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Untersuchung Isabelle^©

(*  Title:      HOL/Probability/Probability_Mass_Function.thy
    Author:     Johannes Hölzl, TU München
    Author:     Andreas Lochbihler, ETH Zurich
*)

section \<open> Probability mass function \<close>

theory Probability_Mass_Function
imports
  Giry_Monad
  "HOL-Library.Multiset"
begin

lemma AE_emeasure_singleton:
  assumes x: "emeasure M {x} \ 0" and ae: "AE x in M. P x" shows "P x"
proof -
  from x have x_M: "{x} \ sets M"
    by (auto intro: emeasure_notin_sets)
  from ae obtain N where N: "{x\space M. \ P x} \ N" "emeasure M N = 0" "N \ sets M"
    by (auto elim: AE_E)
  { assume "\ P x"
    with x_M[THEN sets.sets_into_space] N have "emeasure M {x} \ emeasure M N"
      by (intro emeasure_mono) auto
    with x N have False
      by (auto simp:) }
  then show "P x" by auto
qed

lemma AE_measure_singleton: "measure M {x} \ 0 \ AE x in M. P x \ P x"
  by (metis AE_emeasure_singleton measure_def emeasure_empty measure_empty)

lemma (in finite_measure) AE_support_countable:
  assumes [simp]: "sets M = UNIV"
  shows "(AE x in M. measure M {x} \ 0) \ (\S. countable S \ (AE x in M. x \ S))"
proof
  assume "\S. countable S \ (AE x in M. x \ S)"
  then obtain S where S[intro]: "countable S" and ae: "AE x in M. x \ S"
    by auto
  then have "emeasure M (\x\{x\S. emeasure M {x} \ 0}. {x}) =
    (\<integral>\<^sup>+ x. emeasure M {x} * indicator {x\<in>S. emeasure M {x} \<noteq> 0} x \<partial>count_space UNIV)"
    by (subst emeasure_UN_countable)
       (auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space)
  also have "\ = (\\<^sup>+ x. emeasure M {x} * indicator S x \count_space UNIV)"
    by (auto intro!: nn_integral_cong split: split_indicator)
  also have "\ = emeasure M (\x\S. {x})"
    by (subst emeasure_UN_countable)
       (auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space)
  also have "\ = emeasure M (space M)"
    using ae by (intro emeasure_eq_AE) auto
  finally have "emeasure M {x \ space M. x\S \ emeasure M {x} \ 0} = emeasure M (space M)"
    by (simp add: emeasure_single_in_space cong: rev_conj_cong)
  with finite_measure_compl[of "{x \ space M. x\S \ emeasure M {x} \ 0}"]
  have "AE x in M. x \ S \ emeasure M {x} \ 0"
    by (intro AE_I[OF order_refl]) (auto simp: emeasure_eq_measure measure_nonneg set_diff_eq cong: conj_cong)
  then show "AE x in M. measure M {x} \ 0"
    by (auto simp: emeasure_eq_measure)
qed (auto intro!: exI[of _ "{x. measure M {x} \ 0}"] countable_support)

subsection \<open> PMF as measure \<close>

typedef 'a pmf = "{M :: 'a measure. prob_space M \<and> sets M = UNIV \<and> (AE x in M. measure M {x} \<noteq> 0)}"
  morphisms measure_pmf Abs_pmf
  by (intro exI[of _ "uniform_measure (count_space UNIV) {undefined}"])
     (auto intro!: prob_space_uniform_measure AE_uniform_measureI)

declare [[coercion measure_pmf]]

lemma prob_space_measure_pmf: "prob_space (measure_pmf p)"
  using pmf.measure_pmf[of p] by auto

interpretation measure_pmf: prob_space "measure_pmf M" for M
  by (rule prob_space_measure_pmf)

interpretation measure_pmf: subprob_space "measure_pmf M" for M
  by (rule prob_space_imp_subprob_space) unfold_locales

lemma subprob_space_measure_pmf: "subprob_space (measure_pmf x)"
  by unfold_locales

locale pmf_as_measure
begin

setup_lifting type_definition_pmf

end

context
begin

interpretation pmf_as_measure .

lemma sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV"
  by transfer blast

lemma sets_measure_pmf_count_space[measurable_cong]:
  "sets (measure_pmf M) = sets (count_space UNIV)"
  by simp

lemma space_measure_pmf[simp]: "space (measure_pmf p) = UNIV"
  using sets_eq_imp_space_eq[of "measure_pmf p" "count_space UNIV"] by simp

lemma measure_pmf_UNIV [simp]: "measure (measure_pmf p) UNIV = 1"
using measure_pmf.prob_space[of p] by simp

lemma measure_pmf_in_subprob_algebra[measurable (raw)]: "measure_pmf x \ space (subprob_algebra (count_space UNIV))"
  by (simp add: space_subprob_algebra subprob_space_measure_pmf)

lemma measurable_pmf_measure1[simp]: "measurable (M :: 'a pmf) N = UNIV \ space N"
  by (auto simp: measurable_def)

lemma measurable_pmf_measure2[simp]: "measurable N (M :: 'a pmf) = measurable N (count_space UNIV)"
  by (intro measurable_cong_sets) simp_all

lemma measurable_pair_restrict_pmf2:
  assumes "countable A"
  assumes [measurable]: "\y. y \ A \ (\x. f (x, y)) \ measurable M L"
  shows "f \ measurable (M \\<^sub>M restrict_space (measure_pmf N) A) L" (is "f \ measurable ?M _")
proof -
  have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)"
    by (simp add: restrict_count_space)

  show ?thesis
    by (intro measurable_compose_countable'[where f="\a b. f (fst b, a)" and g=snd and I=A,
                                            unfolded prod.collapse] assms)
        measurable
qed

lemma measurable_pair_restrict_pmf1:
  assumes "countable A"
  assumes [measurable]: "\x. x \ A \ (\y. f (x, y)) \ measurable N L"
  shows "f \ measurable (restrict_space (measure_pmf M) A \\<^sub>M N) L"
proof -
  have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)"
    by (simp add: restrict_count_space)

  show ?thesis
    by (intro measurable_compose_countable'[where f="\a b. f (a, snd b)" and g=fst and I=A,
                                            unfolded prod.collapse] assms)
        measurable
qed

lift_definition pmf :: "'a pmf \ 'a \ real" is "\M x. measure M {x}" .

lift_definition set_pmf :: "'a pmf \ 'a set" is "\M. {x. measure M {x} \ 0}" .
declare [[coercion set_pmf]]

lemma AE_measure_pmf: "AE x in (M::'a pmf). x \ M"
  by transfer simp

lemma emeasure_pmf_single_eq_zero_iff:
  fixes M :: "'a pmf"
  shows "emeasure M {y} = 0 \ y \ M"
  unfolding set_pmf.rep_eq by (simp add: measure_pmf.emeasure_eq_measure)

lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \ (\y\M. P y)"
  using AE_measure_singleton[of M] AE_measure_pmf[of M]
  by (auto simp: set_pmf.rep_eq)

lemma AE_pmfI: "(\y. y \ set_pmf M \ P y) \ almost_everywhere (measure_pmf M) P"
by(simp add: AE_measure_pmf_iff)

lemma countable_set_pmf [simp]: "countable (set_pmf p)"
  by transfer (metis prob_space.finite_measure finite_measure.countable_support)

lemma pmf_positive: "x \ set_pmf p \ 0 < pmf p x"
  by transfer (simp add: less_le)

lemma pmf_nonneg[simp]: "0 \ pmf p x"
  by transfer simp

lemma pmf_not_neg [simp]: "\pmf p x < 0"
  by (simp add: not_less pmf_nonneg)

lemma pmf_pos [simp]: "pmf p x \ 0 \ pmf p x > 0"
  using pmf_nonneg[of p x] by linarith

lemma pmf_le_1: "pmf p x \ 1"
  by (simp add: pmf.rep_eq)

lemma set_pmf_not_empty: "set_pmf M \ {}"
  using AE_measure_pmf[of M] by (intro notI) simp

lemma set_pmf_iff: "x \ set_pmf M \ pmf M x \ 0"
  by transfer simp

lemma pmf_positive_iff: "0 < pmf p x \ x \ set_pmf p"
  unfolding less_le by (simp add: set_pmf_iff)

lemma set_pmf_eq: "set_pmf M = {x. pmf M x \ 0}"
  by (auto simp: set_pmf_iff)

lemma set_pmf_eq': "set_pmf p = {x. pmf p x > 0}"
proof safe
  fix x assume "x \ set_pmf p"
  hence "pmf p x \ 0" by (auto simp: set_pmf_eq)
  with pmf_nonneg[of p x] show "pmf p x > 0" by simp
qed (auto simp: set_pmf_eq)

lemma emeasure_pmf_single:
  fixes M :: "'a pmf"
  shows "emeasure M {x} = pmf M x"
  by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])

lemma measure_pmf_single: "measure (measure_pmf M) {x} = pmf M x"
  using emeasure_pmf_single[of M x] by(simp add: measure_pmf.emeasure_eq_measure pmf_nonneg measure_nonneg)

