diff --git a/src/HOL/Probability/ex/Koepf_Duermuth_Countermeasure.thy b/src/HOL/Probability/ex/Koepf_Duermuth_Countermeasure.thy
--- a/src/HOL/Probability/ex/Koepf_Duermuth_Countermeasure.thy
+++ b/src/HOL/Probability/ex/Koepf_Duermuth_Countermeasure.thy
@@ -1,480 +1,484 @@
 (* Author: Johannes Hölzl, TU München *)
 
 section \<open>Formalization of a Countermeasure by Koepf \& Duermuth 2009\<close>
 
 theory Koepf_Duermuth_Countermeasure
-  imports "HOL-Probability.Information" "HOL-Library.List_Permutation"
+  imports "HOL-Probability.Information"
 begin
 
 lemma SIGMA_image_vimage:
   "snd ` (SIGMA x:f`X. f -` {x} \<inter> X) = X"
   by (auto simp: image_iff)
 
 declare inj_split_Cons[simp]
 
 definition extensionalD :: "'b \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> 'b) set" where
   "extensionalD d A = {f. \<forall>x. x \<notin> A \<longrightarrow> f x = d}"
 
 lemma extensionalD_empty[simp]: "extensionalD d {} = {\<lambda>x. d}"
   unfolding extensionalD_def by (simp add: set_eq_iff fun_eq_iff)
 
 lemma funset_eq_UN_fun_upd_I:
   assumes *: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f(a := d) \<in> F A"
   and **: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f a \<in> G (f(a:=d))"
   and ***: "\<And>f x. \<lbrakk> f \<in> F A ; x \<in> G f \<rbrakk> \<Longrightarrow> f(a:=x) \<in> F (insert a A)"
   shows "F (insert a A) = (\<Union>f\<in>F A. fun_upd f a ` (G f))"
 proof safe
   fix f assume f: "f \<in> F (insert a A)"
   show "f \<in> (\<Union>f\<in>F A. fun_upd f a ` G f)"
   proof (rule UN_I[of "f(a := d)"])
     show "f(a := d) \<in> F A" using *[OF f] .
     show "f \<in> fun_upd (f(a:=d)) a ` G (f(a:=d))"
     proof (rule image_eqI[of _ _ "f a"])
       show "f a \<in> G (f(a := d))" using **[OF f] .
     qed simp
   qed
 next
   fix f x assume "f \<in> F A" "x \<in> G f"
   from ***[OF this] show "f(a := x) \<in> F (insert a A)" .
 qed
 
 lemma extensionalD_insert[simp]:
   assumes "a \<notin> A"
   shows "extensionalD d (insert a A) \<inter> (insert a A \<rightarrow> B) = (\<Union>f \<in> extensionalD d A \<inter> (A \<rightarrow> B). (\<lambda>b. f(a := b)) ` B)"
   apply (rule funset_eq_UN_fun_upd_I)
   using assms
   by (auto intro!: inj_onI dest: inj_onD split: if_split_asm simp: extensionalD_def)
 
 lemma finite_extensionalD_funcset[simp, intro]:
   assumes "finite A" "finite B"
   shows "finite (extensionalD d A \<inter> (A \<rightarrow> B))"
   using assms by induct auto
 
 lemma fun_upd_eq_iff: "f(a := b) = g(a := b') \<longleftrightarrow> b = b' \<and> f(a := d) = g(a := d)"
   by (auto simp: fun_eq_iff)
 
