lean2/library/data/nat/power.lean
2015-07-04 00:37:09 -07:00

77 lines
3.2 KiB
Text
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

/-
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.group_power
namespace nat
section migrate_algebra
open [classes] algebra
local attribute nat.comm_semiring [instance]
local attribute nat.linear_ordered_semiring [instance]
definition pow (a : ) (n : ) : := algebra.pow a n
infix ^ := pow
migrate from algebra with nat
replacing dvd → dvd, has_le.ge → ge, has_lt.gt → gt, pow → pow
hiding add_pos_of_pos_of_nonneg, add_pos_of_nonneg_of_pos,
add_eq_zero_iff_eq_zero_and_eq_zero_of_nonneg_of_nonneg, le_add_of_nonneg_of_le,
le_add_of_le_of_nonneg, lt_add_of_nonneg_of_lt, lt_add_of_lt_of_nonneg,
lt_of_mul_lt_mul_left, lt_of_mul_lt_mul_right, pos_of_mul_pos_left, pos_of_mul_pos_right
end migrate_algebra
-- TODO: eventually this will be subsumed under the algebraic theorems
theorem mul_self_eq_pow_2 (a : nat) : a * a = pow a 2 :=
show a * a = pow a (succ (succ zero)), from
by rewrite [*pow_succ, *pow_zero, one_mul]
theorem pow_cancel_left : ∀ {a b c : nat}, a > 1 → pow a b = pow a c → b = c
| a 0 0 h₁ h₂ := rfl
| a (succ b) 0 h₁ h₂ :=
assert aeq1 : a = 1, by rewrite [pow_succ' at h₂, pow_zero at h₂]; exact (eq_one_of_mul_eq_one_right h₂),
assert h₁ : 1 < 1, by rewrite [aeq1 at h₁]; exact h₁,
absurd h₁ !lt.irrefl
| a 0 (succ c) h₁ h₂ :=
assert aeq1 : a = 1, by rewrite [pow_succ' at h₂, pow_zero at h₂]; exact (eq_one_of_mul_eq_one_right (eq.symm h₂)),
assert h₁ : 1 < 1, by rewrite [aeq1 at h₁]; exact h₁,
absurd h₁ !lt.irrefl
| a (succ b) (succ c) h₁ h₂ :=
assert ane0 : a ≠ 0, from assume aeq0, by rewrite [aeq0 at h₁]; exact (absurd h₁ dec_trivial),
assert beqc : pow a b = pow a c, by rewrite [*pow_succ' at h₂]; exact (eq_of_mul_eq_mul_left (pos_of_ne_zero ane0) h₂),
by rewrite [pow_cancel_left h₁ beqc]
theorem pow_div_cancel : ∀ {a b : nat}, a ≠ 0 → pow a (succ b) div a = pow a b
| a 0 h := by rewrite [pow_succ', pow_zero, mul_one, div_self (pos_of_ne_zero h)]
| a (succ b) h := by rewrite [pow_succ', 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) mod i = 0 :=
iff.mp !dvd_iff_mod_eq_zero (dvd_pow i h)
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 :=
λ n h, coprime_swap (coprime_pow_right n (coprime_swap h))
end nat