lemma emeasure_measure_pmf_finite: "finite S \ emeasure (measure_pmf M) S = (\s\S. pmf M s)"
  by (subst emeasure_eq_sum_singleton) (auto simp: emeasure_pmf_single pmf_nonneg)

lemma measure_measure_pmf_finite: "finite S \ measure (measure_pmf M) S = sum (pmf M) S"
  using emeasure_measure_pmf_finite[of S M]
  by (simp add: measure_pmf.emeasure_eq_measure measure_nonneg sum_nonneg pmf_nonneg)

lemma sum_pmf_eq_1:
  assumes "finite A" "set_pmf p \ A"
  shows   "(\x\A. pmf p x) = 1"
proof -
  have "(\x\A. pmf p x) = measure_pmf.prob p A"
    by (simp add: measure_measure_pmf_finite assms)
  also from assms have "\ = 1"
    by (subst measure_pmf.prob_eq_1) (auto simp: AE_measure_pmf_iff)
  finally show ?thesis .
qed

lemma nn_integral_measure_pmf_support:
  fixes f :: "'a \ ennreal"
  assumes f: "finite A" and nn: "\x. x \ A \ 0 \ f x" "\x. x \ set_pmf M \ x \ A \ f x = 0"
  shows "(\\<^sup>+x. f x \measure_pmf M) = (\x\A. f x * pmf M x)"
proof -
  have "(\\<^sup>+x. f x \M) = (\\<^sup>+x. f x * indicator A x \M)"
    using nn by (intro nn_integral_cong_AE) (auto simp: AE_measure_pmf_iff split: split_indicator)
  also have "\ = (\x\A. f x * emeasure M {x})"
    using assms by (intro nn_integral_indicator_finite) auto
  finally show ?thesis
    by (simp add: emeasure_measure_pmf_finite)
qed

lemma nn_integral_measure_pmf_finite:
  fixes f :: "'a \ ennreal"
  assumes f: "finite (set_pmf M)" and nn: "\x. x \ set_pmf M \ 0 \ f x"
  shows "(\\<^sup>+x. f x \measure_pmf M) = (\x\set_pmf M. f x * pmf M x)"
  using assms by (intro nn_integral_measure_pmf_support) auto

lemma integrable_measure_pmf_finite:
  fixes f :: "'a \ 'b::{banach, second_countable_topology}"
  shows "finite (set_pmf M) \ integrable M f"
  by (auto intro!: integrableI_bounded simp: nn_integral_measure_pmf_finite ennreal_mult_less_top)

lemma integral_measure_pmf_real:
  assumes [simp]: "finite A" and "\a. a \ set_pmf M \ f a \ 0 \ a \ A"
  shows "(\x. f x \measure_pmf M) = (\a\A. f a * pmf M a)"
proof -
  have "(\x. f x \measure_pmf M) = (\x. f x * indicator A x \measure_pmf M)"
    using assms(2) by (intro integral_cong_AE) (auto split: split_indicator simp: AE_measure_pmf_iff)
  also have "\ = (\a\A. f a * pmf M a)"
    by (subst integral_indicator_finite_real)
       (auto simp: measure_def emeasure_measure_pmf_finite pmf_nonneg)
  finally show ?thesis .
qed

lemma integrable_pmf: "integrable (count_space X) (pmf M)"
proof -
  have " (\\<^sup>+ x. pmf M x \count_space X) = (\\<^sup>+ x. pmf M x \count_space (M \ X))"
    by (auto simp add: nn_integral_count_space_indicator set_pmf_iff intro!: nn_integral_cong split: split_indicator)
  then have "integrable (count_space X) (pmf M) = integrable (count_space (M \ X)) (pmf M)"
    by (simp add: integrable_iff_bounded pmf_nonneg)
  then show ?thesis
    by (simp add: pmf.rep_eq measure_pmf.integrable_measure disjoint_family_on_def)
qed

lemma integral_pmf: "(\x. pmf M x \count_space X) = measure M X"
proof -
  have "(\x. pmf M x \count_space X) = (\\<^sup>+x. pmf M x \count_space X)"
    by (simp add: pmf_nonneg integrable_pmf nn_integral_eq_integral)
  also have "\ = (\\<^sup>+x. emeasure M {x} \count_space (X \ M))"
    by (auto intro!: nn_integral_cong_AE split: split_indicator
             simp: pmf.rep_eq measure_pmf.emeasure_eq_measure nn_integral_count_space_indicator
                   AE_count_space set_pmf_iff)
  also have "\ = emeasure M (X \ M)"
    by (rule emeasure_countable_singleton[symmetric]) (auto intro: countable_set_pmf)
  also have "\ = emeasure M X"
    by (auto intro!: emeasure_eq_AE simp: AE_measure_pmf_iff)
  finally show ?thesis
    by (simp add: measure_pmf.emeasure_eq_measure measure_nonneg integral_nonneg pmf_nonneg)
qed

lemma integral_pmf_restrict:
  "(f::'a \ 'b::{banach, second_countable_topology}) \ borel_measurable (count_space UNIV) \
    (\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x \<partial>restrict_space M M)"
  by (auto intro!: integral_cong_AE simp add: integral_restrict_space AE_measure_pmf_iff)

lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1"
proof -
  have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)"
    by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf)
  then show ?thesis
    using measure_pmf.emeasure_space_1 by simp
qed

lemma emeasure_pmf_UNIV [simp]: "emeasure (measure_pmf M) UNIV = 1"
using measure_pmf.emeasure_space_1[of M] by simp

lemma in_null_sets_measure_pmfI:
  "A \ set_pmf p = {} \ A \ null_sets (measure_pmf p)"
using emeasure_eq_0_AE[where ?P="\x. x \ A" and M="measure_pmf p"]
by(auto simp add: null_sets_def AE_measure_pmf_iff)

lemma measure_subprob: "measure_pmf M \ space (subprob_algebra (count_space UNIV))"
  by (simp add: space_subprob_algebra subprob_space_measure_pmf)

subsection \<open> Monad Interpretation \<close>

lemma measurable_measure_pmf[measurable]:
  "(\x. measure_pmf (M x)) \ measurable (count_space UNIV) (subprob_algebra (count_space UNIV))"
  by (auto simp: space_subprob_algebra intro!: prob_space_imp_subprob_space) unfold_locales

lemma bind_measure_pmf_cong:
  assumes "\x. A x \ space (subprob_algebra N)" "\x. B x \ space (subprob_algebra N)"
  assumes "\i. i \ set_pmf x \ A i = B i"
  shows "bind (measure_pmf x) A = bind (measure_pmf x) B"
proof (rule measure_eqI)
  show "sets (measure_pmf x \ A) = sets (measure_pmf x \ B)"
    using assms by (subst (1 2) sets_bind) (auto simp: space_subprob_algebra)
next
  fix X assume "X \ sets (measure_pmf x \ A)"
  then have X: "X \ sets N"
    using assms by (subst (asm) sets_bind) (auto simp: space_subprob_algebra)
  show "emeasure (measure_pmf x \ A) X = emeasure (measure_pmf x \ B) X"
    using assms
    by (subst (1 2) emeasure_bind[where N=N, OF _ _ X])
       (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
qed

lift_definition bind_pmf :: "'a pmf \ ('a \ 'b pmf ) \ 'b pmf" is bind
proof (clarify, intro conjI)
  fix f :: "'a measure" and g :: "'a \ 'b measure"
  assume "prob_space f"
  then interpret f: prob_space f .
  assume "sets f = UNIV" and ae_f: "AE x in f. measure f {x} \ 0"
  then have s_f[simp]: "sets f = sets (count_space UNIV)"
    by simp
  assume g: "\x. prob_space (g x) \ sets (g x) = UNIV \ (AE y in g x. measure (g x) {y} \ 0)"
  then have g: "\x. prob_space (g x)" and s_g[simp]: "\x. sets (g x) = sets (count_space UNIV)"
    and ae_g: "\x. AE y in g x. measure (g x) {y} \ 0"
    by auto

  have [measurable]: "g \ measurable f (subprob_algebra (count_space UNIV))"
    by (auto simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space g)

  show "prob_space (f \ g)"
    using g by (intro f.prob_space_bind[where S="count_space UNIV"]) auto
  then interpret fg: prob_space "f \ g" .
  show [simp]: "sets (f \ g) = UNIV"
    using sets_eq_imp_space_eq[OF s_f]
    by (subst sets_bind[where N="count_space UNIV"]) auto
  show "AE x in f \ g. measure (f \ g) {x} \ 0"
    apply (simp add: fg.prob_eq_0 AE_bind[where B="count_space UNIV"])
    using ae_f
    apply eventually_elim
    using ae_g
    apply eventually_elim
    apply (auto dest: AE_measure_singleton)
    done
qed

adhoc_overloading Monad_Syntax.bind bind_pmf

lemma ennreal_pmf_bind: "pmf (bind_pmf N f) i = (\\<^sup>+x. pmf (f x) i \measure_pmf N)"
  unfolding pmf.rep_eq bind_pmf.rep_eq
  by (auto simp: measure_pmf.measure_bind[where N="count_space UNIV"] measure_subprob measure_nonneg
           intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])

lemma pmf_bind: "pmf (bind_pmf N f) i = (\x. pmf (f x) i \measure_pmf N)"
  using ennreal_pmf_bind[of N f i]
  by (subst (asm) nn_integral_eq_integral)
     (auto simp: pmf_nonneg pmf_le_1 pmf_nonneg integral_nonneg
           intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])