 lemma card_funcset:
   assumes "finite A" "finite B"
   shows "card (extensionalD d A \<inter> (A \<rightarrow> B)) = card B ^ card A"
 using \<open>finite A\<close> proof induct
   case (insert a A) thus ?case unfolding extensionalD_insert[OF \<open>a \<notin> A\<close>]
   proof (subst card_UN_disjoint, safe, simp_all)
     show "finite (extensionalD d A \<inter> (A \<rightarrow> B))" "\<And>f. finite (fun_upd f a ` B)"
       using \<open>finite B\<close> \<open>finite A\<close> by simp_all
   next
     fix f g b b' assume "f \<noteq> g" and eq: "f(a := b) = g(a := b')" and
       ext: "f \<in> extensionalD d A" "g \<in> extensionalD d A"
     have "f a = d" "g a = d"
       using ext \<open>a \<notin> A\<close> by (auto simp add: extensionalD_def)
     with \<open>f \<noteq> g\<close> eq show False unfolding fun_upd_eq_iff[of _ _ b _ _ d]
       by (auto simp: fun_upd_idem fun_upd_eq_iff)
   next
     { fix f assume "f \<in> extensionalD d A \<inter> (A \<rightarrow> B)"
       have "card (fun_upd f a ` B) = card B"
       proof (auto intro!: card_image inj_onI)
         fix b b' assume "f(a := b) = f(a := b')"
         from fun_upd_eq_iff[THEN iffD1, OF this] show "b = b'" by simp
       qed }
     then show "(\<Sum>i\<in>extensionalD d A \<inter> (A \<rightarrow> B). card (fun_upd i a ` B)) = card B * card B ^ card A"
       using insert by simp
   qed
 qed simp
 
 lemma prod_sum_distrib_lists:
   fixes P and S :: "'a set" and f :: "'a \<Rightarrow> _::semiring_0" assumes "finite S"
   shows "(\<Sum>ms\<in>{ms. set ms \<subseteq> S \<and> length ms = n \<and> (\<forall>i<n. P i (ms!i))}. \<Prod>x<n. f (ms ! x)) =
          (\<Prod>i<n. \<Sum>m\<in>{m\<in>S. P i m}. f m)"
 proof (induct n arbitrary: P)
   case 0 then show ?case by (simp cong: conj_cong)
 next
   case (Suc n)
   have *: "{ms. set ms \<subseteq> S \<and> length ms = Suc n \<and> (\<forall>i<Suc n. P i (ms ! i))} =
     (\<lambda>(xs, x). x#xs) ` ({ms. set ms \<subseteq> S \<and> length ms = n \<and> (\<forall>i<n. P (Suc i) (ms ! i))} \<times> {m\<in>S. P 0 m})"
     apply (auto simp: image_iff length_Suc_conv)
     apply force+
     apply (case_tac i)
     by force+
   show ?case unfolding *
     using Suc[of "\<lambda>i. P (Suc i)"]
     by (simp add: sum.reindex sum_cartesian_product'
       lessThan_Suc_eq_insert_0 prod.reindex sum_distrib_right[symmetric] sum_distrib_left[symmetric])
 qed
 
 declare space_point_measure[simp]
 
 declare sets_point_measure[simp]
 
 lemma measure_point_measure:
   "finite \<Omega> \<Longrightarrow> A \<subseteq> \<Omega> \<Longrightarrow> (\<And>x. x \<in> \<Omega> \<Longrightarrow> 0 \<le> p x) \<Longrightarrow>
     measure (point_measure \<Omega> (\<lambda>x. ennreal (p x))) A = (\<Sum>i\<in>A. p i)"
   unfolding measure_def
   by (subst emeasure_point_measure_finite) (auto simp: subset_eq sum_nonneg)
 
 locale finite_information =
   fixes \<Omega> :: "'a set"
     and p :: "'a \<Rightarrow> real"
     and b :: real
   assumes finite_space[simp, intro]: "finite \<Omega>"
   and p_sums_1[simp]: "(\<Sum>\<omega>\<in>\<Omega>. p \<omega>) = 1"
   and positive_p[simp, intro]: "\<And>x. 0 \<le> p x"
   and b_gt_1[simp, intro]: "1 < b"
 
 lemma (in finite_information) positive_p_sum[simp]: "0 \<le> sum p X"
    by (auto intro!: sum_nonneg)
 
 sublocale finite_information \<subseteq> prob_space "point_measure \<Omega> p"
   by standard (simp add: one_ereal_def emeasure_point_measure_finite)
 
 sublocale finite_information \<subseteq> information_space "point_measure \<Omega> p" b
   by standard simp
 
 lemma (in finite_information) \<mu>'_eq: "A \<subseteq> \<Omega> \<Longrightarrow> prob A = sum p A"
   by (auto simp: measure_point_measure)
 
