Module Ident <: OrderedTypeFull
with Definition t := positive
with Definition lt := Plt
with Definition compare := fun x y => Pcompare x y Eq
with Definition le := Ple.
Definition t := positive.
Definition eq : t -> t -> Prop := @Logic.eq positive.
Definition eq_equiv: Equivalence eq := @eq_equivalence t. (*apply eq_equivalence. Qed.*)
Definition lt : t -> t -> Prop := Plt.
apply Positive_as_OT.lt_compat. Qed.
Definition compare: t -> t -> comparison := fun x y => Pcompare x y Eq.
apply Positive_as_OT.compare_spec. Qed.
Definition eq_dec: forall x y, {eq x y}+{~eq x y} := positive_eq_dec.
Definition le : t -> t -> Prop := Ple.

Lemma le_lteq : forall p q : t, le p q <-> lt p q \/ p = q.