lean2/library/algebra/ordered_ring.lean

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/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Module: algebra.ordered_ring
Authors: Jeremy Avigad
Here an "ordered_ring" is partially ordered ring, which is ordered with respect to both a weak
order and an associated strict order. Our numeric structures (int, rat, and real) will be instances
of "linear_ordered_comm_ring". This development is modeled after Isabelle's library.
-/
import algebra.ordered_group algebra.ring
open eq eq.ops
namespace algebra
variable {A : Type}
definition absurd_a_lt_a {B : Type} {a : A} [s : strict_order A] (H : a < a) : B :=
absurd H (lt.irrefl a)
structure ordered_semiring [class] (A : Type)
extends has_mul A, has_zero A, has_lt A, -- TODO: remove hack for improving performance
semiring A, ordered_cancel_comm_monoid A, zero_ne_one_class A :=
(mul_le_mul_of_nonneg_left: ∀a b c, le a b → le zero c → le (mul c a) (mul c b))
(mul_le_mul_of_nonneg_right: ∀a b c, le a b → le zero c → le (mul a c) (mul b c))
(mul_lt_mul_of_pos_left: ∀a b c, lt a b → lt zero c → lt (mul c a) (mul c b))
(mul_lt_mul_of_pos_right: ∀a b c, lt a b → lt zero c → lt (mul a c) (mul b c))
section
variable [s : ordered_semiring A]
variables (a b c d e : A)
include s
theorem mul_le_mul_of_nonneg_left {a b c : A} (Hab : a ≤ b) (Hc : 0 ≤ c) :
c * a ≤ c * b := !ordered_semiring.mul_le_mul_of_nonneg_left Hab Hc
theorem mul_le_mul_of_nonneg_right {a b c : A} (Hab : a ≤ b) (Hc : 0 ≤ c) :
a * c ≤ b * c := !ordered_semiring.mul_le_mul_of_nonneg_right Hab Hc
-- TODO: there are four variations, depending on which variables we assume to be nonneg
theorem mul_le_mul {a b c d : A} (Hac : a ≤ c) (Hbd : b ≤ d) (nn_b : 0 ≤ b) (nn_c : 0 ≤ c) :
a * b ≤ c * d :=
calc
a * b ≤ c * b : mul_le_mul_of_nonneg_right Hac nn_b
... ≤ c * d : mul_le_mul_of_nonneg_left Hbd nn_c
theorem mul_nonneg {a b : A} (Ha : a ≥ 0) (Hb : b ≥ 0) : a * b ≥ 0 :=
begin
have H : 0 * b ≤ a * b, from mul_le_mul_of_nonneg_right Ha Hb,
rewrite zero_mul at H,
exact H
end
theorem mul_nonpos_of_nonneg_of_nonpos {a b : A} (Ha : a ≥ 0) (Hb : b ≤ 0) : a * b ≤ 0 :=
begin
have H : a * b ≤ a * 0, from mul_le_mul_of_nonneg_left Hb Ha,
rewrite mul_zero at H,
exact H
end
theorem mul_nonpos_of_nonpos_of_nonneg {a b : A} (Ha : a ≤ 0) (Hb : b ≥ 0) : a * b ≤ 0 :=
begin
have H : a * b ≤ 0 * b, from mul_le_mul_of_nonneg_right Ha Hb,
rewrite zero_mul at H,
exact H
end
theorem mul_lt_mul_of_pos_left {a b c : A} (Hab : a < b) (Hc : 0 < c) :
c * a < c * b := !ordered_semiring.mul_lt_mul_of_pos_left Hab Hc
theorem mul_lt_mul_of_pos_right {a b c : A} (Hab : a < b) (Hc : 0 < c) :