 locale koepf_duermuth = K: finite_information keys K b + M: finite_information messages M b
     for b :: real
     and keys :: "'key set" and K :: "'key \<Rightarrow> real"
     and messages :: "'message set" and M :: "'message \<Rightarrow> real" +
   fixes observe :: "'key \<Rightarrow> 'message \<Rightarrow> 'observation"
     and n :: nat
 begin
 
 definition msgs :: "('key \<times> 'message list) set" where
   "msgs = keys \<times> {ms. set ms \<subseteq> messages \<and> length ms = n}"
 
 definition P :: "('key \<times> 'message list) \<Rightarrow> real" where
   "P = (\<lambda>(k, ms). K k * (\<Prod>i<n. M (ms ! i)))"
 
 end
 
 sublocale koepf_duermuth \<subseteq> finite_information msgs P b
 proof
   show "finite msgs" unfolding msgs_def
     using finite_lists_length_eq[OF M.finite_space, of n]
     by auto
 
   have [simp]: "\<And>A. inj_on (\<lambda>(xs, n). n # xs) A" by (force intro!: inj_onI)
 
   note sum_distrib_left[symmetric, simp]
   note sum_distrib_right[symmetric, simp]
   note sum_cartesian_product'[simp]
 
   have "(\<Sum>ms | set ms \<subseteq> messages \<and> length ms = n. \<Prod>x<length ms. M (ms ! x)) = 1"
   proof (induct n)
     case 0 then show ?case by (simp cong: conj_cong)
   next
     case (Suc n)
     then show ?case
       by (simp add: lists_length_Suc_eq lessThan_Suc_eq_insert_0
                     sum.reindex prod.reindex)
   qed
   then show "sum P msgs = 1"
     unfolding msgs_def P_def by simp
   fix x
   have "\<And> A f. 0 \<le> (\<Prod>x\<in>A. M (f x))" by (auto simp: prod_nonneg)
   then show "0 \<le> P x"
     unfolding P_def by (auto split: prod.split simp: zero_le_mult_iff)
 qed auto
 
 context koepf_duermuth
 begin
 
 definition observations :: "'observation set" where
   "observations = (\<Union>f\<in>observe ` keys. f ` messages)"
 
 lemma finite_observations[simp, intro]: "finite observations"
   unfolding observations_def by auto
 
 definition OB :: "'key \<times> 'message list \<Rightarrow> 'observation list" where
   "OB = (\<lambda>(k, ms). map (observe k) ms)"
 
 definition t :: "'observation list \<Rightarrow> 'observation \<Rightarrow> nat" where
   t_def2: "t seq obs = card { i. i < length seq \<and> seq ! i = obs}"
 
 lemma t_def: "t seq obs = length (filter ((=) obs) seq)"
   unfolding t_def2 length_filter_conv_card by (subst eq_commute) simp
 
 lemma card_T_bound: "card ((t\<circ>OB)`msgs) \<le> (n+1)^card observations"
 proof -
   have "(t\<circ>OB)`msgs \<subseteq> extensionalD 0 observations \<inter> (observations \<rightarrow> {.. n})"
     unfolding observations_def extensionalD_def OB_def msgs_def
     by (auto simp add: t_def comp_def image_iff subset_eq)
   with finite_extensionalD_funcset
   have "card ((t\<circ>OB)`msgs) \<le> card (extensionalD 0 observations \<inter> (observations \<rightarrow> {.. n}))"
     by (rule card_mono) auto
   also have "\<dots> = (n + 1) ^ card observations"
     by (subst card_funcset) auto
   finally show ?thesis .
 qed
 
 abbreviation
  "p A \<equiv> sum P A"
 
 abbreviation
   "\<mu> \<equiv> point_measure msgs P"
 
 abbreviation probability ("\<P>'(_') _") where
   "\<P>(X) x \<equiv> measure \<mu> (X -` x \<inter> msgs)"
 
 abbreviation joint_probability ("\<P>'(_ ; _') _") where
   "\<P>(X ; Y) x \<equiv> \<P>(\<lambda>x. (X x, Y x)) x"
 
