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(** Constructions of ordered types, for use with the [FSet] functors
for finite sets. *)
Require Import FSets.
Require Import Coqlib.
Require Import Maps.
(** The ordered type of positive numbers *)
Module OrderedPositive <: OrderedType.
Definition t := positive.
Definition eq (x y: t) := x = y.
Definition lt := Plt.
Lemma eq_refl : forall x : t, eq x x.
Proof (@refl_equal t).
Lemma eq_sym : forall x y : t, eq x y -> eq y x.
Proof (@sym_equal t).
Lemma eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z.
Proof (@trans_equal t).
Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.
Proof Plt_trans.
Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.
Proof Plt_ne.
Lemma compare : forall x y : t, Compare lt eq x y.
Proof.
intros. case (plt x y); intro.
apply LT. auto.
case (peq x y); intro.
apply EQ. auto.
apply GT. red; unfold Plt in *.
assert (Zpos x <> Zpos y). congruence. omega.
Qed.
End OrderedPositive.
(** Indexed types (those that inject into [positive]) are ordered. *)
Module OrderedIndexed(A: INDEXED_TYPE) <: OrderedType.
Definition t := A.t.
Definition eq (x y: t) := x = y.
Definition lt (x y: t) := Plt (A.index x) (A.index y).
Lemma eq_refl : forall x : t, eq x x.
Proof (@refl_equal t).
Lemma eq_sym : forall x y : t, eq x y -> eq y x.
Proof (@sym_equal t).
Lemma eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z.
Proof (@trans_equal t).
Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.
Proof.
unfold lt; intros. eapply Plt_trans; eauto.
Qed.
Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.
Proof.
unfold lt; unfold eq; intros.
red; intro. subst y. apply Plt_strict with (A.index x). auto.
Qed.
Lemma compare : forall x y : t, Compare lt eq x y.
Proof.
intros. case (OrderedPositive.compare (A.index x) (A.index y)); intro.
apply LT. exact l.
apply EQ. red; red in e. apply A.index_inj; auto.
apply GT. exact l.
Qed.
End OrderedIndexed.
(** The product of two ordered types is ordered. *)
Module OrderedPair (A B: OrderedType) <: OrderedType.
Definition t := (A.t * B.t)%type.
Definition eq (x y: t) :=
A.eq (fst x) (fst y) /\ B.eq (snd x) (snd y).
Lemma eq_refl : forall x : t, eq x x.
Proof.
intros; split; auto.
Qed.
Lemma eq_sym : forall x y : t, eq x y -> eq y x.
Proof.
unfold eq; intros. intuition auto.
Qed.
Lemma eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z.
Proof.
unfold eq; intros. intuition eauto.
Qed.
Definition lt (x y: t) :=
A.lt (fst x) (fst y) \/
(A.eq (fst x) (fst y) /\ B.lt (snd x) (snd y)).
Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.
Proof.
unfold lt; intros.
elim H; elim H0; intros.
left. apply A.lt_trans with (fst y); auto.
left. elim H1; intros.
case (A.compare (fst x) (fst z)); intro.
assumption.
generalize (A.lt_not_eq H2); intro. elim H5.
apply A.eq_trans with (fst z). auto. auto.
generalize (@A.lt_not_eq (fst z) (fst y)); intro.
elim H5. apply A.lt_trans with (fst x); auto.
apply A.eq_sym; auto.
left. elim H2; intros.
case (A.compare (fst x) (fst z)); intro.
assumption.
generalize (A.lt_not_eq H1); intro. elim H5.
apply A.eq_trans with (fst x).
apply A.eq_sym. auto. auto.
generalize (@A.lt_not_eq (fst y) (fst x)); intro.
elim H5. apply A.lt_trans with (fst z); auto.
apply A.eq_sym; auto.
right. elim H1; elim H2; intros.
split. apply A.eq_trans with (fst y); auto.
apply B.lt_trans with (snd y); auto.
Qed.
Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.
Proof.
unfold lt, eq, not; intros.
elim H0; intros.
elim H; intro.
apply (@A.lt_not_eq _ _ H3 H1).
elim H3; intros.
apply (@B.lt_not_eq _ _ H5 H2).
Qed.
Lemma compare : forall x y : t, Compare lt eq x y.
Proof.
intros.
case (A.compare (fst x) (fst y)); intro.
apply LT. red. left. auto.
case (B.compare (snd x) (snd y)); intro.
apply LT. red. right. tauto.
apply EQ. red. tauto.
apply GT. red. right. split. apply A.eq_sym. auto. auto.
apply GT. red. left. auto.
Qed.
End OrderedPair.
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