a * c < b * c := !ordered_semiring.mul_lt_mul_of_pos_right Hab Hc
-- TODO: once again, there are variations
theorem mul_lt_mul {a b c d : A} (Hac : a < c) (Hbd : b ≤ d) (pos_b : 0 < b) (nn_c : 0 ≤ c) :
a * b < c * d :=
calc
a * b < c * b : mul_lt_mul_of_pos_right Hac pos_b
... ≤ c * d : mul_le_mul_of_nonneg_left Hbd nn_c
theorem mul_pos {a b : A} (Ha : a > 0) (Hb : b > 0) : a * b > 0 :=
begin
have H : 0 * b < a * b, from mul_lt_mul_of_pos_right Ha Hb,
rewrite zero_mul at H,
exact H
end
theorem mul_neg_of_pos_of_neg {a b : A} (Ha : a > 0) (Hb : b < 0) : a * b < 0 :=
begin
have H : a * b < a * 0, from mul_lt_mul_of_pos_left Hb Ha,
rewrite mul_zero at H,
exact H
end
theorem mul_neg_of_neg_of_pos {a b : A} (Ha : a < 0) (Hb : b > 0) : a * b < 0 :=
begin
have H : a * b < 0 * b, from mul_lt_mul_of_pos_right Ha Hb,
rewrite zero_mul at H,
exact H
end
end
structure linear_ordered_semiring [class] (A : Type)
extends ordered_semiring A, linear_strong_order_pair A
section
variable [s : linear_ordered_semiring A]
variables {a b c : A}
include s
theorem lt_of_mul_lt_mul_left (H : c * a < c * b) (Hc : c ≥ 0) : a < b :=
lt_of_not_le
(assume H1 : b ≤ a,
have H2 : c * b ≤ c * a, from mul_le_mul_of_nonneg_left H1 Hc,
not_lt_of_le H2 H)
theorem lt_of_mul_lt_mul_right (H : a * c < b * c) (Hc : c ≥ 0) : a < b :=
lt_of_not_le
(assume H1 : b ≤ a,
have H2 : b * c ≤ a * c, from mul_le_mul_of_nonneg_right H1 Hc,
not_lt_of_le H2 H)
theorem le_of_mul_le_mul_left (H : c * a ≤ c * b) (Hc : c > 0) : a ≤ b :=
le_of_not_lt
(assume H1 : b < a,
have H2 : c * b < c * a, from mul_lt_mul_of_pos_left H1 Hc,
not_le_of_lt H2 H)
theorem le_of_mul_le_mul_right (H : a * c ≤ b * c) (Hc : c > 0) : a ≤ b :=
le_of_not_lt
(assume H1 : b < a,
have H2 : b * c < a * c, from mul_lt_mul_of_pos_right H1 Hc,
not_le_of_lt H2 H)
theorem pos_of_mul_pos_left (H : 0 < a * b) (H1 : 0 ≤ a) : 0 < b :=
lt_of_not_le
(assume H2 : b ≤ 0,
have H3 : a * b ≤ 0, from mul_nonpos_of_nonneg_of_nonpos H1 H2,
not_lt_of_le H3 H)
theorem pos_of_mul_pos_right (H : 0 < a * b) (H1 : 0 ≤ b) : 0 < a :=
lt_of_not_le
(assume H2 : a ≤ 0,
have H3 : a * b ≤ 0, from mul_nonpos_of_nonpos_of_nonneg H2 H1,
not_lt_of_le H3 H)
end
structure ordered_ring [class] (A : Type) extends ring A, ordered_comm_group A, zero_ne_one_class A :=
(mul_nonneg : ∀a b, le zero a → le zero b → le zero (mul a b))
(mul_pos : ∀a b, lt zero a → lt zero b → lt zero (mul a b))
theorem ordered_ring.mul_le_mul_of_nonneg_left [s : ordered_ring A] {a b c : A}
(Hab : a ≤ b) (Hc : 0 ≤ c) : c * a ≤ c * b :=
have H1 : 0 ≤ b - a, from iff.elim_right !sub_nonneg_iff_le Hab,
assert H2 : 0 ≤ c * (b - a), from ordered_ring.mul_nonneg _ _ Hc H1,
