     using measurable_space[OF M]
```
```   614     by (simp add: sets_eq_N space_pair_measure space_subprob_algebra)
```
```   615   fix X assume "X \<in> sets ?L"
```
```   616   then have X[measurable]: "X \<in> sets N"
```
```   617     unfolding sets_eq_N .
```
```   618   then show "emeasure ?L X = emeasure ?R X"
```
```   619     apply (simp add: emeasure_bind[OF _ M' X])
```
```   620     apply (simp add: nn_integral_bind[where B="count_space UNIV"] pair_pmf_def measure_pmf_bind[of A]
```
```   621                      nn_integral_measure_pmf_finite)
```
```   622     apply (subst emeasure_bind[OF _ _ X])
```
```   623     apply measurable
```
```   624     apply (subst emeasure_bind[OF _ _ X])
```
```   625     apply measurable
```
```   626     done
```
```   627 qed
```
```   628
```
```   629 lemma map_fst_pair_pmf: "map_pmf fst (pair_pmf A B) = A"
```
```   630   by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
```
```   631
```
```   632 lemma map_snd_pair_pmf: "map_pmf snd (pair_pmf A B) = B"
```
```   633   by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
```
```   634
```
```   635 lemma nn_integral_pmf':
```
```   636   "inj_on f A \<Longrightarrow> (\<integral>\<^sup>+x. pmf p (f x) \<partial>count_space A) = emeasure p (f ` A)"
```
```   637   by (subst nn_integral_bij_count_space[where g=f and B="f`A"])
```
```   638      (auto simp: bij_betw_def nn_integral_pmf)
```
```   639
```
```   640 lemma pmf_le_0_iff[simp]: "pmf M p \<le> 0 \<longleftrightarrow> pmf M p = 0"
```
```   641   using pmf_nonneg[of M p] by arith
```
```   642
```
```   643 lemma min_pmf_0[simp]: "min (pmf M p) 0 = 0" "min 0 (pmf M p) = 0"
```
```   644   using pmf_nonneg[of M p] by arith+
```
```   645
```
```   646 lemma pmf_eq_0_set_pmf: "pmf M p = 0 \<longleftrightarrow> p \<notin> set_pmf M"
```
```   647   unfolding set_pmf_iff by simp
```
```   648
```
```   649 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"
```
```   650   by (auto simp: pmf.rep_eq map_pmf_rep_eq measure_distr AE_measure_pmf_iff inj_onD
```
```   651            intro!: measure_pmf.finite_measure_eq_AE)
```
```   652
```
```   653 lemma pmf_map_inj': "inj f \<Longrightarrow> pmf (map_pmf f M) (f x) = pmf M x"
```
```   654 apply(cases "x \<in> set_pmf M")
```
```   655  apply(simp add: pmf_map_inj[OF subset_inj_on])
```
```   656 apply(simp add: pmf_eq_0_set_pmf[symmetric])
```
```   657 apply(auto simp add: pmf_eq_0_set_pmf dest: injD)
```
```   658 done
```
```   659
```
```   660 lemma pmf_map_outside: "x \<notin> f ` set_pmf M \<Longrightarrow> pmf (map_pmf f M) x = 0"
```
```   661   unfolding pmf_eq_0_set_pmf by simp
```
```   662
```
```   663 lemma measurable_set_pmf[measurable]: "Measurable.pred (count_space UNIV) (\<lambda>x. x \<in> set_pmf M)"
```
```   664   by simp
```
```   665
```
```   666
```
```   667 subsection \<open> PMFs as function \<close>
```
```   668
```
```   669 context
```
```   670   fixes f :: "'a \<Rightarrow> real"
```
```   671   assumes nonneg: "\<And>x. 0 \<le> f x"
```
```   672   assumes prob: "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
```
```   673 begin
```
```   674
```
```   675 lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ennreal \<circ> f)"
```
```   676 proof (intro conjI)
```
```   677   have *[simp]: "\<And>x y. ennreal (f y) * indicator {x} y = ennreal (f x) * indicator {x} y"
```
```   678     by (simp split: split_indicator)
```
```   679   show "AE x in density (count_space UNIV) (ennreal \<circ> f).
```
```   680     measure (density (count_space UNIV) (ennreal \<circ> f)) {x} \<noteq> 0"
```
```   681     by (simp add: AE_density nonneg measure_def emeasure_density max_def)
```
```   682   show "prob_space (density (count_space UNIV) (ennreal \<circ> f))"
```
```   683     by standard (simp add: emeasure_density prob)
```
```   684 qed simp
```
```   685
```
```   686 lemma pmf_embed_pmf: "pmf embed_pmf x = f x"
```
```   687 proof transfer
```
```   688   have *[simp]: "\<And>x y. ennreal (f y) * indicator {x} y = ennreal (f x) * indicator {x} y"
```
```   689     by (simp split: split_indicator)
```
```   690   fix x show "measure (density (count_space UNIV) (ennreal \<circ> f)) {x} = f x"
```
```   691     by transfer (simp add: measure_def emeasure_density nonneg max_def)
```
```   692 qed
```
```   693
```
```   694 lemma set_embed_pmf: "set_pmf embed_pmf = {x. f x \<noteq> 0}"
```
```   695 by(auto simp add: set_pmf_eq pmf_embed_pmf)
```
```   696
```
```   697 end
```
```   698
```
```   699 lemma embed_pmf_transfer:
```
```   700   "rel_fun (eq_onp (\<lambda>f. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ennreal (f x) \<partial>count_space UNIV) = 1)) pmf_as_measure.cr_pmf (\<lambda>f. density (count_space UNIV) (ennreal \<circ> f)) embed_pmf"
```
```   701   by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer)
```
```   702
```
```   703 lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)"
```
```   704 proof (transfer, elim conjE)
```
```   705   fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
```
```   706   assume "prob_space M" then interpret prob_space M .
```
```   707   show "M = density (count_space UNIV) (\<lambda>x. ennreal (measure M {x}))"
```
```   708   proof (rule measure_eqI)
```
```   709     fix A :: "'a set"
```
```   710     have "(\<integral>\<^sup>+ x. ennreal (measure M {x}) * indicator A x \<partial>count_space UNIV) =
```
```   711       (\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)"
```
```   712       by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator)
```
```   713     also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))"
```
```   714       by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space)
```
```   715     also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})"
```
```   716       by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support)
```
```   717          (auto simp: disjoint_family_on_def)
```
```   718     also have "\<dots> = emeasure M A"
```
```   719       using ae by (intro emeasure_eq_AE) auto
```
```   720     finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ennreal (measure M {x}))) A"
```
```   721       using emeasure_space_1 by (simp add: emeasure_density)
```
```   722   qed simp
```
```   723 qed
```
```   724
```
```   725 lemma td_pmf_embed_pmf:
```
```   726   "type_definition pmf embed_pmf {f::'a \<Rightarrow> real. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ennreal (f x) \<partial>count_space UNIV) = 1}"
```
```   727   unfolding type_definition_def
```
```   728 proof safe
```
```   729   fix p :: "'a pmf"
```
```   730   have "(\<integral>\<^sup>+ x. 1 \<partial>measure_pmf p) = 1"
```
```   731     using measure_pmf.emeasure_space_1[of p] by simp
```
```   732   then show *: "(\<integral>\<^sup>+ x. ennreal (pmf p x) \<partial>count_space UNIV) = 1"
```
```   733     by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const)
```
```   734
```
```   735   show "embed_pmf (pmf p) = p"
```
```   736     by (intro measure_pmf_inject[THEN iffD1])
```
```   737        (simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def)
```
```   738 next
```
```   739   fix f :: "'a \<Rightarrow> real" assume "\<forall>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
```
```   740   then show "pmf (embed_pmf f) = f"
```
```   741     by (auto intro!: pmf_embed_pmf)
```
```   742 qed (rule pmf_nonneg)
```
```   743
```
```   744 end
```
```   745
```
```   746 lemma nn_integral_measure_pmf: "(\<integral>\<^sup>+ x. f x \<partial>measure_pmf p) = \<integral>\<^sup>+ x. ennreal (pmf p x) * f x \<partial>count_space UNIV"
```
```   747 by(simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg)
```
```   748
```
```   749 lemma integral_measure_pmf:
```
```   750   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
```
```   751   assumes A: "finite A"
```
```   752   shows "(\<And>a. a \<in> set_pmf M \<Longrightarrow> f a \<noteq> 0 \<Longrightarrow> a \<in> A) \<Longrightarrow> (LINT x|M. f x) = (\<Sum>a\<in>A. pmf M a *\<^sub>R f a)"
```
```   753   unfolding measure_pmf_eq_density
```
```   754   apply (simp add: integral_density)
```
```   755   apply (subst lebesgue_integral_count_space_finite_support)
```
```   756   apply (auto intro!: finite_subset[OF _ \<open>finite A\<close>] sum.mono_neutral_left simp: pmf_eq_0_set_pmf)
```
```   757   done
```
```   758
```
```   759 lemma expectation_return_pmf [simp]:
```
```   760   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
```
```   761   shows "measure_pmf.expectation (return_pmf x) f = f x"
```
```   762   by (subst integral_measure_pmf[of "{x}"]) simp_all
```
```   763
```
```   764 lemma pmf_expectation_bind:
```
```   765   fixes p :: "'a pmf" and f :: "'a \<Rightarrow> 'b pmf"
```
```   766     and  h :: "'b \<Rightarrow> 'c::{banach, second_countable_topology}"
```
```   767   assumes "finite A" "\<And>x. x \<in> A \<Longrightarrow> finite (set_pmf (f x))" "set_pmf p \<subseteq> A"
```
```   768   shows "measure_pmf.expectation (p \<bind> f) h =
```
```   769            (\<Sum>a\<in>A. pmf p a *\<^sub>R measure_pmf.expectation (f a) h)"
```
```   770 proof -
```
```   771   have "measure_pmf.expectation (p \<bind> f) h = (\<Sum>a\<in>(\<Union>x\<in>A. set_pmf (f x)). pmf (p \<bind> f) a *\<^sub>R h a)"
```
```   772     using assms by (intro integral_measure_pmf) auto
```
```   773   also have "\<dots> = (\<Sum>x\<in>(\<Union>x\<in>A. set_pmf (f x)). (\<Sum>a\<in>A. (pmf p a * pmf (f a) x) *\<^sub>R h x))"
```
```   774   proof (intro sum.cong refl, goal_cases)
```
```   775     case (1 x)
```
```   776     thus ?case
```
```   777       by (subst pmf_bind, subst integral_measure_pmf[of A])
```
```   778          (insert assms, auto simp: scaleR_sum_left)
```
```   779   qed
```
```   780   also have "\<dots> = (\<Sum>j\<in>A. pmf p j *\<^sub>R (\<Sum>i\<in>(\<Union>x\<in>A. set_pmf (f x)). pmf (f j) i *\<^sub>R h i))"
```
```   781     by (subst sum.commute) (simp add: scaleR_sum_right)
```
```   782   also have "\<dots> = (\<Sum>j\<in>A. pmf p j *\<^sub>R measure_pmf.expectation (f j) h)"
```
```   783   proof (intro sum.cong refl, goal_cases)
```
```   784     case (1 x)
```
```   785     thus ?case
```
```   786       by (subst integral_measure_pmf[of "(\<Union>x\<in>A. set_pmf (f x))"])
```
```   787          (insert assms, auto simp: scaleR_sum_left)
```
```   788   qed
```
```   789   finally show ?thesis .