lemma bind_pmf_const[simp]: "bind_pmf M (\x. c) = c"
  by transfer (simp add: bind_const' prob_space_imp_subprob_space)

lemma set_bind_pmf[simp]: "set_pmf (bind_pmf M N) = (\M\set_pmf M. set_pmf (N M))"
proof -
  have "set_pmf (bind_pmf M N) = {x. ennreal (pmf (bind_pmf M N) x) \ 0}"
    by (simp add: set_pmf_eq pmf_nonneg)
  also have "\ = (\M\set_pmf M. set_pmf (N M))"
    unfolding ennreal_pmf_bind
    by (subst nn_integral_0_iff_AE) (auto simp: AE_measure_pmf_iff pmf_nonneg set_pmf_eq)
  finally show ?thesis .
qed

lemma bind_pmf_cong [fundef_cong]:
  assumes "p = q"
  shows "(\x. x \ set_pmf q \ f x = g x) \ bind_pmf p f = bind_pmf q g"
  unfolding \<open>p = q\<close>[symmetric] measure_pmf_inject[symmetric] bind_pmf.rep_eq
  by (auto simp: AE_measure_pmf_iff Pi_iff space_subprob_algebra subprob_space_measure_pmf
                 sets_bind[where N="count_space UNIV"] emeasure_bind[where N="count_space UNIV"]
           intro!: nn_integral_cong_AE measure_eqI)

lemma bind_pmf_cong_simp:
  "p = q \ (\x. x \ set_pmf q =simp=> f x = g x) \ bind_pmf p f = bind_pmf q g"
  by (simp add: simp_implies_def cong: bind_pmf_cong)

lemma measure_pmf_bind: "measure_pmf (bind_pmf M f) = (measure_pmf M \ (\x. measure_pmf (f x)))"
  by transfer simp

lemma nn_integral_bind_pmf[simp]: "(\\<^sup>+x. f x \bind_pmf M N) = (\\<^sup>+x. \\<^sup>+y. f y \N x \M)"
  using measurable_measure_pmf[of N]
  unfolding measure_pmf_bind
  apply (intro nn_integral_bind[where B="count_space UNIV"])
  apply auto
  done

lemma emeasure_bind_pmf[simp]: "emeasure (bind_pmf M N) X = (\\<^sup>+x. emeasure (N x) X \M)"
  using measurable_measure_pmf[of N]
  unfolding measure_pmf_bind
  by (subst emeasure_bind[where N="count_space UNIV"]) auto

lift_definition return_pmf :: "'a \ 'a pmf" is "return (count_space UNIV)"
  by (auto intro!: prob_space_return simp: AE_return measure_return)

lemma bind_return_pmf: "bind_pmf (return_pmf x) f = f x"
  by transfer
     (auto intro!: prob_space_imp_subprob_space bind_return[where N="count_space UNIV"]
           simp: space_subprob_algebra)

lemma set_return_pmf[simp]: "set_pmf (return_pmf x) = {x}"
  by transfer (auto simp add: measure_return split: split_indicator)

lemma bind_return_pmf': "bind_pmf N return_pmf = N"
proof (transfer, clarify)
  fix N :: "'a measure" assume "sets N = UNIV" then show "N \ return (count_space UNIV) = N"
    by (subst return_sets_cong[where N=N]) (simp_all add: bind_return')
qed

lemma bind_assoc_pmf: "bind_pmf (bind_pmf A B) C = bind_pmf A (\x. bind_pmf (B x) C)"
  by transfer
     (auto intro!: bind_assoc[where N="count_space UNIV" and R="count_space UNIV"]
           simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space)

definition "map_pmf f M = bind_pmf M (\x. return_pmf (f x))"

lemma map_bind_pmf: "map_pmf f (bind_pmf M g) = bind_pmf M (\x. map_pmf f (g x))"
  by (simp add: map_pmf_def bind_assoc_pmf)

lemma bind_map_pmf: "bind_pmf (map_pmf f M) g = bind_pmf M (\x. g (f x))"
  by (simp add: map_pmf_def bind_assoc_pmf bind_return_pmf)

lemma map_pmf_transfer[transfer_rule]:
  "rel_fun (=) (rel_fun cr_pmf cr_pmf) (\f M. distr M (count_space UNIV) f) map_pmf"
proof -
  have "rel_fun (=) (rel_fun pmf_as_measure.cr_pmf pmf_as_measure.cr_pmf)
     (\<lambda>f M. M \<bind> (return (count_space UNIV) o f)) map_pmf"
    unfolding map_pmf_def[abs_def] comp_def by transfer_prover
  then show ?thesis
    by (force simp: rel_fun_def cr_pmf_def bind_return_distr)
qed

lemma map_pmf_rep_eq:
  "measure_pmf (map_pmf f M) = distr (measure_pmf M) (count_space UNIV) f"
  unfolding map_pmf_def bind_pmf.rep_eq comp_def return_pmf.rep_eq
  using bind_return_distr[of M f "count_space UNIV"] by (simp add: comp_def)

lemma map_pmf_id[simp]: "map_pmf id = id"
  by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI)

lemma map_pmf_ident[simp]: "map_pmf (\x. x) = (\x. x)"
  using map_pmf_id unfolding id_def .

lemma map_pmf_compose: "map_pmf (f \ g) = map_pmf f \ map_pmf g"
  by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def)

lemma map_pmf_comp: "map_pmf f (map_pmf g M) = map_pmf (\x. f (g x)) M"
  using map_pmf_compose[of f g] by (simp add: comp_def)

lemma map_pmf_cong: "p = q \ (\x. x \ set_pmf q \ f x = g x) \ map_pmf f p = map_pmf g q"
  unfolding map_pmf_def by (rule bind_pmf_cong) auto

lemma pmf_set_map: "set_pmf \ map_pmf f = (`) f \ set_pmf"
  by (auto simp add: comp_def fun_eq_iff map_pmf_def)

lemma set_map_pmf[simp]: "set_pmf (map_pmf f M) = f`set_pmf M"
  using pmf_set_map[of f] by (auto simp: comp_def fun_eq_iff)

lemma emeasure_map_pmf[simp]: "emeasure (map_pmf f M) X = emeasure M (f -` X)"
  unfolding map_pmf_rep_eq by (subst emeasure_distr) auto

lemma measure_map_pmf[simp]: "measure (map_pmf f M) X = measure M (f -` X)"
using emeasure_map_pmf[of f M X] by(simp add: measure_pmf.emeasure_eq_measure measure_nonneg)

lemma nn_integral_map_pmf[simp]: "(\\<^sup>+x. f x \map_pmf g M) = (\\<^sup>+x. f (g x) \M)"
  unfolding map_pmf_rep_eq by (intro nn_integral_distr) auto

lemma ennreal_pmf_map: "pmf (map_pmf f p) x = (\\<^sup>+ y. indicator (f -` {x}) y \measure_pmf p)"
proof (transfer fixing: f x)
  fix p :: "'b measure"
  presume "prob_space p"
  then interpret prob_space p .
  presume "sets p = UNIV"
  then show "ennreal (measure (distr p (count_space UNIV) f) {x}) = integral\<^sup>N p (indicator (f -` {x}))"
    by(simp add: measure_distr measurable_def emeasure_eq_measure)
qed simp_all

lemma pmf_map: "pmf (map_pmf f p) x = measure p (f -` {x})"
proof (transfer fixing: f x)
  fix p :: "'b measure"
  presume "prob_space p"
  then interpret prob_space p .
  presume "sets p = UNIV"
  then show "measure (distr p (count_space UNIV) f) {x} = measure p (f -` {x})"
    by(simp add: measure_distr measurable_def emeasure_eq_measure)
qed simp_all

lemma nn_integral_pmf: "(\\<^sup>+ x. pmf p x \count_space A) = emeasure (measure_pmf p) A"
proof -
  have "(\\<^sup>+ x. pmf p x \count_space A) = (\\<^sup>+ x. pmf p x \count_space (A \ set_pmf p))"
    by(auto simp add: nn_integral_count_space_indicator indicator_def set_pmf_iff intro: nn_integral_cong)
  also have "\ = emeasure (measure_pmf p) (\x\A \ set_pmf p. {x})"
    by(subst emeasure_UN_countable)(auto simp add: emeasure_pmf_single disjoint_family_on_def)
  also have "\ = emeasure (measure_pmf p) ((\x\A \ set_pmf p. {x}) \ {x. x \ A \ x \ set_pmf p})"
    by(rule emeasure_Un_null_set[symmetric])(auto intro: in_null_sets_measure_pmfI)
  also have "\ = emeasure (measure_pmf p) A"
    by(auto intro: arg_cong2[where f=emeasure])
  finally show ?thesis .
qed

lemma integral_map_pmf[simp]:
  fixes f :: "'a \ 'b::{banach, second_countable_topology}"
  shows "integral\<^sup>L (map_pmf g p) f = integral\<^sup>L p (\x. f (g x))"
  by (simp add: integral_distr map_pmf_rep_eq)