begin
rewrite mul_sub_left_distrib at H2,
exact (iff.mp !sub_nonneg_iff_le H2)
end
theorem ordered_ring.mul_le_mul_of_nonneg_right [s : ordered_ring A] {a b c : A}
(Hab : a ≤ b) (Hc : 0 ≤ c) : a * c ≤ b * c :=
have H1 : 0 ≤ b - a, from iff.elim_right !sub_nonneg_iff_le Hab,
assert H2 : 0 ≤ (b - a) * c, from ordered_ring.mul_nonneg _ _ H1 Hc,
begin
rewrite mul_sub_right_distrib at H2,
exact (iff.mp !sub_nonneg_iff_le H2)
end
theorem ordered_ring.mul_lt_mul_of_pos_left [s : ordered_ring A] {a b c : A}
(Hab : a < b) (Hc : 0 < c) : c * a < c * b :=
have H1 : 0 < b - a, from iff.elim_right !sub_pos_iff_lt Hab,
assert H2 : 0 < c * (b - a), from ordered_ring.mul_pos _ _ Hc H1,
begin
rewrite mul_sub_left_distrib at H2,
exact (iff.mp !sub_pos_iff_lt H2)
end
theorem ordered_ring.mul_lt_mul_of_pos_right [s : ordered_ring A] {a b c : A}
(Hab : a < b) (Hc : 0 < c) : a * c < b * c :=
have H1 : 0 < b - a, from iff.elim_right !sub_pos_iff_lt Hab,
assert H2 : 0 < (b - a) * c, from ordered_ring.mul_pos _ _ H1 Hc,
begin
rewrite mul_sub_right_distrib at H2,
exact (iff.mp !sub_pos_iff_lt H2)
end
definition ordered_ring.to_ordered_semiring [instance] [coercion] [reducible] [s : ordered_ring A] :
ordered_semiring A :=
⦃ ordered_semiring, s,
mul_zero := mul_zero,
zero_mul := zero_mul,
add_left_cancel := @add.left_cancel A s,
add_right_cancel := @add.right_cancel A s,
le_of_add_le_add_left := @le_of_add_le_add_left A s,
mul_le_mul_of_nonneg_left := @ordered_ring.mul_le_mul_of_nonneg_left A s,
mul_le_mul_of_nonneg_right := @ordered_ring.mul_le_mul_of_nonneg_right A s,
mul_lt_mul_of_pos_left := @ordered_ring.mul_lt_mul_of_pos_left A s,
mul_lt_mul_of_pos_right := @ordered_ring.mul_lt_mul_of_pos_right A s ⦄
section
variable [s : ordered_ring A]
variables {a b c : A}
include s
theorem mul_le_mul_of_nonpos_left (H : b ≤ a) (Hc : c ≤ 0) : c * a ≤ c * b :=
have Hc' : -c ≥ 0, from iff.mp' !neg_nonneg_iff_nonpos Hc,
assert H1 : -c * b ≤ -c * a, from mul_le_mul_of_nonneg_left H Hc',
have H2 : -(c * b) ≤ -(c * a),
begin
rewrite [-*neg_mul_eq_neg_mul at H1],
exact H1
end,
iff.mp !neg_le_neg_iff_le H2
theorem mul_le_mul_of_nonpos_right (H : b ≤ a) (Hc : c ≤ 0) : a * c ≤ b * c :=
have Hc' : -c ≥ 0, from iff.mp' !neg_nonneg_iff_nonpos Hc,
assert H1 : b * -c ≤ a * -c, from mul_le_mul_of_nonneg_right H Hc',
have H2 : -(b * c) ≤ -(a * c),
begin
rewrite [-*neg_mul_eq_mul_neg at H1],
exact H1
end,
iff.mp !neg_le_neg_iff_le H2
theorem mul_nonneg_of_nonpos_of_nonpos (Ha : a ≤ 0) (Hb : b ≤ 0) : 0 ≤ a * b :=
begin
have H : 0 * b ≤ a * b, from mul_le_mul_of_nonpos_right Ha Hb,
rewrite zero_mul at H,
exact H
end
