Normalisation by Calculation

I outline a new proof of strong normalisation for system F, by direct calculation of the size of a normalisation tree. The system F itself is used as a formulism for representing these calculations, and any reasonably sensible model of the calculus can be used to verify that the correctness of the calculations. The method works for a variety of calculi, and seems reasonably general. Discussion Proofs of strong normalisation for typed-calculi, such as system F, are well known. The standard proof, developed by Tait and Girard using a notion called reducibility, proceeds by deening sets of certain normalising terms in such a way that it can be shown that every term is a member of the appropriate set. This proof is probably best understood in a model theoretic manner: the sets of normalising terms form a model of some sort. A term t may be interpreted in this model by some term (t]), which is necessarily normalising, and either actually t = (t]), or t and (t]) are suuciently similar so that we may infer that t is normalising. All previous proofs of normalisation for system F follow this pattern, there being some diierences in the exact machinery used. We present here a novel proof, which does not rely on a model based on sets of terms. Instead, we present a translation t 7 ! (t]) on terms, such that (t]) is a program computing a witness for the 0 1 statement that t is strongly normalising. By interpreting (t]) in a model, or by otherwise extracting an actual value from (t]), we can verify that t is in fact strongly normalising.