src/HOL/Probability/Probability_Mass_Function.thy
 author Andreas Lochbihler Fri Jan 09 09:16:51 2015 +0100 (2015-01-09) changeset 59325 922d31f5c3f5 parent 59134 a71f2e256ee2 child 59327 8a779359df67 permissions -rw-r--r--
simplify construction for distribution of rel_pmf over op OO
```     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/Number_Theory/Binomial"
```
```    12   "~~/src/HOL/Library/Multiset"
```
```    13 begin
```
```    14
```
```    15 lemma bind_return'': "sets M = sets N \<Longrightarrow> M \<guillemotright>= return N = M"
```
```    16    by (cases "space M = {}")
```
```    17       (simp_all add: bind_empty space_empty[symmetric] bind_nonempty join_return'
```
```    18                 cong: subprob_algebra_cong)
```
```    19
```
```    20
```
```    21 lemma (in prob_space) distr_const[simp]:
```
```    22   "c \<in> space N \<Longrightarrow> distr M N (\<lambda>x. c) = return N c"
```
```    23   by (rule measure_eqI) (auto simp: emeasure_distr emeasure_space_1)
```
```    24
```
```    25 lemma (in finite_measure) countable_support:
```
```    26   "countable {x. measure M {x} \<noteq> 0}"
```
```    27 proof cases
```
```    28   assume "measure M (space M) = 0"
```
```    29   with bounded_measure measure_le_0_iff have "{x. measure M {x} \<noteq> 0} = {}"
```
```    30     by auto
```
```    31   then show ?thesis
```
```    32     by simp
```
```    33 next
```
```    34   let ?M = "measure M (space M)" and ?m = "\<lambda>x. measure M {x}"
```
```    35   assume "?M \<noteq> 0"
```
```    36   then have *: "{x. ?m x \<noteq> 0} = (\<Union>n. {x. ?M / Suc n < ?m x})"
```
```    37     using reals_Archimedean[of "?m x / ?M" for x]
```
```    38     by (auto simp: field_simps not_le[symmetric] measure_nonneg divide_le_0_iff measure_le_0_iff)
```
```    39   have **: "\<And>n. finite {x. ?M / Suc n < ?m x}"
```
```    40   proof (rule ccontr)
```
```    41     fix n assume "infinite {x. ?M / Suc n < ?m x}" (is "infinite ?X")
```
```    42     then obtain X where "finite X" "card X = Suc (Suc n)" "X \<subseteq> ?X"
```
```    43       by (metis infinite_arbitrarily_large)
```
```    44     from this(3) have *: "\<And>x. x \<in> X \<Longrightarrow> ?M / Suc n \<le> ?m x"
```
```    45       by auto
```
```    46     { fix x assume "x \<in> X"
```
```    47       from `?M \<noteq> 0` *[OF this] have "?m x \<noteq> 0" by (auto simp: field_simps measure_le_0_iff)
```
```    48       then have "{x} \<in> sets M" by (auto dest: measure_notin_sets) }
```
```    49     note singleton_sets = this
```
```    50     have "?M < (\<Sum>x\<in>X. ?M / Suc n)"
```
```    51       using `?M \<noteq> 0`
```
```    52       by (simp add: `card X = Suc (Suc n)` real_eq_of_nat[symmetric] real_of_nat_Suc field_simps less_le measure_nonneg)
```
```    53     also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"
```
```    54       by (rule setsum_mono) fact
```
```    55     also have "\<dots> = measure M (\<Union>x\<in>X. {x})"
```
```    56       using singleton_sets `finite X`
```
```    57       by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def)
```
```    58     finally have "?M < measure M (\<Union>x\<in>X. {x})" .
```
```    59     moreover have "measure M (\<Union>x\<in>X. {x}) \<le> ?M"
```
```    60       using singleton_sets[THEN sets.sets_into_space] by (intro finite_measure_mono) auto
```
```    61     ultimately show False by simp
```
```    62   qed
```
```    63   show ?thesis
```
```    64     unfolding * by (intro countable_UN countableI_type countable_finite[OF **])
```
```    65 qed
```
```    66
```
```    67 lemma (in finite_measure) AE_support_countable:
```
```    68   assumes [simp]: "sets M = UNIV"
```
```    69   shows "(AE x in M. measure M {x} \<noteq> 0) \<longleftrightarrow> (\<exists>S. countable S \<and> (AE x in M. x \<in> S))"
```
```    70 proof
```
```    71   assume "\<exists>S. countable S \<and> (AE x in M. x \<in> S)"
```
```    72   then obtain S where S[intro]: "countable S" and ae: "AE x in M. x \<in> S"
```
```    73     by auto
```
```    74   then have "emeasure M (\<Union>x\<in>{x\<in>S. emeasure M {x} \<noteq> 0}. {x}) =
```
```    75     (\<integral>\<^sup>+ x. emeasure M {x} * indicator {x\<in>S. emeasure M {x} \<noteq> 0} x \<partial>count_space UNIV)"
```
```    76     by (subst emeasure_UN_countable)
```
```    77        (auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space)
```
```    78   also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} * indicator S x \<partial>count_space UNIV)"
```
```    79     by (auto intro!: nn_integral_cong split: split_indicator)
```
```    80   also have "\<dots> = emeasure M (\<Union>x\<in>S. {x})"
```
```    81     by (subst emeasure_UN_countable)
```
```    82        (auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space)
```
```    83   also have "\<dots> = emeasure M (space M)"
```
```    84     using ae by (intro emeasure_eq_AE) auto
```
```    85   finally have "emeasure M {x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0} = emeasure M (space M)"
```
```    86     by (simp add: emeasure_single_in_space cong: rev_conj_cong)
```
```    87   with finite_measure_compl[of "{x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0}"]
```
```    88   have "AE x in M. x \<in> S \<and> emeasure M {x} \<noteq> 0"
```
```    89     by (intro AE_I[OF order_refl]) (auto simp: emeasure_eq_measure set_diff_eq cong: conj_cong)
```
```    90   then show "AE x in M. measure M {x} \<noteq> 0"
```
```    91     by (auto simp: emeasure_eq_measure)
```
```    92 qed (auto intro!: exI[of _ "{x. measure M {x} \<noteq> 0}"] countable_support)
```
```    93
```
```    94 subsection {* PMF as measure *}
```
```    95
```
```    96 typedef 'a pmf = "{M :: 'a measure. prob_space M \<and> sets M = UNIV \<and> (AE x in M. measure M {x} \<noteq> 0)}"
```
```    97   morphisms measure_pmf Abs_pmf
```
```    98   by (intro exI[of _ "uniform_measure (count_space UNIV) {undefined}"])
```
```    99      (auto intro!: prob_space_uniform_measure AE_uniform_measureI)
```
```   100
```
```   101 declare [[coercion measure_pmf]]
```
```   102
```
```   103 lemma prob_space_measure_pmf: "prob_space (measure_pmf p)"
```
```   104   using pmf.measure_pmf[of p] by auto
```
```   105
```
```   106 interpretation measure_pmf!: prob_space "measure_pmf M" for M
```
```   107   by (rule prob_space_measure_pmf)
```
```   108
```
```   109 interpretation measure_pmf!: subprob_space "measure_pmf M" for M
```
```   110   by (rule prob_space_imp_subprob_space) unfold_locales
```
```   111
```
```   112 lemma subprob_space_measure_pmf: "subprob_space (measure_pmf x)"
```
```   113   by unfold_locales
```
```   114
```
```   115 locale pmf_as_measure
```
```   116 begin
```
```   117
```
```   118 setup_lifting type_definition_pmf
```
```   119
```
```   120 end
```
```   121
```
```   122 context
```
```   123 begin
```
```   124
```
```   125 interpretation pmf_as_measure .
```
```   126
```
```   127 lift_definition pmf :: "'a pmf \<Rightarrow> 'a \<Rightarrow> real" is "\<lambda>M x. measure M {x}" .
```
```   128
```
```   129 lift_definition set_pmf :: "'a pmf \<Rightarrow> 'a set" is "\<lambda>M. {x. measure M {x} \<noteq> 0}" .