theorem mul_lt_mul_of_neg_left (H : b < a) (Hc : c < 0) : c * a < c * b :=
have Hc' : -c > 0, from iff.mp' !neg_pos_iff_neg Hc,
assert H1 : -c * b < -c * a, from mul_lt_mul_of_pos_left H Hc',
have H2 : -(c * b) < -(c * a),
begin
rewrite [-*neg_mul_eq_neg_mul at H1],
exact H1
end,
iff.mp !neg_lt_neg_iff_lt H2
theorem mul_lt_mul_of_neg_right (H : b < a) (Hc : c < 0) : a * c < b * c :=
have Hc' : -c > 0, from iff.mp' !neg_pos_iff_neg Hc,
assert H1 : b * -c < a * -c, from mul_lt_mul_of_pos_right H Hc',
have H2 : -(b * c) < -(a * c),
begin
rewrite [-*neg_mul_eq_mul_neg at H1],
exact H1
end,
iff.mp !neg_lt_neg_iff_lt H2
theorem mul_pos_of_neg_of_neg (Ha : a < 0) (Hb : b < 0) : 0 < a * b :=
begin
have H : 0 * b < a * b, from mul_lt_mul_of_neg_right Ha Hb,
rewrite zero_mul at H,
exact H
end
end
-- TODO: we can eliminate mul_pos_of_pos, but now it is not worth the effort to redeclare the
-- class instance
structure linear_ordered_ring [class] (A : Type) extends ordered_ring A, linear_strong_order_pair A
-- print fields linear_ordered_semiring
definition linear_ordered_ring.to_linear_ordered_semiring [instance] [coercion] [reducible]
[s : linear_ordered_ring A] :
linear_ordered_semiring A :=
⦃ linear_ordered_semiring, s,
mul_zero := mul_zero,
zero_mul := zero_mul,
add_left_cancel := @add.left_cancel A s,
add_right_cancel := @add.right_cancel A s,
le_of_add_le_add_left := @le_of_add_le_add_left A s,
mul_le_mul_of_nonneg_left := @mul_le_mul_of_nonneg_left A s,
mul_le_mul_of_nonneg_right := @mul_le_mul_of_nonneg_right A s,
mul_lt_mul_of_pos_left := @mul_lt_mul_of_pos_left A s,
mul_lt_mul_of_pos_right := @mul_lt_mul_of_pos_right A s,
le_total := linear_ordered_ring.le_total ⦄
structure linear_ordered_comm_ring [class] (A : Type) extends linear_ordered_ring A, comm_monoid A
theorem linear_ordered_comm_ring.eq_zero_or_eq_zero_of_mul_eq_zero [s : linear_ordered_comm_ring A]
{a b : A} (H : a * b = 0) : a = 0 b = 0 :=
lt.by_cases
(assume Ha : 0 < a,
lt.by_cases
(assume Hb : 0 < b,
begin
have H1 : 0 < a * b, from mul_pos Ha Hb,
rewrite H at H1,
apply (absurd_a_lt_a H1)
end)
(assume Hb : 0 = b, or.inr (Hb⁻¹))
(assume Hb : 0 > b,
begin
have H1 : 0 > a * b, from mul_neg_of_pos_of_neg Ha Hb,
rewrite H at H1,
apply (absurd_a_lt_a H1)
end))
(assume Ha : 0 = a, or.inl (Ha⁻¹))
(assume Ha : 0 > a,
lt.by_cases
(assume Hb : 0 < b,
begin
have H1 : 0 > a * b, from mul_neg_of_neg_of_pos Ha Hb,
rewrite H at H1,
apply (absurd_a_lt_a H1)
end)
(assume Hb : 0 = b, or.inr (Hb⁻¹))
(assume Hb : 0 > b,
begin
have H1 : 0 < a * b, from mul_pos_of_neg_of_neg Ha Hb,
rewrite H at H1,
apply (absurd_a_lt_a H1)
end))
-- Linearity implies no zero divisors. Doesn't need commutativity.
