# 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.