# HG changeset patch
# User hoelzl
# Date 1416933005 -3600
# Node ID 43e07797269b37c08f098698ed7d7aeede6f3cf6
# Parent  a05c8305781e3275fa173f6148c5f5782d967321
tuned proof that pmfs are bnfs

diff -r a05c8305781e -r 43e07797269b src/HOL/Probability/Probability_Mass_Function.thy
--- a/src/HOL/Probability/Probability_Mass_Function.thy	Tue Nov 25 17:29:39 2014 +0100
+++ b/src/HOL/Probability/Probability_Mass_Function.thy	Tue Nov 25 17:30:05 2014 +0100
@@ -297,6 +297,9 @@
 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 (\<lambda>x. x) = (\<lambda>x. x)"
+  using map_pmf_id unfolding id_def .
+
 lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g"
   by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def) 
 
@@ -844,6 +847,24 @@
 lemma map_snd_pair_pmf: "map_pmf snd (pair_pmf A B) = B"
   by (simp add: pair_pmf_def bind_return_pmf''[symmetric] bind_assoc_pmf bind_return_pmf bind_return_pmf')
 
+lemma nn_integral_pmf':
+  "inj_on f A \<Longrightarrow> (\<integral>\<^sup>+x. pmf p (f x) \<partial>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 \<le> 0 \<longleftrightarrow> pmf M p = 0"
+  using pmf_nonneg[of M p] by simp
+
+lemma min_pmf_0[simp]: "min (pmf M p) 0 = 0" "min 0 (pmf M p) = 0"
+  using pmf_nonneg[of M p] by simp_all
+
+lemma pmf_eq_0_set_pmf: "pmf M p = 0 \<longleftrightarrow> p \<notin> set_pmf M"
+  unfolding set_pmf_iff by simp
+
+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"
+  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)
+
 inductive rel_pmf :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf \<Rightarrow> bool"
 for R p q
 where
@@ -864,7 +885,7 @@
   { fix p :: "'s pmf"
     have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq"
       by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"])
-         (auto intro: countable_set_pmf inj_on_to_nat_on)
+         (auto intro: countable_set_pmf)
     also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq"
       by (metis Field_natLeq card_of_least natLeq_Well_order)
     finally show "(card_of (set_pmf p), natLeq) \<in> ordLeq" . }
@@ -888,118 +909,80 @@
     from qr obtain qr where qr: "\<And>y z. (y, z) \<in> set_pmf qr \<Longrightarrow> S y z"
       and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto
 
-    have support_subset: "set_pmf pq O set_pmf qr \<subseteq> set_pmf p \<times> set_pmf r"
-      by(auto simp add: p r set_map_pmf intro: rev_image_eqI)
+    note pmf_nonneg[intro, simp]
 
-    let ?A = "\<lambda>y. {x. (x, y) \<in> set_pmf pq}"
-      and ?B = "\<lambda>y. {z. (y, z) \<in> set_pmf qr}"
-
+    def A \<equiv> "\<lambda>y. {x. (x, y) \<in> set_pmf pq}"
+    then have "\<And>y. A y \<subseteq> set_pmf p" by (auto simp add: p set_map_pmf intro: rev_image_eqI)
+    then have [simp]: "\<And>y. countable (A y)" by (rule countable_subset) simp
+    have A: "\<And>x y. (x, y) \<in> set_pmf pq \<longleftrightarrow> x \<in> A y"
+      by (simp add: A_def)
 
-    def ppp \<equiv> "\<lambda>A. \<lambda>f :: 'a \<Rightarrow> real. \<lambda>n. if n \<in> to_nat_on A ` A then f (from_nat_into A n) else 0"
-    have [simp]: "\<And>A f n. (\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x) \<Longrightarrow> 0 \<le> ppp A f n"
-                 "\<And>A f n x. \<lbrakk> x \<in> A; countable A \<rbrakk> \<Longrightarrow> ppp A f (to_nat_on A x) = f x"
-                 "\<And>A f n. n \<notin> to_nat_on A ` A \<Longrightarrow> ppp A f n = 0"
-      by(auto simp add: ppp_def intro: from_nat_into)
-    def rrr \<equiv> "\<lambda>A. \<lambda>f :: 'c \<Rightarrow> real. \<lambda>n. if n \<in> to_nat_on A ` A then f (from_nat_into A n) else 0"
-    have [simp]: "\<And>A f n. (\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x) \<Longrightarrow> 0 \<le> rrr A f n"
-                 "\<And>A f n x. \<lbrakk> x \<in> A; countable A \<rbrakk> \<Longrightarrow> rrr A f (to_nat_on A x) = f x"
-                 "\<And>A f n. n \<notin> to_nat_on A ` A \<Longrightarrow> rrr A f n = 0"
-      by(auto simp add: rrr_def intro: from_nat_into)
-
-    def pp \<equiv> "\<lambda>y. ppp (?A y) (\<lambda>x. pmf pq (x, y))"
-     and rr \<equiv> "\<lambda>y. rrr (?B y) (\<lambda>z. pmf qr (y, z))"
+    let ?P = "\<lambda>y. to_nat_on (A y)"
+    def pp \<equiv> "map_pmf (\<lambda>(x, y). (y, ?P y x)) pq"
+    let ?pp = "\<lambda>y x. pmf pp (y, x)"
+    { fix x y have "x \<in> A y \<Longrightarrow> pmf pp (y, ?P y x) = pmf pq (x, y)"
+        unfolding pp_def
+        by (intro pmf_map_inj[of "\<lambda>(x, y). (y, ?P y x)" pq "(x, y)", simplified])
+           (auto simp: inj_on_def A) }
+    note pmf_pp = this
 
-    have pos_p [simp]: "\<And>y n. 0 \<le> pp y n"
-      and pos_r [simp]: "\<And>y n. 0 \<le> rr y n"
-      by(simp_all add: pmf_nonneg pp_def rr_def)
-    { fix y n
-      have "pp y n \<le> 0 \<longleftrightarrow> pp y n = 0" "\<not> 0 < pp y n \<longleftrightarrow> pp y n = 0"
-        and "min (pp y n) 0 = 0" "min 0 (pp y n) = 0"
-        using pos_p[of y n] by(auto simp del: pos_p) }
-    note pp_convs [simp] = this
-    { fix y n
-      have "rr y n \<le> 0 \<longleftrightarrow> rr y n = 0" "\<not> 0 < rr y n \<longleftrightarrow> rr y n = 0"
-        and "min (rr y n) 0 = 0" "min 0 (rr y n) = 0"
-        using pos_r[of y n] by(auto simp del: pos_r) }
-    note r_convs [simp] = this
+    def B \<equiv> "\<lambda>y. {z. (y, z) \<in> set_pmf qr}"
+    then have "\<And>y. B y \<subseteq> set_pmf r" by (auto simp add: r set_map_pmf intro: rev_image_eqI)
+    then have [simp]: "\<And>y. countable (B y)" by (rule countable_subset) simp
+    have B: "\<And>y z. (y, z) \<in> set_pmf qr \<longleftrightarrow> z \<in> B y"
+      by (simp add: B_def)
 
-    have "\<And>y. ?A y \<subseteq> set_pmf p" by(auto simp add: p set_map_pmf intro: rev_image_eqI)
-    then have [simp]: "\<And>y. countable (?A y)" by(rule countable_subset) simp
+    let ?R = "\<lambda>y. to_nat_on (B y)"
+    def rr \<equiv> "map_pmf (\<lambda>(y, z). (y, ?R y z)) qr"
+    let ?rr = "\<lambda>y z. pmf rr (y, z)"
+    { fix y z have "z \<in> B y \<Longrightarrow> pmf rr (y, ?R y z) = pmf qr (y, z)"
+        unfolding rr_def
+        by (intro pmf_map_inj[of "\<lambda>(y, z). (y, ?R y z)" qr "(y, z)", simplified])
+           (auto simp: inj_on_def B) }
+    note pmf_rr = this
 
