pow

@[simp]

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_pos {n : Int} {m : Nat} : 0 < n → 0 < n ^ m

theorem Int.pow_nonneg {n : Int} {m : Nat} : 0 ≤ n → 0 ≤ n ^ m

theorem Int.pow_ne_zero {n : Int} {m : Nat} : n ≠ 0 → n ^ m ≠ 0

@[reducible, inline, deprecated Nat.pow_le_pow_left (since := "2025-02-17")]

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

Equations

• @Int.pow_le_pow_of_le_left = @Nat.pow_le_pow_left

@[reducible, inline, deprecated Nat.pow_le_pow_right (since := "2025-02-17")]

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

Equations

• @Int.pow_le_pow_of_le_right = @Nat.pow_le_pow_right

@[reducible, inline, deprecated Nat.pow_pos (since := "2025-02-17")]

abbrev Int.pos_pow_of_pos {a n : Nat} (h : 0 < a) : 0 < a ^ n

Equations

• @Int.pos_pow_of_pos = @Nat.pow_pos

@[simp]

theorem Int.natCast_pow (b 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)

theorem Int.pow_lt_pow_of_lt {a : Int} {b c : Nat} (ha : 1 < a) (hbc : b < c) : a ^ b < a ^ c

@[simp]

theorem Int.natAbs_pow (n : Int) (k : Nat) : (n ^ k).natAbs = n.natAbs ^ k

theorem Int.toNat_pow_of_nonneg {x : Int} (h : 0 ≤ x) (k : Nat) : (x ^ k).toNat = x.toNat ^ k