A notation for lambda terms is described that is useful in contexts where the intensions of these terms need to be manipulated. This notation uses the scheme of de Bruijn for eliminating variable names, thus obviating ff-conversion in comparing terms. This notation also provides for a class of terms that can encode other terms together with substitutions to be performed on them. The notion of an environment is used to realize this `delaying' of substitutions. The precise mechanism employed here is, however, more complex than the usual environment mechanism because it has to support the ability to examine subterms embedded under abstractions. The representation presented permits a fi-contraction to be realized via an atomic step that generates a substitution and associated steps that percolate this substitution over the structure of a term. The operations on terms that are described also include ones for combining substitutions so that they might be performed simultaneously. Our notatio...

A language of constructions for minimal logic is the -calculus, where cut-elimination is encoded as fi-reduction. We examine corresponding languages for the minimal version of the modal logic S4, with notions of reduction that encodes cut-elimination for the corresponding sequent system. It turns out that a natural interpretation of the latter constructions is a -calculus extended by an idealized version of Lisp's eval and quote constructs. In this Part IIIa, we examine the termination and confluence properties of the evQ and evQ H -calculi. Most results are negative: the typed calculi do not terminate, the subsystems \Sigma and \Sigma H that propagate substitutions, quotations and evaluations downwards do not terminate either in the untyped case, and the untyped evQ H -calculus is not confluent. However, the typed versions of \Sigma and \Sigma H do terminate, so the typed evQ-calculus is confluent. It follows that the typed evQ-calculus is a conservative extension of the typed S4-cal...

Abstract. This paper shows how a previously specified theory for Abstract Reduction Systems (ARSs) in which noetherianity was defined by the notion of wellfoundness over binary relations is used in order to prove results such as the wellknown Newman’s Lemma and the Yokouchi’s Lemma. The former one known as the diamond lemma and the later which states a property of commutation between ARSs. Thears theory was specified in the Prototype Verification System (PVS) for which to the best of our knowledge there are no available theory for dealing with rewriting techniques in general. In addition to proof techniques available in PVS the verification of these lemmas implies an elaborated use of natural as well as noetherian induction. 1.