-    have "\<And>y. ?B y \<subseteq> set_pmf r" by(auto simp add: r set_map_pmf intro: rev_image_eqI)
-    then have [simp]: "\<And>y. countable (?B y)" by(rule countable_subset) simp
-
-    let ?P = "\<lambda>y. to_nat_on (?A y)"
-      and ?R = "\<lambda>y. to_nat_on (?B y)"
-
-    have eq: "\<And>y. (\<integral>\<^sup>+ x. pp y x \<partial>count_space UNIV) = \<integral>\<^sup>+ z. rr y z \<partial>count_space UNIV"
+    have eq: "\<And>y. (\<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV) = (\<integral>\<^sup>+ z. ?rr y z \<partial>count_space UNIV)"
     proof -
       fix y
-      have "(\<integral>\<^sup>+ x. pp y x \<partial>count_space UNIV) = (\<integral>\<^sup>+ x. pp y x \<partial>count_space (?P y ` ?A y))"
-        by(auto simp add: pp_def nn_integral_count_space_indicator indicator_def intro!: nn_integral_cong)
-      also have "\<dots> = (\<integral>\<^sup>+ x. pp y (?P y x) \<partial>count_space (?A y))"
-        by(intro nn_integral_bij_count_space[symmetric] inj_on_imp_bij_betw inj_on_to_nat_on) simp
-      also have "\<dots> = (\<integral>\<^sup>+ x. pmf pq (x, y) \<partial>count_space (?A y))"
-        by(rule nn_integral_cong)(simp add: pp_def)
-      also have "\<dots> = \<integral>\<^sup>+ x. emeasure (measure_pmf pq) {(x, y)} \<partial>count_space (?A y)"
-        by(simp add: emeasure_pmf_single)
-      also have "\<dots> = emeasure (measure_pmf pq) (\<Union>x\<in>?A y. {(x, y)})"
-        by(subst emeasure_UN_countable)(simp_all add: disjoint_family_on_def)
-      also have "\<dots> = emeasure (measure_pmf pq) ((\<Union>x\<in>?A y. {(x, y)}) \<union> {(x, y'). x \<notin> ?A y \<and> y' = y})"
-        by(rule emeasure_Un_null_set[symmetric])+
-          (auto simp add: q set_map_pmf split_beta intro!: in_null_sets_measure_pmfI intro: rev_image_eqI)
-      also have "\<dots> = emeasure (measure_pmf pq) (snd -` {y})"
-        by(rule arg_cong2[where f=emeasure])+auto
-      also have "\<dots> = pmf q y" by(simp add: q ereal_pmf_map)
-      also have "\<dots> = emeasure (measure_pmf qr) (fst -` {y})"
-        by(simp add: q' ereal_pmf_map)
-      also have "\<dots> = emeasure (measure_pmf qr) ((\<Union>z\<in>?B y. {(y, z)}) \<union> {(y', z). z \<notin> ?B y \<and> y' = y})"
-        by(rule arg_cong2[where f=emeasure])+auto
-      also have "\<dots> = emeasure (measure_pmf qr) (\<Union>z\<in>?B y. {(y, z)})"
-        by(rule emeasure_Un_null_set)
-          (auto simp add: q' set_map_pmf split_beta intro!: in_null_sets_measure_pmfI intro: rev_image_eqI)
-      also have "\<dots> = \<integral>\<^sup>+ z. emeasure (measure_pmf qr) {(y, z)} \<partial>count_space (?B y)"
-        by(subst emeasure_UN_countable)(simp_all add: disjoint_family_on_def)
-      also have "\<dots> = (\<integral>\<^sup>+ z. pmf qr (y, z) \<partial>count_space (?B y))"
-        by(simp add: emeasure_pmf_single)
-      also have "\<dots> = (\<integral>\<^sup>+ z. rr y (?R y z) \<partial>count_space (?B y))"
-        by(rule nn_integral_cong)(simp add: rr_def)
-      also have "\<dots> = (\<integral>\<^sup>+ z. rr y z \<partial>count_space (?R y ` ?B y))"
-        by(intro nn_integral_bij_count_space inj_on_imp_bij_betw inj_on_to_nat_on) simp
-      also have "\<dots> = \<integral>\<^sup>+ z. rr y z \<partial>count_space UNIV"
-        by(auto simp add: rr_def nn_integral_count_space_indicator indicator_def intro!: nn_integral_cong)
+      have "(\<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV) = pmf q y"
+        by (simp add: nn_integral_pmf' inj_on_def pp_def q)
+           (auto simp add: ereal_pmf_map intro!: arg_cong2[where f=emeasure])
+      also have "\<dots> = (\<integral>\<^sup>+ x. ?rr y x \<partial>count_space UNIV)"
+        by (simp add: nn_integral_pmf' inj_on_def rr_def q')
+           (auto simp add: ereal_pmf_map intro!: arg_cong2[where f=emeasure])
       finally show "?thesis y" .
     qed
 
     def assign_aux \<equiv> "\<lambda>y remainder start weight z.
        if z < start then 0
        else if z = start then min weight remainder
-       else if remainder + setsum (rr y) {Suc start ..<z} < weight then min (weight - remainder - setsum (rr y) {Suc start..<z}) (rr y z) else 0"
+       else if remainder + setsum (?rr y) {Suc start ..<z} < weight then min (weight - remainder - setsum (?rr y) {Suc start..<z}) (?rr y z) else 0"
     hence assign_aux_alt_def: "\<And>y remainder start weight z. assign_aux y remainder start weight z = 
        (if z < start then 0
         else if z = start then min weight remainder
-        else if remainder + setsum (rr y) {Suc start ..<z} < weight then min (weight - remainder - setsum (rr y) {Suc start..<z}) (rr y z) else 0)"
+        else if remainder + setsum (?rr y) {Suc start ..<z} < weight then min (weight - remainder - setsum (?rr y) {Suc start..<z}) (?rr y z) else 0)"
        by simp
     { fix y and remainder :: real and start and weight :: real
       assume weight_nonneg: "0 \<le> weight"
       let ?assign_aux = "assign_aux y remainder start weight"
       { fix z
         have "setsum ?assign_aux {..<z} =
-           (if z \<le> start then 0 else if remainder + setsum (rr y) {Suc start..<z} < weight then remainder + setsum (rr y) {Suc start..<z} else weight)"
+           (if z \<le> start then 0 else if remainder + setsum (?rr y) {Suc start..<z} < weight then remainder + setsum (?rr y) {Suc start..<z} else weight)"
         proof(induction z)
           case (Suc z) show ?case
-            by(auto simp add: Suc.IH assign_aux_alt_def[where z=z] not_less)(metis add.commute add.left_commute add_increasing pos_r)
+            by (auto simp add: Suc.IH assign_aux_alt_def[where z=z] not_less)
+               (metis add.commute add.left_commute add_increasing pmf_nonneg)
         qed(auto simp add: assign_aux_def) }
       note setsum_start_assign_aux = this
       moreover {
         assume remainder_nonneg: "0 \<le> remainder"
         have [simp]: "\<And>z. 0 \<le> ?assign_aux z"
           by(simp add: assign_aux_def weight_nonneg remainder_nonneg)
-        moreover have "\<And>z. \<lbrakk> rr y z = 0; remainder \<le> rr y start \<rbrakk> \<Longrightarrow> ?assign_aux z = 0"
+        moreover have "\<And>z. \<lbrakk> ?rr y z = 0; remainder \<le> ?rr y start \<rbrakk> \<Longrightarrow> ?assign_aux z = 0"
           using remainder_nonneg weight_nonneg
           by(auto simp add: assign_aux_def min_def)
         moreover have "(\<integral>\<^sup>+ z. ?assign_aux z \<partial>count_space UNIV) = 
-          min weight (\<integral>\<^sup>+ z. (if z < start then 0 else if z = start then remainder else rr y z) \<partial>count_space UNIV)"
+          min weight (\<integral>\<^sup>+ z. (if z < start then 0 else if z = start then remainder else ?rr y z) \<partial>count_space UNIV)"
           (is "?lhs = ?rhs" is "_ = min _ (\<integral>\<^sup>+ y. ?f y \<partial>_)")
         proof -
           have "?lhs = (SUP n. \<Sum>z<n. ereal (?assign_aux z))"
@@ -1028,27 +1011,27 @@
       and assign_aux_eq_0_outside = this(3)
       and nn_integral_assign_aux = this(4)
     { fix y and remainder :: real and start target
-      have "setsum (rr y) {Suc start..<target} \<ge> 0" by(simp add: setsum_nonneg)
+      have "setsum (?rr y) {Suc start..<target} \<ge> 0" by (simp add: setsum_nonneg)
       moreover assume "0 \<le> remainder"
       ultimately have "assign_aux y remainder start 0 target = 0"
         by(auto simp add: assign_aux_def min_def) }
     note assign_aux_weight_0 [simp] = this
 