```
```   130
```
```   131 lift_definition map_pmf :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf" is
```
```   132   "\<lambda>f M. distr M (count_space UNIV) f"
```
```   133 proof safe
```
```   134   fix M and f :: "'a \<Rightarrow> 'b"
```
```   135   let ?D = "distr M (count_space UNIV) f"
```
```   136   assume "prob_space M" and [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
```
```   137   interpret prob_space M by fact
```
```   138   from ae have "AE x in M. measure M (f -` {f x}) \<noteq> 0"
```
```   139   proof eventually_elim
```
```   140     fix x
```
```   141     have "measure M {x} \<le> measure M (f -` {f x})"
```
```   142       by (intro finite_measure_mono) auto
```
```   143     then show "measure M {x} \<noteq> 0 \<Longrightarrow> measure M (f -` {f x}) \<noteq> 0"
```
```   144       using measure_nonneg[of M "{x}"] by auto
```
```   145   qed
```
```   146   then show "AE x in ?D. measure ?D {x} \<noteq> 0"
```
```   147     by (simp add: AE_distr_iff measure_distr measurable_def)
```
```   148 qed (auto simp: measurable_def prob_space.prob_space_distr)
```
```   149
```
```   150 declare [[coercion set_pmf]]
```
```   151
```
```   152 lemma countable_set_pmf [simp]: "countable (set_pmf p)"
```
```   153   by transfer (metis prob_space.finite_measure finite_measure.countable_support)
```
```   154
```
```   155 lemma sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV"
```
```   156   by transfer metis
```
```   157
```
```   158 lemma sets_measure_pmf_count_space[measurable_cong]:
```
```   159   "sets (measure_pmf M) = sets (count_space UNIV)"
```
```   160   by simp
```
```   161
```
```   162 lemma space_measure_pmf[simp]: "space (measure_pmf p) = UNIV"
```
```   163   using sets_eq_imp_space_eq[of "measure_pmf p" "count_space UNIV"] by simp
```
```   164
```
```   165 lemma measure_pmf_in_subprob_algebra[measurable (raw)]: "measure_pmf x \<in> space (subprob_algebra (count_space UNIV))"
```
```   166   by (simp add: space_subprob_algebra subprob_space_measure_pmf)
```
```   167
```
```   168 lemma measurable_pmf_measure1[simp]: "measurable (M :: 'a pmf) N = UNIV \<rightarrow> space N"
```
```   169   by (auto simp: measurable_def)
```
```   170
```
```   171 lemma measurable_pmf_measure2[simp]: "measurable N (M :: 'a pmf) = measurable N (count_space UNIV)"
```
```   172   by (intro measurable_cong_sets) simp_all
```
```   173
```
```   174 lemma pmf_positive: "x \<in> set_pmf p \<Longrightarrow> 0 < pmf p x"
```
```   175   by transfer (simp add: less_le measure_nonneg)
```
```   176
```
```   177 lemma pmf_nonneg: "0 \<le> pmf p x"
```
```   178   by transfer (simp add: measure_nonneg)
```
```   179
```
```   180 lemma pmf_le_1: "pmf p x \<le> 1"
```
```   181   by (simp add: pmf.rep_eq)
```
```   182
```
```   183 lemma emeasure_pmf_single:
```
```   184   fixes M :: "'a pmf"
```
```   185   shows "emeasure M {x} = pmf M x"
```
```   186   by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
```
```   187
```
```   188 lemma AE_measure_pmf: "AE x in (M::'a pmf). x \<in> M"
```
```   189   by transfer simp
```
```   190
```
```   191 lemma emeasure_pmf_single_eq_zero_iff:
```
```   192   fixes M :: "'a pmf"
```
```   193   shows "emeasure M {y} = 0 \<longleftrightarrow> y \<notin> M"
```
```   194   by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
```
```   195
```
```   196 lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \<longleftrightarrow> (\<forall>y\<in>M. P y)"
```
```   197 proof -
```
```   198   { fix y assume y: "y \<in> M" and P: "AE x in M. P x" "\<not> P y"
```
```   199     with P have "AE x in M. x \<noteq> y"
```
```   200       by auto
```
```   201     with y have False
```
```   202       by (simp add: emeasure_pmf_single_eq_zero_iff AE_iff_measurable[OF _ refl]) }
```
```   203   then show ?thesis
```
```   204     using AE_measure_pmf[of M] by auto
```
```   205 qed
```
```   206
```
```   207 lemma set_pmf_not_empty: "set_pmf M \<noteq> {}"
```
```   208   using AE_measure_pmf[of M] by (intro notI) simp
```
```   209
```
```   210 lemma set_pmf_iff: "x \<in> set_pmf M \<longleftrightarrow> pmf M x \<noteq> 0"
```
```   211   by transfer simp
```
```   212
```
```   213 lemma emeasure_measure_pmf_finite: "finite S \<Longrightarrow> emeasure (measure_pmf M) S = (\<Sum>s\<in>S. pmf M s)"
```
```   214   by (subst emeasure_eq_setsum_singleton) (auto simp: emeasure_pmf_single)
```
```   215
```
```   216 lemma measure_measure_pmf_finite: "finite S \<Longrightarrow> measure (measure_pmf M) S = setsum (pmf M) S"
```
```   217 using emeasure_measure_pmf_finite[of S M]
```
```   218 by(simp add: measure_pmf.emeasure_eq_measure)
```
```   219
```
```   220 lemma nn_integral_measure_pmf_support:
```
```   221   fixes f :: "'a \<Rightarrow> ereal"
```
```   222   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"
```
```   223   shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>A. f x * pmf M x)"
```
```   224 proof -
```
```   225   have "(\<integral>\<^sup>+x. f x \<partial>M) = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)"
```
```   226     using nn by (intro nn_integral_cong_AE) (auto simp: AE_measure_pmf_iff split: split_indicator)
```
```   227   also have "\<dots> = (\<Sum>x\<in>A. f x * emeasure M {x})"
```
```   228     using assms by (intro nn_integral_indicator_finite) auto
```
```   229   finally show ?thesis
```
```   230     by (simp add: emeasure_measure_pmf_finite)
```
```   231 qed
```
```   232
```
```   233 lemma nn_integral_measure_pmf_finite:
```
```   234   fixes f :: "'a \<Rightarrow> ereal"
```
```   235   assumes f: "finite (set_pmf M)" and nn: "\<And>x. x \<in> set_pmf M \<Longrightarrow> 0 \<le> f x"
```
```   236   shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>set_pmf M. f x * pmf M x)"
```
```   237   using assms by (intro nn_integral_measure_pmf_support) auto
```
```   238 lemma integrable_measure_pmf_finite:
```
```   239   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
```
```   240   shows "finite (set_pmf M) \<Longrightarrow> integrable M f"
```
```   241   by (auto intro!: integrableI_bounded simp: nn_integral_measure_pmf_finite)
```
```   242
```
```   243 lemma integral_measure_pmf:
```
```   244   assumes [simp]: "finite A" and "\<And>a. a \<in> set_pmf M \<Longrightarrow> f a \<noteq> 0 \<Longrightarrow> a \<in> A"
```
```   245   shows "(\<integral>x. f x \<partial>measure_pmf M) = (\<Sum>a\<in>A. f a * pmf M a)"
```
```   246 proof -
```
```   247   have "(\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x * indicator A x \<partial>measure_pmf M)"
```
```   248     using assms(2) by (intro integral_cong_AE) (auto split: split_indicator simp: AE_measure_pmf_iff)
```
```   249   also have "\<dots> = (\<Sum>a\<in>A. f a * pmf M a)"
```
```   250     by (subst integral_indicator_finite_real) (auto simp: measure_def emeasure_measure_pmf_finite)
```
```   251   finally show ?thesis .
```
```   252 qed
```
```   253
```
```   254 lemma integrable_pmf: "integrable (count_space X) (pmf M)"
```
```   255 proof -
```
```   256   have " (\<integral>\<^sup>+ x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+ x. pmf M x \<partial>count_space (M \<inter> X))"
```
```   257     by (auto simp add: nn_integral_count_space_indicator set_pmf_iff intro!: nn_integral_cong split: split_indicator)
```
```   258   then have "integrable (count_space X) (pmf M) = integrable (count_space (M \<inter> X)) (pmf M)"
```
```   259     by (simp add: integrable_iff_bounded pmf_nonneg)
```
```   260   then show ?thesis
```
```   261     by (simp add: pmf.rep_eq measure_pmf.integrable_measure disjoint_family_on_def)
```
```   262 qed
```
```   263
```
```   264 lemma integral_pmf: "(\<integral>x. pmf M x \<partial>count_space X) = measure M X"
```
```   265 proof -
```
```   266   have "(\<integral>x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+x. pmf M x \<partial>count_space X)"
```
```   267     by (simp add: pmf_nonneg integrable_pmf nn_integral_eq_integral)
```
```   268   also have "\<dots> = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space (X \<inter> M))"
```
```   269     by (auto intro!: nn_integral_cong_AE split: split_indicator
```
```   270              simp: pmf.rep_eq measure_pmf.emeasure_eq_measure nn_integral_count_space_indicator
```
```   271                    AE_count_space set_pmf_iff)
```
```   272   also have "\<dots> = emeasure M (X \<inter> M)"
```
```   273     by (rule emeasure_countable_singleton[symmetric]) (auto intro: countable_set_pmf)
```
```   274   also have "\<dots> = emeasure M X"
```
```   275     by (auto intro!: emeasure_eq_AE simp: AE_measure_pmf_iff)
```
```   276   finally show ?thesis
```
```   277     by (simp add: measure_pmf.emeasure_eq_measure)
```
```   278 qed
```
```   279
```
```   280 lemma integral_pmf_restrict:
```
```   281   "(f::'a \<Rightarrow> 'b::{banach, second_countable_topology}) \<in> borel_measurable (count_space UNIV) \<Longrightarrow>
```
```   282     (\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x \<partial>restrict_space M M)"
```
```   283   by (auto intro!: integral_cong_AE simp add: integral_restrict_space AE_measure_pmf_iff)
```
```   284
```
```   285 lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1"
```
```   286 proof -
```
```   287   have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)"
```
```   288     by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf)
```
```   289   then show ?thesis
```
```   290     using measure_pmf.emeasure_space_1 by simp
```
```   291 qed
```
```   292
```
```   293 lemma in_null_sets_measure_pmfI:
```
```   294   "A \<inter> set_pmf p = {} \<Longrightarrow> A \<in> null_sets (measure_pmf p)"
```
```   295 using emeasure_eq_0_AE[where ?P="\<lambda>x. x \<in> A" and M="measure_pmf p"]
```
```   296 by(auto simp add: null_sets_def AE_measure_pmf_iff)
```
```   297
```
```   298 lemma map_pmf_id[simp]: "map_pmf id = id"
```
```   299   by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI)
```
```   300
```
```   301 lemma map_pmf_ident[simp]: "map_pmf (\<lambda>x. x) = (\<lambda>x. x)"
```
```   302   using map_pmf_id unfolding id_def .
```
```   303
```
```   304 lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g"
```
```   305   by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def)
```
```   306
```
```   307 lemma map_pmf_comp: "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M"
```
```   308   using map_pmf_compose[of f g] by (simp add: comp_def)
```
```   309
```
```   310 lemma map_pmf_cong:
```
```   311   assumes "p = q"
```
```   312   shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> map_pmf f p = map_pmf g q"
```
```   313   unfolding `p = q`[symmetric] measure_pmf_inject[symmetric] map_pmf.rep_eq
```
```   314   by (auto simp add: emeasure_distr AE_measure_pmf_iff intro!: emeasure_eq_AE measure_eqI)
```
```   315
```
```   316 lemma emeasure_map_pmf[simp]: "emeasure (map_pmf f M) X = emeasure M (f -` X)"
```
```   317   unfolding map_pmf.rep_eq by (subst emeasure_distr) auto
```
```   318
```
```   319 lemma nn_integral_map_pmf[simp]: "(\<integral>\<^sup>+x. f x \<partial>map_pmf g M) = (\<integral>\<^sup>+x. f (g x) \<partial>M)"
```
```   320   unfolding map_pmf.rep_eq by (intro nn_integral_distr) auto
```
```   321
```
```   322 lemma ereal_pmf_map: "pmf (map_pmf f p) x = (\<integral>\<^sup>+ y. indicator (f -` {x}) y \<partial>measure_pmf p)"
```
```   323 proof(transfer fixing: f x)
```
```   324   fix p :: "'b measure"
```
```   325   presume "prob_space p"
```
```   326   then interpret prob_space p .
