Module OrderedPositive <: OrderedType.
Definition t := positive.
Definition eq (x y: t) := x = y.
Definition lt := Plt.
Proof (@refl_equal t).
Proof (@sym_equal t).
Proof (@trans_equal t).
Proof Plt_trans.
Proof Plt_ne.

Lemma eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z.