Quelle  Ballot.thy

(*   Title: HOL/ex/Ballot.thy
     Author: Lukas Bulwahn <lukas.bulwahn-at-gmail.com>
     Author: Johannes Hölzl <hoelzl@in.tum.de>
*)

section ‹Bertrand's Ballot Theorem›

theory Ballot
imports
  Complex_Main
  "HOL-Library.FuncSet"
begin

subsection ‹Preliminaries›

lemma card_bij':
  assumes "f ∈ A → B" "∧x. x ∈ A ==> g (f x) = x"
    and "g ∈ B → A" "∧x. x ∈ B ==> f (g x) = x"
  shows "card A = card B"
  apply (rule bij_betw_same_card)
  apply (rule bij_betwI)
  apply fact+
  done

subsection ‹Formalization of Problem Statement›

subsubsection ‹Basic Definitions›

datatype vote = A | B

definition
  "all_countings a b = card {f ∈ {1 .. a + b} →_E {A, B}.
      card {x ∈ {1 .. a + b}. f x = A} = a ∧ card {x ∈ {1 .. a + b}. f x = B} = b}"

definition
  "valid_countings a b =
    card {f∈{1..a+b} →_E {A, B}.
      card {x∈{1..a+b}. f x = A} = a ∧ card {x∈{1..a+b}. f x = B} = b ∧
      (∀m∈{1..a+b}. card {x∈{1..m}. f x = A} > card {x∈{1..m}. f x = B})}"

subsubsection ‹Equivalence with Set Cardinality›

lemma Collect_on_transfer:
  assumes "rel_set R X Y"
  shows "rel_fun (rel_fun R (=)) (rel_set R) (λP. {x∈X. P x}) (λP. {y∈Y. P y})"
  using assms unfolding rel_fun_def rel_set_def by fast

lemma rel_fun_trans:
  "rel_fun P Q g g' ==> rel_fun R P f f' ==> rel_fun R Q (λx. g (f x)) (λy. g' (f' y))"
  by (auto simp: rel_fun_def)

lemma rel_fun_trans2:
  "rel_fun P1 (rel_fun P2 Q) g g' ==> rel_fun R P1 f1 f1' ==> rel_fun R P2 f2 f2' ==>
    rel_fun R Q (λx. g (f1 x) (f2 x)) (λy. g' (f1' y) (f2' y))"
  by (auto simp: rel_fun_def) 

lemma rel_fun_trans2':
  "rel_fun R (=) f1 f1' ==> rel_fun R (=) f2 f2' ==>
    rel_fun R (=) (λx. g (f1 x) (f2 x)) (λy. g (f1' y) (f2' y))"
  by (auto simp: rel_fun_def)

lemma rel_fun_const: "rel_fun R (=) (λx. a) (λy. a)"
  by auto

lemma rel_fun_conj:
  "rel_fun R (=) f f' ==> rel_fun R (=) g g' ==> rel_fun R (=) (λx. f x ∧ g x) (λy. f' y ∧ g' y)"
  by (auto simp: rel_fun_def)

lemma rel_fun_ball:
  "(∧i. i ∈ I ==> rel_fun R (=) (f i) (f' i)) ==> rel_fun R (=) (λx. ∀i∈I. f i x) (λy. ∀i∈I. f' i y)"
  by (auto simp: rel_fun_def rel_set_def)

lemma
  shows all_countings_set: "all_countings a b = card {V∈Pow {0..<a+b}. card V = a}"
      (is "_ = card ?A")
    and valid_countings_set: "valid_countings a b =
      card {V∈Pow {0..<a+b}. card V = a ∧ (∀m∈{1..a+b}. card ({0..<m} ∩ V) > m - card ({0..<m} ∩ V))}"
      (is "_ = card ?V")
proof -
  define P where "P j i ⟷ i < a + b ∧ j = Suc i" for j i
  have unique_P: "bi_unique P" and total_P: "∧m. m ≤ a + b ==> rel_set P {1..m} {0..<m}"
    by (auto simp add: bi_unique_def rel_set_def P_def Suc_le_eq gr0_conv_Suc)
  have rel_fun_P: "∧R f g. (∧i. i < a+b ==> R (f (Suc i)) (g i)) ==> rel_fun P R f g"
    by (simp add: rel_fun_def P_def)
    
