Definition lt_eq_lt_dec : forall a b : A, {lt a b} + {a = b} + {lt b a}.Definition A := nat.Definition le := le.Definition lt := lt.

(* Specifications: *)
Theorem ordered : order A le.
Theorem lt_le_weak : forall a b : A, lt a b -> le a b.
Theorem lt_diff : forall a b : A, lt a b -> a <> b.
Theorem le_lt_or_eq : forall a b : A, le a b -> lt a b \/ a = b.