517 lines
19 KiB
Text
517 lines
19 KiB
Text
/-
|
|
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 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 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 (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 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 (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 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 : ∃ 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 : ∃ 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 (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 (div_pos_of_pos Hc)
|
|
... = b / c : div_eq_mul_one_div
|
|
|
|
theorem mul_le_of_ge_div_neg (Hc : c < 0) (H : a ≥ b / c) : a * c ≤ b :=
|
|
div_mul_cancel (ne_of_lt Hc) ▸ mul_le_mul_of_nonpos_right H (le_of_lt Hc)
|
|
|
|
theorem ge_div_of_mul_le_neg (Hc : c < 0) (H : a * c ≤ b) : a ≥ b / c :=
|
|
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 (div_neg_of_neg Hc))
|
|
... = b / c : div_eq_mul_one_div
|
|
|
|
theorem mul_lt_of_gt_div_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 gt_div_of_mul_gt_neg (Hc : c < 0) (H : a * c < b) : a > b / c :=
|
|
calc
|
|
a = a * c * (1 / c) : mul_mul_div (ne_of_lt Hc)
|
|
... > b * (1 / c) : mul_lt_mul_of_neg_right H (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 (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 pos_div_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 div_pos_of_pos,
|
|
exact Hb
|
|
end
|
|
|
|
theorem nonneg_div_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 div_pos_of_pos,
|
|
exact Hb
|
|
end
|
|
|
|
theorem neg_div_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 div_pos_of_pos,
|
|
exact Hb
|
|
end
|
|
|
|
theorem nonpos_div_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 div_pos_of_pos,
|
|
exact Hb
|
|
end
|
|
|
|
theorem neg_div_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 div_neg_of_neg,
|
|
exact Hb
|
|
end
|
|
|
|
theorem nonpos_div_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 div_neg_of_neg,
|
|
exact Hb
|
|
end
|
|
|
|
theorem pos_div_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 div_neg_of_neg,
|
|
exact Hb
|
|
end
|
|
|
|
|
|
theorem nonneg_div_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 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 (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 (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 (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 (div_neg_of_neg Hc))
|
|
end
|
|
|
|
theorem two_ne_zero : (1 : A) + 1 ≠ 0 :=
|
|
ne.symm (ne_of_lt (add_pos zero_lt_one zero_lt_one))
|
|
|
|
notation 2 := 1 + 1
|
|
|
|
theorem add_halves : 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 / 2 = a / 2 :=
|
|
by rewrite [-{a}add_halves at {1}, add_sub_cancel]
|
|
|
|
theorem div_two_sub_self : a / 2 - a = - (a / 2) :=
|
|
by rewrite [-{a}add_halves at {2}, sub_add_eq_sub_sub, sub_self, zero_sub]
|
|
|
|
theorem nonneg_le_nonneg_of_squares_le (Ha : a ≥ 0) (Hb : b ≥ 0) (H : a * a ≤ b * b) : a ≤ b :=
|
|
begin
|
|
apply le_of_not_gt,
|
|
intro Hab,
|
|
let Hposa := lt_of_le_of_lt Hb Hab,
|
|
let H' := calc
|
|
b * b ≤ a * b : mul_le_mul_of_nonneg_right (le_of_lt Hab) Hb
|
|
... < a * a : mul_lt_mul_of_pos_left Hab Hposa,
|
|
apply (not_le_of_gt H') H
|
|
end
|
|
|
|
theorem div_two : (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 pos_div_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 [div_mul_eq_div_mul_one_div Hb (ne_of_gt He), 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 div_pos_of_pos He
|
|
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_div_pos (H : 0 < 1 / a) : 0 < a :=
|
|
have H1 : 0 < 1 / (1 / a), from 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))),
|
|
(div_div (ne_zero_of_one_div_ne_zero H2)) ▸ H1
|
|
|
|
theorem neg_of_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 (one_div_neg_eq_neg_one_div Ha)⁻¹ ▸ H1,
|
|
have H3 : 0 < -a, from pos_of_div_pos H2,
|
|
neg_of_neg_pos H3
|
|
|
|
-- why is mul_le_mul under ordered_ring namespace?