definition linear_ordered_comm_ring.to_integral_domain [instance] [coercion] [reducible]
[s: linear_ordered_comm_ring A] : integral_domain A :=
⦃ integral_domain, s,
eq_zero_or_eq_zero_of_mul_eq_zero :=
@linear_ordered_comm_ring.eq_zero_or_eq_zero_of_mul_eq_zero A s ⦄
section
variable [s : linear_ordered_ring A]
variables (a b c : A)
include s
theorem mul_self_nonneg : a * a ≥ 0 :=
or.elim (le.total 0 a)
(assume H : a ≥ 0, mul_nonneg H H)
(assume H : a ≤ 0, mul_nonneg_of_nonpos_of_nonpos H H)
theorem zero_le_one : 0 ≤ 1 := one_mul 1 ▸ mul_self_nonneg 1
theorem zero_lt_one : 0 < 1 := lt_of_le_of_ne zero_le_one zero_ne_one
theorem pos_and_pos_or_neg_and_neg_of_mul_pos {a b : A} (Hab : a * b > 0) :
(a > 0 ∧ b > 0) (a < 0 ∧ b < 0) :=
lt.by_cases
(assume Ha : 0 < a,
lt.by_cases
(assume Hb : 0 < b, or.inl (and.intro Ha Hb))
(assume Hb : 0 = b,
begin
rewrite [-Hb at Hab, mul_zero at Hab],
apply (absurd_a_lt_a Hab)
end)
(assume Hb : b < 0,
absurd Hab (lt.asymm (mul_neg_of_pos_of_neg Ha Hb))))
(assume Ha : 0 = a,
begin
rewrite [-Ha at Hab, zero_mul at Hab],
apply (absurd_a_lt_a Hab)
end)
(assume Ha : a < 0,
lt.by_cases
(assume Hb : 0 < b,
absurd Hab (lt.asymm (mul_neg_of_neg_of_pos Ha Hb)))
(assume Hb : 0 = b,
begin
rewrite [-Hb at Hab, mul_zero at Hab],
apply (absurd_a_lt_a Hab)
end)
(assume Hb : b < 0, or.inr (and.intro Ha Hb)))
theorem gt_of_mul_lt_mul_neg_left {a b c} (H : c * a < c * b) (Hc : c ≤ 0) : a > b :=
have nhc : -c ≥ 0, from neg_nonneg_of_nonpos Hc,
have H2 : -(c * b) < -(c * a), from iff.mp' (neg_lt_neg_iff_lt _ _) H,
have H3 : (-c) * b < (-c) * a, from calc
(-c) * b = - (c * b) : neg_mul_eq_neg_mul
... < -(c * a) : H2
... = (-c) * a : neg_mul_eq_neg_mul,
lt_of_mul_lt_mul_left H3 nhc
theorem zero_gt_neg_one : -1 < 0 :=
neg_zero ▸ (neg_lt_neg zero_lt_one)
end
/- TODO: Isabelle's library has all kinds of cancelation rules for the simplifier.
Search on mult_le_cancel_right1 in Rings.thy. -/
structure decidable_linear_ordered_comm_ring [class] (A : Type) extends linear_ordered_comm_ring A,
decidable_linear_ordered_comm_group A
section
variable [s : decidable_linear_ordered_comm_ring A]
variables {a b c : A}
include s
definition sign (a : A) : A := lt.cases a 0 (-1) 0 1
theorem sign_of_neg (H : a < 0) : sign a = -1 := lt.cases_of_lt H
theorem sign_zero : sign 0 = 0 := lt.cases_of_eq rfl
theorem sign_of_pos (H : a > 0) : sign a = 1 := lt.cases_of_gt H
theorem sign_one : sign 1 = 1 := sign_of_pos zero_lt_one
theorem sign_neg_one : sign (-1) = -1 := sign_of_neg (neg_neg_of_pos zero_lt_one)
theorem sign_sign (a : A) : sign (sign a) = sign a :=
lt.by_cases
(assume H : a > 0,
calc
sign (sign a) = sign 1 : by rewrite (sign_of_pos H)
... = 1 : by rewrite sign_one
