theory Dining_Cryptographers
imports "~~/src/HOL/Probability/Information"
begin
locale finite_space =
fixes S :: "'a set"
assumes finite[simp]: "finite S"
and not_empty[simp]: "S \<noteq> {}"
definition (in finite_space) "M = \<lparr> space = S, sets = Pow S,
measure = \<lambda>A. ereal (card A / card S) \<rparr>"
lemma (in finite_space)
shows space_M[simp]: "space M = S"
and sets_M[simp]: "sets M = Pow S"
by (simp_all add: M_def)
sublocale finite_space \<subseteq> finite_measure_space M
proof (rule finite_measure_spaceI)
fix A B :: "'a set" assume "A \<inter> B = {}" "A \<subseteq> space M" "B \<subseteq> space M"
then show "measure M (A \<union> B) = measure M A + measure M B"
by (simp add: M_def card_Un_disjoint finite_subset[OF _ finite] field_simps)
qed (auto simp: M_def divide_nonneg_nonneg)
sublocale finite_space \<subseteq> information_space M 2
by default (simp_all add: M_def one_ereal_def)
lemma (in finite_space) \<mu>'_eq[simp]: "\<mu>' A = (if A \<subseteq> S then card A / card S else 0)"
unfolding \<mu>'_def by (auto simp: M_def)
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)"
(is "?lists (Suc n) = _")
proof
show "(\<lambda>(xs, n). n#xs) ` (?lists n \<times> A) \<subseteq> ?lists (Suc n)" by auto
show "?lists (Suc n) \<subseteq> (\<lambda>(xs, n). n#xs) ` (?lists n \<times> A)"
proof
fix x assume "x \<in> ?lists (Suc n)"
moreover
hence "x \<noteq> []" by auto
then obtain t h where "x = h # t" by (cases x) auto
ultimately show "x \<in> (\<lambda>(xs, n). n#xs) ` (?lists n \<times> A)"
by (auto intro!: image_eqI[where x="(t, h)"])
qed
qed
lemma card_finite_list_length:
assumes "finite A"
shows "(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)}"
(is "card (?lists n) = _ \<and> _")
proof (induct n)
case 0 have "{xs. length xs = 0 \<and> (\<forall>x\<in>set xs. x \<in> A)} = {[]}" by auto
thus ?case by simp
next
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
lemma
assumes "finite A"
shows finite_lists: "finite {xs. length xs = n \<and> (\<forall>x\<in>set xs. x \<in> A)}"
and card_list_length: "card {xs. length xs = n \<and> (\<forall>x\<in>set xs. x \<in> A)} = (card A)^n"
using card_finite_list_length[OF assms, of n] by auto
section "Define the state space"
text {*
We introduce the state space on which the algorithm operates.
This contains:
\begin{description}
\item[n]
The number of cryptographers on the table.
\item[payer]
Either one of the cryptographers or the NSA.
\item[coin]
The result of the coin flipping for each cryptographer.
\item[inversion]
The public result for each cryptographer, e.g. the sum of the coin flipping
for the cryptographer, its right neighbour and the information if he paid or
not.
\end{description}
The observables are the \emph{inversions}
*}
locale dining_cryptographers_space =
fixes n :: nat
assumes n_gt_3: "n \<ge> 3"
begin
definition "dining_cryptographers =
({None} \<union> Some ` {0..<n}) \<times> {xs :: bool list. length xs = n}"
definition "payer dc = fst dc"
definition coin :: "(nat option \<times> bool list) => nat \<Rightarrow> bool" where
"coin dc c = snd dc ! (c mod n)"
definition "inversion dc =
map (\<lambda>c. (payer dc = Some c) \<noteq> (coin dc c \<noteq> coin dc (c + 1))) [0..<n]"
definition "result dc = foldl (\<lambda> a b. a \<noteq> b) False (inversion dc)"
lemma coin_n[simp]: "coin dc n = coin dc 0"
unfolding coin_def by simp
theorem correctness:
assumes "dc \<in> dining_cryptographers"
shows "result dc \<longleftrightarrow> (payer dc \<noteq> None)"
proof -
let "?XOR f l" = "foldl (op \<noteq>) False (map f [0..<l])"
have foldl_coin:
"\<not> ?XOR (\<lambda>c. coin dc c \<noteq> coin dc (c + 1)) n"
proof -
def n' \<equiv> n -- "Need to hide n, as it is hidden in coin"
have "?XOR (\<lambda>c. coin dc c \<noteq> coin dc (c + 1)) n'
= (coin dc 0 \<noteq> coin dc n')"
by (induct n') auto
thus ?thesis using `n' \<equiv> n` by simp
qed
from assms have "payer dc = None \<or> (\<exists>k<n. payer dc = Some k)"
unfolding dining_cryptographers_def payer_def by auto
thus ?thesis
proof (rule disjE)
assume "payer dc = None"
thus ?thesis unfolding result_def inversion_def
using foldl_coin by simp
next
assume "\<exists>k<n. payer dc = Some k"
then obtain k where "k < n" and "payer dc = Some k" by auto
def l \<equiv> n -- "Need to hide n, as it is hidden in coin, payer etc."
