lean2/library/data/nat/power.lean

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/-
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 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)]
end nat