src/HOL/Probability/Probability_Mass_Function.thy
 author hoelzl Fri Feb 19 13:40:50 2016 +0100 (2016-02-19) changeset 62378 85ed00c1fe7c parent 62324 ae44f16dcea5 child 62390 842917225d56 permissions -rw-r--r--
generalize more theorems to support enat and ennreal
```     1 (*  Title:      HOL/Probability/Probability_Mass_Function.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3     Author:     Andreas Lochbihler, ETH Zurich
```
```     4 *)
```
```     5
```
```     6 section \<open> Probability mass function \<close>
```
```     7
```
```     8 theory Probability_Mass_Function
```
```     9 imports
```
```    10   Giry_Monad
```
```    11   "~~/src/HOL/Library/Multiset"
```
```    12 begin
```
```    13
```
```    14 lemma AE_emeasure_singleton:
```
```    15   assumes x: "emeasure M {x} \<noteq> 0" and ae: "AE x in M. P x" shows "P x"
```
```    16 proof -
```
```    17   from x have x_M: "{x} \<in> sets M"
```
```    18     by (auto intro: emeasure_notin_sets)
```
```    19   from ae obtain N where N: "{x\<in>space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"
```
```    20     by (auto elim: AE_E)
```
```    21   { assume "\<not> P x"
```
```    22     with x_M[THEN sets.sets_into_space] N have "emeasure M {x} \<le> emeasure M N"
```
```    23       by (intro emeasure_mono) auto
```
```    24     with x N have False
```
```    25       by (auto simp: emeasure_le_0_iff) }
```
```    26   then show "P x" by auto
```
```    27 qed
```
```    28
```
```    29 lemma AE_measure_singleton: "measure M {x} \<noteq> 0 \<Longrightarrow> AE x in M. P x \<Longrightarrow> P x"
```
```    30   by (metis AE_emeasure_singleton measure_def emeasure_empty measure_empty)
```
```    31
```
```    32 lemma ereal_divide': "b \<noteq> 0 \<Longrightarrow> ereal (a / b) = ereal a / ereal b"
```
```    33   using ereal_divide[of a b] by simp
```
```    34
```
```    35 lemma (in finite_measure) AE_support_countable:
```
```    36   assumes [simp]: "sets M = UNIV"
```
```    37   shows "(AE x in M. measure M {x} \<noteq> 0) \<longleftrightarrow> (\<exists>S. countable S \<and> (AE x in M. x \<in> S))"
```
```    38 proof
```
```    39   assume "\<exists>S. countable S \<and> (AE x in M. x \<in> S)"
```
```    40   then obtain S where S[intro]: "countable S" and ae: "AE x in M. x \<in> S"
```
```    41     by auto
```
```    42   then have "emeasure M (\<Union>x\<in>{x\<in>S. emeasure M {x} \<noteq> 0}. {x}) =
```
```    43     (\<integral>\<^sup>+ x. emeasure M {x} * indicator {x\<in>S. emeasure M {x} \<noteq> 0} x \<partial>count_space UNIV)"
```
```    44     by (subst emeasure_UN_countable)
```
```    45        (auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space)
```
```    46   also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} * indicator S x \<partial>count_space UNIV)"
```
```    47     by (auto intro!: nn_integral_cong split: split_indicator)
```
```    48   also have "\<dots> = emeasure M (\<Union>x\<in>S. {x})"
```
```    49     by (subst emeasure_UN_countable)
```
```    50        (auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space)
```
```    51   also have "\<dots> = emeasure M (space M)"
```
```    52     using ae by (intro emeasure_eq_AE) auto
```
```    53   finally have "emeasure M {x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0} = emeasure M (space M)"
```
```    54     by (simp add: emeasure_single_in_space cong: rev_conj_cong)
```
```    55   with finite_measure_compl[of "{x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0}"]
```
```    56   have "AE x in M. x \<in> S \<and> emeasure M {x} \<noteq> 0"
```
```    57     by (intro AE_I[OF order_refl]) (auto simp: emeasure_eq_measure set_diff_eq cong: conj_cong)
```
```    58   then show "AE x in M. measure M {x} \<noteq> 0"
```
```    59     by (auto simp: emeasure_eq_measure)
```
```    60 qed (auto intro!: exI[of _ "{x. measure M {x} \<noteq> 0}"] countable_support)
```
```    61
```
```    62 subsection \<open> PMF as measure \<close>
```
```    63
```
```    64 typedef 'a pmf = "{M :: 'a measure. prob_space M \<and> sets M = UNIV \<and> (AE x in M. measure M {x} \<noteq> 0)}"
```
```    65   morphisms measure_pmf Abs_pmf
```
```    66   by (intro exI[of _ "uniform_measure (count_space UNIV) {undefined}"])
```
```    67      (auto intro!: prob_space_uniform_measure AE_uniform_measureI)
```
```    68
```
```    69 declare [[coercion measure_pmf]]
```
```    70
```
```    71 lemma prob_space_measure_pmf: "prob_space (measure_pmf p)"
```
```    72   using pmf.measure_pmf[of p] by auto
```
```    73
```
```    74 interpretation measure_pmf: prob_space "measure_pmf M" for M
```
```    75   by (rule prob_space_measure_pmf)
```
```    76
```
```    77 interpretation measure_pmf: subprob_space "measure_pmf M" for M
```
```    78   by (rule prob_space_imp_subprob_space) unfold_locales
```
```    79
```
```    80 lemma subprob_space_measure_pmf: "subprob_space (measure_pmf x)"
```
```    81   by unfold_locales
```
```    82
```
```    83 locale pmf_as_measure
```
```    84 begin
```
```    85
```
```    86 setup_lifting type_definition_pmf
```
```    87
```
```    88 end
```
```    89
```
```    90 context
```
```    91 begin
```
```    92
```
```    93 interpretation pmf_as_measure .
```
```    94
```
```    95 lemma sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV"
```
```    96   by transfer blast
```
```    97
```
```    98 lemma sets_measure_pmf_count_space[measurable_cong]:
```
```    99   "sets (measure_pmf M) = sets (count_space UNIV)"
```
```   100   by simp
```
```   101
```
```   102 lemma space_measure_pmf[simp]: "space (measure_pmf p) = UNIV"
```
```   103   using sets_eq_imp_space_eq[of "measure_pmf p" "count_space UNIV"] by simp
```
```   104
```
```   105 lemma measure_pmf_UNIV [simp]: "measure (measure_pmf p) UNIV = 1"
```
```   106 using measure_pmf.prob_space[of p] by simp
```
```   107
```
```   108 lemma measure_pmf_in_subprob_algebra[measurable (raw)]: "measure_pmf x \<in> space (subprob_algebra (count_space UNIV))"
```
```   109   by (simp add: space_subprob_algebra subprob_space_measure_pmf)
```
```   110
```
```   111 lemma measurable_pmf_measure1[simp]: "measurable (M :: 'a pmf) N = UNIV \<rightarrow> space N"
```
```   112   by (auto simp: measurable_def)
```
```   113
```
```   114 lemma measurable_pmf_measure2[simp]: "measurable N (M :: 'a pmf) = measurable N (count_space UNIV)"
```
```   115   by (intro measurable_cong_sets) simp_all
```
```   116
```
```   117 lemma measurable_pair_restrict_pmf2:
```
```   118   assumes "countable A"
```
```   119   assumes [measurable]: "\<And>y. y \<in> A \<Longrightarrow> (\<lambda>x. f (x, y)) \<in> measurable M L"
```
```   120   shows "f \<in> measurable (M \<Otimes>\<^sub>M restrict_space (measure_pmf N) A) L" (is "f \<in> measurable ?M _")
```
```   121 proof -
```
```   122   have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)"
```
```   123     by (simp add: restrict_count_space)
```
```   124
```
```   125   show ?thesis
```
```   126     by (intro measurable_compose_countable'[where f="\<lambda>a b. f (fst b, a)" and g=snd and I=A,
```
```   127                                             unfolded prod.collapse] assms)
```
```   128         measurable
```
```   129 qed
```
```   130
```
```   131 lemma measurable_pair_restrict_pmf1:
```
```   132   assumes "countable A"
```
```   133   assumes [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N L"
```
```   134   shows "f \<in> measurable (restrict_space (measure_pmf M) A \<Otimes>\<^sub>M N) L"
```
```   135 proof -
```
```   136   have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)"
```
```   137     by (simp add: restrict_count_space)
```
```   138
```
```   139   show ?thesis
```
```   140     by (intro measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A,
```
```   141                                             unfolded prod.collapse] assms)
```
```   142         measurable
```
```   143 qed
```
```   144
```
```   145 lift_definition pmf :: "'a pmf \<Rightarrow> 'a \<Rightarrow> real" is "\<lambda>M x. measure M {x}" .
```
```   146
```
```   147 lift_definition set_pmf :: "'a pmf \<Rightarrow> 'a set" is "\<lambda>M. {x. measure M {x} \<noteq> 0}" .
```
```   148 declare [[coercion set_pmf]]
```
```   149
```
```   150 lemma AE_measure_pmf: "AE x in (M::'a pmf). x \<in> M"
```
```   151   by transfer simp
```
```   152
```
```   153 lemma emeasure_pmf_single_eq_zero_iff:
```
```   154   fixes M :: "'a pmf"
```
```   155   shows "emeasure M {y} = 0 \<longleftrightarrow> y \<notin> M"
```
```   156   by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
```
```   157
```
```   158 lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \<longleftrightarrow> (\<forall>y\<in>M. P y)"
```
```   159   using AE_measure_singleton[of M] AE_measure_pmf[of M]
```
```   160   by (auto simp: set_pmf.rep_eq)
```
```   161
```
```   162 lemma AE_pmfI: "(\<And>y. y \<in> set_pmf M \<Longrightarrow> P y) \<Longrightarrow> almost_everywhere (measure_pmf M) P"
```
```   163 by(simp add: AE_measure_pmf_iff)
```
```   164
```
```   165 lemma countable_set_pmf [simp]: "countable (set_pmf p)"
```
```   166   by transfer (metis prob_space.finite_measure finite_measure.countable_support)
```
```   167
```
```   168 lemma pmf_positive: "x \<in> set_pmf p \<Longrightarrow> 0 < pmf p x"
```
```   169   by transfer (simp add: less_le measure_nonneg)
```
```   170
```
```   171 lemma pmf_nonneg: "0 \<le> pmf p x"
```
```   172   by transfer (simp add: measure_nonneg)
```
```   173
```
```   174 lemma pmf_le_1: "pmf p x \<le> 1"
```
```   175   by (simp add: pmf.rep_eq)
```
```   176
```
```   177 lemma set_pmf_not_empty: "set_pmf M \<noteq> {}"
```
```   178   using AE_measure_pmf[of M] by (intro notI) simp
```
```   179
```
```   180 lemma set_pmf_iff: "x \<in> set_pmf M \<longleftrightarrow> pmf M x \<noteq> 0"
```
```   181   by transfer simp
```
```   182
```
```   183 lemma set_pmf_eq: "set_pmf M = {x. pmf M x \<noteq> 0}"
```
```   184   by (auto simp: set_pmf_iff)
```
```   185
```
```   186 lemma emeasure_pmf_single:
```
```   187   fixes M :: "'a pmf"
```
```   188   shows "emeasure M {x} = pmf M x"
```
```   189   by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
```
```   190
```
```   191 lemma measure_pmf_single: "measure (measure_pmf M) {x} = pmf M x"
```
```   192 using emeasure_pmf_single[of M x] by(simp add: measure_pmf.emeasure_eq_measure)
```
```   193
```
```   194 lemma emeasure_measure_pmf_finite: "finite S \<Longrightarrow> emeasure (measure_pmf M) S = (\<Sum>s\<in>S. pmf M s)"
```
```   195   by (subst emeasure_eq_setsum_singleton) (auto simp: emeasure_pmf_single)
```
```   196
```
```   197 lemma measure_measure_pmf_finite: "finite S \<Longrightarrow> measure (measure_pmf M) S = setsum (pmf M) S"
```
```   198   using emeasure_measure_pmf_finite[of S M] by(simp add: measure_pmf.emeasure_eq_measure)
```
```   199
```
```   200 lemma nn_integral_measure_pmf_support:
```
```   201   fixes f :: "'a \<Rightarrow> ereal"
```
```   202   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"
```
```   203   shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>A. f x * pmf M x)"
```
```   204 proof -
```
```   205   have "(\<integral>\<^sup>+x. f x \<partial>M) = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)"
```
```   206     using nn by (intro nn_integral_cong_AE) (auto simp: AE_measure_pmf_iff split: split_indicator)
```
```   207   also have "\<dots> = (\<Sum>x\<in>A. f x * emeasure M {x})"
```
```   208     using assms by (intro nn_integral_indicator_finite) auto
```
```   209   finally show ?thesis
```
```   210     by (simp add: emeasure_measure_pmf_finite)
```
```   211 qed
```
```   212
```
```   213 lemma nn_integral_measure_pmf_finite:
```
```   214   fixes f :: "'a \<Rightarrow> ereal"
```
```   215   assumes f: "finite (set_pmf M)" and nn: "\<And>x. x \<in> set_pmf M \<Longrightarrow> 0 \<le> f x"
```
```   216   shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>set_pmf M. f x * pmf M x)"
```
```   217   using assms by (intro nn_integral_measure_pmf_support) auto
```
```   218 lemma integrable_measure_pmf_finite:
```
```   219   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
```
```   220   shows "finite (set_pmf M) \<Longrightarrow> integrable M f"
```
```   221   by (auto intro!: integrableI_bounded simp: nn_integral_measure_pmf_finite)
```
```   222
```
```   223 lemma integral_measure_pmf:
```
```   224   assumes [simp]: "finite A" and "\<And>a. a \<in> set_pmf M \<Longrightarrow> f a \<noteq> 0 \<Longrightarrow> a \<in> A"
```
```   225   shows "(\<integral>x. f x \<partial>measure_pmf M) = (\<Sum>a\<in>A. f a * pmf M a)"
```
```   226 proof -
```
```   227   have "(\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x * indicator A x \<partial>measure_pmf M)"
```
```   228     using assms(2) by (intro integral_cong_AE) (auto split: split_indicator simp: AE_measure_pmf_iff)
```
```   229   also have "\<dots> = (\<Sum>a\<in>A. f a * pmf M a)"
```
```   230     by (subst integral_indicator_finite_real) (auto simp: measure_def emeasure_measure_pmf_finite)
```
```   231   finally show ?thesis .