lemma pmf_abs_summable [intro]: "pmf p abs_summable_on A"
  by (rule abs_summable_on_subset[OF _ subset_UNIV])
     (auto simp:  abs_summable_on_def integrable_iff_bounded nn_integral_pmf)

lemma measure_pmf_conv_infsetsum: "measure (measure_pmf p) A = infsetsum (pmf p) A"
  unfolding infsetsum_def by (simp add: integral_eq_nn_integral nn_integral_pmf measure_def)

lemma infsetsum_pmf_eq_1:
  assumes "set_pmf p \ A"
  shows   "infsetsum (pmf p) A = 1"
proof -
  have "infsetsum (pmf p) A = lebesgue_integral (count_space UNIV) (pmf p)"
    using assms unfolding infsetsum_altdef set_lebesgue_integral_def
    by (intro Bochner_Integration.integral_cong) (auto simp: indicator_def set_pmf_eq)
  also have "\ = 1"
    by (subst integral_eq_nn_integral) (auto simp: nn_integral_pmf)
  finally show ?thesis .
qed

lemma map_return_pmf [simp]: "map_pmf f (return_pmf x) = return_pmf (f x)"
  by transfer (simp add: distr_return)

lemma map_pmf_const[simp]: "map_pmf (\_. c) M = return_pmf c"
  by transfer (auto simp: prob_space.distr_const)

lemma pmf_return [simp]: "pmf (return_pmf x) y = indicator {y} x"
  by transfer (simp add: measure_return)

lemma nn_integral_return_pmf[simp]: "0 \ f x \ (\\<^sup>+x. f x \return_pmf x) = f x"
  unfolding return_pmf.rep_eq by (intro nn_integral_return) auto

lemma emeasure_return_pmf[simp]: "emeasure (return_pmf x) X = indicator X x"
  unfolding return_pmf.rep_eq by (intro emeasure_return) auto

lemma measure_return_pmf [simp]: "measure_pmf.prob (return_pmf x) A = indicator A x"
proof -
  have "ennreal (measure_pmf.prob (return_pmf x) A) =
          emeasure (measure_pmf (return_pmf x)) A"
    by (simp add: measure_pmf.emeasure_eq_measure)
  also have "\ = ennreal (indicator A x)" by (simp add: ennreal_indicator)
  finally show ?thesis by simp
qed

lemma return_pmf_inj[simp]: "return_pmf x = return_pmf y \ x = y"
  by (metis insertI1 set_return_pmf singletonD)

lemma map_pmf_eq_return_pmf_iff:
  "map_pmf f p = return_pmf x \ (\y \ set_pmf p. f y = x)"
proof
  assume "map_pmf f p = return_pmf x"
  then have "set_pmf (map_pmf f p) = set_pmf (return_pmf x)" by simp
  then show "\y \ set_pmf p. f y = x" by auto
next
  assume "\y \ set_pmf p. f y = x"
  then show "map_pmf f p = return_pmf x"
    unfolding map_pmf_const[symmetric, of _ p] by (intro map_pmf_cong) auto
qed

definition "pair_pmf A B = bind_pmf A (\x. bind_pmf B (\y. return_pmf (x, y)))"

lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b"
  unfolding pair_pmf_def pmf_bind pmf_return
  apply (subst integral_measure_pmf_real[where A="{b}"])
  apply (auto simp: indicator_eq_0_iff)
  apply (subst integral_measure_pmf_real[where A="{a}"])
  apply (auto simp: indicator_eq_0_iff sum_nonneg_eq_0_iff pmf_nonneg)
  done

lemma set_pair_pmf[simp]: "set_pmf (pair_pmf A B) = set_pmf A \ set_pmf B"
  unfolding pair_pmf_def set_bind_pmf set_return_pmf by auto

lemma measure_pmf_in_subprob_space[measurable (raw)]:
  "measure_pmf M \ space (subprob_algebra (count_space UNIV))"
  by (simp add: space_subprob_algebra) intro_locales

lemma nn_integral_pair_pmf': "(\\<^sup>+x. f x \pair_pmf A B) = (\\<^sup>+a. \\<^sup>+b. f (a, b) \B \A)"
proof -
  have "(\\<^sup>+x. f x \pair_pmf A B) = (\\<^sup>+x. f x * indicator (A \ B) x \pair_pmf A B)"
    by (auto simp: AE_measure_pmf_iff intro!: nn_integral_cong_AE)
  also have "\ = (\\<^sup>+a. \\<^sup>+b. f (a, b) * indicator (A \ B) (a, b) \B \A)"
    by (simp add: pair_pmf_def)
  also have "\ = (\\<^sup>+a. \\<^sup>+b. f (a, b) \B \A)"
    by (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
  finally show ?thesis .
qed

lemma bind_pair_pmf:
  assumes M[measurable]: "M \ measurable (count_space UNIV \\<^sub>M count_space UNIV) (subprob_algebra N)"
  shows "measure_pmf (pair_pmf A B) \ M = (measure_pmf A \ (\x. measure_pmf B \ (\y. M (x, y))))"
    (is "?L = ?R")
proof (rule measure_eqI)
  have M'[measurable]: "M \ measurable (pair_pmf A B) (subprob_algebra N)"
    using M[THEN measurable_space] by (simp_all add: space_pair_measure)

  note measurable_bind[where N="count_space UNIV", measurable]
  note measure_pmf_in_subprob_space[simp]

  have sets_eq_N: "sets ?L = N"
    by (subst sets_bind[OF sets_kernel[OF M']]) auto
  show "sets ?L = sets ?R"
    using measurable_space[OF M]
    by (simp add: sets_eq_N space_pair_measure space_subprob_algebra)
  fix X assume "X \ sets ?L"
  then have X[measurable]: "X \ sets N"
    unfolding sets_eq_N .
  then show "emeasure ?L X = emeasure ?R X"
    apply (simp add: emeasure_bind[OF _ M' X])
    apply (simp add: nn_integral_bind[where B="count_space UNIV"] pair_pmf_def measure_pmf_bind[of A]
                     nn_integral_measure_pmf_finite)
    apply (subst emeasure_bind[OF _ _ X])
    apply measurable
    apply (subst emeasure_bind[OF _ _ X])
    apply measurable
    done
qed

lemma map_fst_pair_pmf: "map_pmf fst (pair_pmf A B) = A"
  by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')

lemma map_snd_pair_pmf: "map_pmf snd (pair_pmf A B) = B"
  by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf')

lemma nn_integral_pmf':
  "inj_on f A \ (\\<^sup>+x. pmf p (f x) \count_space A) = emeasure p (f ` A)"
  by (subst nn_integral_bij_count_space[where g=f and B="f`A"])
     (auto simp: bij_betw_def nn_integral_pmf)

lemma pmf_le_0_iff[simp]: "pmf M p \ 0 \ pmf M p = 0"
  using pmf_nonneg[of M p] by arith

lemma min_pmf_0[simp]: "min (pmf M p) 0 = 0" "min 0 (pmf M p) = 0"
  using pmf_nonneg[of M p] by arith+

lemma pmf_eq_0_set_pmf: "pmf M p = 0 \ p \ set_pmf M"
  unfolding set_pmf_iff by simp

lemma pmf_map_inj: "inj_on f (set_pmf M) \ x \ set_pmf M \ pmf (map_pmf f M) (f x) = pmf M x"
  by (auto simp: pmf.rep_eq map_pmf_rep_eq measure_distr AE_measure_pmf_iff inj_onD
           intro!: measure_pmf.finite_measure_eq_AE)

lemma pmf_map_inj': "inj f \ pmf (map_pmf f M) (f x) = pmf M x"
apply(cases "x \ set_pmf M")
apply(simp add: pmf_map_inj[OF subset_inj_on])
apply(simp add: pmf_eq_0_set_pmf[symmetric])
apply(auto simp add: pmf_eq_0_set_pmf dest: injD)
done

lemma pmf_map_outside: "x \ f ` set_pmf M \ pmf (map_pmf f M) x = 0"
  unfolding pmf_eq_0_set_pmf by simp

lemma measurable_set_pmf[measurable]: "Measurable.pred (count_space UNIV) (\x. x \ set_pmf M)"
  by simp

subsection \<open> PMFs as function \<close>

context
  fixes f :: "'a \ real"
  assumes nonneg: "\x. 0 \ f x"
  assumes prob: "(\\<^sup>+x. f x \count_space UNIV) = 1"
begin

lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ennreal \ f)"
proof (intro conjI)
  have *[simp]: "\x y. ennreal (f y) * indicator {x} y = ennreal (f x) * indicator {x} y"
    by (simp split: split_indicator)
  show "AE x in density (count_space UNIV) (ennreal \ f).
    measure (density (count_space UNIV) (ennreal \<circ> f)) {x} \<noteq> 0"
    by (simp add: AE_density nonneg measure_def emeasure_density max_def)
  show "prob_space (density (count_space UNIV) (ennreal \ f))"
    by standard (simp add: emeasure_density prob)
qed simp

lemma pmf_embed_pmf: "pmf embed_pmf x = f x"
proof transfer
  have *[simp]: "\x y. ennreal (f y) * indicator {x} y = ennreal (f x) * indicator {x} y"
    by (simp split: split_indicator)
  fix x show "measure (density (count_space UNIV) (ennreal \ f)) {x} = f x"
    by transfer (simp add: measure_def emeasure_density nonneg max_def)
qed

