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fix(library/data/rat/order): adjust file to recent changes

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Leonardo de Moura 2015-06-10 16:56:17 -07:00
parent 3c1f5f4e33
commit 226a5800dd
1 changed files with 8 additions and 2 deletions
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10
library/data/rat/order.lean
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@ -229,17 +229,23 @@ iff.intro
(assume H1 : a < b, and.left (iff.mp !lt_iff_le_and_ne H1))
(assume H1 : a = b, H1 ▸ !le.refl))
theorem to_nonneg : a ≥ 0 → nonneg a :=
by intros; rewrite -sub_zero; eassumption
theorem add_le_add_left (H : a ≤ b) (c : ℚ) : c + a ≤ c + b :=
have H1 : c + b - (c + a) = b - a,
by rewrite [↑sub, neg_add, -add.assoc, add.comm c, add_neg_cancel_right],
show nonneg (c + b - (c + a)), from H1⁻¹ ▸ H
theorem mul_nonneg (H1 : a ≥ (0 : ℚ)) (H2 : b ≥ (0 : ℚ)) : a * b ≥ (0 : ℚ) :=
have H : nonneg (a * b), from nonneg_mul (!sub_zero ▸ H1) (!sub_zero ▸ H2),
have H : nonneg (a * b), from nonneg_mul (to_nonneg H1) (to_nonneg H2),
!sub_zero⁻¹ ▸ H
theorem to_pos : a > 0 → pos a :=
by intros; rewrite -sub_zero; eassumption
theorem mul_pos (H1 : a > (0 : ℚ)) (H2 : b > (0 : ℚ)) : a * b > (0 : ℚ) :=
have H : pos (a * b), from pos_mul (!sub_zero ▸ H1) (!sub_zero ▸ H2),
have H : pos (a * b), from pos_mul (to_pos H1) (to_pos H2),
!sub_zero⁻¹ ▸ H
definition decidable_lt [instance] : decidable_rel rat.lt :=