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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 *)
(***********************************************************************)
Require Import OrderedType.
Require Import ZArith.
Require Import Omega.
Require Import NArith Ndec.
Require Import Compare_dec.
(** * Examples of Ordered Type structures. *)
(** First, a particular case of [OrderedType] where
the equality is the usual one of Coq. *)
Module Type UsualOrderedType.
Parameter Inline t : Type.
Definition eq := @eq t.
Parameter Inline lt : t -> t -> Prop.
Definition eq_refl := @eq_refl t.
Definition eq_sym := @eq_sym t.
Definition eq_trans := @eq_trans t.
Axiom lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.
Axiom lt_not_eq : forall x y : t, lt x y -> ~ eq x y.
Parameter compare : forall x y : t, Compare lt eq x y.
Parameter eq_dec : forall x y : t, { eq x y } + { ~ eq x y }.
End UsualOrderedType.
(** a [UsualOrderedType] is in particular an [OrderedType]. *)
Module UOT_to_OT (U:UsualOrderedType) <: OrderedType := U.
(** [nat] is an ordered type with respect to the usual order on natural numbers. *)
Module Nat_as_OT <: UsualOrderedType.
Definition t := nat.
Definition eq := @eq nat.
Definition eq_refl := @eq_refl t.
Definition eq_sym := @eq_sym t.
Definition eq_trans := @eq_trans t.
Definition lt := lt.
Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.
Proof. unfold lt; intros; apply lt_trans with y; auto. Qed.
Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.
Proof. unfold lt, eq; intros; omega. Qed.
Definition compare x y : Compare lt eq x y.
Proof.
case_eq (nat_compare x y); intro.
- apply EQ. now apply nat_compare_eq.
- apply LT. now apply nat_compare_Lt_lt.
- apply GT. now apply nat_compare_Gt_gt.
Defined.
Definition eq_dec := eq_nat_dec.
End Nat_as_OT.
(** [Z] is an ordered type with respect to the usual order on integers. *)
Local Open Scope Z_scope.
Module Z_as_OT <: UsualOrderedType.
Definition t := Z.
Definition eq := @eq Z.
Definition eq_refl := @eq_refl t.
Definition eq_sym := @eq_sym t.
Definition eq_trans := @eq_trans t.
Definition lt (x y:Z) := (x<y).
Lemma lt_trans : forall x y z, x<y -> y<z -> x<z.
Proof. intros; omega. Qed.
Lemma lt_not_eq : forall x y, x<y -> ~ x=y.
Proof. intros; omega. Qed.
Definition compare x y : Compare lt eq x y.
Proof.
case_eq (x ?= y); intro.
- apply EQ. now apply Z.compare_eq.
- apply LT. assumption.
- apply GT. now apply Z.gt_lt.
Defined.
Definition eq_dec := Z.eq_dec.
End Z_as_OT.
(** [positive] is an ordered type with respect to the usual order on natural numbers. *)
Local Open Scope positive_scope.
Module Positive_as_OT <: UsualOrderedType.
Definition t:=positive.
Definition eq:=@eq positive.
Definition eq_refl := @eq_refl t.
Definition eq_sym := @eq_sym t.
Definition eq_trans := @eq_trans t.
Definition lt := Pos.lt.
Definition lt_trans := Pos.lt_trans.
Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.
Proof.
intros x y H. contradict H. rewrite H. apply Pos.lt_irrefl.
Qed.
Definition compare x y : Compare lt eq x y.
Proof.
case_eq (x ?= y); intros H.
- apply EQ. now apply Pos.compare_eq.
- apply LT; assumption.
- apply GT. now apply Pos.gt_lt.
Defined.
Definition eq_dec := Pos.eq_dec.
End Positive_as_OT.
(** [N] is an ordered type with respect to the usual order on natural numbers. *)
Module N_as_OT <: UsualOrderedType.
Definition t:=N.
Definition eq:=@eq N.
Definition eq_refl := @eq_refl t.
Definition eq_sym := @eq_sym t.
Definition eq_trans := @eq_trans t.
Definition lt := N.lt.
Definition lt_trans := N.lt_trans.
Definition lt_not_eq := N.lt_neq.
Definition compare x y : Compare lt eq x y.
Proof.
case_eq (x ?= y)%N; intro.
- apply EQ. now apply N.compare_eq.
- apply LT. assumption.
- apply GT. now apply N.gt_lt.
Defined.
Definition eq_dec := N.eq_dec.
End N_as_OT.
(** From two ordered types, we can build a new OrderedType
over their cartesian product, using the lexicographic order. *)
Module PairOrderedType(O1 O2:OrderedType) <: OrderedType.
Module MO1:=OrderedTypeFacts(O1).
