lean2/library/data/int/power.lean
2015-11-08 14:04:54 -08:00

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/-
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 : ) (a : ) : := algebra.nmul n a
infix [priority int.prio] ⬝ := nmul
definition imul (i : ) (a : ) : := algebra.imul i a
-/
open nat
theorem of_nat_pow (a 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