Definition compare x y : Compare lt eq x y.Definition eq_dec := Z.eq_dec.Definition t:=positive.Definition eq:=@eq positive.Definition eq_refl := @eq_refl t.Definition eq_sym := @eq_sym t.Definition eq_trans := @eq_trans t.Definition lt := Pos.lt.Definition lt_trans := Pos.lt_trans.

Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.