lean2/library/algebra/ordered_field.lean

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/-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis
-/
import algebra.ordered_ring algebra.field
open eq eq.ops
namespace algebra
structure linear_ordered_field [class] (A : Type) extends linear_ordered_ring A, field A
section linear_ordered_field
variable {A : Type}
variables [s : linear_ordered_field A] {a b c d : A}
include s
-- helpers for following
theorem mul_zero_lt_mul_inv_of_pos (H : 0 < a) : a * 0 < a * (1 / a) :=
calc
a * 0 = 0 : mul_zero
... < 1 : zero_lt_one
... = a * a⁻¹ : mul_inv_cancel (ne.symm (ne_of_lt H))
... = a * (1 / a) : inv_eq_one_div
theorem mul_zero_lt_mul_inv_of_neg (H : a < 0) : a * 0 < a * (1 / a) :=
calc
a * 0 = 0 : mul_zero
... < 1 : zero_lt_one
... = a * a⁻¹ : mul_inv_cancel (ne_of_lt H)
... = a * (1 / a) : inv_eq_one_div
theorem one_div_pos_of_pos (H : 0 < a) : 0 < 1 / a :=
lt_of_mul_lt_mul_left (mul_zero_lt_mul_inv_of_pos H) (le_of_lt H)
theorem one_div_neg_of_neg (H : a < 0) : 1 / a < 0 :=
gt_of_mul_lt_mul_neg_left (mul_zero_lt_mul_inv_of_neg H) (le_of_lt H)
theorem le_mul_of_ge_one_right (Hb : b ≥ 0) (H : a ≥ 1) : b ≤ b * a :=
mul_one _ ▸ (mul_le_mul_of_nonneg_left H Hb)
theorem lt_mul_of_gt_one_right (Hb : b > 0) (H : a > 1) : b < b * a :=
mul_one _ ▸ (mul_lt_mul_of_pos_left H Hb)
theorem one_le_div_iff_le (a : A) {b : A} (Hb : b > 0) : 1 ≤ a / b ↔ b ≤ a :=
have Hb' : b ≠ 0, from ne.symm (ne_of_lt Hb),
iff.intro
(assume H : 1 ≤ a / b,
calc
b = b : refl
... ≤ b * (a / b) : le_mul_of_ge_one_right (le_of_lt Hb) H
... = a : mul_div_cancel' Hb')
(assume H : b ≤ a,
have Hbinv : 1 / b > 0, from one_div_pos_of_pos Hb, calc
1 = b * (1 / b) : mul_one_div_cancel Hb'
... ≤ a * (1 / b) : mul_le_mul_of_nonneg_right H (le_of_lt Hbinv)
... = a / b : div_eq_mul_one_div)
theorem le_of_one_le_div (Hb : b > 0) (H : 1 ≤ a / b) : b ≤ a :=
(iff.mp (!one_le_div_iff_le Hb)) H
theorem one_le_div_of_le (Hb : b > 0) (H : b ≤ a) : 1 ≤ a / b :=
(iff.mpr (!one_le_div_iff_le Hb)) H
theorem one_lt_div_iff_lt (a : A) {b : A} (Hb : b > 0) : 1 < a / b ↔ b < a :=
have Hb' : b ≠ 0, from ne.symm (ne_of_lt Hb),
iff.intro
(assume H : 1 < a / b,
calc
b < b * (a / b) : lt_mul_of_gt_one_right Hb H
... = a : mul_div_cancel' Hb')
(assume H : b < a,
have Hbinv : 1 / b > 0, from one_div_pos_of_pos Hb, calc
1 = b * (1 / b) : mul_one_div_cancel Hb'
... < a * (1 / b) : mul_lt_mul_of_pos_right H Hbinv
... = a / b : div_eq_mul_one_div)
theorem lt_of_one_lt_div (Hb : b > 0) (H : 1 < a / b) : b < a :=
(iff.mp (!one_lt_div_iff_lt Hb)) H
theorem one_lt_div_of_lt (Hb : b > 0) (H : b < a) : 1 < a / b :=
(iff.mpr (!one_lt_div_iff_lt Hb)) H
theorem exists_lt (a : A) : ∃ x, x < a :=
have H : a - 1 < a, from add_lt_of_le_of_neg (le.refl _) zero_gt_neg_one,
exists.intro _ H
theorem exists_gt (a : A) : ∃ x, x > a :=
have H : a + 1 > a, from lt_add_of_le_of_pos (le.refl _) zero_lt_one,
exists.intro _ H
-- the following theorems amount to four iffs, for <, ≤, ≥, >.
