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-(************************************************************************)
-(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011     *)
-(*   \VV/  **************************************************************)
-(*    //   *      This file is distributed under the terms of the       *)
-(*         *       GNU Lesser General Public License Version 2.1        *)
-(************************************************************************)
-
-Require Import Lt Peano_dec Compare_dec EqNat
- Equalities Orders OrdersTac.
-
-
-(** * DecidableType structure for Peano numbers *)
-
-Module Nat_as_UBE <: UsualBoolEq.
- Definition t := nat.
- Definition eq := @eq nat.
- Definition eqb := beq_nat.
- Definition eqb_eq := beq_nat_true_iff.
-End Nat_as_UBE.
-
-Module Nat_as_DT <: UsualDecidableTypeFull := Make_UDTF Nat_as_UBE.
-
-(** Note that the last module fulfills by subtyping many other
-    interfaces, such as [DecidableType] or [EqualityType]. *)
-
-
-
-(** * OrderedType structure for Peano numbers *)
-
-Module Nat_as_OT <: OrderedTypeFull.
- Include Nat_as_DT.
- Definition lt := lt.
- Definition le := le.
- Definition compare := nat_compare.
-
- Instance lt_strorder : StrictOrder lt.
- Proof. split; [ exact lt_irrefl | exact lt_trans ]. Qed.
-
- Instance lt_compat : Proper (Logic.eq==>Logic.eq==>iff) lt.
- Proof. repeat red; intros; subst; auto. Qed.
-
- Definition le_lteq := le_lt_or_eq_iff.
- Definition compare_spec := nat_compare_spec.
-
-End Nat_as_OT.
-
-(** Note that [Nat_as_OT] can also be seen as a [UsualOrderedType]
-   and a [OrderedType] (and also as a [DecidableType]). *)
-
-
-
-(** * An [order] tactic for Peano numbers *)
-
-Module NatOrder := OTF_to_OrderTac Nat_as_OT.
-Ltac nat_order := NatOrder.order.
-
-(** Note that [nat_order] is domain-agnostic: it will not prove
-    [1<=2] or [x<=x+x], but rather things like [x<=y -> y<=x -> x=y]. *)
-
-Section Test.
-Let test : forall x y : nat, x<=y -> y<=x -> x=y.
-Proof. nat_order. Qed.
-End Test.