lemma set_embed_pmf: "set_pmf embed_pmf = {x. f x \ 0}"
by(auto simp add: set_pmf_eq pmf_embed_pmf)

end

lemma embed_pmf_transfer:
  "rel_fun (eq_onp (\f. (\x. 0 \ f x) \ (\\<^sup>+x. ennreal (f x) \count_space UNIV) = 1)) pmf_as_measure.cr_pmf (\f. density (count_space UNIV) (ennreal \ f)) embed_pmf"
  by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer)

lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)"
proof (transfer, elim conjE)
  fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \ 0"
  assume "prob_space M" then interpret prob_space M .
  show "M = density (count_space UNIV) (\x. ennreal (measure M {x}))"
  proof (rule measure_eqI)
    fix A :: "'a set"
    have "(\\<^sup>+ x. ennreal (measure M {x}) * indicator A x \count_space UNIV) =
      (\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)"
      by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator)
    also have "\ = (\\<^sup>+ x. emeasure M {x} \count_space (A \ {x. measure M {x} \ 0}))"
      by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space)
    also have "\ = emeasure M (\x\(A \ {x. measure M {x} \ 0}). {x})"
      by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support)
         (auto simp: disjoint_family_on_def)
    also have "\ = emeasure M A"
      using ae by (intro emeasure_eq_AE) auto
    finally show " emeasure M A = emeasure (density (count_space UNIV) (\x. ennreal (measure M {x}))) A"
      using emeasure_space_1 by (simp add: emeasure_density)
  qed simp
qed

lemma td_pmf_embed_pmf:
  "type_definition pmf embed_pmf {f::'a \ real. (\x. 0 \ f x) \ (\\<^sup>+x. ennreal (f x) \count_space UNIV) = 1}"
  unfolding type_definition_def
proof safe
  fix p :: "'a pmf"
  have "(\\<^sup>+ x. 1 \measure_pmf p) = 1"
    using measure_pmf.emeasure_space_1[of p] by simp
  then show *: "(\\<^sup>+ x. ennreal (pmf p x) \count_space UNIV) = 1"
    by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const)

  show "embed_pmf (pmf p) = p"
    by (intro measure_pmf_inject[THEN iffD1])
       (simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def)
next
  fix f :: "'a \ real" assume "\x. 0 \ f x" "(\\<^sup>+x. f x \count_space UNIV) = 1"
  then show "pmf (embed_pmf f) = f"
    by (auto intro!: pmf_embed_pmf)
qed (rule pmf_nonneg)

end

lemma nn_integral_measure_pmf: "(\\<^sup>+ x. f x \measure_pmf p) = \\<^sup>+ x. ennreal (pmf p x) * f x \count_space UNIV"
by(simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg)

lemma integral_measure_pmf:
  fixes f :: "'a \ 'b::{banach, second_countable_topology}"
  assumes A: "finite A"
  shows "(\a. a \ set_pmf M \ f a \ 0 \ a \ A) \ (LINT x|M. f x) = (\a\A. pmf M a *\<^sub>R f a)"
  unfolding measure_pmf_eq_density
  apply (simp add: integral_density)
  apply (subst lebesgue_integral_count_space_finite_support)
  apply (auto intro!: finite_subset[OF _ \<open>finite A\<close>] sum.mono_neutral_left simp: pmf_eq_0_set_pmf)
  done

lemma expectation_return_pmf [simp]:
  fixes f :: "'a \ 'b::{banach, second_countable_topology}"
  shows "measure_pmf.expectation (return_pmf x) f = f x"
  by (subst integral_measure_pmf[of "{x}"]) simp_all

lemma pmf_expectation_bind:
  fixes p :: "'a pmf" and f :: "'a \ 'b pmf"
    and  h :: "'b \ 'c::{banach, second_countable_topology}"
  assumes "finite A" "\x. x \ A \ finite (set_pmf (f x))" "set_pmf p \ A"
  shows "measure_pmf.expectation (p \ f) h =
           (\<Sum>a\<in>A. pmf p a *\<^sub>R measure_pmf.expectation (f a) h)"
proof -
  have "measure_pmf.expectation (p \ f) h = (\a\(\x\A. set_pmf (f x)). pmf (p \ f) a *\<^sub>R h a)"
    using assms by (intro integral_measure_pmf) auto
  also have "\ = (\x\(\x\A. set_pmf (f x)). (\a\A. (pmf p a * pmf (f a) x) *\<^sub>R h x))"
  proof (intro sum.cong refl, goal_cases)
    case (1 x)
    thus ?case
      by (subst pmf_bind, subst integral_measure_pmf[of A])
         (insert assms, auto simp: scaleR_sum_left)
  qed
  also have "\ = (\j\A. pmf p j *\<^sub>R (\i\(\x\A. set_pmf (f x)). pmf (f j) i *\<^sub>R h i))"
    by (subst sum.swap) (simp add: scaleR_sum_right)
  also have "\ = (\j\A. pmf p j *\<^sub>R measure_pmf.expectation (f j) h)"
  proof (intro sum.cong refl, goal_cases)
    case (1 x)
    thus ?case
      by (subst integral_measure_pmf[of "(\x\A. set_pmf (f x))"])
         (insert assms, auto simp: scaleR_sum_left)
  qed
  finally show ?thesis .
qed

lemma continuous_on_LINT_pmf: \<comment> \<open>This is dominated convergence!?\<close>
  fixes f :: "'i \ 'a::topological_space \ 'b::{banach, second_countable_topology}"
  assumes f: "\i. i \ set_pmf M \ continuous_on A (f i)"
    and bnd: "\a i. a \ A \ i \ set_pmf M \ norm (f i a) \ B"
  shows "continuous_on A (\a. LINT i|M. f i a)"
proof cases
  assume "finite M" with f show ?thesis
    using integral_measure_pmf[OF \<open>finite M\<close>]
    by (subst integral_measure_pmf[OF \<open>finite M\<close>])
       (auto intro!: continuous_on_sum continuous_on_scaleR continuous_on_const)
next
  assume "infinite M"
  let ?f = "\i x. pmf (map_pmf (to_nat_on M) M) i *\<^sub>R f (from_nat_into M i) x"

  show ?thesis
  proof (rule uniform_limit_theorem)
    show "\\<^sub>F n in sequentially. continuous_on A (\a. \i
      by (intro always_eventually allI continuous_on_sum continuous_on_scaleR continuous_on_const f
                from_nat_into set_pmf_not_empty)
    show "uniform_limit A (\n a. \ia. LINT i|M. f i a) sequentially"
    proof (subst uniform_limit_cong[where g="\n a. \i
      fix a assume "a \ A"
      have 1: "(LINT i|M. f i a) = (LINT i|map_pmf (to_nat_on M) M. f (from_nat_into M i) a)"
        by (auto intro!: integral_cong_AE AE_pmfI)
      have 2: "\ = (LINT i|count_space UNIV. pmf (map_pmf (to_nat_on M) M) i *\<^sub>R f (from_nat_into M i) a)"
        by (simp add: measure_pmf_eq_density integral_density)
      have "(\n. ?f n a) sums (LINT i|M. f i a)"
        unfolding 1 2
      proof (intro sums_integral_count_space_nat)
        have A: "integrable M (\i. f i a)"
          using \<open>a\<in>A\<close> by (auto intro!: measure_pmf.integrable_const_bound AE_pmfI bnd)
        have "integrable (map_pmf (to_nat_on M) M) (\i. f (from_nat_into M i) a)"
          by (auto simp add: map_pmf_rep_eq integrable_distr_eq intro!: AE_pmfI integrable_cong_AE_imp[OF A])
        then show "integrable (count_space UNIV) (\n. ?f n a)"
          by (simp add: measure_pmf_eq_density integrable_density)
      qed
      then show "(LINT i|M. f i a) = (\ n. ?f n a)"
        by (simp add: sums_unique)
    next
      show "uniform_limit A (\n a. \ia. (\ n. ?f n a)) sequentially"
      proof (rule Weierstrass_m_test)
        fix n a assume "a\A"
        then show "norm (?f n a) \ pmf (map_pmf (to_nat_on M) M) n * B"
          using bnd by (auto intro!: mult_mono simp: from_nat_into set_pmf_not_empty)
      next
        have "integrable (map_pmf (to_nat_on M) M) (\n. B)"
          by auto
        then show "summable (\n. pmf (map_pmf (to_nat_on (set_pmf M)) M) n * B)"
          by (fastforce simp add: measure_pmf_eq_density integrable_density integrable_count_space_nat_iff summable_mult2)
      qed
    qed simp
  qed simp
qed

lemma continuous_on_LBINT:
  fixes f :: "real \ real"
  assumes f: "\b. a \ b \ set_integrable lborel {a..b} f"
  shows "continuous_on UNIV (\b. LBINT x:{a..b}. f x)"
proof (subst set_borel_integral_eq_integral)
  { fix b :: real assume "a \ b"
    from f[OF this] have "continuous_on {a..b} (\b. integral {a..b} f)"
      by (intro indefinite_integral_continuous_1 set_borel_integral_eq_integral) }
  note * = this

