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+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+Require Import QArith.
+
+Lemma Qopp_lt_compat: forall p q : Q, p < q -> - q < - p.
+Proof.
+intros (a1,a2) (b1,b2); unfold Qle, Qlt; simpl.
+do 2 rewrite <- Zopp_mult_distr_l; omega.
+Qed.
+
+Hint Resolve Qopp_lt_compat : qarith.
+
+(************)
+
+Coercion Local inject_Z : Z >-> Q.
+
+Definition Qfloor (x:Q) := let (n,d) := x in Zdiv n (Zpos d).
+Definition Qceiling (x:Q) := (-(Qfloor (-x)))%Z.
+
+Lemma Qfloor_Z : forall z:Z, Qfloor z = z.
+Proof.
+intros z.
+simpl.
+auto with *.
+Qed.
+
+Lemma Qceiling_Z : forall z:Z, Qceiling z = z.
+Proof.
+intros z.
+unfold Qceiling.
+simpl.
+rewrite Zdiv_1_r.
+auto with *.
+Qed.
+
+Lemma Qfloor_le : forall x, Qfloor x <= x.
+Proof.
+intros [n d].
+simpl.
+unfold Qle.
+simpl.
+replace (n*1)%Z with n by ring.
+rewrite Zmult_comm.
+apply Z_mult_div_ge.
+auto with *.
+Qed.
+
+Hint Resolve Qfloor_le : qarith.
+
+Lemma Qle_ceiling : forall x, x <= Qceiling x.
+Proof.
+intros x.
+apply Qle_trans with (- - x).
+ rewrite Qopp_involutive.
+ auto with *.
+change (Qceiling x:Q) with (-(Qfloor(-x))).
+auto with *.
+Qed.
+
+Hint Resolve Qle_ceiling : qarith.
+
+Lemma Qle_floor_ceiling : forall x, Qfloor x <= Qceiling x.
+Proof.
+eauto with qarith.
+Qed.
+
+Lemma Qlt_floor : forall x, x < (Qfloor x+1)%Z.
+Proof.
+intros [n d].
+simpl.
+unfold Qlt.
+simpl.
+replace (n*1)%Z with n by ring.
+ring_simplify.
+replace (n / ' d * ' d + ' d)%Z with
+  (('d * (n / 'd) + n mod 'd) + 'd - n mod 'd)%Z by ring.
+rewrite <- Z_div_mod_eq; auto with*.
+rewrite <- Zlt_plus_swap.
+destruct (Z_mod_lt n ('d)); auto with *.
+Qed.
+
+Hint Resolve Qlt_floor : qarith.
+
+Lemma Qceiling_lt : forall x, (Qceiling x-1)%Z < x.
+Proof.
+intros x.
+unfold Qceiling.
+replace (- Qfloor (- x) - 1)%Z with (-(Qfloor (-x) + 1))%Z by ring.
+change ((- (Qfloor (- x) + 1))%Z:Q) with (-(Qfloor (- x) + 1)%Z).
+apply Qlt_le_trans with (- - x); auto with *.
+rewrite Qopp_involutive.
+auto with *.
+Qed.
+
+Hint Resolve Qceiling_lt : qarith.
+
+Lemma Qfloor_resp_le : forall x y, x <= y -> (Qfloor x <= Qfloor y)%Z.
+Proof.
+intros [xn xd] [yn yd] Hxy.
+unfold Qle in *.
+simpl in *.
+rewrite <- (Zdiv_mult_cancel_r xn ('xd) ('yd)); auto with *.
+rewrite <- (Zdiv_mult_cancel_r yn ('yd) ('xd)); auto with *.
+rewrite (Zmult_comm ('yd) ('xd)).
+apply Z_div_le; auto with *.
+Qed.
+
+Hint Resolve Qfloor_resp_le : qarith.
+
+Lemma Qceiling_resp_le : forall x y, x <= y -> (Qceiling x <= Qceiling y)%Z.
+Proof.
+intros x y Hxy.
+unfold Qceiling.
+cut (Qfloor (-y) <= Qfloor (-x))%Z; auto with *.
+Qed.
+
+Hint Resolve Qceiling_resp_le : qarith.
+
+Add Morphism Qfloor with signature Qeq ==> @eq _ as Qfloor_comp.
+Proof.
+intros x y H.
+apply Zle_antisym.
+ auto with *.
+symmetry in H; auto with *.
+Qed.
+
+Add Morphism Qceiling with signature Qeq ==> @eq _ as Qceiling_comp.
+Proof.
+intros x y H.
+apply Zle_antisym.
+ auto with *.
+symmetry in H; auto with *.
+Qed.
+\ No newline at end of file