```
```   790 qed
```
```   791
```
```   792 lemma continuous_on_LINT_pmf: -- \<open>This is dominated convergence!?\<close>
```
```   793   fixes f :: "'i \<Rightarrow> 'a::topological_space \<Rightarrow> 'b::{banach, second_countable_topology}"
```
```   794   assumes f: "\<And>i. i \<in> set_pmf M \<Longrightarrow> continuous_on A (f i)"
```
```   795     and bnd: "\<And>a i. a \<in> A \<Longrightarrow> i \<in> set_pmf M \<Longrightarrow> norm (f i a) \<le> B"
```
```   796   shows "continuous_on A (\<lambda>a. LINT i|M. f i a)"
```
```   797 proof cases
```
```   798   assume "finite M" with f show ?thesis
```
```   799     using integral_measure_pmf[OF \<open>finite M\<close>]
```
```   800     by (subst integral_measure_pmf[OF \<open>finite M\<close>])
```
```   801        (auto intro!: continuous_on_sum continuous_on_scaleR continuous_on_const)
```
```   802 next
```
```   803   assume "infinite M"
```
```   804   let ?f = "\<lambda>i x. pmf (map_pmf (to_nat_on M) M) i *\<^sub>R f (from_nat_into M i) x"
```
```   805
```
```   806   show ?thesis
```
```   807   proof (rule uniform_limit_theorem)
```
```   808     show "\<forall>\<^sub>F n in sequentially. continuous_on A (\<lambda>a. \<Sum>i<n. ?f i a)"
```
```   809       by (intro always_eventually allI continuous_on_sum continuous_on_scaleR continuous_on_const f
```
```   810                 from_nat_into set_pmf_not_empty)
```
```   811     show "uniform_limit A (\<lambda>n a. \<Sum>i<n. ?f i a) (\<lambda>a. LINT i|M. f i a) sequentially"
```
```   812     proof (subst uniform_limit_cong[where g="\<lambda>n a. \<Sum>i<n. ?f i a"])
```
```   813       fix a assume "a \<in> A"
```
```   814       have 1: "(LINT i|M. f i a) = (LINT i|map_pmf (to_nat_on M) M. f (from_nat_into M i) a)"
```
```   815         by (auto intro!: integral_cong_AE AE_pmfI)
```
```   816       have 2: "\<dots> = (LINT i|count_space UNIV. pmf (map_pmf (to_nat_on M) M) i *\<^sub>R f (from_nat_into M i) a)"
```
```   817         by (simp add: measure_pmf_eq_density integral_density)
```
```   818       have "(\<lambda>n. ?f n a) sums (LINT i|M. f i a)"
```
```   819         unfolding 1 2
```
```   820       proof (intro sums_integral_count_space_nat)
```
```   821         have A: "integrable M (\<lambda>i. f i a)"
```
```   822           using \<open>a\<in>A\<close> by (auto intro!: measure_pmf.integrable_const_bound AE_pmfI bnd)
```
```   823         have "integrable (map_pmf (to_nat_on M) M) (\<lambda>i. f (from_nat_into M i) a)"
```
```   824           by (auto simp add: map_pmf_rep_eq integrable_distr_eq intro!: AE_pmfI integrable_cong_AE_imp[OF A])
```
```   825         then show "integrable (count_space UNIV) (\<lambda>n. ?f n a)"
```
```   826           by (simp add: measure_pmf_eq_density integrable_density)
```
```   827       qed
```
```   828       then show "(LINT i|M. f i a) = (\<Sum> n. ?f n a)"
```
```   829         by (simp add: sums_unique)
```
```   830     next
```
```   831       show "uniform_limit A (\<lambda>n a. \<Sum>i<n. ?f i a) (\<lambda>a. (\<Sum> n. ?f n a)) sequentially"
```
```   832       proof (rule weierstrass_m_test)
```
```   833         fix n a assume "a\<in>A"
```
```   834         then show "norm (?f n a) \<le> pmf (map_pmf (to_nat_on M) M) n * B"
```
```   835           using bnd by (auto intro!: mult_mono simp: from_nat_into set_pmf_not_empty)
```
```   836       next
```
```   837         have "integrable (map_pmf (to_nat_on M) M) (\<lambda>n. B)"
```
```   838           by auto
```
```   839         then show "summable (\<lambda>n. pmf (map_pmf (to_nat_on (set_pmf M)) M) n * B)"
```
```   840           by (simp add: measure_pmf_eq_density integrable_density integrable_count_space_nat_iff summable_rabs_cancel)
```
```   841       qed
```
```   842     qed simp
```
```   843   qed simp
```
```   844 qed
```
```   845
```
```   846 lemma continuous_on_LBINT:
```
```   847   fixes f :: "real \<Rightarrow> real"
```
```   848   assumes f: "\<And>b. a \<le> b \<Longrightarrow> set_integrable lborel {a..b} f"
```
```   849   shows "continuous_on UNIV (\<lambda>b. LBINT x:{a..b}. f x)"
```
```   850 proof (subst set_borel_integral_eq_integral)
```
```   851   { fix b :: real assume "a \<le> b"
```
```   852     from f[OF this] have "continuous_on {a..b} (\<lambda>b. integral {a..b} f)"
```
```   853       by (intro indefinite_integral_continuous set_borel_integral_eq_integral) }
```
```   854   note * = this
```
```   855
```
```   856   have "continuous_on (\<Union>b\<in>{a..}. {a <..< b}) (\<lambda>b. integral {a..b} f)"
```
```   857   proof (intro continuous_on_open_UN)
```
```   858     show "b \<in> {a..} \<Longrightarrow> continuous_on {a<..<b} (\<lambda>b. integral {a..b} f)" for b
```
```   859       using *[of b] by (rule continuous_on_subset) auto
```
```   860   qed simp
```
```   861   also have "(\<Union>b\<in>{a..}. {a <..< b}) = {a <..}"
```
```   862     by (auto simp: lt_ex gt_ex less_imp_le) (simp add: Bex_def less_imp_le gt_ex cong: rev_conj_cong)
```
```   863   finally have "continuous_on {a+1 ..} (\<lambda>b. integral {a..b} f)"
```
```   864     by (rule continuous_on_subset) auto
```
```   865   moreover have "continuous_on {a..a+1} (\<lambda>b. integral {a..b} f)"
```
```   866     by (rule *) simp
```
```   867   moreover
```
```   868   have "x \<le> a \<Longrightarrow> {a..x} = (if a = x then {a} else {})" for x
```
```   869     by auto
```
```   870   then have "continuous_on {..a} (\<lambda>b. integral {a..b} f)"
```
```   871     by (subst continuous_on_cong[OF refl, where g="\<lambda>x. 0"]) (auto intro!: continuous_on_const)
```
```   872   ultimately have "continuous_on ({..a} \<union> {a..a+1} \<union> {a+1 ..}) (\<lambda>b. integral {a..b} f)"
```
```   873     by (intro continuous_on_closed_Un) auto
```
```   874   also have "{..a} \<union> {a..a+1} \<union> {a+1 ..} = UNIV"
```
```   875     by auto
```
```   876   finally show "continuous_on UNIV (\<lambda>b. integral {a..b} f)"
```
```   877     by auto
```
```   878 next
```
```   879   show "set_integrable lborel {a..b} f" for b
```
```   880     using f by (cases "a \<le> b") auto
```
```   881 qed
```
```   882
```
```   883 locale pmf_as_function
```
```   884 begin
```
```   885
```
```   886 setup_lifting td_pmf_embed_pmf
```
```   887
```
```   888 lemma set_pmf_transfer[transfer_rule]:
```
```   889   assumes "bi_total A"
```
```   890   shows "rel_fun (pcr_pmf A) (rel_set A) (\<lambda>f. {x. f x \<noteq> 0}) set_pmf"
```
```   891   using \<open>bi_total A\<close>
```
```   892   by (auto simp: pcr_pmf_def cr_pmf_def rel_fun_def rel_set_def bi_total_def Bex_def set_pmf_iff)
```
```   893      metis+
```
```   894
```
```   895 end
```
```   896
```
```   897 context
```
```   898 begin
```
```   899
```
```   900 interpretation pmf_as_function .
```
```   901
```
```   902 lemma pmf_eqI: "(\<And>i. pmf M i = pmf N i) \<Longrightarrow> M = N"
```
```   903   by transfer auto
```
```   904
```
```   905 lemma pmf_eq_iff: "M = N \<longleftrightarrow> (\<forall>i. pmf M i = pmf N i)"
```
```   906   by (auto intro: pmf_eqI)
```
```   907
```
```   908 lemma pmf_neq_exists_less:
```
```   909   assumes "M \<noteq> N"
```
```   910   shows   "\<exists>x. pmf M x < pmf N x"
```
```   911 proof (rule ccontr)
```
```   912   assume "\<not>(\<exists>x. pmf M x < pmf N x)"
```
```   913   hence ge: "pmf M x \<ge> pmf N x" for x by (auto simp: not_less)
```
```   914   from assms obtain x where "pmf M x \<noteq> pmf N x" by (auto simp: pmf_eq_iff)
```
```   915   with ge[of x] have gt: "pmf M x > pmf N x" by simp
```
```   916   have "1 = measure (measure_pmf M) UNIV" by simp
```
```   917   also have "\<dots> = measure (measure_pmf N) {x} + measure (measure_pmf N) (UNIV - {x})"
```
```   918     by (subst measure_pmf.finite_measure_Union [symmetric]) simp_all
```
```   919   also from gt have "measure (measure_pmf N) {x} < measure (measure_pmf M) {x}"
```
```   920     by (simp add: measure_pmf_single)
```
```   921   also have "measure (measure_pmf N) (UNIV - {x}) \<le> measure (measure_pmf M) (UNIV - {x})"
```
```   922     by (subst (1 2) integral_pmf [symmetric])
```
```   923        (intro integral_mono integrable_pmf, simp_all add: ge)
```
```   924   also have "measure (measure_pmf M) {x} + \<dots> = 1"
```
```   925     by (subst measure_pmf.finite_measure_Union [symmetric]) simp_all
```
```   926   finally show False by simp_all
```
```   927 qed
```
```   928
```
```   929 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))"
```
```   930   unfolding pmf_eq_iff pmf_bind
```
```   931 proof
```
```   932   fix i
```
```   933   interpret B: prob_space "restrict_space B B"
```
```   934     by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
```
```   935        (auto simp: AE_measure_pmf_iff)
```
```   936   interpret A: prob_space "restrict_space A A"
```
```   937     by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
```
```   938        (auto simp: AE_measure_pmf_iff)
```
```   939
```
```   940   interpret AB: pair_prob_space "restrict_space A A" "restrict_space B B"
```
```   941     by unfold_locales
```
```   942
```
```   943   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)"
```
```   944     by (rule Bochner_Integration.integral_cong) (auto intro!: integral_pmf_restrict)
```
```   945   also have "\<dots> = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>restrict_space A A)"
```
```   946     by (intro integral_pmf_restrict B.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
```
```   947               countable_set_pmf borel_measurable_count_space)
```
```   948   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>restrict_space B B)"
```
```   949     by (rule AB.Fubini_integral[symmetric])
```
```   950        (auto intro!: AB.integrable_const_bound[where B=1] measurable_pair_restrict_pmf2
```
```   951              simp: pmf_nonneg pmf_le_1 measurable_restrict_space1)
```
```   952   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>B)"
```
```   953     by (intro integral_pmf_restrict[symmetric] A.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
```
```   954               countable_set_pmf borel_measurable_count_space)
```
```   955   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>A \<partial>B)"
```
```   956     by (rule Bochner_Integration.integral_cong) (auto intro!: integral_pmf_restrict[symmetric])
```
```   957   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)" .
```
```   958 qed
```
```   959
```
```   960 lemma pair_map_pmf1: "pair_pmf (map_pmf f A) B = map_pmf (apfst f) (pair_pmf A B)"
```
```   961 proof (safe intro!: pmf_eqI)
```
```   962   fix a :: "'a" and b :: "'b"
```
```   963   have [simp]: "\<And>c d. indicator (apfst f -` {(a, b)}) (c, d) = indicator (f -` {a}) c * (indicator {b} d::ennreal)"
```
```   964     by (auto split: split_indicator)
```
```   965
```
```   966   have "ennreal (pmf (pair_pmf (map_pmf f A) B) (a, b)) =
```
```   967          ennreal (pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b))"
```
```   968     unfolding pmf_pair ennreal_pmf_map
```
```   969     by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_multc pmf_nonneg
```
```   970                   emeasure_map_pmf[symmetric] ennreal_mult del: emeasure_map_pmf)
```
```   971   then show "pmf (pair_pmf (map_pmf f A) B) (a, b) = pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b)"
```
```   972     by (simp add: pmf_nonneg)
```
```   973 qed
```
```   974
```
```   975 lemma pair_map_pmf2: "pair_pmf A (map_pmf f B) = map_pmf (apsnd f) (pair_pmf A B)"
```
```   976 proof (safe intro!: pmf_eqI)
```
```   977   fix a :: "'a" and b :: "'b"
```
```   978   have [simp]: "\<And>c d. indicator (apsnd f -` {(a, b)}) (c, d) = indicator {a} c * (indicator (f -` {b}) d::ennreal)"
```
```   979     by (auto split: split_indicator)
```
```   980
```
```   981   have "ennreal (pmf (pair_pmf A (map_pmf f B)) (a, b)) =
```
```   982          ennreal (pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b))"
```
```   983     unfolding pmf_pair ennreal_pmf_map
```
```   984     by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_cmult nn_integral_multc pmf_nonneg
```
```   985                   emeasure_map_pmf[symmetric] ennreal_mult del: emeasure_map_pmf)
```
```   986   then show "pmf (pair_pmf A (map_pmf f B)) (a, b) = pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b)"
```
```   987     by (simp add: pmf_nonneg)
```
```   988 qed
```
```   989
```
```   990 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)"
```
```   991   by (simp add: pair_map_pmf2 pair_map_pmf1 map_pmf_comp split_beta')
```
```   992
```
```   993 end
```
```   994
```
```   995 lemma pair_return_pmf1: "pair_pmf (return_pmf x) y = map_pmf (Pair x) y"
```
```   996 by(simp add: pair_pmf_def bind_return_pmf map_pmf_def)
```
```   997
```
```   998 lemma pair_return_pmf2: "pair_pmf x (return_pmf y) = map_pmf (\<lambda>x. (x, y)) x"
```
```   999 by(simp add: pair_pmf_def bind_return_pmf map_pmf_def)
```
```  1000
```
```  1001 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))"
```
```  1002 by(simp add: pair_pmf_def bind_return_pmf map_pmf_def bind_assoc_pmf)
```
```  1003
```
```  1004 lemma pair_commute_pmf: "pair_pmf x y = map_pmf (\<lambda>(x, y). (y, x)) (pair_pmf y x)"
```
```  1005 unfolding pair_pmf_def by(subst bind_commute_pmf)(simp add: map_pmf_def bind_assoc_pmf bind_return_pmf)
```
```  1006
```
```  1007 lemma set_pmf_subset_singleton: "set_pmf p \<subseteq> {x} \<longleftrightarrow> p = return_pmf x"
```
```  1008 proof(intro iffI pmf_eqI)
```
```  1009   fix i
```
```  1010   assume x: "set_pmf p \<subseteq> {x}"
```
```  1011   hence *: "set_pmf p = {x}" using set_pmf_not_empty[of p] by auto
```
```  1012   have "ennreal (pmf p x) = \<integral>\<^sup>+ i. indicator {x} i \<partial>p" by(simp add: emeasure_pmf_single)
```
```  1013   also have "\<dots> = \<integral>\<^sup>+ i. 1 \<partial>p" by(rule nn_integral_cong_AE)(simp add: AE_measure_pmf_iff * )
```
```  1014   also have "\<dots> = 1" by simp
```
```  1015   finally show "pmf p i = pmf (return_pmf x) i" using x
```
```  1016     by(auto split: split_indicator simp add: pmf_eq_0_set_pmf)
```
```  1017 qed auto
```
```  1018
```
```  1019 lemma bind_eq_return_pmf:
```
```  1020   "bind_pmf p f = return_pmf x \<longleftrightarrow> (\<forall>y\<in>set_pmf p. f y = return_pmf x)"
```
```  1021   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1022 proof(intro iffI strip)
```
```  1023   fix y
```
```  1024   assume y: "y \<in> set_pmf p"
```
```  1025   assume "?lhs"
```
```  1026   hence "set_pmf (bind_pmf p f) = {x}" by simp
```
```  1027   hence "(\<Union>y\<in>set_pmf p. set_pmf (f y)) = {x}" by simp
```
```  1028   hence "set_pmf (f y) \<subseteq> {x}" using y by auto
```
```  1029   thus "f y = return_pmf x" by(simp add: set_pmf_subset_singleton)
```
```  1030 next
```
```  1031   assume *: ?rhs
```
```  1032   show ?lhs
```
```  1033   proof(rule pmf_eqI)
```
```  1034     fix i
```
```  1035     have "ennreal (pmf (bind_pmf p f) i) = \<integral>\<^sup>+ y. ennreal (pmf (f y) i) \<partial>p"
```
```  1036       by (simp add: ennreal_pmf_bind)
```
```  1037     also have "\<dots> = \<integral>\<^sup>+ y. ennreal (pmf (return_pmf x) i) \<partial>p"
```
```  1038       by(rule nn_integral_cong_AE)(simp add: AE_measure_pmf_iff * )
```
```  1039     also have "\<dots> = ennreal (pmf (return_pmf x) i)"
```
```  1040       by simp
```
```  1041     finally show "pmf (bind_pmf p f) i = pmf (return_pmf x) i"
```
```  1042       by (simp add: pmf_nonneg)
```
```  1043   qed
```
```  1044 qed
```
```  1045
```
```  1046 lemma pmf_False_conv_True: "pmf p False = 1 - pmf p True"
```
```  1047 proof -
```
```  1048   have "pmf p False + pmf p True = measure p {False} + measure p {True}"
```
```  1049     by(simp add: measure_pmf_single)
```
```  1050   also have "\<dots> = measure p ({False} \<union> {True})"
```
```  1051     by(subst measure_pmf.finite_measure_Union) simp_all
```
```  1052   also have "{False} \<union> {True} = space p" by auto
```
```  1053   finally show ?thesis by simp
```
```  1054 qed
```
```  1055
```
```  1056 lemma pmf_True_conv_False: "pmf p True = 1 - pmf p False"
```
```  1057 by(simp add: pmf_False_conv_True)
```
```  1058
```
```  1059 subsection \<open> Conditional Probabilities \<close>
```
```  1060
```
```  1061 lemma measure_pmf_zero_iff: "measure (measure_pmf p) s = 0 \<longleftrightarrow> set_pmf p \<inter> s = {}"
```
```  1062   by (subst measure_pmf.prob_eq_0) (auto simp: AE_measure_pmf_iff)
```
```  1063
```
```  1064 context
```
```  1065   fixes p :: "'a pmf" and s :: "'a set"
```
```  1066   assumes not_empty: "set_pmf p \<inter> s \<noteq> {}"
```
```  1067 begin
```
```  1068
```
```  1069 interpretation pmf_as_measure .