have "?XOR (\<lambda>c. (payer dc = Some c) \<noteq> (coin dc c \<noteq> coin dc (c + 1))) l =
((k < l) \<noteq> ?XOR (\<lambda>c. (coin dc c \<noteq> coin dc (c + 1))) l)"
using `payer dc = Some k` by (induct l) auto
thus ?thesis
unfolding result_def inversion_def l_def
using `payer dc = Some k` foldl_coin `k < n` by simp
qed
qed
text {*
We now restrict the state space for the dining cryptographers to the cases when
one of the cryptographer pays.
*}
definition
"dc_crypto = dining_cryptographers - {None}\<times>UNIV"
lemma dc_crypto: "dc_crypto = Some ` {0..<n} \<times> {xs :: bool list. length xs = n}"
unfolding dc_crypto_def dining_cryptographers_def by auto
lemma image_payer_dc_crypto: "payer ` dc_crypto = Some ` {0..<n}"
proof -
have *: "{xs. length xs = n} \<noteq> {}"
by (auto intro!: exI[of _ "replicate n undefined"])
show ?thesis
unfolding payer_def_raw dc_crypto fst_image_times if_not_P[OF *] ..
qed
lemma image_ex1_eq: "inj_on f A \<Longrightarrow> (b \<in> f ` A) \<longleftrightarrow> (\<exists>!x \<in> A. b = f x)"
by (unfold inj_on_def) blast
lemma Ex1_eq: "\<exists>! x. P x \<Longrightarrow> P x \<Longrightarrow> P y \<Longrightarrow> x = y"
by auto
lemma card_payer_and_inversion:
assumes "xs \<in> inversion ` dc_crypto" and "i < n"
shows "card {dc \<in> dc_crypto. payer dc = Some i \<and> inversion dc = xs} = 2"
(is "card ?S = 2")
proof -
obtain ys j where xs_inv: "inversion (Some j, ys) = xs" and
"j < n" and "(Some j, ys) \<in> dc_crypto"
using assms(1) by (auto simp: dc_crypto)
hence "length ys = n" by (simp add: dc_crypto)
have [simp]: "length xs = n" using xs_inv[symmetric] by (simp add: inversion_def)
{ fix b
have "inj_on (\<lambda>x. inversion (Some i, x)) {ys. ys ! 0 = b \<and> length ys = length xs}"
proof (rule inj_onI)
fix x y
assume "x \<in> {ys. ys ! 0 = b \<and> length ys = length xs}"
and "y \<in> {ys. ys ! 0 = b \<and> length ys = length xs}"
and inv: "inversion (Some i, x) = inversion (Some i, y)"
hence [simp]: "x ! 0 = y ! 0" "length y = n" "length x = n"
using `length xs = n` by simp_all
have *: "\<And>j. j < n \<Longrightarrow>
(x ! j = x ! (Suc j mod n)) = (y ! j = y ! (Suc j mod n))"
using inv unfolding inversion_def map_eq_conv payer_def coin_def
by fastforce
show "x = y"
proof (rule nth_equalityI, simp, rule allI, rule impI)
fix j assume "j < length x" hence "j < n" using `length xs = n` by simp
thus "x ! j = y ! j"
proof (induct j)
case (Suc j)
moreover hence "j < n" by simp
ultimately show ?case using *[OF `j < n`]
by (cases "y ! j") simp_all
qed simp
qed
qed }
note inj_inv = this
txt {*
We now construct the possible inversions for @{term xs} when the payer is
@{term i}.
