111 lines
4.5 KiB
Text
111 lines
4.5 KiB
Text
/-
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Copyright (c) 2015 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Leonardo de Moura, Jeremy Avigad
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The power function on the natural numbers.
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-/
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import data.nat.basic data.nat.order data.nat.div data.nat.gcd algebra.ring_power
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open algebra
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namespace nat
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definition nat_has_pow_nat [instance] [reducible] [priority nat.prio] : has_pow_nat nat :=
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has_pow_nat.mk has_pow_nat.pow_nat
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theorem pow_le_pow_of_le {x y : ℕ} (i : ℕ) (H : x ≤ y) : x^i ≤ y^i :=
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algebra.pow_le_pow_of_le i !zero_le H
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theorem eq_zero_of_pow_eq_zero {a m : ℕ} (H : a^m = 0) : a = 0 :=
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or.elim (eq_zero_or_pos m)
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(suppose m = 0,
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by rewrite [`m = 0` at H, pow_zero at H]; contradiction)
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(suppose m > 0,
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have h₁ : ∀ m, a^succ m = 0 → a = 0,
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begin
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intro m,
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induction m with m ih,
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{rewrite pow_one; intros; assumption},
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rewrite pow_succ,
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intro H,
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cases eq_zero_or_eq_zero_of_mul_eq_zero H with h₃ h₄,
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assumption,
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exact ih h₄
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end,
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obtain m' (h₂ : m = succ m'), from exists_eq_succ_of_pos `m > 0`,
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show a = 0, by rewrite h₂ at H; apply h₁ m' H)
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-- generalize to semirings?
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theorem le_pow_self {x : ℕ} (H : x > 1) : ∀ i, i ≤ x^i
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| 0 := !zero_le
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| (succ j) := have x > 0, from lt.trans zero_lt_one H,
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have h₁ : x^j ≥ 1, from succ_le_of_lt (pow_pos_of_pos _ this),
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have x ≥ 2, from succ_le_of_lt H,
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calc
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succ j = j + 1 : rfl
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... ≤ x^j + 1 : add_le_add_right (le_pow_self j)
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... ≤ x^j + x^j : add_le_add_left h₁
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... = x^j * (1 + 1) : by rewrite [left_distrib, *mul_one]
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... = x^j * 2 : rfl
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... ≤ x^j * x : mul_le_mul_left _ `x ≥ 2`
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... = x^(succ j) : pow_succ'
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-- TODO: eventually this will be subsumed under the algebraic theorems
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theorem mul_self_eq_pow_2 (a : nat) : a * a = a ^ 2 :=
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show a * a = a ^ (succ (succ zero)), from
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by rewrite [*pow_succ, *pow_zero, mul_one]
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theorem pow_cancel_left : ∀ {a b c : nat}, a > 1 → a^b = a^c → b = c
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| a 0 0 h₁ h₂ := rfl
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| a (succ b) 0 h₁ h₂ :=
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assert a = 1, by rewrite [pow_succ at h₂, pow_zero at h₂]; exact (eq_one_of_mul_eq_one_right h₂),
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assert 1 < 1, by rewrite [this at h₁]; exact h₁,
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absurd `1 <[nat] 1` !lt.irrefl
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| a 0 (succ c) h₁ h₂ :=
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assert a = 1, by rewrite [pow_succ at h₂, pow_zero at h₂]; exact (eq_one_of_mul_eq_one_right (eq.symm h₂)),
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assert 1 < 1, by rewrite [this at h₁]; exact h₁,
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absurd `1 <[nat] 1` !lt.irrefl
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| a (succ b) (succ c) h₁ h₂ :=
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assert a ≠ 0, from assume aeq0, by rewrite [aeq0 at h₁]; exact (absurd h₁ dec_trivial),
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assert a^b = a^c, by rewrite [*pow_succ at h₂]; exact (eq_of_mul_eq_mul_left (pos_of_ne_zero this) h₂),
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by rewrite [pow_cancel_left h₁ this]
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theorem pow_div_cancel : ∀ {a b : nat}, a ≠ 0 → (a ^ succ b) div a = a ^ b
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| a 0 h := by rewrite [pow_succ, pow_zero, mul_one, div_self (pos_of_ne_zero h)]
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| a (succ b) h := by rewrite [pow_succ, mul_div_cancel_left _ (pos_of_ne_zero h)]
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lemma dvd_pow : ∀ (i : nat) {n : nat}, n > 0 → i ∣ i^n
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| i 0 h := absurd h !lt.irrefl
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| i (succ n) h := by rewrite [pow_succ']; apply dvd_mul_left
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lemma dvd_pow_of_dvd_of_pos : ∀ {i j n : nat}, i ∣ j → n > 0 → i ∣ j^n
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| i j 0 h₁ h₂ := absurd h₂ !lt.irrefl
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| i j (succ n) h₁ h₂ := by rewrite [pow_succ']; apply dvd_mul_of_dvd_right h₁
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lemma pow_mod_eq_zero (i : nat) {n : nat} (h : n > 0) : (i^n) mod i = 0 :=
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iff.mp !dvd_iff_mod_eq_zero (dvd_pow i h)
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lemma pow_dvd_of_pow_succ_dvd {p i n : nat} : p^(succ i) ∣ n → p^i ∣ n :=
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suppose p^(succ i) ∣ n,
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assert p^i ∣ p^(succ i), from by rewrite [pow_succ']; apply dvd_of_eq_mul; apply rfl,
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dvd.trans `p^i ∣ p^(succ i)` `p^(succ i) ∣ n`
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lemma dvd_of_pow_succ_dvd_mul_pow {p i n : nat} (Ppos : p > 0) :
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p^(succ i) ∣ (n * p^i) → p ∣ n :=
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by rewrite [pow_succ]; apply dvd_of_mul_dvd_mul_right; apply pow_pos_of_pos _ Ppos
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lemma coprime_pow_right {a b} : ∀ n, coprime b a → coprime b (a^n)
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| 0 h := !comprime_one_right
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| (succ n) h :=
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begin
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rewrite [pow_succ'],
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apply coprime_mul_right,
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exact coprime_pow_right n h,
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exact h
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end
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lemma coprime_pow_left {a b} : ∀ n, coprime b a → coprime (b^n) a :=
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take n, suppose coprime b a,
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coprime_swap (coprime_pow_right n (coprime_swap this))
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end nat
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