```
```  1070
```
```  1071 lemma emeasure_measure_pmf_not_zero: "emeasure (measure_pmf p) s \<noteq> 0"
```
```  1072 proof
```
```  1073   assume "emeasure (measure_pmf p) s = 0"
```
```  1074   then have "AE x in measure_pmf p. x \<notin> s"
```
```  1075     by (rule AE_I[rotated]) auto
```
```  1076   with not_empty show False
```
```  1077     by (auto simp: AE_measure_pmf_iff)
```
```  1078 qed
```
```  1079
```
```  1080 lemma measure_measure_pmf_not_zero: "measure (measure_pmf p) s \<noteq> 0"
```
```  1081   using emeasure_measure_pmf_not_zero by (simp add: measure_pmf.emeasure_eq_measure measure_nonneg)
```
```  1082
```
```  1083 lift_definition cond_pmf :: "'a pmf" is
```
```  1084   "uniform_measure (measure_pmf p) s"
```
```  1085 proof (intro conjI)
```
```  1086   show "prob_space (uniform_measure (measure_pmf p) s)"
```
```  1087     by (intro prob_space_uniform_measure) (auto simp: emeasure_measure_pmf_not_zero)
```
```  1088   show "AE x in uniform_measure (measure_pmf p) s. measure (uniform_measure (measure_pmf p) s) {x} \<noteq> 0"
```
```  1089     by (simp add: emeasure_measure_pmf_not_zero measure_measure_pmf_not_zero AE_uniform_measure
```
```  1090                   AE_measure_pmf_iff set_pmf.rep_eq less_top[symmetric])
```
```  1091 qed simp
```
```  1092
```
```  1093 lemma pmf_cond: "pmf cond_pmf x = (if x \<in> s then pmf p x / measure p s else 0)"
```
```  1094   by transfer (simp add: emeasure_measure_pmf_not_zero pmf.rep_eq)
```
```  1095
```
```  1096 lemma set_cond_pmf[simp]: "set_pmf cond_pmf = set_pmf p \<inter> s"
```
```  1097   by (auto simp add: set_pmf_iff pmf_cond measure_measure_pmf_not_zero split: if_split_asm)
```
```  1098
```
```  1099 end
```
```  1100
```
```  1101 lemma measure_pmf_posI: "x \<in> set_pmf p \<Longrightarrow> x \<in> A \<Longrightarrow> measure_pmf.prob p A > 0"
```
```  1102   using measure_measure_pmf_not_zero[of p A] by (subst zero_less_measure_iff) blast
```
```  1103
```
```  1104 lemma cond_map_pmf:
```
```  1105   assumes "set_pmf p \<inter> f -` s \<noteq> {}"
```
```  1106   shows "cond_pmf (map_pmf f p) s = map_pmf f (cond_pmf p (f -` s))"
```
```  1107 proof -
```
```  1108   have *: "set_pmf (map_pmf f p) \<inter> s \<noteq> {}"
```
```  1109     using assms by auto
```
```  1110   { fix x
```
```  1111     have "ennreal (pmf (map_pmf f (cond_pmf p (f -` s))) x) =
```
```  1112       emeasure p (f -` s \<inter> f -` {x}) / emeasure p (f -` s)"
```
```  1113       unfolding ennreal_pmf_map cond_pmf.rep_eq[OF assms] by (simp add: nn_integral_uniform_measure)
```
```  1114     also have "f -` s \<inter> f -` {x} = (if x \<in> s then f -` {x} else {})"
```
```  1115       by auto
```
```  1116     also have "emeasure p (if x \<in> s then f -` {x} else {}) / emeasure p (f -` s) =
```
```  1117       ennreal (pmf (cond_pmf (map_pmf f p) s) x)"
```
```  1118       using measure_measure_pmf_not_zero[OF *]
```
```  1119       by (simp add: pmf_cond[OF *] ennreal_pmf_map measure_pmf.emeasure_eq_measure
```
```  1120                     divide_ennreal pmf_nonneg measure_nonneg zero_less_measure_iff pmf_map)
```
```  1121     finally have "ennreal (pmf (cond_pmf (map_pmf f p) s) x) = ennreal (pmf (map_pmf f (cond_pmf p (f -` s))) x)"
```
```  1122       by simp }
```
```  1123   then show ?thesis
```
```  1124     by (intro pmf_eqI) (simp add: pmf_nonneg)
```
```  1125 qed
```
```  1126
```
```  1127 lemma bind_cond_pmf_cancel:
```
```  1128   assumes [simp]: "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}"
```
```  1129   assumes [simp]: "\<And>y. y \<in> set_pmf q \<Longrightarrow> set_pmf p \<inter> {x. R x y} \<noteq> {}"
```
```  1130   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}"
```
```  1131   shows "bind_pmf p (\<lambda>x. cond_pmf q {y. R x y}) = q"
```
```  1132 proof (rule pmf_eqI)
```
```  1133   fix i
```
```  1134   have "ennreal (pmf (bind_pmf p (\<lambda>x. cond_pmf q {y. R x y})) i) =
```
```  1135     (\<integral>\<^sup>+x. ennreal (pmf q i / measure p {x. R x i}) * ennreal (indicator {x. R x i} x) \<partial>p)"
```
```  1136     by (auto simp add: ennreal_pmf_bind AE_measure_pmf_iff pmf_cond pmf_eq_0_set_pmf pmf_nonneg measure_nonneg
```
```  1137              intro!: nn_integral_cong_AE)
```
```  1138   also have "\<dots> = (pmf q i * measure p {x. R x i}) / measure p {x. R x i}"
```
```  1139     by (simp add: pmf_nonneg measure_nonneg zero_ennreal_def[symmetric] ennreal_indicator
```
```  1140                   nn_integral_cmult measure_pmf.emeasure_eq_measure ennreal_mult[symmetric])
```
```  1141   also have "\<dots> = pmf q i"
```
```  1142     by (cases "pmf q i = 0")
```
```  1143        (simp_all add: pmf_eq_0_set_pmf measure_measure_pmf_not_zero pmf_nonneg)
```
```  1144   finally show "pmf (bind_pmf p (\<lambda>x. cond_pmf q {y. R x y})) i = pmf q i"
```
```  1145     by (simp add: pmf_nonneg)
```
```  1146 qed
```
```  1147
```
```  1148 subsection \<open> Relator \<close>
```
```  1149
```
```  1150 inductive rel_pmf :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf \<Rightarrow> bool"
```
```  1151 for R p q
```
```  1152 where
```
```  1153   "\<lbrakk> \<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y;
```
```  1154      map_pmf fst pq = p; map_pmf snd pq = q \<rbrakk>
```
```  1155   \<Longrightarrow> rel_pmf R p q"
```
```  1156
```
```  1157 lemma rel_pmfI:
```
```  1158   assumes R: "rel_set R (set_pmf p) (set_pmf q)"
```
```  1159   assumes eq: "\<And>x y. x \<in> set_pmf p \<Longrightarrow> y \<in> set_pmf q \<Longrightarrow> R x y \<Longrightarrow>
```
```  1160     measure p {x. R x y} = measure q {y. R x y}"
```
```  1161   shows "rel_pmf R p q"
```
```  1162 proof
```
```  1163   let ?pq = "bind_pmf p (\<lambda>x. bind_pmf (cond_pmf q {y. R x y}) (\<lambda>y. return_pmf (x, y)))"
```
```  1164   have "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}"
```
```  1165     using R by (auto simp: rel_set_def)
```
```  1166   then show "\<And>x y. (x, y) \<in> set_pmf ?pq \<Longrightarrow> R x y"
```
```  1167     by auto
```
```  1168   show "map_pmf fst ?pq = p"
```
```  1169     by (simp add: map_bind_pmf bind_return_pmf')
```
```  1170
```
```  1171   show "map_pmf snd ?pq = q"
```
```  1172     using R eq
```
```  1173     apply (simp add: bind_cond_pmf_cancel map_bind_pmf bind_return_pmf')
```
```  1174     apply (rule bind_cond_pmf_cancel)
```
```  1175     apply (auto simp: rel_set_def)
```
```  1176     done
```
```  1177 qed
```
```  1178
```
```  1179 lemma rel_pmf_imp_rel_set: "rel_pmf R p q \<Longrightarrow> rel_set R (set_pmf p) (set_pmf q)"
```
```  1180   by (force simp add: rel_pmf.simps rel_set_def)
```
```  1181
```
```  1182 lemma rel_pmfD_measure:
```
```  1183   assumes rel_R: "rel_pmf R p q" and R: "\<And>a b. R a b \<Longrightarrow> R a y \<longleftrightarrow> R x b"
```
```  1184   assumes "x \<in> set_pmf p" "y \<in> set_pmf q"
```
```  1185   shows "measure p {x. R x y} = measure q {y. R x y}"
```
```  1186 proof -
```
```  1187   from rel_R obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
```
```  1188     and eq: "p = map_pmf fst pq" "q = map_pmf snd pq"
```
```  1189     by (auto elim: rel_pmf.cases)
```
```  1190   have "measure p {x. R x y} = measure pq {x. R (fst x) y}"
```
```  1191     by (simp add: eq map_pmf_rep_eq measure_distr)
```
```  1192   also have "\<dots> = measure pq {y. R x (snd y)}"
```
```  1193     by (intro measure_pmf.finite_measure_eq_AE)
```
```  1194        (auto simp: AE_measure_pmf_iff R dest!: pq)
```
```  1195   also have "\<dots> = measure q {y. R x y}"
```
```  1196     by (simp add: eq map_pmf_rep_eq measure_distr)
```
```  1197   finally show "measure p {x. R x y} = measure q {y. R x y}" .
