  The notion of computability closure has been introduced for proving the
termination of the combination of higher-order rewriting and beta-reduction. It
is also used for strengthening the higher-order recursive path ordering. In the
present paper, we study in more details the relations between the computability
closure and the (higher-order) recursive path ordering. We show that the
first-order recursive path ordering is equal to an ordering naturally defined
from the computability closure. In the higher-order case, we get an ordering
containing the higher-order recursive path ordering whose well-foundedness
relies on the correctness of the computability closure. This provides a simple
way to extend the higher-order recursive path ordering to richer type systems.
