516 lines
20 KiB
Text
516 lines
20 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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Authors: Robert Lewis
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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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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 d : A}
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include s
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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 one_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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theorem one_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 le_mul_of_ge_one_right (Hb : b ≥ 0) (H : a ≥ 1) : b ≤ b * a :=
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mul_one _ ▸ (mul_le_mul_of_nonneg_left H Hb)
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theorem lt_mul_of_gt_one_right (Hb : b > 0) (H : a > 1) : b < b * a :=
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mul_one _ ▸ (mul_lt_mul_of_pos_left H Hb)
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theorem one_le_div_iff_le (a : A) {b : A} (Hb : b > 0) : 1 ≤ a / b ↔ b ≤ a :=
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have Hb' : b ≠ 0, from ne.symm (ne_of_lt Hb),
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iff.intro
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(assume H : 1 ≤ a / b,
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calc
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b = b : refl
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... ≤ b * (a / b) : le_mul_of_ge_one_right (le_of_lt Hb) H
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... = a : mul_div_cancel' Hb')
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(assume H : b ≤ a,
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have Hbinv : 1 / b > 0, from one_div_pos_of_pos Hb, calc
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1 = b * (1 / b) : mul_one_div_cancel Hb'
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... ≤ a * (1 / b) : mul_le_mul_of_nonneg_right H (le_of_lt Hbinv)
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... = a / b : div_eq_mul_one_div)
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theorem le_of_one_le_div (Hb : b > 0) (H : 1 ≤ a / b) : b ≤ a :=
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(iff.mp (!one_le_div_iff_le Hb)) H
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theorem one_le_div_of_le (Hb : b > 0) (H : b ≤ a) : 1 ≤ a / b :=
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(iff.mpr (!one_le_div_iff_le Hb)) H
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theorem one_lt_div_iff_lt (a : A) {b : A} (Hb : b > 0) : 1 < a / b ↔ b < a :=
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have Hb' : b ≠ 0, from ne.symm (ne_of_lt Hb),
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iff.intro
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(assume H : 1 < a / b,
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calc
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b < b * (a / b) : lt_mul_of_gt_one_right Hb H
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... = a : mul_div_cancel' Hb')
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(assume H : b < a,
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have Hbinv : 1 / b > 0, from one_div_pos_of_pos Hb, calc
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1 = b * (1 / b) : mul_one_div_cancel Hb'
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... < a * (1 / b) : mul_lt_mul_of_pos_right H Hbinv
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... = a / b : div_eq_mul_one_div)
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theorem lt_of_one_lt_div (Hb : b > 0) (H : 1 < a / b) : b < a :=
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(iff.mp (!one_lt_div_iff_lt Hb)) H
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theorem one_lt_div_of_lt (Hb : b > 0) (H : b < a) : 1 < a / b :=
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(iff.mpr (!one_lt_div_iff_lt Hb)) H
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theorem exists_lt (a : A) : ∃ x, x < a :=
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have H : a - 1 < a, from add_lt_of_le_of_neg (le.refl _) zero_gt_neg_one,
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exists.intro _ H
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theorem exists_gt (a : A) : ∃ x, x > a :=
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have H : a + 1 > a, from lt_add_of_le_of_pos (le.refl _) zero_lt_one,
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exists.intro _ H
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-- the following theorems amount to four iffs, for <, ≤, ≥, >.