*}
def zs \<equiv> "map (\<lambda>p. if p \<in> {min i j<..max i j} then \<not> ys ! p else ys ! p) [0..<n]"
hence [simp]: "length zs = n" by simp
hence [simp]: "0 < length zs" using n_gt_3 by simp
have "\<And>l. l < max i j \<Longrightarrow> Suc l mod n = Suc l"
using `i < n` `j < n` by auto
{ fix l assume "l < n"
hence "(((l < min i j \<or> l = min i j) \<or> (min i j < l \<and> l < max i j)) \<or> l = max i j) \<or> max i j < l" by auto
hence "((i = l) = (zs ! l = zs ! (Suc l mod n))) = ((j = l) = (ys ! l = ys ! (Suc l mod n)))"
apply - proof ((erule disjE)+)
assume "l < min i j"
hence "l \<noteq> i" and "l \<noteq> j" and "zs ! l = ys ! l" and
"zs ! (Suc l mod n) = ys ! (Suc l mod n)" using `i < n` `j < n` unfolding zs_def by auto
thus ?thesis by simp
next
assume "l = min i j"
show ?thesis
proof (cases rule: linorder_cases)
assume "i < j"
hence "l = i" and "Suc l < n" and "i \<noteq> j" and "Suc l \<le> max i j" using `l = min i j` using `j < n` by auto
hence "zs ! l = ys ! l" and "zs ! (Suc l mod n) = (\<not> ys ! (Suc l mod n))"
using `l = min i j`[symmetric] by (simp_all add: zs_def)
thus ?thesis using `l = i` `i \<noteq> j` by simp
next
assume "j < i"
hence "l = j" and "Suc l < n" and "i \<noteq> j" and "Suc l \<le> max i j" using `l = min i j` using `i < n` by auto
hence "zs ! l = ys ! l" and "zs ! (Suc l mod n) = (\<not> ys ! (Suc l mod n))"
using `l = min i j`[symmetric] by (simp_all add: zs_def)
thus ?thesis using `l = j` `i \<noteq> j` by simp
next
assume "i = j"
hence "i = j" and "max i j = l" and "min i j = l" and "zs = ys"
using `l = min i j` by (simp_all add: zs_def `length ys = n`[symmetric] map_nth)
thus ?thesis by simp
qed
next
assume "min i j < l \<and> l < max i j"
hence "i \<noteq> l" and "j \<noteq> l" and "zs ! l = (\<not> ys ! l)"
"zs ! (Suc l mod n) = (\<not> ys ! (Suc l mod n))"
using `i < n` `j < n` by (auto simp: zs_def)
thus ?thesis by simp
next
assume "l = max i j"
show ?thesis
proof (cases rule: linorder_cases)
assume "i < j"
hence "l = j" and "i \<noteq> j" using `l = max i j` using `j < n` by auto
have "zs ! (Suc l mod n) = ys ! (Suc l mod n)"
using `j < n` `i < j` `l = j` by (cases "Suc l = n") (auto simp add: zs_def)
moreover have "zs ! l = (\<not> ys ! l)"
using `j < n` `i < j` by (auto simp add: `l = j` zs_def)
ultimately show ?thesis using `l = j` `i \<noteq> j` by simp
next
assume "j < i"
hence "l = i" and "i \<noteq> j" using `l = max i j` by auto
have "zs ! (Suc l mod n) = ys ! (Suc l mod n)"
using `i < n` `j < i` `l = i` by (cases "Suc l = n") (auto simp add: zs_def)
moreover have "zs ! l = (\<not> ys ! l)"
using `i < n` `j < i` by (auto simp add: `l = i` zs_def)
ultimately show ?thesis using `l = i` `i \<noteq> j` by auto
next
assume "i = j"
hence "i = j" and "max i j = l" and "min i j = l" and "zs = ys"
using `l = max i j` by (simp_all add: zs_def `length ys = n`[symmetric] map_nth)
thus ?thesis by simp
qed
next
assume "max i j < l"
hence "j \<noteq> l" and "i \<noteq> l" by simp_all
have "zs ! (Suc l mod n) = ys ! (Suc l mod n)"
using `l < n` `max i j < l` by (cases "Suc l = n") (auto simp add: zs_def)
moreover have "zs ! l = ys ! l"
using `l < n` `max i j < l` by (auto simp add: zs_def)
ultimately show ?thesis using `j \<noteq> l` `i \<noteq> l` by auto
qed }
hence zs: "inversion (Some i, zs) = xs"
by (simp add: xs_inv[symmetric] inversion_def coin_def payer_def)
moreover
hence Not_zs: "inversion (Some i, (map Not zs)) = xs"
by (simp add: xs_inv[symmetric] inversion_def coin_def payer_def)
ultimately
have "{dc \<in> dc_crypto. payer dc = Some i \<and> inversion dc = xs} =
{(Some i, zs), (Some i, map Not zs)}"
using `i < n`
proof (safe, simp_all add:dc_crypto payer_def)
fix b assume [simp]: "length b = n"
and *: "inversion (Some i, b) = xs" and "b \<noteq> zs"
show "b = map Not zs"
proof (cases "b ! 0 = zs ! 0")
case True
hence zs: "zs \<in> {ys. ys ! 0 = b ! 0 \<and> length ys = length xs} \<and> xs = inversion (Some i, zs)"
using zs by simp
have b: "b \<in> {ys. ys ! 0 = b ! 0 \<and> length ys = length xs} \<and> xs = inversion (Some i, b)"
using * by simp
hence "b \<in> {ys. ys ! 0 = b ! 0 \<and> length ys = length xs}" ..
