# Library Coq.Structures.OrdersEx

Require Import Orders BoolOrder PeanoNat POrderedType BinNat BinInt
RelationPairs EqualitiesFacts.

# Examples of Ordered Type structures.

Ordered Type for bool, nat, Positive, N, Z with the usual order.

Module Bool_as_OT := BoolOrder.BoolOrd.
Module Nat_as_OT := PeanoNat.Nat.
Module Positive_as_OT := BinPos.Pos.
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 bool, nat, positive, N, Z

Module Bool_as_DT <: UsualDecidableType := Bool_as_OT.
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.

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 MSetPositive).

Local Open Scope positive.

Module PositiveOrderedTypeBits <: UsualOrderedType.
Definition t:=positive.
Include HasUsualEq <+ UsualIsEq.
Definition eqb := Pos.eqb.
Definition eqb_eq := Pos.eqb_eq.
Include HasEqBool2Dec.

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_antirefl : forall x : positive, ~ bits_lt x x.

Lemma bits_lt_trans :
forall x y z : positive, bits_lt x y -> bits_lt y z -> bits_lt x z.

Instance lt_compat : Proper (eq==>eq==>iff) lt.

Instance lt_strorder : StrictOrder lt.

Fixpoint compare x y :=
match x, y with
| x~1, y~1 => compare x y
| x~1, _ => Gt
| x~0, y~0 => compare x y
| x~0, _ => Lt
| 1, y~1 => Lt
| 1, 1 => Eq
| 1, y~0 => Gt
end.

Lemma compare_spec : forall x y, CompSpec eq lt x y (compare x y).

End PositiveOrderedTypeBits.