theorem mul_le_of_le_div (Hc : 0 < c) (H : a ≤ b / c) : a * c ≤ b :=
!div_mul_cancel (ne.symm (ne_of_lt Hc)) ▸ mul_le_mul_of_nonneg_right H (le_of_lt Hc)
theorem le_div_of_mul_le (Hc : 0 < c) (H : a * c ≤ b) : a ≤ b / c :=
calc
a = a * c * (1 / c) : !mul_mul_div (ne.symm (ne_of_lt Hc))
... ≤ b * (1 / c) : mul_le_mul_of_nonneg_right H (le_of_lt (one_div_pos_of_pos Hc))
... = b / c : div_eq_mul_one_div
theorem mul_lt_of_lt_div (Hc : 0 < c) (H : a < b / c) : a * c < b :=
!div_mul_cancel (ne.symm (ne_of_lt Hc)) ▸ mul_lt_mul_of_pos_right H Hc
theorem lt_div_of_mul_lt (Hc : 0 < c) (H : a * c < b) : a < b / c :=
calc
a = a * c * (1 / c) : !mul_mul_div (ne.symm (ne_of_lt Hc))
... < b * (1 / c) : mul_lt_mul_of_pos_right H (one_div_pos_of_pos Hc)
... = b / c : div_eq_mul_one_div
theorem mul_le_of_div_le_of_neg (Hc : c < 0) (H : b / c ≤ a) : a * c ≤ b :=
!div_mul_cancel (ne_of_lt Hc) ▸ mul_le_mul_of_nonpos_right H (le_of_lt Hc)
theorem div_le_of_mul_le_of_neg (Hc : c < 0) (H : a * c ≤ b) : b / c ≤ a :=
calc
a = a * c * (1 / c) : !mul_mul_div (ne_of_lt Hc)
... ≥ b * (1 / c) : mul_le_mul_of_nonpos_right H (le_of_lt (one_div_neg_of_neg Hc))
... = b / c : div_eq_mul_one_div
theorem mul_lt_of_gt_div_of_neg (Hc : c < 0) (H : a > b / c) : a * c < b :=
!div_mul_cancel (ne_of_lt Hc) ▸ mul_lt_mul_of_neg_right H Hc
theorem div_lt_of_mul_gt_of_neg (Hc : c < 0) (H : a * c < b) : b / c < a :=
calc
a = a * c * (1 / c) : !mul_mul_div (ne_of_lt Hc)
... > b * (1 / c) : mul_lt_mul_of_neg_right H (one_div_neg_of_neg Hc)
... = b / c : div_eq_mul_one_div
theorem div_le_of_le_mul (Hb : b > 0) (H : a ≤ b * c) : a / b ≤ c :=
calc
a / b = a * (1 / b) : div_eq_mul_one_div
... ≤ (b * c) * (1 / b) : mul_le_mul_of_nonneg_right H (le_of_lt (one_div_pos_of_pos Hb))
... = (b * c) / b : div_eq_mul_one_div
... = c : mul_div_cancel_left (ne.symm (ne_of_lt Hb))
theorem le_mul_of_div_le (Hc : c > 0) (H : a / c ≤ b) : a ≤ b * c :=
calc
a = a / c * c : !div_mul_cancel (ne.symm (ne_of_lt Hc))
... ≤ b * c : mul_le_mul_of_nonneg_right H (le_of_lt Hc)
-- following these in the isabelle file, there are 8 biconditionals for the above with - signs
-- skipping for now
theorem mul_sub_mul_div_mul_neg (Hc : c ≠ 0) (Hd : d ≠ 0) (H : a / c < b / d) :
(a * d - b * c) / (c * d) < 0 :=
have H1 : a / c - b / d < 0, from calc
a / c - b / d < b / d - b / d : sub_lt_sub_right H
... = 0 : sub_self,
calc
0 > a / c - b / d : H1
... = (a * d - c * b) / (c * d) : !div_sub_div Hc Hd
... = (a * d - b * c) / (c * d) : mul.comm
theorem mul_sub_mul_div_mul_nonpos (Hc : c ≠ 0) (Hd : d ≠ 0) (H : a / c ≤ b / d) :
(a * d - b * c) / (c * d) ≤ 0 :=
have H1 : a / c - b / d ≤ 0, from calc
a / c - b / d ≤ b / d - b / d : sub_le_sub_right H
... = 0 : sub_self,
calc
0 ≥ a / c - b / d : H1
... = (a * d - c * b) / (c * d) : !div_sub_div Hc Hd
... = (a * d - b * c) / (c * d) : mul.comm
