src/HOL/Probability/ex/Koepf_Duermuth_Countermeasure.thy

reworked Probability theory: measures are not type restricted to positive extended reals

(* Author: Johannes Hölzl, TU München *)

header {* Formalization of a Countermeasure by Koepf \& Duermuth 2009 *}

theory Koepf_Duermuth_Countermeasure
  imports "~~/src/HOL/Probability/Information" "~~/src/HOL/Library/Permutation"
begin

lemma
  fixes p u :: "'a \<Rightarrow> real"
  assumes "1 < b"
  assumes sum: "(\<Sum>i\<in>S. p i) = (\<Sum>i\<in>S. u i)"
  and pos: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> p i" "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> u i"
  and cont: "\<And>i. i \<in> S \<Longrightarrow> 0 < p i \<Longrightarrow> 0 < u i"
  shows "(\<Sum>i\<in>S. p i * log b (u i)) \<le> (\<Sum>i\<in>S. p i * log b (p i))"
proof -
  have "(\<Sum>i\<in>S. p i * ln (u i) - p i * ln (p i)) \<le> (\<Sum>i\<in>S. u i - p i)"
  proof (intro setsum_mono)
    fix i assume [intro, simp]: "i \<in> S"
    show "p i * ln (u i) - p i * ln (p i) \<le> u i - p i"
    proof cases
      assume "p i = 0" with pos[of i] show ?thesis by simp
    next
      assume "p i \<noteq> 0"
      then have "0 < p i" "0 < u i" using pos[of i] cont[of i] by auto
      then have *: "0 < u i / p i" by (blast intro: divide_pos_pos cont)
      from `0 < p i` `0 < u i`
      have "p i * ln (u i) - p i * ln (p i) = p i * ln (u i / p i)"
        by (simp add: ln_div field_simps)
      also have "\<dots> \<le> p i * (u i / p i - 1)"
        using exp_ge_add_one_self[of "ln (u i / p i)"] pos[of i] exp_ln[OF *]
        by (auto intro!: mult_left_mono)
      also have "\<dots> = u i - p i"
        using `p i \<noteq> 0` by (simp add: field_simps)
      finally show ?thesis by simp
    qed
  qed
  also have "\<dots> = 0" using sum by (simp add: setsum_subtractf)
  finally show ?thesis using `1 < b` unfolding log_def setsum_subtractf
    by (auto intro!: divide_right_mono
             simp: times_divide_eq_right setsum_divide_distrib[symmetric])
qed

definition (in prob_space)
  "ordered_variable_partition X = (SOME f. bij_betw f {0..<card (X`space M)} (X`space M) \<and>
    (\<forall>i<card (X`space M). \<forall>j<card (X`space M). i \<le> j \<longrightarrow> distribution X {f i} \<le> distribution X {f j}))"

lemma ex_ordered_bij_betw_nat_finite:
  fixes order :: "nat \<Rightarrow> 'a\<Colon>linorder"
  assumes "finite S"
  shows "\<exists>f. bij_betw f {0..<card S} S \<and> (\<forall>i<card S. \<forall>j<card S. i \<le> j \<longrightarrow> order (f i) \<le> order (f j))"
proof -
  from ex_bij_betw_nat_finite [OF `finite S`] guess f .. note f = this
  let ?xs = "sort_key order (map f [0 ..< card S])"

  have "?xs <~~> map f [0 ..< card S]"
    unfolding multiset_of_eq_perm[symmetric] by (rule multiset_of_sort)
  from permutation_Ex_bij [OF this]
  obtain g where g: "bij_betw g {0..<card S} {0..<card S}" and
    map: "\<And>i. i<card S \<Longrightarrow> ?xs ! i = map f [0 ..< card S] ! g i"
    by (auto simp: atLeast0LessThan)

  { fix i assume "i < card S"
    then have "g i < card S" using g by (auto simp: bij_betw_def)
    with map [OF `i < card S`] have "f (g i) = ?xs ! i" by simp }
  note this[simp]

  show ?thesis
  proof (intro exI allI conjI impI)
    show "bij_betw (f\<circ>g) {0..<card S} S"
      using g f by (rule bij_betw_trans)
    fix i j assume [simp]: "i < card S" "j < card S" "i \<le> j"
    from sorted_nth_mono[of "map order ?xs" i j]
    show "order ((f\<circ>g) i) \<le> order ((f\<circ>g) j)" by simp
  qed
qed