```
```   327   presume "sets p = UNIV"
```
```   328   then show "ereal (measure (distr p (count_space UNIV) f) {x}) = integral\<^sup>N p (indicator (f -` {x}))"
```
```   329     by(simp add: measure_distr measurable_def emeasure_eq_measure)
```
```   330 qed simp_all
```
```   331
```
```   332 lemma pmf_set_map:
```
```   333   fixes f :: "'a \<Rightarrow> 'b"
```
```   334   shows "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
```
```   335 proof (rule, transfer, clarsimp simp add: measure_distr measurable_def)
```
```   336   fix f :: "'a \<Rightarrow> 'b" and M :: "'a measure"
```
```   337   assume "prob_space M" and ae: "AE x in M. measure M {x} \<noteq> 0" and [simp]: "sets M = UNIV"
```
```   338   interpret prob_space M by fact
```
```   339   show "{x. measure M (f -` {x}) \<noteq> 0} = f ` {x. measure M {x} \<noteq> 0}"
```
```   340   proof safe
```
```   341     fix x assume "measure M (f -` {x}) \<noteq> 0"
```
```   342     moreover have "measure M (f -` {x}) = measure M {y. f y = x \<and> measure M {y} \<noteq> 0}"
```
```   343       using ae by (intro finite_measure_eq_AE) auto
```
```   344     ultimately have "{y. f y = x \<and> measure M {y} \<noteq> 0} \<noteq> {}"
```
```   345       by (metis measure_empty)
```
```   346     then show "x \<in> f ` {x. measure M {x} \<noteq> 0}"
```
```   347       by auto
```
```   348   next
```
```   349     fix x assume "measure M {x} \<noteq> 0"
```
```   350     then have "0 < measure M {x}"
```
```   351       using measure_nonneg[of M "{x}"] by auto
```
```   352     also have "measure M {x} \<le> measure M (f -` {f x})"
```
```   353       by (intro finite_measure_mono) auto
```
```   354     finally show "measure M (f -` {f x}) = 0 \<Longrightarrow> False"
```
```   355       by simp
```
```   356   qed
```
```   357 qed
```
```   358
```
```   359 lemma set_map_pmf: "set_pmf (map_pmf f M) = f`set_pmf M"
```
```   360   using pmf_set_map[of f] by (auto simp: comp_def fun_eq_iff)
```
```   361
```
```   362 lemma nn_integral_pmf: "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = emeasure (measure_pmf p) A"
```
```   363 proof -
```
```   364   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))"
```
```   365     by(auto simp add: nn_integral_count_space_indicator indicator_def set_pmf_iff intro: nn_integral_cong)
```
```   366   also have "\<dots> = emeasure (measure_pmf p) (\<Union>x\<in>A \<inter> set_pmf p. {x})"
```
```   367     by(subst emeasure_UN_countable)(auto simp add: emeasure_pmf_single disjoint_family_on_def)
```
```   368   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})"
```
```   369     by(rule emeasure_Un_null_set[symmetric])(auto intro: in_null_sets_measure_pmfI)
```
```   370   also have "\<dots> = emeasure (measure_pmf p) A"
```
```   371     by(auto intro: arg_cong2[where f=emeasure])
```
```   372   finally show ?thesis .
```
```   373 qed
```
```   374
```
```   375 subsection {* PMFs as function *}
```
```   376
```
```   377 context
```
```   378   fixes f :: "'a \<Rightarrow> real"
```
```   379   assumes nonneg: "\<And>x. 0 \<le> f x"
```
```   380   assumes prob: "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
```
```   381 begin
```
```   382
```
```   383 lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ereal \<circ> f)"
```
```   384 proof (intro conjI)
```
```   385   have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
```
```   386     by (simp split: split_indicator)
```
```   387   show "AE x in density (count_space UNIV) (ereal \<circ> f).
```
```   388     measure (density (count_space UNIV) (ereal \<circ> f)) {x} \<noteq> 0"
```
```   389     by (simp add: AE_density nonneg measure_def emeasure_density max_def)
```
```   390   show "prob_space (density (count_space UNIV) (ereal \<circ> f))"
```
```   391     by default (simp add: emeasure_density prob)
```
```   392 qed simp
```
```   393
```
```   394 lemma pmf_embed_pmf: "pmf embed_pmf x = f x"
```
```   395 proof transfer
```
```   396   have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
```
```   397     by (simp split: split_indicator)
```
```   398   fix x show "measure (density (count_space UNIV) (ereal \<circ> f)) {x} = f x"
```
```   399     by transfer (simp add: measure_def emeasure_density nonneg max_def)
```
```   400 qed
```
```   401
```
```   402 end
```
```   403
```
```   404 lemma embed_pmf_transfer:
```
```   405   "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"
```
```   406   by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer)
```
```   407
```
```   408 lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)"
```
```   409 proof (transfer, elim conjE)
```
```   410   fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
```
```   411   assume "prob_space M" then interpret prob_space M .
```
```   412   show "M = density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))"
```
```   413   proof (rule measure_eqI)
```
```   414     fix A :: "'a set"
```
```   415     have "(\<integral>\<^sup>+ x. ereal (measure M {x}) * indicator A x \<partial>count_space UNIV) =
```
```   416       (\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)"
```
```   417       by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator)
```
```   418     also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))"
```
```   419       by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space)
```
```   420     also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})"
```
```   421       by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support)
```
```   422          (auto simp: disjoint_family_on_def)
```
```   423     also have "\<dots> = emeasure M A"
```
```   424       using ae by (intro emeasure_eq_AE) auto
```
```   425     finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))) A"
```
```   426       using emeasure_space_1 by (simp add: emeasure_density)
```
```   427   qed simp
```
```   428 qed
```
```   429
```
```   430 lemma td_pmf_embed_pmf:
```
```   431   "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}"
```
```   432   unfolding type_definition_def
```
```   433 proof safe
```
```   434   fix p :: "'a pmf"
```
```   435   have "(\<integral>\<^sup>+ x. 1 \<partial>measure_pmf p) = 1"
```
```   436     using measure_pmf.emeasure_space_1[of p] by simp
```
```   437   then show *: "(\<integral>\<^sup>+ x. ereal (pmf p x) \<partial>count_space UNIV) = 1"
```
```   438     by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const)
```
```   439
```
```   440   show "embed_pmf (pmf p) = p"
```
```   441     by (intro measure_pmf_inject[THEN iffD1])
```
```   442        (simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def)
```
```   443 next
```
```   444   fix f :: "'a \<Rightarrow> real" assume "\<forall>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
```
```   445   then show "pmf (embed_pmf f) = f"
```
```   446     by (auto intro!: pmf_embed_pmf)
```
```   447 qed (rule pmf_nonneg)
```
```   448
```
```   449 end
```
```   450
```
```   451 locale pmf_as_function
```
```   452 begin
```
```   453
```
```   454 setup_lifting td_pmf_embed_pmf
```
```   455
```
```   456 lemma set_pmf_transfer[transfer_rule]:
```
```   457   assumes "bi_total A"
```
```   458   shows "rel_fun (pcr_pmf A) (rel_set A) (\<lambda>f. {x. f x \<noteq> 0}) set_pmf"
```
```   459   using `bi_total A`
```
```   460   by (auto simp: pcr_pmf_def cr_pmf_def rel_fun_def rel_set_def bi_total_def Bex_def set_pmf_iff)
```
```   461      metis+
```
```   462
```
```   463 end
```
```   464
```
```   465 context
```
```   466 begin
```
```   467
```
```   468 interpretation pmf_as_function .
```
```   469
```
```   470 lemma pmf_eqI: "(\<And>i. pmf M i = pmf N i) \<Longrightarrow> M = N"
```
```   471   by transfer auto
```
```   472
```
```   473 lemma pmf_eq_iff: "M = N \<longleftrightarrow> (\<forall>i. pmf M i = pmf N i)"
```
```   474   by (auto intro: pmf_eqI)
```
```   475
```
```   476 end
```
```   477
```
```   478 context
```
```   479 begin
```
```   480
```
```   481 interpretation pmf_as_function .