theorem div_lt_div_of_mul_sub_mul_div_neg (Hc : c ≠ 0) (Hd : d ≠ 0)
(H : (a * d - b * c) / (c * d) < 0) : a / c < b / d :=
assert H1 : (a * d - c * b) / (c * d) < 0, by rewrite [mul.comm c b]; exact H,
assert H2 : a / c - b / d < 0, by rewrite [!div_sub_div Hc Hd]; exact H1,
assert H3 : a / c - b / d + b / d < 0 + b / d, from add_lt_add_right H2 _,
begin rewrite [zero_add at H3, neg_add_cancel_right at H3], exact H3 end
theorem div_le_div_of_mul_sub_mul_div_nonpos (Hc : c ≠ 0) (Hd : d ≠ 0)
(H : (a * d - b * c) / (c * d) ≤ 0) : a / c ≤ b / d :=
assert H1 : (a * d - c * b) / (c * d) ≤ 0, by rewrite [mul.comm c b]; exact H,
assert H2 : a / c - b / d ≤ 0, by rewrite [!div_sub_div Hc Hd]; exact H1,
assert H3 : a / c - b / d + b / d ≤ 0 + b / d, from add_le_add_right H2 _,
begin rewrite [zero_add at H3, neg_add_cancel_right at H3], exact H3 end
theorem div_pos_of_pos_of_pos (Ha : 0 < a) (Hb : 0 < b) : 0 < a / b :=
begin
rewrite div_eq_mul_one_div,
apply mul_pos,
exact Ha,
apply one_div_pos_of_pos,
exact Hb
end
theorem div_nonneg_of_nonneg_of_pos (Ha : 0 ≤ a) (Hb : 0 < b) : 0 ≤ a / b :=
begin
rewrite div_eq_mul_one_div,
apply mul_nonneg,
exact Ha,
apply le_of_lt,
apply one_div_pos_of_pos,
exact Hb
end
theorem div_neg_of_neg_of_pos (Ha : a < 0) (Hb : 0 < b) : a / b < 0:=
begin
rewrite div_eq_mul_one_div,
apply mul_neg_of_neg_of_pos,
exact Ha,
apply one_div_pos_of_pos,
exact Hb
end
theorem div_nonpos_of_nonpos_of_pos (Ha : a ≤ 0) (Hb : 0 < b) : a / b ≤ 0 :=
begin
rewrite div_eq_mul_one_div,
apply mul_nonpos_of_nonpos_of_nonneg,
exact Ha,
apply le_of_lt,
apply one_div_pos_of_pos,
exact Hb
end
theorem div_neg_of_pos_of_neg (Ha : 0 < a) (Hb : b < 0) : a / b < 0 :=
begin
rewrite div_eq_mul_one_div,
apply mul_neg_of_pos_of_neg,
exact Ha,
apply one_div_neg_of_neg,
exact Hb
end
theorem div_nonpos_of_nonneg_of_neg (Ha : 0 ≤ a) (Hb : b < 0) : a / b ≤ 0 :=
begin
rewrite div_eq_mul_one_div,
apply mul_nonpos_of_nonneg_of_nonpos,
exact Ha,
apply le_of_lt,
apply one_div_neg_of_neg,
exact Hb
end
theorem div_pos_of_neg_of_neg (Ha : a < 0) (Hb : b < 0) : 0 < a / b :=
begin
rewrite div_eq_mul_one_div,
apply mul_pos_of_neg_of_neg,
exact Ha,
apply one_div_neg_of_neg,
exact Hb
end
theorem div_nonneg_of_nonpos_of_neg (Ha : a ≤ 0) (Hb : b < 0) : 0 ≤ a / b :=
begin
rewrite div_eq_mul_one_div,
apply mul_nonneg_of_nonpos_of_nonpos,
exact Ha,
apply le_of_lt,
apply one_div_neg_of_neg,
exact Hb
end
theorem div_lt_div_of_lt_of_pos (H : a < b) (Hc : 0 < c) : a / c < b / c :=
begin
rewrite [{a/c}div_eq_mul_one_div, {b/c}div_eq_mul_one_div],
exact mul_lt_mul_of_pos_right H (one_div_pos_of_pos Hc)
end
theorem div_le_div_of_le_of_pos (H : a ≤ b) (Hc : 0 < c) : a / c ≤ b / c :=
begin
rewrite [{a/c}div_eq_mul_one_div, {b/c}div_eq_mul_one_div],
exact mul_le_mul_of_nonneg_right H (le_of_lt (one_div_pos_of_pos Hc))