-    def find_start \<equiv> "\<lambda>y weight. if \<exists>n. weight \<le> setsum (rr y)  {..n} then Some (LEAST n. weight \<le> setsum (rr y) {..n}) else None"
+    def find_start \<equiv> "\<lambda>y weight. if \<exists>n. weight \<le> setsum (?rr y)  {..n} then Some (LEAST n. weight \<le> setsum (?rr y) {..n}) else None"
     have find_start_eq_Some_above:
-      "\<And>y weight n. find_start y weight = Some n \<Longrightarrow> weight \<le> setsum (rr y) {..n}"
+      "\<And>y weight n. find_start y weight = Some n \<Longrightarrow> weight \<le> setsum (?rr y) {..n}"
       by(drule sym)(auto simp add: find_start_def split: split_if_asm intro: LeastI)
     { fix y weight n
       assume find_start: "find_start y weight = Some n"
       and weight: "0 \<le> weight"
-      have "setsum (rr y) {..n} \<le> rr y n + weight"
+      have "setsum (?rr y) {..n} \<le> ?rr y n + weight"
       proof(rule ccontr)
         assume "\<not> ?thesis"
-        hence "rr y n + weight < setsum (rr y) {..n}" by simp
+        hence "?rr y n + weight < setsum (?rr y) {..n}" by simp
         moreover with weight obtain n' where "n = Suc n'" by(cases n) auto
-        ultimately have "weight \<le> setsum (rr y) {..n'}" by simp
-        hence "(LEAST n. weight \<le> setsum (rr y) {..n}) \<le> n'" by(rule Least_le)
-        moreover from find_start have "n = (LEAST n. weight \<le> setsum (rr y) {..n})"
+        ultimately have "weight \<le> setsum (?rr y) {..n'}" by simp
+        hence "(LEAST n. weight \<le> setsum (?rr y) {..n}) \<le> n'" by(rule Least_le)
+        moreover from find_start have "n = (LEAST n. weight \<le> setsum (?rr y) {..n})"
           by(auto simp add: find_start_def split: split_if_asm)
         ultimately show False using \<open>n = Suc n'\<close> by auto
       qed }
@@ -1056,41 +1039,41 @@
     have find_start_0 [simp]: "\<And>y. find_start y 0 = Some 0"
       by(auto simp add: find_start_def intro!: exI[where x=0])
     { fix y and weight :: real
-      assume "weight < \<integral>\<^sup>+ z. rr y z \<partial>count_space UNIV"
-      also have "(\<integral>\<^sup>+ z. rr y z \<partial>count_space UNIV) = (SUP n. \<Sum>z<n. ereal (rr y z))"
+      assume "weight < \<integral>\<^sup>+ z. ?rr y z \<partial>count_space UNIV"
+      also have "(\<integral>\<^sup>+ z. ?rr y z \<partial>count_space UNIV) = (SUP n. \<Sum>z<n. ereal (?rr y z))"
         by(simp add: nn_integral_count_space_nat suminf_ereal_eq_SUP)
-      finally obtain n where "weight < (\<Sum>z<n. rr y z)" by(auto simp add: less_SUP_iff)
+      finally obtain n where "weight < (\<Sum>z<n. ?rr y z)" by(auto simp add: less_SUP_iff)
       hence "weight \<in> dom (find_start y)"
-        by(auto simp add: find_start_def)(meson atMost_iff finite_atMost lessThan_iff less_imp_le order_trans pos_r setsum_mono3 subsetI) }
+        by(auto simp add: find_start_def)(meson atMost_iff finite_atMost lessThan_iff less_imp_le order_trans pmf_nonneg setsum_mono3 subsetI) }
     note in_dom_find_startI = this
     { fix y and w w' :: real and m
-      let ?m' = "LEAST m. w' \<le> setsum (rr y) {..m}"
+      let ?m' = "LEAST m. w' \<le> setsum (?rr y) {..m}"
       assume "w' \<le> w"
       also  assume "find_start y w = Some m"
-      hence "w \<le> setsum (rr y) {..m}" by(rule find_start_eq_Some_above)
+      hence "w \<le> setsum (?rr y) {..m}" by(rule find_start_eq_Some_above)
       finally have "find_start y w' = Some ?m'" by(auto simp add: find_start_def)
-      moreover from \<open>w' \<le> setsum (rr y) {..m}\<close> have "?m' \<le> m" by(rule Least_le)
+      moreover from \<open>w' \<le> setsum (?rr y) {..m}\<close> have "?m' \<le> m" by(rule Least_le)
       ultimately have "\<exists>m'. find_start y w' = Some m' \<and> m' \<le> m" by blast }
     note find_start_mono = this[rotated]
 
-    def assign \<equiv> "\<lambda>y x z. let used = setsum (pp y) {..<x}
+    def assign \<equiv> "\<lambda>y x z. let used = setsum (?pp y) {..<x}
       in case find_start y used of None \<Rightarrow> 0
-         | Some start \<Rightarrow> assign_aux y (setsum (rr y) {..start} - used) start (pp y x) z"
+         | Some start \<Rightarrow> assign_aux y (setsum (?rr y) {..start} - used) start (?pp y x) z"
     hence assign_alt_def: "\<And>y x z. assign y x z = 
-      (let used = setsum (pp y) {..<x}
+      (let used = setsum (?pp y) {..<x}
        in case find_start y used of None \<Rightarrow> 0
-          | Some start \<Rightarrow> assign_aux y (setsum (rr y) {..start} - used) start (pp y x) z)"
+          | Some start \<Rightarrow> assign_aux y (setsum (?rr y) {..start} - used) start (?pp y x) z)"
       by simp
     have assign_nonneg [simp]: "\<And>y x z. 0 \<le> assign y x z"
-      by(simp add: assign_def diff_le_iff find_start_eq_Some_above split: option.split)
-    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"
-      by(auto simp add: assign_def assign_aux_eq_0_outside diff_le_iff find_start_eq_Some_above find_start_eq_Some_least setsum_nonneg split: option.split)
+      by(simp add: assign_def diff_le_iff find_start_eq_Some_above Let_def split: option.split)
+    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"
+      by(auto simp add: assign_def assign_aux_eq_0_outside diff_le_iff find_start_eq_Some_above find_start_eq_Some_least setsum_nonneg Let_def split: option.split)
 
