An Ordinal Calculus for Proving Termination inTerm

In this article, we are concerned with the proofs of termination of rewrite systems by the method of interpretations. We propose a modiied approach to subrecursive hierarchies of functions by means of a syntactical approach to ordinal recursion. Our method appears to be appropriate for nding interpretations in a systematic way. We provide several examples of applications. It is shown that three usual recursion schemas over the natural numbers, recursion with parameter substitution , simple nested recursion and unnested multiple recursion can be encoded directly in our system. As the corresponding ordinal terms are primitive recursively closed, we get a concise and intuitive proof for the closure of the class of primitive recursive functions under these schemes.