with *[symmetric] have "xs \<in> (\<lambda>x. inversion (Some i, x)) ` {ys. ys ! 0 = b ! 0 \<and> length ys = length xs}"
by (rule image_eqI)
from this[unfolded image_ex1_eq[OF inj_inv]] b zs
have "b = zs" by (rule Ex1_eq)
thus ?thesis using `b \<noteq> zs` by simp
next
case False
hence zs: "map Not zs \<in> {ys. ys ! 0 = b ! 0 \<and> length ys = length xs} \<and> xs = inversion (Some i, map Not zs)"
using Not_zs by (simp add: nth_map[OF `0 < length zs`])
have b: "b \<in> {ys. ys ! 0 = b ! 0 \<and> length ys = length xs} \<and> xs = inversion (Some i, b)"
using * by simp
hence "b \<in> {ys. ys ! 0 = b ! 0 \<and> length ys = length xs}" ..
with *[symmetric] have "xs \<in> (\<lambda>x. inversion (Some i, x)) ` {ys. ys ! 0 = b ! 0 \<and> length ys = length xs}"
by (rule image_eqI)
from this[unfolded image_ex1_eq[OF inj_inv]] b zs
show "b = map Not zs" by (rule Ex1_eq)
qed
qed
moreover
have "zs \<noteq> map Not zs"
using `0 < length zs` by (cases zs) simp_all
ultimately show ?thesis by simp
qed
lemma finite_dc_crypto: "finite dc_crypto"
using finite_lists[where A="UNIV :: bool set"]
unfolding dc_crypto by simp
lemma card_inversion:
assumes "xs \<in> inversion ` dc_crypto"
shows "card {dc \<in> dc_crypto. inversion dc = xs} = 2 * n"
proof -
let "?set i" = "{dc \<in> dc_crypto. payer dc = Some i \<and> inversion dc = xs}"
let "?sets" = "{?set i | i. i < n}"
have [simp]: "length xs = n" using assms
by (auto simp: dc_crypto inversion_def_raw)
have "{dc \<in> dc_crypto. inversion dc = xs} = (\<Union> i < n. ?set i)"
unfolding dc_crypto payer_def by auto
also have "\<dots> = (\<Union> ?sets)" by auto
finally have eq_Union: "{dc \<in> dc_crypto. inversion dc = xs} = (\<Union> ?sets)" by simp
have card_double: "2 * card ?sets = card (\<Union> ?sets)"
proof (rule card_partition)
show "finite ?sets" by simp
{ fix i assume "i < n"
have "?set i \<subseteq> dc_crypto" by auto
have "finite (?set i)" using finite_dc_crypto by auto }
thus "finite (\<Union>?sets)" by auto
next
fix c assume "c \<in> ?sets"
thus "card c = 2" using card_payer_and_inversion[OF assms] by auto
next
fix x y assume "x \<in> ?sets" and "y \<in> ?sets" "x \<noteq> y"
then obtain i j where xy: "x = ?set i" "y = ?set j" by auto
hence "i \<noteq> j" using `x \<noteq> y` by auto
thus "x \<inter> y = {}" using xy by auto
qed
have sets: "?sets = ?set ` {..< n}"
unfolding image_def by auto
{ fix i j :: nat assume asm: "i \<noteq> j" "i < n" "j < n"
{ assume iasm: "?set i = {}"
have "card (?set i) = 2"
using card_payer_and_inversion[OF assms `i < n`] by auto
hence "False"
using iasm by auto }
then obtain c where ci: "c \<in> ?set i" by blast
hence cj: "c \<notin> ?set j" using asm by auto
{ assume "?set i = ?set j"
hence "False" using ci cj by auto }
hence "?set i \<noteq> ?set j" by auto }
