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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 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.
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