30 lines
801 B
Text
30 lines
801 B
Text
/-
|
||
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
Author: Jeremy Avigad
|
||
|
||
The power function on the integers.
|
||
-/
|
||
import data.int.basic data.int.order data.int.div algebra.group_power data.nat.power
|
||
|
||
namespace int
|
||
open - [notations] algebra
|
||
|
||
definition int_has_pow_nat [reducible] [instance] [priority int.prio] : has_pow_nat int :=
|
||
has_pow_nat.mk has_pow_nat.pow_nat
|
||
|
||
/-
|
||
definition nmul (n : ℕ) (a : ℤ) : ℤ := algebra.nmul n a
|
||
infix [priority int.prio] ⬝ := nmul
|
||
definition imul (i : ℤ) (a : ℤ) : ℤ := algebra.imul i a
|
||
-/
|
||
|
||
open nat
|
||
theorem of_nat_pow (a n : ℕ) : of_nat (a^n) = (of_nat a)^n :=
|
||
begin
|
||
induction n with n ih,
|
||
apply eq.refl,
|
||
rewrite [pow_succ, pow_succ, of_nat_mul, ih]
|
||
end
|
||
|
||
end int
|