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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 Lt Peano_dec Compare_dec EqNat
DecidableType2 OrderedType2 OrderedType2Facts.
(** * DecidableType structure for Peano numbers *)
Module Nat_as_MiniDT <: MiniDecidableType.
Definition t := nat.
Definition eq_dec := eq_nat_dec.
End Nat_as_MiniDT.
Module Nat_as_DT <: UsualDecidableType.
Include Make_UDT Nat_as_MiniDT.
(** The next fields aren't mandatory but allow more subtyping. *)
Definition eqb := beq_nat.
Definition eqb_eq := beq_nat_true_iff.
End Nat_as_DT.
(** Note that [Nat_as_DT] can also be seen as a [DecidableType],
or a [DecidableTypeOrig] or a [BooleanEqualityType]. *)
(** * 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 :=
change (@eq nat) with NatOrder.OrderElts.eq in *;
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]. *)
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