```
```   232 qed
```
```   233
```
```   234 lemma integrable_pmf: "integrable (count_space X) (pmf M)"
```
```   235 proof -
```
```   236   have " (\<integral>\<^sup>+ x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+ x. pmf M x \<partial>count_space (M \<inter> X))"
```
```   237     by (auto simp add: nn_integral_count_space_indicator set_pmf_iff intro!: nn_integral_cong split: split_indicator)
```
```   238   then have "integrable (count_space X) (pmf M) = integrable (count_space (M \<inter> X)) (pmf M)"
```
```   239     by (simp add: integrable_iff_bounded pmf_nonneg)
```
```   240   then show ?thesis
```
```   241     by (simp add: pmf.rep_eq measure_pmf.integrable_measure disjoint_family_on_def)
```
```   242 qed
```
```   243
```
```   244 lemma integral_pmf: "(\<integral>x. pmf M x \<partial>count_space X) = measure M X"
```
```   245 proof -
```
```   246   have "(\<integral>x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+x. pmf M x \<partial>count_space X)"
```
```   247     by (simp add: pmf_nonneg integrable_pmf nn_integral_eq_integral)
```
```   248   also have "\<dots> = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space (X \<inter> M))"
```
```   249     by (auto intro!: nn_integral_cong_AE split: split_indicator
```
```   250              simp: pmf.rep_eq measure_pmf.emeasure_eq_measure nn_integral_count_space_indicator
```
```   251                    AE_count_space set_pmf_iff)
```
```   252   also have "\<dots> = emeasure M (X \<inter> M)"
```
```   253     by (rule emeasure_countable_singleton[symmetric]) (auto intro: countable_set_pmf)
```
```   254   also have "\<dots> = emeasure M X"
```
```   255     by (auto intro!: emeasure_eq_AE simp: AE_measure_pmf_iff)
```
```   256   finally show ?thesis
```
```   257     by (simp add: measure_pmf.emeasure_eq_measure)
```
```   258 qed
```
```   259
```
```   260 lemma integral_pmf_restrict:
```
```   261   "(f::'a \<Rightarrow> 'b::{banach, second_countable_topology}) \<in> borel_measurable (count_space UNIV) \<Longrightarrow>
```
```   262     (\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x \<partial>restrict_space M M)"
```
```   263   by (auto intro!: integral_cong_AE simp add: integral_restrict_space AE_measure_pmf_iff)
```
```   264
```
```   265 lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1"
```
```   266 proof -
```
```   267   have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)"
```
```   268     by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf)
```
```   269   then show ?thesis
```
```   270     using measure_pmf.emeasure_space_1 by simp
```
```   271 qed
```
```   272
```
```   273 lemma emeasure_pmf_UNIV [simp]: "emeasure (measure_pmf M) UNIV = 1"
```
```   274 using measure_pmf.emeasure_space_1[of M] by simp
```
```   275
```
```   276 lemma in_null_sets_measure_pmfI:
```
```   277   "A \<inter> set_pmf p = {} \<Longrightarrow> A \<in> null_sets (measure_pmf p)"
```
```   278 using emeasure_eq_0_AE[where ?P="\<lambda>x. x \<in> A" and M="measure_pmf p"]
```
```   279 by(auto simp add: null_sets_def AE_measure_pmf_iff)
```
```   280
```
```   281 lemma measure_subprob: "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
```
```   282   by (simp add: space_subprob_algebra subprob_space_measure_pmf)
```
```   283
```
```   284 subsection \<open> Monad Interpretation \<close>
```
```   285
```
```   286 lemma measurable_measure_pmf[measurable]:
```
```   287   "(\<lambda>x. measure_pmf (M x)) \<in> measurable (count_space UNIV) (subprob_algebra (count_space UNIV))"
```
```   288   by (auto simp: space_subprob_algebra intro!: prob_space_imp_subprob_space) unfold_locales
```
```   289
```
```   290 lemma bind_measure_pmf_cong:
```
```   291   assumes "\<And>x. A x \<in> space (subprob_algebra N)" "\<And>x. B x \<in> space (subprob_algebra N)"
```
```   292   assumes "\<And>i. i \<in> set_pmf x \<Longrightarrow> A i = B i"
```
```   293   shows "bind (measure_pmf x) A = bind (measure_pmf x) B"
```
```   294 proof (rule measure_eqI)
```
```   295   show "sets (measure_pmf x \<bind> A) = sets (measure_pmf x \<bind> B)"
```
```   296     using assms by (subst (1 2) sets_bind) (auto simp: space_subprob_algebra)
```
```   297 next
```
```   298   fix X assume "X \<in> sets (measure_pmf x \<bind> A)"
```
```   299   then have X: "X \<in> sets N"
```
```   300     using assms by (subst (asm) sets_bind) (auto simp: space_subprob_algebra)
```
```   301   show "emeasure (measure_pmf x \<bind> A) X = emeasure (measure_pmf x \<bind> B) X"
```
```   302     using assms
```
```   303     by (subst (1 2) emeasure_bind[where N=N, OF _ _ X])
```
```   304        (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
```
```   305 qed
```
```   306
```
```   307 lift_definition bind_pmf :: "'a pmf \<Rightarrow> ('a \<Rightarrow> 'b pmf ) \<Rightarrow> 'b pmf" is bind
```
```   308 proof (clarify, intro conjI)
```
```   309   fix f :: "'a measure" and g :: "'a \<Rightarrow> 'b measure"
```
```   310   assume "prob_space f"
```
```   311   then interpret f: prob_space f .
```
```   312   assume "sets f = UNIV" and ae_f: "AE x in f. measure f {x} \<noteq> 0"
```
```   313   then have s_f[simp]: "sets f = sets (count_space UNIV)"
```
```   314     by simp
```
```   315   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)"
```
```   316   then have g: "\<And>x. prob_space (g x)" and s_g[simp]: "\<And>x. sets (g x) = sets (count_space UNIV)"
```
```   317     and ae_g: "\<And>x. AE y in g x. measure (g x) {y} \<noteq> 0"
```
```   318     by auto
```
```   319
```
```   320   have [measurable]: "g \<in> measurable f (subprob_algebra (count_space UNIV))"
```
```   321     by (auto simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space g)
```
```   322
```
```   323   show "prob_space (f \<bind> g)"
```
```   324     using g by (intro f.prob_space_bind[where S="count_space UNIV"]) auto
```
```   325   then interpret fg: prob_space "f \<bind> g" .
```
```   326   show [simp]: "sets (f \<bind> g) = UNIV"
```
```   327     using sets_eq_imp_space_eq[OF s_f]
```
```   328     by (subst sets_bind[where N="count_space UNIV"]) auto
```
```   329   show "AE x in f \<bind> g. measure (f \<bind> g) {x} \<noteq> 0"
```
```   330     apply (simp add: fg.prob_eq_0 AE_bind[where B="count_space UNIV"])
```
```   331     using ae_f
```
```   332     apply eventually_elim
```
```   333     using ae_g
```
```   334     apply eventually_elim
```
```   335     apply (auto dest: AE_measure_singleton)
```
```   336     done
```
```   337 qed
```
```   338
```
```   339 lemma ereal_pmf_bind: "pmf (bind_pmf N f) i = (\<integral>\<^sup>+x. pmf (f x) i \<partial>measure_pmf N)"
```
```   340   unfolding pmf.rep_eq bind_pmf.rep_eq
```
```   341   by (auto simp: measure_pmf.measure_bind[where N="count_space UNIV"] measure_subprob measure_nonneg
```
```   342            intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])
```
```   343
```
```   344 lemma pmf_bind: "pmf (bind_pmf N f) i = (\<integral>x. pmf (f x) i \<partial>measure_pmf N)"
```
```   345   using ereal_pmf_bind[of N f i]
```
```   346   by (subst (asm) nn_integral_eq_integral)
```
```   347      (auto simp: pmf_nonneg pmf_le_1
```
```   348            intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])
```
```   349
```
```   350 lemma bind_pmf_const[simp]: "bind_pmf M (\<lambda>x. c) = c"
```
```   351   by transfer (simp add: bind_const' prob_space_imp_subprob_space)
```
```   352
```
```   353 lemma set_bind_pmf[simp]: "set_pmf (bind_pmf M N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))"
```
```   354   unfolding set_pmf_eq ereal_eq_0(1)[symmetric] ereal_pmf_bind
```
```   355   by (auto simp add: nn_integral_0_iff_AE AE_measure_pmf_iff set_pmf_eq not_le less_le pmf_nonneg)
```
```   356
```
```   357 lemma bind_pmf_cong:
```
```   358   assumes "p = q"
```
```   359   shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g"
```
```   360   unfolding \<open>p = q\<close>[symmetric] measure_pmf_inject[symmetric] bind_pmf.rep_eq
```
```   361   by (auto simp: AE_measure_pmf_iff Pi_iff space_subprob_algebra subprob_space_measure_pmf
```
```   362                  sets_bind[where N="count_space UNIV"] emeasure_bind[where N="count_space UNIV"]
```
```   363            intro!: nn_integral_cong_AE measure_eqI)
```
```   364
```
```   365 lemma bind_pmf_cong_simp:
```
```   366   "p = q \<Longrightarrow> (\<And>x. x \<in> set_pmf q =simp=> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g"
```
```   367   by (simp add: simp_implies_def cong: bind_pmf_cong)
```
```   368
```
```   369 lemma measure_pmf_bind: "measure_pmf (bind_pmf M f) = (measure_pmf M \<bind> (\<lambda>x. measure_pmf (f x)))"
```
```   370   by transfer simp
```
```   371
```
```   372 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)"
```
```   373   using measurable_measure_pmf[of N]
```
```   374   unfolding measure_pmf_bind
```
```   375   apply (subst (1 3) nn_integral_max_0[symmetric])
```
```   376   apply (intro nn_integral_bind[where B="count_space UNIV"])
```
```   377   apply auto
```
```   378   done
```
```   379
```
```   380 lemma emeasure_bind_pmf[simp]: "emeasure (bind_pmf M N) X = (\<integral>\<^sup>+x. emeasure (N x) X \<partial>M)"
```
```   381   using measurable_measure_pmf[of N]
```
```   382   unfolding measure_pmf_bind
```
```   383   by (subst emeasure_bind[where N="count_space UNIV"]) auto
```
```   384
```
```   385 lift_definition return_pmf :: "'a \<Rightarrow> 'a pmf" is "return (count_space UNIV)"
```
```   386   by (auto intro!: prob_space_return simp: AE_return measure_return)
```
```   387
```
```   388 lemma bind_return_pmf: "bind_pmf (return_pmf x) f = f x"
```
```   389   by transfer
```
```   390      (auto intro!: prob_space_imp_subprob_space bind_return[where N="count_space UNIV"]
```
```   391            simp: space_subprob_algebra)
```
```   392
```
```   393 lemma set_return_pmf[simp]: "set_pmf (return_pmf x) = {x}"
```
```   394   by transfer (auto simp add: measure_return split: split_indicator)
```
```   395
```
```   396 lemma bind_return_pmf': "bind_pmf N return_pmf = N"
```
```   397 proof (transfer, clarify)
```
```   398   fix N :: "'a measure" assume "sets N = UNIV" then show "N \<bind> return (count_space UNIV) = N"
```
```   399     by (subst return_sets_cong[where N=N]) (simp_all add: bind_return')
```
```   400 qed
```
```   401
```
```   402 lemma bind_assoc_pmf: "bind_pmf (bind_pmf A B) C = bind_pmf A (\<lambda>x. bind_pmf (B x) C)"
```
```   403   by transfer
```
```   404      (auto intro!: bind_assoc[where N="count_space UNIV" and R="count_space UNIV"]
```
```   405            simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space)
```
```   406
```
```   407 definition "map_pmf f M = bind_pmf M (\<lambda>x. return_pmf (f x))"
```
```   408
```
```   409 lemma map_bind_pmf: "map_pmf f (bind_pmf M g) = bind_pmf M (\<lambda>x. map_pmf f (g x))"
```
```   410   by (simp add: map_pmf_def bind_assoc_pmf)
```
```   411
```
```   412 lemma bind_map_pmf: "bind_pmf (map_pmf f M) g = bind_pmf M (\<lambda>x. g (f x))"
```
```   413   by (simp add: map_pmf_def bind_assoc_pmf bind_return_pmf)
```
```   414
```
```   415 lemma map_pmf_transfer[transfer_rule]:
```
```   416   "rel_fun op = (rel_fun cr_pmf cr_pmf) (\<lambda>f M. distr M (count_space UNIV) f) map_pmf"
```
```   417 proof -
```
```   418   have "rel_fun op = (rel_fun pmf_as_measure.cr_pmf pmf_as_measure.cr_pmf)
```
```   419      (\<lambda>f M. M \<bind> (return (count_space UNIV) o f)) map_pmf"
```
```   420     unfolding map_pmf_def[abs_def] comp_def by transfer_prover
```
```   421   then show ?thesis
```
```   422     by (force simp: rel_fun_def cr_pmf_def bind_return_distr)
```
```   423 qed
```
```   424
```
```   425 lemma map_pmf_rep_eq:
```
```   426   "measure_pmf (map_pmf f M) = distr (measure_pmf M) (count_space UNIV) f"
```
```   427   unfolding map_pmf_def bind_pmf.rep_eq comp_def return_pmf.rep_eq
```
```   428   using bind_return_distr[of M f "count_space UNIV"] by (simp add: comp_def)
```
```   429
```
```   430 lemma map_pmf_id[simp]: "map_pmf id = id"
```
```   431   by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI)
```
```   432
```
```   433 lemma map_pmf_ident[simp]: "map_pmf (\<lambda>x. x) = (\<lambda>x. x)"
```
```   434   using map_pmf_id unfolding id_def .