... = sign a : by rewrite (sign_of_pos H))
(assume H : 0 = a,
calc
sign (sign a) = sign (sign 0) : by rewrite H
... = sign 0 : by rewrite sign_zero at {1}
... = sign a : by rewrite -H)
(assume H : a < 0,
calc
sign (sign a) = sign (-1) : by rewrite (sign_of_neg H)
... = -1 : by rewrite sign_neg_one
... = sign a : by rewrite (sign_of_neg H))
theorem pos_of_sign_eq_one (H : sign a = 1) : a > 0 :=
lt.by_cases
(assume H1 : 0 < a, H1)
(assume H1 : 0 = a,
begin
rewrite [-H1 at H, sign_zero at H],
apply (absurd H zero_ne_one)
end)
(assume H1 : 0 > a,
have H2 : -1 = 1, from (sign_of_neg H1)⁻¹ ⬝ H,
absurd ((eq_zero_of_neg_eq H2)⁻¹) zero_ne_one)
theorem eq_zero_of_sign_eq_zero (H : sign a = 0) : a = 0 :=
lt.by_cases
(assume H1 : 0 < a,
absurd (H⁻¹ ⬝ sign_of_pos H1) zero_ne_one)
(assume H1 : 0 = a, H1⁻¹)
(assume H1 : 0 > a,
have H2 : 0 = -1, from H⁻¹ ⬝ sign_of_neg H1,
have H3 : 1 = 0, from eq_neg_of_eq_neg H2 ⬝ neg_zero,
absurd (H3⁻¹) zero_ne_one)
theorem neg_of_sign_eq_neg_one (H : sign a = -1) : a < 0 :=
lt.by_cases
(assume H1 : 0 < a,
have H2 : -1 = 1, from H⁻¹ ⬝ (sign_of_pos H1),
absurd ((eq_zero_of_neg_eq H2)⁻¹) zero_ne_one)
(assume H1 : 0 = a,
have H2 : 0 = -1,
begin
rewrite [-H1 at H, sign_zero at H],
exact H
end,
have H3 : 1 = 0, from eq_neg_of_eq_neg H2 ⬝ neg_zero,
absurd (H3⁻¹) zero_ne_one)
(assume H1 : 0 > a, H1)
theorem sign_neg (a : A) : sign (-a) = -(sign a) :=
lt.by_cases
(assume H1 : 0 < a,
calc
sign (-a) = -1 : sign_of_neg (neg_neg_of_pos H1)
... = -(sign a) : by rewrite (sign_of_pos H1))
(assume H1 : 0 = a,
calc
sign (-a) = sign (-0) : by rewrite H1
... = sign 0 : by rewrite neg_zero
... = 0 : by rewrite sign_zero
... = -0 : by rewrite neg_zero
... = -(sign 0) : by rewrite sign_zero
... = -(sign a) : by rewrite -H1)
(assume H1 : 0 > a,
calc
sign (-a) = 1 : sign_of_pos (neg_pos_of_neg H1)
... = -(-1) : by rewrite neg_neg
... = -(sign a) : sign_of_neg H1)
theorem sign_mul (a b : A) : sign (a * b) = sign a * sign b :=
lt.by_cases
(assume z_lt_a : 0 < a,
lt.by_cases
(assume z_lt_b : 0 < b,
by rewrite [sign_of_pos z_lt_a, sign_of_pos z_lt_b,
sign_of_pos (mul_pos z_lt_a z_lt_b), one_mul])
(assume z_eq_b : 0 = b, by rewrite [-z_eq_b, mul_zero, *sign_zero, mul_zero])
(assume z_gt_b : 0 > b,
by rewrite [sign_of_pos z_lt_a, sign_of_neg z_gt_b,
sign_of_neg (mul_neg_of_pos_of_neg z_lt_a z_gt_b), one_mul]))
(assume z_eq_a : 0 = a, by rewrite [-z_eq_a, zero_mul, *sign_zero, zero_mul])
(assume z_gt_a : 0 > a,
lt.by_cases
(assume z_lt_b : 0 < b,
by rewrite [sign_of_neg z_gt_a, sign_of_pos z_lt_b,
sign_of_neg (mul_neg_of_neg_of_pos z_gt_a z_lt_b), mul_one])
(assume z_eq_b : 0 = b, by rewrite [-z_eq_b, mul_zero, *sign_zero, mul_zero])
(assume z_gt_b : 0 > b,