  define R where "R f V ⟷
    V ⊆ {0..<a+b} ∧ f ∈ extensional {1..a+b} ∧ (∀i<a+b. i ∈ V ⟷ f (Suc i) = A)" for f V
  { fix f g :: "nat ==> vote" assume "f ∈ extensional {1..a + b}" "g ∈ extensional {1..a + b}" 
    moreover assume "∀i<a + b. (f (Suc i) = A) = (g (Suc i) = A)"
    then have "∀i<a + b. f (Suc i) = g (Suc i)"
      by (metis vote.nchotomy)
    ultimately have "f i = g i" for i
      by (cases "i ∈ {1..a+b}") (auto simp: extensional_def Suc_le_eq gr0_conv_Suc) }
  then have unique_R: "bi_unique R"
    by (auto simp: bi_unique_def R_def)

  have "f ∈ extensional {1..a + b} ==> ∃V∈Pow {0..<a + b}. R f V" for f
    by (intro bexI[of _ "{i. i < a+b ∧ f (Suc i) = A}"]) (auto simp add: R_def PiE_def)
  moreover have "V ∈ Pow {0..<a + b} ==> ∃f∈extensional {1..a+b}. R f V" for V
    by (intro bexI[of _ "λi∈{1..a+b}. if i - 1 ∈ V then A else B"]) (auto simp add: R_def PiE_def)
  ultimately have total_R: "rel_set R (extensional {1..a+b}) (Pow {0..<a+b})"
    by (auto simp: rel_set_def)

  have P: "rel_fun R (rel_fun P (=)) (λf x. f x = A) (λV y. y ∈ V)"
    by (auto simp: P_def R_def Suc_le_eq gr0_conv_Suc rel_fun_def)

  have eq_B: "x = B ⟷ x ≠ A" for x
    by (cases x; simp)

  { fix f and m :: nat
    have "card {x∈{1..m}. f x = B} = card ({1..m} - {x∈{1..m}. f x = A})"
      by (simp add: eq_B set_diff_eq cong: conj_cong)
    also have "… = m - card {x∈{1..m}. f x = A}"
      by (subst card_Diff_subset) auto
    finally have "card {x∈{1..m}. f x = B} = m - card {x∈{1..m}. f x = A}" . }
  note card_B = this

  note transfers = rel_fun_const card_transfer[THEN rel_funD, OF unique_R] rel_fun_conj rel_fun_ball
    Collect_on_transfer[THEN rel_funD, OF total_R] Collect_on_transfer[THEN rel_funD, OF total_P]
    rel_fun_trans[OF card_transfer, OF unique_P] rel_fun_trans[OF Collect_on_transfer[OF total_P]]
    rel_fun_trans2'[where g="(=)"] rel_fun_trans2'[where g="(<)"] rel_fun_trans2'[where g="(-)"]

  have "all_countings a b = card {f ∈ extensional {1..a + b}. card {x ∈ {1..a + b}. f x = A} = a}"
    using card_B by (simp add: all_countings_def PiE_iff vote.nchotomy cong: conj_cong)
  also have "… = card {V∈Pow {0..<a+b}. card ({x∈{0 ..< a + b}. x ∈ V}) = a}"
    by (intro P order_refl transfers)
  finally show "all_countings a b = card ?A"
    unfolding Int_def[symmetric] by (simp add: Int_absorb1 cong: conj_cong)

  have "valid_countings a b = card {f∈extensional {1..a+b}.
      card {x∈{1..a+b}. f x = A} = a ∧ (∀m∈{1..a+b}. card {x∈{1..m}. f x = A} > m - card {x∈{1..m}. f x = A})}"
    using card_B by (simp add: valid_countings_def PiE_iff vote.nchotomy cong: conj_cong)
  also have "… = card {V∈Pow {0..<a+b}. card {x∈{0..<a+b}. x∈V} = a ∧
    (∀m∈{1..a+b}. card {x∈{0..<m}. x∈V} > m - card {x∈{0..<m}. x∈V})}"
    by (intro P order_refl transfers) auto
  finally show "valid_countings a b = card ?V"
    unfolding Int_def[symmetric] by (simp add: Int_absorb1 cong: conj_cong)
qed

lemma all_countings: "all_countings a b = (a + b) choose a"
  unfolding all_countings_set by (simp add: n_subsets)

subsection ‹Facts About term‹valid_countings››

subsubsection ‹Non-Recursive Cases›

lemma card_V_eq_a: "V ⊆ {0..<a} ==> card V = a ⟷ V = {0..<a}"
  using card_subset_eq[of "{0..<a}" V] by auto