end
theorem div_lt_div_of_lt_of_neg (H : b < a) (Hc : c < 0) : a / c < b / c :=
begin
rewrite [{a/c}div_eq_mul_one_div, {b/c}div_eq_mul_one_div],
exact mul_lt_mul_of_neg_right H (one_div_neg_of_neg Hc)
end
theorem div_le_div_of_le_of_neg (H : b ≤ a) (Hc : c < 0) : a / c ≤ b / c :=
begin
rewrite [{a/c}div_eq_mul_one_div, {b/c}div_eq_mul_one_div],
exact mul_le_mul_of_nonpos_right H (le_of_lt (one_div_neg_of_neg Hc))
end
theorem two_pos : (1 : A) + 1 > 0 :=
add_pos zero_lt_one zero_lt_one
theorem two_ne_zero : (1 : A) + 1 ≠ 0 :=
ne.symm (ne_of_lt two_pos)
notation 2 := 1 + 1
theorem add_halves (a : A) : a / 2 + a / 2 = a :=
calc
a / 2 + a / 2 = (a + a) / 2 : by rewrite div_add_div_same
... = (a * 1 + a * 1) / 2 : by rewrite mul_one
... = (a * 2) / 2 : by rewrite left_distrib
... = a : by rewrite [@mul_div_cancel A _ _ _ two_ne_zero]
theorem sub_self_div_two (a : A) : a - a / 2 = a / 2 :=
by rewrite [-{a}add_halves at {1}, add_sub_cancel]
theorem div_two_sub_self (a : A) : a / 2 - a = - (a / 2) :=
by rewrite [-{a}add_halves at {2}, sub_add_eq_sub_sub, sub_self, zero_sub]
theorem add_self_div_two (a : A) : (a + a) / 2 = a :=
symm (iff.mpr (!eq_div_iff_mul_eq (ne_of_gt (add_pos zero_lt_one zero_lt_one)))
(by rewrite [left_distrib, *mul_one]))
theorem two_ge_one : (2 : A) ≥ 1 :=
by rewrite -(add_zero 1) at {3}; apply add_le_add_left; apply zero_le_one
theorem mul_le_mul_of_mul_div_le (H : a * (b / c) ≤ d) (Hc : c > 0) : b * a ≤ d * c :=
begin
rewrite [-mul_div_assoc at H, mul.comm b],
apply le_mul_of_div_le Hc H
end
theorem div_two_lt_of_pos (H : a > 0) : a / (1 + 1) < a :=
have Ha : a / (1 + 1) > 0, from div_pos_of_pos_of_pos H (add_pos zero_lt_one zero_lt_one),
calc
a / (1 + 1) < a / (1 + 1) + a / (1 + 1) : lt_add_of_pos_left Ha
... = a : add_halves
theorem div_mul_le_div_mul_of_div_le_div_pos {e : A} (Hb : b ≠ 0) (Hd : d ≠ 0) (H : a / b ≤ c / d)
(He : e > 0) : a / (b * e) ≤ c / (d * e) :=
begin
rewrite [!field.div_mul_eq_div_mul_one_div Hb (ne_of_gt He),
!field.div_mul_eq_div_mul_one_div Hd (ne_of_gt He)],
apply mul_le_mul_of_nonneg_right H,
apply le_of_lt,
apply one_div_pos_of_pos He
end
theorem exists_add_lt_and_pos_of_lt (H : b < a) : ∃ c : A, b + c < a ∧ c > 0 :=
exists.intro ((a - b) / (1 + 1))
(and.intro (assert H2 : a + a > (b + b) + (a - b), from calc
a + a > b + a : add_lt_add_right H
... = b + a + b - b : add_sub_cancel
... = b + b + a - b : add.right_comm
... = (b + b) + (a - b) : add_sub,
assert H3 : (a + a) / (1 + 1) > ((b + b) + (a - b)) / (1 + 1),
from div_lt_div_of_lt_of_pos H2 two_pos,
by rewrite [add_self_div_two at H3, -div_add_div_same at H3, add_self_div_two at H3];
exact H3)
(div_pos_of_pos_of_pos (iff.mpr !sub_pos_iff_lt H) two_pos))
theorem ge_of_forall_ge_sub {a b : A} (H : ∀ ε : A, ε > 0 → a ≥ b - ε) : a ≥ b :=
begin
apply le_of_not_gt,
