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(***********************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
(* \VV/ *************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
(* Finite sets library.
* Authors: Pierre Letouzey and Jean-Christophe Filliâtre
* Institution: LRI, CNRS UMR 8623 - Université Paris Sud
* 91405 Orsay, France *)
Require Import OrderedType Orders.
Set Implicit Arguments.
(** * Some alternative (but equivalent) presentations for an Ordered Type
inferface. *)
(** ** The original interface *)
Module Type OrderedTypeOrig := OrderedType.OrderedType.
(** ** An interface based on compare *)
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 OrderedTypeOrig to OrderedType. *)
Module Update_OT (O:OrderedTypeOrig) <: OrderedType.
Include Update_DT O. (* Provides : t eq eq_equiv eq_dec *)
Definition lt := O.lt.
Instance lt_strorder : StrictOrder lt.
Proof.
split.
intros x Hx. apply (O.lt_not_eq Hx); auto with *.
exact O.lt_trans.
Qed.
Instance lt_compat : Proper (eq==>eq==>iff) lt.
Proof.
apply proper_sym_impl_iff_2; auto with *.
intros x x' Hx y y' Hy H.
assert (H0 : lt x' y).
destruct (O.compare x' y) as [H'|H'|H']; auto.
elim (O.lt_not_eq H). transitivity x'; auto with *.
elim (O.lt_not_eq (O.lt_trans H H')); auto.
destruct (O.compare x' y') as [H'|H'|H']; auto.
elim (O.lt_not_eq H).
transitivity x'; auto with *. transitivity y'; auto with *.
elim (O.lt_not_eq (O.lt_trans H' H0)); auto with *.
Qed.
Definition compare x y :=
match O.compare x y with
| EQ _ => Eq
| LT _ => Lt
| GT _ => Gt
end.
Lemma compare_spec : forall x y, CompSpec eq lt x y (compare x y).
Proof.
intros; unfold compare; destruct O.compare; auto.
Qed.
End Update_OT.
(** ** From OrderedType to OrderedTypeOrig. *)
Module Backport_OT (O:OrderedType) <: OrderedTypeOrig.
Include Backport_DT O. (* Provides : t eq eq_refl eq_sym eq_trans eq_dec *)
Definition lt := O.lt.
Lemma lt_not_eq : forall x y, lt x y -> ~eq x y.
Proof.
intros x y L E; rewrite E in L. apply (StrictOrder_Irreflexive y); auto.
Qed.
Lemma lt_trans : Transitive lt.
Proof. apply O.lt_strorder. Qed.
Definition compare : forall x y, Compare lt eq x y.
Proof.
intros x y; destruct (CompSpec2Type (O.compare_spec x y));
[apply EQ|apply LT|apply GT]; auto.
Defined.
End Backport_OT.
(** ** From OrderedTypeAlt to OrderedType. *)
Module OT_from_Alt (Import O:OrderedTypeAlt) <: OrderedType.
Definition t := t.
Definition eq x y := (x?=y) = Eq.
Definition lt x y := (x?=y) = Lt.
Instance eq_equiv : Equivalence eq.
Proof.
split; red.
(* refl *)
unfold eq; intros x.
assert (H:=compare_sym x x).
destruct (x ?= x); simpl in *; auto; discriminate.
(* sym *)
unfold eq; intros x y H.
rewrite compare_sym, H; simpl; auto.
(* trans *)
apply compare_trans.
Qed.
Instance lt_strorder : StrictOrder lt.
Proof.
split; repeat red; unfold lt; try apply compare_trans.
intros x H.
assert (eq x x) by reflexivity.
unfold eq in *; congruence.
Qed.
Lemma lt_eq : forall x y z, lt x y -> eq y z -> lt x z.
Proof.
unfold lt, eq; intros x y z Hxy Hyz.
destruct (compare x z) eqn:Hxz; auto.
rewrite compare_sym, CompOpp_iff in Hyz. simpl in Hyz.
rewrite (compare_trans Hxz Hyz) in Hxy; discriminate.
rewrite compare_sym, CompOpp_iff in Hxy. simpl in Hxy.
rewrite (compare_trans Hxy Hxz) in Hyz; discriminate.
Qed.
Lemma eq_lt : forall x y z, eq x y -> lt y z -> lt x z.
Proof.
unfold lt, eq; intros x y z Hxy Hyz.
destruct (compare x z) eqn:Hxz; auto.
rewrite compare_sym, CompOpp_iff in Hxy. simpl in Hxy.
rewrite (compare_trans Hxy Hxz) in Hyz; discriminate.
rewrite compare_sym, CompOpp_iff in Hyz. simpl in Hyz.
rewrite (compare_trans Hxz Hyz) in Hxy; discriminate.
Qed.
Instance lt_compat : Proper (eq==>eq==>iff) lt.
Proof.
apply proper_sym_impl_iff_2; auto with *.
repeat red; intros.
eapply lt_eq; eauto. eapply eq_lt; eauto. symmetry; auto.
Qed.
Definition compare := O.compare.
Lemma compare_spec : forall x y, CompSpec eq lt x y (compare x y).
Proof.
unfold eq, lt, compare; intros.
destruct (O.compare x y) eqn:H; auto.
apply CompGt.
rewrite compare_sym, H; auto.
Qed.
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 OT_from_Alt.
(** From the original presentation to this alternative one. *)
Module OT_to_Alt (Import O:OrderedType) <: OrderedTypeAlt.
Definition t := t.
Definition compare := compare.
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 (compare_spec x y) as [U|U|U];
destruct (compare_spec y x) as [V|V|V]; auto.
rewrite U in V. elim (StrictOrder_Irreflexive y); auto.
rewrite U in V. elim (StrictOrder_Irreflexive y); auto.
rewrite V in U. elim (StrictOrder_Irreflexive x); auto.
rewrite V in U. elim (StrictOrder_Irreflexive x); auto.
rewrite V in U. elim (StrictOrder_Irreflexive x); auto.
rewrite V in U. elim (StrictOrder_Irreflexive y); auto.
Qed.
Lemma compare_Eq : forall x y, compare x y = Eq <-> eq x y.
Proof.
unfold compare.
intros x y; destruct (compare_spec x y); intuition;
try discriminate.
rewrite H0 in H. elim (StrictOrder_Irreflexive y); auto.
rewrite H0 in H. elim (StrictOrder_Irreflexive y); auto.
Qed.
Lemma compare_Lt : forall x y, compare x y = Lt <-> lt x y.
Proof.
unfold compare.
intros x y; destruct (compare_spec x y); intuition;
try discriminate.
rewrite H in H0. elim (StrictOrder_Irreflexive y); auto.
rewrite H in H0. elim (StrictOrder_Irreflexive x); auto.
Qed.
Lemma compare_Gt : forall x y, compare x y = Gt <-> lt y x.
Proof.
intros x y. rewrite compare_sym, CompOpp_iff. apply compare_Lt.
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;
rewrite ?compare_Eq, ?compare_Lt, ?compare_Gt;
transitivity y; auto.
Qed.
End OT_to_Alt.
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