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theorem mul_le_of_le_div (Hc : 0 < c) (H : a ≤ b / c) : a * c ≤ b :=
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!div_mul_cancel (ne.symm (ne_of_lt Hc)) ▸ mul_le_mul_of_nonneg_right H (le_of_lt Hc)
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theorem le_div_of_mul_le (Hc : 0 < c) (H : a * c ≤ b) : a ≤ b / c :=
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calc
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a = a * c * (1 / c) : !mul_mul_div (ne.symm (ne_of_lt Hc))
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... ≤ b * (1 / c) : mul_le_mul_of_nonneg_right H (le_of_lt (one_div_pos_of_pos Hc))
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... = b / c : div_eq_mul_one_div
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theorem mul_lt_of_lt_div (Hc : 0 < c) (H : a < b / c) : a * c < b :=
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!div_mul_cancel (ne.symm (ne_of_lt Hc)) ▸ mul_lt_mul_of_pos_right H Hc
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theorem lt_div_of_mul_lt (Hc : 0 < c) (H : a * c < b) : a < b / c :=
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calc
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a = a * c * (1 / c) : !mul_mul_div (ne.symm (ne_of_lt Hc))
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... < b * (1 / c) : mul_lt_mul_of_pos_right H (one_div_pos_of_pos Hc)
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... = b / c : div_eq_mul_one_div
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theorem mul_le_of_div_le_of_neg (Hc : c < 0) (H : b / c ≤ a) : a * c ≤ b :=
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!div_mul_cancel (ne_of_lt Hc) ▸ mul_le_mul_of_nonpos_right H (le_of_lt Hc)
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theorem div_le_of_mul_le_of_neg (Hc : c < 0) (H : a * c ≤ b) : b / c ≤ a :=
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calc
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a = a * c * (1 / c) : !mul_mul_div (ne_of_lt Hc)
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... ≥ b * (1 / c) : mul_le_mul_of_nonpos_right H (le_of_lt (one_div_neg_of_neg Hc))
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... = b / c : div_eq_mul_one_div
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theorem mul_lt_of_gt_div_of_neg (Hc : c < 0) (H : a > b / c) : a * c < b :=
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!div_mul_cancel (ne_of_lt Hc) ▸ mul_lt_mul_of_neg_right H Hc
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theorem div_lt_of_mul_gt_of_neg (Hc : c < 0) (H : a * c < b) : b / c < a :=
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calc
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a = a * c * (1 / c) : !mul_mul_div (ne_of_lt Hc)
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... > b * (1 / c) : mul_lt_mul_of_neg_right H (one_div_neg_of_neg Hc)
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... = b / c : div_eq_mul_one_div
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theorem div_le_of_le_mul (Hb : b > 0) (H : a ≤ b * c) : a / b ≤ c :=
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calc
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a / b = a * (1 / b) : div_eq_mul_one_div
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... ≤ (b * c) * (1 / b) : mul_le_mul_of_nonneg_right H (le_of_lt (one_div_pos_of_pos Hb))
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... = (b * c) / b : div_eq_mul_one_div
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... = c : mul_div_cancel_left (ne.symm (ne_of_lt Hb))
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theorem le_mul_of_div_le (Hc : c > 0) (H : a / c ≤ b) : a ≤ b * c :=
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calc
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a = a / c * c : !div_mul_cancel (ne.symm (ne_of_lt Hc))
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... ≤ b * c : mul_le_mul_of_nonneg_right H (le_of_lt Hc)
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-- following these in the isabelle file, there are 8 biconditionals for the above with - signs
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-- skipping for now
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theorem mul_sub_mul_div_mul_neg (Hc : c ≠ 0) (Hd : d ≠ 0) (H : a / c < b / d) :
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(a * d - b * c) / (c * d) < 0 :=
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have H1 : a / c - b / d < 0, from calc
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a / c - b / d < b / d - b / d : sub_lt_sub_right H
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... = 0 : sub_self,
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calc
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0 > a / c - b / d : H1
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... = (a * d - c * b) / (c * d) : !div_sub_div Hc Hd
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... = (a * d - b * c) / (c * d) : mul.comm
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theorem mul_sub_mul_div_mul_nonpos (Hc : c ≠ 0) (Hd : d ≠ 0) (H : a / c ≤ b / d) :
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(a * d - b * c) / (c * d) ≤ 0 :=
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have H1 : a / c - b / d ≤ 0, from calc
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a / c - b / d ≤ b / d - b / d : sub_le_sub_right H
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... = 0 : sub_self,
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calc
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0 ≥ a / c - b / d : H1
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... = (a * d - c * b) / (c * d) : !div_sub_div Hc Hd
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... = (a * d - b * c) / (c * d) : mul.comm
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theorem div_lt_div_of_mul_sub_mul_div_neg (Hc : c ≠ 0) (Hd : d ≠ 0)
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(H : (a * d - b * c) / (c * d) < 0) : a / c < b / d :=
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assert H1 : (a * d - c * b) / (c * d) < 0, by rewrite [mul.comm c b]; exact H,
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assert H2 : a / c - b / d < 0, by rewrite [!div_sub_div Hc Hd]; exact H1,
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assert H3 : a / c - b / d + b / d < 0 + b / d, from add_lt_add_right H2 _,