```
```   482
```
```   483 subsubsection \<open> Bernoulli Distribution \<close>
```
```   484
```
```   485 lift_definition bernoulli_pmf :: "real \<Rightarrow> bool pmf" is
```
```   486   "\<lambda>p b. ((\<lambda>p. if b then p else 1 - p) \<circ> min 1 \<circ> max 0) p"
```
```   487   by (auto simp: nn_integral_count_space_finite[where A="{False, True}"] UNIV_bool
```
```   488            split: split_max split_min)
```
```   489
```
```   490 lemma pmf_bernoulli_True[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) True = p"
```
```   491   by transfer simp
```
```   492
```
```   493 lemma pmf_bernoulli_False[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) False = 1 - p"
```
```   494   by transfer simp
```
```   495
```
```   496 lemma set_pmf_bernoulli: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (bernoulli_pmf p) = UNIV"
```
```   497   by (auto simp add: set_pmf_iff UNIV_bool)
```
```   498
```
```   499 lemma nn_integral_bernoulli_pmf[simp]:
```
```   500   assumes [simp]: "0 \<le> p" "p \<le> 1" "\<And>x. 0 \<le> f x"
```
```   501   shows "(\<integral>\<^sup>+x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
```
```   502   by (subst nn_integral_measure_pmf_support[of UNIV])
```
```   503      (auto simp: UNIV_bool field_simps)
```
```   504
```
```   505 lemma integral_bernoulli_pmf[simp]:
```
```   506   assumes [simp]: "0 \<le> p" "p \<le> 1"
```
```   507   shows "(\<integral>x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)"
```
```   508   by (subst integral_measure_pmf[of UNIV]) (auto simp: UNIV_bool)
```
```   509
```
```   510 subsubsection \<open> Geometric Distribution \<close>
```
```   511
```
```   512 lift_definition geometric_pmf :: "nat pmf" is "\<lambda>n. 1 / 2^Suc n"
```
```   513 proof
```
```   514   note geometric_sums[of "1 / 2"]
```
```   515   note sums_mult[OF this, of "1 / 2"]
```
```   516   from sums_suminf_ereal[OF this]
```
```   517   show "(\<integral>\<^sup>+ x. ereal (1 / 2 ^ Suc x) \<partial>count_space UNIV) = 1"
```
```   518     by (simp add: nn_integral_count_space_nat field_simps)
```
```   519 qed simp
```
```   520
```
```   521 lemma pmf_geometric[simp]: "pmf geometric_pmf n = 1 / 2^Suc n"
```
```   522   by transfer rule
```
```   523
```
```   524 lemma set_pmf_geometric[simp]: "set_pmf geometric_pmf = UNIV"
```
```   525   by (auto simp: set_pmf_iff)
```
```   526
```
```   527 subsubsection \<open> Uniform Multiset Distribution \<close>
```
```   528
```
```   529 context
```
```   530   fixes M :: "'a multiset" assumes M_not_empty: "M \<noteq> {#}"
```
```   531 begin
```
```   532
```
```   533 lift_definition pmf_of_multiset :: "'a pmf" is "\<lambda>x. count M x / size M"
```
```   534 proof
```
```   535   show "(\<integral>\<^sup>+ x. ereal (real (count M x) / real (size M)) \<partial>count_space UNIV) = 1"
```
```   536     using M_not_empty
```
```   537     by (simp add: zero_less_divide_iff nn_integral_count_space nonempty_has_size
```
```   538                   setsum_divide_distrib[symmetric])
```
```   539        (auto simp: size_multiset_overloaded_eq intro!: setsum.cong)
```
```   540 qed simp
```
```   541
```
```   542 lemma pmf_of_multiset[simp]: "pmf pmf_of_multiset x = count M x / size M"
```
```   543   by transfer rule
```
```   544
```
```   545 lemma set_pmf_of_multiset[simp]: "set_pmf pmf_of_multiset = set_of M"
```
```   546   by (auto simp: set_pmf_iff)
```
```   547
```
```   548 end
```
```   549
```
```   550 subsubsection \<open> Uniform Distribution \<close>
```
```   551
```
```   552 context
```
```   553   fixes S :: "'a set" assumes S_not_empty: "S \<noteq> {}" and S_finite: "finite S"
```
```   554 begin
```
```   555
```
```   556 lift_definition pmf_of_set :: "'a pmf" is "\<lambda>x. indicator S x / card S"
```
```   557 proof
```
```   558   show "(\<integral>\<^sup>+ x. ereal (indicator S x / real (card S)) \<partial>count_space UNIV) = 1"
```
```   559     using S_not_empty S_finite by (subst nn_integral_count_space'[of S]) auto
```
```   560 qed simp
```
```   561
```
```   562 lemma pmf_of_set[simp]: "pmf pmf_of_set x = indicator S x / card S"
```
```   563   by transfer rule
```
```   564
```
```   565 lemma set_pmf_of_set[simp]: "set_pmf pmf_of_set = S"
```
```   566   using S_finite S_not_empty by (auto simp: set_pmf_iff)
```
```   567
```
```   568 lemma emeasure_pmf_of_set[simp]: "emeasure pmf_of_set S = 1"
```
```   569   by (rule measure_pmf.emeasure_eq_1_AE) (auto simp: AE_measure_pmf_iff)
```
```   570
```
```   571 end
```
```   572
```
```   573 subsubsection \<open> Poisson Distribution \<close>
```
```   574
```
```   575 context
```
```   576   fixes rate :: real assumes rate_pos: "0 < rate"
```
```   577 begin
```
```   578
```
```   579 lift_definition poisson_pmf :: "nat pmf" is "\<lambda>k. rate ^ k / fact k * exp (-rate)"
```
```   580 proof
```
```   581   (* Proof by Manuel Eberl *)
```
```   582
```
```   583   have summable: "summable (\<lambda>x::nat. rate ^ x / fact x)" using summable_exp
```
```   584     by (simp add: field_simps field_divide_inverse[symmetric])
```
```   585   have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x * exp (-rate) \<partial>count_space UNIV) =
```
```   586           exp (-rate) * (\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV)"
```
```   587     by (simp add: field_simps nn_integral_cmult[symmetric])
```
```   588   also from rate_pos have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV) = (\<Sum>x. rate ^ x / fact x)"
```
```   589     by (simp_all add: nn_integral_count_space_nat suminf_ereal summable suminf_ereal_finite)
```
```   590   also have "... = exp rate" unfolding exp_def
```
```   591     by (simp add: field_simps field_divide_inverse[symmetric] transfer_int_nat_factorial)
```
```   592   also have "ereal (exp (-rate)) * ereal (exp rate) = 1"
```
```   593     by (simp add: mult_exp_exp)
```
```   594   finally show "(\<integral>\<^sup>+ x. ereal (rate ^ x / real (fact x) * exp (- rate)) \<partial>count_space UNIV) = 1" .
```
```   595 qed (simp add: rate_pos[THEN less_imp_le])
```
```   596
```
```   597 lemma pmf_poisson[simp]: "pmf poisson_pmf k = rate ^ k / fact k * exp (-rate)"
```
```   598   by transfer rule
```
```   599
```
```   600 lemma set_pmf_poisson[simp]: "set_pmf poisson_pmf = UNIV"
```
```   601   using rate_pos by (auto simp: set_pmf_iff)
```
```   602
```
```   603 end
```
```   604
```
```   605 subsubsection \<open> Binomial Distribution \<close>
```
```   606
```
```   607 context
```
```   608   fixes n :: nat and p :: real assumes p_nonneg: "0 \<le> p" and p_le_1: "p \<le> 1"
```
```   609 begin
```
```   610
```
```   611 lift_definition binomial_pmf :: "nat pmf" is "\<lambda>k. (n choose k) * p^k * (1 - p)^(n - k)"
```
```   612 proof
```
```   613   have "(\<integral>\<^sup>+k. ereal (real (n choose k) * p ^ k * (1 - p) ^ (n - k)) \<partial>count_space UNIV) =
```
```   614     ereal (\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k))"
```
```   615     using p_le_1 p_nonneg by (subst nn_integral_count_space') auto
```
```   616   also have "(\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k)) = (p + (1 - p)) ^ n"
```
```   617     by (subst binomial_ring) (simp add: atLeast0AtMost real_of_nat_def)
```
```   618   finally show "(\<integral>\<^sup>+ x. ereal (real (n choose x) * p ^ x * (1 - p) ^ (n - x)) \<partial>count_space UNIV) = 1"
```
```   619     by simp
```
```   620 qed (insert p_nonneg p_le_1, simp)
```
```   621
```
```   622 lemma pmf_binomial[simp]: "pmf binomial_pmf k = (n choose k) * p^k * (1 - p)^(n - k)"
```
```   623   by transfer rule
```
```   624
```
```   625 lemma set_pmf_binomial_eq: "set_pmf binomial_pmf = (if p = 0 then {0} else if p = 1 then {n} else {.. n})"
```
```   626   using p_nonneg p_le_1 unfolding set_eq_iff set_pmf_iff pmf_binomial by (auto simp: set_pmf_iff)
```
```   627
```
```   628 end
```
```   629
```
```   630 end
```
```   631
```
```   632 lemma set_pmf_binomial_0[simp]: "set_pmf (binomial_pmf n 0) = {0}"
```
```   633   by (simp add: set_pmf_binomial_eq)
```
```   634
```
```   635 lemma set_pmf_binomial_1[simp]: "set_pmf (binomial_pmf n 1) = {n}"
```
```   636   by (simp add: set_pmf_binomial_eq)
```
```   637
```
```   638 lemma set_pmf_binomial[simp]: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (binomial_pmf n p) = {..n}"
```
```   639   by (simp add: set_pmf_binomial_eq)
```
```   640
```
```   641 subsection \<open> Monad Interpretation \<close>
```
```   642
```
```   643 lemma measurable_measure_pmf[measurable]:
```
```   644   "(\<lambda>x. measure_pmf (M x)) \<in> measurable (count_space UNIV) (subprob_algebra (count_space UNIV))"
```
```   645   by (auto simp: space_subprob_algebra intro!: prob_space_imp_subprob_space) unfold_locales
```
```   646
```
```   647 lemma bind_pmf_cong:
```
```   648   assumes "\<And>x. A x \<in> space (subprob_algebra N)" "\<And>x. B x \<in> space (subprob_algebra N)"
```
```   649   assumes "\<And>i. i \<in> set_pmf x \<Longrightarrow> A i = B i"
```
```   650   shows "bind (measure_pmf x) A = bind (measure_pmf x) B"
```
```   651 proof (rule measure_eqI)
```
```   652   show "sets (measure_pmf x \<guillemotright>= A) = sets (measure_pmf x \<guillemotright>= B)"
```
```   653     using assms by (subst (1 2) sets_bind) (auto simp: space_subprob_algebra)
```
```   654 next
```
```   655   fix X assume "X \<in> sets (measure_pmf x \<guillemotright>= A)"
```
```   656   then have X: "X \<in> sets N"
```
```   657     using assms by (subst (asm) sets_bind) (auto simp: space_subprob_algebra)
```
```   658   show "emeasure (measure_pmf x \<guillemotright>= A) X = emeasure (measure_pmf x \<guillemotright>= B) X"
```
```   659     using assms
```
```   660     by (subst (1 2) emeasure_bind[where N=N, OF _ _ X])
```
```   661        (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
```
```   662 qed
```
```   663
```
```   664 context
```
```   665 begin
```
```   666
```
```   667 interpretation pmf_as_measure .