lemma valid_countings_a_0: "valid_countings a 0 = 1"
  by (simp add: valid_countings_set card_V_eq_a cong: conj_cong)

lemma valid_countings_eq_zero:
  "a ≤ b ==> 0 < b ==> valid_countings a b = 0"
  by (auto simp add: valid_countings_set Int_absorb1 intro!: bexI[of _ "a + b"])

lemma Ico_subset_finite: "i ⊆ {a ..< b::nat} ==> finite i"
  by (auto dest: finite_subset)

lemma Icc_Suc2: "a ≤ b ==> {a..Suc b} = insert (Suc b) {a..b}"
  by auto

lemma Ico_Suc2: "a ≤ b ==> {a..<Suc b} = insert b {a..<b}"
  by auto

lemma valid_countings_Suc_Suc:
  assumes "b < a"
  shows "valid_countings (Suc a) (Suc b) = valid_countings a (Suc b) + valid_countings (Suc a) b"
proof -
  let ?l = "Suc (a + b)"
  let ?Q = "λV c. ∀m∈{1..c}. m - card ({0..<m} ∩ V) < card ({0..<m} ∩ V)"
  let ?V = "λP. {V. (V ∈ Pow {0..<Suc ?l} ∧ P V) ∧ card V = Suc a ∧ ?Q V (Suc ?l)}"
  have "valid_countings (Suc a) (Suc b) = card (?V (λV. ?l ∉ V)) + card (?V (λV. ?l ∈ V))"
    unfolding valid_countings_set
    by (subst card_Un_disjoint[symmetric]) (auto simp add: set_eq_iff intro!: arg_cong[where f=card])
  also have "card (?V (λV. ?l ∈ V)) = valid_countings a (Suc b)"
    unfolding valid_countings_set
  proof (rule card_bij'[where f="λV. V - {?l}" and g="insert ?l"])
    have *: "∧m V. m ∈ {1..a + Suc b} ==> {0..<m} ∩ (V - {?l}) = {0..<m} ∩ V"
      by auto
    show "(λV. V - {?l}) ∈ ?V (λV. ?l ∈ V) → {V ∈ Pow {0..<a + Suc b}. card V = a ∧ ?Q V (a + Suc b)}"
      by (auto simp: Ico_subset_finite *)
    { fix V assume V: "V ⊆ {0..<?l}"
      then have "finite V" "?l ∉ V" "{0..<Suc ?l} ∩ V = V"
        by (auto dest: finite_subset)
      with V have "card (insert ?l V) = Suc (card V)"
        "card ({0..<m} ∩ insert ?l V) = (if m = Suc ?l then Suc (card V) else card ({0..<m} ∩ V))"
        if "m ≤ Suc ?l" for m
        using that by auto }
    then show "insert ?l ∈ {V ∈ Pow {0..<a + Suc b}. card V = a ∧ ?Q V (a + Suc b)} → ?V (λV. ?l ∈ V)"
      using ‹b < a› by auto
  qed auto
  also have "card (?V (λV. ?l ∉ V)) = valid_countings (Suc a) b"
    unfolding valid_countings_set
  proof (intro arg_cong[where f="λP. card {x. P x}"] ext conj_cong)
    fix V assume "V ∈ Pow {0..<Suc a + b}" and [simp]: "card V = Suc a"
    then have [simp]: "V ⊆ {0..<Suc ?l}"
      by auto
    show "?Q V (Suc ?l) = ?Q V (Suc a + b)"
      using ‹b<a\› by (simp add: Int_absorb1 Icc_Suc2)
  qed (auto simp: subset_eq less_Suc_eq)
  finally show ?thesis
    by simp
qed

lemma valid_countings:
  "(a + b) * valid_countings a b = (a - b) * ((a + b) choose a)"
proof (induct a arbitrary: b)
  case 0 show ?case
    by (cases b) (simp_all add: valid_countings_eq_zero)
next
  case (Suc a) note Suc_a = this
  show ?case
  proof (induct b)
    case (Suc b) note Suc_b = this
    show ?case
    proof cases
      assume "a ≤ b" then show ?thesis
        by (simp add: valid_countings_eq_zero)
    next
      assume "¬ a ≤ b"
      then have "b < a" by simp

