mathlib documentation

data.nat.choose.basic

Binomial coefficients #

This file defines binomial coefficients and proves simple lemmas (i.e. those not requiring more imports).

Main definition and results #

def nat.choose  :

choose n k is the number of k-element subsets in an n-element set. Also known as binomial coefficients.

Equations
@[simp]
theorem nat.choose_zero_right (n : ) :
n.choose 0 = 1
@[simp]
theorem nat.choose_zero_succ (k : ) :
0.choose k.succ = 0
theorem nat.choose_succ_succ (n k : ) :
theorem nat.choose_eq_zero_of_lt {n k : } :
n < kn.choose k = 0
@[simp]
theorem nat.choose_self (n : ) :
n.choose n = 1
@[simp]
theorem nat.choose_succ_self (n : ) :
n.choose n.succ = 0
@[simp]
theorem nat.choose_one_right (n : ) :
n.choose 1 = n
theorem nat.triangle_succ (n : ) :
(n + 1) * (n + 1 - 1) / 2 = n * (n - 1) / 2 + n
theorem nat.choose_two_right (n : ) :
n.choose 2 = n * (n - 1) / 2

choose n 2 is the n-th triangle number.

theorem nat.choose_pos {n k : } :
k n0 < n.choose k
theorem nat.choose_eq_zero_iff {n k : } :
n.choose k = 0 n < k
theorem nat.succ_mul_choose_eq (n k : ) :
theorem nat.choose_mul {n k s : } (hkn : k n) (hsk : s k) :
n.choose k * k.choose s = n.choose s * (n - s).choose (k - s)
theorem nat.choose_eq_factorial_div_factorial {n k : } (hk : k n) :
theorem nat.add_choose (i j : ) :
(i + j).choose j = (i + j).factorial / (i.factorial * j.factorial)
@[simp]
theorem nat.choose_symm {n k : } (hk : k n) :
n.choose (n - k) = n.choose k
theorem nat.choose_symm_of_eq_add {n a b : } (h : n = a + b) :
n.choose a = n.choose b
theorem nat.choose_symm_add {a b : } :
(a + b).choose a = (a + b).choose b
theorem nat.choose_symm_half (m : ) :
(2 * m + 1).choose (m + 1) = (2 * m + 1).choose m
theorem nat.choose_succ_right_eq (n k : ) :
n.choose (k + 1) * (k + 1) = n.choose k * (n - k)
@[simp]
theorem nat.choose_succ_self_right (n : ) :
(n + 1).choose n = n + 1
theorem nat.choose_mul_succ_eq (n k : ) :
n.choose k * (n + 1) = (n + 1).choose k * (n + 1 - k)

Inequalities #

theorem nat.choose_le_succ_of_lt_half_left {r n : } (h : r < n / 2) :
n.choose r n.choose (r + 1)

Show that nat.choose is increasing for small values of the right argument.

theorem nat.choose_le_middle (r n : ) :
n.choose r n.choose (n / 2)

choose n r is maximised when r is n/2.

Inequalities about increasing the first argument #

theorem nat.choose_le_succ (a c : ) :
theorem nat.choose_le_add (a b c : ) :
a.choose c (a + b).choose c
theorem nat.choose_le_choose {a b : } (c : ) (h : a b) :
a.choose c b.choose c
theorem nat.choose_mono (b : ) :
monotone (λ (a : ), a.choose b)