Module MO2:=OrderedTypeFacts(O2).
Definition t := prod O1.t O2.t.
Definition eq x y := O1.eq (fst x) (fst y) /\ O2.eq (snd x) (snd y).
Definition lt x y :=
O1.lt (fst x) (fst y) \/
(O1.eq (fst x) (fst y) /\ O2.lt (snd x) (snd y)).
Lemma eq_refl : forall x : t, eq x x.
Proof.
intros (x1,x2); red; simpl; auto.
Qed.
Lemma eq_sym : forall x y : t, eq x y -> eq y x.
Proof.
intros (x1,x2) (y1,y2); unfold eq; simpl; intuition.
Qed.
Lemma eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z.
Proof.
intros (x1,x2) (y1,y2) (z1,z2); unfold eq; simpl; intuition eauto.
Qed.
Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.
Proof.
intros (x1,x2) (y1,y2) (z1,z2); unfold eq, lt; simpl; intuition.
left; eauto.
left; eapply MO1.lt_eq; eauto.
left; eapply MO1.eq_lt; eauto.
right; split; eauto.
Qed.
Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.
Proof.
intros (x1,x2) (y1,y2); unfold eq, lt; simpl; intuition.
apply (O1.lt_not_eq H0 H1).
apply (O2.lt_not_eq H3 H2).
Qed.
Definition compare : forall x y : t, Compare lt eq x y.
intros (x1,x2) (y1,y2).
destruct (O1.compare x1 y1).
apply LT; unfold lt; auto.
destruct (O2.compare x2 y2).
apply LT; unfold lt; auto.
apply EQ; unfold eq; auto.
apply GT; unfold lt; auto.
apply GT; unfold lt; auto.
Defined.
Definition eq_dec : forall x y : t, {eq x y} + {~ eq x y}.
Proof.
intros; elim (compare x y); intro H; [ right | left | right ]; auto.
auto using lt_not_eq.
assert (~ eq y x); auto using lt_not_eq, eq_sym.
Defined.
End PairOrderedType.
(** Even if [positive] can be seen as an ordered type with respect to the
usual order (see above), we can also use a lexicographic order over bits
(lower bits are considered first). This is more natural when using
[positive] as indexes for sets or maps (see FSetPositive and FMapPositive. *)
Module PositiveOrderedTypeBits <: UsualOrderedType.
Definition t:=positive.
Definition eq:=@eq positive.
Definition eq_refl := @eq_refl t.
Definition eq_sym := @eq_sym t.
Definition eq_trans := @eq_trans t.
Fixpoint bits_lt (p q:positive) : Prop :=
match p, q with
| xH, xI _ => True
| xH, _ => False
| xO p, xO q => bits_lt p q
| xO _, _ => True
| xI p, xI q => bits_lt p q
| xI _, _ => False
end.
Definition lt:=bits_lt.
Lemma bits_lt_trans :
forall x y z : positive, bits_lt x y -> bits_lt y z -> bits_lt x z.
Proof.
induction x.
induction y; destruct z; simpl; eauto; intuition.
induction y; destruct z; simpl; eauto; intuition.
induction y; destruct z; simpl; eauto; intuition.
Qed.
Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.
Proof.
exact bits_lt_trans.
Qed.
Lemma bits_lt_antirefl : forall x : positive, ~ bits_lt x x.
Proof.
induction x; simpl; auto.
Qed.
Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.
Proof.
intros; intro.
rewrite <- H0 in H; clear H0 y.
unfold lt in H.
exact (bits_lt_antirefl x H).
Qed.
Definition compare : forall x y : t, Compare lt eq x y.
Proof.
induction x; destruct y.
- (* I I *)
destruct (IHx y) as [l|e|g].
apply LT; auto.
apply EQ; rewrite e; red; auto.
apply GT; auto.
- (* I O *)
apply GT; simpl; auto.
- (* I H *)
apply GT; simpl; auto.
- (* O I *)
apply LT; simpl; auto.
- (* O O *)
destruct (IHx y) as [l|e|g].
apply LT; auto.
apply EQ; rewrite e; red; auto.
apply GT; auto.
- (* O H *)
apply LT; simpl; auto.
- (* H I *)
apply LT; simpl; auto.
- (* H O *)
apply GT; simpl; auto.
- (* H H *)
apply EQ; red; auto.
Qed.
Lemma eq_dec (x y: positive): {x = y} + {x <> y}.
Proof.
intros. case_eq (x ?= y); intros.
- left. now apply Pos.compare_eq.
- right. intro. subst y. now rewrite (Pos.compare_refl x) in *.
- right. intro. subst y. now rewrite (Pos.compare_refl x) in *.
Qed.
End PositiveOrderedTypeBits.
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