```
```   435
```
```   436 lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g"
```
```   437   by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def)
```
```   438
```
```   439 lemma map_pmf_comp: "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M"
```
```   440   using map_pmf_compose[of f g] by (simp add: comp_def)
```
```   441
```
```   442 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"
```
```   443   unfolding map_pmf_def by (rule bind_pmf_cong) auto
```
```   444
```
```   445 lemma pmf_set_map: "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
```
```   446   by (auto simp add: comp_def fun_eq_iff map_pmf_def)
```
```   447
```
```   448 lemma set_map_pmf[simp]: "set_pmf (map_pmf f M) = f`set_pmf M"
```
```   449   using pmf_set_map[of f] by (auto simp: comp_def fun_eq_iff)
```
```   450
```
```   451 lemma emeasure_map_pmf[simp]: "emeasure (map_pmf f M) X = emeasure M (f -` X)"
```
```   452   unfolding map_pmf_rep_eq by (subst emeasure_distr) auto
```
```   453
```
```   454 lemma measure_map_pmf[simp]: "measure (map_pmf f M) X = measure M (f -` X)"
```
```   455 using emeasure_map_pmf[of f M X] by(simp add: measure_pmf.emeasure_eq_measure)
```
```   456
```
```   457 lemma nn_integral_map_pmf[simp]: "(\<integral>\<^sup>+x. f x \<partial>map_pmf g M) = (\<integral>\<^sup>+x. f (g x) \<partial>M)"
```
```   458   unfolding map_pmf_rep_eq by (intro nn_integral_distr) auto
```
```   459
```
```   460 lemma ereal_pmf_map: "pmf (map_pmf f p) x = (\<integral>\<^sup>+ y. indicator (f -` {x}) y \<partial>measure_pmf p)"
```
```   461 proof (transfer fixing: f x)
```
```   462   fix p :: "'b measure"
```
```   463   presume "prob_space p"
```
```   464   then interpret prob_space p .
```
```   465   presume "sets p = UNIV"
```
```   466   then show "ereal (measure (distr p (count_space UNIV) f) {x}) = integral\<^sup>N p (indicator (f -` {x}))"
```
```   467     by(simp add: measure_distr measurable_def emeasure_eq_measure)
```
```   468 qed simp_all
```
```   469
```
```   470 lemma nn_integral_pmf: "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = emeasure (measure_pmf p) A"
```
```   471 proof -
```
```   472   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))"
```
```   473     by(auto simp add: nn_integral_count_space_indicator indicator_def set_pmf_iff intro: nn_integral_cong)
```
```   474   also have "\<dots> = emeasure (measure_pmf p) (\<Union>x\<in>A \<inter> set_pmf p. {x})"
```
```   475     by(subst emeasure_UN_countable)(auto simp add: emeasure_pmf_single disjoint_family_on_def)
```
```   476   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})"
```
```   477     by(rule emeasure_Un_null_set[symmetric])(auto intro: in_null_sets_measure_pmfI)
```
```   478   also have "\<dots> = emeasure (measure_pmf p) A"
```
```   479     by(auto intro: arg_cong2[where f=emeasure])
```
```   480   finally show ?thesis .
```
```   481 qed
```
```   482
```
```   483 lemma map_return_pmf [simp]: "map_pmf f (return_pmf x) = return_pmf (f x)"
```
```   484   by transfer (simp add: distr_return)
```
```   485
```
```   486 lemma map_pmf_const[simp]: "map_pmf (\<lambda>_. c) M = return_pmf c"
```
```   487   by transfer (auto simp: prob_space.distr_const)
```
```   488
```
```   489 lemma pmf_return [simp]: "pmf (return_pmf x) y = indicator {y} x"
```
```   490   by transfer (simp add: measure_return)
```
```   491
```
```   492 lemma nn_integral_return_pmf[simp]: "0 \<le> f x \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>return_pmf x) = f x"
```
```   493   unfolding return_pmf.rep_eq by (intro nn_integral_return) auto
```
```   494
```
```   495 lemma emeasure_return_pmf[simp]: "emeasure (return_pmf x) X = indicator X x"
```
```   496   unfolding return_pmf.rep_eq by (intro emeasure_return) auto
```
```   497
```
```   498 lemma return_pmf_inj[simp]: "return_pmf x = return_pmf y \<longleftrightarrow> x = y"
```
```   499   by (metis insertI1 set_return_pmf singletonD)
```
```   500
```
```   501 lemma map_pmf_eq_return_pmf_iff:
```
```   502   "map_pmf f p = return_pmf x \<longleftrightarrow> (\<forall>y \<in> set_pmf p. f y = x)"
```
```   503 proof
```
```   504   assume "map_pmf f p = return_pmf x"
```
```   505   then have "set_pmf (map_pmf f p) = set_pmf (return_pmf x)" by simp
```
```   506   then show "\<forall>y \<in> set_pmf p. f y = x" by auto
```
```   507 next
```
```   508   assume "\<forall>y \<in> set_pmf p. f y = x"
```
```   509   then show "map_pmf f p = return_pmf x"
```
```   510     unfolding map_pmf_const[symmetric, of _ p] by (intro map_pmf_cong) auto
```
```   511 qed
```
```   512
```
```   513 definition "pair_pmf A B = bind_pmf A (\<lambda>x. bind_pmf B (\<lambda>y. return_pmf (x, y)))"
```
```   514
```
```   515 lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b"
```
```   516   unfolding pair_pmf_def pmf_bind pmf_return
```
```   517   apply (subst integral_measure_pmf[where A="{b}"])
```
```   518   apply (auto simp: indicator_eq_0_iff)
```
```   519   apply (subst integral_measure_pmf[where A="{a}"])
```
```   520   apply (auto simp: indicator_eq_0_iff setsum_nonneg_eq_0_iff pmf_nonneg)
```
```   521   done
```
```   522
```
```   523 lemma set_pair_pmf[simp]: "set_pmf (pair_pmf A B) = set_pmf A \<times> set_pmf B"
```
```   524   unfolding pair_pmf_def set_bind_pmf set_return_pmf by auto
```
```   525
```
```   526 lemma measure_pmf_in_subprob_space[measurable (raw)]:
```
```   527   "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
```
```   528   by (simp add: space_subprob_algebra) intro_locales
```
```   529
```
```   530 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)"
```
```   531 proof -
```
```   532   have "(\<integral>\<^sup>+x. f x \<partial>pair_pmf A B) = (\<integral>\<^sup>+x. max 0 (f x) * indicator (A \<times> B) x \<partial>pair_pmf A B)"
```
```   533     by (subst nn_integral_max_0[symmetric])
```
```   534        (auto simp: AE_measure_pmf_iff intro!: nn_integral_cong_AE)
```
```   535   also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) * indicator (A \<times> B) (a, b) \<partial>B \<partial>A)"
```
```   536     by (simp add: pair_pmf_def)
```
```   537   also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) \<partial>B \<partial>A)"
```
```   538     by (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
```
```   539   finally show ?thesis
```
```   540     unfolding nn_integral_max_0 .
```
```   541 qed
```
```   542
```
```   543 lemma bind_pair_pmf:
```
```   544   assumes M[measurable]: "M \<in> measurable (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) (subprob_algebra N)"
```
```   545   shows "measure_pmf (pair_pmf A B) \<bind> M = (measure_pmf A \<bind> (\<lambda>x. measure_pmf B \<bind> (\<lambda>y. M (x, y))))"
```
```   546     (is "?L = ?R")
```
```   547 proof (rule measure_eqI)
```
```   548   have M'[measurable]: "M \<in> measurable (pair_pmf A B) (subprob_algebra N)"
```
```   549     using M[THEN measurable_space] by (simp_all add: space_pair_measure)
```
```   550
```
```   551   note measurable_bind[where N="count_space UNIV", measurable]
```
```   552   note measure_pmf_in_subprob_space[simp]
```
```   553
```
```   554   have sets_eq_N: "sets ?L = N"
```
```   555     by (subst sets_bind[OF sets_kernel[OF M']]) auto
```
```   556   show "sets ?L = sets ?R"
```
```   557     using measurable_space[OF M]
```
```   558     by (simp add: sets_eq_N space_pair_measure space_subprob_algebra)
```
```   559   fix X assume "X \<in> sets ?L"
```
```   560   then have X[measurable]: "X \<in> sets N"
```
```   561     unfolding sets_eq_N .
```
```   562   then show "emeasure ?L X = emeasure ?R X"
```
```   563     apply (simp add: emeasure_bind[OF _ M' X])
```
```   564     apply (simp add: nn_integral_bind[where B="count_space UNIV"] pair_pmf_def measure_pmf_bind[of A]
```
```   565                      nn_integral_measure_pmf_finite emeasure_nonneg one_ereal_def[symmetric])
```
```   566     apply (subst emeasure_bind[OF _ _ X])
```
```   567     apply measurable
```
```   568     apply (subst emeasure_bind[OF _ _ X])
```
```   569     apply measurable
```
```   570     done
```
```   571 qed
```
```   572
```
```   573 lemma map_fst_pair_pmf: "map_pmf fst (pair_pmf A B) = A"
```
```   574   by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
```
```   575
```
```   576 lemma map_snd_pair_pmf: "map_pmf snd (pair_pmf A B) = B"
```
```   577   by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
```
```   578
```
```   579 lemma nn_integral_pmf':
```
```   580   "inj_on f A \<Longrightarrow> (\<integral>\<^sup>+x. pmf p (f x) \<partial>count_space A) = emeasure p (f ` A)"
```
```   581   by (subst nn_integral_bij_count_space[where g=f and B="f`A"])
```
```   582      (auto simp: bij_betw_def nn_integral_pmf)
```
```   583
```
```   584 lemma pmf_le_0_iff[simp]: "pmf M p \<le> 0 \<longleftrightarrow> pmf M p = 0"
```
```   585   using pmf_nonneg[of M p] by simp
```
```   586
```
```   587 lemma min_pmf_0[simp]: "min (pmf M p) 0 = 0" "min 0 (pmf M p) = 0"
```
```   588   using pmf_nonneg[of M p] by simp_all
```
```   589
```
```   590 lemma pmf_eq_0_set_pmf: "pmf M p = 0 \<longleftrightarrow> p \<notin> set_pmf M"
```
```   591   unfolding set_pmf_iff by simp
```
```   592
```
```   593 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"
```
```   594   by (auto simp: pmf.rep_eq map_pmf_rep_eq measure_distr AE_measure_pmf_iff inj_onD
```
```   595            intro!: measure_pmf.finite_measure_eq_AE)
```
```   596
```
```   597 lemma pmf_map_inj': "inj f \<Longrightarrow> pmf (map_pmf f M) (f x) = pmf M x"
```
```   598 apply(cases "x \<in> set_pmf M")
```
```   599  apply(simp add: pmf_map_inj[OF subset_inj_on])
```
```   600 apply(simp add: pmf_eq_0_set_pmf[symmetric])
```
```   601 apply(auto simp add: pmf_eq_0_set_pmf dest: injD)
```
```   602 done
```
```   603
```
```   604 lemma pmf_map_outside: "x \<notin> f ` set_pmf M \<Longrightarrow> pmf (map_pmf f M) x = 0"
```
```   605 unfolding pmf_eq_0_set_pmf by simp
```
```   606
```
```   607 subsection \<open> PMFs as function \<close>
```
```   608
```
```   609 context
```
```   610   fixes f :: "'a \<Rightarrow> real"
```
```   611   assumes nonneg: "\<And>x. 0 \<le> f x"
```
```   612   assumes prob: "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
```
```   613 begin
```
```   614
```
```   615 lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ereal \<circ> f)"
```
```   616 proof (intro conjI)
```
```   617   have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
```
```   618     by (simp split: split_indicator)
```
```   619   show "AE x in density (count_space UNIV) (ereal \<circ> f).