     { fix y x z
       have "(\<Sum>n<Suc x. assign y n z) =
-            (case find_start y (setsum (pp y) {..<x}) of None \<Rightarrow> rr y z
-             | Some m \<Rightarrow> if z < m then rr y z 
-                         else min (rr y z) (max 0 (setsum (pp y) {..<x} + pp y x - setsum (rr y) {..<z})))"
+            (case find_start y (setsum (?pp y) {..<x}) of None \<Rightarrow> ?rr y z
+             | Some m \<Rightarrow> if z < m then ?rr y z 
+                         else min (?rr y z) (max 0 (setsum (?pp y) {..<x} + ?pp y x - setsum (?rr y) {..<z})))"
         (is "?lhs x = ?rhs x")
       proof(induction x)
         case 0 thus ?case 
@@ -1100,46 +1083,46 @@
         have "?lhs (Suc x) = ?lhs x + assign y (Suc x) z" by simp
         also have "?lhs x = ?rhs x" by(rule Suc.IH)
         also have "?rhs x + assign y (Suc x) z = ?rhs (Suc x)"
-        proof(cases "find_start y (setsum (pp y) {..<Suc x})")
+        proof(cases "find_start y (setsum (?pp y) {..<Suc x})")
           case None
           thus ?thesis
             by(auto split: option.split simp add: assign_def min_def max_def diff_le_iff setsum_nonneg not_le field_simps)
               (metis add.commute add_increasing find_start_def lessThan_Suc_atMost less_imp_le option.distinct(1) setsum_lessThan_Suc)+
-        next
+        next 
           case (Some m)
-          have [simp]: "setsum (rr y) {..m} = rr y m + setsum (rr y) {..<m}"
+          have [simp]: "setsum (?rr y) {..m} = ?rr y m + setsum (?rr y) {..<m}"
             by(simp add: ivl_disj_un(2)[symmetric])
-          from Some obtain m' where m': "find_start y (setsum (pp y) {..<x}) = Some m'" "m' \<le> m"
-            by(auto dest: find_start_mono[where w'2="setsum (pp y) {..<x}"])
+          from Some obtain m' where m': "find_start y (setsum (?pp y) {..<x}) = Some m'" "m' \<le> m"
+            by(auto dest: find_start_mono[where w'2="setsum (?pp y) {..<x}"])
           moreover {
             assume "z < m"
-            then have "setsum (rr y) {..z} \<le> setsum (rr y) {..<m}"
+            then have "setsum (?rr y) {..z} \<le> setsum (?rr y) {..<m}"
               by(auto intro: setsum_mono3)
-            also have "\<dots> \<le> setsum (pp y) {..<Suc x}" using find_start_eq_Some_least[OF Some]
+            also have "\<dots> \<le> setsum (?pp y) {..<Suc x}" using find_start_eq_Some_least[OF Some]
               by(simp add: ivl_disj_un(2)[symmetric] setsum_nonneg)
-            finally have "rr y z \<le> max 0 (setsum (pp y) {..<x} + pp y x - setsum (rr y) {..<z})"
-              by(auto simp add: ivl_disj_un(2)[symmetric] max_def diff_le_iff simp del: r_convs)
+            finally have "?rr y z \<le> max 0 (setsum (?pp y) {..<x} + ?pp y x - setsum (?rr y) {..<z})"
+              by(auto simp add: ivl_disj_un(2)[symmetric] max_def diff_le_iff simp del: pmf_le_0_iff)
           } moreover {
             assume "m \<le> z"
-            have "setsum (pp y) {..<Suc x} \<le> setsum (rr y) {..m}"
+            have "setsum (?pp y) {..<Suc x} \<le> setsum (?rr y) {..m}"
               using Some by(rule find_start_eq_Some_above)
-            also have "\<dots> \<le> setsum (rr y) {..<Suc z}" using \<open>m \<le> z\<close> by(intro setsum_mono3) auto
-            finally have "max 0 (setsum (pp y) {..<x} + pp y x - setsum (rr y) {..<z}) \<le> rr y z" by simp
-            moreover have "z \<noteq> m \<Longrightarrow> setsum (rr y) {..m} + setsum (rr y) {Suc m..<z} = setsum (rr y) {..<z}"
+            also have "\<dots> \<le> setsum (?rr y) {..<Suc z}" using \<open>m \<le> z\<close> by(intro setsum_mono3) auto
+            finally have "max 0 (setsum (?pp y) {..<x} + ?pp y x - setsum (?rr y) {..<z}) \<le> ?rr y z" by simp
+            moreover have "z \<noteq> m \<Longrightarrow> setsum (?rr y) {..m} + setsum (?rr y) {Suc m..<z} = setsum (?rr y) {..<z}"
               using \<open>m \<le> z\<close>
               by(subst ivl_disj_un(8)[where l="Suc m", symmetric])
                 (simp_all add: setsum_Un ivl_disj_un(2)[symmetric] setsum.neutral)
             moreover note calculation
           } moreover {
             assume "m < z"
-            have "setsum (pp y) {..<Suc x} \<le> setsum (rr y) {..m}"
+            have "setsum (?pp y) {..<Suc x} \<le> setsum (?rr y) {..m}"
               using Some by(rule find_start_eq_Some_above)
-            also have "\<dots> \<le> setsum (rr y) {..<z}" using \<open>m < z\<close> by(intro setsum_mono3) auto
-            finally have "max 0 (setsum (pp y) {..<Suc x} - setsum (rr y) {..<z}) = 0" by simp }
-          moreover have "setsum (pp y) {..<Suc x} \<ge> setsum (rr y) {..<m}"
+            also have "\<dots> \<le> setsum (?rr y) {..<z}" using \<open>m < z\<close> by(intro setsum_mono3) auto
+            finally have "max 0 (setsum (?pp y) {..<Suc x} - setsum (?rr y) {..<z}) = 0" by simp }
+          moreover have "setsum (?pp y) {..<Suc x} \<ge> setsum (?rr y) {..<m}"
             using find_start_eq_Some_least[OF Some]
             by(simp add: setsum_nonneg ivl_disj_un(2)[symmetric])
-          moreover hence "setsum (pp y) {..<Suc (Suc x)} \<ge> setsum (rr y) {..<m}"
+          moreover hence "setsum (?pp y) {..<Suc (Suc x)} \<ge> setsum (?rr y) {..<m}"
             by(fastforce intro: order_trans)
           ultimately show ?thesis using Some
             by(auto simp add: assign_def assign_aux_def Let_def field_simps max_def)
@@ -1148,60 +1131,63 @@
       qed }
     note setsum_assign = this
 
-    have nn_integral_assign1: "\<And>y z. (\<integral>\<^sup>+ x. assign y x z \<partial>count_space UNIV) = rr y z"
+    have nn_integral_assign1: "\<And>y z. (\<integral>\<^sup>+ x. assign y x z \<partial>count_space UNIV) = ?rr y z"
     proof -
       fix y z
       have "(\<integral>\<^sup>+ x. assign y x z \<partial>count_space UNIV) = (SUP n. ereal (\<Sum>x<n. assign y x z))"
         by(simp add: nn_integral_count_space_nat suminf_ereal_eq_SUP)
-      also have "\<dots> = rr y z"
+      also have "\<dots> = ?rr y z"
       proof(rule antisym)
-        show "(SUP n. ereal (\<Sum>x<n. assign y x z)) \<le> rr y z"
+        show "(SUP n. ereal (\<Sum>x<n. assign y x z)) \<le> ?rr y z"
         proof(rule SUP_least)
           fix n
-          show "ereal (\<Sum>x<n. (assign y x z)) \<le> rr y z"
+          show "ereal (\<Sum>x<n. (assign y x z)) \<le> ?rr y z"
             using setsum_assign[of y z "n - 1"]
             by(cases n)(simp_all split: option.split)
         qed
-        show "rr y z \<le> (SUP n. ereal (\<Sum>x<n. assign y x z))"
-        proof(cases "setsum (rr y) {..z} < \<integral>\<^sup>+ x. pp y x \<partial>count_space UNIV")
+        show "?rr y z \<le> (SUP n. ereal (\<Sum>x<n. assign y x z))"
+        proof(cases "setsum (?rr y) {..z} < \<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV")
           case True
-          then obtain n where "setsum (rr y) {..z} < setsum (pp y) {..<n}"
+          then obtain n where "setsum (?rr y) {..z} < setsum (?pp y) {..<n}"
             by(auto simp add: nn_integral_count_space_nat suminf_ereal_eq_SUP less_SUP_iff)
-          moreover have "\<And>k. k < z \<Longrightarrow> setsum (rr y) {..k} \<le> setsum (rr y) {..<z}"
+          moreover have "\<And>k. k < z \<Longrightarrow> setsum (?rr y) {..k} \<le> setsum (?rr y) {..<z}"
             by(auto intro: setsum_mono3)
-          ultimately have "rr y z \<le> (\<Sum>x<Suc n. assign y x z)"
+          ultimately have "?rr y z \<le> (\<Sum>x<Suc n. assign y x z)"
             by(subst setsum_assign)(auto split: option.split dest!: find_start_eq_Some_above simp add: ivl_disj_un(2)[symmetric] add.commute add_increasing le_diff_eq le_max_iff_disj)
           also have "\<dots> \<le> (SUP n. ereal (\<Sum>x<n. assign y x z))" 
             by(rule SUP_upper) simp
           finally show ?thesis by simp
         next
           case False
-          have "setsum (rr y) {..z} = \<integral>\<^sup>+ z. rr y z \<partial>count_space {..z}"
+          have "setsum (?rr y) {..z} = \<integral>\<^sup>+ z. ?rr y z \<partial>count_space {..z}"
             by(simp add: nn_integral_count_space_finite max_def)
-          also have "\<dots> \<le> \<integral>\<^sup>+ z. rr y z \<partial>count_space UNIV"
+          also have "\<dots> \<le> \<integral>\<^sup>+ z. ?rr y z \<partial>count_space UNIV"
             by(auto simp add: nn_integral_count_space_indicator indicator_def intro: nn_integral_mono)
-          also have "\<dots> = \<integral>\<^sup>+ x. pp y x \<partial>count_space UNIV" by(simp add: eq)
-          finally have *: "setsum (rr y) {..z} = \<dots>" using False by simp
-          also have "\<dots> = (SUP n. ereal (\<Sum>x<n. pp y x))"
+          also have "\<dots> = \<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV" by(simp add: eq)
+          finally have *: "setsum (?rr y) {..z} = \<dots>" using False by simp
+          also have "\<dots> = (SUP n. ereal (\<Sum>x<n. ?pp y x))"
             by(simp add: nn_integral_count_space_nat suminf_ereal_eq_SUP)
-          also have "\<dots> \<le> (SUP n. ereal (\<Sum>x<n. assign y x z)) + setsum (rr y) {..<z}"
+          also have "\<dots> \<le> (SUP n. ereal (\<Sum>x<n. assign y x z)) + setsum (?rr y) {..<z}"
           proof(rule SUP_least)
             fix n
-            have "setsum (pp y) {..<n} = \<integral>\<^sup>+ x. pp y x \<partial>count_space {..<n}"
+            have "setsum (?pp y) {..<n} = \<integral>\<^sup>+ x. ?pp y x \<partial>count_space {..<n}"
               by(simp add: nn_integral_count_space_finite max_def)
-            also have "\<dots> \<le> \<integral>\<^sup>+ x. pp y x \<partial>count_space UNIV"
+            also have "\<dots> \<le> \<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV"
               by(auto simp add: nn_integral_count_space_indicator indicator_def intro: nn_integral_mono)
-            also have "\<dots> = setsum (rr y) {..z}" using * by simp
-            finally obtain k where k: "find_start y (setsum (pp y) {..<n}) = Some k"
+            also have "\<dots> = setsum (?rr y) {..z}" using * by simp
+            finally obtain k where k: "find_start y (setsum (?pp y) {..<n}) = Some k"
               by(fastforce simp add: find_start_def)
-            with \<open>ereal (setsum (pp y) {..<n}) \<le> setsum (rr y) {..z}\<close>
+            with \<open>ereal (setsum (?pp y) {..<n}) \<le> setsum (?rr y) {..z}\<close>
             have "k \<le> z" by(auto simp add: find_start_def split: split_if_asm intro: Least_le)
-            then have "setsum (pp y) {..<n} - setsum (rr y) {..<z} \<le> ereal (\<Sum>x<Suc n. assign y x z)"
-              using \<open>ereal (setsum (pp y) {..<n}) \<le> setsum (rr y) {..z}\<close>
-              by(subst setsum_assign)(auto simp add: field_simps max_def k ivl_disj_un(2)[symmetric], metis le_add_same_cancel2 max.bounded_iff max_def pos_p)
+            then have "setsum (?pp y) {..<n} - setsum (?rr y) {..<z} \<le> ereal (\<Sum>x<Suc n. assign y x z)"
+              using \<open>ereal (setsum (?pp y) {..<n}) \<le> setsum (?rr y) {..z}\<close>
+              apply (subst setsum_assign)
+              apply (auto simp add: field_simps max_def k ivl_disj_un(2)[symmetric])
+              apply (meson add_increasing le_cases pmf_nonneg)
+              done
             also have "\<dots> \<le> (SUP n. ereal (\<Sum>x<n. assign y x z))"
               by(rule SUP_upper) simp
-            finally show "ereal (\<Sum>x<n. pp y x) \<le> \<dots> + setsum (rr y) {..<z}" 
+            finally show "ereal (\<Sum>x<n. ?pp y x) \<le> \<dots> + setsum (?rr y) {..<z}" 
               by(simp add: ereal_minus(1)[symmetric] ereal_minus_le del: ereal_minus(1))
           qed
           finally show ?thesis
@@ -1212,249 +1198,113 @@
     qed
 
