Polynomial Interpretations for Higher-Order Rewriting

Formalizing Higher-Order Termination in Coq

“…Within the higher-order framework, several methods have been developed, and our focus is on the so-called semantical methods: to prove an AFS is terminating we need to find a well-founded interpretation domain such that s > t , whenever s → R t. This is achieved by orienting each rule in R, that is, ℓ > r for all rules ℓ → r in R. The idea was first introduced by van de Pol [11] (in the context of HRS) as an extension of the first-order semantic interpretation method. Later, these semantic methods got extended to AFSs by Fuhs and Kop [5] with a special focus on implementation, and it is part of Wanda, a termination tool developed by Kop [8].…”

“…Given two extended well-founded sets X and Y , we can construct another well-founded set whose elements are weakly monotonic functions. Using those, we can define the interpretation WM A of a simple type A the same way as Fuhs and Kop [5]. For a full interpretation, we also should interpret terms.…”

“…For a full interpretation, we also should interpret terms. The necessary data for that is more complicated, and for the details we refer the reader to [5].…”

“…In this section, we define the notion of AFS, and we describe how we formalized this notion in Coq. The definitions given here correspond to those usually given in the literature [5] while the definitions in Coq deviate slightly. Note that since we want to extract our formalization to an OCaml program, we use a deep embedding, which is similar to the approach taken in CeTA and CoLoR [2,13].…”

Size-based termination of higher-order rewriting

“…Gandy's approach was later extended by van de Pol (van de Pol, 1993;van de Pol, 1996) and Kahrs (Kahrs, 1995) to arbitrary higher-order rewriting à la Nipkow (Nipkow, 1991;Mayr & Nipkow, 1998), that is, to rewriting on terms in β -normal η-long form with higherorder pattern-matching (Miller, 1991). But this approach has been implemented only recently (Fuhs & Kop, 2012).…”