 no_notation disj (infixr "|" 30)
 
 abbreviation conditional_probability ("\<P>'(_ | _') _") where
   "\<P>(X | Y) x \<equiv> (\<P>(X ; Y) x) / \<P>(Y) (snd`x)"
 
 notation
   entropy_Pow ("\<H>'( _ ')")
 
 notation
   conditional_entropy_Pow ("\<H>'( _ | _ ')")
 
 notation
   mutual_information_Pow ("\<I>'( _ ; _ ')")
 
 lemma t_eq_imp_bij_func:
   assumes "t xs = t ys"
   shows "\<exists>f. bij_betw f {..<length xs} {..<length ys} \<and> (\<forall>i<length xs. xs ! i = ys ! (f i))"
 proof -
-  have "count (mset xs) = count (mset ys)"
-    using assms by (simp add: fun_eq_iff count_mset t_def)
-  then have "xs <~~> ys" unfolding mset_eq_perm count_inject .
-  then show ?thesis by (rule permutation_Ex_bij)
+  from assms have \<open>mset xs = mset ys\<close>
+    using assms by (simp add: fun_eq_iff t_def multiset_eq_iff flip: count_mset)
+  then obtain p where \<open>p permutes {..<length ys}\<close> \<open>permute_list p ys = xs\<close>
+    by (rule mset_eq_permutation)
+  then have \<open>bij_betw p {..<length xs} {..<length ys}\<close> \<open>\<forall>i<length xs. xs ! i = ys ! p i\<close>
+    by (auto dest: permutes_imp_bij simp add: permute_list_nth)
+  then show ?thesis
+    by blast
 qed
 
 lemma \<P>_k: assumes "k \<in> keys" shows "\<P>(fst) {k} = K k"
 proof -
   from assms have *:
       "fst -` {k} \<inter> msgs = {k}\<times>{ms. set ms \<subseteq> messages \<and> length ms = n}"
     unfolding msgs_def by auto
   show "(\<P>(fst) {k}) = K k"
     apply (simp add: \<mu>'_eq)
     apply (simp add: * P_def)
     apply (simp add: sum_cartesian_product')
     using prod_sum_distrib_lists[OF M.finite_space, of M n "\<lambda>x x. True"] \<open>k \<in> keys\<close>
     by (auto simp add: sum_distrib_left[symmetric] subset_eq prod.neutral_const)
 qed
 
 lemma fst_image_msgs[simp]: "fst`msgs = keys"
 proof -
   from M.not_empty obtain m where "m \<in> messages" by auto
   then have *: "{m. set m \<subseteq> messages \<and> length m = n} \<noteq> {}"
     by (auto intro!: exI[of _ "replicate n m"])
   then show ?thesis
     unfolding msgs_def fst_image_times if_not_P[OF *] by simp
 qed
 
 lemma sum_distribution_cut:
   "\<P>(X) {x} = (\<Sum>y \<in> Y`space \<mu>. \<P>(X ; Y) {(x, y)})"
   by (subst finite_measure_finite_Union[symmetric])
      (auto simp add: disjoint_family_on_def inj_on_def
            intro!: arg_cong[where f=prob])
 
 lemma prob_conj_imp1:
   "prob ({x. Q x} \<inter> msgs) = 0 \<Longrightarrow> prob ({x. Pr x \<and> Q x} \<inter> msgs) = 0"
   using finite_measure_mono[of "{x. Pr x \<and> Q x} \<inter> msgs" "{x. Q x} \<inter> msgs"]
   using measure_nonneg[of \<mu> "{x. Pr x \<and> Q x} \<inter> msgs"]
   by (simp add: subset_eq)
 
 lemma prob_conj_imp2:
   "prob ({x. Pr x} \<inter> msgs) = 0 \<Longrightarrow> prob ({x. Pr x \<and> Q x} \<inter> msgs) = 0"
   using finite_measure_mono[of "{x. Pr x \<and> Q x} \<inter> msgs" "{x. Pr x} \<inter> msgs"]
   using measure_nonneg[of \<mu> "{x. Pr x \<and> Q x} \<inter> msgs"]
   by (simp add: subset_eq)
 