intro Hb,
cases exists_add_lt_and_pos_of_lt Hb with [c, Hc],
let Hc' := H c (and.right Hc),
apply (not_le_of_gt (and.left Hc)) (iff.mpr !le_add_iff_sub_right_le Hc')
end
end linear_ordered_field
structure discrete_linear_ordered_field [class] (A : Type) extends linear_ordered_field A,
decidable_linear_ordered_comm_ring A :=
(inv_zero : inv zero = zero)
section discrete_linear_ordered_field
variable {A : Type}
variables [s : discrete_linear_ordered_field A] {a b c : A}
include s
definition dec_eq_of_dec_lt : ∀ x y : A, decidable (x = y) :=
take x y,
decidable.by_cases
(assume H : x < y, decidable.inr (ne_of_lt H))
(assume H : ¬ x < y,
decidable.by_cases
(assume H' : y < x, decidable.inr (ne.symm (ne_of_lt H')))
(assume H' : ¬ y < x,
decidable.inl (le.antisymm (le_of_not_gt H') (le_of_not_gt H))))
definition discrete_linear_ordered_field.to_discrete_field [trans-instance] [reducible] [coercion]
: discrete_field A :=
⦃ discrete_field, s, has_decidable_eq := dec_eq_of_dec_lt⦄
theorem pos_of_one_div_pos (H : 0 < 1 / a) : 0 < a :=
have H1 : 0 < 1 / (1 / a), from one_div_pos_of_pos H,
have H2 : 1 / a ≠ 0, from
(assume H3 : 1 / a = 0,
have H4 : 1 / (1 / a) = 0, from H3⁻¹ ▸ !div_zero,
absurd H4 (ne.symm (ne_of_lt H1))),
(division_ring.one_div_one_div (ne_zero_of_one_div_ne_zero H2)) ▸ H1
theorem neg_of_one_div_neg (H : 1 / a < 0) : a < 0 :=
have H1 : 0 < - (1 / a), from neg_pos_of_neg H,
have Ha : a ≠ 0, from ne_zero_of_one_div_ne_zero (ne_of_lt H),
have H2 : 0 < 1 / (-a), from (division_ring.one_div_neg_eq_neg_one_div Ha)⁻¹ ▸ H1,
have H3 : 0 < -a, from pos_of_one_div_pos H2,
neg_of_neg_pos H3
theorem le_of_one_div_le_one_div (H : 0 < a) (Hl : 1 / a ≤ 1 / b) : b ≤ a :=
have Hb : 0 < b, from pos_of_one_div_pos (calc
0 < 1 / a : one_div_pos_of_pos H
... ≤ 1 / b : Hl),
have H' : 1 ≤ a / b, from (calc
1 = a / a : div_self (ne.symm (ne_of_lt H))
... = a * (1 / a) : div_eq_mul_one_div
... ≤ a * (1 / b) : mul_le_mul_of_nonneg_left Hl (le_of_lt H)
... = a / b : div_eq_mul_one_div
), le_of_one_le_div Hb H'
theorem le_of_one_div_le_one_div_of_neg (H : b < 0) (Hl : 1 / a ≤ 1 / b) : b ≤ a :=
assert Ha : a ≠ 0, from ne_of_lt (neg_of_one_div_neg (calc
1 / a ≤ 1 / b : Hl
... < 0 : one_div_neg_of_neg H)),
have H' : -b > 0, from neg_pos_of_neg H,
have Hl' : - (1 / b) ≤ - (1 / a), from neg_le_neg Hl,
have Hl'' : 1 / - b ≤ 1 / - a, from calc
1 / -b = - (1 / b) : by rewrite [division_ring.one_div_neg_eq_neg_one_div (ne_of_lt H)]
... ≤ - (1 / a) : Hl'
... = 1 / -a : by rewrite [division_ring.one_div_neg_eq_neg_one_div Ha],
le_of_neg_le_neg (le_of_one_div_le_one_div H' Hl'')
theorem lt_of_one_div_lt_one_div (H : 0 < a) (Hl : 1 / a < 1 / b) : b < a :=
have Hb : 0 < b, from pos_of_one_div_pos (calc
0 < 1 / a : one_div_pos_of_pos H
... < 1 / b : Hl),
have H : 1 < a / b, from (calc
1 = a / a : div_self (ne.symm (ne_of_lt H))
