R : Type ?u.164552
n : ℕ
⊢ 4 ^ n = (2 ^ 2) ^ n
R : Type ?u.164552
n : ℕ
⊢ 4 ^ n = (1 + 1) ^ (2 * n)
simp only [pow_mul, one_add_one_eq_two]
no goals
R : Type ?u.164552
n : ℕ
⊢ 4 ^ n = (2 ^ 2) ^ n
rfl
no goals
R : Type ?u.164552
n : ℕ
⊢ (1 + 1) ^ (2 * n) = ∑ m in range (2 * n + 1), choose (2 * n) m
simp [add_pow]
case h
R : Type ?u.164552
n i✝ : ℕ
a✝ : i✝ ∈ range (2 * n + 1)
⊢ choose (2 * n) i✝ ≤ choose (2 * n) (2 * n / 2)
R : Type ?u.164552
n : ℕ
⊢ ∑ m in range (2 * n + 1), choose (2 * n) m ≤ ∑ m in range (2 * n + 1), choose (2 * n) (2 * n / 2)
gcongr
no goals
case h
R : Type ?u.164552
n i✝ : ℕ
a✝ : i✝ ∈ range (2 * n + 1)
⊢ choose (2 * n) i✝ ≤ choose (2 * n) (2 * n / 2)
apply choose_le_middle
no goals
R : Type ?u.164552
n : ℕ
⊢ ∑ m in range (2 * n + 1), choose (2 * n) (2 * n / 2) = (2 * n + 1) * choose (2 * n) n
simp