 lemma simple_function_finite: "simple_function \<mu> f"
   by (simp add: simple_function_def)
 
 lemma entropy_commute: "\<H>(\<lambda>x. (X x, Y x)) = \<H>(\<lambda>x. (Y x, X x))"
   apply (subst (1 2) entropy_simple_distributed[OF simple_distributedI[OF simple_function_finite _ refl]])
   unfolding space_point_measure
 proof -
   have eq: "(\<lambda>x. (X x, Y x)) ` msgs = (\<lambda>(x, y). (y, x)) ` (\<lambda>x. (Y x, X x)) ` msgs"
     by auto
   show "- (\<Sum>x\<in>(\<lambda>x. (X x, Y x)) ` msgs. (\<P>(X ; Y) {x}) * log b (\<P>(X ; Y) {x})) =
     - (\<Sum>x\<in>(\<lambda>x. (Y x, X x)) ` msgs. (\<P>(Y ; X) {x}) * log b (\<P>(Y ; X) {x}))"
     unfolding eq
     apply (subst sum.reindex)
     apply (auto simp: vimage_def inj_on_def intro!: sum.cong arg_cong[where f="\<lambda>x. prob x * log b (prob x)"])
     done
 qed simp_all
 
 lemma (in -) measure_eq_0I: "A = {} \<Longrightarrow> measure M A = 0" by simp
 
 lemma conditional_entropy_eq_ce_with_hypothesis:
   "\<H>(X | Y) = -(\<Sum>y\<in>Y`msgs. (\<P>(Y) {y}) * (\<Sum>x\<in>X`msgs. (\<P>(X ; Y) {(x,y)}) / (\<P>(Y) {y}) *
      log b ((\<P>(X ; Y) {(x,y)}) / (\<P>(Y) {y}))))"
   apply (subst conditional_entropy_eq[OF
     simple_distributedI[OF simple_function_finite _ refl]
     simple_distributedI[OF simple_function_finite _ refl]])
   unfolding space_point_measure
 proof -
   have "- (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` msgs. (\<P>(X ; Y) {(x, y)}) * log b ((\<P>(X ; Y) {(x, y)}) / (\<P>(Y) {y}))) =
     - (\<Sum>x\<in>X`msgs. (\<Sum>y\<in>Y`msgs. (\<P>(X ; Y) {(x, y)}) * log b ((\<P>(X ; Y) {(x, y)}) / (\<P>(Y) {y}))))"
     unfolding sum.cartesian_product
     apply (intro arg_cong[where f=uminus] sum.mono_neutral_left)
     apply (auto simp: vimage_def image_iff intro!: measure_eq_0I)
     apply metis
     done
   also have "\<dots> = - (\<Sum>y\<in>Y`msgs. (\<Sum>x\<in>X`msgs. (\<P>(X ; Y) {(x, y)}) * log b ((\<P>(X ; Y) {(x, y)}) / (\<P>(Y) {y}))))"
     by (subst sum.swap) rule
   also have "\<dots> = -(\<Sum>y\<in>Y`msgs. (\<P>(Y) {y}) * (\<Sum>x\<in>X`msgs. (\<P>(X ; Y) {(x,y)}) / (\<P>(Y) {y}) * log b ((\<P>(X ; Y) {(x,y)}) / (\<P>(Y) {y}))))"
     by (auto simp add: sum_distrib_left vimage_def intro!: sum.cong prob_conj_imp1)
   finally show "- (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` msgs. (\<P>(X ; Y) {(x, y)}) * log b ((\<P>(X ; Y) {(x, y)}) / (\<P>(Y) {y}))) =
     -(\<Sum>y\<in>Y`msgs. (\<P>(Y) {y}) * (\<Sum>x\<in>X`msgs. (\<P>(X ; Y) {(x,y)}) / (\<P>(Y) {y}) * log b ((\<P>(X ; Y) {(x,y)}) / (\<P>(Y) {y}))))" .
 qed simp_all
 
 lemma ce_OB_eq_ce_t: "\<I>(fst ; OB) = \<I>(fst ; t\<circ>OB)"
 proof -
   txt \<open>Lemma 2\<close>
 
