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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import QArith_base Equalities Orders OrdersTac.
Local Open Scope Q_scope.
(** * DecidableType structure for rational numbers *)
Module Q_as_DT <: DecidableTypeFull.
Definition t := Q.
Definition eq := Qeq.
Definition eq_equiv := Q_Setoid.
Definition eqb := Qeq_bool.
Definition eqb_eq := Qeq_bool_iff.
Include BackportEq. (** eq_refl, eq_sym, eq_trans *)
Include HasEqBool2Dec. (** eq_dec *)
End Q_as_DT.
(** Note that the last module fulfills by subtyping many other
interfaces, such as [DecidableType] or [EqualityType]. *)
(** * 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.
Definition compare_spec := Qcompare_spec.
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]. *)
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