```
```   668
```
```   669 lift_definition join_pmf :: "'a pmf pmf \<Rightarrow> 'a pmf" is "\<lambda>M. measure_pmf M \<guillemotright>= measure_pmf"
```
```   670 proof (intro conjI)
```
```   671   fix M :: "'a pmf pmf"
```
```   672
```
```   673   interpret bind: prob_space "measure_pmf M \<guillemotright>= measure_pmf"
```
```   674     apply (intro measure_pmf.prob_space_bind[where S="count_space UNIV"] AE_I2)
```
```   675     apply (auto intro!: subprob_space_measure_pmf simp: space_subprob_algebra)
```
```   676     apply unfold_locales
```
```   677     done
```
```   678   show "prob_space (measure_pmf M \<guillemotright>= measure_pmf)"
```
```   679     by intro_locales
```
```   680   show "sets (measure_pmf M \<guillemotright>= measure_pmf) = UNIV"
```
```   681     by (subst sets_bind) auto
```
```   682   have "AE x in measure_pmf M \<guillemotright>= measure_pmf. emeasure (measure_pmf M \<guillemotright>= measure_pmf) {x} \<noteq> 0"
```
```   683     by (auto simp: AE_bind[where B="count_space UNIV"] measure_pmf_in_subprob_algebra
```
```   684                    emeasure_bind[where N="count_space UNIV"] AE_measure_pmf_iff nn_integral_0_iff_AE
```
```   685                    measure_pmf.emeasure_eq_measure measure_le_0_iff set_pmf_iff pmf.rep_eq)
```
```   686   then show "AE x in measure_pmf M \<guillemotright>= measure_pmf. measure (measure_pmf M \<guillemotright>= measure_pmf) {x} \<noteq> 0"
```
```   687     unfolding bind.emeasure_eq_measure by simp
```
```   688 qed
```
```   689
```
```   690 lemma pmf_join: "pmf (join_pmf N) i = (\<integral>M. pmf M i \<partial>measure_pmf N)"
```
```   691 proof (transfer fixing: N i)
```
```   692   have N: "subprob_space (measure_pmf N)"
```
```   693     by (rule prob_space_imp_subprob_space) intro_locales
```
```   694   show "measure (measure_pmf N \<guillemotright>= measure_pmf) {i} = integral\<^sup>L (measure_pmf N) (\<lambda>M. measure M {i})"
```
```   695     using measurable_measure_pmf[of "\<lambda>x. x"]
```
```   696     by (intro subprob_space.measure_bind[where N="count_space UNIV", OF N]) auto
```
```   697 qed (auto simp: Transfer.Rel_def rel_fun_def cr_pmf_def)
```
```   698
```
```   699 lemma set_pmf_join_pmf: "set_pmf (join_pmf f) = (\<Union>p\<in>set_pmf f. set_pmf p)"
```
```   700 apply(simp add: set_eq_iff set_pmf_iff pmf_join)
```
```   701 apply(subst integral_nonneg_eq_0_iff_AE)
```
```   702 apply(auto simp add: pmf_le_1 pmf_nonneg AE_measure_pmf_iff intro!: measure_pmf.integrable_const_bound[where B=1])
```
```   703 done
```
```   704
```
```   705 lift_definition return_pmf :: "'a \<Rightarrow> 'a pmf" is "return (count_space UNIV)"
```
```   706   by (auto intro!: prob_space_return simp: AE_return measure_return)
```
```   707
```
```   708 lemma join_return_pmf: "join_pmf (return_pmf M) = M"
```
```   709   by (simp add: integral_return pmf_eq_iff pmf_join return_pmf.rep_eq)
```
```   710
```
```   711 lemma map_return_pmf: "map_pmf f (return_pmf x) = return_pmf (f x)"
```
```   712   by transfer (simp add: distr_return)
```
```   713
```
```   714 lemma map_pmf_const[simp]: "map_pmf (\<lambda>_. c) M = return_pmf c"
```
```   715   by transfer (auto simp: prob_space.distr_const)
```
```   716
```
```   717 lemma set_return_pmf: "set_pmf (return_pmf x) = {x}"
```
```   718   by transfer (auto simp add: measure_return split: split_indicator)
```
```   719
```
```   720 lemma pmf_return: "pmf (return_pmf x) y = indicator {y} x"
```
```   721   by transfer (simp add: measure_return)
```
```   722
```
```   723 lemma nn_integral_return_pmf[simp]: "0 \<le> f x \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>return_pmf x) = f x"
```
```   724   unfolding return_pmf.rep_eq by (intro nn_integral_return) auto
```
```   725
```
```   726 lemma emeasure_return_pmf[simp]: "emeasure (return_pmf x) X = indicator X x"
```
```   727   unfolding return_pmf.rep_eq by (intro emeasure_return) auto
```
```   728
```
```   729 end
```
```   730
```
```   731 definition "bind_pmf M f = join_pmf (map_pmf f M)"
```
```   732
```
```   733 lemma (in pmf_as_measure) bind_transfer[transfer_rule]:
```
```   734   "rel_fun pmf_as_measure.cr_pmf (rel_fun (rel_fun op = pmf_as_measure.cr_pmf) pmf_as_measure.cr_pmf) op \<guillemotright>= bind_pmf"
```
```   735 proof (auto simp: pmf_as_measure.cr_pmf_def rel_fun_def bind_pmf_def join_pmf.rep_eq map_pmf.rep_eq)
```
```   736   fix M f and g :: "'a \<Rightarrow> 'b pmf" assume "\<forall>x. f x = measure_pmf (g x)"
```
```   737   then have f: "f = (\<lambda>x. measure_pmf (g x))"
```
```   738     by auto
```
```   739   show "measure_pmf M \<guillemotright>= f = distr (measure_pmf M) (count_space UNIV) g \<guillemotright>= measure_pmf"
```
```   740     unfolding f by (subst bind_distr[OF _ measurable_measure_pmf]) auto
```
```   741 qed
```
```   742
```
```   743 lemma pmf_bind: "pmf (bind_pmf N f) i = (\<integral>x. pmf (f x) i \<partial>measure_pmf N)"
```
```   744   by (auto intro!: integral_distr simp: bind_pmf_def pmf_join map_pmf.rep_eq)
```
```   745
```
```   746 lemma bind_return_pmf: "bind_pmf (return_pmf x) f = f x"
```
```   747   unfolding bind_pmf_def map_return_pmf join_return_pmf ..
```
```   748
```
```   749 lemma join_eq_bind_pmf: "join_pmf M = bind_pmf M id"
```
```   750   by (simp add: bind_pmf_def)
```
```   751
```
```   752 lemma bind_pmf_const[simp]: "bind_pmf M (\<lambda>x. c) = c"
```
```   753   unfolding bind_pmf_def map_pmf_const join_return_pmf ..
```
```   754
```
```   755 lemma set_bind_pmf: "set_pmf (bind_pmf M N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))"
```
```   756   apply (simp add: set_eq_iff set_pmf_iff pmf_bind)
```
```   757   apply (subst integral_nonneg_eq_0_iff_AE)
```
```   758   apply (auto simp: pmf_nonneg pmf_le_1 AE_measure_pmf_iff
```
```   759               intro!: measure_pmf.integrable_const_bound[where B=1])
```
```   760   done
```
```   761
```
```   762 lemma measurable_pair_restrict_pmf2:
```
```   763   assumes "countable A"
```
```   764   assumes "\<And>y. y \<in> A \<Longrightarrow> (\<lambda>x. f (x, y)) \<in> measurable M L"
```
```   765   shows "f \<in> measurable (M \<Otimes>\<^sub>M restrict_space (measure_pmf N) A) L"
```
```   766   apply (subst measurable_cong_sets)
```
```   767   apply (rule sets_pair_measure_cong sets_restrict_space_cong sets_measure_pmf_count_space refl)+
```
```   768   apply (simp_all add: restrict_count_space)
```
```   769   apply (subst split_eta[symmetric])
```
```   770   unfolding measurable_split_conv
```
```   771   apply (rule measurable_compose_countable'[OF _ measurable_snd `countable A`])
```
```   772   apply (rule measurable_compose[OF measurable_fst])
```
```   773   apply fact
```
```   774   done
```
```   775
```
```   776 lemma measurable_pair_restrict_pmf1:
```
```   777   assumes "countable A"
```
```   778   assumes "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N L"
```
```   779   shows "f \<in> measurable (restrict_space (measure_pmf M) A \<Otimes>\<^sub>M N) L"
```
```   780   apply (subst measurable_cong_sets)
```
```   781   apply (rule sets_pair_measure_cong sets_restrict_space_cong sets_measure_pmf_count_space refl)+
```
```   782   apply (simp_all add: restrict_count_space)
```
```   783   apply (subst split_eta[symmetric])
```
```   784   unfolding measurable_split_conv
```
```   785   apply (rule measurable_compose_countable'[OF _ measurable_fst `countable A`])
```
```   786   apply (rule measurable_compose[OF measurable_snd])
```
```   787   apply fact
```
```   788   done
```
```   789
```
```   790 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))"
```
```   791   unfolding pmf_eq_iff pmf_bind
```
```   792 proof
```
```   793   fix i
```
```   794   interpret B: prob_space "restrict_space B B"
```
```   795     by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
```
```   796        (auto simp: AE_measure_pmf_iff)
```
```   797   interpret A: prob_space "restrict_space A A"
```
```   798     by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
```
```   799        (auto simp: AE_measure_pmf_iff)
```
```   800
```
```   801   interpret AB: pair_prob_space "restrict_space A A" "restrict_space B B"
```
```   802     by unfold_locales
```
```   803
```
```   804   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)"
```
```   805     by (rule integral_cong) (auto intro!: integral_pmf_restrict)
```
```   806   also have "\<dots> = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>restrict_space A A)"
```
```   807     by (intro integral_pmf_restrict B.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
```
```   808               countable_set_pmf borel_measurable_count_space)
```
```   809   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>restrict_space B B)"
```
```   810     by (rule AB.Fubini_integral[symmetric])
```
```   811        (auto intro!: AB.integrable_const_bound[where B=1] measurable_pair_restrict_pmf2
```
```   812              simp: pmf_nonneg pmf_le_1 measurable_restrict_space1)
```
```   813   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>B)"
```
```   814     by (intro integral_pmf_restrict[symmetric] A.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
```
```   815               countable_set_pmf borel_measurable_count_space)
```
```   816   also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>A \<partial>B)"
```
```   817     by (rule integral_cong) (auto intro!: integral_pmf_restrict[symmetric])
```
```   818   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)" .
```
```   819 qed
```
```   820
```
```   821
```
```   822 context
```
```   823 begin
```
```   824
```
```   825 interpretation pmf_as_measure .
