# Library Coq.QArith.Qabs

Require Export QArith.
Require Export Qreduction.
Hint Resolve Qlt_le_weak : qarith.
Definition Qabs (x:Q) := let (n,d):=x in (Z.abs n # d).
Lemma Qabs_case : forall (x:Q) (P : Q -> Type), (0 <= x -> P x) -> (x <= 0 -> P (- x)) -> P (Qabs x).
Add Morphism Qabs with signature Qeq ==> Qeq as Qabs_wd.
Qed.
Lemma Qabs_pos : forall x, 0 <= x -> Qabs x == x.
Lemma Qabs_neg : forall x, x <= 0 -> Qabs x == - x.
Lemma Qabs_nonneg : forall x, 0 <= ( Qabs x ).
Lemma Zabs_Qabs : forall n d, ( Z.abs n # d )== Qabs (n # d).
Lemma Qabs_opp : forall x, Qabs (- x) == Qabs x.
Lemma Qabs_triangle : forall x y, Qabs (x + y) <= Qabs x + Qabs y.
Lemma Qabs_Qmult : forall a b, Qabs (a * b) == ( Qabs a )*( Qabs b ).
Lemma Qabs_Qminus x y: Qabs (x - y) = Qabs (y - x).
Lemma Qle_Qabs : forall a, a <= Qabs a.
Lemma Qabs_triangle_reverse : forall x y, Qabs x - Qabs y <= Qabs (x - y).
Lemma Qabs_Qle_condition x y: Qabs x <= y <-> - y <= x <= y.
Lemma Qabs_diff_Qle_condition x y r: Qabs (x - y) <= r <-> x - r <= y <= x + r.