# Theory Dining_Cryptographers

theory Dining_Cryptographers
imports "HOL-Probability.Information"
begin

lemma image_ex1_eq: "inj_on f A ⟹ (b ∈ f  A) ⟷ (∃!x ∈ A. b = f x)"
by (unfold inj_on_def) blast

lemma Ex1_eq: "∃!x. P x ⟹ P x ⟹ P y ⟹ x = y"
by auto

subsection ‹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 ≥ 3"
begin

definition "dining_cryptographers =
({None} ∪ Some  {0..<n}) × {xs :: bool list. length xs = n}"
definition "payer dc = fst dc"
definition coin :: "(nat option × bool list) ⇒ nat ⇒ bool" where
"coin dc c = snd dc ! (c mod n)"
definition "inversion dc =
map (λc. (payer dc = Some c) ≠ (coin dc c ≠ coin dc (c + 1))) [0..<n]"

definition "result dc = foldl (λ a b. a ≠ b) False (inversion dc)"

lemma coin_n[simp]: "coin dc n = coin dc 0"
unfolding coin_def by simp

theorem correctness:
assumes "dc ∈ dining_cryptographers"
shows "result dc ⟷ (payer dc ≠ None)"
proof -
let ?XOR = "λf l. foldl (≠) False (map f [0..<l])"

have foldl_coin:
"¬ ?XOR (λc. coin dc c ≠ coin dc (c + 1)) n"
proof -
define n' where "n' = n" ― ‹Need to hide n, as it is hidden in coin›
have "?XOR (λc. coin dc c ≠ coin dc (c + 1)) n'
= (coin dc 0 ≠ coin dc n')"
by (induct n') auto
thus ?thesis using ‹n' ≡ n› by simp
qed

from assms have "payer dc = None ∨ (∃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 "∃k<n. payer dc = Some k"
then obtain k where "k < n" and "payer dc = Some k" by auto
define l where "l = n" ― ‹Need to hide n, as it is hidden in coin, payer etc.›
have "?XOR (λc. (payer dc = Some c) ≠ (coin dc c ≠ coin dc (c + 1))) l =
((k < l) ≠ ?XOR (λc. (coin dc c ≠ 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}×UNIV"

lemma dc_crypto: "dc_crypto = Some  {0..<n} × {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} ≠ {}"
by (auto intro!: exI[of _ "replicate n undefined"])
show ?thesis
unfolding payer_def [abs_def] dc_crypto fst_image_times if_not_P[OF *] ..
qed

lemma card_payer_and_inversion:
assumes "xs ∈ inversion  dc_crypto" and "i < n"
shows "card {dc ∈ dc_crypto. payer dc = Some i ∧ 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) ∈ 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 (λx. inversion (Some i, x)) {ys. ys ! 0 = b ∧ length ys = length xs}"
proof (rule inj_onI)
fix x y
assume "x ∈ {ys. ys ! 0 = b ∧ length ys = length xs}"
and "y ∈ {ys. ys ! 0 = b ∧ 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 *: "⋀j. j < n ⟹
(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)
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)
hence "j < n" by simp
with Suc 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›.
›

