Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
(* Author: Johannes Hölzl, TU München *)
section {* Formalization of a Countermeasure by Koepf \& Duermuth 2009 *}
theory Koepf_Duermuth_Countermeasure
imports "~~/src/HOL/Probability/Information" "~~/src/HOL/Library/Permutation"
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: split_if_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 `finite A` proof induct
case (insert a A) thus ?case unfolding extensionalD_insert[OF `a \<notin> A`]
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 `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> extensionalD d A" "g \<in> extensionalD d A"
have "f a = d" "g a = d"
using ext `a \<notin> A` by (auto simp add: extensionalD_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> 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 zero_notin_Suc_image[simp]: "0 \<notin> Suc ` A"
by auto
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. 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: setsum.reindex split_conv setsum_cartesian_product'
lessThan_Suc_eq_insert_0 setprod.reindex setsum_left_distrib[symmetric] setsum_right_distrib[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. ereal (p x))) A = (\<Sum>i\<in>A. p i)"
unfolding measure_def by (subst emeasure_point_measure_finite) auto
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> 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 = setsum 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 setsum_right_distrib[symmetric, simp]
note setsum_left_distrib[symmetric, simp]
note setsum_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
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
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 (op = 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> setsum 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)
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: 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.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 setsum_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 setsum.reindex)
apply (auto simp: vimage_def inj_on_def intro!: setsum.cong arg_cong[where f="\<lambda>x. prob x * log b (prob x)"])
done
qed
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 setsum.cartesian_product
apply (intro arg_cong[where f=uminus] setsum.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 setsum.commute) 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: setsum_right_distrib vimage_def intro!: setsum.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
lemma ce_OB_eq_ce_t: "\<I>(fst ; OB) = \<I>(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: \<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]) ..
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 = "\<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!: 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])
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 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)
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: setsum_distribution_cut[of "t\<circ>OB" "t obs" fst])
apply (auto intro!: setsum.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 `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_cut[of OB obs fst]
by (auto intro!: setsum.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!: 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 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 (fastforce intro!: inj_onI)
apply simp
apply (subst setsum.Sigma[symmetric, unfolded split_def])
using finite_space apply fastforce
using finite_space apply fastforce
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
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] sign_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 power_real_of_nat 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