In this paper, we show the correspondence existing between normalization in calculi with explicit substitution and cut elimination in sequent calculus for Linear Logic, via Proof Nets. This correspondence allows us to prove that a typed version of the #x-calculus [30, 29] is strongly normalizing, as well as of all the calculi isomorphic to it such as # # [24], # s [19], # d [21], and # f [11]. In order to achieve this result, we introduce a new notion of reduction in Proof Nets: this extended reduction is still confluent and strongly normalizing, and is of interest of its own, as it correspond to more identifications of proofs in Linear Logic that differ by inessential details. These results show that calculi with explicit substitutions are really an intermediate formalism between lambda calculus and proof nets, and suggest a completely new way to look at the problems still open in the field of explicit substitutions.

This paper studies confluence of extensional and non-extensional -calculi with explicit substitutions, where extensionality is interpreted by j-expansion. For that, we propose a scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence. Our method makes it possible to treat at the same time many well-known calculi such as oe , oe * , OE , s , AE , f , d and dn . Keywords: functional programming, -calculi, explicit substitutions, confluence, extensionality. 1 Introduction The -calculus is a convenient framework to study functional programming, where the evaluation process is modeled by fi-reduction. The main mechanism used to perform fi-reduction is substitution, which consists of the replacement of formal parameters by actual arguments. The correctness of substitution is guaranteed by a systematic renaming of bound variables, inconvenient which can be simply avoided in the -calculus `a la de Bruijn by using natur...

Calculi of explicit substitutions have almost always been presented using de Bruijn indices with the aim of avoiding ff-conversion and being as close to machines as possible. De Bruijn indices however, though very suitable for the machine, are difficult to human users. This is the reason for a renewed interest in systems of explicit substitutions using variable names. Formal systems of explicit substitutions using variable names is a new area however and we believe, it should not develop without being well-tied to existing work on explicit substitutions. The aim of this paper is to establish a bridge between explicit substitutions using de Bruijn indices and using variable names. In our aim to do so, we provide the t-calculus: a -calculus `a la de Bruijn which can be translated into a -calculus with explicit substitutions written with variables names. We present explicitly this translation and use it to obtain preservation of strong normalisation for t. Moreover, we show several prope...

Typed #-terms are used as a compact and linear representation of proofs in intuitionistic logic. This is possible since the Curry-Howard isomorphism relates proof trees with typed #-terms. The proofs-as-terms principle can be used to check a proof by type checking the #-term extracted from the complete proof tree. However, proof trees and typed #-terms are built differently. Usually, an auxiliary representation of unfinished proofs is needed, where type checking is possible only on complete proofs. In this paper we present a proof synthesis method for dependent-type systems where typed open terms are built incrementally at the same time as proofs are done. This way, every construction step, not just the last one, may be type checked. The method is based on a suitable calculus where substitutions as well as meta-variables are first-class objects.

We refine the simulation technique introduced in [10] to show strong normalization of -calculi with explicit substitutions via termination of cut elimination in proof nets [12]. We first propose a notion of equivalence relation for proof nets that extends the one in [9], and we show that cut elimination modulo this equivalence relation is terminating. We then show strong normalization of the typed version of the l - calculus with de Bruijn indices (a calculus with full composition defined in [8]) using a translation from typed l to proof nets. Finally, we propose a version of typed l with named variables which helps to better understand the complex mechanism of the explicit weakening notation introduced in the l -calculus with de Bruijn indices [8]. 1

The past decade has seen an explosion of work on calculi of explicit substitutions. Numerous work has illustrated the usefulness of these calculi for practical notions like the implementation of typed functional programming languages and higher order proof assistants. It has also been shown that eta reduction is useful for adapting substitution calculi for practical problems like higher order uni cation. This paper concentrates on rewrite rules for eta reduction in three dierent styles of explicit substitution calculi: , se and the suspension calculus. Both and se when extended with eta reduction, have proved useful for solving higher order uni cation. We enlarge the suspension calculus with an adequate eta-reduction which we show to preserve termination and conuence of the associated substitution calculus and to correspond to the eta-reductions of the other two calculi. We prove that and se as well as and the suspension calculus are non comparable while se is more adequate than the suspension calculus in simulating one step of beta-contraction.

Contrary to all expectations, the lambda-sigma-calculus, the canonical simply-typed lambda-calculus with explicit substitutions, is not strongly normalising. This result has led to a proliferation of calculi with explicit substitutions. This paper shows that the reducibility method provides a general criterion when a calculus of explicit substitution is strongly normalising for all untyped lambda-terms that are strongly normalising. This result is general enough to imply preservation of strong normalisation of the calculi considered in the literature. We also propose a version of the lambda-sigma-calculus with explicit substitutions which is strongly normalising for strongly normalising lambda-terms.

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.

We propose a solution to the standing open problem of finding a calculus of explicit substitution with the following four properties: 1. simulates one-step β-reduction, 2. is confluent on meta-terms (also known as "open terms"), 3. has a strongly normalizing substitution sub-calculus, and 4. preserves β-strong normalization. Our solution, λxci, is based on insights gained by studying the critical pair between two meta-terms that makes calculi without substitution composition non-confluent (on meta-terms). The insight is closely tied to the fact that this critical pair is essentially an explicit representation of the "substitution lemma" of λ-calculus, and the missing link in the solution is to express finiteness of all reductions starting from any reachable development of the source term. We give an encoding of the system as a first order system using de Bruijn's explicit variable indexing idea, and show that it enjoys the same properties by an easy equivalence.

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.