      have "Suc a * (a - Suc b) + (Suc a - b) * Suc b =
        (Suc a * a - Suc a * Suc b) + (Suc a * Suc b - Suc b * b)"
        by (simp add: algebra_simps)
      also have "… = (Suc a * a + (Suc a * Suc b - Suc b * b)) - Suc a * Suc b"
        using ‹b<a\› by (intro add_diff_assoc2 mult_mono) auto
      also have "… = (Suc a * a + Suc a * Suc b) - Suc b * b - Suc a * Suc b"
        using ‹b<a\› by (intro arg_cong2[where f="(-)"] add_diff_assoc mult_mono) auto
      also have "… = (Suc a * Suc (a + b)) - (Suc b * Suc (a + b))"
        by (simp add: algebra_simps)
      finally have rearrange: "Suc a * (a - Suc b) + (Suc a - b) * Suc b = (Suc a - Suc b) * Suc (a + b)"
        unfolding diff_mult_distrib by simp

      have "(Suc a * Suc (a + b)) * ((Suc a + Suc b) * valid_countings (Suc a) (Suc b)) =
        (Suc a + Suc b) * Suc a * ((a + Suc b) * valid_countings a (Suc b) + (Suc a + b) * valid_countings (Suc a) b)"
        unfolding valid_countings_Suc_Suc[OF ‹b < a›] by (simp add: field_simps)
      also have "... = (Suc a + Suc b) * ((a - Suc b) * (Suc a * (Suc (a + b) choose a)) +
        (Suc a - b) * (Suc a * (Suc (a + b) choose Suc a)))"
        unfolding Suc_a Suc_b by (simp add: field_simps)
      also have "... = (Suc a * (a - Suc b) + (Suc a - b) * Suc b) * (Suc (Suc a + b) * (Suc a + b choose a))"
        unfolding Suc_times_binomial_add by (simp add: field_simps)
      also have "... = Suc a * (Suc a * (a - Suc b) + (Suc a - b) * Suc b) * (Suc a + Suc b choose Suc a)"
        unfolding Suc_times_binomial_eq by (simp add: field_simps)
      also have "... = (Suc a * Suc (a + b)) * ((Suc a - Suc b) * (Suc a + Suc b choose Suc a))"
        unfolding rearrange by (simp only: mult_ac)
      finally show ?thesis
        unfolding mult_cancel1 by simp
    qed
  qed (simp add: valid_countings_a_0)
qed

lemma valid_countings_eq[code]:
  "valid_countings a b = (if a + b = 0 then 1 else ((a - b) * ((a + b) choose a)) div (a + b))"
  by (simp add: valid_countings[symmetric] valid_countings_a_0)

subsection ‹Relation Between term‹valid_countings› and term‹all_countings››

lemma main_nat: "(a + b) * valid_countings a b = (a - b) * all_countings a b"
  unfolding valid_countings all_countings ..

lemma main_real:
  assumes "b < a"
  shows "valid_countings a b = (a - b) / (a + b) * all_countings a b"
using assms
proof -
  from main_nat[of a b] ‹b < a› have
    "(real a + real b) * real (valid_countings a b) = (real a - real b) * real (all_countings a b)"
    by (simp only: of_nat_add[symmetric] of_nat_mult[symmetric]) auto
  from this ‹b < a› show ?thesis
    by (subst mult_left_cancel[of "real a + real b", symmetric]) auto
qed

lemma
  "valid_countings a b = (if a ≤ b then (if b = 0 then 1 else 0) else (a - b) / (a + b) * all_countings a b)"
proof (cases "a ≤ b")
  case False
    from this show ?thesis by (simp add: main_real)
next
  case True
    from this show ?thesis
      by (auto simp add: valid_countings_a_0 all_countings valid_countings_eq_zero)
qed

subsubsection ‹Executable Definition›

declare all_countings_def [code del]
declare all_countings[code]

value "all_countings 1 0"
value "all_countings 0 1"
value "all_countings 1 1"
value "all_countings 2 1"
value "all_countings 1 2"
value "all_countings 2 4"
value "all_countings 4 2"

subsubsection ‹Executable Definition›

declare valid_countings_def [code del]

value "valid_countings 1 0"
value "valid_countings 0 1"
value "valid_countings 1 1"
value "valid_countings 2 1"
value "valid_countings 1 2"
value "valid_countings 2 4"
value "valid_countings 4 2"

end