We show how to mechanise equational proofs about higher-order languages by using the primitive proof principles of first-order abstract syntax over one-sorted variable names. We illustrate the method here by proving (in Isabelle/HOL) a technical property which makes the method widely applicable for the λ-calculus: the residual theory of β is renaming-free up-to an initiality condition akin to the so-called Barendregt Variable Convention. We use our results to give a new diagram-based proof of the development part of the strong finite development property for the λ-calculus. The proof has the same equational implications (e.g., confluence) as the proof of the full property but without the need to prove SN. We account for two other uses of the proof method, as presented elsewhere. One has been mechanised in full in Isabelle/HOL.

An intersection type assignment system for the extension LJ of the untyped l-calculus, introduced by Joachimski and Matthes, is given and proven to characterize the strongly normalizing terms of LJ. Since LJ's generalized applications naturally allow permutative/commuting conversions, this is the first analysis of a term rewrite system with permutative conversions by help of intersection types. Two proofs are given for the fact that the typable terms are strongly normalizing: One by the computability predicates method a la Tait and one showing directly that strongly normalizing typable terms are closed under (generalized) application and substitution. It is also shown that a straightforward extension of the type assignment for l-calculus fails to capture the strongly normalizing terms. Keywords Intersection Types, Strong Normalization, Permutative Conversions, Saturated Sets. 1 Introduction In [5] an extension LJ of l-calculus with generalized applications inspired by vo...

It is shown how the sequent calculus LJ can be embedded into a simple extension of the -calculus by generalized applications, called J. The reduction rules of cut elimination and normalization can be precisely correlated, if explicit substitutions are added to J. The resulting system J2 is proved strongly normalizing, thus showing strong normalization for Gentzen's cut elimination steps. This re nes previous results by Zucker, Pottinger and Herbelin on the isomorphism between natural deduction and sequent calculus.

Abstract. The intuitionistic fragment of the call-by-name version of Curien and Herbelin’s λµ˜µ-calculus is isolated and proved strongly normalising by means of an embedding into the simply-typed λ-calculus. Our embedding is a continuation-and-garbage-passing style translation, the inspiring idea coming from Ikeda and Nakazawa’s translation of Parigot’s λµ-calculus. The embedding simulates reductions while usual continuation-passing-style transformations erase permutative reduction steps. For our intuitionistic sequent calculus, we even only need “units of garbage ” to be passed. We apply the same method to other calculi, namely successive extensions of the simply-typed λ-calculus leading to our intuitionistic system, and already for the simplest extension we consider (λ-calculus with generalised application), this yields the first proof of strong normalisation through a reduction-preserving embedding. 1

In this talk we overview our study on an extension of the λ-calculus introduced in [1], exhibiting the features of multiarity and generality. The former feature is the ability of applying a term to a list of arguments. The latter is the ability of specifying a future use, or “continuation”, for a (possibly multiary) application. The calculus was named the generalised multiary λ-calculus, or the λJ m-calculus for short. In its simply typed version, the calculus corresponds to a sequent calculus, and the novelty of the mentioned features relates to the novelty of the left introduction rule relatively to a natural deduction format. The calculus was introduced in [1] as a calculus of multiary sequent terms (in the sense of [4]) for a study of permutative conversions in sequent calculus. The main lesson of this study is the existence of a relationship between permutative conversions, subsystems of λJ m and the features of generality and multiarity. Moreover, some subsystems turn out to be isomorphic to fragments of natural deduction, taking natural deduction in the extended sense of von Plato [5]. Later work [2] observes an overlap between the features of generality and multiarity. In [2] the overlap is mainly used to transfer results about reduction from ΛJ to λJ m, where ΛJ is the λ-calculus with generalised application introduced in [3]. In addition this overlap suggests a refinement of the original view of λJ m as obtained from λ by modularly adding the two new features. This refinement is ongoing work. Another issue we are considering is that of obtaining a λ-term in β-normal form out of a λJ m-term. This requires the combination of reduction and permutative conversion and, in particular, raises questions of termination. These questions relate to the problem of preservation of strong normalisation in calculi of explicit substitutions.

Abstract. We consider formal provability with structural induction and related proof principles in the λ-calculus presented with first-order abstract syntax over onesorted variable names. As well as summarising and elaborating on earlier, formally verified proofs (in Isabelle/HOL) of the relative renaming-freeness of β-residual theory and β-confluence, we also present proofs of η-confluence, βη-confluence, the strong weakly-finite β-development (aka residual-completion) property, residual β-confluence, η-over-β-postponement, and notably β-standardisation. In the latter case, the known proofs fail in instructive ways. Interestingly, our uniform proof methodology, which has relevance beyond the λ-calculus, properly contains pen-and-paper proof practices in a precise sense. The proof methodology also makes precise what is the full algebraic proof burden of the considered results, which we, moreover, appear to be the first to resolve. 1