```
```   826
```
```   827 lemma measure_pmf_bind: "measure_pmf (bind_pmf M f) = (measure_pmf M \<guillemotright>= (\<lambda>x. measure_pmf (f x)))"
```
```   828   by transfer simp
```
```   829
```
```   830 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)"
```
```   831   using measurable_measure_pmf[of N]
```
```   832   unfolding measure_pmf_bind
```
```   833   apply (subst (1 3) nn_integral_max_0[symmetric])
```
```   834   apply (intro nn_integral_bind[where B="count_space UNIV"])
```
```   835   apply auto
```
```   836   done
```
```   837
```
```   838 lemma emeasure_bind_pmf[simp]: "emeasure (bind_pmf M N) X = (\<integral>\<^sup>+x. emeasure (N x) X \<partial>M)"
```
```   839   using measurable_measure_pmf[of N]
```
```   840   unfolding measure_pmf_bind
```
```   841   by (subst emeasure_bind[where N="count_space UNIV"]) auto
```
```   842
```
```   843 lemma bind_return_pmf': "bind_pmf N return_pmf = N"
```
```   844 proof (transfer, clarify)
```
```   845   fix N :: "'a measure" assume "sets N = UNIV" then show "N \<guillemotright>= return (count_space UNIV) = N"
```
```   846     by (subst return_sets_cong[where N=N]) (simp_all add: bind_return')
```
```   847 qed
```
```   848
```
```   849 lemma bind_return_pmf'': "bind_pmf N (\<lambda>x. return_pmf (f x)) = map_pmf f N"
```
```   850 proof (transfer, clarify)
```
```   851   fix N :: "'b measure" and f :: "'b \<Rightarrow> 'a" assume "prob_space N" "sets N = UNIV"
```
```   852   then show "N \<guillemotright>= (\<lambda>x. return (count_space UNIV) (f x)) = distr N (count_space UNIV) f"
```
```   853     by (subst bind_return_distr[symmetric])
```
```   854        (auto simp: prob_space.not_empty measurable_def comp_def)
```
```   855 qed
```
```   856
```
```   857 lemma bind_assoc_pmf: "bind_pmf (bind_pmf A B) C = bind_pmf A (\<lambda>x. bind_pmf (B x) C)"
```
```   858   by transfer
```
```   859      (auto intro!: bind_assoc[where N="count_space UNIV" and R="count_space UNIV"]
```
```   860            simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space)
```
```   861
```
```   862 end
```
```   863
```
```   864 lemma map_join_pmf: "map_pmf f (join_pmf AA) = join_pmf (map_pmf (map_pmf f) AA)"
```
```   865   unfolding bind_pmf_def[symmetric]
```
```   866   unfolding bind_return_pmf''[symmetric] join_eq_bind_pmf bind_assoc_pmf
```
```   867   by (simp add: bind_return_pmf'')
```
```   868
```
```   869 definition "pair_pmf A B = bind_pmf A (\<lambda>x. bind_pmf B (\<lambda>y. return_pmf (x, y)))"
```
```   870
```
```   871 lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b"
```
```   872   unfolding pair_pmf_def pmf_bind pmf_return
```
```   873   apply (subst integral_measure_pmf[where A="{b}"])
```
```   874   apply (auto simp: indicator_eq_0_iff)
```
```   875   apply (subst integral_measure_pmf[where A="{a}"])
```
```   876   apply (auto simp: indicator_eq_0_iff setsum_nonneg_eq_0_iff pmf_nonneg)
```
```   877   done
```
```   878
```
```   879 lemma set_pair_pmf: "set_pmf (pair_pmf A B) = set_pmf A \<times> set_pmf B"
```
```   880   unfolding pair_pmf_def set_bind_pmf set_return_pmf by auto
```
```   881
```
```   882 lemma measure_pmf_in_subprob_space[measurable (raw)]:
```
```   883   "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
```
```   884   by (simp add: space_subprob_algebra) intro_locales
```
```   885
```
```   886 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)"
```
```   887 proof -
```
```   888   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)"
```
```   889     by (subst nn_integral_max_0[symmetric])
```
```   890        (auto simp: AE_measure_pmf_iff set_pair_pmf intro!: nn_integral_cong_AE)
```
```   891   also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) * indicator (A \<times> B) (a, b) \<partial>B \<partial>A)"
```
```   892     by (simp add: pair_pmf_def)
```
```   893   also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) \<partial>B \<partial>A)"
```
```   894     by (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
```
```   895   finally show ?thesis
```
```   896     unfolding nn_integral_max_0 .
```
```   897 qed
```
```   898
```
```   899 lemma pair_map_pmf1: "pair_pmf (map_pmf f A) B = map_pmf (apfst f) (pair_pmf A B)"
```
```   900 proof (safe intro!: pmf_eqI)
```
```   901   fix a :: "'a" and b :: "'b"
```
```   902   have [simp]: "\<And>c d. indicator (apfst f -` {(a, b)}) (c, d) = indicator (f -` {a}) c * (indicator {b} d::ereal)"
```
```   903     by (auto split: split_indicator)
```
```   904
```
```   905   have "ereal (pmf (pair_pmf (map_pmf f A) B) (a, b)) =
```
```   906          ereal (pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b))"
```
```   907     unfolding pmf_pair ereal_pmf_map
```
```   908     by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_multc pmf_nonneg
```
```   909                   emeasure_map_pmf[symmetric] del: emeasure_map_pmf)
```
```   910   then show "pmf (pair_pmf (map_pmf f A) B) (a, b) = pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b)"
```
```   911     by simp
```
```   912 qed
```
```   913
```
```   914 lemma pair_map_pmf2: "pair_pmf A (map_pmf f B) = map_pmf (apsnd f) (pair_pmf A B)"
```
```   915 proof (safe intro!: pmf_eqI)
```
```   916   fix a :: "'a" and b :: "'b"
```
```   917   have [simp]: "\<And>c d. indicator (apsnd f -` {(a, b)}) (c, d) = indicator {a} c * (indicator (f -` {b}) d::ereal)"
```
```   918     by (auto split: split_indicator)
```
```   919
```
```   920   have "ereal (pmf (pair_pmf A (map_pmf f B)) (a, b)) =
```
```   921          ereal (pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b))"
```
```   922     unfolding pmf_pair ereal_pmf_map
```
```   923     by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_cmult nn_integral_multc pmf_nonneg
```
```   924                   emeasure_map_pmf[symmetric] del: emeasure_map_pmf)
```
```   925   then show "pmf (pair_pmf A (map_pmf f B)) (a, b) = pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b)"
```
```   926     by simp
```
```   927 qed
```
```   928
```
```   929 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)"
```
```   930   by (simp add: pair_map_pmf2 pair_map_pmf1 map_pmf_comp split_beta')
```
```   931
```
```   932 lemma bind_pair_pmf:
```
```   933   assumes M[measurable]: "M \<in> measurable (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) (subprob_algebra N)"
```
```   934   shows "measure_pmf (pair_pmf A B) \<guillemotright>= M = (measure_pmf A \<guillemotright>= (\<lambda>x. measure_pmf B \<guillemotright>= (\<lambda>y. M (x, y))))"
```
```   935     (is "?L = ?R")
```
```   936 proof (rule measure_eqI)
```
```   937   have M'[measurable]: "M \<in> measurable (pair_pmf A B) (subprob_algebra N)"
```
```   938     using M[THEN measurable_space] by (simp_all add: space_pair_measure)
```
```   939
```
```   940   note measurable_bind[where N="count_space UNIV", measurable]
```
```   941   note measure_pmf_in_subprob_space[simp]
```
```   942
```
```   943   have sets_eq_N: "sets ?L = N"
```
```   944     by (subst sets_bind[OF sets_kernel[OF M']]) auto
```
```   945   show "sets ?L = sets ?R"
```
```   946     using measurable_space[OF M]
```
```   947     by (simp add: sets_eq_N space_pair_measure space_subprob_algebra)
```
```   948   fix X assume "X \<in> sets ?L"
```
```   949   then have X[measurable]: "X \<in> sets N"
```
```   950     unfolding sets_eq_N .
```
```   951   then show "emeasure ?L X = emeasure ?R X"
```
```   952     apply (simp add: emeasure_bind[OF _ M' X])
```
```   953     apply (simp add: nn_integral_bind[where B="count_space UNIV"] pair_pmf_def measure_pmf_bind[of A]
```
```   954       nn_integral_measure_pmf_finite set_return_pmf emeasure_nonneg pmf_return one_ereal_def[symmetric])
```
```   955     apply (subst emeasure_bind[OF _ _ X])
```
```   956     apply measurable
```
```   957     apply (subst emeasure_bind[OF _ _ X])
```
```   958     apply measurable
```
```   959     done
```
```   960 qed
```
```   961
```
```   962 lemma join_map_return_pmf: "join_pmf (map_pmf return_pmf A) = A"
```
```   963   unfolding bind_pmf_def[symmetric] bind_return_pmf' ..
