128 lines
3.6 KiB
Text
128 lines
3.6 KiB
Text
/-
|
||
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
Author: Jeremy Avigad
|
||
|
||
Properties of the power operation in an ordered ring or field.
|
||
|
||
(Right now, this file is just a stub. More soon.)
|
||
-/
|
||
import .group_power .ordered_field
|
||
open nat
|
||
|
||
namespace algebra
|
||
|
||
variable {A : Type}
|
||
|
||
section semiring
|
||
variable [s : semiring A]
|
||
include s
|
||
|
||
theorem zero_pow {m : ℕ} (mpos : m > 0) : 0^m = (0 : A) :=
|
||
have h₁ : ∀ m, 0^succ m = (0 : A),
|
||
from take m, nat.induction_on m
|
||
(show 0^1 = 0, by rewrite pow_one)
|
||
(take m, suppose 0^(succ m) = 0,
|
||
show 0^(succ (succ m)) = 0, from !zero_mul),
|
||
obtain m' (h₂ : m = succ m'), from exists_eq_succ_of_pos mpos,
|
||
show 0^m = 0, by rewrite h₂; apply h₁
|
||
|
||
end semiring
|
||
|
||
section integral_domain
|
||
variable [s : integral_domain A]
|
||
include s
|
||
|
||
theorem eq_zero_of_pow_eq_zero {a : A} {m : ℕ} (H : a^m = 0) : a = 0 :=
|
||
or.elim (eq_zero_or_pos m)
|
||
(suppose m = 0,
|
||
by rewrite [`m = 0` at H, pow_zero at H]; apply absurd H (ne.symm zero_ne_one))
|
||
(suppose m > 0,
|
||
have h₁ : ∀ m, a^succ m = 0 → a = 0,
|
||
begin
|
||
intro m,
|
||
induction m with m ih,
|
||
{rewrite pow_one; intros; assumption},
|
||
rewrite pow_succ,
|
||
intro H,
|
||
cases eq_zero_or_eq_zero_of_mul_eq_zero H with h₃ h₄,
|
||
assumption,
|
||
exact ih h₄
|
||
end,
|
||
obtain m' (h₂ : m = succ m'), from exists_eq_succ_of_pos `m > 0`,
|
||
show a = 0, by rewrite h₂ at H; apply h₁ m' H)
|
||
|
||
end integral_domain
|
||
|
||
section linear_ordered_semiring
|
||
variable [s : linear_ordered_semiring A]
|
||
include s
|
||
|
||
theorem pow_pos_of_pos {x : A} (i : ℕ) (H : x > 0) : x^i > 0 :=
|
||
begin
|
||
induction i with [j, ih],
|
||
{show (1 : A) > 0, from zero_lt_one},
|
||
{show x^(succ j) > 0, from mul_pos H ih}
|
||
end
|
||
|
||
theorem pow_nonneg_of_nonneg {x : A} (i : ℕ) (H : x ≥ 0) : x^i ≥ 0 :=
|
||
begin
|
||
induction i with j ih,
|
||
{show (1 : A) ≥ 0, from le_of_lt zero_lt_one},
|
||
{show x^(succ j) ≥ 0, from mul_nonneg H ih}
|
||
end
|
||
|
||
theorem pow_le_pow_of_le {x y : A} (i : ℕ) (H₁ : 0 ≤ x) (H₂ : x ≤ y) : x^i ≤ y^i :=
|
||
begin
|
||
induction i with i ih,
|
||
{rewrite *pow_zero, apply le.refl},
|
||
rewrite *pow_succ,
|
||
have H : 0 ≤ x^i, from pow_nonneg_of_nonneg i H₁,
|
||
apply mul_le_mul H₂ ih H (le.trans H₁ H₂)
|
||
end
|
||
|
||
theorem pow_ge_one {x : A} (i : ℕ) (xge1 : x ≥ 1) : x^i ≥ 1 :=
|
||
assert H : x^i ≥ 1^i, from pow_le_pow_of_le i (le_of_lt zero_lt_one) xge1,
|
||
by rewrite one_pow at H; exact H
|
||
|
||
theorem pow_gt_one {x : A} {i : ℕ} (xgt1 : x > 1) (ipos : i > 0) : x^i > 1 :=
|
||
assert xpos : x > 0, from lt.trans zero_lt_one xgt1,
|
||
begin
|
||
induction i with [i, ih],
|
||
{exfalso, exact !nat.lt.irrefl ipos},
|
||
have xige1 : x^i ≥ 1, from pow_ge_one _ (le_of_lt xgt1),
|
||
rewrite [pow_succ, -mul_one 1, ↑has_lt.gt],
|
||
apply mul_lt_mul xgt1 xige1 zero_lt_one,
|
||
apply le_of_lt xpos
|
||
end
|
||
|
||
end linear_ordered_semiring
|
||
|
||
section decidable_linear_ordered_comm_ring
|
||
variable [s : decidable_linear_ordered_comm_ring A]
|
||
include s
|
||
|
||
theorem abs_pow (a : A) (n : ℕ) : abs (a^n) = abs a^n :=
|
||
begin
|
||
induction n with n ih,
|
||
rewrite [*pow_zero, (abs_of_nonneg zero_le_one : abs (1 : A) = 1)],
|
||
rewrite [*pow_succ, abs_mul, ih]
|
||
end
|
||
|
||
end decidable_linear_ordered_comm_ring
|
||
|
||
section discrete_field
|
||
variable [s : discrete_field A]
|
||
include s
|
||
|
||
theorem div_pow (a : A) {b : A} {n : ℕ} (bnz : b ≠ 0) : (a / b)^n = a^n / b^n :=
|
||
begin
|
||
induction n with n ih,
|
||
rewrite [*pow_zero, div_one],
|
||
have bnnz : b^n ≠ 0, from suppose b^n = 0, bnz (eq_zero_of_pow_eq_zero this),
|
||
rewrite [*pow_succ, ih, div_mul_div bnz bnnz]
|
||
end
|
||
|
||
end discrete_field
|
||
|
||
end algebra
|