|
|
theorem le_of_div_le (H : 0 < a) (Hl : 1 / a ≤ 1 / b) : b ≤ a :=
|
|
have Hb : 0 < b, from pos_of_div_pos (calc
|
|
0 < 1 / a : 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) : ordered_ring.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_div_le_neg (H : b < 0) (Hl : 1 / a ≤ 1 / b) : b ≤ a :=
|
|
assert Ha : a ≠ 0, from ne_of_lt (neg_of_div_neg (calc
|
|
1 / a ≤ 1 / b : Hl
|
|
... < 0 : 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 [one_div_neg_eq_neg_one_div (ne_of_lt H)]
|
|
... ≤ - (1 / a) : Hl'
|
|
... = 1 / -a : by rewrite [one_div_neg_eq_neg_one_div Ha],
|
|
le_of_neg_le_neg (le_of_div_le H' Hl'')
|
|
|
|
theorem lt_of_div_lt (H : 0 < a) (Hl : 1 / a < 1 / b) : b < a :=
|
|
have Hb : 0 < b, from pos_of_div_pos (calc
|
|
0 < 1 / a : 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_div_lt_neg (H : b < 0) (Hl : 1 / a < 1 / b) : b < a :=
|
|
have H1 : b ≤ a, from le_of_div_le_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_div_le 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_div_lt 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_div_le_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_div_lt_neg Hb H')))
|
|
|
|
|
|
theorem one_lt_div (H1 : 0 < a) (H2 : a < 1) : 1 < 1 / a :=
|
|
one_div_one ▸ div_lt_div_of_lt H1 H2
|
|
|
|
theorem one_le_div (H1 : 0 < a) (H2 : a ≤ 1) : 1 ≤ 1 / a :=
|
|
one_div_one ▸ div_le_div_of_le H1 H2
|
|
|
|
theorem neg_one_lt_div_neg (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 neg_one_le_div_neg (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],
|
|
rewrite -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,
|
|
exact Hb, exact H
|
|
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 div_pos_of_pos He
|
|
end
|
|
|
|
theorem abs_one_div : abs (1 / a) = 1 / abs a :=
|
|
if H : a > 0 then
|
|
by rewrite [abs_of_pos H, abs_of_pos (div_pos_of_pos H)]
|
|
else
|
|
(if H' : a < 0 then
|
|
by rewrite [abs_of_neg H', abs_of_neg (div_neg_of_neg H'),
|
|
-(one_div_neg_eq_neg_one_div (ne_of_lt H'))]
|
|
else
|
|
have Heq [visible] : 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 ge_sub_of_abs_sub_le_left (H : abs (a - b) ≤ c) : a ≥ b - c :=
|
|
if Hz : 0 ≤ a - b then
|
|
(calc
|
|
a ≥ b : (iff.mp !sub_nonneg_iff_le) Hz
|
|
... ≥ b - c : sub_le_of_nonneg _ _ (le.trans !abs_nonneg H))
|
|
else
|
|
(have Habs : b - a ≤ c, by rewrite [abs_of_neg (lt_of_not_ge Hz) at H, neg_sub at H]; apply H,
|
|
have Habs' : b ≤ c + a, from (iff.mpr !le_add_iff_sub_right_le) Habs,
|
|
(iff.mp !le_add_iff_sub_left_le) Habs')
|
|
|
|
theorem ge_sub_of_abs_sub_le_right (H : abs (a - b) ≤ c) : b ≥ a - c :=
|
|
ge_sub_of_abs_sub_le_left (!abs_sub ▸ H)
|
|
|
|
theorem abs_sub_square : abs (a - b) * abs (a - b) = a * a + b * b - 2 * a * b :=
|
|
by rewrite [abs_mul_self, *mul_sub_left_distrib, *mul_sub_right_distrib,
|
|
sub_add_eq_sub_sub, sub_neg_eq_add, *right_distrib, sub_add_eq_sub_sub, *one_mul,
|
|
*add.assoc, {_ + b * b}add.comm, {_ + (b * b + _)}add.comm, mul.comm b a, *add.assoc]
|
|
|
|
end discrete_linear_ordered_field
|
|
end algebra
|