... = a * (1 / a) : div_eq_mul_one_div
... < a * (1 / b) : mul_lt_mul_of_pos_left Hl H
... = a / b : div_eq_mul_one_div),
lt_of_one_lt_div Hb H
theorem lt_of_one_div_lt_one_div_of_neg (H : b < 0) (Hl : 1 / a < 1 / b) : b < a :=
have H1 : b ≤ a, from le_of_one_div_le_one_div_of_neg H (le_of_lt Hl),
have Hn : b ≠ a, from
(assume Hn' : b = a,
have Hl' : 1 / a = 1 / b, from Hn' ▸ refl _,
absurd Hl' (ne_of_lt Hl)),
lt_of_le_of_ne H1 Hn
theorem div_lt_div_of_lt (Ha : 0 < a) (H : a < b) : 1 / b < 1 / a :=
lt_of_not_ge
(assume H',
absurd H (not_lt_of_ge (le_of_one_div_le_one_div Ha H')))
theorem div_le_div_of_le (Ha : 0 < a) (H : a ≤ b) : 1 / b ≤ 1 / a :=
le_of_not_gt
(assume H',
absurd H (not_le_of_gt (lt_of_one_div_lt_one_div Ha H')))
theorem div_lt_div_of_lt_neg (Hb : b < 0) (H : a < b) : 1 / b < 1 / a :=
lt_of_not_ge
(assume H',
absurd H (not_lt_of_ge (le_of_one_div_le_one_div_of_neg Hb H')))
theorem div_le_div_of_le_neg (Hb : b < 0) (H : a ≤ b) : 1 / b ≤ 1 / a :=
le_of_not_gt
(assume H',
absurd H (not_le_of_gt (lt_of_one_div_lt_one_div_of_neg Hb H')))
theorem one_lt_one_div (H1 : 0 < a) (H2 : a < 1) : 1 < 1 / a :=
one_div_one ▸ div_lt_div_of_lt H1 H2
theorem one_le_one_div (H1 : 0 < a) (H2 : a ≤ 1) : 1 ≤ 1 / a :=
one_div_one ▸ div_le_div_of_le H1 H2
theorem one_div_lt_neg_one (H1 : a < 0) (H2 : -1 < a) : 1 / a < -1 :=
one_div_neg_one_eq_neg_one ▸ div_lt_div_of_lt_neg H1 H2
theorem one_div_le_neg_one (H1 : a < 0) (H2 : -1 ≤ a) : 1 / a ≤ -1 :=
one_div_neg_one_eq_neg_one ▸ div_le_div_of_le_neg H1 H2
theorem div_lt_div_of_pos_of_lt_of_pos (Hb : 0 < b) (H : b < a) (Hc : 0 < c) : c / a < c / b :=
begin
apply iff.mp !sub_neg_iff_lt,
rewrite [div_eq_mul_one_div, {c / b}div_eq_mul_one_div, -mul_sub_left_distrib],
apply mul_neg_of_pos_of_neg,
exact Hc,
apply iff.mpr !sub_neg_iff_lt,
apply div_lt_div_of_lt,
repeat assumption
end
theorem div_mul_le_div_mul_of_div_le_div_pos' {d e : A} (H : a / b ≤ c / d)
(He : e > 0) : a / (b * e) ≤ c / (d * e) :=
begin
rewrite [2 div_mul_eq_div_mul_one_div],
apply mul_le_mul_of_nonneg_right H,
apply le_of_lt,
apply one_div_pos_of_pos He
end
theorem abs_one_div (a : A) : abs (1 / a) = 1 / abs a :=
if H : a > 0 then
by rewrite [abs_of_pos H, abs_of_pos (one_div_pos_of_pos H)]
else
(if H' : a < 0 then
by rewrite [abs_of_neg H', abs_of_neg (one_div_neg_of_neg H'),
-(division_ring.one_div_neg_eq_neg_one_div (ne_of_lt H'))]
else
assert Heq : a = 0, from eq_of_le_of_ge (le_of_not_gt H) (le_of_not_gt H'),
by rewrite [Heq, div_zero, *abs_zero, div_zero])
theorem sign_eq_div_abs (a : A) : sign a = a / (abs a) :=
decidable.by_cases
(suppose a = 0, by subst a; rewrite [zero_div, sign_zero])
(suppose a ≠ 0,
have abs a ≠ 0, from assume H, this (eq_zero_of_abs_eq_zero H),
!eq_div_of_mul_eq this !eq_sign_mul_abs⁻¹)
end discrete_linear_ordered_field
end algebra