Library Coq.Structures.OrderedTypeEx


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.

  Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.

  Definition compare x y : Compare lt eq x y.

  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.

  Lemma lt_not_eq : forall x y, x<y -> ~ x=y.

  Definition compare x y : Compare lt eq x y.

  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.

  Definition compare x y : Compare lt eq x y.

  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.

  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.

 Lemma eq_sym : forall x y : t, eq x y -> eq y x.

 Lemma eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z.

 Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.

 Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.

 Definition compare : forall x y : t, Compare lt eq x y.
 Defined.

 Definition eq_dec : forall x y : t, {eq x y} + {~ eq x y}.

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.

  Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.

  Lemma bits_lt_antirefl : forall x : positive, ~ bits_lt x x.

  Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.

  Definition compare : forall x y : t, Compare lt eq x y.

  Lemma eq_dec (x y: positive): {x = y} + {x <> y}.

End PositiveOrderedTypeBits.