hence "inj_on ?set {..< n}" unfolding inj_on_def by auto
from card_image[OF this]
have "card (?set ` {..< n}) = n" by auto
hence "card ?sets = n" using sets by auto
thus ?thesis using eq_Union card_double by auto
qed
lemma card_dc_crypto:
"card dc_crypto = n * 2^n"
unfolding dc_crypto
using card_list_length[of "UNIV :: bool set"]
by (simp add: card_cartesian_product card_image)
lemma card_image_inversion:
"card (inversion ` dc_crypto) = 2^(n - 1)"
proof -
let ?P = "{inversion -` {x} \<inter> dc_crypto |x. x \<in> inversion ` dc_crypto}"
have "\<Union>?P = dc_crypto" by auto
{ fix a b assume *: "(a, b) \<in> dc_crypto"
have inv_SOME: "inversion (SOME x. inversion x = inversion (a, b) \<and> x \<in> dc_crypto) = inversion (a, b)"
apply (rule someI2)
by (auto simp: *) }
note inv_SOME = this
{ fix a b assume *: "(a, b) \<in> dc_crypto"
have "(SOME x. inversion x = inversion (a, b) \<and> x \<in> dc_crypto) \<in> dc_crypto"
by (rule someI2) (auto simp: *) }
note SOME_inv_dc = this
have "bij_betw (\<lambda>s. inversion (SOME x. x \<in> s \<and> x \<in> dc_crypto))
{inversion -` {x} \<inter> dc_crypto |x. x \<in> inversion ` dc_crypto}
(inversion ` dc_crypto)"
unfolding bij_betw_def
by (auto intro!: inj_onI image_eqI simp: inv_SOME SOME_inv_dc)
hence card_eq: "card {inversion -` {x} \<inter> dc_crypto |x. x \<in> inversion ` dc_crypto} = card (inversion ` dc_crypto)"
by (rule bij_betw_same_card)
have "(2*n) * card (inversion ` dc_crypto) = card (\<Union>?P)"
unfolding card_eq[symmetric]
proof (rule card_partition)
have "\<Union>?P \<subseteq> dc_crypto" by auto
thus "finite (\<Union>?P)" using finite_dc_crypto by (auto intro: finite_subset)
have "?P = (\<lambda>x. inversion -` {x} \<inter> dc_crypto) ` (inversion ` dc_crypto)"
by auto
thus "finite ?P" using finite_dc_crypto by auto
next
fix c assume "c \<in> {inversion -` {x} \<inter> dc_crypto |x. x \<in> inversion ` dc_crypto}"
then obtain x where "c = inversion -` {x} \<inter> dc_crypto" and x: "x \<in> inversion ` dc_crypto" by auto
hence "c = {dc \<in> dc_crypto. inversion dc = x}" by auto
thus "card c = 2 * n" using card_inversion[OF x] by simp
next
fix x y assume "x \<in> ?P" "y \<in> ?P" and "x \<noteq> y"
then obtain i j where
x: "x = inversion -` {i} \<inter> dc_crypto" and i: "i \<in> inversion ` dc_crypto" and
y: "y = inversion -` {j} \<inter> dc_crypto" and j: "j \<in> inversion ` dc_crypto" by auto
show "x \<inter> y = {}" using x y `x \<noteq> y` by auto
qed
hence "2 * card (inversion ` dc_crypto) = 2 ^ n" unfolding `\<Union>?P = dc_crypto` card_dc_crypto
using n_gt_3 by auto
thus ?thesis by (cases n) auto
qed
end
sublocale
dining_cryptographers_space \<subseteq> finite_space "dc_crypto"
proof
show "finite dc_crypto" using finite_dc_crypto .