```
```  1198 qed
```
```  1199
```
```  1200 lemma rel_pmf_measureD:
```
```  1201   assumes "rel_pmf R p q"
```
```  1202   shows "measure (measure_pmf p) A \<le> measure (measure_pmf q) {y. \<exists>x\<in>A. R x y}" (is "?lhs \<le> ?rhs")
```
```  1203 using assms
```
```  1204 proof cases
```
```  1205   fix pq
```
```  1206   assume R: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
```
```  1207     and p[symmetric]: "map_pmf fst pq = p"
```
```  1208     and q[symmetric]: "map_pmf snd pq = q"
```
```  1209   have "?lhs = measure (measure_pmf pq) (fst -` A)" by(simp add: p)
```
```  1210   also have "\<dots> \<le> measure (measure_pmf pq) {y. \<exists>x\<in>A. R x (snd y)}"
```
```  1211     by(rule measure_pmf.finite_measure_mono_AE)(auto 4 3 simp add: AE_measure_pmf_iff dest: R)
```
```  1212   also have "\<dots> = ?rhs" by(simp add: q)
```
```  1213   finally show ?thesis .
```
```  1214 qed
```
```  1215
```
```  1216 lemma rel_pmf_iff_measure:
```
```  1217   assumes "symp R" "transp R"
```
```  1218   shows "rel_pmf R p q \<longleftrightarrow>
```
```  1219     rel_set R (set_pmf p) (set_pmf q) \<and>
```
```  1220     (\<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})"
```
```  1221   by (safe intro!: rel_pmf_imp_rel_set rel_pmfI)
```
```  1222      (auto intro!: rel_pmfD_measure dest: sympD[OF \<open>symp R\<close>] transpD[OF \<open>transp R\<close>])
```
```  1223
```
```  1224 lemma quotient_rel_set_disjoint:
```
```  1225   "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 = {}"
```
```  1226   using in_quotient_imp_closed[of UNIV "{(x, y). R x y}" C]
```
```  1227   by (auto 0 0 simp: equivp_equiv rel_set_def set_eq_iff elim: equivpE)
```
```  1228      (blast dest: equivp_symp)+
```
```  1229
```
```  1230 lemma quotientD: "equiv X R \<Longrightarrow> A \<in> X // R \<Longrightarrow> x \<in> A \<Longrightarrow> A = R `` {x}"
```
```  1231   by (metis Image_singleton_iff equiv_class_eq_iff quotientE)
```
```  1232
```
```  1233 lemma rel_pmf_iff_equivp:
```
```  1234   assumes "equivp R"
```
```  1235   shows "rel_pmf R p q \<longleftrightarrow> (\<forall>C\<in>UNIV // {(x, y). R x y}. measure p C = measure q C)"
```
```  1236     (is "_ \<longleftrightarrow>   (\<forall>C\<in>_//?R. _)")
```
```  1237 proof (subst rel_pmf_iff_measure, safe)
```
```  1238   show "symp R" "transp R"
```
```  1239     using assms by (auto simp: equivp_reflp_symp_transp)
```
```  1240 next
```
```  1241   fix C assume C: "C \<in> UNIV // ?R" and R: "rel_set R (set_pmf p) (set_pmf q)"
```
```  1242   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}"
```
```  1243
```
```  1244   show "measure p C = measure q C"
```
```  1245   proof (cases "p \<inter> C = {}")
```
```  1246     case True
```
```  1247     then have "q \<inter> C = {}"
```
```  1248       using quotient_rel_set_disjoint[OF assms C R] by simp
```
```  1249     with True show ?thesis
```
```  1250       unfolding measure_pmf_zero_iff[symmetric] by simp
```
```  1251   next
```
```  1252     case False
```
```  1253     then have "q \<inter> C \<noteq> {}"
```
```  1254       using quotient_rel_set_disjoint[OF assms C R] by simp
```
```  1255     with False 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"
```
```  1256       by auto
```
```  1257     then have "R x y"
```
```  1258       using in_quotient_imp_in_rel[of UNIV ?R C x y] C assms
```
```  1259       by (simp add: equivp_equiv)
```
```  1260     with in_set eq have "measure p {x. R x y} = measure q {y. R x y}"
```
```  1261       by auto
```
```  1262     moreover have "{y. R x y} = C"
```
```  1263       using assms \<open>x \<in> C\<close> C quotientD[of UNIV ?R C x] by (simp add: equivp_equiv)
```
```  1264     moreover have "{x. R x y} = C"
```
```  1265       using assms \<open>y \<in> C\<close> C quotientD[of UNIV "?R" C y] sympD[of R]
```
```  1266       by (auto simp add: equivp_equiv elim: equivpE)
```
```  1267     ultimately show ?thesis
```
```  1268       by auto
```
```  1269   qed
```
```  1270 next
```
```  1271   assume eq: "\<forall>C\<in>UNIV // ?R. measure p C = measure q C"
```
```  1272   show "rel_set R (set_pmf p) (set_pmf q)"
```
```  1273     unfolding rel_set_def
```
```  1274   proof safe
```
```  1275     fix x assume x: "x \<in> set_pmf p"
```
```  1276     have "{y. R x y} \<in> UNIV // ?R"
```
```  1277       by (auto simp: quotient_def)
```
```  1278     with eq have *: "measure q {y. R x y} = measure p {y. R x y}"
```
```  1279       by auto
```
```  1280     have "measure q {y. R x y} \<noteq> 0"
```
```  1281       using x assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp)
```
```  1282     then show "\<exists>y\<in>set_pmf q. R x y"
```
```  1283       unfolding measure_pmf_zero_iff by auto
```
```  1284   next
```
```  1285     fix y assume y: "y \<in> set_pmf q"
```
```  1286     have "{x. R x y} \<in> UNIV // ?R"
```
```  1287       using assms by (auto simp: quotient_def dest: equivp_symp)
```
```  1288     with eq have *: "measure p {x. R x y} = measure q {x. R x y}"
```
```  1289       by auto
```
```  1290     have "measure p {x. R x y} \<noteq> 0"
```
```  1291       using y assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp)
```
```  1292     then show "\<exists>x\<in>set_pmf p. R x y"
```
```  1293       unfolding measure_pmf_zero_iff by auto
```
```  1294   qed
```
```  1295
```
```  1296   fix x y assume "x \<in> set_pmf p" "y \<in> set_pmf q" "R x y"
```
```  1297   have "{y. R x y} \<in> UNIV // ?R" "{x. R x y} = {y. R x y}"
```
```  1298     using assms \<open>R x y\<close> by (auto simp: quotient_def dest: equivp_symp equivp_transp)
```
```  1299   with eq show "measure p {x. R x y} = measure q {y. R x y}"
```
```  1300     by auto
```
```  1301 qed
```
```  1302
```
```  1303 bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: rel_pmf
```
```  1304 proof -
```
```  1305   show "map_pmf id = id" by (rule map_pmf_id)
```
```  1306   show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose)
```
```  1307   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"
```
```  1308     by (intro map_pmf_cong refl)
```
```  1309
```
```  1310   show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
```
```  1311     by (rule pmf_set_map)
```
```  1312
```
```  1313   show "(card_of (set_pmf p), natLeq) \<in> ordLeq" for p :: "'s pmf"
```
```  1314   proof -
```
```  1315     have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq"
```
```  1316       by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"])
```
```  1317          (auto intro: countable_set_pmf)
```
```  1318     also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq"
```
```  1319       by (metis Field_natLeq card_of_least natLeq_Well_order)
```
```  1320     finally show ?thesis .
```
```  1321   qed
```
```  1322
```
```  1323   show "\<And>R. rel_pmf R = (\<lambda>x y. \<exists>z. set_pmf z \<subseteq> {(x, y). R x y} \<and>
```
```  1324     map_pmf fst z = x \<and> map_pmf snd z = y)"
```
```  1325      by (auto simp add: fun_eq_iff rel_pmf.simps)
```
```  1326
```
```  1327   show "rel_pmf R OO rel_pmf S \<le> rel_pmf (R OO S)"
```
```  1328     for R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
```
```  1329   proof -
```
```  1330     { fix p q r
```
```  1331       assume pq: "rel_pmf R p q"
```
```  1332         and qr:"rel_pmf S q r"
```
```  1333       from pq obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
```
```  1334         and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto
```
```  1335       from qr obtain qr where qr: "\<And>y z. (y, z) \<in> set_pmf qr \<Longrightarrow> S y z"
```
```  1336         and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto
```
```  1337
```
```  1338       define pr where "pr =
```
```  1339         bind_pmf pq (\<lambda>xy. bind_pmf (cond_pmf qr {yz. fst yz = snd xy})
```
```  1340           (\<lambda>yz. return_pmf (fst xy, snd yz)))"
```
```  1341       have pr_welldefined: "\<And>y. y \<in> q \<Longrightarrow> qr \<inter> {yz. fst yz = y} \<noteq> {}"
```
```  1342         by (force simp: q')
```
```  1343
```
```  1344       have "rel_pmf (R OO S) p r"
```
```  1345       proof (rule rel_pmf.intros)
```
```  1346         fix x z assume "(x, z) \<in> pr"
```
```  1347         then have "\<exists>y. (x, y) \<in> pq \<and> (y, z) \<in> qr"
```
```  1348           by (auto simp: q pr_welldefined pr_def split_beta)
```
```  1349         with pq qr show "(R OO S) x z"
```
```  1350           by blast
```
```  1351       next
```
```  1352         have "map_pmf snd pr = map_pmf snd (bind_pmf q (\<lambda>y. cond_pmf qr {yz. fst yz = y}))"
```
```  1353           by (simp add: pr_def q split_beta bind_map_pmf map_pmf_def[symmetric] map_bind_pmf map_pmf_comp)
```
```  1354         then show "map_pmf snd pr = r"
```
```  1355           unfolding r q' bind_map_pmf by (subst (asm) bind_cond_pmf_cancel) (auto simp: eq_commute)
```
```  1356       qed (simp add: pr_def map_bind_pmf split_beta map_pmf_def[symmetric] p map_pmf_comp)
```
```  1357     }
```
```  1358     then show ?thesis
```
```  1359       by(auto simp add: le_fun_def)
```
```  1360   qed
```
```  1361 qed (fact natLeq_card_order natLeq_cinfinite)+
```
```  1362
```
```  1363 lemma map_pmf_idI: "(\<And>x. x \<in> set_pmf p \<Longrightarrow> f x = x) \<Longrightarrow> map_pmf f p = p"
```
```  1364 by(simp cong: pmf.map_cong)
```
```  1365
```
```  1366 lemma rel_pmf_conj[simp]:
```
```  1367   "rel_pmf (\<lambda>x y. P \<and> Q x y) x y \<longleftrightarrow> P \<and> rel_pmf Q x y"
```
```  1368   "rel_pmf (\<lambda>x y. Q x y \<and> P) x y \<longleftrightarrow> P \<and> rel_pmf Q x y"
```
```  1369   using set_pmf_not_empty by (fastforce simp: pmf.in_rel subset_eq)+
```
```  1370
```
```  1371 lemma rel_pmf_top[simp]: "rel_pmf top = top"
```
```  1372   by (auto simp: pmf.in_rel[abs_def] fun_eq_iff map_fst_pair_pmf map_snd_pair_pmf
```
```  1373            intro: exI[of _ "pair_pmf x y" for x y])
```
```  1374
```
```  1375 lemma rel_pmf_return_pmf1: "rel_pmf R (return_pmf x) M \<longleftrightarrow> (\<forall>a\<in>M. R x a)"
```
```  1376 proof safe
```
```  1377   fix a assume "a \<in> M" "rel_pmf R (return_pmf x) M"
```
```  1378   then obtain pq where *: "\<And>a b. (a, b) \<in> set_pmf pq \<Longrightarrow> R a b"
```
```  1379     and eq: "return_pmf x = map_pmf fst pq" "M = map_pmf snd pq"
```
```  1380     by (force elim: rel_pmf.cases)
```
```  1381   moreover have "set_pmf (return_pmf x) = {x}"
```
```  1382     by simp
```
```  1383   with \<open>a \<in> M\<close> have "(x, a) \<in> pq"
```
```  1384     by (force simp: eq)
```
```  1385   with * show "R x a"
```
```  1386     by auto
```
```  1387 qed (auto intro!: rel_pmf.intros[where pq="pair_pmf (return_pmf x) M"]
```
```  1388           simp: map_fst_pair_pmf map_snd_pair_pmf)
```
```  1389
```
```  1390 lemma rel_pmf_return_pmf2: "rel_pmf R M (return_pmf x) \<longleftrightarrow> (\<forall>a\<in>M. R a x)"
```
```  1391   by (subst pmf.rel_flip[symmetric]) (simp add: rel_pmf_return_pmf1)
```
```  1392
```
```  1393 lemma rel_return_pmf[simp]: "rel_pmf R (return_pmf x1) (return_pmf x2) = R x1 x2"
```
```  1394   unfolding rel_pmf_return_pmf2 set_return_pmf by simp
```
```  1395
```
```  1396 lemma rel_pmf_False[simp]: "rel_pmf (\<lambda>x y. False) x y = False"
```
```  1397   unfolding pmf.in_rel fun_eq_iff using set_pmf_not_empty by fastforce
```
```  1398
```
```  1399 lemma rel_pmf_rel_prod:
```
```  1400   "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'"
```
```  1401 proof safe
```
```  1402   assume "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
```
```  1403   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"
```
```  1404     and eq: "map_pmf fst pq = pair_pmf A A'" "map_pmf snd pq = pair_pmf B B'"
```
```  1405     by (force elim: rel_pmf.cases)
```
```  1406   show "rel_pmf R A B"
```
```  1407   proof (rule rel_pmf.intros)
```
```  1408     let ?f = "\<lambda>(a, b). (fst a, fst b)"
```
```  1409     have [simp]: "(\<lambda>x. fst (?f x)) = fst o fst" "(\<lambda>x. snd (?f x)) = fst o snd"
```
```  1410       by auto
```
```  1411
```
```  1412     show "map_pmf fst (map_pmf ?f pq) = A"
```
```  1413       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
```
```  1414     show "map_pmf snd (map_pmf ?f pq) = B"
```
```  1415       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
```
```  1416
```
```  1417     fix a b assume "(a, b) \<in> set_pmf (map_pmf ?f pq)"
```
```  1418     then obtain c d where "((a, c), (b, d)) \<in> set_pmf pq"
```
```  1419       by auto
```
```  1420     from pq[OF this] show "R a b" ..