define zs where "zs = map (λp. if p ∈ {min i j<..max i j} then ¬ 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 "⋀l. l < max i j ⟹ Suc l mod n = Suc l"
using ‹i < n› ‹j < n› by auto
{ fix l assume "l < n"
hence "(((l < min i j ∨ l = min i j) ∨ (min i j < l ∧ l < max i j)) ∨ l = max i j) ∨ 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 ≠ i" and "l ≠ 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 ≠ j" and "Suc l ≤ max i j" using ‹l = min i j› using ‹j < n› by auto
hence "zs ! l = ys ! l" and "zs ! (Suc l mod n) = (¬ ys ! (Suc l mod n))"
using ‹l = min i j›[symmetric] by (simp_all add: zs_def)
thus ?thesis using ‹l = i› ‹i ≠ j› by simp
next
assume "j < i"
hence "l = j" and "Suc l < n" and "i ≠ j" and "Suc l ≤ max i j" using ‹l = min i j› using ‹i < n› by auto
hence "zs ! l = ys ! l" and "zs ! (Suc l mod n) = (¬ ys ! (Suc l mod n))"
using ‹l = min i j›[symmetric] by (simp_all add: zs_def)
thus ?thesis using ‹l = j› ‹i ≠ 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 ∧ l < max i j"
hence "i ≠ l" and "j ≠ l" and "zs ! l = (¬ ys ! l)"
"zs ! (Suc l mod n) = (¬ 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 ≠ 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 = (¬ ys ! l)"
using ‹j < n› ‹i < j› by (auto simp add: ‹l = j› zs_def)
ultimately show ?thesis using ‹l = j› ‹i ≠ j› by simp
next
assume "j < i"
hence "l = i" and "i ≠ 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 = (¬ ys ! l)"
using ‹i < n› ‹j < i› by (auto simp add: ‹l = i› zs_def)
ultimately show ?thesis using ‹l = i› ‹i ≠ 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 ≠ l" and "i ≠ 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 ≠ l› ‹i ≠ l› by auto
qed }
hence zs: "inversion (Some i, zs) = xs"
by (simp add: xs_inv[symmetric] inversion_def coin_def payer_def)
moreover
from zs have Not_zs: "inversion (Some i, (map Not zs)) = xs"
by (simp add: xs_inv[symmetric] inversion_def coin_def payer_def)
ultimately
have "{dc ∈ dc_crypto. payer dc = Some i ∧ inversion dc = xs} =
{(Some i, zs), (Some i, map Not zs)}"
using ‹i < n› [[ hypsubst_thin = true ]]
fix b assume [simp]: "length b = n"
and *: "inversion (Some i, b) = xs" and "b ≠ zs"
show "b = map Not zs"
proof (cases "b ! 0 = zs ! 0")
case True
hence zs: "zs ∈ {ys. ys ! 0 = b ! 0 ∧ length ys = length xs} ∧ xs = inversion (Some i, zs)"
using zs by simp
have b: "b ∈ {ys. ys ! 0 = b ! 0 ∧ length ys = length xs} ∧ xs = inversion (Some i, b)"
using * by simp
hence "b ∈ {ys. ys ! 0 = b ! 0 ∧ length ys = length xs}" ..
with *[symmetric] have "xs ∈ (λx. inversion (Some i, x))  {ys. ys ! 0 = b ! 0 ∧ 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 ≠ zs› by simp
next
case False
hence zs: "map Not zs ∈ {ys. ys ! 0 = b ! 0 ∧ length ys = length xs} ∧ xs = inversion (Some i, map Not zs)"
using Not_zs by (simp add: nth_map[OF ‹0 < length zs›])
have b: "b ∈ {ys. ys ! 0 = b ! 0 ∧ length ys = length xs} ∧ xs = inversion (Some i, b)"
using * by simp
hence "b ∈ {ys. ys ! 0 = b ! 0 ∧ length ys = length xs}" ..
with *[symmetric] have "xs ∈ (λx. inversion (Some i, x))  {ys. ys ! 0 = b ! 0 ∧ 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 ≠ 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_length_eq[where A="UNIV :: bool set"]
unfolding dc_crypto by simp

lemma card_inversion:
assumes "xs ∈ inversion  dc_crypto"
shows "card {dc ∈ dc_crypto. inversion dc = xs} = 2 * n"
proof -
let ?set = "λi. {dc ∈ dc_crypto. payer dc = Some i ∧ inversion dc = xs}"
let ?sets = "{?set i | i. i < n}"

have [simp]: "length xs = n" using assms
by (auto simp: dc_crypto inversion_def [abs_def])

have "{dc ∈ dc_crypto. inversion dc = xs} = (⋃i < n. ?set i)"
unfolding dc_crypto payer_def by auto
also have "… = (⋃?sets)" by auto
finally have eq_Union: "{dc ∈ dc_crypto. inversion dc = xs} = (⋃?sets)" by simp