show "dc_crypto \<noteq> {}"
unfolding dc_crypto
using n_gt_3 by (auto intro: exI[of _ "replicate n True"])
qed
notation (in dining_cryptographers_space)
mutual_information_Pow ("\<I>'( _ ; _ ')")
notation (in dining_cryptographers_space)
entropy_Pow ("\<H>'( _ ')")
notation (in dining_cryptographers_space)
conditional_entropy_Pow ("\<H>'( _ | _ ')")
theorem (in dining_cryptographers_space)
"\<I>( inversion ; payer ) = 0"
proof -
have n: "0 < n" using n_gt_3 by auto
have lists: "{xs. length xs = n} \<noteq> {}" using Ex_list_of_length by auto
have card_image_inversion:
"real (card (inversion ` dc_crypto)) = 2^n / 2"
unfolding card_image_inversion using `0 < n` by (cases n) auto
let ?dIP = "distribution (\<lambda>x. (inversion x, payer x))"
let ?dP = "distribution payer"
let ?dI = "distribution inversion"
{ have "\<H>(inversion|payer) =
- (\<Sum>x\<in>inversion`dc_crypto. (\<Sum>z\<in>Some ` {0..<n}. ?dIP {(x, z)} * log 2 (?dIP {(x, z)} / ?dP {z})))"
unfolding conditional_entropy_eq[OF simple_function_finite simple_function_finite]
by (simp add: image_payer_dc_crypto setsum_Sigma)
also have "... =
- (\<Sum>x\<in>inversion`dc_crypto. (\<Sum>z\<in>Some ` {0..<n}. 2 / (real n * 2^n) * (1 - real n)))"
unfolding neg_equal_iff_equal
proof (rule setsum_cong[OF refl], rule setsum_cong[OF refl])
fix x z assume x: "x \<in> inversion`dc_crypto" and z: "z \<in> Some ` {0..<n}"
hence "(\<lambda>x. (inversion x, payer x)) -` {(x, z)} \<inter> dc_crypto =
{dc \<in> dc_crypto. payer dc = Some (the z) \<and> inversion dc = x}"
by (auto simp add: payer_def)
moreover from x z obtain i where "z = Some i" and "i < n" by auto
moreover from x have "length x = n" by (auto simp: inversion_def_raw dc_crypto)
ultimately
have "?dIP {(x, z)} = 2 / (real n * 2^n)" using x
by (auto simp add: card_dc_crypto card_payer_and_inversion distribution_def)
moreover
from z have "payer -` {z} \<inter> dc_crypto = {z} \<times> {xs. length xs = n}"
by (auto simp: dc_crypto payer_def)
hence "card (payer -` {z} \<inter> dc_crypto) = 2^n"
using card_list_length[where A="UNIV::bool set"]
by (simp add: card_cartesian_product_singleton)
hence "?dP {z} = 1 / real n"
by (simp add: distribution_def card_dc_crypto)
ultimately
show "?dIP {(x,z)} * log 2 (?dIP {(x,z)} / ?dP {z}) =
2 / (real n * 2^n) * (1 - real n)"
by (simp add: log_divide log_nat_power[of 2])
qed
also have "... = real n - 1"
using n finite_space
by (simp add: card_image_inversion card_image[OF inj_Some] field_simps real_eq_of_nat[symmetric])
finally have "\<H>(inversion|payer) = real n - 1" . }
moreover
{ have "\<H>(inversion) = - (\<Sum>x \<in> inversion`dc_crypto. ?dI {x} * log 2 (?dI {x}))"
unfolding entropy_eq[OF simple_function_finite] by simp
also have "... = - (\<Sum>x \<in> inversion`dc_crypto. 2 * (1 - real n) / 2^n)"
unfolding neg_equal_iff_equal
proof (rule setsum_cong[OF refl])
fix x assume x_inv: "x \<in> inversion ` dc_crypto"
hence "length x = n" by (auto simp: inversion_def_raw dc_crypto)
moreover have "inversion -` {x} \<inter> dc_crypto = {dc \<in> dc_crypto. inversion dc = x}" by auto
ultimately have "?dI {x} = 2 / 2^n" using `0 < n`
by (auto simp: card_inversion[OF x_inv] card_dc_crypto distribution_def)
thus "?dI {x} * log 2 (?dI {x}) = 2 * (1 - real n) / 2^n"
by (simp add: log_divide log_nat_power power_le_zero_eq inverse_eq_divide)
qed
also have "... = real n - 1"
by (simp add: card_image_inversion real_of_nat_def[symmetric] field_simps)
finally have "\<H>(inversion) = real n - 1" .
}
ultimately show ?thesis
unfolding mutual_information_eq_entropy_conditional_entropy[OF simple_function_finite simple_function_finite]
by simp
qed
end