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begin rewrite [zero_add at H3, sub_eq_add_neg at H3, neg_add_cancel_right at H3], exact H3 end
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theorem div_le_div_of_mul_sub_mul_div_nonpos (Hc : c ≠ 0) (Hd : d ≠ 0)
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(H : (a * d - b * c) / (c * d) ≤ 0) : a / c ≤ b / d :=
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assert H1 : (a * d - c * b) / (c * d) ≤ 0, by rewrite [mul.comm c b]; exact H,
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assert H2 : a / c - b / d ≤ 0, by rewrite [!div_sub_div Hc Hd]; exact H1,
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assert H3 : a / c - b / d + b / d ≤ 0 + b / d, from add_le_add_right H2 _,
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begin rewrite [zero_add at H3, sub_eq_add_neg at H3, neg_add_cancel_right at H3], exact H3 end
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theorem div_pos_of_pos_of_pos (Ha : 0 < a) (Hb : 0 < b) : 0 < a / b :=
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begin
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rewrite div_eq_mul_one_div,
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apply mul_pos,
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exact Ha,
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apply one_div_pos_of_pos,
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exact Hb
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end
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theorem div_nonneg_of_nonneg_of_pos (Ha : 0 ≤ a) (Hb : 0 < b) : 0 ≤ a / b :=
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begin
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rewrite div_eq_mul_one_div,
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apply mul_nonneg,
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exact Ha,
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apply le_of_lt,
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apply one_div_pos_of_pos,
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exact Hb
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end
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theorem div_neg_of_neg_of_pos (Ha : a < 0) (Hb : 0 < b) : a / b < 0:=
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begin
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rewrite div_eq_mul_one_div,
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apply mul_neg_of_neg_of_pos,
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exact Ha,
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apply one_div_pos_of_pos,
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exact Hb
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end
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theorem div_nonpos_of_nonpos_of_pos (Ha : a ≤ 0) (Hb : 0 < b) : a / b ≤ 0 :=
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begin
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rewrite div_eq_mul_one_div,
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apply mul_nonpos_of_nonpos_of_nonneg,
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exact Ha,
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apply le_of_lt,
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apply one_div_pos_of_pos,
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exact Hb
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end
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theorem div_neg_of_pos_of_neg (Ha : 0 < a) (Hb : b < 0) : a / b < 0 :=
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begin
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rewrite div_eq_mul_one_div,
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apply mul_neg_of_pos_of_neg,
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exact Ha,
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apply one_div_neg_of_neg,
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exact Hb
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end
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theorem div_nonpos_of_nonneg_of_neg (Ha : 0 ≤ a) (Hb : b < 0) : a / b ≤ 0 :=
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begin
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rewrite div_eq_mul_one_div,
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apply mul_nonpos_of_nonneg_of_nonpos,
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exact Ha,
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apply le_of_lt,
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apply one_div_neg_of_neg,
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exact Hb
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end
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theorem div_pos_of_neg_of_neg (Ha : a < 0) (Hb : b < 0) : 0 < a / b :=
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begin
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rewrite div_eq_mul_one_div,
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apply mul_pos_of_neg_of_neg,
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exact Ha,
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apply one_div_neg_of_neg,
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exact Hb
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end
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theorem div_nonneg_of_nonpos_of_neg (Ha : a ≤ 0) (Hb : b < 0) : 0 ≤ a / b :=
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begin
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rewrite div_eq_mul_one_div,
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apply mul_nonneg_of_nonpos_of_nonpos,
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exact Ha,
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apply le_of_lt,
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apply one_div_neg_of_neg,
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exact Hb
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end
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theorem div_lt_div_of_lt_of_pos (H : a < b) (Hc : 0 < c) : a / c < b / c :=
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begin
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rewrite [{a/c}div_eq_mul_one_div, {b/c}div_eq_mul_one_div],
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exact mul_lt_mul_of_pos_right H (one_div_pos_of_pos Hc)
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end
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theorem div_le_div_of_le_of_pos (H : a ≤ b) (Hc : 0 < c) : a / c ≤ b / c :=