   { fix k obs obs'
     assume "k \<in> keys" "K k \<noteq> 0" and obs': "obs' \<in> OB ` msgs" and obs: "obs \<in> OB ` msgs"
     assume "t obs = t obs'"
     from t_eq_imp_bij_func[OF this]
     obtain t_f where "bij_betw t_f {..<n} {..<n}" and
       obs_t_f: "\<And>i. i<n \<Longrightarrow> obs!i = obs' ! t_f i"
       using obs obs' unfolding OB_def msgs_def by auto
     then have t_f: "inj_on t_f {..<n}" "{..<n} = t_f`{..<n}" unfolding bij_betw_def by auto
 
     { fix obs assume "obs \<in> OB`msgs"
       then have **: "\<And>ms. length ms = n \<Longrightarrow> OB (k, ms) = obs \<longleftrightarrow> (\<forall>i<n. observe k (ms!i) = obs ! i)"
         unfolding OB_def msgs_def by (simp add: image_iff list_eq_iff_nth_eq)
 
       have "(\<P>(OB ; fst) {(obs, k)}) / K k =
           p ({k}\<times>{ms. (k,ms) \<in> msgs \<and> OB (k,ms) = obs}) / K k"
         apply (simp add: \<mu>'_eq) by (auto intro!: arg_cong[where f=p])
       also have "\<dots> =
           (\<Prod>i<n. \<Sum>m\<in>{m\<in>messages. observe k m = obs ! i}. M m)"
         unfolding P_def using \<open>K k \<noteq> 0\<close> \<open>k \<in> keys\<close>
         apply (simp add: sum_cartesian_product' sum_divide_distrib msgs_def ** cong: conj_cong)
         apply (subst prod_sum_distrib_lists[OF M.finite_space]) ..
       finally have "(\<P>(OB ; fst) {(obs, k)}) / K k =
             (\<Prod>i<n. \<Sum>m\<in>{m\<in>messages. observe k m = obs ! i}. M m)" . }
     note * = this
 
     have "(\<P>(OB ; fst) {(obs, k)}) / K k = (\<P>(OB ; fst) {(obs', k)}) / K k"
       unfolding *[OF obs] *[OF obs']
       using t_f(1) obs_t_f by (subst (2) t_f(2)) (simp add: prod.reindex)
     then have "(\<P>(OB ; fst) {(obs, k)}) = (\<P>(OB ; fst) {(obs', k)})"
       using \<open>K k \<noteq> 0\<close> by auto }
   note t_eq_imp = this
 
   let ?S = "\<lambda>obs. t -`{t obs} \<inter> OB`msgs"
   { fix k obs assume "k \<in> keys" "K k \<noteq> 0" and obs: "obs \<in> OB`msgs"
     have *: "((\<lambda>x. (t (OB x), fst x)) -` {(t obs, k)} \<inter> msgs) =
       (\<Union>obs'\<in>?S obs. ((\<lambda>x. (OB x, fst x)) -` {(obs', k)} \<inter> msgs))" by auto
     have df: "disjoint_family_on (\<lambda>obs'. (\<lambda>x. (OB x, fst x)) -` {(obs', k)} \<inter> msgs) (?S obs)"
       unfolding disjoint_family_on_def by auto
     have "\<P>(t\<circ>OB ; fst) {(t obs, k)} = (\<Sum>obs'\<in>?S obs. \<P>(OB ; fst) {(obs', k)})"
       unfolding comp_def
       using finite_measure_finite_Union[OF _ _ df]
       by (auto simp add: * intro!: sum_nonneg)
     also have "(\<Sum>obs'\<in>?S obs. \<P>(OB ; fst) {(obs', k)}) = real (card (?S obs)) * \<P>(OB ; fst) {(obs, k)}"
       by (simp add: t_eq_imp[OF \<open>k \<in> keys\<close> \<open>K k \<noteq> 0\<close> obs])
     finally have "\<P>(t\<circ>OB ; fst) {(t obs, k)} = real (card (?S obs)) * \<P>(OB ; fst) {(obs, k)}" .}
   note P_t_eq_P_OB = this
 
