# Library Coq.QArith.Qround

Require Import QArith.
Lemma Qopp_lt_compat: forall p q : Q, p < q -> - q < - p.
Hint Resolve Qopp_lt_compat : qarith.
Local Coercion inject_Z : Z >-> Q.
Definition Qfloor (x:Q) := let (n,d) := x in Z.div n (Zpos d).
Definition Qceiling (x:Q) := (-(Qfloor (-x)))%Z.
Lemma Qfloor_Z : forall z:Z, Qfloor z = z.
Lemma Qceiling_Z : forall z:Z, Qceiling z = z.
Lemma Qfloor_le : forall x, Qfloor x <= x.
Hint Resolve Qfloor_le : qarith.
Lemma Qle_ceiling : forall x, x <= Qceiling x.
Hint Resolve Qle_ceiling : qarith.
Lemma Qle_floor_ceiling : forall x, Qfloor x <= Qceiling x.
Lemma Qlt_floor : forall x, x < (Qfloor x+1)%Z.
Hint Resolve Qlt_floor : qarith.
Lemma Qceiling_lt : forall x, (Qceiling x-1)%Z < x.
Hint Resolve Qceiling_lt : qarith.
Lemma Qfloor_resp_le : forall x y, x <= y -> (Qfloor x <= Qfloor y)%Z.
Hint Resolve Qfloor_resp_le : qarith.
Lemma Qceiling_resp_le : forall x y, x <= y -> (Qceiling x <= Qceiling y)%Z.
Hint Resolve Qceiling_resp_le : qarith.
Add Morphism Qfloor with signature Qeq ==> eq as Qfloor_comp.
Add Morphism Qceiling with signature Qeq ==> eq as Qceiling_comp.
Lemma Zdiv_Qdiv (n m: Z): (n / m)%Z = Qfloor (n / m).