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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) *)
(************************************************************************)
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.
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