  have "continuous_on (\b\{a..}. {a <..< b}) (\b. integral {a..b} f)"
  proof (intro continuous_on_open_UN)
    show "b \ {a..} \ continuous_on {a<..b. integral {a..b} f)" for b
      using *[of b] by (rule continuous_on_subset) auto
  qed simp
  also have "(\b\{a..}. {a <..< b}) = {a <..}"
    by (auto simp: lt_ex gt_ex less_imp_le) (simp add: Bex_def less_imp_le gt_ex cong: rev_conj_cong)
  finally have "continuous_on {a+1 ..} (\b. integral {a..b} f)"
    by (rule continuous_on_subset) auto
  moreover have "continuous_on {a..a+1} (\b. integral {a..b} f)"
    by (rule *) simp
  moreover
  have "x \ a \ {a..x} = (if a = x then {a} else {})" for x
    by auto
  then have "continuous_on {..a} (\b. integral {a..b} f)"
    by (subst continuous_on_cong[OF refl, where g="\x. 0"]) (auto intro!: continuous_on_const)
  ultimately have "continuous_on ({..a} \ {a..a+1} \ {a+1 ..}) (\b. integral {a..b} f)"
    by (intro continuous_on_closed_Un) auto
  also have "{..a} \ {a..a+1} \ {a+1 ..} = UNIV"
    by auto
  finally show "continuous_on UNIV (\b. integral {a..b} f)"
    by auto
next
  show "set_integrable lborel {a..b} f" for b
    using f by (cases "a \ b") auto
qed

locale pmf_as_function
begin

setup_lifting td_pmf_embed_pmf

lemma set_pmf_transfer[transfer_rule]:
  assumes "bi_total A"
  shows "rel_fun (pcr_pmf A) (rel_set A) (\f. {x. f x \ 0}) set_pmf"
  using \<open>bi_total A\<close>
  by (auto simp: pcr_pmf_def cr_pmf_def rel_fun_def rel_set_def bi_total_def Bex_def set_pmf_iff)
     metis+

end

context
begin

interpretation pmf_as_function .

lemma pmf_eqI: "(\i. pmf M i = pmf N i) \ M = N"
  by transfer auto

lemma pmf_eq_iff: "M = N \ (\i. pmf M i = pmf N i)"
  by (auto intro: pmf_eqI)

lemma pmf_neq_exists_less:
  assumes "M \ N"
  shows   "\x. pmf M x < pmf N x"
proof (rule ccontr)
  assume "\(\x. pmf M x < pmf N x)"
  hence ge: "pmf M x \ pmf N x" for x by (auto simp: not_less)
  from assms obtain x where "pmf M x \ pmf N x" by (auto simp: pmf_eq_iff)
  with ge[of x] have gt: "pmf M x > pmf N x" by simp
  have "1 = measure (measure_pmf M) UNIV" by simp
  also have "\ = measure (measure_pmf N) {x} + measure (measure_pmf N) (UNIV - {x})"
    by (subst measure_pmf.finite_measure_Union [symmetric]) simp_all
  also from gt have "measure (measure_pmf N) {x} < measure (measure_pmf M) {x}"
    by (simp add: measure_pmf_single)
  also have "measure (measure_pmf N) (UNIV - {x}) \ measure (measure_pmf M) (UNIV - {x})"
    by (subst (1 2) integral_pmf [symmetric])
       (intro integral_mono integrable_pmf, simp_all add: ge)
  also have "measure (measure_pmf M) {x} + \ = 1"
    by (subst measure_pmf.finite_measure_Union [symmetric]) simp_all
  finally show False by simp_all
qed

lemma bind_commute_pmf: "bind_pmf A (\x. bind_pmf B (C x)) = bind_pmf B (\y. bind_pmf A (\x. C x y))"
  unfolding pmf_eq_iff pmf_bind
proof
  fix i
  interpret B: prob_space "restrict_space B B"
    by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
       (auto simp: AE_measure_pmf_iff)
  interpret A: prob_space "restrict_space A A"
    by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
       (auto simp: AE_measure_pmf_iff)

  interpret AB: pair_prob_space "restrict_space A A" "restrict_space B B"
    by unfold_locales

  have "(\ x. \ y. pmf (C x y) i \B \A) = (\ x. (\ y. pmf (C x y) i \restrict_space B B) \A)"
    by (rule Bochner_Integration.integral_cong) (auto intro!: integral_pmf_restrict)
  also have "\ = (\ x. (\ y. pmf (C x y) i \restrict_space B B) \restrict_space A A)"
    by (intro integral_pmf_restrict B.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
              countable_set_pmf borel_measurable_count_space)
  also have "\ = (\ y. \ x. pmf (C x y) i \restrict_space A A \restrict_space B B)"
    by (rule AB.Fubini_integral[symmetric])
       (auto intro!: AB.integrable_const_bound[where B=1] measurable_pair_restrict_pmf2
             simp: pmf_nonneg pmf_le_1 measurable_restrict_space1)
  also have "\ = (\ y. \ x. pmf (C x y) i \restrict_space A A \B)"
    by (intro integral_pmf_restrict[symmetric] A.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2
              countable_set_pmf borel_measurable_count_space)
  also have "\ = (\ y. \ x. pmf (C x y) i \A \B)"
    by (rule Bochner_Integration.integral_cong) (auto intro!: integral_pmf_restrict[symmetric])
  finally show "(\ x. \ y. pmf (C x y) i \B \A) = (\ y. \ x. pmf (C x y) i \A \B)" .
qed

lemma pair_map_pmf1: "pair_pmf (map_pmf f A) B = map_pmf (apfst f) (pair_pmf A B)"
proof (safe intro!: pmf_eqI)
  fix a :: "'a" and b :: "'b"
  have [simp]: "\c d. indicator (apfst f -` {(a, b)}) (c, d) = indicator (f -` {a}) c * (indicator {b} d::ennreal)"
    by (auto split: split_indicator)

  have "ennreal (pmf (pair_pmf (map_pmf f A) B) (a, b)) =
         ennreal (pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b))"
    unfolding pmf_pair ennreal_pmf_map
    by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_multc pmf_nonneg
                  emeasure_map_pmf[symmetric] ennreal_mult del: emeasure_map_pmf)
  then show "pmf (pair_pmf (map_pmf f A) B) (a, b) = pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b)"
    by (simp add: pmf_nonneg)
qed

lemma pair_map_pmf2: "pair_pmf A (map_pmf f B) = map_pmf (apsnd f) (pair_pmf A B)"
proof (safe intro!: pmf_eqI)
  fix a :: "'a" and b :: "'b"
  have [simp]: "\c d. indicator (apsnd f -` {(a, b)}) (c, d) = indicator {a} c * (indicator (f -` {b}) d::ennreal)"
    by (auto split: split_indicator)

  have "ennreal (pmf (pair_pmf A (map_pmf f B)) (a, b)) =
         ennreal (pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b))"
    unfolding pmf_pair ennreal_pmf_map
    by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_cmult nn_integral_multc pmf_nonneg
                  emeasure_map_pmf[symmetric] ennreal_mult del: emeasure_map_pmf)
  then show "pmf (pair_pmf A (map_pmf f B)) (a, b) = pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b)"
    by (simp add: pmf_nonneg)
qed

lemma map_pair: "map_pmf (\(a, b). (f a, g b)) (pair_pmf A B) = pair_pmf (map_pmf f A) (map_pmf g B)"
  by (simp add: pair_map_pmf2 pair_map_pmf1 map_pmf_comp split_beta')

end

lemma pair_return_pmf1: "pair_pmf (return_pmf x) y = map_pmf (Pair x) y"
by(simp add: pair_pmf_def bind_return_pmf map_pmf_def)

lemma pair_return_pmf2: "pair_pmf x (return_pmf y) = map_pmf (\x. (x, y)) x"
by(simp add: pair_pmf_def bind_return_pmf map_pmf_def)

lemma pair_pair_pmf: "pair_pmf (pair_pmf u v) w = map_pmf (\(x, (y, z)). ((x, y), z)) (pair_pmf u (pair_pmf v w))"
by(simp add: pair_pmf_def bind_return_pmf map_pmf_def bind_assoc_pmf)

lemma pair_commute_pmf: "pair_pmf x y = map_pmf (\(x, y). (y, x)) (pair_pmf y x)"
unfolding pair_pmf_def by(subst bind_commute_pmf)(simp add: map_pmf_def bind_assoc_pmf bind_return_pmf)

lemma set_pmf_subset_singleton: "set_pmf p \ {x} \ p = return_pmf x"
proof(intro iffI pmf_eqI)
  fix i
  assume x: "set_pmf p \ {x}"
  hence *: "set_pmf p = {x}" using set_pmf_not_empty[of p] by auto
  have "ennreal (pmf p x) = \\<^sup>+ i. indicator {x} i \p" by(simp add: emeasure_pmf_single)
  also have "\ = \\<^sup>+ i. 1 \p" by(rule nn_integral_cong_AE)(simp add: AE_measure_pmf_iff * )
  also have "\ = 1" by simp
  finally show "pmf p i = pmf (return_pmf x) i" using x
    by(auto split: split_indicator simp add: pmf_eq_0_set_pmf)
qed auto