```
```   620     measure (density (count_space UNIV) (ereal \<circ> f)) {x} \<noteq> 0"
```
```   621     by (simp add: AE_density nonneg measure_def emeasure_density max_def)
```
```   622   show "prob_space (density (count_space UNIV) (ereal \<circ> f))"
```
```   623     by standard (simp add: emeasure_density prob)
```
```   624 qed simp
```
```   625
```
```   626 lemma pmf_embed_pmf: "pmf embed_pmf x = f x"
```
```   627 proof transfer
```
```   628   have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
```
```   629     by (simp split: split_indicator)
```
```   630   fix x show "measure (density (count_space UNIV) (ereal \<circ> f)) {x} = f x"
```
```   631     by transfer (simp add: measure_def emeasure_density nonneg max_def)
```
```   632 qed
```
```   633
```
```   634 lemma set_embed_pmf: "set_pmf embed_pmf = {x. f x \<noteq> 0}"
```
```   635 by(auto simp add: set_pmf_eq assms pmf_embed_pmf)
```
```   636
```
```   637 end
```
```   638
```
```   639 lemma embed_pmf_transfer:
```
```   640   "rel_fun (eq_onp (\<lambda>f. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ereal (f x) \<partial>count_space UNIV) = 1)) pmf_as_measure.cr_pmf (\<lambda>f. density (count_space UNIV) (ereal \<circ> f)) embed_pmf"
```
```   641   by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer)
```
```   642
```
```   643 lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)"
```
```   644 proof (transfer, elim conjE)
```
```   645   fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
```
```   646   assume "prob_space M" then interpret prob_space M .
```
```   647   show "M = density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))"
```
```   648   proof (rule measure_eqI)
```
```   649     fix A :: "'a set"
```
```   650     have "(\<integral>\<^sup>+ x. ereal (measure M {x}) * indicator A x \<partial>count_space UNIV) =
```
```   651       (\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)"
```
```   652       by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator)
```
```   653     also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))"
```
```   654       by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space)
```
```   655     also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})"
```
```   656       by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support)
```
```   657          (auto simp: disjoint_family_on_def)
```
```   658     also have "\<dots> = emeasure M A"
```
```   659       using ae by (intro emeasure_eq_AE) auto
```
```   660     finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))) A"
```
```   661       using emeasure_space_1 by (simp add: emeasure_density)
```
```   662   qed simp
```
```   663 qed
```
```   664
```
```   665 lemma td_pmf_embed_pmf:
```
```   666   "type_definition pmf embed_pmf {f::'a \<Rightarrow> real. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ereal (f x) \<partial>count_space UNIV) = 1}"
```
```   667   unfolding type_definition_def
```
```   668 proof safe
```
```   669   fix p :: "'a pmf"
```
```   670   have "(\<integral>\<^sup>+ x. 1 \<partial>measure_pmf p) = 1"
```
```   671     using measure_pmf.emeasure_space_1[of p] by simp
```
```   672   then show *: "(\<integral>\<^sup>+ x. ereal (pmf p x) \<partial>count_space UNIV) = 1"
```
```   673     by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const)
```
```   674
```
```   675   show "embed_pmf (pmf p) = p"
```
```   676     by (intro measure_pmf_inject[THEN iffD1])
```
```   677        (simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def)
```
```   678 next
```
```   679   fix f :: "'a \<Rightarrow> real" assume "\<forall>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
```
```   680   then show "pmf (embed_pmf f) = f"
```
```   681     by (auto intro!: pmf_embed_pmf)
```
```   682 qed (rule pmf_nonneg)
```
```   683
```
```   684 end
```
```   685
```
```   686 lemma nn_integral_measure_pmf: "(\<integral>\<^sup>+ x. f x \<partial>measure_pmf p) = \<integral>\<^sup>+ x. ereal (pmf p x) * f x \<partial>count_space UNIV"
```
```   687 by(simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg)
```
```   688
```
```   689 locale pmf_as_function
```
```   690 begin
```
```   691
```
```   692 setup_lifting td_pmf_embed_pmf
```
```   693
```
```   694 lemma set_pmf_transfer[transfer_rule]:
```
```   695   assumes "bi_total A"
```
```   696   shows "rel_fun (pcr_pmf A) (rel_set A) (\<lambda>f. {x. f x \<noteq> 0}) set_pmf"
```
```   697   using \<open>bi_total A\<close>
```
```   698   by (auto simp: pcr_pmf_def cr_pmf_def rel_fun_def rel_set_def bi_total_def Bex_def set_pmf_iff)
```
```   699      metis+
```
```   700
```
```   701 end
```
```   702
```
```   703 context
```
```   704 begin
```
```   705
```
```   706 interpretation pmf_as_function .
```
```   707
```
```   708 lemma pmf_eqI: "(\<And>i. pmf M i = pmf N i) \<Longrightarrow> M = N"
```
```   709   by transfer auto
```
```   710
```
```   711 lemma pmf_eq_iff: "M = N \<longleftrightarrow> (\<forall>i. pmf M i = pmf N i)"
```
```   712   by (auto intro: pmf_eqI)
```
```   713
```
```   714 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))"
```
```   715   unfolding pmf_eq_iff pmf_bind
```
```   716 proof
```
```   717   fix i
```
```   718   interpret B: prob_space "restrict_space B B"
```
```   719     by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
```
```   720        (auto simp: AE_measure_pmf_iff)
```
```   721   interpret A: prob_space "restrict_space A A"
```
```   722     by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
```
```   723        (auto simp: AE_measure_pmf_iff)
```
```   724
```
```   725   interpret AB: pair_prob_space "restrict_space A A" "restrict_space B B"
```
```   726     by unfold_locales
```
```   727
```
```   728   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)"
```
```   729     by (rule integral_cong) (auto intro!: integral_pmf_restrict)
```
```   730   also have "\<dots> = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>restrict_space A A)"
```
```   731     by (intro integral_pmf_restrict B.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
```
```   732               countable_set_pmf borel_measurable_count_space)
```
```   733   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>restrict_space B B)"
```
```   734     by (rule AB.Fubini_integral[symmetric])
```
```   735        (auto intro!: AB.integrable_const_bound[where B=1] measurable_pair_restrict_pmf2
```
```   736              simp: pmf_nonneg pmf_le_1 measurable_restrict_space1)
```
```   737   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>B)"
```
```   738     by (intro integral_pmf_restrict[symmetric] A.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
```
```   739               countable_set_pmf borel_measurable_count_space)
```
```   740   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>A \<partial>B)"
```
```   741     by (rule integral_cong) (auto intro!: integral_pmf_restrict[symmetric])
```
```   742   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)" .
```
```   743 qed
```
```   744
```
```   745 lemma pair_map_pmf1: "pair_pmf (map_pmf f A) B = map_pmf (apfst f) (pair_pmf A B)"
```
```   746 proof (safe intro!: pmf_eqI)
```
```   747   fix a :: "'a" and b :: "'b"
```
```   748   have [simp]: "\<And>c d. indicator (apfst f -` {(a, b)}) (c, d) = indicator (f -` {a}) c * (indicator {b} d::ereal)"
```
```   749     by (auto split: split_indicator)
```
```   750
```
```   751   have "ereal (pmf (pair_pmf (map_pmf f A) B) (a, b)) =
```
```   752          ereal (pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b))"
```
```   753     unfolding pmf_pair ereal_pmf_map
```
```   754     by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_multc pmf_nonneg
```
```   755                   emeasure_map_pmf[symmetric] del: emeasure_map_pmf)
```
```   756   then show "pmf (pair_pmf (map_pmf f A) B) (a, b) = pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b)"
```
```   757     by simp
```
```   758 qed
```
```   759
```
```   760 lemma pair_map_pmf2: "pair_pmf A (map_pmf f B) = map_pmf (apsnd f) (pair_pmf A B)"
```
```   761 proof (safe intro!: pmf_eqI)
```
```   762   fix a :: "'a" and b :: "'b"
```
```   763   have [simp]: "\<And>c d. indicator (apsnd f -` {(a, b)}) (c, d) = indicator {a} c * (indicator (f -` {b}) d::ereal)"
```
```   764     by (auto split: split_indicator)
```
```   765
```
```   766   have "ereal (pmf (pair_pmf A (map_pmf f B)) (a, b)) =
```
```   767          ereal (pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b))"
```
```   768     unfolding pmf_pair ereal_pmf_map
```
```   769     by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_cmult nn_integral_multc pmf_nonneg
```
```   770                   emeasure_map_pmf[symmetric] del: emeasure_map_pmf)
```
```   771   then show "pmf (pair_pmf A (map_pmf f B)) (a, b) = pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b)"
```
```   772     by simp
```
```   773 qed
```
```   774
```
```   775 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)"
```
```   776   by (simp add: pair_map_pmf2 pair_map_pmf1 map_pmf_comp split_beta')
```
```   777
```
```   778 end
```
```   779
```
```   780 lemma pair_return_pmf1: "pair_pmf (return_pmf x) y = map_pmf (Pair x) y"
```
```   781 by(simp add: pair_pmf_def bind_return_pmf map_pmf_def)
```
```   782
```
```   783 lemma pair_return_pmf2: "pair_pmf x (return_pmf y) = map_pmf (\<lambda>x. (x, y)) x"
```
```   784 by(simp add: pair_pmf_def bind_return_pmf map_pmf_def)
```
```   785
```
```   786 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))"
```
```   787 by(simp add: pair_pmf_def bind_return_pmf map_pmf_def bind_assoc_pmf)
```
```   788
```
```   789 lemma pair_commute_pmf: "pair_pmf x y = map_pmf (\<lambda>(x, y). (y, x)) (pair_pmf y x)"
```
```   790 unfolding pair_pmf_def by(subst bind_commute_pmf)(simp add: map_pmf_def bind_assoc_pmf bind_return_pmf)
```
```   791
```
```   792 lemma set_pmf_subset_singleton: "set_pmf p \<subseteq> {x} \<longleftrightarrow> p = return_pmf x"
```
```   793 proof(intro iffI pmf_eqI)
```
```   794   fix i
```
```   795   assume x: "set_pmf p \<subseteq> {x}"
```
```   796   hence *: "set_pmf p = {x}" using set_pmf_not_empty[of p] by auto
```
```   797   have "ereal (pmf p x) = \<integral>\<^sup>+ i. indicator {x} i \<partial>p" by(simp add: emeasure_pmf_single)
```
```   798   also have "\<dots> = \<integral>\<^sup>+ i. 1 \<partial>p" by(rule nn_integral_cong_AE)(simp add: AE_measure_pmf_iff * )
```
```   799   also have "\<dots> = 1" by simp
```
```   800   finally show "pmf p i = pmf (return_pmf x) i" using x
```
```   801     by(auto split: split_indicator simp add: pmf_eq_0_set_pmf)
```
```   802 qed auto
```
```   803
```
```   804 lemma bind_eq_return_pmf:
```
```   805   "bind_pmf p f = return_pmf x \<longleftrightarrow> (\<forall>y\<in>set_pmf p. f y = return_pmf x)"
```
```   806   (is "?lhs \<longleftrightarrow> ?rhs")
```
```   807 proof(intro iffI strip)
```
```   808   fix y
```
```   809   assume y: "y \<in> set_pmf p"
```
```   810   assume "?lhs"
```
```   811   hence "set_pmf (bind_pmf p f) = {x}" by simp
```
```   812   hence "(\<Union>y\<in>set_pmf p. set_pmf (f y)) = {x}" by simp
```
```   813   hence "set_pmf (f y) \<subseteq> {x}" using y by auto
```
```   814   thus "f y = return_pmf x" by(simp add: set_pmf_subset_singleton)
```
```   815 next
```
```   816   assume *: ?rhs
```
```   817   show ?lhs
```
```   818   proof(rule pmf_eqI)
```
```   819     fix i
```
```   820     have "ereal (pmf (bind_pmf p f) i) = \<integral>\<^sup>+ y. ereal (pmf (f y) i) \<partial>p" by(simp add: ereal_pmf_bind)
```
```   821     also have "\<dots> = \<integral>\<^sup>+ y. ereal (pmf (return_pmf x) i) \<partial>p"
```
```   822       by(rule nn_integral_cong_AE)(simp add: AE_measure_pmf_iff * )
```
```   823     also have "\<dots> = ereal (pmf (return_pmf x) i)" by simp
```
```   824     finally show "pmf (bind_pmf p f) i = pmf (return_pmf x) i" by simp
```
```   825   qed
```
```   826 qed
```
```   827
```
```   828 lemma pmf_False_conv_True: "pmf p False = 1 - pmf p True"
```
```   829 proof -
```
```   830   have "pmf p False + pmf p True = measure p {False} + measure p {True}"
```
```   831     by(simp add: measure_pmf_single)
```
```   832   also have "\<dots> = measure p ({False} \<union> {True})"
```
```   833     by(subst measure_pmf.finite_measure_Union) simp_all
```
```   834   also have "{False} \<union> {True} = space p" by auto
```
```   835   finally show ?thesis by simp
```
```   836 qed
```
```   837
```
```   838 lemma pmf_True_conv_False: "pmf p True = 1 - pmf p False"
```
```   839 by(simp add: pmf_False_conv_True)
```
```   840
```
```   841 subsection \<open> Conditional Probabilities \<close>
```
```   842
```
```   843 lemma measure_pmf_zero_iff: "measure (measure_pmf p) s = 0 \<longleftrightarrow> set_pmf p \<inter> s = {}"
```
```   844   by (subst measure_pmf.prob_eq_0) (auto simp: AE_measure_pmf_iff)
```
```   845
```
```   846 context
```
```   847   fixes p :: "'a pmf" and s :: "'a set"
```
```   848   assumes not_empty: "set_pmf p \<inter> s \<noteq> {}"
```
```   849 begin
```
```   850
```
```   851 interpretation pmf_as_measure .