lemma (in prob_space)
  assumes "finite (X`space M)"
  shows "bij_betw (ordered_variable_partition X) {0..<card (X`space M)} (X`space M)"
  and "\<And>i j. \<lbrakk> i < card (X`space M) ; j < card (X`space M) ; i \<le> j \<rbrakk> \<Longrightarrow>
    distribution X {ordered_variable_partition X i} \<le> distribution X {ordered_variable_partition X j}"
  oops

definition (in prob_space)
  "order_distribution X i = real (distribution X {ordered_variable_partition X i})"

definition (in prob_space)
  "guessing_entropy b X = (\<Sum>i<card(X`space M). real i * log b (order_distribution X i))"

abbreviation (in information_space)
  finite_guessing_entropy ("\<G>'(_')") where
  "\<G>(X) \<equiv> guessing_entropy b X"

lemma zero_notin_Suc_image[simp]: "0 \<notin> Suc ` A"
  by auto

definition extensional :: "'b \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> 'b) set" where
  "extensional d A = {f. \<forall>x. x \<notin> A \<longrightarrow> f x = d}"

lemma extensional_empty[simp]: "extensional d {} = {\<lambda>x. d}"
  unfolding extensional_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 extensional_insert[simp]:
  assumes "a \<notin> A"
  shows "extensional d (insert a A) \<inter> (insert a A \<rightarrow> B) = (\<Union>f \<in> extensional 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: split_if_asm simp: extensional_def)

lemma finite_extensional_funcset[simp, intro]:
  assumes "finite A" "finite B"
  shows "finite (extensional 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 (extensional d A \<inter> (A \<rightarrow> B)) = card B ^ card A"
using `finite A` proof induct
  case (insert a A) thus ?case unfolding extensional_insert[OF `a \<notin> A`]
  proof (subst card_UN_disjoint, safe, simp_all)
    show "finite (extensional d A \<inter> (A \<rightarrow> B))" "\<And>f. finite (fun_upd f a ` B)"
      using `finite B` `finite A` 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> extensional d A" "g \<in> extensional d A"
    have "f a = d" "g a = d"
      using ext `a \<notin> A` by (auto simp add: extensional_def)
    with `f \<noteq> g` 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> extensional 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>extensional 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 set_of_list_extend:
  "{xs. length xs = Suc n \<and> (\<forall>x\<in>set xs. x \<in> A)} =
    (\<lambda>(xs, n). n#xs) ` ({xs. length xs = n \<and> (\<forall>x\<in>set xs. x \<in> A)} \<times> A)"
  by (auto simp: length_Suc_conv)

lemma
  assumes "finite A"
  shows finite_lists:
    "finite {xs. length xs = n \<and> (\<forall>x\<in>set xs. x \<in> A)}" (is "finite (?lists n)")
  and card_list_length:
    "card {xs. length xs = n \<and> (\<forall>x\<in>set xs. x \<in> A)} = (card A)^n"
proof -
  from `finite A`
  have "(card {xs. length xs = n \<and> (\<forall>x\<in>set xs. x \<in> A)} = (card A)^n) \<and>
         finite {xs. length xs = n \<and> (\<forall>x\<in>set xs. x \<in> A)}"
  proof (induct n)
    case (Suc n)
    moreover note set_of_list_extend[of n A]
    moreover have "inj_on (\<lambda>(xs, n). n#xs) (?lists n \<times> A)"
      by (auto intro!: inj_onI)
    ultimately show ?case using assms by (auto simp: card_image)
  qed (simp cong: conj_cong)
  then show "finite (?lists n)" "card (?lists n) = card A ^ n"
    by auto
qed

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> setsum p X"
   by (auto intro!: setsum_nonneg)

sublocale finite_information \<subseteq> finite_measure_space "\<lparr> space = \<Omega>, sets = Pow \<Omega>, measure = \<lambda>S. extreal (setsum p S)\<rparr>"
  by (rule finite_measure_spaceI) (simp_all add: setsum_Un_disjoint finite_subset)

sublocale finite_information \<subseteq> finite_prob_space "\<lparr> space = \<Omega>, sets = Pow \<Omega>, measure = \<lambda>S. extreal (setsum p S)\<rparr>"
  by default (simp add: one_extreal_def)

sublocale finite_information \<subseteq> information_space "\<lparr> space = \<Omega>, sets = Pow \<Omega>, measure = \<lambda>S. extreal (setsum p S)\<rparr>" b
  by default simp

lemma (in finite_information) \<mu>'_eq: "A \<subseteq> \<Omega> \<Longrightarrow> \<mu>' A = setsum p A"
  unfolding \<mu>'_def by auto