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begin
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rewrite [{a/c}div_eq_mul_one_div, {b/c}div_eq_mul_one_div],
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exact mul_le_mul_of_nonneg_right H (le_of_lt (one_div_pos_of_pos Hc))
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end
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theorem div_lt_div_of_lt_of_neg (H : b < a) (Hc : c < 0) : a / c < b / c :=
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begin
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rewrite [{a/c}div_eq_mul_one_div, {b/c}div_eq_mul_one_div],
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exact mul_lt_mul_of_neg_right H (one_div_neg_of_neg Hc)
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end
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theorem div_le_div_of_le_of_neg (H : b ≤ a) (Hc : c < 0) : a / c ≤ b / c :=
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begin
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rewrite [{a/c}div_eq_mul_one_div, {b/c}div_eq_mul_one_div],
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exact mul_le_mul_of_nonpos_right H (le_of_lt (one_div_neg_of_neg Hc))
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end
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theorem two_pos : (1 : A) + 1 > 0 :=
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add_pos zero_lt_one zero_lt_one
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theorem one_add_one_ne_zero : 1 + 1 ≠ (0:A) :=
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ne.symm (ne_of_lt two_pos)
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theorem two_ne_zero : 2 ≠ (0:A) :=
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by unfold bit0; apply one_add_one_ne_zero
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theorem add_halves (a : A) : a / 2 + a / 2 = a :=
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calc
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a / 2 + a / 2 = (a + a) / 2 : by rewrite div_add_div_same
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... = (a * 1 + a * 1) / 2 : by rewrite mul_one
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... = (a * (1 + 1)) / 2 : by rewrite left_distrib
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... = (a * 2) / 2 : by rewrite one_add_one_eq_two
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... = a : by rewrite [@mul_div_cancel A _ _ _ two_ne_zero]
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theorem sub_self_div_two (a : A) : a - a / 2 = a / 2 :=
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by rewrite [-{a}add_halves at {1}, add_sub_cancel]
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theorem add_midpoint {a b : A} (H : a < b) : a + (b - a) / 2 < b :=
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begin
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rewrite [-div_sub_div_same, sub_eq_add_neg, {b / 2 + _}add.comm, -add.assoc, -sub_eq_add_neg],
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apply add_lt_of_lt_sub_right,
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rewrite *sub_self_div_two,
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apply div_lt_div_of_lt_of_pos H two_pos
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end
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theorem div_two_sub_self (a : A) : a / 2 - a = - (a / 2) :=
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by rewrite [-{a}add_halves at {2}, sub_add_eq_sub_sub, sub_self, zero_sub]
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theorem add_self_div_two (a : A) : (a + a) / 2 = a :=
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symm (iff.mpr (!eq_div_iff_mul_eq (ne_of_gt (add_pos zero_lt_one zero_lt_one)))
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(by krewrite [left_distrib, *mul_one]))
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theorem two_ge_one : (2:A) ≥ 1 :=
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calc (2:A) = 1+1 : one_add_one_eq_two
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... ≥ 1+0 : add_le_add_left (le_of_lt zero_lt_one)
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... = 1 : add_zero
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theorem mul_le_mul_of_mul_div_le (H : a * (b / c) ≤ d) (Hc : c > 0) : b * a ≤ d * c :=
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begin
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rewrite [-mul_div_assoc at H, mul.comm b],
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apply le_mul_of_div_le Hc H
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end
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theorem div_two_lt_of_pos (H : a > 0) : a / (1 + 1) < a :=
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have Ha : a / (1 + 1) > 0, from div_pos_of_pos_of_pos H (add_pos zero_lt_one zero_lt_one),
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calc
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a / (1 + 1) < a / (1 + 1) + a / (1 + 1) : lt_add_of_pos_left Ha
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... = a : add_halves
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theorem div_mul_le_div_mul_of_div_le_div_pos {e : A} (Hb : b ≠ 0) (Hd : d ≠ 0) (H : a / b ≤ c / d)
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(He : e > 0) : a / (b * e) ≤ c / (d * e) :=
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begin
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rewrite [!field.div_mul_eq_div_mul_one_div Hb (ne_of_gt He),
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!field.div_mul_eq_div_mul_one_div Hd (ne_of_gt He)],
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apply mul_le_mul_of_nonneg_right H,
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apply le_of_lt,
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apply one_div_pos_of_pos He
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end
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theorem exists_add_lt_and_pos_of_lt (H : b < a) : ∃ c : A, b + c < a ∧ c > 0 :=
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exists.intro ((a - b) / (1 + 1))
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(and.intro (assert H2 : a + a > (b + b) + (a - b), from calc
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a + a > b + a : add_lt_add_right H
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... = b + a + b - b : add_sub_cancel