```
```   964
```
```   965 lemma map_fst_pair_pmf: "map_pmf fst (pair_pmf A B) = A"
```
```   966   by (simp add: pair_pmf_def bind_return_pmf''[symmetric] bind_assoc_pmf bind_return_pmf bind_return_pmf')
```
```   967
```
```   968 lemma map_snd_pair_pmf: "map_pmf snd (pair_pmf A B) = B"
```
```   969   by (simp add: pair_pmf_def bind_return_pmf''[symmetric] bind_assoc_pmf bind_return_pmf bind_return_pmf')
```
```   970
```
```   971 lemma nn_integral_pmf':
```
```   972   "inj_on f A \<Longrightarrow> (\<integral>\<^sup>+x. pmf p (f x) \<partial>count_space A) = emeasure p (f ` A)"
```
```   973   by (subst nn_integral_bij_count_space[where g=f and B="f`A"])
```
```   974      (auto simp: bij_betw_def nn_integral_pmf)
```
```   975
```
```   976 lemma pmf_le_0_iff[simp]: "pmf M p \<le> 0 \<longleftrightarrow> pmf M p = 0"
```
```   977   using pmf_nonneg[of M p] by simp
```
```   978
```
```   979 lemma min_pmf_0[simp]: "min (pmf M p) 0 = 0" "min 0 (pmf M p) = 0"
```
```   980   using pmf_nonneg[of M p] by simp_all
```
```   981
```
```   982 lemma pmf_eq_0_set_pmf: "pmf M p = 0 \<longleftrightarrow> p \<notin> set_pmf M"
```
```   983   unfolding set_pmf_iff by simp
```
```   984
```
```   985 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"
```
```   986   by (auto simp: pmf.rep_eq map_pmf.rep_eq measure_distr AE_measure_pmf_iff inj_onD
```
```   987            intro!: measure_pmf.finite_measure_eq_AE)
```
```   988
```
```   989 inductive rel_pmf :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf \<Rightarrow> bool"
```
```   990 for R p q
```
```   991 where
```
```   992   "\<lbrakk> \<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y;
```
```   993      map_pmf fst pq = p; map_pmf snd pq = q \<rbrakk>
```
```   994   \<Longrightarrow> rel_pmf R p q"
```
```   995
```
```   996 bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: rel_pmf
```
```   997 proof -
```
```   998   show "map_pmf id = id" by (rule map_pmf_id)
```
```   999   show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose)
```
```  1000   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"
```
```  1001     by (intro map_pmf_cong refl)
```
```  1002
```
```  1003   show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
```
```  1004     by (rule pmf_set_map)
```
```  1005
```
```  1006   { fix p :: "'s pmf"
```
```  1007     have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq"
```
```  1008       by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"])
```
```  1009          (auto intro: countable_set_pmf)
```
```  1010     also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq"
```
```  1011       by (metis Field_natLeq card_of_least natLeq_Well_order)
```
```  1012     finally show "(card_of (set_pmf p), natLeq) \<in> ordLeq" . }
```
```  1013
```
```  1014   show "\<And>R. rel_pmf R =
```
```  1015          (BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf fst))\<inverse>\<inverse> OO
```
```  1016          BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf snd)"
```
```  1017      by (auto simp add: fun_eq_iff BNF_Def.Grp_def OO_def rel_pmf.simps)
```
```  1018
```
```  1019   { fix p :: "'a pmf" and f :: "'a \<Rightarrow> 'b" and g x
```
```  1020     assume p: "\<And>z. z \<in> set_pmf p \<Longrightarrow> f z = g z"
```
```  1021       and x: "x \<in> set_pmf p"
```
```  1022     thus "f x = g x" by simp }
```
```  1023
```
```  1024   fix R :: "'a => 'b \<Rightarrow> bool" and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
```
```  1025   { fix p q r
```
```  1026     assume pq: "rel_pmf R p q"
```
```  1027       and qr:"rel_pmf S q r"
```
```  1028     from pq obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
```
```  1029       and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto
```
```  1030     from qr obtain qr where qr: "\<And>y z. (y, z) \<in> set_pmf qr \<Longrightarrow> S y z"
```
```  1031       and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto
```
```  1032
```
```  1033     note pmf_nonneg[intro, simp]
```
```  1034
```
```  1035     def A \<equiv> "\<lambda>y. {x. (x, y) \<in> set_pmf pq}"
```
```  1036     then have "\<And>y. A y \<subseteq> set_pmf p" by (auto simp add: p set_map_pmf intro: rev_image_eqI)
```
```  1037     then have [simp]: "\<And>y. countable (A y)" by (rule countable_subset) simp
```
```  1038     have A: "\<And>x y. (x, y) \<in> set_pmf pq \<longleftrightarrow> x \<in> A y"
```
```  1039       by (simp add: A_def)
```
```  1040
```
```  1041     let ?P = "\<lambda>y. to_nat_on (A y)"
```
```  1042     def pp \<equiv> "map_pmf (\<lambda>(x, y). (y, ?P y x)) pq"
```
```  1043     let ?pp = "\<lambda>y x. pmf pp (y, x)"
```
```  1044     { fix x y have "x \<in> A y \<Longrightarrow> ?pp y (?P y x) = pmf pq (x, y)"
```
```  1045         unfolding pp_def
```
```  1046         by (intro pmf_map_inj[of "\<lambda>(x, y). (y, ?P y x)" pq "(x, y)", simplified])
```
```  1047            (auto simp: inj_on_def A) }
```
```  1048     note pmf_pp = this
```
```  1049     have pp_0: "\<And>y x. pmf q y = 0 \<Longrightarrow> ?pp y x = 0"
```
```  1050     proof(erule contrapos_pp)
```
```  1051       fix y x
```
```  1052       assume "?pp y x \<noteq> 0"
```
```  1053       hence "(y, x) \<in> set_pmf pp" by(simp add: set_pmf_iff)
```
```  1054       hence "y \<in> set_pmf q" by(auto simp add: pp_def q set_map_pmf intro: rev_image_eqI)
```
```  1055       thus "pmf q y \<noteq> 0" by(simp add: set_pmf_iff)
```
```  1056     qed
```
```  1057
```
```  1058     def B \<equiv> "\<lambda>y. {z. (y, z) \<in> set_pmf qr}"
```
```  1059     then have "\<And>y. B y \<subseteq> set_pmf r" by (auto simp add: r set_map_pmf intro: rev_image_eqI)
```
```  1060     then have [simp]: "\<And>y. countable (B y)" by (rule countable_subset) simp
```
```  1061     have B: "\<And>y z. (y, z) \<in> set_pmf qr \<longleftrightarrow> z \<in> B y"
```
```  1062       by (simp add: B_def)
```
```  1063
```
```  1064     let ?R = "\<lambda>y. to_nat_on (B y)"
```
```  1065     def rr \<equiv> "map_pmf (\<lambda>(y, z). (y, ?R y z)) qr"
```
```  1066     let ?rr = "\<lambda>y z. pmf rr (y, z)"
```
```  1067     { fix y z have "z \<in> B y \<Longrightarrow> ?rr y (?R y z) = pmf qr (y, z)"
```
```  1068         unfolding rr_def
```
```  1069         by (intro pmf_map_inj[of "\<lambda>(y, z). (y, ?R y z)" qr "(y, z)", simplified])
```
```  1070            (auto simp: inj_on_def B) }
```
```  1071     note pmf_rr = this
```
```  1072     have rr_0: "\<And>y z. pmf q y = 0 \<Longrightarrow> ?rr y z = 0"
```
```  1073     proof(erule contrapos_pp)
```
```  1074       fix y z
```
```  1075       assume "?rr y z \<noteq> 0"
```
```  1076       hence "(y, z) \<in> set_pmf rr" by(simp add: set_pmf_iff)
```
```  1077       hence "y \<in> set_pmf q" by(auto simp add: rr_def q' set_map_pmf intro: rev_image_eqI)
```
```  1078       thus "pmf q y \<noteq> 0" by(simp add: set_pmf_iff)
```
```  1079     qed
```
```  1080
```
```  1081     have nn_integral_pp2: "\<And>y. (\<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV) = pmf q y"
```
```  1082       by (simp add: nn_integral_pmf' inj_on_def pp_def q)
```
```  1083          (auto simp add: ereal_pmf_map intro!: arg_cong2[where f=emeasure])
```
```  1084     have nn_integral_rr1: "\<And>y. (\<integral>\<^sup>+ x. ?rr y x \<partial>count_space UNIV) = pmf q y"
```
```  1085       by (simp add: nn_integral_pmf' inj_on_def rr_def q')
```
```  1086          (auto simp add: ereal_pmf_map intro!: arg_cong2[where f=emeasure])
```
```  1087     have eq: "\<And>y. (\<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV) = (\<integral>\<^sup>+ z. ?rr y z \<partial>count_space UNIV)"
```
```  1088       by(simp add: nn_integral_pp2 nn_integral_rr1)
```
```  1089
```
```  1090     def assign \<equiv> "\<lambda>y x z. ?pp y x * ?rr y z / pmf q y"
```
```  1091     have assign_nonneg [simp]: "\<And>y x z. 0 \<le> assign y x z" by(simp add: assign_def)
```
```  1092     have assign_eq_0_outside: "\<And>y x z. \<lbrakk> ?pp y x = 0 \<or> ?rr y z = 0 \<rbrakk> \<Longrightarrow> assign y x z = 0"
```
```  1093       by(auto simp add: assign_def)
```
```  1094     have nn_integral_assign1: "\<And>y z. (\<integral>\<^sup>+ x. assign y x z \<partial>count_space UNIV) = ?rr y z"
```
```  1095     proof -
```
```  1096       fix y z
```
```  1097       have "(\<integral>\<^sup>+ x. assign y x z \<partial>count_space UNIV) =
```
```  1098             (\<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV) * (?rr y z / pmf q y)"
```
```  1099         by(simp add: assign_def nn_integral_multc times_ereal.simps(1)[symmetric] divide_real_def mult.assoc del: times_ereal.simps(1))
```
```  1100       also have "\<dots> = ?rr y z" by(simp add: rr_0 nn_integral_pp2)
```
```  1101       finally show "?thesis y z" .
```
```  1102     qed
```
```  1103     have nn_integral_assign2: "\<And>y x. (\<integral>\<^sup>+ z. assign y x z \<partial>count_space UNIV) = ?pp y x"
```
```  1104     proof -
```
```  1105       fix x y
```
```  1106       have "(\<integral>\<^sup>+ z. assign y x z \<partial>count_space UNIV) = (\<integral>\<^sup>+ z. ?rr y z \<partial>count_space UNIV) * (?pp y x / pmf q y)"
```
```  1107         by(simp add: assign_def divide_real_def mult.commute[where a="?pp y x"] mult.assoc nn_integral_multc times_ereal.simps(1)[symmetric] del: times_ereal.simps(1))
```
```  1108       also have "\<dots> = ?pp y x" by(simp add: nn_integral_rr1 pp_0)
```
```  1109       finally show "?thesis y x" .