     { fix y x
-      have "(\<integral>\<^sup>+ z. assign y x z \<partial>count_space UNIV) = pp y x"
-      proof(cases "setsum (pp y) {..<x} = \<integral>\<^sup>+ x. pp y x \<partial>count_space UNIV")
+      have "(\<integral>\<^sup>+ z. assign y x z \<partial>count_space UNIV) = ?pp y x"
+      proof(cases "setsum (?pp y) {..<x} = \<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV")
         case False
-        let ?used = "setsum (pp y) {..<x}"
-        have "?used = \<integral>\<^sup>+ x. pp y x \<partial>count_space {..<x}"
+        let ?used = "setsum (?pp y) {..<x}"
+        have "?used = \<integral>\<^sup>+ x. ?pp y x \<partial>count_space {..<x}"
           by(simp add: nn_integral_count_space_finite max_def)
-        also have "\<dots> \<le> \<integral>\<^sup>+ x. pp y x \<partial>count_space UNIV"
+        also have "\<dots> \<le> \<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV"
           by(auto simp add: nn_integral_count_space_indicator indicator_def intro!: nn_integral_mono)
         finally have "?used < \<dots>" using False by auto
         also note eq finally have "?used \<in> dom (find_start y)" by(rule in_dom_find_startI)
         then obtain k where k: "find_start y ?used = Some k" by auto
-        let ?f = "\<lambda>z. if z < k then 0 else if z = k then setsum (rr y) {..k} - ?used else rr y z"
-        let ?g = "\<lambda>x'. if x' < x then 0 else pp y x'"
-        have "pp y x = ?g x" by simp
+        let ?f = "\<lambda>z. if z < k then 0 else if z = k then setsum (?rr y) {..k} - ?used else ?rr y z"
+        let ?g = "\<lambda>x'. if x' < x then 0 else ?pp y x'"
+        have "?pp y x = ?g x" by simp
         also have "?g x \<le> \<integral>\<^sup>+ x'. ?g x' \<partial>count_space UNIV" by(rule nn_integral_ge_point) simp
         also {
-          have "?used = \<integral>\<^sup>+ x. pp y x \<partial>count_space {..<x}"
+          have "?used = \<integral>\<^sup>+ x. ?pp y x \<partial>count_space {..<x}"
             by(simp add: nn_integral_count_space_finite max_def)
-          also have "\<dots> = \<integral>\<^sup>+ x'. (if x' < x then pp y x' else 0) \<partial>count_space UNIV"
-            by(simp add: nn_integral_count_space_indicator indicator_def if_distrib zero_ereal_def cong: if_cong)
-          also have "(\<integral>\<^sup>+ x'. ?g x' \<partial>count_space UNIV) + \<dots> = \<integral>\<^sup>+ x. pp y x \<partial>count_space UNIV"
+          also have "\<dots> = \<integral>\<^sup>+ x'. (if x' < x then ?pp y x' else 0) \<partial>count_space UNIV"
+            by(simp add: nn_integral_count_space_indicator indicator_def if_distrib zero_ereal_def cong del: if_cong)
+          also have "(\<integral>\<^sup>+ x'. ?g x' \<partial>count_space UNIV) + \<dots> = \<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV"
             by(subst nn_integral_add[symmetric])(auto intro: nn_integral_cong)
           also note calculation }
-        ultimately have "ereal (pp y x) + ?used \<le> \<integral>\<^sup>+ x. pp y x \<partial>count_space UNIV"
+        ultimately have "ereal (?pp y x) + ?used \<le> \<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV"
           by (metis (no_types, lifting) ereal_add_mono order_refl)
         also note eq
-        also have "(\<integral>\<^sup>+ z. rr y z \<partial>count_space UNIV) = (\<integral>\<^sup>+ z. ?f z \<partial>count_space UNIV) + (\<integral>\<^sup>+ z. (if z < k then rr y z else if z = k then ?used - setsum (rr y) {..<k} else 0) \<partial>count_space UNIV)"
+        also have "(\<integral>\<^sup>+ z. ?rr y z \<partial>count_space UNIV) = (\<integral>\<^sup>+ z. ?f z \<partial>count_space UNIV) + (\<integral>\<^sup>+ z. (if z < k then ?rr y z else if z = k then ?used - setsum (?rr y) {..<k} else 0) \<partial>count_space UNIV)"
           using k by(subst nn_integral_add[symmetric])(auto intro!: nn_integral_cong simp add: ivl_disj_un(2)[symmetric] setsum_nonneg dest: find_start_eq_Some_least find_start_eq_Some_above)
-        also have "(\<integral>\<^sup>+ z. (if z < k then rr y z else if z = k then ?used - setsum (rr y) {..<k} else 0) \<partial>count_space UNIV) =
-          (\<integral>\<^sup>+ z. (if z < k then rr y z else if z = k then ?used - setsum (rr y) {..<k} else 0) \<partial>count_space {..k})"
+        also have "(\<integral>\<^sup>+ z. (if z < k then ?rr y z else if z = k then ?used - setsum (?rr y) {..<k} else 0) \<partial>count_space UNIV) =
+          (\<integral>\<^sup>+ z. (if z < k then ?rr y z else if z = k then ?used - setsum (?rr y) {..<k} else 0) \<partial>count_space {..k})"
           by(auto simp add: nn_integral_count_space_indicator indicator_def intro: nn_integral_cong)
         also have "\<dots> = ?used" 
           using k by(auto simp add: nn_integral_count_space_finite max_def ivl_disj_un(2)[symmetric] diff_le_iff setsum_nonneg dest: find_start_eq_Some_least)
-        finally have "pp y x \<le> (\<integral>\<^sup>+ z. ?f z \<partial>count_space UNIV)"
+        finally have "?pp y x \<le> (\<integral>\<^sup>+ z. ?f z \<partial>count_space UNIV)"
           by(cases "\<integral>\<^sup>+ z. ?f z \<partial>count_space UNIV") simp_all
         then show ?thesis using k
           by(simp add: assign_def nn_integral_assign_aux diff_le_iff find_start_eq_Some_above min_def)
       next
         case True
-        have "setsum (pp y) {..x} = \<integral>\<^sup>+ x. pp y x \<partial>count_space {..x}"
+        have "setsum (?pp y) {..x} = \<integral>\<^sup>+ x. ?pp y x \<partial>count_space {..x}"
           by(simp add: nn_integral_count_space_finite max_def)
-        also have "\<dots> \<le> \<integral>\<^sup>+ x. pp y x \<partial>count_space UNIV"
+        also have "\<dots> \<le> \<integral>\<^sup>+ x. ?pp y x \<partial>count_space UNIV"
           by(auto simp add: nn_integral_count_space_indicator indicator_def intro: nn_integral_mono)
-        also have "\<dots> = setsum (pp y) {..<x}" by(simp add: True)
-        finally have "pp y x = 0" by(simp add: ivl_disj_un(2)[symmetric] eq_iff del: pp_convs)
+        also have "\<dots> = setsum (?pp y) {..<x}" by(simp add: True)
+        finally have "?pp y x = 0" by(simp add: ivl_disj_un(2)[symmetric] eq_iff del: pmf_le_0_iff)
         thus ?thesis
-          by(cases "find_start y (setsum (pp y) {..<x})")(simp_all add: assign_def diff_le_iff find_start_eq_Some_above)
+          by(cases "find_start y (setsum (?pp y) {..<x})")(simp_all add: assign_def diff_le_iff find_start_eq_Some_above)
       qed }
     note nn_integral_assign2 = this
 