lemma bind_eq_return_pmf:
  "bind_pmf p f = return_pmf x \ (\y\set_pmf p. f y = return_pmf x)"
  (is "?lhs \ ?rhs")
proof(intro iffI strip)
  fix y
  assume y: "y \ set_pmf p"
  assume "?lhs"
  hence "set_pmf (bind_pmf p f) = {x}" by simp
  hence "(\y\set_pmf p. set_pmf (f y)) = {x}" by simp
  hence "set_pmf (f y) \ {x}" using y by auto
  thus "f y = return_pmf x" by(simp add: set_pmf_subset_singleton)
next
  assume *: ?rhs
  show ?lhs
  proof(rule pmf_eqI)
    fix i
    have "ennreal (pmf (bind_pmf p f) i) = \\<^sup>+ y. ennreal (pmf (f y) i) \p"
      by (simp add: ennreal_pmf_bind)
    also have "\ = \\<^sup>+ y. ennreal (pmf (return_pmf x) i) \p"
      by(rule nn_integral_cong_AE)(simp add: AE_measure_pmf_iff * )
    also have "\ = ennreal (pmf (return_pmf x) i)"
      by simp
    finally show "pmf (bind_pmf p f) i = pmf (return_pmf x) i"
      by (simp add: pmf_nonneg)
  qed
qed

lemma pmf_False_conv_True: "pmf p False = 1 - pmf p True"
proof -
  have "pmf p False + pmf p True = measure p {False} + measure p {True}"
    by(simp add: measure_pmf_single)
  also have "\ = measure p ({False} \ {True})"
    by(subst measure_pmf.finite_measure_Union) simp_all
  also have "{False} \ {True} = space p" by auto
  finally show ?thesis by simp
qed

lemma pmf_True_conv_False: "pmf p True = 1 - pmf p False"
by(simp add: pmf_False_conv_True)

subsection \<open> Conditional Probabilities \<close>

lemma measure_pmf_zero_iff: "measure (measure_pmf p) s = 0 \ set_pmf p \ s = {}"
  by (subst measure_pmf.prob_eq_0) (auto simp: AE_measure_pmf_iff)

context
  fixes p :: "'a pmf" and s :: "'a set"
  assumes not_empty: "set_pmf p \ s \ {}"
begin

interpretation pmf_as_measure .

lemma emeasure_measure_pmf_not_zero: "emeasure (measure_pmf p) s \ 0"
proof
  assume "emeasure (measure_pmf p) s = 0"
  then have "AE x in measure_pmf p. x \ s"
    by (rule AE_I[rotated]) auto
  with not_empty show False
    by (auto simp: AE_measure_pmf_iff)
qed

lemma measure_measure_pmf_not_zero: "measure (measure_pmf p) s \ 0"
  using emeasure_measure_pmf_not_zero by (simp add: measure_pmf.emeasure_eq_measure measure_nonneg)

lift_definition cond_pmf :: "'a pmf" is
  "uniform_measure (measure_pmf p) s"
proof (intro conjI)
  show "prob_space (uniform_measure (measure_pmf p) s)"
    by (intro prob_space_uniform_measure) (auto simp: emeasure_measure_pmf_not_zero)
  show "AE x in uniform_measure (measure_pmf p) s. measure (uniform_measure (measure_pmf p) s) {x} \ 0"
    by (simp add: emeasure_measure_pmf_not_zero measure_measure_pmf_not_zero AE_uniform_measure
                  AE_measure_pmf_iff set_pmf.rep_eq less_top[symmetric])
qed simp

lemma pmf_cond: "pmf cond_pmf x = (if x \ s then pmf p x / measure p s else 0)"
  by transfer (simp add: emeasure_measure_pmf_not_zero pmf.rep_eq)

lemma set_cond_pmf[simp]: "set_pmf cond_pmf = set_pmf p \ s"
  by (auto simp add: set_pmf_iff pmf_cond measure_measure_pmf_not_zero split: if_split_asm)

end

lemma measure_pmf_posI: "x \ set_pmf p \ x \ A \ measure_pmf.prob p A > 0"
  using measure_measure_pmf_not_zero[of p A] by (subst zero_less_measure_iff) blast

lemma cond_map_pmf:
  assumes "set_pmf p \ f -` s \ {}"
  shows "cond_pmf (map_pmf f p) s = map_pmf f (cond_pmf p (f -` s))"
proof -
  have *: "set_pmf (map_pmf f p) \ s \ {}"
    using assms by auto
  { fix x
    have "ennreal (pmf (map_pmf f (cond_pmf p (f -` s))) x) =
      emeasure p (f -` s \<inter> f -` {x}) / emeasure p (f -` s)"
      unfolding ennreal_pmf_map cond_pmf.rep_eq[OF assms] by (simp add: nn_integral_uniform_measure)
    also have "f -` s \ f -` {x} = (if x \ s then f -` {x} else {})"
      by auto
    also have "emeasure p (if x \ s then f -` {x} else {}) / emeasure p (f -` s) =
      ennreal (pmf (cond_pmf (map_pmf f p) s) x)"
      using measure_measure_pmf_not_zero[OF *]
      by (simp add: pmf_cond[OF *] ennreal_pmf_map measure_pmf.emeasure_eq_measure
                    divide_ennreal pmf_nonneg measure_nonneg zero_less_measure_iff pmf_map)
    finally have "ennreal (pmf (cond_pmf (map_pmf f p) s) x) = ennreal (pmf (map_pmf f (cond_pmf p (f -` s))) x)"
      by simp }
  then show ?thesis
    by (intro pmf_eqI) (simp add: pmf_nonneg)
qed

lemma bind_cond_pmf_cancel:
  assumes [simp]: "\x. x \ set_pmf p \ set_pmf q \ {y. R x y} \ {}"
  assumes [simp]: "\y. y \ set_pmf q \ set_pmf p \ {x. R x y} \ {}"
  assumes [simp]: "\x y. x \ set_pmf p \ y \ set_pmf q \ R x y \ measure q {y. R x y} = measure p {x. R x y}"
  shows "bind_pmf p (\x. cond_pmf q {y. R x y}) = q"
proof (rule pmf_eqI)
  fix i
  have "ennreal (pmf (bind_pmf p (\x. cond_pmf q {y. R x y})) i) =
    (\<integral>\<^sup>+x. ennreal (pmf q i / measure p {x. R x i}) * ennreal (indicator {x. R x i} x) \<partial>p)"
    by (auto simp add: ennreal_pmf_bind AE_measure_pmf_iff pmf_cond pmf_eq_0_set_pmf pmf_nonneg measure_nonneg
             intro!: nn_integral_cong_AE)
  also have "\ = (pmf q i * measure p {x. R x i}) / measure p {x. R x i}"
    by (simp add: pmf_nonneg measure_nonneg zero_ennreal_def[symmetric] ennreal_indicator
                  nn_integral_cmult measure_pmf.emeasure_eq_measure ennreal_mult[symmetric])
  also have "\ = pmf q i"
    by (cases "pmf q i = 0")
       (simp_all add: pmf_eq_0_set_pmf measure_measure_pmf_not_zero pmf_nonneg)
  finally show "pmf (bind_pmf p (\x. cond_pmf q {y. R x y})) i = pmf q i"
    by (simp add: pmf_nonneg)
qed

subsection \<open> Relator \<close>

inductive rel_pmf :: "('a \ 'b \ bool) \ 'a pmf \ 'b pmf \ bool"
for R p q
where
  "\ \x y. (x, y) \ set_pmf pq \ R x y;
     map_pmf fst pq = p; map_pmf snd pq = q \<rbrakk>
  \<Longrightarrow> rel_pmf R p q"

lemma rel_pmfI:
  assumes R: "rel_set R (set_pmf p) (set_pmf q)"
  assumes eq: "\x y. x \ set_pmf p \ y \ set_pmf q \ R x y \
    measure p {x. R x y} = measure q {y. R x y}"
  shows "rel_pmf R p q"
proof
  let ?pq = "bind_pmf p (\x. bind_pmf (cond_pmf q {y. R x y}) (\y. return_pmf (x, y)))"
  have "\x. x \ set_pmf p \ set_pmf q \ {y. R x y} \ {}"
    using R by (auto simp: rel_set_def)
  then show "\x y. (x, y) \ set_pmf ?pq \ R x y"
    by auto
  show "map_pmf fst ?pq = p"
    by (simp add: map_bind_pmf bind_return_pmf')

  show "map_pmf snd ?pq = q"
    using R eq
    apply (simp add: bind_cond_pmf_cancel map_bind_pmf bind_return_pmf')
    apply (rule bind_cond_pmf_cancel)
    apply (auto simp: rel_set_def)
    done
qed

lemma rel_pmf_imp_rel_set: "rel_pmf R p q \ rel_set R (set_pmf p) (set_pmf q)"
  by (force simp add: rel_pmf.simps rel_set_def)

