lean2/library/data/nat/power.lean

/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad

The power function on the natural numbers.
-/
import data.nat.basic data.nat.order data.nat.div data.nat.gcd algebra.ring_power

namespace nat

definition nat_has_pow_nat [instance] [reducible] [priority nat.prio] : has_pow_nat nat :=
has_pow_nat.mk has_pow_nat.pow_nat

theorem pow_le_pow_of_le {x y : ℕ} (i : ℕ) (H : x ≤ y) : x^i ≤ y^i :=
pow_le_pow_of_le i !zero_le H

theorem eq_zero_of_pow_eq_zero {a m : ℕ} (H : a^m = 0) : a = 0 :=
or.elim (eq_zero_or_pos m)
  (suppose m = 0,
    by rewrite [`m = 0` at H, pow_zero at H]; contradiction)
  (suppose m > 0,
    have h₁ : ∀ m, a^succ m = 0 → a = 0,
      begin
        intro m,
        induction m with m ih,
          {rewrite pow_one; intros; assumption},
        rewrite pow_succ,
        intro H,
        cases eq_zero_or_eq_zero_of_mul_eq_zero H with h₃ h₄,
          assumption,
        exact ih h₄
      end,
    obtain m' (h₂ : m = succ m'), from exists_eq_succ_of_pos `m > 0`,
    show a = 0, by rewrite h₂ at H; apply h₁ m' H)

-- generalize to semirings?
theorem le_pow_self {x : ℕ} (H : x > 1) : ∀ i, i ≤ x^i
| 0        := !zero_le
| (succ j) := have x > 0,        from lt.trans zero_lt_one H,
              have h₁ : x^j ≥ 1, from succ_le_of_lt (pow_pos_of_pos _ this),
              have x ≥ 2,        from succ_le_of_lt H,
              calc
                succ j = j + 1         : rfl
                   ... ≤ x^j + 1       : add_le_add_right (le_pow_self j)
                   ... ≤ x^j + x^j     : add_le_add_left h₁
                   ... = x^j * (1 + 1) : by rewrite [left_distrib, *mul_one]
                   ... = x^j * 2       : rfl
                   ... ≤ x^j * x       : mul_le_mul_left _ `x ≥ 2`
                   ... = x^(succ j)    : pow_succ'

-- TODO: eventually this will be subsumed under the algebraic theorems

theorem mul_self_eq_pow_2 (a : nat) : a * a = a ^ 2 :=
show a * a = a ^ (succ (succ zero)), from
by rewrite [*pow_succ, *pow_zero, mul_one]

theorem pow_cancel_left : ∀ {a b c : nat}, a > 1 → a ^ b = a ^ c → b = c
| a 0        0        h₁ h₂ := rfl
| a (succ b) 0        h₁ h₂ :=
  assert a = 1, by rewrite [pow_succ at h₂, pow_zero at h₂]; exact (eq_one_of_mul_eq_one_right h₂),
  assert 1 < 1, by rewrite [this at h₁]; exact h₁,
  absurd `1 <[nat] 1` !lt.irrefl
| a 0        (succ c) h₁ h₂ :=
  assert a = 1, by rewrite [pow_succ at h₂, pow_zero at h₂]; exact (eq_one_of_mul_eq_one_right (eq.symm h₂)),
  assert 1 < 1, by rewrite [this at h₁]; exact h₁,
  absurd `1 <[nat] 1` !lt.irrefl
| a (succ b) (succ c) h₁ h₂ :=
  assert a ≠ 0, from assume aeq0, by rewrite [aeq0 at h₁]; exact (absurd h₁ dec_trivial),
  assert a^b = a^c, by rewrite [*pow_succ at h₂]; exact (eq_of_mul_eq_mul_left (pos_of_ne_zero this) h₂),
  by rewrite [pow_cancel_left h₁ this]

theorem pow_div_cancel : ∀ {a b : nat}, a ≠ 0 → (a ^ succ b) / a = a ^ b
| a 0        h := by rewrite [pow_succ, pow_zero, mul_one, nat.div_self (pos_of_ne_zero h)]
| a (succ b) h := by rewrite [pow_succ, nat.mul_div_cancel_left _ (pos_of_ne_zero h)]

lemma dvd_pow : ∀ (i : nat) {n : nat}, n > 0 → i ∣ i^n
| i 0        h := absurd h !lt.irrefl
| i (succ n) h := by rewrite [pow_succ']; apply dvd_mul_left

lemma dvd_pow_of_dvd_of_pos : ∀ {i j n : nat}, i ∣ j → n > 0 → i ∣ j^n
| i j 0        h₁ h₂ := absurd h₂ !lt.irrefl
| i j (succ n) h₁ h₂ := by rewrite [pow_succ']; apply dvd_mul_of_dvd_right h₁

lemma pow_mod_eq_zero (i : nat) {n : nat} (h : n > 0) : (i ^ n) % i = 0 :=
iff.mp !dvd_iff_mod_eq_zero (dvd_pow i h)

lemma pow_dvd_of_pow_succ_dvd {p i n : nat} : p^(succ i) ∣ n → p^i ∣ n :=
suppose p^(succ i) ∣ n,
assert p^i ∣ p^(succ i),
  by rewrite [pow_succ']; apply nat.dvd_of_eq_mul; apply rfl,
dvd.trans `p^i ∣ p^(succ i)` `p^(succ i) ∣ n`

lemma dvd_of_pow_succ_dvd_mul_pow {p i n : nat} (Ppos : p > 0) :
  p^(succ i) ∣ (n * p^i) → p ∣ n :=
by rewrite [pow_succ]; apply nat.dvd_of_mul_dvd_mul_right; apply pow_pos_of_pos _ Ppos

lemma coprime_pow_right {a b} : ∀ n, coprime b a → coprime b (a^n)
| 0        h := !comprime_one_right
| (succ n) h :=
  begin
    rewrite [pow_succ'],
    apply coprime_mul_right,
      exact coprime_pow_right n h,
      exact h
  end

lemma coprime_pow_left {a b} : ∀ n, coprime b a → coprime (b^n) a :=
take n, suppose coprime b a,
coprime_swap (coprime_pow_right n (coprime_swap this))
end nat