Theory Stirling

(*  Author:     Amine Chaieb
    Author:     Florian Haftmann
    Author:     Lukas Bulwahn
    Author:     Manuel Eberl
*)

section ‹Stirling numbers of first and second kind›

theory Stirling
imports Main
begin

subsection ‹Stirling numbers of the second kind›

fun Stirling :: "nat ⇒ nat ⇒ nat"
  where
    "Stirling 0 0 = 1"
  | "Stirling 0 (Suc k) = 0"
  | "Stirling (Suc n) 0 = 0"
  | "Stirling (Suc n) (Suc k) = Suc k * Stirling n (Suc k) + Stirling n k"

lemma Stirling_1 [simp]: "Stirling (Suc n) (Suc 0) = 1"
  by (induct n) simp_all

lemma Stirling_less [simp]: "n < k ⟹ Stirling n k = 0"
  by (induct n k rule: Stirling.induct) simp_all

lemma Stirling_same [simp]: "Stirling n n = 1"
  by (induct n) simp_all

lemma Stirling_2_2: "Stirling (Suc (Suc n)) (Suc (Suc 0)) = 2 ^ Suc n - 1"
proof (induct n)
  case 0
  then show ?case by simp
next
  case (Suc n)
  have "Stirling (Suc (Suc (Suc n))) (Suc (Suc 0)) =
      2 * Stirling (Suc (Suc n)) (Suc (Suc 0)) + Stirling (Suc (Suc n)) (Suc 0)"
    by simp
  also have "… = 2 * (2 ^ Suc n - 1) + 1"
    by (simp only: Suc Stirling_1)
  also have "… = 2 ^ Suc (Suc n) - 1"
  proof -
    have "(2::nat) ^ Suc n - 1 > 0"
      by (induct n) simp_all
    then have "2 * ((2::nat) ^ Suc n - 1) > 0"
      by simp
    then have "2 ≤ 2 * ((2::nat) ^ Suc n)"
      by simp
    with add_diff_assoc2 [of 2 "2 * 2 ^ Suc n" 1]
    have "2 * 2 ^ Suc n - 2 + (1::nat) = 2 * 2 ^ Suc n + 1 - 2" .
    then show ?thesis
      by (simp add: nat_distrib)
  qed
  finally show ?case by simp
qed

lemma Stirling_2: "Stirling (Suc n) (Suc (Suc 0)) = 2 ^ n - 1"
  using Stirling_2_2 by (cases n) simp_all


subsection ‹Stirling numbers of the first kind›

fun stirling :: "nat ⇒ nat ⇒ nat"
  where
    "stirling 0 0 = 1"
  | "stirling 0 (Suc k) = 0"
  | "stirling (Suc n) 0 = 0"
  | "stirling (Suc n) (Suc k) = n * stirling n (Suc k) + stirling n k"

lemma stirling_0 [simp]: "n > 0 ⟹ stirling n 0 = 0"
  by (cases n) simp_all

lemma stirling_less [simp]: "n < k ⟹ stirling n k = 0"
  by (induct n k rule: stirling.induct) simp_all

lemma stirling_same [simp]: "stirling n n = 1"
  by (induct n) simp_all

lemma stirling_Suc_n_1: "stirling (Suc n) (Suc 0) = fact n"
  by (induct n) auto

lemma stirling_Suc_n_n: "stirling (Suc n) n = Suc n choose 2"
  by (induct n) (auto simp add: numerals(2))

lemma stirling_Suc_n_2:
  assumes "n ≥ Suc 0"
  shows "stirling (Suc n) 2 = (∑k=1..n. fact n div k)"
  using assms
proof (induct n)
  case 0
  then show ?case by simp
next
  case (Suc n)
  show ?case
  proof (cases n)
    case 0
    then show ?thesis
      by (simp add: numerals(2))
  next
    case Suc
    then have geq1: "Suc 0 ≤ n"
      by simp
    have "stirling (Suc (Suc n)) 2 = Suc n * stirling (Suc n) 2 + stirling (Suc n) (Suc 0)"
      by (simp only: stirling.simps(4)[of "Suc n"] numerals(2))
    also have "… = Suc n * (∑k=1..n. fact n div k) + fact n"
      using Suc.hyps[OF geq1]
      by (simp only: stirling_Suc_n_1 of_nat_fact of_nat_add of_nat_mult)
    also have "… = Suc n * (∑k=1..n. fact n div k) + Suc n * fact n div Suc n"
      by (metis nat.distinct(1) nonzero_mult_div_cancel_left)
    also have "… = (∑k=1..n. fact (Suc n) div k) + fact (Suc n) div Suc n"
      by (simp add: sum_distrib_left div_mult_swap dvd_fact)
    also have "… = (∑k=1..Suc n. fact (Suc n) div k)"
      by simp
    finally show ?thesis .
  qed
qed