-    let ?f = "\<lambda>y x z. if x \<in> ?A y \<and> z \<in> ?B y then assign y (?P y x) (?R y z) else 0"
-    def f \<equiv> "\<lambda>y x z. ereal (?f y x z)"
-
-    have pos: "\<And>y x z. 0 \<le> f y x z" by(simp add: f_def)
+    def a \<equiv> "embed_pmf (\<lambda>(y, x, z). assign y x z)"
     { fix y x z
-      have "f y x z \<le> 0 \<longleftrightarrow> f y x z = 0" using pos[of y x z] by simp }
-    note f [simp] = this
-    have support:
-      "\<And>x y z. (x, y) \<notin> set_pmf pq \<Longrightarrow> f y x z = 0"
-      "\<And>x y z. (y, z) \<notin> set_pmf qr \<Longrightarrow> f y x z = 0"
-      by(auto simp add: f_def)
-
-    from pos support have support':
-      "\<And>x z. x \<notin> set_pmf p \<Longrightarrow> (\<integral>\<^sup>+ y. f y x z \<partial>count_space UNIV) = 0"
-      "\<And>x z. z \<notin> set_pmf r \<Longrightarrow> (\<integral>\<^sup>+ y. f y x z \<partial>count_space UNIV) = 0"
-    and support'':
-      "\<And>x y z. x \<notin> set_pmf p \<Longrightarrow> f y x z = 0"
-      "\<And>x y z. y \<notin> set_pmf q \<Longrightarrow> f y x z = 0"
-      "\<And>x y z. z \<notin> set_pmf r \<Longrightarrow> f y x z = 0"
-      by(auto simp add: nn_integral_0_iff_AE AE_count_space p q r set_map_pmf image_iff)(metis fst_conv snd_conv)+
+      have "assign y x z = pmf a (y, x, z)"
+        unfolding a_def
+      proof (subst pmf_embed_pmf)
+        have "(\<integral>\<^sup>+ x. ereal ((\<lambda>(y, x, z). assign y x z) x) \<partial>count_space UNIV) =
+          (\<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))))"
+          by (force simp add: nn_integral_count_space_indicator pmf_eq_0_set_pmf split: split_indicator
+                    intro!: nn_integral_cong assign_eq_0_outside)
+        also have "\<dots> = (\<integral>\<^sup>+ x. ereal ((\<lambda>((y, x), z). assign y x z) x) \<partial>(count_space (pp \<times> UNIV)))"
+          by (subst nn_integral_bij_count_space[OF inj_on_imp_bij_betw, symmetric])
+             (auto simp: inj_on_def intro!: nn_integral_cong)
+        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)"
+          by (subst sigma_finite_measure.nn_integral_fst)
+             (auto simp: pair_measure_countable sigma_finite_measure_count_space_countable)
+        also have "\<dots> = (\<integral>\<^sup>+ z. ?pp (fst z) (snd z) \<partial>count_space pp)"
+          by (subst nn_integral_assign2[symmetric]) (auto intro!: nn_integral_cong)
+        finally show "(\<integral>\<^sup>+ x. ereal ((\<lambda>(y, x, z). assign y x z) x) \<partial>count_space UNIV) = 1"
+          by (simp add: nn_integral_pmf emeasure_pmf)
+      qed auto }
+    note a = this
 
-    have f_x: "\<And>y z. (\<integral>\<^sup>+ x. f y x z \<partial>count_space (set_pmf p)) = pmf qr (y, z)"
-    proof(case_tac "z \<in> ?B y")
-      fix y z
-      assume z: "z \<in> ?B y"
-      have "(\<integral>\<^sup>+ x. f y x z \<partial>count_space (set_pmf p)) = (\<integral>\<^sup>+ x. ?f y x z \<partial>count_space (?A y))"
-        using support''(1)[of _ y z]
-        by(fastforce simp add: f_def nn_integral_count_space_indicator indicator_def intro!: nn_integral_cong)
-      also have "\<dots> = \<integral>\<^sup>+ x. assign y (?P y x) (?R y z) \<partial>count_space (?A y)"
-        using z by(intro nn_integral_cong) simp
-      also have "\<dots> = \<integral>\<^sup>+ x. assign y x (?R y z) \<partial>count_space (?P y ` ?A y)"
-        by(intro nn_integral_bij_count_space inj_on_imp_bij_betw inj_on_to_nat_on) simp
-      also have "\<dots> = \<integral>\<^sup>+ x. assign y x (?R y z) \<partial>count_space UNIV"
-        by(auto simp add: nn_integral_count_space_indicator indicator_def assign_eq_0_outside pp_def intro!: nn_integral_cong)
-      also have "\<dots> = rr y (?R y z)" by(rule nn_integral_assign1)
-      also have "\<dots> = pmf qr (y, z)" using z by(simp add: rr_def)
-      finally show "?thesis y z" .
-    qed(auto simp add: f_def zero_ereal_def[symmetric] set_pmf_iff)
-
-    have f_z: "\<And>x y. (\<integral>\<^sup>+ z. f y x z \<partial>count_space (set_pmf r)) = pmf pq (x, y)"
-    proof(case_tac "x \<in> ?A y")
-      fix x y
-      assume x: "x \<in> ?A y"
-      have "(\<integral>\<^sup>+ z. f y x z \<partial>count_space (set_pmf r)) = (\<integral>\<^sup>+ z. ?f y x z \<partial>count_space (?B y))"
-        using support''(3)[of _ y x]
-        by(fastforce simp add: f_def nn_integral_count_space_indicator indicator_def intro!: nn_integral_cong)
-      also have "\<dots> = \<integral>\<^sup>+ z. assign y (?P y x) (?R y z) \<partial>count_space (?B y)"
-        using x by(intro nn_integral_cong) simp
-      also have "\<dots> = \<integral>\<^sup>+ z. assign y (?P y x) z \<partial>count_space (?R y ` ?B y)"
-        by(intro nn_integral_bij_count_space inj_on_imp_bij_betw inj_on_to_nat_on) simp
-      also have "\<dots> = \<integral>\<^sup>+ z. assign y (?P y x) z \<partial>count_space UNIV"
-        by(auto simp add: nn_integral_count_space_indicator indicator_def assign_eq_0_outside rr_def intro!: nn_integral_cong)
-      also have "\<dots> = pp y (?P y x)" by(rule nn_integral_assign2)
-      also have "\<dots> = pmf pq (x, y)" using x by(simp add: pp_def)
-      finally show "?thesis x y" .
-    qed(auto simp add: f_def zero_ereal_def[symmetric] set_pmf_iff)
-
-    let ?pr = "\<lambda>(x, z). \<integral>\<^sup>+ y. f y x z \<partial>count_space UNIV"
-
-    have pr_pos: "\<And>xz. 0 \<le> ?pr xz"
-      by(auto simp add: nn_integral_nonneg)
+    def pr \<equiv> "map_pmf (\<lambda>(y, x, z). (from_nat_into (A y) x, from_nat_into (B y) z)) a"
 
