59 lines
1.9 KiB
Text
59 lines
1.9 KiB
Text
/-
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Copyright (c) 2016 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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Authors: Jeremy Avigad, Deniz Aydin, Floris van Doorn, Mario Carneiro
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-/
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prelude
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import Init.Data.Int.Lemmas
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import Init.Data.Nat.Lemmas
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namespace Int
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/-! # pow -/
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protected theorem pow_zero (b : Int) : b^0 = 1 := rfl
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protected theorem pow_succ (b : Int) (e : Nat) : b ^ (e+1) = (b ^ e) * b := rfl
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protected theorem pow_succ' (b : Int) (e : Nat) : b ^ (e+1) = b * (b ^ e) := by
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rw [Int.mul_comm, Int.pow_succ]
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theorem pow_le_pow_of_le_left {n m : Nat} (h : n ≤ m) : ∀ (i : Nat), n^i ≤ m^i
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| 0 => Nat.le_refl _
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| i + 1 => Nat.mul_le_mul (pow_le_pow_of_le_left h i) h
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theorem pow_le_pow_of_le_right {n : Nat} (hx : n > 0) {i : Nat} : ∀ {j}, i ≤ j → n^i ≤ n^j
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| 0, h =>
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have : i = 0 := Nat.eq_zero_of_le_zero h
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this.symm ▸ Nat.le_refl _
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| j + 1, h =>
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match Nat.le_or_eq_of_le_succ h with
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| Or.inl h => show n^i ≤ n^j * n from
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have : n^i * 1 ≤ n^j * n := Nat.mul_le_mul (pow_le_pow_of_le_right hx h) hx
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Nat.mul_one (n^i) ▸ this
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| Or.inr h =>
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h.symm ▸ Nat.le_refl _
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theorem pos_pow_of_pos {n : Nat} (m : Nat) (h : 0 < n) : 0 < n^m :=
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pow_le_pow_of_le_right h (Nat.zero_le _)
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@[norm_cast]
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theorem natCast_pow (b n : Nat) : ((b^n : Nat) : Int) = (b : Int) ^ n := by
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match n with
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| 0 => rfl
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| n + 1 =>
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simp only [Nat.pow_succ, Int.pow_succ, natCast_mul, natCast_pow _ n]
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@[simp]
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protected theorem two_pow_pred_sub_two_pow {w : Nat} (h : 0 < w) :
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((2 ^ (w - 1) : Nat) - (2 ^ w : Nat) : Int) = - ((2 ^ (w - 1) : Nat) : Int) := by
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rw [← Nat.two_pow_pred_add_two_pow_pred h]
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omega
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@[simp]
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protected theorem two_pow_pred_sub_two_pow' {w : Nat} (h : 0 < w) :
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(2 : Int) ^ (w - 1) - (2 : Int) ^ w = - (2 : Int) ^ (w - 1) := by
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norm_cast
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rw [← Nat.two_pow_pred_add_two_pow_pred h]
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simp [h]
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end Int
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