lemma of_nat_stirling_Suc_n_2:
  assumes "n ≥ Suc 0"
  shows "(of_nat (stirling (Suc n) 2)::'a::field_char_0) = fact n * (∑k=1..n. (1 / of_nat k))"
  using assms
proof (induct n)
  case 0
  then show ?case by simp
next
  case (Suc n)
  show ?case
  proof (cases n)
    case 0
    then show ?thesis
      by (auto simp add: numerals(2))
  next
    case Suc
    then have geq1: "Suc 0 ≤ n"
      by simp
    have "(of_nat (stirling (Suc (Suc n)) 2)::'a) =
        of_nat (Suc n * stirling (Suc n) 2 + stirling (Suc n) (Suc 0))"
      by (simp only: stirling.simps(4)[of "Suc n"] numerals(2))
    also have "… = of_nat (Suc n) * (fact n * (∑k = 1..n. 1 / of_nat k)) + fact n"
      using Suc.hyps[OF geq1]
      by (simp only: stirling_Suc_n_1 of_nat_fact of_nat_add of_nat_mult)
    also have "… = fact (Suc n) * (∑k = 1..n. 1 / of_nat k) + fact (Suc n) * (1 / of_nat (Suc n))"
      using of_nat_neq_0 by auto
    also have "… = fact (Suc n) * (∑k = 1..Suc n. 1 / of_nat k)"
      by (simp add: distrib_left)
    finally show ?thesis .
  qed
qed

lemma sum_stirling: "(∑k≤n. stirling n k) = fact n"
proof (induct n)
  case 0
  then show ?case by simp
next
  case (Suc n)
  have "(∑k≤Suc n. stirling (Suc n) k) = stirling (Suc n) 0 + (∑k≤n. stirling (Suc n) (Suc k))"
    by (simp only: sum.atMost_Suc_shift)
  also have "… = (∑k≤n. stirling (Suc n) (Suc k))"
    by simp
  also have "… = (∑k≤n. n * stirling n (Suc k) + stirling n k)"
    by simp
  also have "… = n * (∑k≤n. stirling n (Suc k)) + (∑k≤n. stirling n k)"
    by (simp add: sum.distrib sum_distrib_left)
  also have "… = n * fact n + fact n"
  proof -
    have "n * (∑k≤n. stirling n (Suc k)) = n * ((∑k≤Suc n. stirling n k) - stirling n 0)"
      by (metis add_diff_cancel_left' sum.atMost_Suc_shift)
    also have "… = n * (∑k≤n. stirling n k)"
      by (cases n) simp_all
    also have "… = n * fact n"
      using Suc.hyps by simp
    finally have "n * (∑k≤n. stirling n (Suc k)) = n * fact n" .
    moreover have "(∑k≤n. stirling n k) = fact n"
      using Suc.hyps .
    ultimately show ?thesis by simp
  qed
  also have "… = fact (Suc n)" by simp
  finally show ?case .
qed

lemma stirling_pochhammer:
  "(∑k≤n. of_nat (stirling n k) * x ^ k) = (pochhammer x n :: 'a::comm_semiring_1)"
proof (induct n)
  case 0
  then show ?case by simp
next
  case (Suc n)
  have "of_nat (n * stirling n 0) = (0 :: 'a)" by (cases n) simp_all
  then have "(∑k≤Suc n. of_nat (stirling (Suc n) k) * x ^ k) =
      (of_nat (n * stirling n 0) * x ^ 0 +
      (∑i≤n. of_nat (n * stirling n (Suc i)) * (x ^ Suc i))) +
      (∑i≤n. of_nat (stirling n i) * (x ^ Suc i))"
    by (subst sum.atMost_Suc_shift) (simp add: sum.distrib ring_distribs)
  also have "… = pochhammer x (Suc n)"
    by (subst sum.atMost_Suc_shift [symmetric])
      (simp add: algebra_simps sum.distrib sum_distrib_left pochhammer_Suc flip: Suc)
  finally show ?case .
qed