   { fix k obs assume "k \<in> keys" and obs: "obs \<in> OB`msgs"
     have "\<P>(t\<circ>OB | fst) {(t obs, k)} =
       real (card (t -` {t obs} \<inter> OB ` msgs)) * \<P>(OB | fst) {(obs, k)}"
       using \<P>_k[OF \<open>k \<in> keys\<close>] P_t_eq_P_OB[OF \<open>k \<in> keys\<close> _ obs] by auto }
   note CP_t_K = this
 
   { fix k obs assume "k \<in> keys" and obs: "obs \<in> OB`msgs"
     then have "t -`{t obs} \<inter> OB`msgs \<noteq> {}" (is "?S \<noteq> {}") by auto
     then have "real (card ?S) \<noteq> 0" by auto
 
     have "\<P>(fst | t\<circ>OB) {(k, t obs)} = \<P>(t\<circ>OB | fst) {(t obs, k)} * \<P>(fst) {k} / \<P>(t\<circ>OB) {t obs}"
       using finite_measure_mono[of "{x. fst x = k \<and> t (OB x) = t obs} \<inter> msgs" "{x. fst x = k} \<inter> msgs"]
       using measure_nonneg[of \<mu> "{x. fst x = k \<and> t (OB x) = t obs} \<inter> msgs"]
       by (auto simp add: vimage_def conj_commute subset_eq simp del: measure_nonneg)
     also have "(\<P>(t\<circ>OB) {t obs}) = (\<Sum>k'\<in>keys. (\<P>(t\<circ>OB|fst) {(t obs, k')}) * (\<P>(fst) {k'}))"
       using finite_measure_mono[of "{x. t (OB x) = t obs \<and> fst x = k} \<inter> msgs" "{x. fst x = k} \<inter> msgs"]
       using measure_nonneg[of \<mu> "{x. fst x = k \<and> t (OB x) = t obs} \<inter> msgs"]
       apply (simp add: sum_distribution_cut[of "t\<circ>OB" "t obs" fst])
       apply (auto intro!: sum.cong simp: subset_eq vimage_def prob_conj_imp1)
       done
     also have "\<P>(t\<circ>OB | fst) {(t obs, k)} * \<P>(fst) {k} / (\<Sum>k'\<in>keys. \<P>(t\<circ>OB|fst) {(t obs, k')} * \<P>(fst) {k'}) =
       \<P>(OB | fst) {(obs, k)} * \<P>(fst) {k} / (\<Sum>k'\<in>keys. \<P>(OB|fst) {(obs, k')} * \<P>(fst) {k'})"
       using CP_t_K[OF \<open>k\<in>keys\<close> obs] CP_t_K[OF _ obs] \<open>real (card ?S) \<noteq> 0\<close>
       by (simp only: sum_distrib_left[symmetric] ac_simps
           mult_divide_mult_cancel_left[OF \<open>real (card ?S) \<noteq> 0\<close>]
         cong: sum.cong)
     also have "(\<Sum>k'\<in>keys. \<P>(OB|fst) {(obs, k')} * \<P>(fst) {k'}) = \<P>(OB) {obs}"
       using sum_distribution_cut[of OB obs fst]
       by (auto intro!: sum.cong simp: prob_conj_imp1 vimage_def)
     also have "\<P>(OB | fst) {(obs, k)} * \<P>(fst) {k} / \<P>(OB) {obs} = \<P>(fst | OB) {(k, obs)}"
       by (auto simp: vimage_def conj_commute prob_conj_imp2)
     finally have "\<P>(fst | t\<circ>OB) {(k, t obs)} = \<P>(fst | OB) {(k, obs)}" . }
   note CP_T_eq_CP_O = this
 
   let ?H = "\<lambda>obs. (\<Sum>k\<in>keys. \<P>(fst|OB) {(k, obs)} * log b (\<P>(fst|OB) {(k, obs)})) :: real"
   let ?Ht = "\<lambda>obs. (\<Sum>k\<in>keys. \<P>(fst|t\<circ>OB) {(k, obs)} * log b (\<P>(fst|t\<circ>OB) {(k, obs)})) :: real"
 