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... = b + b + a - b : add.right_comm
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... = (b + b) + (a - b) : add_sub,
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assert H3 : (a + a) / 2 > ((b + b) + (a - b)) / 2,
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from div_lt_div_of_lt_of_pos H2 two_pos,
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by rewrite [one_add_one_eq_two, sub_eq_add_neg, add_self_div_two at H3, -div_add_div_same at H3, add_self_div_two at H3];
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exact H3)
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(div_pos_of_pos_of_pos (iff.mpr !sub_pos_iff_lt H) two_pos))
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theorem ge_of_forall_ge_sub {a b : A} (H : ∀ ε : A, ε > 0 → a ≥ b - ε) : a ≥ b :=
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begin
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apply le_of_not_gt,
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intro Hb,
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cases exists_add_lt_and_pos_of_lt Hb with [c, Hc],
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let Hc' := H c (and.right Hc),
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apply (not_le_of_gt (and.left Hc)) (iff.mpr !le_add_iff_sub_right_le Hc')
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end
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end linear_ordered_field
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structure discrete_linear_ordered_field [class] (A : Type) extends linear_ordered_field A,
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decidable_linear_ordered_comm_ring A :=
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(inv_zero : inv zero = zero)
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section discrete_linear_ordered_field
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variable {A : Type}
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variables [s : discrete_linear_ordered_field A] {a b c : A}
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include s
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definition dec_eq_of_dec_lt : ∀ x y : A, decidable (x = y) :=
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take x y,
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decidable.by_cases
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(assume H : x < y, decidable.inr (ne_of_lt H))
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(assume H : ¬ x < y,
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decidable.by_cases
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(assume H' : y < x, decidable.inr (ne.symm (ne_of_lt H')))
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(assume H' : ¬ y < x,
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decidable.inl (le.antisymm (le_of_not_gt H') (le_of_not_gt H))))
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definition discrete_linear_ordered_field.to_discrete_field [trans_instance] [reducible]
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: discrete_field A :=
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⦃ discrete_field, s, has_decidable_eq := dec_eq_of_dec_lt⦄
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theorem pos_of_one_div_pos (H : 0 < 1 / a) : 0 < a :=
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have H1 : 0 < 1 / (1 / a), from one_div_pos_of_pos H,
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have H2 : 1 / a ≠ 0, from
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(assume H3 : 1 / a = 0,
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have H4 : 1 / (1 / a) = 0, from H3⁻¹ ▸ !div_zero,
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absurd H4 (ne.symm (ne_of_lt H1))),
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(division_ring.one_div_one_div (ne_zero_of_one_div_ne_zero H2)) ▸ H1
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theorem neg_of_one_div_neg (H : 1 / a < 0) : a < 0 :=
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have H1 : 0 < - (1 / a), from neg_pos_of_neg H,
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have Ha : a ≠ 0, from ne_zero_of_one_div_ne_zero (ne_of_lt H),
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have H2 : 0 < 1 / (-a), from (division_ring.one_div_neg_eq_neg_one_div Ha)⁻¹ ▸ H1,
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have H3 : 0 < -a, from pos_of_one_div_pos H2,
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neg_of_neg_pos H3
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theorem le_of_one_div_le_one_div (H : 0 < a) (Hl : 1 / a ≤ 1 / b) : b ≤ a :=
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have Hb : 0 < b, from pos_of_one_div_pos (calc
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0 < 1 / a : one_div_pos_of_pos H
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... ≤ 1 / b : Hl),
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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_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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), le_of_one_le_div Hb H'
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theorem le_of_one_div_le_one_div_of_neg (H : b < 0) (Hl : 1 / a ≤ 1 / b) : b ≤ a :=
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assert Ha : a ≠ 0, from ne_of_lt (neg_of_one_div_neg (calc
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1 / a ≤ 1 / b : Hl
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... < 0 : one_div_neg_of_neg H)),
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have H' : -b > 0, from neg_pos_of_neg H,
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have Hl' : - (1 / b) ≤ - (1 / a), from neg_le_neg Hl,
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have Hl'' : 1 / - b ≤ 1 / - a, from calc
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1 / -b = - (1 / b) : by rewrite [division_ring.one_div_neg_eq_neg_one_div (ne_of_lt H)]
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... ≤ - (1 / a) : Hl'
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... = 1 / -a : by rewrite [division_ring.one_div_neg_eq_neg_one_div Ha],
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le_of_neg_le_neg (le_of_one_div_le_one_div H' Hl'')
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theorem lt_of_one_div_lt_one_div (H : 0 < a) (Hl : 1 / a < 1 / b) : b < a :=
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have Hb : 0 < b, from pos_of_one_div_pos (calc