text ‹A row of the Stirling number triangle›

definition stirling_row :: "nat ⇒ nat list"
  where "stirling_row n = [stirling n k. k ← [0..<Suc n]]"

lemma nth_stirling_row: "k ≤ n ⟹ stirling_row n ! k = stirling n k"
  by (simp add: stirling_row_def del: upt_Suc)

lemma length_stirling_row [simp]: "length (stirling_row n) = Suc n"
  by (simp add: stirling_row_def)

lemma stirling_row_nonempty [simp]: "stirling_row n ≠ []"
  using length_stirling_row[of n] by (auto simp del: length_stirling_row)


subsubsection ‹Efficient code›

text ‹
  Naively using the defining equations of the Stirling numbers of the first
  kind to compute them leads to exponential run time due to repeated
  computations. We can use memoisation to compute them row by row without
  repeating computations, at the cost of computing a few unneeded values.

  As a bonus, this is very efficient for applications where an entire row of
  Stirling numbers is needed.
›

definition zip_with_prev :: "('a ⇒ 'a ⇒ 'b) ⇒ 'a ⇒ 'a list ⇒ 'b list"
  where "zip_with_prev f x xs = map2 f (x # xs) xs"

lemma zip_with_prev_altdef:
  "zip_with_prev f x xs =
    (if xs = [] then [] else f x (hd xs) # [f (xs!i) (xs!(i+1)). i ← [0..<length xs - 1]])"
proof (cases xs)
  case Nil
  then show ?thesis
    by (simp add: zip_with_prev_def)
next
  case (Cons y ys)
  then have "zip_with_prev f x xs = f x (hd xs) # zip_with_prev f y ys"
    by (simp add: zip_with_prev_def)
  also have "zip_with_prev f y ys = map (λi. f (xs ! i) (xs ! (i + 1))) [0..<length xs - 1]"
    unfolding Cons
    by (induct ys arbitrary: y)
      (simp_all add: zip_with_prev_def upt_conv_Cons flip: map_Suc_upt del: upt_Suc)
  finally show ?thesis
    using Cons by simp
qed


primrec stirling_row_aux
  where
    "stirling_row_aux n y [] = [1]"
  | "stirling_row_aux n y (x#xs) = (y + n * x) # stirling_row_aux n x xs"

lemma stirling_row_aux_correct:
  "stirling_row_aux n y xs = zip_with_prev (λa b. a + n * b) y xs @ [1]"
  by (induct xs arbitrary: y) (simp_all add: zip_with_prev_def)

lemma stirling_row_code [code]:
  "stirling_row 0 = [1]"
  "stirling_row (Suc n) = stirling_row_aux n 0 (stirling_row n)"
proof goal_cases
  case 1
  show ?case by (simp add: stirling_row_def)
next
  case 2
  have "stirling_row (Suc n) =
    0 # [stirling_row n ! i + stirling_row n ! (i+1) * n. i ← [0..<n]] @ [1]"
  proof (rule nth_equalityI, goal_cases length nth)
    case (nth i)
    from nth have "i ≤ Suc n"
      by simp
    then consider "i = 0 ∨ i = Suc n" | "i > 0" "i ≤ n"
      by linarith
    then show ?case
    proof cases
      case 1
      then show ?thesis
        by (auto simp: nth_stirling_row nth_append)
    next
      case 2
      then show ?thesis
        by (cases i) (simp_all add: nth_append nth_stirling_row)
    qed
  next
    case length
    then show ?case by simp
  qed
  also have "0 # [stirling_row n ! i + stirling_row n ! (i+1) * n. i ← [0..<n]] @ [1] =
      zip_with_prev (λa b. a + n * b) 0 (stirling_row n) @ [1]"
    by (cases n) (auto simp add: zip_with_prev_altdef stirling_row_def hd_map simp del: upt_Suc)
  also have "… = stirling_row_aux n 0 (stirling_row n)"
    by (simp add: stirling_row_aux_correct)
  finally show ?case .
qed

lemma stirling_code [code]:
  "stirling n k =
    (if k = 0 then (if n = 0 then 1 else 0)
     else if k > n then 0
     else if k = n then 1
     else stirling_row n ! k)"
  by (simp add: nth_stirling_row)

end