```
```   852
```
```   853 lemma emeasure_measure_pmf_not_zero: "emeasure (measure_pmf p) s \<noteq> 0"
```
```   854 proof
```
```   855   assume "emeasure (measure_pmf p) s = 0"
```
```   856   then have "AE x in measure_pmf p. x \<notin> s"
```
```   857     by (rule AE_I[rotated]) auto
```
```   858   with not_empty show False
```
```   859     by (auto simp: AE_measure_pmf_iff)
```
```   860 qed
```
```   861
```
```   862 lemma measure_measure_pmf_not_zero: "measure (measure_pmf p) s \<noteq> 0"
```
```   863   using emeasure_measure_pmf_not_zero unfolding measure_pmf.emeasure_eq_measure by simp
```
```   864
```
```   865 lift_definition cond_pmf :: "'a pmf" is
```
```   866   "uniform_measure (measure_pmf p) s"
```
```   867 proof (intro conjI)
```
```   868   show "prob_space (uniform_measure (measure_pmf p) s)"
```
```   869     by (intro prob_space_uniform_measure) (auto simp: emeasure_measure_pmf_not_zero)
```
```   870   show "AE x in uniform_measure (measure_pmf p) s. measure (uniform_measure (measure_pmf p) s) {x} \<noteq> 0"
```
```   871     by (simp add: emeasure_measure_pmf_not_zero measure_measure_pmf_not_zero AE_uniform_measure
```
```   872                   AE_measure_pmf_iff set_pmf.rep_eq)
```
```   873 qed simp
```
```   874
```
```   875 lemma pmf_cond: "pmf cond_pmf x = (if x \<in> s then pmf p x / measure p s else 0)"
```
```   876   by transfer (simp add: emeasure_measure_pmf_not_zero pmf.rep_eq)
```
```   877
```
```   878 lemma set_cond_pmf[simp]: "set_pmf cond_pmf = set_pmf p \<inter> s"
```
```   879   by (auto simp add: set_pmf_iff pmf_cond measure_measure_pmf_not_zero split: split_if_asm)
```
```   880
```
```   881 end
```
```   882
```
```   883 lemma cond_map_pmf:
```
```   884   assumes "set_pmf p \<inter> f -` s \<noteq> {}"
```
```   885   shows "cond_pmf (map_pmf f p) s = map_pmf f (cond_pmf p (f -` s))"
```
```   886 proof -
```
```   887   have *: "set_pmf (map_pmf f p) \<inter> s \<noteq> {}"
```
```   888     using assms by auto
```
```   889   { fix x
```
```   890     have "ereal (pmf (map_pmf f (cond_pmf p (f -` s))) x) =
```
```   891       emeasure p (f -` s \<inter> f -` {x}) / emeasure p (f -` s)"
```
```   892       unfolding ereal_pmf_map cond_pmf.rep_eq[OF assms] by (simp add: nn_integral_uniform_measure)
```
```   893     also have "f -` s \<inter> f -` {x} = (if x \<in> s then f -` {x} else {})"
```
```   894       by auto
```
```   895     also have "emeasure p (if x \<in> s then f -` {x} else {}) / emeasure p (f -` s) =
```
```   896       ereal (pmf (cond_pmf (map_pmf f p) s) x)"
```
```   897       using measure_measure_pmf_not_zero[OF *]
```
```   898       by (simp add: pmf_cond[OF *] ereal_divide' ereal_pmf_map measure_pmf.emeasure_eq_measure[symmetric]
```
```   899                del: ereal_divide)
```
```   900     finally have "ereal (pmf (cond_pmf (map_pmf f p) s) x) = ereal (pmf (map_pmf f (cond_pmf p (f -` s))) x)"
```
```   901       by simp }
```
```   902   then show ?thesis
```
```   903     by (intro pmf_eqI) simp
```
```   904 qed
```
```   905
```
```   906 lemma bind_cond_pmf_cancel:
```
```   907   assumes [simp]: "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}"
```
```   908   assumes [simp]: "\<And>y. y \<in> set_pmf q \<Longrightarrow> set_pmf p \<inter> {x. R x y} \<noteq> {}"
```
```   909   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}"
```
```   910   shows "bind_pmf p (\<lambda>x. cond_pmf q {y. R x y}) = q"
```
```   911 proof (rule pmf_eqI)
```
```   912   fix i
```
```   913   have "ereal (pmf (bind_pmf p (\<lambda>x. cond_pmf q {y. R x y})) i) =
```
```   914     (\<integral>\<^sup>+x. ereal (pmf q i / measure p {x. R x i}) * ereal (indicator {x. R x i} x) \<partial>p)"
```
```   915     by (auto simp add: ereal_pmf_bind AE_measure_pmf_iff pmf_cond pmf_eq_0_set_pmf intro!: nn_integral_cong_AE)
```
```   916   also have "\<dots> = (pmf q i * measure p {x. R x i}) / measure p {x. R x i}"
```
```   917     by (simp add: pmf_nonneg measure_nonneg zero_ereal_def[symmetric] ereal_indicator
```
```   918                   nn_integral_cmult measure_pmf.emeasure_eq_measure)
```
```   919   also have "\<dots> = pmf q i"
```
```   920     by (cases "pmf q i = 0") (simp_all add: pmf_eq_0_set_pmf measure_measure_pmf_not_zero)
```
```   921   finally show "pmf (bind_pmf p (\<lambda>x. cond_pmf q {y. R x y})) i = pmf q i"
```
```   922     by simp
```
```   923 qed
```
```   924
```
```   925 subsection \<open> Relator \<close>
```
```   926
```
```   927 inductive rel_pmf :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf \<Rightarrow> bool"
```
```   928 for R p q
```
```   929 where
```
```   930   "\<lbrakk> \<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y;
```
```   931      map_pmf fst pq = p; map_pmf snd pq = q \<rbrakk>
```
```   932   \<Longrightarrow> rel_pmf R p q"
```
```   933
```
```   934 lemma rel_pmfI:
```
```   935   assumes R: "rel_set R (set_pmf p) (set_pmf q)"
```
```   936   assumes eq: "\<And>x y. x \<in> set_pmf p \<Longrightarrow> y \<in> set_pmf q \<Longrightarrow> R x y \<Longrightarrow>
```
```   937     measure p {x. R x y} = measure q {y. R x y}"
```
```   938   shows "rel_pmf R p q"
```
```   939 proof
```
```   940   let ?pq = "bind_pmf p (\<lambda>x. bind_pmf (cond_pmf q {y. R x y}) (\<lambda>y. return_pmf (x, y)))"
```
```   941   have "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}"
```
```   942     using R by (auto simp: rel_set_def)
```
```   943   then show "\<And>x y. (x, y) \<in> set_pmf ?pq \<Longrightarrow> R x y"
```
```   944     by auto
```
```   945   show "map_pmf fst ?pq = p"
```
```   946     by (simp add: map_bind_pmf bind_return_pmf')
```
```   947
```
```   948   show "map_pmf snd ?pq = q"
```
```   949     using R eq
```
```   950     apply (simp add: bind_cond_pmf_cancel map_bind_pmf bind_return_pmf')
```
```   951     apply (rule bind_cond_pmf_cancel)
```
```   952     apply (auto simp: rel_set_def)
```
```   953     done
```
```   954 qed
```
```   955
```
```   956 lemma rel_pmf_imp_rel_set: "rel_pmf R p q \<Longrightarrow> rel_set R (set_pmf p) (set_pmf q)"
```
```   957   by (force simp add: rel_pmf.simps rel_set_def)
```
```   958
```
```   959 lemma rel_pmfD_measure:
```
```   960   assumes rel_R: "rel_pmf R p q" and R: "\<And>a b. R a b \<Longrightarrow> R a y \<longleftrightarrow> R x b"
```
```   961   assumes "x \<in> set_pmf p" "y \<in> set_pmf q"
```
```   962   shows "measure p {x. R x y} = measure q {y. R x y}"
```
```   963 proof -
```
```   964   from rel_R obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
```
```   965     and eq: "p = map_pmf fst pq" "q = map_pmf snd pq"
```
```   966     by (auto elim: rel_pmf.cases)
```
```   967   have "measure p {x. R x y} = measure pq {x. R (fst x) y}"
```
```   968     by (simp add: eq map_pmf_rep_eq measure_distr)
```
```   969   also have "\<dots> = measure pq {y. R x (snd y)}"
```
```   970     by (intro measure_pmf.finite_measure_eq_AE)
```
```   971        (auto simp: AE_measure_pmf_iff R dest!: pq)
```
```   972   also have "\<dots> = measure q {y. R x y}"
```
```   973     by (simp add: eq map_pmf_rep_eq measure_distr)
```
```   974   finally show "measure p {x. R x y} = measure q {y. R x y}" .
```
```   975 qed
```
```   976
```
```   977 lemma rel_pmf_measureD:
```
```   978   assumes "rel_pmf R p q"
```
```   979   shows "measure (measure_pmf p) A \<le> measure (measure_pmf q) {y. \<exists>x\<in>A. R x y}" (is "?lhs \<le> ?rhs")
```
```   980 using assms
```
```   981 proof cases
```
```   982   fix pq
```
```   983   assume R: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
```
```   984     and p[symmetric]: "map_pmf fst pq = p"
```
```   985     and q[symmetric]: "map_pmf snd pq = q"
```
```   986   have "?lhs = measure (measure_pmf pq) (fst -` A)" by(simp add: p)
```
```   987   also have "\<dots> \<le> measure (measure_pmf pq) {y. \<exists>x\<in>A. R x (snd y)}"
```
```   988     by(rule measure_pmf.finite_measure_mono_AE)(auto 4 3 simp add: AE_measure_pmf_iff dest: R)
```
```   989   also have "\<dots> = ?rhs" by(simp add: q)
```
```   990   finally show ?thesis .
