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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import BinNat Equalities Orders OrdersTac.
Local Open Scope N_scope.
(** * DecidableType structure for [N] binary natural numbers *)
Module N_as_UBE <: UsualBoolEq.
Definition t := N.
Definition eq := @eq N.
Definition eqb := Neqb.
Definition eqb_eq := Neqb_eq.
End N_as_UBE.
Module N_as_DT <: UsualDecidableTypeFull := Make_UDTF N_as_UBE.
(** Note that the last module fulfills by subtyping many other
interfaces, such as [DecidableType] or [EqualityType]. *)
(** * OrderedType structure for [N] numbers *)
Module N_as_OT <: OrderedTypeFull.
Include N_as_DT.
Definition lt := Nlt.
Definition le := Nle.
Definition compare := Ncompare.
Instance lt_strorder : StrictOrder Nlt.
Proof. split; [ exact Nlt_irrefl | exact Nlt_trans ]. Qed.
Instance lt_compat : Proper (Logic.eq==>Logic.eq==>iff) Nlt.
Proof. repeat red; intros; subst; auto. Qed.
Definition le_lteq := Nle_lteq.
Definition compare_spec := Ncompare_spec.
End N_as_OT.
(** Note that [N_as_OT] can also be seen as a [UsualOrderedType]
and a [OrderedType] (and also as a [DecidableType]). *)
(** * An [order] tactic for [N] numbers *)
Module NOrder := OTF_to_OrderTac N_as_OT.
Ltac n_order := NOrder.order.
(** Note that [n_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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