have card_double: "2 * card ?sets = card (⋃?sets)"
proof (rule card_partition)
show "finite ?sets" by simp
{ fix i assume "i < n"
have "?set i ⊆ dc_crypto" by auto
have "finite (?set i)" using finite_dc_crypto by auto }
thus "finite (⋃?sets)" by auto

next
fix c assume "c ∈ ?sets"
thus "card c = 2" using card_payer_and_inversion[OF assms] by auto

next
fix x y assume "x ∈ ?sets" and "y ∈ ?sets" "x ≠ y"
then obtain i j where xy: "x = ?set i" "y = ?set j" by auto
hence "i ≠ j" using ‹x ≠ y› by auto
thus "x ∩ y = {}" using xy by auto
qed

have sets: "?sets = ?set  {..< n}"
unfolding image_def by auto
{ fix i j :: nat assume asm: "i ≠ 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 ∈ ?set i" by blast
hence cj: "c ∉ ?set j" using asm by auto
{ assume "?set i = ?set j"
hence "False" using ci cj by auto }
hence "?set i ≠ ?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_lists_length_eq[of "UNIV :: bool set"]

lemma card_image_inversion:
"card (inversion  dc_crypto) = 2^(n - 1)"
proof -
let ?P = "{inversion - {x} ∩ dc_crypto |x. x ∈ inversion  dc_crypto}"
have "⋃?P = dc_crypto" by auto

{ fix a b assume *: "(a, b) ∈ dc_crypto"
have inv_SOME: "inversion (SOME x. inversion x = inversion (a, b) ∧ x ∈ dc_crypto) = inversion (a, b)"
apply (rule someI2)
by (auto simp: *) }
note inv_SOME = this

{ fix a b assume *: "(a, b) ∈ dc_crypto"
have "(SOME x. inversion x = inversion (a, b) ∧ x ∈ dc_crypto) ∈ dc_crypto"
by (rule someI2) (auto simp: *) }
note SOME_inv_dc = this

have "bij_betw (λs. inversion (SOME x. x ∈ s ∧ x ∈ dc_crypto))
{inversion - {x} ∩ dc_crypto |x. x ∈ 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} ∩ dc_crypto |x. x ∈ inversion  dc_crypto} = card (inversion  dc_crypto)"
by (rule bij_betw_same_card)

have "(2*n) * card (inversion  dc_crypto) = card (⋃?P)"
unfolding card_eq[symmetric]
proof (rule card_partition)
have "⋃?P ⊆ dc_crypto" by auto
thus "finite (⋃?P)" using finite_dc_crypto by (auto intro: finite_subset)

have "?P = (λx. inversion - {x} ∩ dc_crypto)  (inversion  dc_crypto)"
by auto
thus "finite ?P" using finite_dc_crypto by auto

next
fix c assume "c ∈ {inversion - {x} ∩ dc_crypto |x. x ∈ inversion  dc_crypto}"
then obtain x where "c = inversion - {x} ∩ dc_crypto" and x: "x ∈ inversion  dc_crypto" by auto
hence "c = {dc ∈ 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 ∈ ?P" "y ∈ ?P" and "x ≠ y"
then obtain i j where
x: "x = inversion - {i} ∩ dc_crypto" and i: "i ∈ inversion  dc_crypto" and
y: "y = inversion - {j} ∩ dc_crypto" and j: "j ∈ inversion  dc_crypto" by auto
show "x ∩ y = {}" using x y ‹x ≠ y› by auto
qed
hence "2 * card (inversion  dc_crypto) = 2 ^ n" unfolding ‹⋃?P = dc_crypto› card_dc_crypto
using n_gt_3 by auto
thus ?thesis by (cases n) auto
qed

end

sublocale dining_cryptographers_space ⊆ prob_space "uniform_count_measure dc_crypto"
by (rule prob_space_uniform_count_measure[OF finite_dc_crypto])
(insert n_gt_3, auto simp: dc_crypto intro: exI[of _ "replicate n True"])

sublocale dining_cryptographers_space ⊆ information_space "uniform_count_measure dc_crypto" 2
by standard auto