```
```   991 qed
```
```   992
```
```   993 lemma rel_pmf_iff_measure:
```
```   994   assumes "symp R" "transp R"
```
```   995   shows "rel_pmf R p q \<longleftrightarrow>
```
```   996     rel_set R (set_pmf p) (set_pmf q) \<and>
```
```   997     (\<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})"
```
```   998   by (safe intro!: rel_pmf_imp_rel_set rel_pmfI)
```
```   999      (auto intro!: rel_pmfD_measure dest: sympD[OF \<open>symp R\<close>] transpD[OF \<open>transp R\<close>])
```
```  1000
```
```  1001 lemma quotient_rel_set_disjoint:
```
```  1002   "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 = {}"
```
```  1003   using in_quotient_imp_closed[of UNIV "{(x, y). R x y}" C]
```
```  1004   by (auto 0 0 simp: equivp_equiv rel_set_def set_eq_iff elim: equivpE)
```
```  1005      (blast dest: equivp_symp)+
```
```  1006
```
```  1007 lemma quotientD: "equiv X R \<Longrightarrow> A \<in> X // R \<Longrightarrow> x \<in> A \<Longrightarrow> A = R `` {x}"
```
```  1008   by (metis Image_singleton_iff equiv_class_eq_iff quotientE)
```
```  1009
```
```  1010 lemma rel_pmf_iff_equivp:
```
```  1011   assumes "equivp R"
```
```  1012   shows "rel_pmf R p q \<longleftrightarrow> (\<forall>C\<in>UNIV // {(x, y). R x y}. measure p C = measure q C)"
```
```  1013     (is "_ \<longleftrightarrow>   (\<forall>C\<in>_//?R. _)")
```
```  1014 proof (subst rel_pmf_iff_measure, safe)
```
```  1015   show "symp R" "transp R"
```
```  1016     using assms by (auto simp: equivp_reflp_symp_transp)
```
```  1017 next
```
```  1018   fix C assume C: "C \<in> UNIV // ?R" and R: "rel_set R (set_pmf p) (set_pmf q)"
```
```  1019   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}"
```
```  1020
```
```  1021   show "measure p C = measure q C"
```
```  1022   proof cases
```
```  1023     assume "p \<inter> C = {}"
```
```  1024     moreover then have "q \<inter> C = {}"
```
```  1025       using quotient_rel_set_disjoint[OF assms C R] by simp
```
```  1026     ultimately show ?thesis
```
```  1027       unfolding measure_pmf_zero_iff[symmetric] by simp
```
```  1028   next
```
```  1029     assume "p \<inter> C \<noteq> {}"
```
```  1030     moreover then have "q \<inter> C \<noteq> {}"
```
```  1031       using quotient_rel_set_disjoint[OF assms C R] by simp
```
```  1032     ultimately obtain x y where in_set: "x \<in> set_pmf p" "y \<in> set_pmf q" and in_C: "x \<in> C" "y \<in> C"
```
```  1033       by auto
```
```  1034     then have "R x y"
```
```  1035       using in_quotient_imp_in_rel[of UNIV ?R C x y] C assms
```
```  1036       by (simp add: equivp_equiv)
```
```  1037     with in_set eq have "measure p {x. R x y} = measure q {y. R x y}"
```
```  1038       by auto
```
```  1039     moreover have "{y. R x y} = C"
```
```  1040       using assms \<open>x \<in> C\<close> C quotientD[of UNIV ?R C x] by (simp add: equivp_equiv)
```
```  1041     moreover have "{x. R x y} = C"
```
```  1042       using assms \<open>y \<in> C\<close> C quotientD[of UNIV "?R" C y] sympD[of R]
```
```  1043       by (auto simp add: equivp_equiv elim: equivpE)
```
```  1044     ultimately show ?thesis
```
```  1045       by auto
```
```  1046   qed
```
```  1047 next
```
```  1048   assume eq: "\<forall>C\<in>UNIV // ?R. measure p C = measure q C"
```
```  1049   show "rel_set R (set_pmf p) (set_pmf q)"
```
```  1050     unfolding rel_set_def
```
```  1051   proof safe
```
```  1052     fix x assume x: "x \<in> set_pmf p"
```
```  1053     have "{y. R x y} \<in> UNIV // ?R"
```
```  1054       by (auto simp: quotient_def)
```
```  1055     with eq have *: "measure q {y. R x y} = measure p {y. R x y}"
```
```  1056       by auto
```
```  1057     have "measure q {y. R x y} \<noteq> 0"
```
```  1058       using x assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp)
```
```  1059     then show "\<exists>y\<in>set_pmf q. R x y"
```
```  1060       unfolding measure_pmf_zero_iff by auto
```
```  1061   next
```
```  1062     fix y assume y: "y \<in> set_pmf q"
```
```  1063     have "{x. R x y} \<in> UNIV // ?R"
```
```  1064       using assms by (auto simp: quotient_def dest: equivp_symp)
```
```  1065     with eq have *: "measure p {x. R x y} = measure q {x. R x y}"
```
```  1066       by auto
```
```  1067     have "measure p {x. R x y} \<noteq> 0"
```
```  1068       using y assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp)
```
```  1069     then show "\<exists>x\<in>set_pmf p. R x y"
```
```  1070       unfolding measure_pmf_zero_iff by auto
```
```  1071   qed
```
```  1072
```
```  1073   fix x y assume "x \<in> set_pmf p" "y \<in> set_pmf q" "R x y"
```
```  1074   have "{y. R x y} \<in> UNIV // ?R" "{x. R x y} = {y. R x y}"
```
```  1075     using assms \<open>R x y\<close> by (auto simp: quotient_def dest: equivp_symp equivp_transp)
```
```  1076   with eq show "measure p {x. R x y} = measure q {y. R x y}"
```
```  1077     by auto
```
```  1078 qed
```
```  1079
```
```  1080 bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: rel_pmf
```
```  1081 proof -
```
```  1082   show "map_pmf id = id" by (rule map_pmf_id)
```
```  1083   show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose)
```
```  1084   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"
```
```  1085     by (intro map_pmf_cong refl)
```
```  1086
```
```  1087   show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
```
```  1088     by (rule pmf_set_map)
```
```  1089
```
```  1090   show "(card_of (set_pmf p), natLeq) \<in> ordLeq" for p :: "'s pmf"
```
```  1091   proof -
```
```  1092     have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq"
```
```  1093       by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"])
```
```  1094          (auto intro: countable_set_pmf)
```
```  1095     also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq"
```
```  1096       by (metis Field_natLeq card_of_least natLeq_Well_order)
```
```  1097     finally show ?thesis .
```
```  1098   qed
```
```  1099
```
```  1100   show "\<And>R. rel_pmf R = (\<lambda>x y. \<exists>z. set_pmf z \<subseteq> {(x, y). R x y} \<and>
```
```  1101     map_pmf fst z = x \<and> map_pmf snd z = y)"
```
```  1102      by (auto simp add: fun_eq_iff rel_pmf.simps)
```
```  1103
```
```  1104   show "rel_pmf R OO rel_pmf S \<le> rel_pmf (R OO S)"
```
```  1105     for R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
```
```  1106   proof -
```
```  1107     { fix p q r
```
```  1108       assume pq: "rel_pmf R p q"
```
```  1109         and qr:"rel_pmf S q r"
```
```  1110       from pq obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
```
```  1111         and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto
```
```  1112       from qr obtain qr where qr: "\<And>y z. (y, z) \<in> set_pmf qr \<Longrightarrow> S y z"
```
```  1113         and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto
```
```  1114
```
```  1115       def pr \<equiv> "bind_pmf pq (\<lambda>xy. bind_pmf (cond_pmf qr {yz. fst yz = snd xy}) (\<lambda>yz. return_pmf (fst xy, snd yz)))"
```
```  1116       have pr_welldefined: "\<And>y. y \<in> q \<Longrightarrow> qr \<inter> {yz. fst yz = y} \<noteq> {}"
```
```  1117         by (force simp: q')
```
```  1118
```
```  1119       have "rel_pmf (R OO S) p r"
```
```  1120       proof (rule rel_pmf.intros)
```
```  1121         fix x z assume "(x, z) \<in> pr"
```
```  1122         then have "\<exists>y. (x, y) \<in> pq \<and> (y, z) \<in> qr"
```
```  1123           by (auto simp: q pr_welldefined pr_def split_beta)
```
```  1124         with pq qr show "(R OO S) x z"
```
```  1125           by blast
```
```  1126       next
```
```  1127         have "map_pmf snd pr = map_pmf snd (bind_pmf q (\<lambda>y. cond_pmf qr {yz. fst yz = y}))"
```
```  1128           by (simp add: pr_def q split_beta bind_map_pmf map_pmf_def[symmetric] map_bind_pmf map_pmf_comp)
```
```  1129         then show "map_pmf snd pr = r"
```
```  1130           unfolding r q' bind_map_pmf by (subst (asm) bind_cond_pmf_cancel) (auto simp: eq_commute)
```
```  1131       qed (simp add: pr_def map_bind_pmf split_beta map_pmf_def[symmetric] p map_pmf_comp)
```
```  1132     }
```
```  1133     then show ?thesis
```
```  1134       by(auto simp add: le_fun_def)
```
```  1135   qed
```
```  1136 qed (fact natLeq_card_order natLeq_cinfinite)+
```
```  1137
```
```  1138 lemma map_pmf_idI: "(\<And>x. x \<in> set_pmf p \<Longrightarrow> f x = x) \<Longrightarrow> map_pmf f p = p"
```
```  1139 by(simp cong: pmf.map_cong)
```
```  1140
```
```  1141 lemma rel_pmf_conj[simp]:
```
```  1142   "rel_pmf (\<lambda>x y. P \<and> Q x y) x y \<longleftrightarrow> P \<and> rel_pmf Q x y"
```
```  1143   "rel_pmf (\<lambda>x y. Q x y \<and> P) x y \<longleftrightarrow> P \<and> rel_pmf Q x y"
```
```  1144   using set_pmf_not_empty by (fastforce simp: pmf.in_rel subset_eq)+
```
```  1145
```
```  1146 lemma rel_pmf_top[simp]: "rel_pmf top = top"
```
```  1147   by (auto simp: pmf.in_rel[abs_def] fun_eq_iff map_fst_pair_pmf map_snd_pair_pmf
```
```  1148            intro: exI[of _ "pair_pmf x y" for x y])
```
```  1149
```
```  1150 lemma rel_pmf_return_pmf1: "rel_pmf R (return_pmf x) M \<longleftrightarrow> (\<forall>a\<in>M. R x a)"
```
```  1151 proof safe
```
```  1152   fix a assume "a \<in> M" "rel_pmf R (return_pmf x) M"
```
```  1153   then obtain pq where *: "\<And>a b. (a, b) \<in> set_pmf pq \<Longrightarrow> R a b"
```
```  1154     and eq: "return_pmf x = map_pmf fst pq" "M = map_pmf snd pq"
```
```  1155     by (force elim: rel_pmf.cases)
```
```  1156   moreover have "set_pmf (return_pmf x) = {x}"
```
```  1157     by simp
```
```  1158   with \<open>a \<in> M\<close> have "(x, a) \<in> pq"
```
```  1159     by (force simp: eq)
```
```  1160   with * show "R x a"
```
```  1161     by auto
```
```  1162 qed (auto intro!: rel_pmf.intros[where pq="pair_pmf (return_pmf x) M"]
```
```  1163           simp: map_fst_pair_pmf map_snd_pair_pmf)
```
```  1164
```
```  1165 lemma rel_pmf_return_pmf2: "rel_pmf R M (return_pmf x) \<longleftrightarrow> (\<forall>a\<in>M. R a x)"
```
```  1166   by (subst pmf.rel_flip[symmetric]) (simp add: rel_pmf_return_pmf1)
```
```  1167
```
```  1168 lemma rel_return_pmf[simp]: "rel_pmf R (return_pmf x1) (return_pmf x2) = R x1 x2"
```
```  1169   unfolding rel_pmf_return_pmf2 set_return_pmf by simp
```
```  1170
```
```  1171 lemma rel_pmf_False[simp]: "rel_pmf (\<lambda>x y. False) x y = False"
```
```  1172   unfolding pmf.in_rel fun_eq_iff using set_pmf_not_empty by fastforce
```
```  1173
```
```  1174 lemma rel_pmf_rel_prod:
```
```  1175   "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'"
```
```  1176 proof safe
```
```  1177   assume "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
```
```  1178   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"
```
```  1179     and eq: "map_pmf fst pq = pair_pmf A A'" "map_pmf snd pq = pair_pmf B B'"
```
```  1180     by (force elim: rel_pmf.cases)
```
```  1181   show "rel_pmf R A B"
```
```  1182   proof (rule rel_pmf.intros)
```
```  1183     let ?f = "\<lambda>(a, b). (fst a, fst b)"
```
```  1184     have [simp]: "(\<lambda>x. fst (?f x)) = fst o fst" "(\<lambda>x. snd (?f x)) = fst o snd"
```
```  1185       by auto
```
```  1186
```
```  1187     show "map_pmf fst (map_pmf ?f pq) = A"
```
```  1188       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
```
```  1189     show "map_pmf snd (map_pmf ?f pq) = B"
```
```  1190       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
```
```  1191
```
```  1192     fix a b assume "(a, b) \<in> set_pmf (map_pmf ?f pq)"
```
```  1193     then obtain c d where "((a, c), (b, d)) \<in> set_pmf pq"
```
```  1194       by auto
```
```  1195     from pq[OF this] show "R a b" ..
```
```  1196   qed
```
```  1197   show "rel_pmf S A' B'"
```
```  1198   proof (rule rel_pmf.intros)
```
```  1199     let ?f = "\<lambda>(a, b). (snd a, snd b)"
```
```  1200     have [simp]: "(\<lambda>x. fst (?f x)) = snd o fst" "(\<lambda>x. snd (?f x)) = snd o snd"
```
```  1201       by auto
```
```  1202
```
```  1203     show "map_pmf fst (map_pmf ?f pq) = A'"
```
```  1204       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
```
```  1205     show "map_pmf snd (map_pmf ?f pq) = B'"
```
```  1206       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
```
```  1207
```
```  1208     fix c d assume "(c, d) \<in> set_pmf (map_pmf ?f pq)"
```
```  1209     then obtain a b where "((a, c), (b, d)) \<in> set_pmf pq"
```
```  1210       by auto
```
```  1211     from pq[OF this] show "S c d" ..
```
```  1212   qed
```
```  1213 next
```
```  1214   assume "rel_pmf R A B" "rel_pmf S A' B'"
```
```  1215   then obtain Rpq Spq
```
```  1216     where Rpq: "\<And>a b. (a, b) \<in> set_pmf Rpq \<Longrightarrow> R a b"
```
```  1217         "map_pmf fst Rpq = A" "map_pmf snd Rpq = B"
```
```  1218       and Spq: "\<And>a b. (a, b) \<in> set_pmf Spq \<Longrightarrow> S a b"
```
```  1219         "map_pmf fst Spq = A'" "map_pmf snd Spq = B'"
```
```  1220     by (force elim: rel_pmf.cases)
```
```  1221
```
```  1222   let ?f = "(\<lambda>((a, c), (b, d)). ((a, b), (c, d)))"
```
```  1223   let ?pq = "map_pmf ?f (pair_pmf Rpq Spq)"
```
```  1224   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))"
```
```  1225     by auto
```
```  1226
```
```  1227   show "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
```
```  1228     by (rule rel_pmf.intros[where pq="?pq"])
```
```  1229        (auto simp: map_snd_pair_pmf map_fst_pair_pmf map_pmf_comp Rpq Spq
```
```  1230                    map_pair)
```
```  1231 qed
```
```  1232
```
```  1233 lemma rel_pmf_reflI:
```
```  1234   assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> P x x"
```
```  1235   shows "rel_pmf P p p"
```
```  1236   by (rule rel_pmf.intros[where pq="map_pmf (\<lambda>x. (x, x)) p"])
```
```  1237      (auto simp add: pmf.map_comp o_def assms)
```
```  1238
```
```  1239 lemma rel_pmf_bij_betw:
```
```  1240   assumes f: "bij_betw f (set_pmf p) (set_pmf q)"
```
```  1241   and eq: "\<And>x. x \<in> set_pmf p \<Longrightarrow> pmf p x = pmf q (f x)"
```
```  1242   shows "rel_pmf (\<lambda>x y. f x = y) p q"
```
```  1243 proof(rule rel_pmf.intros)
```
```  1244   let ?pq = "map_pmf (\<lambda>x. (x, f x)) p"
```
```  1245   show "map_pmf fst ?pq = p" by(simp add: pmf.map_comp o_def)
```
```  1246
```
```  1247   have "map_pmf f p = q"
```
```  1248   proof(rule pmf_eqI)
```
```  1249     fix i
```
```  1250     show "pmf (map_pmf f p) i = pmf q i"
```
```  1251     proof(cases "i \<in> set_pmf q")
```
```  1252       case True
```
```  1253       with f obtain j where "i = f j" "j \<in> set_pmf p"
```
```  1254         by(auto simp add: bij_betw_def image_iff)
```
```  1255       thus ?thesis using f by(simp add: bij_betw_def pmf_map_inj eq)
```
```  1256     next
```
```  1257       case False thus ?thesis
```
```  1258         by(subst pmf_map_outside)(auto simp add: set_pmf_iff eq[symmetric])
```
```  1259     qed
```
```  1260   qed
```
```  1261   then show "map_pmf snd ?pq = q" by(simp add: pmf.map_comp o_def)
```
```  1262 qed auto
```
```  1263
```
```  1264 context
```
```  1265 begin
```
```  1266
```
```  1267 interpretation pmf_as_measure .
