Library Coq.NArith.POrderedType


Require Import BinPos Equalities Orders OrdersTac.

Local Open Scope positive_scope.

DecidableType structure for positive numbers


Module Positive_as_UBE <: UsualBoolEq.
 Definition t := positive.
 Definition eq := @eq positive.
 Definition eqb := Peqb.
 Definition eqb_eq := Peqb_eq.
End Positive_as_UBE.

Module Positive_as_DT <: UsualDecidableTypeFull
 := Make_UDTF Positive_as_UBE.

Note that the last module fulfills by subtyping many other interfaces, such as DecidableType or EqualityType.

OrderedType structure for positive numbers


Module Positive_as_OT <: OrderedTypeFull.
 Include Positive_as_DT.
 Definition lt := Plt.
 Definition le := Ple.
 Definition compare p q := Pcompare p q Eq.

 Instance lt_strorder : StrictOrder Plt.

 Instance lt_compat : Proper (Logic.eq==>Logic.eq==>iff) Plt.

 Definition le_lteq := Ple_lteq.
 Definition compare_spec := Pcompare_spec.

End Positive_as_OT.

Note that Positive_as_OT can also be seen as a UsualOrderedType and a OrderedType (and also as a DecidableType).

An order tactic for positive numbers


Module PositiveOrder := OTF_to_OrderTac Positive_as_OT.
Ltac p_order := PositiveOrder.order.

Note that p_order is domain-agnostic: it will not prove 1<=2 or x<=x+x, but rather things like x<=y -> y<=x -> x=y.