```
```  1421   qed
```
```  1422   show "rel_pmf S A' B'"
```
```  1423   proof (rule rel_pmf.intros)
```
```  1424     let ?f = "\<lambda>(a, b). (snd a, snd b)"
```
```  1425     have [simp]: "(\<lambda>x. fst (?f x)) = snd o fst" "(\<lambda>x. snd (?f x)) = snd o snd"
```
```  1426       by auto
```
```  1427
```
```  1428     show "map_pmf fst (map_pmf ?f pq) = A'"
```
```  1429       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
```
```  1430     show "map_pmf snd (map_pmf ?f pq) = B'"
```
```  1431       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
```
```  1432
```
```  1433     fix c d assume "(c, d) \<in> set_pmf (map_pmf ?f pq)"
```
```  1434     then obtain a b where "((a, c), (b, d)) \<in> set_pmf pq"
```
```  1435       by auto
```
```  1436     from pq[OF this] show "S c d" ..
```
```  1437   qed
```
```  1438 next
```
```  1439   assume "rel_pmf R A B" "rel_pmf S A' B'"
```
```  1440   then obtain Rpq Spq
```
```  1441     where Rpq: "\<And>a b. (a, b) \<in> set_pmf Rpq \<Longrightarrow> R a b"
```
```  1442         "map_pmf fst Rpq = A" "map_pmf snd Rpq = B"
```
```  1443       and Spq: "\<And>a b. (a, b) \<in> set_pmf Spq \<Longrightarrow> S a b"
```
```  1444         "map_pmf fst Spq = A'" "map_pmf snd Spq = B'"
```
```  1445     by (force elim: rel_pmf.cases)
```
```  1446
```
```  1447   let ?f = "(\<lambda>((a, c), (b, d)). ((a, b), (c, d)))"
```
```  1448   let ?pq = "map_pmf ?f (pair_pmf Rpq Spq)"
```
```  1449   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))"
```
```  1450     by auto
```
```  1451
```
```  1452   show "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
```
```  1453     by (rule rel_pmf.intros[where pq="?pq"])
```
```  1454        (auto simp: map_snd_pair_pmf map_fst_pair_pmf map_pmf_comp Rpq Spq
```
```  1455                    map_pair)
```
```  1456 qed
```
```  1457
```
```  1458 lemma rel_pmf_reflI:
```
```  1459   assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> P x x"
```
```  1460   shows "rel_pmf P p p"
```
```  1461   by (rule rel_pmf.intros[where pq="map_pmf (\<lambda>x. (x, x)) p"])
```
```  1462      (auto simp add: pmf.map_comp o_def assms)
```
```  1463
```
```  1464 lemma rel_pmf_bij_betw:
```
```  1465   assumes f: "bij_betw f (set_pmf p) (set_pmf q)"
```
```  1466   and eq: "\<And>x. x \<in> set_pmf p \<Longrightarrow> pmf p x = pmf q (f x)"
```
```  1467   shows "rel_pmf (\<lambda>x y. f x = y) p q"
```
```  1468 proof(rule rel_pmf.intros)
```
```  1469   let ?pq = "map_pmf (\<lambda>x. (x, f x)) p"
```
```  1470   show "map_pmf fst ?pq = p" by(simp add: pmf.map_comp o_def)
```
```  1471
```
```  1472   have "map_pmf f p = q"
```
```  1473   proof(rule pmf_eqI)
```
```  1474     fix i
```
```  1475     show "pmf (map_pmf f p) i = pmf q i"
```
```  1476     proof(cases "i \<in> set_pmf q")
```
```  1477       case True
```
```  1478       with f obtain j where "i = f j" "j \<in> set_pmf p"
```
```  1479         by(auto simp add: bij_betw_def image_iff)
```
```  1480       thus ?thesis using f by(simp add: bij_betw_def pmf_map_inj eq)
```
```  1481     next
```
```  1482       case False thus ?thesis
```
```  1483         by(subst pmf_map_outside)(auto simp add: set_pmf_iff eq[symmetric])
```
```  1484     qed
```
```  1485   qed
```
```  1486   then show "map_pmf snd ?pq = q" by(simp add: pmf.map_comp o_def)
```
```  1487 qed auto
```
```  1488
```
```  1489 context
```
```  1490 begin
```
```  1491
```
```  1492 interpretation pmf_as_measure .
```
```  1493
```
```  1494 definition "join_pmf M = bind_pmf M (\<lambda>x. x)"
```
```  1495
```
```  1496 lemma bind_eq_join_pmf: "bind_pmf M f = join_pmf (map_pmf f M)"
```
```  1497   unfolding join_pmf_def bind_map_pmf ..
```
```  1498
```
```  1499 lemma join_eq_bind_pmf: "join_pmf M = bind_pmf M id"
```
```  1500   by (simp add: join_pmf_def id_def)
```
```  1501
```
```  1502 lemma pmf_join: "pmf (join_pmf N) i = (\<integral>M. pmf M i \<partial>measure_pmf N)"
```
```  1503   unfolding join_pmf_def pmf_bind ..
```
```  1504
```
```  1505 lemma ennreal_pmf_join: "ennreal (pmf (join_pmf N) i) = (\<integral>\<^sup>+M. pmf M i \<partial>measure_pmf N)"
```
```  1506   unfolding join_pmf_def ennreal_pmf_bind ..
```
```  1507
```
```  1508 lemma set_pmf_join_pmf[simp]: "set_pmf (join_pmf f) = (\<Union>p\<in>set_pmf f. set_pmf p)"
```
```  1509   by (simp add: join_pmf_def)
```
```  1510
```
```  1511 lemma join_return_pmf: "join_pmf (return_pmf M) = M"
```
```  1512   by (simp add: integral_return pmf_eq_iff pmf_join return_pmf.rep_eq)
```
```  1513
```
```  1514 lemma map_join_pmf: "map_pmf f (join_pmf AA) = join_pmf (map_pmf (map_pmf f) AA)"
```
```  1515   by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf)
```
```  1516
```
```  1517 lemma join_map_return_pmf: "join_pmf (map_pmf return_pmf A) = A"
```
```  1518   by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
```
```  1519
```
```  1520 end
```
```  1521
```
```  1522 lemma rel_pmf_joinI:
```
```  1523   assumes "rel_pmf (rel_pmf P) p q"
```
```  1524   shows "rel_pmf P (join_pmf p) (join_pmf q)"
```
```  1525 proof -
```
```  1526   from assms obtain pq where p: "p = map_pmf fst pq"
```
```  1527     and q: "q = map_pmf snd pq"
```
```  1528     and P: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> rel_pmf P x y"
```
```  1529     by cases auto
```
```  1530   from P obtain PQ
```
```  1531     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"
```
```  1532     and x: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf fst (PQ x y) = x"
```
```  1533     and y: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf snd (PQ x y) = y"
```
```  1534     by(metis rel_pmf.simps)
```
```  1535
```
```  1536   let ?r = "bind_pmf pq (\<lambda>(x, y). PQ x y)"
```
```  1537   have "\<And>a b. (a, b) \<in> set_pmf ?r \<Longrightarrow> P a b" by (auto intro: PQ)
```
```  1538   moreover have "map_pmf fst ?r = join_pmf p" "map_pmf snd ?r = join_pmf q"
```
```  1539     by (simp_all add: p q x y join_pmf_def map_bind_pmf bind_map_pmf split_def cong: bind_pmf_cong)
```
```  1540   ultimately show ?thesis ..
```
```  1541 qed
```
```  1542
```
```  1543 lemma rel_pmf_bindI:
```
```  1544   assumes pq: "rel_pmf R p q"
```
```  1545   and fg: "\<And>x y. R x y \<Longrightarrow> rel_pmf P (f x) (g y)"
```
```  1546   shows "rel_pmf P (bind_pmf p f) (bind_pmf q g)"
```
```  1547   unfolding bind_eq_join_pmf
```
```  1548   by (rule rel_pmf_joinI)
```
```  1549      (auto simp add: pmf.rel_map intro: pmf.rel_mono[THEN le_funD, THEN le_funD, THEN le_boolD, THEN mp, OF _ pq] fg)
```
```  1550
```
```  1551 text \<open>
```
```  1552   Proof that @{const rel_pmf} preserves orders.
```
```  1553   Antisymmetry proof follows Thm. 1 in N. Saheb-Djahromi, Cpo's of measures for nondeterminism,
```
```  1554   Theoretical Computer Science 12(1):19--37, 1980,
```
```  1555   \<^url>\<open>http://dx.doi.org/10.1016/0304-3975(80)90003-1\<close>
```
```  1556 \<close>
```
```  1557
```
```  1558 lemma
```
```  1559   assumes *: "rel_pmf R p q"
```
```  1560   and refl: "reflp R" and trans: "transp R"
```
```  1561   shows measure_Ici: "measure p {y. R x y} \<le> measure q {y. R x y}" (is ?thesis1)
```
```  1562   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)
```
```  1563 proof -
```
```  1564   from * obtain pq
```
```  1565     where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
```
```  1566     and p: "p = map_pmf fst pq"
```
```  1567     and q: "q = map_pmf snd pq"
```
```  1568     by cases auto
```
```  1569   show ?thesis1 ?thesis2 unfolding p q map_pmf_rep_eq using refl trans
```
```  1570     by(auto 4 3 simp add: measure_distr reflpD AE_measure_pmf_iff intro!: measure_pmf.finite_measure_mono_AE dest!: pq elim: transpE)
```
```  1571 qed
```
```  1572
```
```  1573 lemma rel_pmf_inf:
```
```  1574   fixes p q :: "'a pmf"
```
```  1575   assumes 1: "rel_pmf R p q"
```
```  1576   assumes 2: "rel_pmf R q p"
```
```  1577   and refl: "reflp R" and trans: "transp R"
```
```  1578   shows "rel_pmf (inf R R\<inverse>\<inverse>) p q"
```
```  1579 proof (subst rel_pmf_iff_equivp, safe)
```
```  1580   show "equivp (inf R R\<inverse>\<inverse>)"
```
```  1581     using trans refl by (auto simp: equivp_reflp_symp_transp intro: sympI transpI reflpI dest: transpD reflpD)
```
```  1582
```
```  1583   fix C assume "C \<in> UNIV // {(x, y). inf R R\<inverse>\<inverse> x y}"
```
```  1584   then obtain x where C: "C = {y. R x y \<and> R y x}"
```
```  1585     by (auto elim: quotientE)
```
```  1586
```
```  1587   let ?R = "\<lambda>x y. R x y \<and> R y x"
```
```  1588   let ?\<mu>R = "\<lambda>y. measure q {x. ?R x y}"
```
```  1589   have "measure p {y. ?R x y} = measure p ({y. R x y} - {y. R x y \<and> \<not> R y x})"
```
```  1590     by(auto intro!: arg_cong[where f="measure p"])
```
```  1591   also have "\<dots> = measure p {y. R x y} - measure p {y. R x y \<and> \<not> R y x}"
```
```  1592     by (rule measure_pmf.finite_measure_Diff) auto
```
```  1593   also have "measure p {y. R x y \<and> \<not> R y x} = measure q {y. R x y \<and> \<not> R y x}"
```
```  1594     using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ioi)
```
```  1595   also have "measure p {y. R x y} = measure q {y. R x y}"
```
```  1596     using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ici)
```
```  1597   also have "measure q {y. R x y} - measure q {y. R x y \<and> \<not> R y x} =
```
```  1598     measure q ({y. R x y} - {y. R x y \<and> \<not> R y x})"
```
```  1599     by(rule measure_pmf.finite_measure_Diff[symmetric]) auto
```
```  1600   also have "\<dots> = ?\<mu>R x"
```
```  1601     by(auto intro!: arg_cong[where f="measure q"])
```
```  1602   finally show "measure p C = measure q C"
```
```  1603     by (simp add: C conj_commute)
```
```  1604 qed
```
```  1605
```
```  1606 lemma rel_pmf_antisym:
```
```  1607   fixes p q :: "'a pmf"
```
```  1608   assumes 1: "rel_pmf R p q"
```
```  1609   assumes 2: "rel_pmf R q p"
```
```  1610   and refl: "reflp R" and trans: "transp R" and antisym: "antisymp R"
```
```  1611   shows "p = q"
```
```  1612 proof -
```
```  1613   from 1 2 refl trans have "rel_pmf (inf R R\<inverse>\<inverse>) p q" by(rule rel_pmf_inf)
```
```  1614   also have "inf R R\<inverse>\<inverse> = op ="
```
```  1615     using refl antisym by (auto intro!: ext simp add: reflpD dest: antisympD)
```
```  1616   finally show ?thesis unfolding pmf.rel_eq .