```
```  1268
```
```  1269 definition "join_pmf M = bind_pmf M (\<lambda>x. x)"
```
```  1270
```
```  1271 lemma bind_eq_join_pmf: "bind_pmf M f = join_pmf (map_pmf f M)"
```
```  1272   unfolding join_pmf_def bind_map_pmf ..
```
```  1273
```
```  1274 lemma join_eq_bind_pmf: "join_pmf M = bind_pmf M id"
```
```  1275   by (simp add: join_pmf_def id_def)
```
```  1276
```
```  1277 lemma pmf_join: "pmf (join_pmf N) i = (\<integral>M. pmf M i \<partial>measure_pmf N)"
```
```  1278   unfolding join_pmf_def pmf_bind ..
```
```  1279
```
```  1280 lemma ereal_pmf_join: "ereal (pmf (join_pmf N) i) = (\<integral>\<^sup>+M. pmf M i \<partial>measure_pmf N)"
```
```  1281   unfolding join_pmf_def ereal_pmf_bind ..
```
```  1282
```
```  1283 lemma set_pmf_join_pmf[simp]: "set_pmf (join_pmf f) = (\<Union>p\<in>set_pmf f. set_pmf p)"
```
```  1284   by (simp add: join_pmf_def)
```
```  1285
```
```  1286 lemma join_return_pmf: "join_pmf (return_pmf M) = M"
```
```  1287   by (simp add: integral_return pmf_eq_iff pmf_join return_pmf.rep_eq)
```
```  1288
```
```  1289 lemma map_join_pmf: "map_pmf f (join_pmf AA) = join_pmf (map_pmf (map_pmf f) AA)"
```
```  1290   by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf)
```
```  1291
```
```  1292 lemma join_map_return_pmf: "join_pmf (map_pmf return_pmf A) = A"
```
```  1293   by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')
```
```  1294
```
```  1295 end
```
```  1296
```
```  1297 lemma rel_pmf_joinI:
```
```  1298   assumes "rel_pmf (rel_pmf P) p q"
```
```  1299   shows "rel_pmf P (join_pmf p) (join_pmf q)"
```
```  1300 proof -
```
```  1301   from assms obtain pq where p: "p = map_pmf fst pq"
```
```  1302     and q: "q = map_pmf snd pq"
```
```  1303     and P: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> rel_pmf P x y"
```
```  1304     by cases auto
```
```  1305   from P obtain PQ
```
```  1306     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"
```
```  1307     and x: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf fst (PQ x y) = x"
```
```  1308     and y: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf snd (PQ x y) = y"
```
```  1309     by(metis rel_pmf.simps)
```
```  1310
```
```  1311   let ?r = "bind_pmf pq (\<lambda>(x, y). PQ x y)"
```
```  1312   have "\<And>a b. (a, b) \<in> set_pmf ?r \<Longrightarrow> P a b" by (auto intro: PQ)
```
```  1313   moreover have "map_pmf fst ?r = join_pmf p" "map_pmf snd ?r = join_pmf q"
```
```  1314     by (simp_all add: p q x y join_pmf_def map_bind_pmf bind_map_pmf split_def cong: bind_pmf_cong)
```
```  1315   ultimately show ?thesis ..
```
```  1316 qed
```
```  1317
```
```  1318 lemma rel_pmf_bindI:
```
```  1319   assumes pq: "rel_pmf R p q"
```
```  1320   and fg: "\<And>x y. R x y \<Longrightarrow> rel_pmf P (f x) (g y)"
```
```  1321   shows "rel_pmf P (bind_pmf p f) (bind_pmf q g)"
```
```  1322   unfolding bind_eq_join_pmf
```
```  1323   by (rule rel_pmf_joinI)
```
```  1324      (auto simp add: pmf.rel_map intro: pmf.rel_mono[THEN le_funD, THEN le_funD, THEN le_boolD, THEN mp, OF _ pq] fg)
```
```  1325
```
```  1326 text \<open>
```
```  1327   Proof that @{const rel_pmf} preserves orders.
```
```  1328   Antisymmetry proof follows Thm. 1 in N. Saheb-Djahromi, Cpo's of measures for nondeterminism,
```
```  1329   Theoretical Computer Science 12(1):19--37, 1980,
```
```  1330   @{url "http://dx.doi.org/10.1016/0304-3975(80)90003-1"}
```
```  1331 \<close>
```
```  1332
```
```  1333 lemma
```
```  1334   assumes *: "rel_pmf R p q"
```
```  1335   and refl: "reflp R" and trans: "transp R"
```
```  1336   shows measure_Ici: "measure p {y. R x y} \<le> measure q {y. R x y}" (is ?thesis1)
```
```  1337   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)
```
```  1338 proof -
```
```  1339   from * obtain pq
```
```  1340     where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
```
```  1341     and p: "p = map_pmf fst pq"
```
```  1342     and q: "q = map_pmf snd pq"
```
```  1343     by cases auto
```
```  1344   show ?thesis1 ?thesis2 unfolding p q map_pmf_rep_eq using refl trans
```
```  1345     by(auto 4 3 simp add: measure_distr reflpD AE_measure_pmf_iff intro!: measure_pmf.finite_measure_mono_AE dest!: pq elim: transpE)
```
```  1346 qed
```
```  1347
```
```  1348 lemma rel_pmf_inf:
```
```  1349   fixes p q :: "'a pmf"
```
```  1350   assumes 1: "rel_pmf R p q"
```
```  1351   assumes 2: "rel_pmf R q p"
```
```  1352   and refl: "reflp R" and trans: "transp R"
```
```  1353   shows "rel_pmf (inf R R\<inverse>\<inverse>) p q"
```
```  1354 proof (subst rel_pmf_iff_equivp, safe)
```
```  1355   show "equivp (inf R R\<inverse>\<inverse>)"
```
```  1356     using trans refl by (auto simp: equivp_reflp_symp_transp intro: sympI transpI reflpI dest: transpD reflpD)
```
```  1357
```
```  1358   fix C assume "C \<in> UNIV // {(x, y). inf R R\<inverse>\<inverse> x y}"
```
```  1359   then obtain x where C: "C = {y. R x y \<and> R y x}"
```
```  1360     by (auto elim: quotientE)
```
```  1361
```
```  1362   let ?R = "\<lambda>x y. R x y \<and> R y x"
```
```  1363   let ?\<mu>R = "\<lambda>y. measure q {x. ?R x y}"
```
```  1364   have "measure p {y. ?R x y} = measure p ({y. R x y} - {y. R x y \<and> \<not> R y x})"
```
```  1365     by(auto intro!: arg_cong[where f="measure p"])
```
```  1366   also have "\<dots> = measure p {y. R x y} - measure p {y. R x y \<and> \<not> R y x}"
```
```  1367     by (rule measure_pmf.finite_measure_Diff) auto
```
```  1368   also have "measure p {y. R x y \<and> \<not> R y x} = measure q {y. R x y \<and> \<not> R y x}"
```
```  1369     using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ioi)
```
```  1370   also have "measure p {y. R x y} = measure q {y. R x y}"
```
```  1371     using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ici)
```
```  1372   also have "measure q {y. R x y} - measure q {y. R x y \<and> \<not> R y x} =
```
```  1373     measure q ({y. R x y} - {y. R x y \<and> \<not> R y x})"
```
```  1374     by(rule measure_pmf.finite_measure_Diff[symmetric]) auto
```
```  1375   also have "\<dots> = ?\<mu>R x"
```
```  1376     by(auto intro!: arg_cong[where f="measure q"])
```
```  1377   finally show "measure p C = measure q C"
```
```  1378     by (simp add: C conj_commute)
```
```  1379 qed
```
```  1380
```
```  1381 lemma rel_pmf_antisym:
```
```  1382   fixes p q :: "'a pmf"
```
```  1383   assumes 1: "rel_pmf R p q"
```
```  1384   assumes 2: "rel_pmf R q p"
```
```  1385   and refl: "reflp R" and trans: "transp R" and antisym: "antisymP R"
```
```  1386   shows "p = q"
```
```  1387 proof -
```
```  1388   from 1 2 refl trans have "rel_pmf (inf R R\<inverse>\<inverse>) p q" by(rule rel_pmf_inf)
```
```  1389   also have "inf R R\<inverse>\<inverse> = op ="
```
```  1390     using refl antisym by (auto intro!: ext simp add: reflpD dest: antisymD)
```
```  1391   finally show ?thesis unfolding pmf.rel_eq .
```
```  1392 qed
```
```  1393
```
```  1394 lemma reflp_rel_pmf: "reflp R \<Longrightarrow> reflp (rel_pmf R)"
```
```  1395 by(blast intro: reflpI rel_pmf_reflI reflpD)
```
```  1396
```
```  1397 lemma antisymP_rel_pmf:
```
```  1398   "\<lbrakk> reflp R; transp R; antisymP R \<rbrakk>
```
```  1399   \<Longrightarrow> antisymP (rel_pmf R)"
```
```  1400 by(rule antisymI)(blast intro: rel_pmf_antisym)
```
```  1401
```
```  1402 lemma transp_rel_pmf:
```
```  1403   assumes "transp R"
```
```  1404   shows "transp (rel_pmf R)"
```
```  1405 proof (rule transpI)
```
```  1406   fix x y z
```
```  1407   assume "rel_pmf R x y" and "rel_pmf R y z"
```
```  1408   hence "rel_pmf (R OO R) x z" by (simp add: pmf.rel_compp relcompp.relcompI)
```
```  1409   thus "rel_pmf R x z"
```
```  1410     using assms by (metis (no_types) pmf.rel_mono rev_predicate2D transp_relcompp_less_eq)
```
```  1411 qed
```
```  1412
```
```  1413 subsection \<open> Distributions \<close>
```
```  1414
```
```  1415 context
```
```  1416 begin
```
```  1417
```
```  1418 interpretation pmf_as_function .