```
```  1110     qed
```
```  1111
```
```  1112     def a \<equiv> "embed_pmf (\<lambda>(y, x, z). assign y x z)"
```
```  1113     { fix y x z
```
```  1114       have "assign y x z = pmf a (y, x, z)"
```
```  1115         unfolding a_def
```
```  1116       proof (subst pmf_embed_pmf)
```
```  1117         have "(\<integral>\<^sup>+ x. ereal ((\<lambda>(y, x, z). assign y x z) x) \<partial>count_space UNIV) =
```
```  1118           (\<integral>\<^sup>+ x. ereal ((\<lambda>(y, x, z). assign y x z) x) \<partial>(count_space ((\<lambda>((y, x), z). (y, x, z)) ` (pp \<times> UNIV))))"
```
```  1119           by (force simp add: nn_integral_count_space_indicator pmf_eq_0_set_pmf split: split_indicator
```
```  1120                     intro!: nn_integral_cong assign_eq_0_outside)
```
```  1121         also have "\<dots> = (\<integral>\<^sup>+ x. ereal ((\<lambda>((y, x), z). assign y x z) x) \<partial>(count_space (pp \<times> UNIV)))"
```
```  1122           by (subst nn_integral_bij_count_space[OF inj_on_imp_bij_betw, symmetric])
```
```  1123              (auto simp: inj_on_def intro!: nn_integral_cong)
```
```  1124         also have "\<dots> = (\<integral>\<^sup>+ y. \<integral>\<^sup>+z. ereal ((\<lambda>((y, x), z). assign y x z) (y, z)) \<partial>count_space UNIV \<partial>count_space pp)"
```
```  1125           by (subst sigma_finite_measure.nn_integral_fst)
```
```  1126              (auto simp: pair_measure_countable sigma_finite_measure_count_space_countable)
```
```  1127         also have "\<dots> = (\<integral>\<^sup>+ z. ?pp (fst z) (snd z) \<partial>count_space pp)"
```
```  1128           by (subst nn_integral_assign2[symmetric]) (auto intro!: nn_integral_cong)
```
```  1129         finally show "(\<integral>\<^sup>+ x. ereal ((\<lambda>(y, x, z). assign y x z) x) \<partial>count_space UNIV) = 1"
```
```  1130           by (simp add: nn_integral_pmf emeasure_pmf)
```
```  1131       qed auto }
```
```  1132     note a = this
```
```  1133
```
```  1134     def pr \<equiv> "map_pmf (\<lambda>(y, x, z). (from_nat_into (A y) x, from_nat_into (B y) z)) a"
```
```  1135
```
```  1136     have "rel_pmf (R OO S) p r"
```
```  1137     proof
```
```  1138       have pp_eq: "pp = map_pmf (\<lambda>(y, x, z). (y, x)) a"
```
```  1139       proof (rule pmf_eqI)
```
```  1140         fix i
```
```  1141         show "pmf pp i = pmf (map_pmf (\<lambda>(y, x, z). (y, x)) a) i"
```
```  1142           using nn_integral_assign2[of "fst i" "snd i", symmetric]
```
```  1143           by (auto simp add: a nn_integral_pmf' inj_on_def ereal.inject[symmetric] ereal_pmf_map
```
```  1144                    simp del: ereal.inject intro!: arg_cong2[where f=emeasure])
```
```  1145       qed
```
```  1146       moreover have pq_eq: "pq = map_pmf (\<lambda>(y, x). (from_nat_into (A y) x, y)) pp"
```
```  1147         by (simp add: pp_def map_pmf_comp split_beta A[symmetric] cong: map_pmf_cong)
```
```  1148       ultimately show "map_pmf fst pr = p"
```
```  1149         unfolding p pr_def by (simp add: map_pmf_comp split_beta)
```
```  1150
```
```  1151       have rr_eq: "rr = map_pmf (\<lambda>(y, x, z). (y, z)) a"
```
```  1152       proof (rule pmf_eqI)
```
```  1153         fix i show "pmf rr i = pmf (map_pmf (\<lambda>(y, x, z). (y, z)) a) i"
```
```  1154           using nn_integral_assign1[of "fst i" "snd i", symmetric]
```
```  1155           by (auto simp add: a nn_integral_pmf' inj_on_def ereal.inject[symmetric] ereal_pmf_map
```
```  1156                    simp del: ereal.inject intro!: arg_cong2[where f=emeasure])
```
```  1157       qed
```
```  1158       moreover have qr_eq: "qr = map_pmf (\<lambda>(y, z). (y, from_nat_into (B y) z)) rr"
```
```  1159         by (simp add: rr_def map_pmf_comp split_beta B[symmetric] cong: map_pmf_cong)
```
```  1160       ultimately show "map_pmf snd pr = r"
```
```  1161         unfolding r pr_def by (simp add: map_pmf_comp split_beta)
```
```  1162
```
```  1163       fix x z assume "(x, z) \<in> set_pmf pr"
```
```  1164       then have "\<exists>y. (x, y) \<in> set_pmf pq \<and> (y, z) \<in> set_pmf qr"
```
```  1165         by (force simp add: pp_eq pq_eq rr_eq qr_eq set_map_pmf pr_def image_image)
```
```  1166       with pq qr show "(R OO S) x z"
```
```  1167         by blast
```
```  1168     qed }
```
```  1169   then show "rel_pmf R OO rel_pmf S \<le> rel_pmf (R OO S)"
```
```  1170     by(auto simp add: le_fun_def)
```
```  1171 qed (fact natLeq_card_order natLeq_cinfinite)+
```
```  1172
```
```  1173 lemma rel_pmf_return_pmf1: "rel_pmf R (return_pmf x) M \<longleftrightarrow> (\<forall>a\<in>M. R x a)"
```
```  1174 proof safe
```
```  1175   fix a assume "a \<in> M" "rel_pmf R (return_pmf x) M"
```
```  1176   then obtain pq where *: "\<And>a b. (a, b) \<in> set_pmf pq \<Longrightarrow> R a b"
```
```  1177     and eq: "return_pmf x = map_pmf fst pq" "M = map_pmf snd pq"
```
```  1178     by (force elim: rel_pmf.cases)
```
```  1179   moreover have "set_pmf (return_pmf x) = {x}"
```
```  1180     by (simp add: set_return_pmf)
```
```  1181   with `a \<in> M` have "(x, a) \<in> pq"
```
```  1182     by (force simp: eq set_map_pmf)
```
```  1183   with * show "R x a"
```
```  1184     by auto
```
```  1185 qed (auto intro!: rel_pmf.intros[where pq="pair_pmf (return_pmf x) M"]
```
```  1186           simp: map_fst_pair_pmf map_snd_pair_pmf set_pair_pmf set_return_pmf)
```
```  1187
```
```  1188 lemma rel_pmf_return_pmf2: "rel_pmf R M (return_pmf x) \<longleftrightarrow> (\<forall>a\<in>M. R a x)"
```
```  1189   by (subst pmf.rel_flip[symmetric]) (simp add: rel_pmf_return_pmf1)
```
```  1190
```
```  1191 lemma rel_pmf_rel_prod:
```
```  1192   "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'"
```
```  1193 proof safe
```
```  1194   assume "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
```
```  1195   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"
```
```  1196     and eq: "map_pmf fst pq = pair_pmf A A'" "map_pmf snd pq = pair_pmf B B'"
```
```  1197     by (force elim: rel_pmf.cases)
```
```  1198   show "rel_pmf R A B"
```
```  1199   proof (rule rel_pmf.intros)
```
```  1200     let ?f = "\<lambda>(a, b). (fst a, fst b)"
```
```  1201     have [simp]: "(\<lambda>x. fst (?f x)) = fst o fst" "(\<lambda>x. snd (?f x)) = fst o snd"
```
```  1202       by auto
```
```  1203
```
```  1204     show "map_pmf fst (map_pmf ?f pq) = A"
```
```  1205       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
```
```  1206     show "map_pmf snd (map_pmf ?f pq) = B"
```
```  1207       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf)
```
```  1208
```
```  1209     fix a b assume "(a, b) \<in> set_pmf (map_pmf ?f pq)"
```
```  1210     then obtain c d where "((a, c), (b, d)) \<in> set_pmf pq"
```
```  1211       by (auto simp: set_map_pmf)
```
```  1212     from pq[OF this] show "R a b" ..
```
```  1213   qed
```
```  1214   show "rel_pmf S A' B'"
```
```  1215   proof (rule rel_pmf.intros)
```
```  1216     let ?f = "\<lambda>(a, b). (snd a, snd b)"
```
```  1217     have [simp]: "(\<lambda>x. fst (?f x)) = snd o fst" "(\<lambda>x. snd (?f x)) = snd o snd"
```
```  1218       by auto
```
```  1219
```
```  1220     show "map_pmf fst (map_pmf ?f pq) = A'"
```
```  1221       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
```
```  1222     show "map_pmf snd (map_pmf ?f pq) = B'"
```
```  1223       by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf)
```
```  1224
```
```  1225     fix c d assume "(c, d) \<in> set_pmf (map_pmf ?f pq)"
```
```  1226     then obtain a b where "((a, c), (b, d)) \<in> set_pmf pq"
```
```  1227       by (auto simp: set_map_pmf)
```
```  1228     from pq[OF this] show "S c d" ..
```
```  1229   qed
```
```  1230 next
```
```  1231   assume "rel_pmf R A B" "rel_pmf S A' B'"
```
```  1232   then obtain Rpq Spq
```
```  1233     where Rpq: "\<And>a b. (a, b) \<in> set_pmf Rpq \<Longrightarrow> R a b"
```
```  1234         "map_pmf fst Rpq = A" "map_pmf snd Rpq = B"
```
```  1235       and Spq: "\<And>a b. (a, b) \<in> set_pmf Spq \<Longrightarrow> S a b"
```
```  1236         "map_pmf fst Spq = A'" "map_pmf snd Spq = B'"
```
```  1237     by (force elim: rel_pmf.cases)
```
```  1238
```
```  1239   let ?f = "(\<lambda>((a, c), (b, d)). ((a, b), (c, d)))"
```
```  1240   let ?pq = "map_pmf ?f (pair_pmf Rpq Spq)"
```
```  1241   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))"
```
```  1242     by auto
```
```  1243
```
```  1244   show "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')"
```
```  1245     by (rule rel_pmf.intros[where pq="?pq"])
```
```  1246        (auto simp: map_snd_pair_pmf map_fst_pair_pmf set_pair_pmf set_map_pmf map_pmf_comp Rpq Spq
```
```  1247                    map_pair)
```
```  1248 qed
```
```  1249
```
```  1250 end
```
```  1251
```