```
```  1617 qed
```
```  1618
```
```  1619 lemma reflp_rel_pmf: "reflp R \<Longrightarrow> reflp (rel_pmf R)"
```
```  1620   by (fact pmf.rel_reflp)
```
```  1621
```
```  1622 lemma antisymp_rel_pmf:
```
```  1623   "\<lbrakk> reflp R; transp R; antisymp R \<rbrakk>
```
```  1624   \<Longrightarrow> antisymp (rel_pmf R)"
```
```  1625 by(rule antisympI)(blast intro: rel_pmf_antisym)
```
```  1626
```
```  1627 lemma transp_rel_pmf:
```
```  1628   assumes "transp R"
```
```  1629   shows "transp (rel_pmf R)"
```
```  1630   using assms by (fact pmf.rel_transp)
```
```  1631
```
```  1632
```
```  1633 subsection \<open> Distributions \<close>
```
```  1634
```
```  1635 context
```
```  1636 begin
```
```  1637
```
```  1638 interpretation pmf_as_function .
```
```  1639
```
```  1640 subsubsection \<open> Bernoulli Distribution \<close>
```
```  1641
```
```  1642 lift_definition bernoulli_pmf :: "real \<Rightarrow> bool pmf" is
```
```  1643   "\<lambda>p b. ((\<lambda>p. if b then p else 1 - p) \<circ> min 1 \<circ> max 0) p"
```
```  1644   by (auto simp: nn_integral_count_space_finite[where A="{False, True}"] UNIV_bool
```
```  1645            split: split_max split_min)
```
```  1646
```
```  1647 lemma pmf_bernoulli_True[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) True = p"
```
```  1648   by transfer simp
```
```  1649
```
```  1650 lemma pmf_bernoulli_False[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) False = 1 - p"
```
```  1651   by transfer simp
```
```  1652
```
```  1653 lemma set_pmf_bernoulli[simp]: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (bernoulli_pmf p) = UNIV"
```
```  1654   by (auto simp add: set_pmf_iff UNIV_bool)
```
```  1655
```
```  1656 lemma nn_integral_bernoulli_pmf[simp]:
```
```  1657   assumes [simp]: "0 \<le> p" "p \<le> 1" "\<And>x. 0 \<le> f x"
```
```  1658   shows "(\<integral>\<^sup>+x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
```
```  1659   by (subst nn_integral_measure_pmf_support[of UNIV])
```
```  1660      (auto simp: UNIV_bool field_simps)
```
```  1661
```
```  1662 lemma integral_bernoulli_pmf[simp]:
```
```  1663   assumes [simp]: "0 \<le> p" "p \<le> 1"
```
```  1664   shows "(\<integral>x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
```
```  1665   by (subst integral_measure_pmf[of UNIV]) (auto simp: UNIV_bool)
```
```  1666
```
```  1667 lemma pmf_bernoulli_half [simp]: "pmf (bernoulli_pmf (1 / 2)) x = 1 / 2"
```
```  1668 by(cases x) simp_all
```
```  1669
```
```  1670 lemma measure_pmf_bernoulli_half: "measure_pmf (bernoulli_pmf (1 / 2)) = uniform_count_measure UNIV"
```
```  1671   by (rule measure_eqI)
```
```  1672      (simp_all add: nn_integral_pmf[symmetric] emeasure_uniform_count_measure ennreal_divide_numeral[symmetric]
```
```  1673                     nn_integral_count_space_finite sets_uniform_count_measure divide_ennreal_def mult_ac
```
```  1674                     ennreal_of_nat_eq_real_of_nat)
```
```  1675
```
```  1676 subsubsection \<open> Geometric Distribution \<close>
```
```  1677
```
```  1678 context
```
```  1679   fixes p :: real assumes p[arith]: "0 < p" "p \<le> 1"
```
```  1680 begin
```
```  1681
```
```  1682 lift_definition geometric_pmf :: "nat pmf" is "\<lambda>n. (1 - p)^n * p"
```
```  1683 proof
```
```  1684   have "(\<Sum>i. ennreal (p * (1 - p) ^ i)) = ennreal (p * (1 / (1 - (1 - p))))"
```
```  1685     by (intro suminf_ennreal_eq sums_mult geometric_sums) auto
```
```  1686   then show "(\<integral>\<^sup>+ x. ennreal ((1 - p)^x * p) \<partial>count_space UNIV) = 1"
```
```  1687     by (simp add: nn_integral_count_space_nat field_simps)
```
```  1688 qed simp
```
```  1689
```
```  1690 lemma pmf_geometric[simp]: "pmf geometric_pmf n = (1 - p)^n * p"
```
```  1691   by transfer rule
```
```  1692
```
```  1693 end
```
```  1694
```
```  1695 lemma set_pmf_geometric: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (geometric_pmf p) = UNIV"
```
```  1696   by (auto simp: set_pmf_iff)
```
```  1697
```
```  1698 subsubsection \<open> Uniform Multiset Distribution \<close>
```
```  1699
```
```  1700 context
```
```  1701   fixes M :: "'a multiset" assumes M_not_empty: "M \<noteq> {#}"
```
```  1702 begin
```
```  1703
```
```  1704 lift_definition pmf_of_multiset :: "'a pmf" is "\<lambda>x. count M x / size M"
```
```  1705 proof
```
```  1706   show "(\<integral>\<^sup>+ x. ennreal (real (count M x) / real (size M)) \<partial>count_space UNIV) = 1"
```
```  1707     using M_not_empty
```
```  1708     by (simp add: zero_less_divide_iff nn_integral_count_space nonempty_has_size
```
```  1709                   sum_divide_distrib[symmetric])
```
```  1710        (auto simp: size_multiset_overloaded_eq intro!: sum.cong)
```
```  1711 qed simp
```
```  1712
```
```  1713 lemma pmf_of_multiset[simp]: "pmf pmf_of_multiset x = count M x / size M"
```
```  1714   by transfer rule
```
```  1715
```
```  1716 lemma set_pmf_of_multiset[simp]: "set_pmf pmf_of_multiset = set_mset M"
```
```  1717   by (auto simp: set_pmf_iff)
```
```  1718
```
```  1719 end
```
```  1720
```
```  1721 subsubsection \<open> Uniform Distribution \<close>
```
```  1722
```
```  1723 context
```
```  1724   fixes S :: "'a set" assumes S_not_empty: "S \<noteq> {}" and S_finite: "finite S"
```
```  1725 begin
```
```  1726
```
```  1727 lift_definition pmf_of_set :: "'a pmf" is "\<lambda>x. indicator S x / card S"
```
```  1728 proof
```
```  1729   show "(\<integral>\<^sup>+ x. ennreal (indicator S x / real (card S)) \<partial>count_space UNIV) = 1"
```
```  1730     using S_not_empty S_finite
```
```  1731     by (subst nn_integral_count_space'[of S])
```
```  1732        (auto simp: ennreal_of_nat_eq_real_of_nat ennreal_mult[symmetric])
```
```  1733 qed simp
```
```  1734
```
```  1735 lemma pmf_of_set[simp]: "pmf pmf_of_set x = indicator S x / card S"
```
```  1736   by transfer rule
```
```  1737
```
```  1738 lemma set_pmf_of_set[simp]: "set_pmf pmf_of_set = S"
```
```  1739   using S_finite S_not_empty by (auto simp: set_pmf_iff)
```
```  1740
```
```  1741 lemma emeasure_pmf_of_set_space[simp]: "emeasure pmf_of_set S = 1"
```
```  1742   by (rule measure_pmf.emeasure_eq_1_AE) (auto simp: AE_measure_pmf_iff)
```
```  1743
```
```  1744 lemma nn_integral_pmf_of_set: "nn_integral (measure_pmf pmf_of_set) f = sum f S / card S"
```
```  1745   by (subst nn_integral_measure_pmf_finite)
```
```  1746      (simp_all add: sum_distrib_right[symmetric] card_gt_0_iff S_not_empty S_finite divide_ennreal_def
```
```  1747                 divide_ennreal[symmetric] ennreal_of_nat_eq_real_of_nat[symmetric] ennreal_times_divide)
```
```  1748
```
```  1749 lemma integral_pmf_of_set: "integral\<^sup>L (measure_pmf pmf_of_set) f = sum f S / card S"
```
```  1750   by (subst integral_measure_pmf[of S]) (auto simp: S_finite sum_divide_distrib)
```
```  1751
```
```  1752 lemma emeasure_pmf_of_set: "emeasure (measure_pmf pmf_of_set) A = card (S \<inter> A) / card S"
```
```  1753   by (subst nn_integral_indicator[symmetric], simp)
```
```  1754      (simp add: S_finite S_not_empty card_gt_0_iff indicator_def sum.If_cases divide_ennreal
```
```  1755                 ennreal_of_nat_eq_real_of_nat nn_integral_pmf_of_set)
```
```  1756
```
```  1757 lemma measure_pmf_of_set: "measure (measure_pmf pmf_of_set) A = card (S \<inter> A) / card S"
```
```  1758   using emeasure_pmf_of_set[of A]
```
```  1759   by (simp add: measure_nonneg measure_pmf.emeasure_eq_measure)
```
```  1760
```
```  1761 end
```
```  1762
```
```  1763 lemma pmf_expectation_bind_pmf_of_set:
```
```  1764   fixes A :: "'a set" and f :: "'a \<Rightarrow> 'b pmf"
```
```  1765     and  h :: "'b \<Rightarrow> 'c::{banach, second_countable_topology}"
```
```  1766   assumes "A \<noteq> {}" "finite A" "\<And>x. x \<in> A \<Longrightarrow> finite (set_pmf (f x))"
```
```  1767   shows "measure_pmf.expectation (pmf_of_set A \<bind> f) h =
```
```  1768            (\<Sum>a\<in>A. measure_pmf.expectation (f a) h /\<^sub>R real (card A))"
```
```  1769   using assms by (subst pmf_expectation_bind[of A]) (auto simp: divide_simps)
```
```  1770
```
```  1771 lemma map_pmf_of_set:
```
```  1772   assumes "finite A" "A \<noteq> {}"
```
```  1773   shows   "map_pmf f (pmf_of_set A) = pmf_of_multiset (image_mset f (mset_set A))"
```
```  1774     (is "?lhs = ?rhs")
```
```  1775 proof (intro pmf_eqI)
```
```  1776   fix x
```
```  1777   from assms have "ennreal (pmf ?lhs x) = ennreal (pmf ?rhs x)"
```
```  1778     by (subst ennreal_pmf_map)
```
```  1779        (simp_all add: emeasure_pmf_of_set mset_set_empty_iff count_image_mset Int_commute)
```
```  1780   thus "pmf ?lhs x = pmf ?rhs x" by simp
```
```  1781 qed
```
```  1782
```
```  1783 lemma pmf_bind_pmf_of_set:
```
```  1784   assumes "A \<noteq> {}" "finite A"
```
```  1785   shows   "pmf (bind_pmf (pmf_of_set A) f) x =
```
```  1786              (\<Sum>xa\<in>A. pmf (f xa) x) / real_of_nat (card A)" (is "?lhs = ?rhs")
```
```  1787 proof -
```
```  1788   from assms have "card A > 0" by auto
```
```  1789   with assms have "ennreal ?lhs = ennreal ?rhs"
```
```  1790     by (subst ennreal_pmf_bind)
```
```  1791        (simp_all add: nn_integral_pmf_of_set max_def pmf_nonneg divide_ennreal [symmetric]
```
```  1792         sum_nonneg ennreal_of_nat_eq_real_of_nat)
```
```  1793   thus ?thesis by (subst (asm) ennreal_inj) (auto intro!: sum_nonneg divide_nonneg_nonneg)
```
```  1794 qed
```
```  1795
```
```  1796 lemma pmf_of_set_singleton: "pmf_of_set {x} = return_pmf x"
```
```  1797 by(rule pmf_eqI)(simp add: indicator_def)
```
```  1798
```
```  1799 lemma map_pmf_of_set_inj:
```
```  1800   assumes f: "inj_on f A"
```
```  1801   and [simp]: "A \<noteq> {}" "finite A"
```
```  1802   shows "map_pmf f (pmf_of_set A) = pmf_of_set (f ` A)" (is "?lhs = ?rhs")
```
```  1803 proof(rule pmf_eqI)
```
```  1804   fix i
```
```  1805   show "pmf ?lhs i = pmf ?rhs i"
```
```  1806   proof(cases "i \<in> f ` A")
```
```  1807     case True
```
```  1808     then obtain i' where "i = f i'" "i' \<in> A" by auto
```
```  1809     thus ?thesis using f by(simp add: card_image pmf_map_inj)
```
```  1810   next
```
```  1811     case False
```
```  1812     hence "pmf ?lhs i = 0" by(simp add: pmf_eq_0_set_pmf set_map_pmf)
```
```  1813     moreover have "pmf ?rhs i = 0" using False by simp
```
```  1814     ultimately show ?thesis by simp
```
```  1815   qed
```
```  1816 qed
```
```  1817
```
```  1818 lemma map_pmf_of_set_bij_betw:
```
```  1819   assumes "bij_betw f A B" "A \<noteq> {}" "finite A"
```
```  1820   shows   "map_pmf f (pmf_of_set A) = pmf_of_set B"
```
```  1821 proof -
```
```  1822   have "map_pmf f (pmf_of_set A) = pmf_of_set (f ` A)"
```
```  1823     by (intro map_pmf_of_set_inj assms bij_betw_imp_inj_on[OF assms(1)])
```
```  1824   also from assms have "f ` A = B" by (simp add: bij_betw_def)
```
```  1825   finally show ?thesis .
```
```  1826 qed
```
```  1827
```
```  1828 text \<open>
```
```  1829   Choosing an element uniformly at random from the union of a disjoint family
```
```  1830   of finite non-empty sets with the same size is the same as first choosing a set
```
```  1831   from the family uniformly at random and then choosing an element from the chosen set
```
```  1832   uniformly at random.