   { fix obs assume obs: "obs \<in> OB`msgs"
     have "?H obs = (\<Sum>k\<in>keys. \<P>(fst|t\<circ>OB) {(k, t obs)} * log b (\<P>(fst|t\<circ>OB) {(k, t obs)}))"
       using CP_T_eq_CP_O[OF _ obs]
       by simp
     then have "?H obs = ?Ht (t obs)" . }
   note * = this
 
   have **: "\<And>x f A. (\<Sum>y\<in>t-`{x}\<inter>A. f y (t y)) = (\<Sum>y\<in>t-`{x}\<inter>A. f y x)" by auto
 
   { fix x
     have *: "(\<lambda>x. t (OB x)) -` {t (OB x)} \<inter> msgs =
       (\<Union>obs\<in>?S (OB x). OB -` {obs} \<inter> msgs)" by auto
     have df: "disjoint_family_on (\<lambda>obs. OB -` {obs} \<inter> msgs) (?S (OB x))"
       unfolding disjoint_family_on_def by auto
     have "\<P>(t\<circ>OB) {t (OB x)} = (\<Sum>obs\<in>?S (OB x). \<P>(OB) {obs})"
       unfolding comp_def
       using finite_measure_finite_Union[OF _ _ df]
       by (force simp add: * intro!: sum_nonneg) }
   note P_t_sum_P_O = this
 
   txt \<open>Lemma 3\<close>
   have "\<H>(fst | OB) = -(\<Sum>obs\<in>OB`msgs. \<P>(OB) {obs} * ?Ht (t obs))"
     unfolding conditional_entropy_eq_ce_with_hypothesis using * by simp
   also have "\<dots> = -(\<Sum>obs\<in>t`OB`msgs. \<P>(t\<circ>OB) {obs} * ?Ht obs)"
     apply (subst SIGMA_image_vimage[symmetric, of "OB`msgs" t])
     apply (subst sum.reindex)
     apply (fastforce intro!: inj_onI)
     apply simp
     apply (subst sum.Sigma[symmetric, unfolded split_def])
     using finite_space apply fastforce
     using finite_space apply fastforce
     apply (safe intro!: sum.cong)
     using P_t_sum_P_O
     by (simp add: sum_divide_distrib[symmetric] field_simps **
                   sum_distrib_left[symmetric] sum_distrib_right[symmetric])
   also have "\<dots> = \<H>(fst | t\<circ>OB)"
     unfolding conditional_entropy_eq_ce_with_hypothesis
     by (simp add: comp_def image_image[symmetric])
   finally show ?thesis
     by (subst (1 2) mutual_information_eq_entropy_conditional_entropy) simp_all
 qed
 
 theorem "\<I>(fst ; OB) \<le> real (card observations) * log b (real n + 1)"
 proof -
   have "\<I>(fst ; OB) = \<H>(fst) - \<H>(fst | t\<circ>OB)"
     unfolding ce_OB_eq_ce_t
     by (rule mutual_information_eq_entropy_conditional_entropy simple_function_finite)+
   also have "\<dots> = \<H>(t\<circ>OB) - \<H>(t\<circ>OB | fst)"
     unfolding entropy_chain_rule[symmetric, OF simple_function_finite simple_function_finite] algebra_simps
     by (subst entropy_commute) simp
   also have "\<dots> \<le> \<H>(t\<circ>OB)"
     using conditional_entropy_nonneg[of "t\<circ>OB" fst] by simp
   also have "\<dots> \<le> log b (real (card ((t\<circ>OB)`msgs)))"
     using entropy_le_card[of "t\<circ>OB", OF simple_distributedI[OF simple_function_finite _ refl]] by simp
   also have "\<dots> \<le> log b (real (n + 1)^card observations)"
     using card_T_bound not_empty
     by (auto intro!: log_le simp: card_gt_0_iff of_nat_power [symmetric] simp del: of_nat_power of_nat_Suc)
   also have "\<dots> = real (card observations) * log b (real n + 1)"
     by (simp add: log_nat_power add.commute)
   finally show ?thesis  .
 qed
 
 end
 
 end