# Library Coq.QArith.Qround

Require Import QArith.
Import Zdiv.

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.

#[global]
Hint Resolve Qfloor_le : qarith.

Lemma Qle_ceiling : forall x, x <= Qceiling x.

#[global]
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.

#[global]
Hint Resolve Qlt_floor : qarith.

Lemma Qceiling_lt : forall x, (Qceiling x-1)%Z < x.

#[global]
Hint Resolve Qceiling_lt : qarith.

Lemma Qfloor_resp_le : forall x y, x <= y -> (Qfloor x <= Qfloor y)%Z.

#[global]
Hint Resolve Qfloor_resp_le : qarith.

Lemma Qceiling_resp_le : forall x y, x <= y -> (Qceiling x <= Qceiling y)%Z.

#[global]
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).