### Strict functionals for termination proofs [chapter]

Jaco van de Pol, Helmut Schwichtenberg
1995 Lecture Notes in Computer Science
A semantical method to prove termination of higher order rewrite systems (HRS) is presented. Its main tool is the notion of a strict functional, which is a variant of Gandy's notion of a hereditarily monotonic functional 1]. The main advantage of the method is that it makes it possible to transfer ones intuitions about why an HRS should be terminating into a proof: one has to nd a \strict" interpretation of the constants involved in such a way that the left hand side of any rewrite rule gets a
more » ... igger value than the right hand side. The applicability of the method is demonstrated in three examples. An HRS involving map and append. The usual rules for higher order primitive recursion in G odel's T. Derivation terms for natural deduction systems. We prove termination of the rules for {conversion and permutative conversion for logical rules including introduction and elimination rules for the existential quanti er. This has already been proved by Prawitz in 5]; however, our proof seems to be more perspicuous. Technically we build on 7]. There a notion of a strict functional and simultaneously of a strict greater{than relation > str between monotonic functionals is introduced. The main result then is the following. Let M be a term in normal form and 2 FV(M). Then for any strict environment U and all monotonic f and g, one has f > mon g =) M] ] U 7 !f] > str M] ] U 7 !g] . From this van de Pol derives the technique described above for proving termination of higher order term rewrite systems, generalizing a similar approach for rst order rewrite systems (cf. 3, p. 367]). Interesting applications are given in 7]. Here a slight change in the de nition of strictness is exploited (against the original conference paper; cf. 7, Footnote p. 316]). This makes it possible to deal with rewrite rules involving types of level > 2 too, and in particular with proof theoretic applications. In order to do this some theory of strict functionals is developed. We also add product types, which are necessary to treat e.g. the existential quanti er.