lemma rel_pmfD_measure:
  assumes rel_R: "rel_pmf R p q" and R: "\a b. R a b \ R a y \ R x b"
  assumes "x \ set_pmf p" "y \ set_pmf q"
  shows "measure p {x. R x y} = measure q {y. R x y}"
proof -
  from rel_R obtain pq where pq: "\x y. (x, y) \ set_pmf pq \ R x y"
    and eq: "p = map_pmf fst pq" "q = map_pmf snd pq"
    by (auto elim: rel_pmf.cases)
  have "measure p {x. R x y} = measure pq {x. R (fst x) y}"
    by (simp add: eq map_pmf_rep_eq measure_distr)
  also have "\ = measure pq {y. R x (snd y)}"
    by (intro measure_pmf.finite_measure_eq_AE)
       (auto simp: AE_measure_pmf_iff R dest!: pq)
  also have "\ = measure q {y. R x y}"
    by (simp add: eq map_pmf_rep_eq measure_distr)
  finally show "measure p {x. R x y} = measure q {y. R x y}" .
qed

lemma rel_pmf_measureD:
  assumes "rel_pmf R p q"
  shows "measure (measure_pmf p) A \ measure (measure_pmf q) {y. \x\A. R x y}" (is "?lhs \ ?rhs")
using assms
proof cases
  fix pq
  assume R: "\x y. (x, y) \ set_pmf pq \ R x y"
    and p[symmetric]: "map_pmf fst pq = p"
    and q[symmetric]: "map_pmf snd pq = q"
  have "?lhs = measure (measure_pmf pq) (fst -` A)" by(simp add: p)
  also have "\ \ measure (measure_pmf pq) {y. \x\A. R x (snd y)}"
    by(rule measure_pmf.finite_measure_mono_AE)(auto 4 3 simp add: AE_measure_pmf_iff dest: R)
  also have "\ = ?rhs" by(simp add: q)
  finally show ?thesis .
qed

lemma rel_pmf_iff_measure:
  assumes "symp R" "transp R"
  shows "rel_pmf R p q \
    rel_set R (set_pmf p) (set_pmf q) \<and>
    (\<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})"
  by (safe intro!: rel_pmf_imp_rel_set rel_pmfI)
     (auto intro!: rel_pmfD_measure dest: sympD[OF \<open>symp R\<close>] transpD[OF \<open>transp R\<close>])

lemma quotient_rel_set_disjoint:
  "equivp R \ C \ UNIV // {(x, y). R x y} \ rel_set R A B \ A \ C = {} \ B \ C = {}"
  using in_quotient_imp_closed[of UNIV "{(x, y). R x y}" C]
  by (auto 0 0 simp: equivp_equiv rel_set_def set_eq_iff elim: equivpE)
     (blast dest: equivp_symp)+

lemma quotientD: "equiv X R \ A \ X // R \ x \ A \ A = R `` {x}"
  by (metis Image_singleton_iff equiv_class_eq_iff quotientE)

lemma rel_pmf_iff_equivp:
  assumes "equivp R"
  shows "rel_pmf R p q \ (\C\UNIV // {(x, y). R x y}. measure p C = measure q C)"
    (is "_ \ (\C\_//?R. _)")
proof (subst rel_pmf_iff_measure, safe)
  show "symp R" "transp R"
    using assms by (auto simp: equivp_reflp_symp_transp)
next
  fix C assume C: "C \ UNIV // ?R" and R: "rel_set R (set_pmf p) (set_pmf q)"
  assume eq: "\x\set_pmf p. \y\set_pmf q. R x y \ measure p {x. R x y} = measure q {y. R x y}"

  show "measure p C = measure q C"
  proof (cases "p \ C = {}")
    case True
    then have "q \ C = {}"
      using quotient_rel_set_disjoint[OF assms C R] by simp
    with True show ?thesis
      unfolding measure_pmf_zero_iff[symmetric] by simp
  next
    case False
    then have "q \ C \ {}"
      using quotient_rel_set_disjoint[OF assms C R] by simp
    with False obtain x y where in_set: "x \ set_pmf p" "y \ set_pmf q" and in_C: "x \ C" "y \ C"
      by auto
    then have "R x y"
      using in_quotient_imp_in_rel[of UNIV ?R C x y] C assms
      by (simp add: equivp_equiv)
    with in_set eq have "measure p {x. R x y} = measure q {y. R x y}"
      by auto
    moreover have "{y. R x y} = C"
      using assms \<open>x \<in> C\<close> C quotientD[of UNIV ?R C x] by (simp add: equivp_equiv)
    moreover have "{x. R x y} = C"
      using assms \<open>y \<in> C\<close> C quotientD[of UNIV "?R" C y] sympD[of R]
      by (auto simp add: equivp_equiv elim: equivpE)
    ultimately show ?thesis
      by auto
  qed
next
  assume eq: "\C\UNIV // ?R. measure p C = measure q C"
  show "rel_set R (set_pmf p) (set_pmf q)"
    unfolding rel_set_def
  proof safe
    fix x assume x: "x \ set_pmf p"
    have "{y. R x y} \ UNIV // ?R"
      by (auto simp: quotient_def)
    with eq have *: "measure q {y. R x y} = measure p {y. R x y}"
      by auto
    have "measure q {y. R x y} \ 0"
      using x assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp)
    then show "\y\set_pmf q. R x y"
      unfolding measure_pmf_zero_iff by auto
  next
    fix y assume y: "y \ set_pmf q"
    have "{x. R x y} \ UNIV // ?R"
      using assms by (auto simp: quotient_def dest: equivp_symp)
    with eq have *: "measure p {x. R x y} = measure q {x. R x y}"
      by auto
    have "measure p {x. R x y} \ 0"
      using y assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp)
    then show "\x\set_pmf p. R x y"
      unfolding measure_pmf_zero_iff by auto
  qed

  fix x y assume "x \ set_pmf p" "y \ set_pmf q" "R x y"
  have "{y. R x y} \ UNIV // ?R" "{x. R x y} = {y. R x y}"
    using assms \<open>R x y\<close> by (auto simp: quotient_def dest: equivp_symp equivp_transp)
  with eq show "measure p {x. R x y} = measure q {y. R x y}"
    by auto
qed

bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: rel_pmf
proof -
  show "map_pmf id = id" by (rule map_pmf_id)
  show "\f g. map_pmf (f \ g) = map_pmf f \ map_pmf g" by (rule map_pmf_compose)
  show "\f g::'a \ 'b. \p. (\x. x \ set_pmf p \ f x = g x) \ map_pmf f p = map_pmf g p"
    by (intro map_pmf_cong refl)

  show "\f::'a \ 'b. set_pmf \ map_pmf f = (`) f \ set_pmf"
    by (rule pmf_set_map)

  show "(card_of (set_pmf p), natLeq) \ ordLeq" for p :: "'s pmf"
  proof -
    have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \ ordLeq"
      by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"])
         (auto intro: countable_set_pmf)
    also have "(card_of (UNIV :: nat set), natLeq) \ ordLeq"
      by (metis Field_natLeq card_of_least natLeq_Well_order)
    finally show ?thesis .
  qed

  show "\R. rel_pmf R = (\x y. \z. set_pmf z \ {(x, y). R x y} \
    map_pmf fst z = x \<and> map_pmf snd z = y)"
     by (auto simp add: fun_eq_iff rel_pmf.simps)

  show "rel_pmf R OO rel_pmf S \ rel_pmf (R OO S)"
    for R :: "'a \ 'b \ bool" and S :: "'b \ 'c \ bool"
  proof -
    { fix p q r
      assume pq: "rel_pmf R p q"
        and qr:"rel_pmf S q r"
      from pq obtain pq where pq: "\x y. (x, y) \ set_pmf pq \ R x y"
        and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto
      from qr obtain qr where qr: "\y z. (y, z) \ set_pmf qr \ S y z"
        and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto

      define pr where "pr =
        bind_pmf pq (\<lambda>xy. bind_pmf (cond_pmf qr {yz. fst yz = snd xy})
          (\<lambda>yz. return_pmf (fst xy, snd yz)))"
      have pr_welldefined: "\y. y \ q \ qr \ {yz. fst yz = y} \ {}"
        by (force simp: q')

      have "rel_pmf (R OO S) p r"
      proof (rule rel_pmf.intros)
        fix x z assume "(x, z) \ pr"
        then have "\y. (x, y) \ pq \ (y, z) \ qr"
          by (auto simp: q pr_welldefined pr_def split_beta)
        with pq qr show "(R OO S) x z"
          by blast
      next
        have "map_pmf snd pr = map_pmf snd (bind_pmf q (\y. cond_pmf qr {yz. fst yz = y}))"
          by (simp add: pr_def q split_beta bind_map_pmf map_pmf_def[symmetric] map_bind_pmf map_pmf_comp)
        then show "map_pmf snd pr = r"
          unfolding r q' bind_map_pmf by (subst (asm) bind_cond_pmf_cancel) (auto simp: eq_commute)
      qed (simp add: pr_def map_bind_pmf split_beta map_pmf_def[symmetric] p map_pmf_comp)
    }
    then show ?thesis
      by(auto simp add: le_fun_def)
  qed
qed (fact natLeq_card_order natLeq_cinfinite)+

lemma map_pmf_idI: "(\x. x \ set_pmf p \ f x = x) \ map_pmf f p = p"
by(simp cong: pmf.map_cong)

lemma rel_pmf_conj[simp]:
  "rel_pmf (\x y. P \ Q x y) x y \ P \ rel_pmf Q x y"
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