```
```  1419
```
```  1420 subsubsection \<open> Bernoulli Distribution \<close>
```
```  1421
```
```  1422 lift_definition bernoulli_pmf :: "real \<Rightarrow> bool pmf" is
```
```  1423   "\<lambda>p b. ((\<lambda>p. if b then p else 1 - p) \<circ> min 1 \<circ> max 0) p"
```
```  1424   by (auto simp: nn_integral_count_space_finite[where A="{False, True}"] UNIV_bool
```
```  1425            split: split_max split_min)
```
```  1426
```
```  1427 lemma pmf_bernoulli_True[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) True = p"
```
```  1428   by transfer simp
```
```  1429
```
```  1430 lemma pmf_bernoulli_False[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) False = 1 - p"
```
```  1431   by transfer simp
```
```  1432
```
```  1433 lemma set_pmf_bernoulli: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (bernoulli_pmf p) = UNIV"
```
```  1434   by (auto simp add: set_pmf_iff UNIV_bool)
```
```  1435
```
```  1436 lemma nn_integral_bernoulli_pmf[simp]:
```
```  1437   assumes [simp]: "0 \<le> p" "p \<le> 1" "\<And>x. 0 \<le> f x"
```
```  1438   shows "(\<integral>\<^sup>+x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
```
```  1439   by (subst nn_integral_measure_pmf_support[of UNIV])
```
```  1440      (auto simp: UNIV_bool field_simps)
```
```  1441
```
```  1442 lemma integral_bernoulli_pmf[simp]:
```
```  1443   assumes [simp]: "0 \<le> p" "p \<le> 1"
```
```  1444   shows "(\<integral>x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
```
```  1445   by (subst integral_measure_pmf[of UNIV]) (auto simp: UNIV_bool)
```
```  1446
```
```  1447 lemma pmf_bernoulli_half [simp]: "pmf (bernoulli_pmf (1 / 2)) x = 1 / 2"
```
```  1448 by(cases x) simp_all
```
```  1449
```
```  1450 lemma measure_pmf_bernoulli_half: "measure_pmf (bernoulli_pmf (1 / 2)) = uniform_count_measure UNIV"
```
```  1451 by(rule measure_eqI)(simp_all add: nn_integral_pmf[symmetric] emeasure_uniform_count_measure nn_integral_count_space_finite sets_uniform_count_measure)
```
```  1452
```
```  1453 subsubsection \<open> Geometric Distribution \<close>
```
```  1454
```
```  1455 context
```
```  1456   fixes p :: real assumes p[arith]: "0 < p" "p \<le> 1"
```
```  1457 begin
```
```  1458
```
```  1459 lift_definition geometric_pmf :: "nat pmf" is "\<lambda>n. (1 - p)^n * p"
```
```  1460 proof
```
```  1461   have "(\<Sum>i. ereal (p * (1 - p) ^ i)) = ereal (p * (1 / (1 - (1 - p))))"
```
```  1462     by (intro sums_suminf_ereal sums_mult geometric_sums) auto
```
```  1463   then show "(\<integral>\<^sup>+ x. ereal ((1 - p)^x * p) \<partial>count_space UNIV) = 1"
```
```  1464     by (simp add: nn_integral_count_space_nat field_simps)
```
```  1465 qed simp
```
```  1466
```
```  1467 lemma pmf_geometric[simp]: "pmf geometric_pmf n = (1 - p)^n * p"
```
```  1468   by transfer rule
```
```  1469
```
```  1470 end
```
```  1471
```
```  1472 lemma set_pmf_geometric: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (geometric_pmf p) = UNIV"
```
```  1473   by (auto simp: set_pmf_iff)
```
```  1474
```
```  1475 subsubsection \<open> Uniform Multiset Distribution \<close>
```
```  1476
```
```  1477 context
```
```  1478   fixes M :: "'a multiset" assumes M_not_empty: "M \<noteq> {#}"
```
```  1479 begin
```
```  1480
```
```  1481 lift_definition pmf_of_multiset :: "'a pmf" is "\<lambda>x. count M x / size M"
```
```  1482 proof
```
```  1483   show "(\<integral>\<^sup>+ x. ereal (real (count M x) / real (size M)) \<partial>count_space UNIV) = 1"
```
```  1484     using M_not_empty
```
```  1485     by (simp add: zero_less_divide_iff nn_integral_count_space nonempty_has_size
```
```  1486                   setsum_divide_distrib[symmetric])
```
```  1487        (auto simp: size_multiset_overloaded_eq intro!: setsum.cong)
```
```  1488 qed simp
```
```  1489
```
```  1490 lemma pmf_of_multiset[simp]: "pmf pmf_of_multiset x = count M x / size M"
```
```  1491   by transfer rule
```
```  1492
```
```  1493 lemma set_pmf_of_multiset[simp]: "set_pmf pmf_of_multiset = set_mset M"
```
```  1494   by (auto simp: set_pmf_iff)
```
```  1495
```
```  1496 end
```
```  1497
```
```  1498 subsubsection \<open> Uniform Distribution \<close>
```
```  1499
```
```  1500 context
```
```  1501   fixes S :: "'a set" assumes S_not_empty: "S \<noteq> {}" and S_finite: "finite S"
```
```  1502 begin
```
```  1503
```
```  1504 lift_definition pmf_of_set :: "'a pmf" is "\<lambda>x. indicator S x / card S"
```
```  1505 proof
```
```  1506   show "(\<integral>\<^sup>+ x. ereal (indicator S x / real (card S)) \<partial>count_space UNIV) = 1"
```
```  1507     using S_not_empty S_finite by (subst nn_integral_count_space'[of S]) auto
```
```  1508 qed simp
```
```  1509
```
```  1510 lemma pmf_of_set[simp]: "pmf pmf_of_set x = indicator S x / card S"
```
```  1511   by transfer rule
```
```  1512
```
```  1513 lemma set_pmf_of_set[simp]: "set_pmf pmf_of_set = S"
```
```  1514   using S_finite S_not_empty by (auto simp: set_pmf_iff)
```
```  1515
```
```  1516 lemma emeasure_pmf_of_set_space[simp]: "emeasure pmf_of_set S = 1"
```
```  1517   by (rule measure_pmf.emeasure_eq_1_AE) (auto simp: AE_measure_pmf_iff)
```
```  1518
```
```  1519 lemma nn_integral_pmf_of_set':
```
```  1520   "(\<And>x. x \<in> S \<Longrightarrow> f x \<ge> 0) \<Longrightarrow> nn_integral (measure_pmf pmf_of_set) f = setsum f S / card S"
```
```  1521 apply(subst nn_integral_measure_pmf_finite, simp_all add: S_finite)
```
```  1522 apply(simp add: setsum_ereal_left_distrib[symmetric])
```
```  1523 apply(subst ereal_divide', simp add: S_not_empty S_finite)
```
```  1524 apply(simp add: ereal_times_divide_eq one_ereal_def[symmetric])
```
```  1525 done
```
```  1526
```
```  1527 lemma nn_integral_pmf_of_set:
```
```  1528   "nn_integral (measure_pmf pmf_of_set) f = setsum (max 0 \<circ> f) S / card S"
```
```  1529 apply(subst nn_integral_max_0[symmetric])
```
```  1530 apply(subst nn_integral_pmf_of_set')
```
```  1531 apply simp_all
```
```  1532 done
```
```  1533
```
```  1534 lemma integral_pmf_of_set:
```
```  1535   "integral\<^sup>L (measure_pmf pmf_of_set) f = setsum f S / card S"
```
```  1536 apply(simp add: real_lebesgue_integral_def integrable_measure_pmf_finite nn_integral_pmf_of_set S_finite)
```
```  1537 apply(subst real_of_ereal_minus')
```
```  1538  apply(simp add: ereal_max_0 S_finite del: ereal_max)
```
```  1539 apply(simp add: ereal_max_0 S_finite S_not_empty del: ereal_max)
```
```  1540 apply(simp add: field_simps S_finite S_not_empty)
```
```  1541 apply(subst setsum.distrib[symmetric])
```
```  1542 apply(rule setsum.cong; simp_all)
```
```  1543 done
```
```  1544
```
```  1545 lemma emeasure_pmf_of_set:
```
```  1546   "emeasure (measure_pmf pmf_of_set) A = card (S \<inter> A) / card S"
```
```  1547 apply(subst nn_integral_indicator[symmetric], simp)
```
```  1548 apply(subst nn_integral_pmf_of_set)
```
```  1549 apply(simp add: o_def max_def ereal_indicator[symmetric] S_not_empty S_finite real_of_nat_indicator[symmetric] of_nat_setsum[symmetric] setsum_indicator_eq_card del: of_nat_setsum)
```
```  1550 done
```
```  1551
```
```  1552 end
```
```  1553
```
```  1554 lemma pmf_of_set_singleton: "pmf_of_set {x} = return_pmf x"
```
```  1555 by(rule pmf_eqI)(simp add: indicator_def)
```
```  1556
```
```  1557 lemma map_pmf_of_set_inj:
```
```  1558   assumes f: "inj_on f A"
```
```  1559   and [simp]: "A \<noteq> {}" "finite A"
```
```  1560   shows "map_pmf f (pmf_of_set A) = pmf_of_set (f ` A)" (is "?lhs = ?rhs")
```
```  1561 proof(rule pmf_eqI)
```
```  1562   fix i
```
```  1563   show "pmf ?lhs i = pmf ?rhs i"
```
```  1564   proof(cases "i \<in> f ` A")
```
```  1565     case True
```
```  1566     then obtain i' where "i = f i'" "i' \<in> A" by auto
```
```  1567     thus ?thesis using f by(simp add: card_image pmf_map_inj)
```
```  1568   next
```
```  1569     case False
```
```  1570     hence "pmf ?lhs i = 0" by(simp add: pmf_eq_0_set_pmf set_map_pmf)
```
```  1571     moreover have "pmf ?rhs i = 0" using False by simp
```
```  1572     ultimately show ?thesis by simp
```
```  1573   qed
```
```  1574 qed
```
```  1575
```
```  1576 lemma bernoulli_pmf_half_conv_pmf_of_set: "bernoulli_pmf (1 / 2) = pmf_of_set UNIV"
```
```  1577 by(rule pmf_eqI) simp_all
```
```  1578
```
```  1579
```
```  1580
```
```  1581 lemma measure_pmf_of_set:
```
```  1582   assumes "S \<noteq> {}" "finite S"
```
```  1583   shows "measure (measure_pmf (pmf_of_set S)) A = card (S \<inter> A) / card S"
```
```  1584 using emeasure_pmf_of_set[OF assms, of A]
```
```  1585 unfolding measure_pmf.emeasure_eq_measure by simp
```
```  1586
```
```  1587 subsubsection \<open> Poisson Distribution \<close>
```
```  1588
```
```  1589 context
```
```  1590   fixes rate :: real assumes rate_pos: "0 < rate"
```
```  1591 begin
```
```  1592
```
```  1593 lift_definition poisson_pmf :: "nat pmf" is "\<lambda>k. rate ^ k / fact k * exp (-rate)"
```
```  1594 proof  (* by Manuel Eberl *)
```
```  1595   have summable: "summable (\<lambda>x::nat. rate ^ x / fact x)" using summable_exp
```
```  1596     by (simp add: field_simps divide_inverse [symmetric])
```
```  1597   have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x * exp (-rate) \<partial>count_space UNIV) =
```
```  1598           exp (-rate) * (\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV)"
```
```  1599     by (simp add: field_simps nn_integral_cmult[symmetric])
```
```  1600   also from rate_pos have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV) = (\<Sum>x. rate ^ x / fact x)"
```
```  1601     by (simp_all add: nn_integral_count_space_nat suminf_ereal summable suminf_ereal_finite)
```
```  1602   also have "... = exp rate" unfolding exp_def
```
```  1603     by (simp add: field_simps divide_inverse [symmetric])
```
```  1604   also have "ereal (exp (-rate)) * ereal (exp rate) = 1"
```
```  1605     by (simp add: mult_exp_exp)
```
```  1606   finally show "(\<integral>\<^sup>+ x. ereal (rate ^ x / (fact x) * exp (- rate)) \<partial>count_space UNIV) = 1" .
```
```  1607 qed (simp add: rate_pos[THEN less_imp_le])
```
```  1608
```
```  1609 lemma pmf_poisson[simp]: "pmf poisson_pmf k = rate ^ k / fact k * exp (-rate)"
```
```  1610   by transfer rule
```
```  1611
```
```  1612 lemma set_pmf_poisson[simp]: "set_pmf poisson_pmf = UNIV"
```
```  1613   using rate_pos by (auto simp: set_pmf_iff)
```
```  1614
```
```  1615 end
```
```  1616
```
```  1617 subsubsection \<open> Binomial Distribution \<close>
```
```  1618
```
```  1619 context
```
```  1620   fixes n :: nat and p :: real assumes p_nonneg: "0 \<le> p" and p_le_1: "p \<le> 1"
```
```  1621 begin
```
```  1622
```
```  1623 lift_definition binomial_pmf :: "nat pmf" is "\<lambda>k. (n choose k) * p^k * (1 - p)^(n - k)"
```
```  1624 proof
```
```  1625   have "(\<integral>\<^sup>+k. ereal (real (n choose k) * p ^ k * (1 - p) ^ (n - k)) \<partial>count_space UNIV) =
```
```  1626     ereal (\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k))"
```
```  1627     using p_le_1 p_nonneg by (subst nn_integral_count_space') auto
```
```  1628   also have "(\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k)) = (p + (1 - p)) ^ n"
```
```  1629     by (subst binomial_ring) (simp add: atLeast0AtMost)
```
```  1630   finally show "(\<integral>\<^sup>+ x. ereal (real (n choose x) * p ^ x * (1 - p) ^ (n - x)) \<partial>count_space UNIV) = 1"
```
```  1631     by simp
```
```  1632 qed (insert p_nonneg p_le_1, simp)
```
```  1633
```
```  1634 lemma pmf_binomial[simp]: "pmf binomial_pmf k = (n choose k) * p^k * (1 - p)^(n - k)"
```
```  1635   by transfer rule
```
```  1636
```
```  1637 lemma set_pmf_binomial_eq: "set_pmf binomial_pmf = (if p = 0 then {0} else if p = 1 then {n} else {.. n})"
```
```  1638   using p_nonneg p_le_1 unfolding set_eq_iff set_pmf_iff pmf_binomial by (auto simp: set_pmf_iff)
```
```  1639
```
```  1640 end
```
```  1641
```
```  1642 end
```
```  1643
```
```  1644 lemma set_pmf_binomial_0[simp]: "set_pmf (binomial_pmf n 0) = {0}"
```
```  1645   by (simp add: set_pmf_binomial_eq)
```
```  1646
```
```  1647 lemma set_pmf_binomial_1[simp]: "set_pmf (binomial_pmf n 1) = {n}"
```
```  1648   by (simp add: set_pmf_binomial_eq)
```
```  1649
```
```  1650 lemma set_pmf_binomial[simp]: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (binomial_pmf n p) = {..n}"
```
```  1651   by (simp add: set_pmf_binomial_eq)
```
```  1652
```
```  1653 context begin interpretation lifting_syntax .
```
```  1654
```
```  1655 lemma bind_pmf_parametric [transfer_rule]:
```
```  1656   "(rel_pmf A ===> (A ===> rel_pmf B) ===> rel_pmf B) bind_pmf bind_pmf"
```
```  1657 by(blast intro: rel_pmf_bindI dest: rel_funD)
```
```  1658
```
```  1659 lemma return_pmf_parametric [transfer_rule]: "(A ===> rel_pmf A) return_pmf return_pmf"
```
```  1660 by(rule rel_funI) simp
```
```  1661
```
```  1662 end
```
```  1663
```
```  1664 end
```