. This paper presents a system for explicit substitutions in Pure Type Systems (PTS). The system allows to solve type checking, type inhabitation, higher-order unification, and type inference for PTS using purely first-order machinery. A novel feature of our system is that it combines substitutions and variable declarations. This allows as a sideeffect to type check let-bindings. Our treatment of meta-variables is also explicit, such that instantiations of meta-variables is internalized in the calculus. This produces a confluent -calculus with distinguished holes and explicit substitutions that is insensitive to ff-conversion, and allows directly embedding the system into rewriting logic. 1 Introduction Explicit substitutions provide a convenient framework for encoding higher-order typed -calculus using first-order machinery. In particular, this allows to integrate higher-order unification with first-order provers, rewriting logic, and to delay evaluation and resolve scoping...

We present a dependent-type system for a #-calculus with explicit substitutions. In this system, meta-variables, as well as substitutions, are first-class objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization.

Functions play a central role in type theory, logic and computation. We describe how the notions of functionalisation (the way in which functions can be constructed) and instantiation (the process of applying a function to an argument) have been developed in the last century. We explain how both processes were implemented in Frege's Begriffschrift [17], Russell's Ramified Type Theory [42] and the lambda-calculus (originally introduced by Church [12, 13]) showing that the lambda-calculus misses a crucial aspect of functionalisation. We then pay attention to some special forms of function abstraction that do not exist in the lambda-calculus and we show that various logical constructs (e.g., let expressions and definitions and the use of parameters in mathematics), can be seen as forms of the missing part of functionalisation. Our study of the function concept leads...

Explicit substitution calculi are extensions of the lambda-calculus where the substitution mechanism is internalized into the theory. This feature makes them suitable for implementation and theoretical study of logic based tools as strongly typed programming languages and proof assistant systems. In this paper we explore new developments on two of the most successful styles of explicit substitution calculi: the lambdasigma- and lambda_se-calculi.

A unication method based on the se -style of explicit substitution is proposed. This method together with appropriate translations, provide a Higher Order Unication (HOU) procedure for the pure -calculus. Our method is inuenced by the treatment introduced by Dowek, Hardin and Kirchner using the -style of explicit substitution. Correctness and completeness properties of the proposed se-unication method are shown and its advantages, inherited from the qualities of the se -calculus, are pointed out. Our method needs only one sort of objects: terms. And in contrast to the HOU approach based on the -calculus, it avoids the use of substitution objects. This makes our method closer to the syntax of the -calculus. Furthermore, detection of redices depends on the search for solutions of simple arithmetic constraints which makes our method more operational than the one based on the -style of explicit substitution. Keywords: Higher order unication, explicit substitution, lambda-calculi. 1

Calculi with explicit substitutions (ES) are widely used in different areas of computer science. Complex systems with ES were developed these last 15 years to capture the good computational behaviour of the original systems (with meta-level substitutions) they were implementing. In this paper we first survey previous work in the domain by pointing out the motivations and challenges that guided the development of such calculi. Then we use very simple technology to establish a general theory of explicit substitutions for the lambda-calculus which enjoys fundamental properties such as simulation of one-step beta-reduction, confluence on metaterms, preservation of beta-strong normalisation, strong normalisation of typed terms and full composition. The calculus also admits a natural translation into Linear Logic’s proof-nets.

We show the soundness of a -calculus B where de Bruijn indices are used, substitution is explicit, and reduction is step-wise. This is done by interpreting B in the classical calculus where the explicit substitution becomes implicit and de Bruijn indices become named variables. This is the first flat semantics of explicit substitution and step-wise reduction and the first clear account of exactly when ff-reduction is needed. Keywords: Explicit Substitution, de Bruijn indices, Variable names, Soundness. 1. Introduction Variables play a very demanding role in the reduction and substitution of the -calculus. This has lead in many cases to using explicit rather than implicit substitution. Implementations of the -calculus provide their own explicit substitution procedures as in Nuprl 9 and Automath 23 . Furthermore, research on theories of explicit substitution has been striving lately 5;12;13;22;4;18 . In this paper, we extend the calculus of [13] (which is influenced by Automath...

This paper is concerned with strong normalisation of cut-elimination for a standard intuitionistic sequent calculus. The cut-elimination procedure is based on a rewrite system for proof-terms with cut-permutation rules allowing the simulation of β-reduction. Strong normalisation of the typed terms is inferred from that of the simply-typed λ-calculus, using the notions of safe and minimal reductions as well as a simulation in Nederpelt-Klop’s λI-calculus. It is also shown that the type-free terms enjoy the preservation of strong normalisation (PSN) property with respect to β-reduction in an isomorphic image of the type-free λ-calculus.

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. This paper is part of a general programme of treating explicit substitutions as the primary -calculi from the point of view of foundations as well as applications. Here we investigate the property of strong normalization. To date all the proofs of strong normalization of typed calculi of explicit substitutions use a reduction to the strong normalization of classical -calculus via the so-called \preservation of strong normalization" property. This paper develops a new approach, namely a direct proof that the strongly normalizing terms are precisely those typable under the intersection-types discipline. We also dene an eective perpetual strategy for the general calculus, give an inductive denition of the strongly normalizing terms, and furthermore show that normalization properties are essentially unaected by the inclusion of a rule for garbage collection. A key role is played by a certain general combinatorial lemma relating the reduction properties of two interacting abstract r...