Module OrderedType_from_Alt (O:OrderedTypeAlt) <: OrderedType.
Definition t := t.
Definition eq x y := (x?=y) = Eq.
Definition lt x y := (x?=y) = Lt.
Definition eq_trans := (compare_trans Eq).
Definition lt_trans := (compare_trans Lt).

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