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 (rule mutual_information_eq_0_simple)
have n: "0 < n" using n_gt_3 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

show inversion: "simple_distributed (uniform_count_measure dc_crypto) inversion (λx. 2 / 2^n)"
proof (rule simple_distributedI)
show "simple_function (uniform_count_measure dc_crypto) inversion"
using finite_dc_crypto
by (auto simp: simple_function_def space_uniform_count_measure sets_uniform_count_measure)
fix x assume "x ∈ inversion  space (uniform_count_measure dc_crypto)"
moreover have "inversion - {x} ∩ dc_crypto = {dc ∈ dc_crypto. inversion dc = x}" by auto
ultimately show "2 / 2^n = prob (inversion - {x} ∩ space (uniform_count_measure dc_crypto))"
using ‹0 < n›
by (simp add: card_inversion card_dc_crypto finite_dc_crypto
subset_eq space_uniform_count_measure measure_uniform_count_measure)
qed simp

show "simple_distributed (uniform_count_measure dc_crypto) payer (λx. 1 / real n)"
proof (rule simple_distributedI)
show "simple_function (uniform_count_measure dc_crypto) payer"
using finite_dc_crypto
by (auto simp: simple_function_def space_uniform_count_measure sets_uniform_count_measure)
fix z assume "z ∈ payer  space (uniform_count_measure dc_crypto)"
then have "payer - {z} ∩ dc_crypto = {z} × {xs. length xs = n}"
by (auto simp: dc_crypto payer_def space_uniform_count_measure cong del: image_cong_simp)
hence "card (payer - {z} ∩ dc_crypto) = 2^n"
using card_lists_length_eq[where A="UNIV::bool set"]
then show "1 / real n = prob (payer - {z} ∩ space (uniform_count_measure dc_crypto))"
using finite_dc_crypto
by (subst measure_uniform_count_measure) (auto simp add: card_dc_crypto space_uniform_count_measure)
qed simp

show "simple_distributed (uniform_count_measure dc_crypto) (λx. (inversion x, payer x)) (λx. 2 / (real n *2^n))"
proof (rule simple_distributedI)
show "simple_function (uniform_count_measure dc_crypto) (λx. (inversion x, payer x))"
using finite_dc_crypto
by (auto simp: simple_function_def space_uniform_count_measure sets_uniform_count_measure)
fix x assume "x ∈ (λx. (inversion x, payer x))  space (uniform_count_measure dc_crypto)"
then obtain i xs where x: "x = (inversion (Some i, xs), payer (Some i, xs))"
and "i < n" "length xs = n"
by (simp add: image_iff space_uniform_count_measure dc_crypto Bex_def) blast
then have xs: "inversion (Some i, xs) ∈ inversiondc_crypto" and i: "Some i ∈ Some  {0..<n}"
and x: "x = (inversion (Some i, xs), Some i)" by (simp_all add: payer_def dc_crypto)
moreover define ys where "ys = inversion (Some i, xs)"
ultimately have ys: "ys ∈ inversiondc_crypto"
and "Some i ∈ Some  {0..<n}" "x = (ys, Some i)" by simp_all
then have "(λx. (inversion x, payer x)) - {x} ∩ space (uniform_count_measure dc_crypto) =
{dc ∈ dc_crypto. payer dc = Some (the (Some i)) ∧ inversion dc = ys}"
by (auto simp add: payer_def space_uniform_count_measure)
then show "2 / (real n * 2 ^ n) = prob ((λx. (inversion x, payer x)) - {x} ∩ space (uniform_count_measure dc_crypto))"
using ‹i < n› ys
by (simp add: measure_uniform_count_measure card_payer_and_inversion finite_dc_crypto subset_eq card_dc_crypto)
qed simp

show "∀x∈space (uniform_count_measure dc_crypto). 2 / (real n * 2 ^ n) = 2 / 2 ^ n * (1 / real n) "
by simp
qed

end