by rewrite [sign_of_neg z_gt_a, sign_of_neg z_gt_b,
sign_of_pos (mul_pos_of_neg_of_neg z_gt_a z_gt_b),
neg_mul_neg, one_mul]))
theorem abs_eq_sign_mul (a : A) : abs a = sign a * a :=
lt.by_cases
(assume H1 : 0 < a,
calc
abs a = a : abs_of_pos H1
... = 1 * a : by rewrite one_mul
... = sign a * a : by rewrite (sign_of_pos H1))
(assume H1 : 0 = a,
calc
abs a = abs 0 : by rewrite H1
... = 0 : by rewrite abs_zero
... = 0 * a : by rewrite zero_mul
... = sign 0 * a : by rewrite sign_zero
... = sign a * a : by rewrite H1)
(assume H1 : a < 0,
calc
abs a = -a : abs_of_neg H1
... = -1 * a : by rewrite neg_eq_neg_one_mul
... = sign a * a : by rewrite (sign_of_neg H1))
theorem eq_sign_mul_abs (a : A) : a = sign a * abs a :=
lt.by_cases
(assume H1 : 0 < a,
calc
a = abs a : abs_of_pos H1
... = 1 * abs a : by rewrite one_mul
... = sign a * abs a : by rewrite (sign_of_pos H1))
(assume H1 : 0 = a,
calc
a = 0 : H1⁻¹
... = 0 * abs a : by rewrite zero_mul
... = sign 0 * abs a : by rewrite sign_zero
... = sign a * abs a : by rewrite H1)
(assume H1 : a < 0,
calc
a = -(-a) : by rewrite neg_neg
... = -abs a : by rewrite (abs_of_neg H1)
... = -1 * abs a : by rewrite neg_eq_neg_one_mul
... = sign a * abs a : by rewrite (sign_of_neg H1))
theorem abs_dvd_iff_dvd (a b : A) : (abs a | b) ↔ (a | b) :=
abs.by_cases !iff.refl !neg_dvd_iff_dvd
theorem dvd_abs_iff (a b : A) : (a | abs b) ↔ (a | b) :=
abs.by_cases !iff.refl !dvd_neg_iff_dvd
theorem abs_mul (a b : A) : abs (a * b) = abs a * abs b :=
or.elim (le.total 0 a)
(assume H1 : 0 ≤ a,
or.elim (le.total 0 b)
(assume H2 : 0 ≤ b,
calc
abs (a * b) = a * b : abs_of_nonneg (mul_nonneg H1 H2)
... = abs a * b : by rewrite (abs_of_nonneg H1)
... = abs a * abs b : by rewrite (abs_of_nonneg H2))
(assume H2 : b ≤ 0,
calc
abs (a * b) = -(a * b) : abs_of_nonpos (mul_nonpos_of_nonneg_of_nonpos H1 H2)
... = a * -b : by rewrite neg_mul_eq_mul_neg
... = abs a * -b : by rewrite (abs_of_nonneg H1)
... = abs a * abs b : by rewrite (abs_of_nonpos H2)))
(assume H1 : a ≤ 0,
or.elim (le.total 0 b)
(assume H2 : 0 ≤ b,
calc
abs (a * b) = -(a * b) : abs_of_nonpos (mul_nonpos_of_nonpos_of_nonneg H1 H2)
... = -a * b : by rewrite neg_mul_eq_neg_mul
... = abs a * b : by rewrite (abs_of_nonpos H1)
... = abs a * abs b : by rewrite (abs_of_nonneg H2))
(assume H2 : b ≤ 0,
calc
abs (a * b) = a * b : abs_of_nonneg (mul_nonneg_of_nonpos_of_nonpos H1 H2)
... = -a * -b : by rewrite neg_mul_neg
... = abs a * -b : by rewrite (abs_of_nonpos H1)
... = abs a * abs b : by rewrite (abs_of_nonpos H2)))
theorem abs_mul_self (a : A) : abs a * abs a = a * a :=
abs.by_cases rfl !neg_mul_neg
end
/- TODO: Multiplication and one, starting with mult_right_le_one_le. -/
end algebra