-    have pr': "?pr = (\<lambda>(x, z). \<integral>\<^sup>+ y. f y x z \<partial>count_space (set_pmf q))"
-      by(auto simp add: fun_eq_iff nn_integral_count_space_indicator indicator_def support'' intro: nn_integral_cong)
-    
-    have "(\<integral>\<^sup>+ xz. ?pr xz \<partial>count_space UNIV) = (\<integral>\<^sup>+ xz. ?pr xz * indicator (set_pmf p \<times> set_pmf r) xz \<partial>count_space UNIV)"
-      by(rule nn_integral_cong)(auto simp add: indicator_def support' intro: ccontr)
-    also have "\<dots> = (\<integral>\<^sup>+ xz. ?pr xz \<partial>count_space (set_pmf p \<times> set_pmf r))"
-      by(simp add: nn_integral_count_space_indicator)
-    also have "\<dots> = (\<integral>\<^sup>+ xz. ?pr xz \<partial>(count_space (set_pmf p) \<Otimes>\<^sub>M count_space (set_pmf r)))"
-      by(simp add: pair_measure_countable)
-    also have "\<dots> = (\<integral>\<^sup>+ (x, z). \<integral>\<^sup>+ y. f y x z \<partial>count_space (set_pmf q) \<partial>(count_space (set_pmf p) \<Otimes>\<^sub>M count_space (set_pmf r)))"
-      by(simp add: pr')
-    also have "\<dots> = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ z. \<integral>\<^sup>+ y. f y x z \<partial>count_space (set_pmf q) \<partial>count_space (set_pmf r) \<partial>count_space (set_pmf p))"
-      by(subst sigma_finite_measure.nn_integral_fst[symmetric, OF sigma_finite_measure_count_space_countable])(simp_all add: pair_measure_countable)
-    also have "\<dots> = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. \<integral>\<^sup>+ z. f y x z \<partial>count_space (set_pmf r) \<partial>count_space (set_pmf q) \<partial>count_space (set_pmf p))"
-      by(subst (2) pair_sigma_finite.Fubini')(simp_all add: pair_sigma_finite.intro sigma_finite_measure_count_space_countable pair_measure_countable)
-    also have "\<dots> = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. pmf pq (x, y) \<partial>count_space (set_pmf q) \<partial>count_space (set_pmf p))"
-      by(simp add: f_z)
-    also have "\<dots> = (\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. pmf pq (x, y) \<partial>count_space (set_pmf p) \<partial>count_space (set_pmf q))"
-      by(subst pair_sigma_finite.Fubini')(simp_all add: pair_sigma_finite.intro sigma_finite_measure_count_space_countable pair_measure_countable)
-    also have "\<dots> = (\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. emeasure (measure_pmf pq) {(x, y)} \<partial>count_space (set_pmf p) \<partial>count_space (set_pmf q))"
-      by(simp add: emeasure_pmf_single)
-    also have "\<dots> = (\<integral>\<^sup>+ y. emeasure (measure_pmf pq) (\<Union>x\<in>set_pmf p. {(x, y)}) \<partial>count_space (set_pmf q))"
-      by(subst emeasure_UN_countable)(simp_all add: disjoint_family_on_def)
-    also have "\<dots> = (\<integral>\<^sup>+ y. emeasure (measure_pmf pq) ((\<Union>x\<in>set_pmf p. {(x, y)}) \<union> {(x, y'). x \<notin> set_pmf p \<and> y' = y}) \<partial>count_space (set_pmf q))"
-      by(rule nn_integral_cong emeasure_Un_null_set[symmetric])+
-        (auto simp add: p set_map_pmf split_beta intro!: in_null_sets_measure_pmfI intro: rev_image_eqI)
-    also have "\<dots> = (\<integral>\<^sup>+ y. emeasure (measure_pmf pq) (snd -` {y}) \<partial>count_space (set_pmf q))"
-      by(rule nn_integral_cong arg_cong2[where f=emeasure])+auto
-    also have "\<dots> = (\<integral>\<^sup>+ y. pmf q y \<partial>count_space (set_pmf q))"
-      by(simp add: ereal_pmf_map q)
-    also have "\<dots> = (\<integral>\<^sup>+ y. pmf q y \<partial>count_space UNIV)"
-      by(auto simp add: nn_integral_count_space_indicator indicator_def set_pmf_iff intro: nn_integral_cong)
-    also have "\<dots> = 1"
-      by(subst nn_integral_pmf)(simp add: measure_pmf.emeasure_eq_1_AE)
-    finally have pr_prob: "(\<integral>\<^sup>+ xz. ?pr xz \<partial>count_space UNIV) = 1" .
-
-    have pr_bounded: "\<And>xz. ?pr xz \<noteq> \<infinity>"
-    proof -
-      fix xz
-      have "?pr xz \<le> \<integral>\<^sup>+ xz. ?pr xz \<partial>count_space UNIV"
-        by(rule nn_integral_ge_point) simp
-      also have "\<dots> = 1" by(fact pr_prob)
-      finally show "?thesis xz" by auto
-    qed
-
-    def pr \<equiv> "embed_pmf (real \<circ> ?pr)"
-    have pmf_pr: "\<And>xz. pmf pr xz = real (?pr xz)" using pr_pos pr_prob
-      unfolding pr_def by(subst pmf_embed_pmf)(auto simp add: real_of_ereal_pos ereal_real pr_bounded)
-
-    have set_pmf_pr_subset: "set_pmf pr \<subseteq> set_pmf pq O set_pmf qr"
+    have "rel_pmf (R OO S) p r"
     proof
-      fix xz :: "'a \<times> 'c"
-      obtain x z where xz: "xz = (x, z)" by(cases xz)
-      assume "xz \<in> set_pmf pr"
-      with xz have "pmf pr (x, z) \<noteq> 0" by(simp add: set_pmf_iff)
-      hence "\<exists>y. f y x z \<noteq> 0" by(rule contrapos_np)(simp add: pmf_pr)
-      then obtain y where y: "f y x z \<noteq> 0" ..
-      then have "(x, y) \<in> set_pmf pq" "(y, z) \<in> set_pmf qr" 
-        using support by fastforce+
-      then show "xz \<in> set_pmf pq O set_pmf qr" using xz by auto
-    qed
-    hence "\<And>x z. (x, z) \<in> set_pmf pr \<Longrightarrow> (R OO S) x z" using pq qr by blast
-    moreover
-    have "map_pmf fst pr = p"
-    proof(rule pmf_eqI)
-      fix x
-      have "pmf (map_pmf fst pr) x = emeasure (measure_pmf pr) (fst -` {x})"
-        by(simp add: ereal_pmf_map)
-      also have "\<dots> = \<integral>\<^sup>+ xz. pmf pr xz \<partial>count_space (fst -` {x})"
-        by(simp add: nn_integral_pmf)
-      also have "\<dots> = \<integral>\<^sup>+ xz. ?pr xz \<partial>count_space (fst -` {x})"
-        by(simp add: pmf_pr ereal_real pr_bounded pr_pos)
-      also have "\<dots> =  \<integral>\<^sup>+ xz. ?pr xz \<partial>count_space {x} \<Otimes>\<^sub>M count_space (set_pmf r)"
-        by(auto simp add: nn_integral_count_space_indicator indicator_def support' pair_measure_countable intro!: nn_integral_cong)
-      also have "\<dots> = \<integral>\<^sup>+ z. \<integral>\<^sup>+ x. ?pr (x, z) \<partial>count_space {x} \<partial>count_space (set_pmf r)"
-        by(subst pair_sigma_finite.nn_integral_snd[symmetric])(simp_all add: pair_measure_countable pair_sigma_finite.intro sigma_finite_measure_count_space_countable)
-      also have "\<dots> = \<integral>\<^sup>+ z. ?pr (x, z) \<partial>count_space (set_pmf r)"
