119 lines
4 KiB
Text
119 lines
4 KiB
Text
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/-
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Copyright (c) 2015 Jeremy Avigad. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Jeremy Avigad
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Binomial coefficients, "n choose k".
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-/
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import data.nat.div data.nat.fact
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open decidable
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namespace nat
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/- choose -/
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definition choose : ℕ → ℕ → ℕ
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| 0 0 := 1
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| 0 (succ k) := 0
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| (succ n) 0 := 1
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| (succ n) (succ k) := choose n (succ k) + choose n k
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theorem choose_zero_right (n : ℕ) : choose n 0 = 1 :=
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nat.cases_on n rfl (take m, rfl)
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theorem choose_zero_succ (k : ℕ) : choose 0 (succ k) = 0 := rfl
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theorem choose_succ_succ (n k : ℕ) : choose (succ n) (succ k) = choose n (succ k) + choose n k := rfl
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theorem choose_eq_zero_of_lt {n : ℕ} : ∀{k : ℕ}, n < k → choose n k = 0 :=
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nat.induction_on n
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(take k, assume H : 0 < k,
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obtain k' (H : k = succ k'), from exists_eq_succ_of_pos H,
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by rewrite H)
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(take n',
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assume IH: ∀ k, n' < k → choose n' k = 0,
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take k,
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assume H : succ n' < k,
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obtain k' (keq : k = succ k'), from exists_eq_succ_of_lt H,
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assert H' : n' < k', by rewrite keq at H; apply lt_of_succ_lt_succ H,
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by rewrite [keq, choose_succ_succ, IH _ H', IH _ (lt.trans H' !lt_succ_self)])
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theorem choose_self (n : ℕ) : choose n n = 1 :=
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begin
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induction n with [n, ih],
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{apply rfl},
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rewrite [choose_succ_succ, ih, choose_eq_zero_of_lt !lt_succ_self]
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end
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theorem choose_succ_self (n : ℕ) : choose (succ n) n = succ n :=
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begin
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induction n with [n, ih],
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{apply rfl},
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rewrite [choose_succ_succ, ih, choose_self, add.comm]
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end
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theorem choose_one_right (n : ℕ) : choose n 1 = n :=
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begin
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induction n with [n, ih],
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{apply rfl},
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rewrite [choose_succ_succ, ih, choose_zero_right]
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end
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theorem choose_pos {n : ℕ} : ∀ {k : ℕ}, k ≤ n → choose n k > 0 :=
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begin
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induction n with [n, ih],
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{intros [k, H],
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have kz : k = 0, from eq_of_le_of_ge H !zero_le,
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rewrite [kz, choose_zero_right]; apply zero_lt_one},
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intro k,
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cases k with k,
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{intro H, rewrite [choose_zero_right], apply zero_lt_one},
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assume H : succ k ≤ succ n,
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assert H' : k ≤ n, from le_of_succ_le_succ H,
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by rewrite [choose_succ_succ]; apply add_pos_right (ih H')
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end
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-- A key identity. The proof is subtle.
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theorem succ_mul_choose_eq (n : ℕ) : ∀ k, succ n * (choose n k) = choose (succ n) (succ k) * succ k :=
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begin
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induction n with [n, ih],
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{intro k,
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cases k with k',
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{rewrite [*choose_self, one_mul, mul_one]},
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{have H : 1 < succ (succ k'), from succ_lt_succ !zero_lt_succ,
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rewrite [one_mul, choose_zero_succ, choose_eq_zero_of_lt H, zero_mul]}},
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intro k,
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cases k with k',
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{rewrite [choose_zero_right, choose_one_right]},
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rewrite [choose_succ_succ (succ n), mul.right_distrib, -ih (succ k')],
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rewrite [choose_succ_succ at {1}, mul.left_distrib, *succ_mul (succ n), mul_succ, -ih k'],
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rewrite [*add.assoc, add.left_comm (choose n _)]
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end
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theorem choose_mul_fact_mul_fact {n : ℕ} :
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∀ {k : ℕ}, k ≤ n → choose n k * fact k * fact (n - k) = fact n :=
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begin
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induction n using nat.strong_induction_on with [n, ih],
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cases n with n,
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{intro k H, have kz : k = 0, from eq_zero_of_le_zero H, rewrite [kz]},
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intro k,
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intro H, -- k ≤ n,
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cases k with k,
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{rewrite [choose_zero_right, fact_zero, *one_mul]},
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have kle : k ≤ n, from le_of_succ_le_succ H,
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show choose (succ n) (succ k) * fact (succ k) * fact (succ n - succ k) = fact (succ n), from
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begin
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rewrite [succ_sub_succ, fact_succ, -mul.assoc, -succ_mul_choose_eq],
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rewrite [fact_succ n, -ih n !lt_succ_self kle, *mul.assoc]
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end
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end
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theorem choose_def_alt {n k : ℕ} (H : k ≤ n) : choose n k = fact n div (fact k * fact (n -k)) :=
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eq.symm (div_eq_of_eq_mul_left (mul_pos !fact_pos !fact_pos)
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(by rewrite [-mul.assoc, choose_mul_fact_mul_fact H]))
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theorem fact_mul_fact_dvd_fact {n k : ℕ} (H : k ≤ n) : fact k * fact (n - k) ∣ fact n :=
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by rewrite [-choose_mul_fact_mul_fact H, mul.assoc]; apply dvd_mul_left
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end nat
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