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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_base
+ DecidableType2 OrderedType2 OrderedType2Facts.
+
+Local Open Scope Q_scope.
+
+(** * DecidableType structure for rational numbers *)
+
+Module Q_as_DT <: DecidableType.
+ Definition t := Q.
+ Definition eq := Qeq.
+ Instance eq_equiv : Equivalence Qeq.
+ Definition eq_dec := Qeq_dec.
+
+ (** The next fields are not mandatory, but allow [Q_as_DT] to be
+     also a [DecidableTypeOrig]. *)
+ Definition eq_refl := Qeq_refl.
+ Definition eq_sym := Qeq_sym.
+ Definition eq_trans := eq_trans.
+End Q_as_DT.
+
+
+(** * OrderedType structure for rational numbers *)
+
+Module Q_as_OT <: OrderedTypeFull.
+ Include Q_as_DT.
+ Definition lt := Qlt.
+ Definition le := Qle.
+ Definition compare := Qcompare.
+
+ Instance lt_strorder : StrictOrder Qlt.
+ Proof. split; [ exact Qlt_irrefl | exact Qlt_trans ]. Qed.
+
+ Instance lt_compat : Proper (Qeq==>Qeq==>iff) Qlt.
+ Proof. auto with *. Qed.
+
+ Definition le_lteq := Qle_lteq.
+
+ Lemma Qcompare_antisym : forall x y, CompOpp (x ?= y) = (y ?= x).
+ Proof.
+ unfold "?=". intros. apply Zcompare_antisym.
+ Qed.
+
+ Lemma compare_spec : forall x y, Cmp Qeq Qlt x y (Qcompare x y).
+ Proof.
+ intros.
+ destruct (x ?= y) as [ ]_eqn:H; constructor; auto.
+ rewrite Qeq_alt; auto.
+ rewrite Qlt_alt, <- Qcompare_antisym, H; auto.
+ Qed.
+
+End Q_as_OT.
+
+
+(** * An [order] tactic for [Q] numbers *)
+
+Module QOrder := OTF_to_OrderTac Q_as_OT.
+Ltac q_order := QOrder.order.
+
+(** Note that [q_order] is domain-agnostic: it will not prove
+    [1<=2] or [x<=x+x], but rather things like [x<=y -> y<=x -> x==y]. *)