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. length ms = n \<and> (\<forall>M\<in>set ms. M \<in> messages)}"

definition P :: "('key \<times> 'message list) \<Rightarrow> real" where
  "P = (\<lambda>(k, ms). K k * (\<Prod>i<length ms. M (ms ! i)))"

end

sublocale koepf_duermuth \<subseteq> finite_information msgs P b
proof default
  show "finite msgs" unfolding msgs_def
    using finite_lists[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 setsum_right_distrib[symmetric, simp]
  note setsum_left_distrib[symmetric, simp]
  note setsum_cartesian_product'[simp]

  have "(\<Sum>ms | length ms = n \<and> (\<forall>M\<in>set ms. M \<in> messages). \<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: comp_def set_of_list_extend lessThan_Suc_eq_insert_0
                    setsum_reindex setprod_reindex)
  qed
  then show "setsum 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: setprod_nonneg)
  then show "0 \<le> P x"
    unfolding P_def by (auto split: prod.split simp: zero_le_mult_iff)
qed auto

lemma SIGMA_image_vimage:
  "snd ` (SIGMA x:f`X. f -` {x} \<inter> X) = X"
  by (auto simp: image_iff)

lemma inj_Cons[simp]: "\<And>X. inj_on (\<lambda>(xs, x). x#xs) X" by (auto intro!: inj_onI)

lemma setprod_setsum_distrib_lists:
  fixes P and S :: "'a set" and f :: "'a \<Rightarrow> _::semiring_0" assumes "finite S"
  shows "(\<Sum>ms\<in>{ms. length ms = n \<and> set ms \<subseteq> S \<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. length ms = Suc n \<and> set ms \<subseteq> S \<and> (\<forall>i<Suc n. P i (ms ! i))} =
    (\<lambda>(xs, x). x#xs) ` ({ms. length ms = n \<and> set ms \<subseteq> S \<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: setsum_reindex split_conv setsum_cartesian_product'
      lessThan_Suc_eq_insert_0 setprod_reindex setsum_left_distrib[symmetric] setsum_right_distrib[symmetric])
qed

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 seq obs = length (filter (op = obs) seq)"

lemma card_T_bound: "card ((t\<circ>OB)`msgs) \<le> (n+1)^card observations"
proof -
  have "(t\<circ>OB)`msgs \<subseteq> extensional 0 observations \<inter> (observations \<rightarrow> {.. n})"
    unfolding observations_def extensional_def OB_def msgs_def
    by (auto simp add: t_def comp_def image_iff)
  with finite_extensional_funcset
  have "card ((t\<circ>OB)`msgs) \<le> card (extensional 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> setsum P A"

abbreviation probability ("\<P>'(_') _") where
 "\<P>(X) x \<equiv> distribution X x"

abbreviation joint_probability ("\<P>'(_, _') _") where
 "\<P>(X, Y) x \<equiv> joint_distribution X Y x"

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 (multiset_of xs) = count (multiset_of ys)"
    using assms by (simp add: fun_eq_iff count_multiset_of t_def)
  then have "xs <~~> ys" unfolding multiset_of_eq_perm count_inject .
  then show ?thesis by (rule permutation_Ex_bij)
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. length ms = n \<and> (\<forall>M\<in>set ms. M \<in> messages)}"
    unfolding msgs_def by auto
  show "\<P>(fst) {k} = K k"
    apply (simp add: \<mu>'_eq distribution_def)
    apply (simp add: * P_def)
    apply (simp add: setsum_cartesian_product')
    using setprod_setsum_distrib_lists[OF M.finite_space, of M n "\<lambda>x x. True"] `k \<in> keys`
    by (auto simp add: setsum_right_distrib[symmetric] subset_eq setprod_1)
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. length m = n \<and> (\<forall>x\<in>set m. x\<in>messages)} \<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 ce_OB_eq_ce_t: "\<H>(fst | OB) = \<H>(fst | t\<circ>OB)"
proof -
  txt {* Lemma 2 *}