```
```  1833 \<close>
```
```  1834 lemma pmf_of_set_UN:
```
```  1835   assumes "finite (UNION A f)" "A \<noteq> {}" "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> {}"
```
```  1836           "\<And>x. x \<in> A \<Longrightarrow> card (f x) = n" "disjoint_family_on f A"
```
```  1837   shows   "pmf_of_set (UNION A f) = do {x \<leftarrow> pmf_of_set A; pmf_of_set (f x)}"
```
```  1838             (is "?lhs = ?rhs")
```
```  1839 proof (intro pmf_eqI)
```
```  1840   fix x
```
```  1841   from assms have [simp]: "finite A"
```
```  1842     using infinite_disjoint_family_imp_infinite_UNION[of A f] by blast
```
```  1843   from assms have "ereal (pmf (pmf_of_set (UNION A f)) x) =
```
```  1844     ereal (indicator (\<Union>x\<in>A. f x) x / real (card (\<Union>x\<in>A. f x)))"
```
```  1845     by (subst pmf_of_set) auto
```
```  1846   also from assms have "card (\<Union>x\<in>A. f x) = card A * n"
```
```  1847     by (subst card_UN_disjoint) (auto simp: disjoint_family_on_def)
```
```  1848   also from assms
```
```  1849     have "indicator (\<Union>x\<in>A. f x) x / real \<dots> =
```
```  1850               indicator (\<Union>x\<in>A. f x) x / (n * real (card A))"
```
```  1851       by (simp add: sum_divide_distrib [symmetric] mult_ac)
```
```  1852   also from assms have "indicator (\<Union>x\<in>A. f x) x = (\<Sum>y\<in>A. indicator (f y) x)"
```
```  1853     by (intro indicator_UN_disjoint) simp_all
```
```  1854   also from assms have "ereal ((\<Sum>y\<in>A. indicator (f y) x) / (real n * real (card A))) =
```
```  1855                           ereal (pmf ?rhs x)"
```
```  1856     by (subst pmf_bind_pmf_of_set) (simp_all add: sum_divide_distrib)
```
```  1857   finally show "pmf ?lhs x = pmf ?rhs x" by simp
```
```  1858 qed
```
```  1859
```
```  1860 lemma bernoulli_pmf_half_conv_pmf_of_set: "bernoulli_pmf (1 / 2) = pmf_of_set UNIV"
```
```  1861   by (rule pmf_eqI) simp_all
```
```  1862
```
```  1863 subsubsection \<open> Poisson Distribution \<close>
```
```  1864
```
```  1865 context
```
```  1866   fixes rate :: real assumes rate_pos: "0 < rate"
```
```  1867 begin
```
```  1868
```
```  1869 lift_definition poisson_pmf :: "nat pmf" is "\<lambda>k. rate ^ k / fact k * exp (-rate)"
```
```  1870 proof  (* by Manuel Eberl *)
```
```  1871   have summable: "summable (\<lambda>x::nat. rate ^ x / fact x)" using summable_exp
```
```  1872     by (simp add: field_simps divide_inverse [symmetric])
```
```  1873   have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x * exp (-rate) \<partial>count_space UNIV) =
```
```  1874           exp (-rate) * (\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV)"
```
```  1875     by (simp add: field_simps nn_integral_cmult[symmetric] ennreal_mult'[symmetric])
```
```  1876   also from rate_pos have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV) = (\<Sum>x. rate ^ x / fact x)"
```
```  1877     by (simp_all add: nn_integral_count_space_nat suminf_ennreal summable ennreal_suminf_neq_top)
```
```  1878   also have "... = exp rate" unfolding exp_def
```
```  1879     by (simp add: field_simps divide_inverse [symmetric])
```
```  1880   also have "ennreal (exp (-rate)) * ennreal (exp rate) = 1"
```
```  1881     by (simp add: mult_exp_exp ennreal_mult[symmetric])
```
```  1882   finally show "(\<integral>\<^sup>+ x. ennreal (rate ^ x / (fact x) * exp (- rate)) \<partial>count_space UNIV) = 1" .
```
```  1883 qed (simp add: rate_pos[THEN less_imp_le])
```
```  1884
```
```  1885 lemma pmf_poisson[simp]: "pmf poisson_pmf k = rate ^ k / fact k * exp (-rate)"
```
```  1886   by transfer rule
```
```  1887
```
```  1888 lemma set_pmf_poisson[simp]: "set_pmf poisson_pmf = UNIV"
```
```  1889   using rate_pos by (auto simp: set_pmf_iff)
```
```  1890
```
```  1891 end
```
```  1892
```
```  1893 subsubsection \<open> Binomial Distribution \<close>
```
```  1894
```
```  1895 context
```
```  1896   fixes n :: nat and p :: real assumes p_nonneg: "0 \<le> p" and p_le_1: "p \<le> 1"
```
```  1897 begin
```
```  1898
```
```  1899 lift_definition binomial_pmf :: "nat pmf" is "\<lambda>k. (n choose k) * p^k * (1 - p)^(n - k)"
```
```  1900 proof
```
```  1901   have "(\<integral>\<^sup>+k. ennreal (real (n choose k) * p ^ k * (1 - p) ^ (n - k)) \<partial>count_space UNIV) =
```
```  1902     ennreal (\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k))"
```
```  1903     using p_le_1 p_nonneg by (subst nn_integral_count_space') auto
```
```  1904   also have "(\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k)) = (p + (1 - p)) ^ n"
```
```  1905     by (subst binomial_ring) (simp add: atLeast0AtMost)
```
```  1906   finally show "(\<integral>\<^sup>+ x. ennreal (real (n choose x) * p ^ x * (1 - p) ^ (n - x)) \<partial>count_space UNIV) = 1"
```
```  1907     by simp
```
```  1908 qed (insert p_nonneg p_le_1, simp)
```
```  1909
```
```  1910 lemma pmf_binomial[simp]: "pmf binomial_pmf k = (n choose k) * p^k * (1 - p)^(n - k)"
```
```  1911   by transfer rule
```
```  1912
```
```  1913 lemma set_pmf_binomial_eq: "set_pmf binomial_pmf = (if p = 0 then {0} else if p = 1 then {n} else {.. n})"
```
```  1914   using p_nonneg p_le_1 unfolding set_eq_iff set_pmf_iff pmf_binomial by (auto simp: set_pmf_iff)
```
```  1915
```
```  1916 end
```
```  1917
```
```  1918 end
```
```  1919
```
```  1920 lemma set_pmf_binomial_0[simp]: "set_pmf (binomial_pmf n 0) = {0}"
```
```  1921   by (simp add: set_pmf_binomial_eq)
```
```  1922
```
```  1923 lemma set_pmf_binomial_1[simp]: "set_pmf (binomial_pmf n 1) = {n}"
```
```  1924   by (simp add: set_pmf_binomial_eq)
```
```  1925
```
```  1926 lemma set_pmf_binomial[simp]: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (binomial_pmf n p) = {..n}"
```
```  1927   by (simp add: set_pmf_binomial_eq)
```
```  1928
```
```  1929 context includes lifting_syntax
```
```  1930 begin
```
```  1931
```
```  1932 lemma bind_pmf_parametric [transfer_rule]:
```
```  1933   "(rel_pmf A ===> (A ===> rel_pmf B) ===> rel_pmf B) bind_pmf bind_pmf"
```
```  1934 by(blast intro: rel_pmf_bindI dest: rel_funD)
```
```  1935
```
```  1936 lemma return_pmf_parametric [transfer_rule]: "(A ===> rel_pmf A) return_pmf return_pmf"
```
```  1937 by(rule rel_funI) simp
```
```  1938
```
```  1939 end
```
```  1940
```
```  1941
```
```  1942 primrec replicate_pmf :: "nat \<Rightarrow> 'a pmf \<Rightarrow> 'a list pmf" where
```
```  1943   "replicate_pmf 0 _ = return_pmf []"
```
```  1944 | "replicate_pmf (Suc n) p = do {x \<leftarrow> p; xs \<leftarrow> replicate_pmf n p; return_pmf (x#xs)}"
```
```  1945
```
```  1946 lemma replicate_pmf_1: "replicate_pmf 1 p = map_pmf (\<lambda>x. [x]) p"
```
```  1947   by (simp add: map_pmf_def bind_return_pmf)
```
```  1948
```
```  1949 lemma set_replicate_pmf:
```
```  1950   "set_pmf (replicate_pmf n p) = {xs\<in>lists (set_pmf p). length xs = n}"
```
```  1951   by (induction n) (auto simp: length_Suc_conv)
```
```  1952
```
```  1953 lemma replicate_pmf_distrib:
```
```  1954   "replicate_pmf (m + n) p =
```
```  1955      do {xs \<leftarrow> replicate_pmf m p; ys \<leftarrow> replicate_pmf n p; return_pmf (xs @ ys)}"
```
```  1956   by (induction m) (simp_all add: bind_return_pmf bind_return_pmf' bind_assoc_pmf)
```
```  1957
```
```  1958 lemma power_diff':
```
```  1959   assumes "b \<le> a"
```
```  1960   shows   "x ^ (a - b) = (if x = 0 \<and> a = b then 1 else x ^ a / (x::'a::field) ^ b)"
```
```  1961 proof (cases "x = 0")
```
```  1962   case True
```
```  1963   with assms show ?thesis by (cases "a - b") simp_all
```
```  1964 qed (insert assms, simp_all add: power_diff)
```
```  1965
```
```  1966
```
```  1967 lemma binomial_pmf_Suc:
```
```  1968   assumes "p \<in> {0..1}"
```
```  1969   shows   "binomial_pmf (Suc n) p =
```
```  1970              do {b \<leftarrow> bernoulli_pmf p;
```
```  1971                  k \<leftarrow> binomial_pmf n p;
```
```  1972                  return_pmf ((if b then 1 else 0) + k)}" (is "_ = ?rhs")
```
```  1973 proof (intro pmf_eqI)
```
```  1974   fix k
```
```  1975   have A: "indicator {Suc a} (Suc b) = indicator {a} b" for a b
```
```  1976     by (simp add: indicator_def)
```
```  1977   show "pmf (binomial_pmf (Suc n) p) k = pmf ?rhs k"
```
```  1978     by (cases k; cases "k > n")
```
```  1979        (insert assms, auto simp: pmf_bind measure_pmf_single A divide_simps algebra_simps
```
```  1980           not_less less_eq_Suc_le [symmetric] power_diff')
```
```  1981 qed
```
```  1982
```
```  1983 lemma binomial_pmf_0: "p \<in> {0..1} \<Longrightarrow> binomial_pmf 0 p = return_pmf 0"
```
```  1984   by (rule pmf_eqI) (simp_all add: indicator_def)
```
```  1985
```
```  1986 lemma binomial_pmf_altdef:
```
```  1987   assumes "p \<in> {0..1}"
```
```  1988   shows   "binomial_pmf n p = map_pmf (length \<circ> filter id) (replicate_pmf n (bernoulli_pmf p))"
```
```  1989   by (induction n)
```
```  1990      (insert assms, auto simp: binomial_pmf_Suc map_pmf_def bind_return_pmf bind_assoc_pmf
```
```  1991         bind_return_pmf' binomial_pmf_0 intro!: bind_pmf_cong)
```
```  1992
```
```  1993
```
```  1994 subsection \<open>PMFs from assiciation lists\<close>
```
```  1995
```
```  1996 definition pmf_of_list ::" ('a \<times> real) list \<Rightarrow> 'a pmf" where
```
```  1997   "pmf_of_list xs = embed_pmf (\<lambda>x. sum_list (map snd (filter (\<lambda>z. fst z = x) xs)))"
```
```  1998
```
```  1999 definition pmf_of_list_wf where
```
```  2000   "pmf_of_list_wf xs \<longleftrightarrow> (\<forall>x\<in>set (map snd xs) . x \<ge> 0) \<and> sum_list (map snd xs) = 1"
```
```  2001
```
```  2002 lemma pmf_of_list_wfI:
```
```  2003   "(\<And>x. x \<in> set (map snd xs) \<Longrightarrow> x \<ge> 0) \<Longrightarrow> sum_list (map snd xs) = 1 \<Longrightarrow> pmf_of_list_wf xs"
```
```  2004   unfolding pmf_of_list_wf_def by simp
```
```  2005
```
```  2006 context
```
```  2007 begin
```
```  2008
```
```  2009 private lemma pmf_of_list_aux:
```
```  2010   assumes "\<And>x. x \<in> set (map snd xs) \<Longrightarrow> x \<ge> 0"
```
```  2011   assumes "sum_list (map snd xs) = 1"
```
```  2012   shows "(\<integral>\<^sup>+ x. ennreal (sum_list (map snd [z\<leftarrow>xs . fst z = x])) \<partial>count_space UNIV) = 1"
```
```  2013 proof -
```
```  2014   have "(\<integral>\<^sup>+ x. ennreal (sum_list (map snd (filter (\<lambda>z. fst z = x) xs))) \<partial>count_space UNIV) =
```
```  2015             (\<integral>\<^sup>+ x. ennreal (sum_list (map (\<lambda>(x',p). indicator {x'} x * p) xs)) \<partial>count_space UNIV)"
```
```  2016     by (intro nn_integral_cong ennreal_cong, subst sum_list_map_filter') (auto intro: sum_list_cong)
```
```  2017   also have "\<dots> = (\<Sum>(x',p)\<leftarrow>xs. (\<integral>\<^sup>+ x. ennreal (indicator {x'} x * p) \<partial>count_space UNIV))"
```
```  2018     using assms(1)
```
```  2019   proof (induction xs)
```
```  2020     case (Cons x xs)
```
```  2021     from Cons.prems have "snd x \<ge> 0" by simp
```
```  2022     moreover have "b \<ge> 0" if "(a,b) \<in> set xs" for a b
```
```  2023       using Cons.prems[of b] that by force
```
```  2024     ultimately have "(\<integral>\<^sup>+ y. ennreal (\<Sum>(x', p)\<leftarrow>x # xs. indicator {x'} y * p) \<partial>count_space UNIV) =
```
```  2025             (\<integral>\<^sup>+ y. ennreal (indicator {fst x} y * snd x) +
```
```  2026             ennreal (\<Sum>(x', p)\<leftarrow>xs. indicator {x'} y * p) \<partial>count_space UNIV)"
```
```  2027       by (intro nn_integral_cong, subst ennreal_plus [symmetric])
```
```  2028          (auto simp: case_prod_unfold indicator_def intro!: sum_list_nonneg)
```
```  2029     also have "\<dots> = (\<integral>\<^sup>+ y. ennreal (indicator {fst x} y * snd x) \<partial>count_space UNIV) +
```
```  2030                       (\<integral>\<^sup>+ y. ennreal (\<Sum>(x', p)\<leftarrow>xs. indicator {x'} y * p) \<partial>count_space UNIV)"
```
```  2031       by (intro nn_integral_add)
```
```  2032          (force intro!: sum_list_nonneg AE_I2 intro: Cons simp: indicator_def)+
```
```  2033     also have "(\<integral>\<^sup>+ y. ennreal (\<Sum>(x', p)\<leftarrow>xs. indicator {x'} y * p) \<partial>count_space UNIV) =
```
```  2034                (\<Sum>(x', p)\<leftarrow>xs. (\<integral>\<^sup>+ y. ennreal (indicator {x'} y * p) \<partial>count_space UNIV))"
```
```  2035       using Cons(1) by (intro Cons) simp_all
```
```  2036     finally show ?case by (simp add: case_prod_unfold)
```
```  2037   qed simp
```
```  2038   also have "\<dots> = (\<Sum>(x',p)\<leftarrow>xs. ennreal p * (\<integral>\<^sup>+ x. indicator {x'} x \<partial>count_space UNIV))"
```
```  2039     using assms(1)
```
```  2040     by (intro sum_list_cong, simp only: case_prod_unfold, subst nn_integral_cmult [symmetric])
```
```  2041        (auto intro!: assms(1) simp: max_def times_ereal.simps [symmetric] mult_ac ereal_indicator
```
```  2042              simp del: times_ereal.simps)+
```
```  2043   also from assms have "\<dots> = sum_list (map snd xs)" by (simp add: case_prod_unfold sum_list_ennreal)
```
```  2044   also have "\<dots> = 1" using assms(2) by simp
```
```  2045   finally show ?thesis .
