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feat(library/algebra/ordered_field): complete proofs of many theorems. Define discrete linear ordered field

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Rob Lewis 2015-03-18 17:30:35 -04:00 committed by Leonardo de Moura
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library/algebra/ordered_field.lean
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@ -5,9 +5,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
Module: algebra.ordered_field
Authors: Robert Lewis
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_ring algebra.field
@ -30,88 +27,156 @@ section linear_ordered_field
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 = (-1 * c) * b : neg_eq_neg_one_mul
... = -1 * (c * b) : mul.assoc
... = - (c * b) : neg_eq_neg_one_mul
... < -(c * a) : H2
... = -1 * (c * a) : neg_eq_neg_one_mul
... = (-1 * c) * a : mul.assoc
... = (-c) * a : neg_eq_neg_one_mul
... = -1 * (c * b) : mul.assoc
... = - (c * b) : neg_eq_neg_one_mul
... < -(c * a) : H2
... = -1 * (c * a) : neg_eq_neg_one_mul
... = (-1 * c) * a : mul.assoc
... = (-c) * a : neg_eq_neg_one_mul
),
lt_of_mul_lt_mul_left H3 nhc
-- 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 * 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 * 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)
-- this would go in ring, if it worked
theorem ne_zero_of_div_ne_zero (H : 1 / a ≠ 0) : a ≠ 0 :=
assume Ha : a = 0, sorry
theorem pos_of_div_pos (H : 0 < 1 / a) : 0 < a :=
have H1 : 0 < 1 / (1 / a), from div_pos_of_pos H,
-- want a ≠ 0. Can get this with decidable =, from discrete_field.inv_zero_imp_zero
div_div (sorry) ▸ H1
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 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 neg_of_div_neg (H : 1 / a < 0) : a < 0 :=
sorry
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
-- is this theorem (and le_of_div_le which depends on it) classical?
theorem one_le_div_iff_le : 1 ≤ a / b ↔ b ≤ a :=
sorry
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 one_lt_div_iff_lt : 1 < a / b ↔ b < a :=
sorry
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 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 : refl
... < 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)
-- 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 H : 1 ≤ a / b, from (calc
1 = a / a : div_self (ne.symm (ne_of_lt H))
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
), (iff.mp one_le_div_iff_le) H
... = a / b : div_eq_mul_one_div
), iff.mp (one_le_div_iff_le Hb) H'
theorem lt_of_div_lt (H : a > 0) (Hl : 1 / a < 1 / b) : b < a :=
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))
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
), (iff.mp one_lt_div_iff_lt) H
... = a / b : div_eq_mul_one_div),
iff.mp (one_lt_div_iff_lt Hb) H
theorem le_of_div_le_neg (H : b < 0) (Hl : 1 / a ≤ 1 / b) : b ≤ a :=
have Ha : 1 / a < 0, from (calc
have Ha : a ≠ 0, from ne_of_lt (neg_of_div_neg (calc
1 / a ≤ 1 / b : Hl
... < 0 : div_neg_of_neg H
),
have Ha' : a ≠ 0, from ne_of_lt (neg_of_div_neg Ha),
have H : 1 ≤ a / b, from (calc
1 = a / a : div_self Ha'
... ≤ a / b : sorry), sorry
... < 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) : one_div_neg_eq_neg_one_div (ne_of_lt H)
... ≤ - (1 / a) : Hl'
... = 1 / -a : one_div_neg_eq_neg_one_div Ha,
le_of_neg_le_neg (le_of_div_le H' Hl'')
theorem lt_of_div_lt_pos (H : b < 0) (Hl : 1 / a < 1 / b) : b < a :=
sorry
theorem pos_iff_div_pos : a > 0 ↔ 1 / a > 0 :=
sorry
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
end linear_ordered_field
structure discrete_linear_ordered_field [class] (A : Type) extends linear_ordered_field A,
decidable_linear_ordered_comm_ring A
section discrete_linear_ordered_field
variable {A : Type}
variables [s : discrete_linear_ordered_field A] {a b c : A}
include s
theorem 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_lt H') (le_of_not_lt H))))
definition discrete_linear_ordered_field.to_discrete_field [instance] [reducible] [coercion]
[s : discrete_linear_ordered_field A] : discrete_field A :=
⦃ discrete_field, s, decidable_equality := @dec_eq_of_dec_lt A s⦄
end discrete_linear_ordered_field
end algebra