  { 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: distribution_def \<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 `K k \<noteq> 0` `k \<in> keys`
        apply (simp add: setsum_cartesian_product' setsum_divide_distrib msgs_def ** cong: conj_cong)
        apply (subst setprod_setsum_distrib_lists[OF M.finite_space, unfolded subset_eq]) ..
      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: setprod_reindex)
    then have "\<P>(OB, fst) {(obs, k)} = \<P>(OB, fst) {(obs', k)}"
      using `K k \<noteq> 0` by auto }
  note t_eq_imp = this

  let "?S 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 distribution_def comp_def
      using finite_measure_finite_Union[OF _ _ df]
      by (force simp add: * intro!: setsum_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 `k \<in> keys` `K k \<noteq> 0` obs] real_eq_of_nat)
    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 `k \<in> keys`] P_t_eq_P_OB[OF `k \<in> keys` _ 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 distribution_order(7,8)[where X=fst and x=k and Y="t\<circ>OB" and y="t obs"]
      by (subst joint_distribution_commute) auto
    also have "\<P>(t\<circ>OB) {t obs} = (\<Sum>k'\<in>keys. \<P>(t\<circ>OB|fst) {(t obs, k')} * \<P>(fst) {k'})"
      using setsum_distribution(2)[of "t\<circ>OB" fst "t obs", symmetric]
      by (auto intro!: setsum_cong distribution_order(8))
    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 `k\<in>keys` obs] CP_t_K[OF _ obs] `real (card ?S) \<noteq> 0`
      by (simp only: setsum_right_distrib[symmetric] ac_simps
          mult_divide_mult_cancel_left[OF `real (card ?S) \<noteq> 0`]
        cong: setsum_cong)
    also have "(\<Sum>k'\<in>keys. \<P>(OB|fst) {(obs, k')} * \<P>(fst) {k'}) = \<P>(OB) {obs}"
      using setsum_distribution(2)[of OB fst obs, symmetric]
      by (auto intro!: setsum_cong distribution_order(8))
    also have "\<P>(OB | fst) {(obs, k)} * \<P>(fst) {k} / \<P>(OB) {obs} = \<P>(fst | OB) {(k, obs)}"
      by (subst joint_distribution_commute) (auto intro!: distribution_order(8))
    finally have "\<P>(fst | t\<circ>OB) {(k, t obs)} = \<P>(fst | OB) {(k, obs)}" . }
  note CP_T_eq_CP_O = this

  let "?H obs" = "(\<Sum>k\<in>keys. \<P>(fst|OB) {(k, obs)} * log b (\<P>(fst|OB) {(k, obs)})) :: real"
  let "?Ht 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 distribution_def comp_def
      using finite_measure_finite_Union[OF _ _ df]
      by (force simp add: * intro!: setsum_nonneg) }
  note P_t_sum_P_O = this

  txt {* Lemma 3 *}
  have "\<H>(fst | OB) = -(\<Sum>obs\<in>OB`msgs. \<P>(OB) {obs} * ?Ht (t obs))"
    unfolding conditional_entropy_eq_ce_with_hypothesis[OF
      simple_function_finite simple_function_finite] 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 setsum_reindex)
    apply (fastsimp intro!: inj_onI)
    apply simp
    apply (subst setsum_Sigma[symmetric, unfolded split_def])
    using finite_space apply fastsimp
    using finite_space apply fastsimp
    apply (safe intro!: setsum_cong)
    using P_t_sum_P_O
    by (simp add: setsum_divide_distrib[symmetric] field_simps **
                  setsum_right_distrib[symmetric] setsum_left_distrib[symmetric])
  also have "\<dots> = \<H>(fst | t\<circ>OB)"
    unfolding conditional_entropy_eq_ce_with_hypothesis[OF
      simple_function_finite simple_function_finite]
    by (simp add: comp_def image_image[symmetric])
  finally show ?thesis .
qed

theorem "\<I>(fst ; OB) \<le> real (card observations) * log b (real n + 1)"
proof -
  from simple_function_finite simple_function_finite
  have "\<I>(fst ; OB) = \<H>(fst) - \<H>(fst | OB)"
    by (rule mutual_information_eq_entropy_conditional_entropy)
  also have "\<dots> = \<H>(fst) - \<H>(fst | t\<circ>OB)"
    unfolding ce_OB_eq_ce_t ..
  also have "\<dots> = \<H>(t\<circ>OB) - \<H>(t\<circ>OB | fst)"
    unfolding entropy_chain_rule[symmetric, OF simple_function_finite simple_function_finite] sign_simps
    by (subst entropy_commute[OF simple_function_finite simple_function_finite]) simp
  also have "\<dots> \<le> \<H>(t\<circ>OB)"
    using conditional_entropy_positive[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"] 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 power_real_of_nat simp del: real_of_nat_power)
  also have "\<dots> = real (card observations) * log b (real n + 1)"
    by (simp add: log_nat_power real_of_nat_Suc)
  finally show ?thesis  .
qed

end

end