```
```  2046 qed
```
```  2047
```
```  2048 lemma pmf_pmf_of_list:
```
```  2049   assumes "pmf_of_list_wf xs"
```
```  2050   shows   "pmf (pmf_of_list xs) x = sum_list (map snd (filter (\<lambda>z. fst z = x) xs))"
```
```  2051   using assms pmf_of_list_aux[of xs] unfolding pmf_of_list_def pmf_of_list_wf_def
```
```  2052   by (subst pmf_embed_pmf) (auto intro!: sum_list_nonneg)
```
```  2053
```
```  2054 end
```
```  2055
```
```  2056 lemma set_pmf_of_list:
```
```  2057   assumes "pmf_of_list_wf xs"
```
```  2058   shows   "set_pmf (pmf_of_list xs) \<subseteq> set (map fst xs)"
```
```  2059 proof clarify
```
```  2060   fix x assume A: "x \<in> set_pmf (pmf_of_list xs)"
```
```  2061   show "x \<in> set (map fst xs)"
```
```  2062   proof (rule ccontr)
```
```  2063     assume "x \<notin> set (map fst xs)"
```
```  2064     hence "[z\<leftarrow>xs . fst z = x] = []" by (auto simp: filter_empty_conv)
```
```  2065     with A assms show False by (simp add: pmf_pmf_of_list set_pmf_eq)
```
```  2066   qed
```
```  2067 qed
```
```  2068
```
```  2069 lemma finite_set_pmf_of_list:
```
```  2070   assumes "pmf_of_list_wf xs"
```
```  2071   shows   "finite (set_pmf (pmf_of_list xs))"
```
```  2072   using assms by (rule finite_subset[OF set_pmf_of_list]) simp_all
```
```  2073
```
```  2074 lemma emeasure_Int_set_pmf:
```
```  2075   "emeasure (measure_pmf p) (A \<inter> set_pmf p) = emeasure (measure_pmf p) A"
```
```  2076   by (rule emeasure_eq_AE) (auto simp: AE_measure_pmf_iff)
```
```  2077
```
```  2078 lemma measure_Int_set_pmf:
```
```  2079   "measure (measure_pmf p) (A \<inter> set_pmf p) = measure (measure_pmf p) A"
```
```  2080   using emeasure_Int_set_pmf[of p A] by (simp add: Sigma_Algebra.measure_def)
```
```  2081
```
```  2082 lemma emeasure_pmf_of_list:
```
```  2083   assumes "pmf_of_list_wf xs"
```
```  2084   shows   "emeasure (pmf_of_list xs) A = ennreal (sum_list (map snd (filter (\<lambda>x. fst x \<in> A) xs)))"
```
```  2085 proof -
```
```  2086   have "emeasure (pmf_of_list xs) A = nn_integral (measure_pmf (pmf_of_list xs)) (indicator A)"
```
```  2087     by simp
```
```  2088   also from assms
```
```  2089     have "\<dots> = (\<Sum>x\<in>set_pmf (pmf_of_list xs) \<inter> A. ennreal (sum_list (map snd [z\<leftarrow>xs . fst z = x])))"
```
```  2090     by (subst nn_integral_measure_pmf_finite) (simp_all add: finite_set_pmf_of_list pmf_pmf_of_list Int_def)
```
```  2091   also from assms
```
```  2092     have "\<dots> = ennreal (\<Sum>x\<in>set_pmf (pmf_of_list xs) \<inter> A. sum_list (map snd [z\<leftarrow>xs . fst z = x]))"
```
```  2093     by (subst sum_ennreal) (auto simp: pmf_of_list_wf_def intro!: sum_list_nonneg)
```
```  2094   also have "\<dots> = ennreal (\<Sum>x\<in>set_pmf (pmf_of_list xs) \<inter> A.
```
```  2095       indicator A x * pmf (pmf_of_list xs) x)" (is "_ = ennreal ?S")
```
```  2096     using assms by (intro ennreal_cong sum.cong) (auto simp: pmf_pmf_of_list)
```
```  2097   also have "?S = (\<Sum>x\<in>set_pmf (pmf_of_list xs). indicator A x * pmf (pmf_of_list xs) x)"
```
```  2098     using assms by (intro sum.mono_neutral_left set_pmf_of_list finite_set_pmf_of_list) auto
```
```  2099   also have "\<dots> = (\<Sum>x\<in>set (map fst xs). indicator A x * pmf (pmf_of_list xs) x)"
```
```  2100     using assms by (intro sum.mono_neutral_left set_pmf_of_list) (auto simp: set_pmf_eq)
```
```  2101   also have "\<dots> = (\<Sum>x\<in>set (map fst xs). indicator A x *
```
```  2102                       sum_list (map snd (filter (\<lambda>z. fst z = x) xs)))"
```
```  2103     using assms by (simp add: pmf_pmf_of_list)
```
```  2104   also have "\<dots> = (\<Sum>x\<in>set (map fst xs). sum_list (map snd (filter (\<lambda>z. fst z = x \<and> x \<in> A) xs)))"
```
```  2105     by (intro sum.cong) (auto simp: indicator_def)
```
```  2106   also have "\<dots> = (\<Sum>x\<in>set (map fst xs). (\<Sum>xa = 0..<length xs.
```
```  2107                      if fst (xs ! xa) = x \<and> x \<in> A then snd (xs ! xa) else 0))"
```
```  2108     by (intro sum.cong refl, subst sum_list_map_filter', subst sum_list_sum_nth) simp
```
```  2109   also have "\<dots> = (\<Sum>xa = 0..<length xs. (\<Sum>x\<in>set (map fst xs).
```
```  2110                      if fst (xs ! xa) = x \<and> x \<in> A then snd (xs ! xa) else 0))"
```
```  2111     by (rule sum.commute)
```
```  2112   also have "\<dots> = (\<Sum>xa = 0..<length xs. if fst (xs ! xa) \<in> A then
```
```  2113                      (\<Sum>x\<in>set (map fst xs). if x = fst (xs ! xa) then snd (xs ! xa) else 0) else 0)"
```
```  2114     by (auto intro!: sum.cong sum.neutral)
```
```  2115   also have "\<dots> = (\<Sum>xa = 0..<length xs. if fst (xs ! xa) \<in> A then snd (xs ! xa) else 0)"
```
```  2116     by (intro sum.cong refl) (simp_all add: sum.delta)
```
```  2117   also have "\<dots> = sum_list (map snd (filter (\<lambda>x. fst x \<in> A) xs))"
```
```  2118     by (subst sum_list_map_filter', subst sum_list_sum_nth) simp_all
```
```  2119   finally show ?thesis .
```
```  2120 qed
```
```  2121
```
```  2122 lemma measure_pmf_of_list:
```
```  2123   assumes "pmf_of_list_wf xs"
```
```  2124   shows   "measure (pmf_of_list xs) A = sum_list (map snd (filter (\<lambda>x. fst x \<in> A) xs))"
```
```  2125   using assms unfolding pmf_of_list_wf_def Sigma_Algebra.measure_def
```
```  2126   by (subst emeasure_pmf_of_list [OF assms], subst enn2real_ennreal) (auto intro!: sum_list_nonneg)
```
```  2127
```
```  2128 (* TODO Move? *)
```
```  2129 lemma sum_list_nonneg_eq_zero_iff:
```
```  2130   fixes xs :: "'a :: linordered_ab_group_add list"
```
```  2131   shows "(\<And>x. x \<in> set xs \<Longrightarrow> x \<ge> 0) \<Longrightarrow> sum_list xs = 0 \<longleftrightarrow> set xs \<subseteq> {0}"
```
```  2132 proof (induction xs)
```
```  2133   case (Cons x xs)
```
```  2134   from Cons.prems have "sum_list (x#xs) = 0 \<longleftrightarrow> x = 0 \<and> sum_list xs = 0"
```
```  2135     unfolding sum_list_simps by (subst add_nonneg_eq_0_iff) (auto intro: sum_list_nonneg)
```
```  2136   with Cons.IH Cons.prems show ?case by simp
```
```  2137 qed simp_all
```
```  2138
```
```  2139 lemma sum_list_filter_nonzero:
```
```  2140   "sum_list (filter (\<lambda>x. x \<noteq> 0) xs) = sum_list xs"
```
```  2141   by (induction xs) simp_all
```
```  2142 (* END MOVE *)
```
```  2143
```
```  2144 lemma set_pmf_of_list_eq:
```
```  2145   assumes "pmf_of_list_wf xs" "\<And>x. x \<in> snd ` set xs \<Longrightarrow> x > 0"
```
```  2146   shows   "set_pmf (pmf_of_list xs) = fst ` set xs"
```
```  2147 proof
```
```  2148   {
```
```  2149     fix x assume A: "x \<in> fst ` set xs" and B: "x \<notin> set_pmf (pmf_of_list xs)"
```
```  2150     then obtain y where y: "(x, y) \<in> set xs" by auto
```
```  2151     from B have "sum_list (map snd [z\<leftarrow>xs. fst z = x]) = 0"
```
```  2152       by (simp add: pmf_pmf_of_list[OF assms(1)] set_pmf_eq)
```
```  2153     moreover from y have "y \<in> snd ` {xa \<in> set xs. fst xa = x}" by force
```
```  2154     ultimately have "y = 0" using assms(1)
```
```  2155       by (subst (asm) sum_list_nonneg_eq_zero_iff) (auto simp: pmf_of_list_wf_def)
```
```  2156     with assms(2) y have False by force
```
```  2157   }
```
```  2158   thus "fst ` set xs \<subseteq> set_pmf (pmf_of_list xs)" by blast
```
```  2159 qed (insert set_pmf_of_list[OF assms(1)], simp_all)
```
```  2160
```
```  2161 lemma pmf_of_list_remove_zeros:
```
```  2162   assumes "pmf_of_list_wf xs"
```
```  2163   defines "xs' \<equiv> filter (\<lambda>z. snd z \<noteq> 0) xs"
```
```  2164   shows   "pmf_of_list_wf xs'" "pmf_of_list xs' = pmf_of_list xs"
```
```  2165 proof -
```
```  2166   have "map snd [z\<leftarrow>xs . snd z \<noteq> 0] = filter (\<lambda>x. x \<noteq> 0) (map snd xs)"
```
```  2167     by (induction xs) simp_all
```
```  2168   with assms(1) show wf: "pmf_of_list_wf xs'"
```
```  2169     by (auto simp: pmf_of_list_wf_def xs'_def sum_list_filter_nonzero)
```
```  2170   have "sum_list (map snd [z\<leftarrow>xs' . fst z = i]) = sum_list (map snd [z\<leftarrow>xs . fst z = i])" for i
```
```  2171     unfolding xs'_def by (induction xs) simp_all
```
```  2172   with assms(1) wf show "pmf_of_list xs' = pmf_of_list xs"
```
```  2173     by (intro pmf_eqI) (simp_all add: pmf_pmf_of_list)
```
```  2174 qed
```
```  2175
```
```  2176 end
```