117 lines
4 KiB
Text
117 lines
4 KiB
Text
/-
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Copyright (c) 2014 Robert Lewis. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: algebra.ordered_field
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Authors: Robert Lewis
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Here an "ordered_ring" is partially ordered ring, which is ordered with respect to both a weak
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order and an associated strict order. Our numeric structures (int, rat, and real) will be instances
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of "linear_ordered_comm_ring". This development is modeled after Isabelle's library.
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-/
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import algebra.ordered_ring algebra.field
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open eq eq.ops
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namespace algebra
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structure linear_ordered_field [class] (A : Type) extends linear_ordered_ring A, field A
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section linear_ordered_field
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variable {A : Type}
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variables [s : linear_ordered_field A] {a b c : A}
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include s
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-- ordered ring theorem?
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-- split H3 into its own lemma
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theorem gt_of_mul_lt_mul_neg_left (H : c * a < c * b) (Hc : c ≤ 0) : a > b :=
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have nhc : -c ≥ 0, from neg_nonneg_of_nonpos Hc,
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have H2 : -(c * b) < -(c * a), from (iff.mp' (neg_lt_neg_iff_lt _ _) H),
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have H3 : (-c) * b < (-c) * a, from (calc
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(-c) * b = (-1 * c) * b : neg_eq_neg_one_mul
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... = -1 * (c * b) : mul.assoc
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... = - (c * b) : neg_eq_neg_one_mul
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... < -(c * a) : H2
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... = -1 * (c * a) : neg_eq_neg_one_mul
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... = (-1 * c) * a : mul.assoc
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... = (-c) * a : neg_eq_neg_one_mul
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),
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lt_of_mul_lt_mul_left H3 nhc
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-- helpers for following
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theorem mul_zero_lt_mul_inv_of_pos (H : 0 < a) : a * 0 < a * (1 / a) :=
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calc
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a * 0 = 0 : mul_zero
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... < 1 : zero_lt_one
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... = a * a⁻¹ : mul_inv_cancel (ne.symm (ne_of_lt H))
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... = a * (1 / a) : inv_eq_one_div
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theorem mul_zero_lt_mul_inv_of_neg (H : a < 0) : a * 0 < a * (1 / a) :=
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calc
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a * 0 = 0 : mul_zero
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... < 1 : zero_lt_one
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... = a * a⁻¹ : mul_inv_cancel (ne_of_lt H)
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... = a * (1 / a) : inv_eq_one_div
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theorem div_pos_of_pos (H : 0 < a) : 0 < 1 / a :=
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lt_of_mul_lt_mul_left (mul_zero_lt_mul_inv_of_pos H) (le_of_lt H)
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-- this would go in ring, if it worked
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theorem ne_zero_of_div_ne_zero (H : 1 / a ≠ 0) : a ≠ 0 :=
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assume Ha : a = 0, sorry
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theorem pos_of_div_pos (H : 0 < 1 / a) : 0 < a :=
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have H1 : 0 < 1 / (1 / a), from div_pos_of_pos H,
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-- want a ≠ 0. Can get this with decidable =, from discrete_field.inv_zero_imp_zero
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div_div (sorry) ▸ H1
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theorem div_neg_of_neg (H : a < 0) : 1 / a < 0 :=
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gt_of_mul_lt_mul_neg_left (mul_zero_lt_mul_inv_of_neg H) (le_of_lt H)
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theorem neg_of_div_neg (H : 1 / a < 0) : a < 0 :=
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sorry
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-- is this theorem (and le_of_div_le which depends on it) classical?
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theorem one_le_div_iff_le : 1 ≤ a / b ↔ b ≤ a :=
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sorry
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theorem one_lt_div_iff_lt : 1 < a / b ↔ b < a :=
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sorry
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-- why is mul_le_mul under ordered_ring namespace?
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theorem le_of_div_le (H : 0 < a) (Hl : 1 / a ≤ 1 / b) : b ≤ a :=
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have H : 1 ≤ a / b, from (calc
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1 = a / a : div_self (ne.symm (ne_of_lt H))
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... = a * (1 / a) : div_eq_mul_one_div
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... ≤ a * (1 / b) : ordered_ring.mul_le_mul_of_nonneg_left Hl (le_of_lt H)
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... = a / b : div_eq_mul_one_div
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), (iff.mp one_le_div_iff_le) H
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theorem lt_of_div_lt (H : a > 0) (Hl : 1 / a < 1 / b) : b < a :=
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have H : 1 < a / b, from (calc
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1 = a / a : div_self (ne.symm (ne_of_lt H))
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... = a * (1 / a) : div_eq_mul_one_div
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... < a * (1 / b) : mul_lt_mul_of_pos_left Hl H
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... = a / b : div_eq_mul_one_div
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), (iff.mp one_lt_div_iff_lt) H
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theorem le_of_div_le_neg (H : b < 0) (Hl : 1 / a ≤ 1 / b) : b ≤ a :=
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have Ha : 1 / a < 0, from (calc
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1 / a ≤ 1 / b : Hl
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... < 0 : div_neg_of_neg H
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),
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have Ha' : a ≠ 0, from ne_of_lt (neg_of_div_neg Ha),
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have H : 1 ≤ a / b, from (calc
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1 = a / a : div_self Ha'
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... ≤ a / b : sorry), sorry
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theorem lt_of_div_lt_pos (H : b < 0) (Hl : 1 / a < 1 / b) : b < a :=
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sorry
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theorem pos_iff_div_pos : a > 0 ↔ 1 / a > 0 :=
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sorry
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end linear_ordered_field
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end algebra
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