Primitive recursion is a well known syntactic restriction on recursive definitions which guarantees termination. Unfortunately many natural definitions, such as the most common definition of Euclid's GCD algorithm, are not primitive recursive. Walther has recently given a proof system for verifying termination of a broader class of definitions. Although Walther's system is highly automatible, the class of acceptable definitions remains only semi-decidable. Here we simplify Walther's calculus and give a syntactic criterion on definitions which guarantees termination. This syntactic criteria generalizes primitive recursion and handles most of the examples given by Walther. We call the corresponding class of acceptable definitions “Walther recursive”.
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