pow

theorem Int.pow_zero (b : Int) : b ^ 0 = 1

theorem Int.pow_succ (b : Int) (e : Nat) : b ^ (e + 1) = b ^ e * b

theorem Int.pow_succ' (b : Int) (e : Nat) : b ^ (e + 1) = b * b ^ e

theorem Int.pow_le_pow_of_le_left {n : Nat} {m : Nat} (h : n ≤ m) (i : Nat) : n ^ i ≤ m ^ i

theorem Int.pow_le_pow_of_le_right {n : Nat} (hx : n > 0) {i : Nat} {j : Nat} : i ≤ j → n ^ i ≤ n ^ j

theorem Int.pos_pow_of_pos {n : Nat} (m : Nat) (h : 0 < n) : 0 < n ^ m

theorem Int.natCast_pow (b : Nat) (n : Nat) : ↑(b ^ n) = ↑b ^ n

@[simp]

theorem Int.two_pow_pred_sub_two_pow {w : Nat} (h : 0 < w) : ↑(2 ^ (w - 1)) - ↑(2 ^ w) = -↑(2 ^ (w - 1))

@[simp]

theorem Int.two_pow_pred_sub_two_pow' {w : Nat} (h : 0 < w) : 2 ^ (w - 1) - 2 ^ w = -2 ^ (w - 1)