-        using pr_pos by(clarsimp simp add: nn_integral_count_space_finite max_def)
-      also have "\<dots> = \<integral>\<^sup>+ z. \<integral>\<^sup>+ y. f y x z \<partial>count_space (set_pmf q) \<partial>count_space (set_pmf r)"
-        by(simp add: pr')
-      also have "\<dots> =  \<integral>\<^sup>+ y. \<integral>\<^sup>+ z. f y x z \<partial>count_space (set_pmf r) \<partial>count_space (set_pmf q)"
-        by(subst pair_sigma_finite.Fubini')(simp_all add: pair_sigma_finite.intro sigma_finite_measure_count_space_countable pair_measure_countable)
-      also have "\<dots> = \<integral>\<^sup>+ y. pmf pq (x, y) \<partial>count_space (set_pmf q)"
-        by(simp add: f_z)
-      also have "\<dots> = \<integral>\<^sup>+ y. emeasure (measure_pmf pq) {(x, y)} \<partial>count_space (set_pmf q)"
-        by(simp add: emeasure_pmf_single)
-      also have "\<dots> = emeasure (measure_pmf pq) (\<Union>y\<in>set_pmf q. {(x, y)})"
-        by(subst emeasure_UN_countable)(simp_all add: disjoint_family_on_def)
-      also have "\<dots> = emeasure (measure_pmf pq) ((\<Union>y\<in>set_pmf q. {(x, y)}) \<union> {(x', y). y \<notin> set_pmf q \<and> x' = x})"
-        by(rule emeasure_Un_null_set[symmetric])+
-          (auto simp add: q set_map_pmf split_beta intro!: in_null_sets_measure_pmfI intro: rev_image_eqI)
-      also have "\<dots> = emeasure (measure_pmf pq) (fst -` {x})"
-        by(rule arg_cong2[where f=emeasure])+auto
-      also have "\<dots> = pmf p x" by(simp add: ereal_pmf_map p)
-      finally show "pmf (map_pmf fst pr) x = pmf p x" by simp
-    qed
-    moreover
-    have "map_pmf snd pr = r"
-    proof(rule pmf_eqI)
-      fix z
-      have "pmf (map_pmf snd pr) z = emeasure (measure_pmf pr) (snd -` {z})"
-        by(simp add: ereal_pmf_map)
-      also have "\<dots> = \<integral>\<^sup>+ xz. pmf pr xz \<partial>count_space (snd -` {z})"
-        by(simp add: nn_integral_pmf)
-      also have "\<dots> = \<integral>\<^sup>+ xz. ?pr xz \<partial>count_space (snd -` {z})"
-        by(simp add: pmf_pr ereal_real pr_bounded pr_pos)
-      also have "\<dots> =  \<integral>\<^sup>+ xz. ?pr xz \<partial>count_space (set_pmf p) \<Otimes>\<^sub>M count_space {z}"
-        by(auto simp add: nn_integral_count_space_indicator indicator_def support' pair_measure_countable intro!: nn_integral_cong)
-      also have "\<dots> = \<integral>\<^sup>+ x. \<integral>\<^sup>+ z. ?pr (x, z) \<partial>count_space {z} \<partial>count_space (set_pmf p)"
-        by(subst sigma_finite_measure.nn_integral_fst[symmetric])(simp_all add: pair_measure_countable sigma_finite_measure_count_space_countable)
-      also have "\<dots> = \<integral>\<^sup>+ x. ?pr (x, z) \<partial>count_space (set_pmf p)"
-        using pr_pos by(clarsimp simp add: nn_integral_count_space_finite max_def)
-      also have "\<dots> = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f y x z \<partial>count_space (set_pmf q) \<partial>count_space (set_pmf p)"
-        by(simp add: pr')
-      also have "\<dots> =  \<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f y x z \<partial>count_space (set_pmf p) \<partial>count_space (set_pmf q)"
-        by(subst pair_sigma_finite.Fubini')(simp_all add: pair_sigma_finite.intro sigma_finite_measure_count_space_countable pair_measure_countable)
-      also have "\<dots> = \<integral>\<^sup>+ y. pmf qr (y, z) \<partial>count_space (set_pmf q)"
-        by(simp add: f_x)
-      also have "\<dots> = \<integral>\<^sup>+ y. emeasure (measure_pmf qr) {(y, z)} \<partial>count_space (set_pmf q)"
-        by(simp add: emeasure_pmf_single)
-      also have "\<dots> = emeasure (measure_pmf qr) (\<Union>y\<in>set_pmf q. {(y, z)})"
-        by(subst emeasure_UN_countable)(simp_all add: disjoint_family_on_def)
-      also have "\<dots> = emeasure (measure_pmf qr) ((\<Union>y\<in>set_pmf q. {(y, z)}) \<union> {(y, z'). y \<notin> set_pmf q \<and> z' = z})"
-        by(rule emeasure_Un_null_set[symmetric])+
-          (auto simp add: q' set_map_pmf split_beta intro!: in_null_sets_measure_pmfI intro: rev_image_eqI)
-      also have "\<dots> = emeasure (measure_pmf qr) (snd -` {z})"
-        by(rule arg_cong2[where f=emeasure])+auto
-      also have "\<dots> = pmf r z" by(simp add: ereal_pmf_map r)
-      finally show "pmf (map_pmf snd pr) z = pmf r z" by simp
-    qed
-    ultimately have "rel_pmf (R OO S) p r" .. }
+      have pp_eq: "pp = map_pmf (\<lambda>(y, x, z). (y, x)) a"
+      proof (rule pmf_eqI)
+        fix i
+        show "pmf pp i = pmf (map_pmf (\<lambda>(y, x, z). (y, x)) a) i"
+          using nn_integral_assign2[of "fst i" "snd i", symmetric]
+          by (auto simp add: a nn_integral_pmf' inj_on_def ereal.inject[symmetric] ereal_pmf_map 
+                   simp del: ereal.inject intro!: arg_cong2[where f=emeasure])
+      qed
+      moreover have pq_eq: "pq = map_pmf (\<lambda>(y, x). (from_nat_into (A y) x, y)) pp"
+        by (simp add: pp_def map_pmf_comp split_beta A[symmetric] cong: map_pmf_cong)
+      ultimately show "map_pmf fst pr = p"
+        unfolding p pr_def by (simp add: map_pmf_comp split_beta)
+
+      have rr_eq: "rr = map_pmf (\<lambda>(y, x, z). (y, z)) a"
+      proof (rule pmf_eqI)
+        fix i show "pmf rr i = pmf (map_pmf (\<lambda>(y, x, z). (y, z)) a) i"
+          using nn_integral_assign1[of "fst i" "snd i", symmetric]
+          by (auto simp add: a nn_integral_pmf' inj_on_def ereal.inject[symmetric] ereal_pmf_map 
+                   simp del: ereal.inject intro!: arg_cong2[where f=emeasure])
+      qed
+      moreover have qr_eq: "qr = map_pmf (\<lambda>(y, z). (y, from_nat_into (B y) z)) rr"
+        by (simp add: rr_def map_pmf_comp split_beta B[symmetric] cong: map_pmf_cong)
+      ultimately show "map_pmf snd pr = r"
+        unfolding r pr_def by (simp add: map_pmf_comp split_beta)
+
+      fix x z assume "(x, z) \<in> set_pmf pr"
+      then have "\<exists>y. (x, y) \<in> set_pmf pq \<and> (y, z) \<in> set_pmf qr"
+        by (force simp add: pp_eq pq_eq rr_eq qr_eq set_map_pmf pr_def image_image)
+      with pq qr show "(R OO S) x z"
+        by blast
+    qed }
   then show "rel_pmf R OO rel_pmf S \<le> rel_pmf (R OO S)"
     by(auto simp add: le_fun_def)
 qed (fact natLeq_card_order natLeq_cinfinite)+