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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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Require Import OrderedType.

(** * An alternative (but equivalent) presentation for an Ordered Type
   inferface. *)

(** NB: [comparison], defined in [Datatypes.v] is [Eq|Lt|Gt]
whereas [compare], defined in [OrderedType.v] is [EQ _ | LT _ | GT _ ]
*)

Module Type OrderedTypeAlt.

 Parameter t : Type.

 Parameter compare : t -> t -> comparison.

 Infix "?=" := compare (at level 70, no associativity).

 Parameter compare_sym :
   forall x y, (y?=x) = CompOpp (x?=y).
 Parameter compare_trans :
   forall c x y z, (x?=y) = c -> (y?=z) = c -> (x?=z) = c.

End OrderedTypeAlt.

(** From this new presentation to the original one. *)

Module OrderedType_from_Alt (O:OrderedTypeAlt) <: OrderedType.
 Import O.

 Definition t := t.

 Definition eq x y := (x?=y) = Eq.
 Definition lt x y := (x?=y) = Lt.

 Lemma eq_refl : forall x, eq x x.
 Proof.
 intro x.
 unfold eq.
 assert (H:=compare_sym x x).
 destruct (x ?= x); simpl in *; try discriminate; auto.
 Qed.

 Lemma eq_sym : forall x y, eq x y -> eq y x.
 Proof.
 unfold eq; intros.
 rewrite compare_sym.
 rewrite H; simpl; auto.
 Qed.

 Definition eq_trans := (compare_trans Eq).

 Definition lt_trans := (compare_trans Lt).

 Lemma lt_not_eq : forall x y, lt x y -> ~eq x y.
 Proof.
 unfold eq, lt; intros.
 rewrite H; discriminate.
 Qed.

 Definition compare : forall x y, Compare lt eq x y.
 Proof.
 intros.
 case_eq (x ?= y); intros.
 apply EQ; auto.
 apply LT; auto.
 apply GT; red.
 rewrite compare_sym; rewrite H; auto.
 Defined.

 Definition eq_dec : forall x y, { eq x y } + { ~ eq x y }.
 Proof.
 intros; unfold eq.
 case (x ?= y); [ left | right | right ]; auto; discriminate.
 Defined.

End OrderedType_from_Alt.

(** From the original presentation to this alternative one. *)

Module OrderedType_to_Alt (O:OrderedType) <: OrderedTypeAlt.
 Import O.
 Module MO:=OrderedTypeFacts(O).
 Import MO.

 Definition t := t.

 Definition compare x y := match compare x y with
   | LT _ => Lt
   | EQ _ => Eq
   | GT _ => Gt
  end.

 Infix "?=" := compare (at level 70, no associativity).

 Lemma compare_sym :
   forall x y, (y?=x) = CompOpp (x?=y).
 Proof.
 intros x y; unfold compare.
 destruct O.compare; elim_comp; simpl; auto.
 Qed.

 Lemma compare_trans :
   forall c x y z, (x?=y) = c -> (y?=z) = c -> (x?=z) = c.
 Proof.
 intros c x y z.
 destruct c; unfold compare;
 do 2 (destruct O.compare; intros; try discriminate);
 elim_comp; auto.
 Qed.

End OrderedType_to_Alt.