Computability Closure: Ten Years Later

The notion of computability closure has been introduced for proving the termination of higher-order rewriting with first-order matching by Jean-Pierre Jouannaud and Mitsuhiro Okada in a 1997 draft which later served as a basis for the author's PhD. In this paper, we show how this notion can also be used for dealing with beta-normalized rewriting with matching modulo beta-eta (on patterns \`a la Miller), rewriting with matching modulo some equational theory, and higher-order data types (types with constructors having functional recursive arguments). Finally, we show how the computability closure can easily be turned into a reduction ordering which, in the higher-order case, contains Jean-Pierre Jouannaud and Albert Rubio's higher-order recursive path ordering and, in the first-order case, is equal to the usual first-order recursive path ordering.