Parameter le_lt_eq_dec : forall a b : A, le a b -> {lt a b} + {a = b}.
End MORE_DEC_ORDERS.
(* A functor for getting some useful derived properties on decidable orders *)
Module More_Dec_Orders (D: DEC_ORDER) : MORE_DEC_ORDERS with Definition
A := D.A with Definition le := D.le with Definition lt := D.lt.
Definition A := D.A.
Definition le := D.le.
Definition lt := D.lt.

Theorem lt_not_le : forall a b : A, lt a b -> ~ le b a.