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0 < 1 / a : one_div_pos_of_pos H
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... < 1 / b : Hl),
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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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lt_of_one_lt_div Hb H
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theorem lt_of_one_div_lt_one_div_of_neg (H : b < 0) (Hl : 1 / a < 1 / b) : b < a :=
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have H1 : b ≤ a, from le_of_one_div_le_one_div_of_neg H (le_of_lt Hl),
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have Hn : b ≠ a, from
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(assume Hn' : b = a,
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have Hl' : 1 / a = 1 / b, from Hn' ▸ refl _,
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absurd Hl' (ne_of_lt Hl)),
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lt_of_le_of_ne H1 Hn
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theorem one_div_lt_one_div_of_lt (Ha : 0 < a) (H : a < b) : 1 / b < 1 / a :=
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lt_of_not_ge
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(assume H',
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absurd H (not_lt_of_ge (le_of_one_div_le_one_div Ha H')))
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theorem one_div_le_one_div_of_le (Ha : 0 < a) (H : a ≤ b) : 1 / b ≤ 1 / a :=
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le_of_not_gt
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(assume H',
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absurd H (not_le_of_gt (lt_of_one_div_lt_one_div Ha H')))
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theorem one_div_lt_one_div_of_lt_of_neg (Hb : b < 0) (H : a < b) : 1 / b < 1 / a :=
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lt_of_not_ge
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(assume H',
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absurd H (not_lt_of_ge (le_of_one_div_le_one_div_of_neg Hb H')))
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theorem one_div_le_one_div_of_le_of_neg (Hb : b < 0) (H : a ≤ b) : 1 / b ≤ 1 / a :=
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le_of_not_gt
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(assume H',
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absurd H (not_le_of_gt (lt_of_one_div_lt_one_div_of_neg Hb H')))
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theorem one_lt_one_div (H1 : 0 < a) (H2 : a < 1) : 1 < 1 / a :=
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one_div_one ▸ one_div_lt_one_div_of_lt H1 H2
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theorem one_le_one_div (H1 : 0 < a) (H2 : a ≤ 1) : 1 ≤ 1 / a :=
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one_div_one ▸ one_div_le_one_div_of_le H1 H2
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theorem one_div_lt_neg_one (H1 : a < 0) (H2 : -1 < a) : 1 / a < -1 :=
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one_div_neg_one_eq_neg_one ▸ one_div_lt_one_div_of_lt_of_neg H1 H2
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theorem one_div_le_neg_one (H1 : a < 0) (H2 : -1 ≤ a) : 1 / a ≤ -1 :=
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one_div_neg_one_eq_neg_one ▸ one_div_le_one_div_of_le_of_neg H1 H2
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theorem div_lt_div_of_pos_of_lt_of_pos (Hb : 0 < b) (H : b < a) (Hc : 0 < c) : c / a < c / b :=
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begin
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apply iff.mp !sub_neg_iff_lt,
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rewrite [div_eq_mul_one_div, {c / b}div_eq_mul_one_div, -mul_sub_left_distrib],
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apply mul_neg_of_pos_of_neg,
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exact Hc,
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apply iff.mpr !sub_neg_iff_lt,
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apply one_div_lt_one_div_of_lt,
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repeat assumption
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end
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theorem div_mul_le_div_mul_of_div_le_div_pos' {d e : A} (H : a / b ≤ c / d)
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(He : e > 0) : a / (b * e) ≤ c / (d * e) :=
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begin
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rewrite [2 div_mul_eq_div_mul_one_div],
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apply mul_le_mul_of_nonneg_right H,
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apply le_of_lt,
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apply one_div_pos_of_pos He
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end
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theorem abs_one_div (a : A) : abs (1 / a) = 1 / abs a :=
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if H : a > 0 then
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by rewrite [abs_of_pos H, abs_of_pos (one_div_pos_of_pos H)]
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else
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(if H' : a < 0 then
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by rewrite [abs_of_neg H', abs_of_neg (one_div_neg_of_neg H'),
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-(division_ring.one_div_neg_eq_neg_one_div (ne_of_lt H'))]
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else
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assert Heq : a = 0, from eq_of_le_of_ge (le_of_not_gt H) (le_of_not_gt H'),
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by rewrite [Heq, div_zero, *abs_zero, div_zero])
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theorem sign_eq_div_abs (a : A) : sign a = a / (abs a) :=
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decidable.by_cases
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(suppose a = 0, by subst a; rewrite [zero_div, sign_zero])
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(suppose a ≠ 0,
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have abs a ≠ 0, from assume H, this (eq_zero_of_abs_eq_zero H),
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!eq_div_of_mul_eq this !eq_sign_mul_abs⁻¹)
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end discrete_linear_ordered_field
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