# Library Coq.Structures.OrdersEx

Require Import Orders NPeano POrderedType NArith
ZArith RelationPairs EqualitiesFacts.

# Examples of Ordered Type structures.

Ordered Type for nat, Positive, N, Z with the usual order.
Module Nat_as_OT := NPeano.Nat.
Module Positive_as_OT := POrderedType.Positive_as_OT.
Module N_as_OT := BinNat.N.
Module Z_as_OT := BinInt.Z.
An OrderedType can now directly be seen as a DecidableType
Module OT_as_DT (O:OrderedType) <: DecidableType := O.
(Usual) Decidable Type for nat, positive, N, Z
Module Nat_as_DT <: UsualDecidableType := Nat_as_OT.
Module Positive_as_DT <: UsualDecidableType := Positive_as_OT.
Module N_as_DT <: UsualDecidableType := N_as_OT.
Module Z_as_DT <: UsualDecidableType := Z_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.
Include PairDecidableType O1 O2.
Definition lt :=
(relation_disjunction (O1.lt @@1) (O1.eq * O2.lt))%signature.
Instance lt_strorder : StrictOrder lt.
Instance lt_compat : Proper (eq==>eq==>iff) lt.
Definition compare x y :=
match O1.compare (fst x) (fst y) with
| Eq => O2.compare (snd x) (snd y)
| Lt => Lt
| Gt => Gt
end.
Lemma compare_spec : forall x y, CompSpec eq lt x y (compare x y).
End PairOrderedType.