Categorical combinators [12, 21, 43] and more recently oe-calculus [1, 23], have been introduced to provide an explicit treatment of substitutions in the -calculus. We reintroduce here the ingredients of these calculi in a self-contained and stepwise way, with a special emphasis on confluence properties. The main new results of the paper w.r.t. [12, 21, 1, 23] are the following: 1. We present a confluent weak calculus of substitutions, where no variable clashes can be feared. 2. We solve a conjecture raised in [1]: oe-calculus is not confluent (it is confluent on ground terms only). This unfortunate result is "repaired" by presenting a confluent version of oe-calculus, named the Env-calculus in [23], called here the confluent oe-calculus.

The aim of this paper is to present the s-calculus which is a very simple -calculus with explicit substitutions and to prove its confluence on closed terms and the preservation of strong normalisation of -terms. We shall prove strong normalisation of the corresponding calculus of substitution by translating it into the oe-calculus [ACCL91], and therefore the relation between both calculi will be made explicit. The confluence of the s-calculus is obtained by the "interpretation method" ([Har89], [CHL92]). The proof of the preservation of normalisation follows the lines of an analogous result for the AE-calculus (cf. [BBLRD95]). The relation between s and AE is also studied.

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...

In this article we show how we proved correctness of the translation from a small applicative language with recursive definitions (Mini-ML) to the Categorical abstract machine (CAM) using the Coq system. Our aim was to mechanise the proof of J. Despeyroux [10]. Like her, we use natural semantics to axiomatise the semantics of our languages. The axiomatisations of inferences systems and of the languages is nicely performed by the mechanism of inductive definitions in the Coq system. Unfortunately both the source and the target semantics involve nested structures that cannot be formalised inductively. We have overcome this problem by making some slight modifications of both the source and target semantics and show how the changes in the source and target semantics are related. For the remaining tranlation we explain how we can use the Coq system to formalize non-terminating programs and incorrect programs, objects that are impossible to explain with only the formalism of natural semantic...

This article is a brief review of the type free λ-calculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λ-calculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentation of the λ-calculus with de Bruijn indices, we illustrate how a calculus of explicit substitutions can be obtained. In addition, de Bruijn's notation for the λ-calculus is introduced and some of its advantages are outlined.

We give an alternative, proof-theory inspired proof of the well-known result that the untyped -calculus presented with variable names and `a la de Bruijn are isomorphic. The two presentations of the -calculus come about from two isomorphic logic formalisations by observing that, for the logic in question, the Curry-Howard correspondence is formulaindependent. We identify the exchange rule as the the proof-theoretical difference between the two representations of the systems. 1 Introduction The Curry-Howard correspondence relates formal inference systems of symbolic logic to typed -like calculi. An inference system for formal, symbolic logic is said to be in Hilbert-style if, 1 no logical rule (i.e., excluding cut, weakening, etc.) change the set of assumptions. Such systems are also referred to as combinatory logics, in that they typically consist of a set of tautologies (or combinators) which are combined by the, so-called, Modus Ponens rule: A ! B A (Modus Ponens) B For...

Machine). Notre objectif a 'et'e de m'ecaniser une preuve pr'esent'ee dans l'article de J. Despeyroux [9] et 'ecrite `a l'aide du langage Typol. Nous utilisons des s'emantiques naturelles pour mod'eliser l"evaluation de nos langages. Nous ne sommes parvenus que partiellement `a m'ecaniser cette preuve de correction. En effet, les sp'ecifications naturelles des langages source et cible contiennent des termes rationnels difficiles `a axiomatiser dans l"etat actuel du syst`eme. Nous proposons un d'ecoupage de la preuve isolant cette difficult'e. (Abstract: pto) Samuel.Boutin@inria.fr Unit'e de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) T'el'ephone : (33 1) 39 63 55 11 -- T'el'ecopie : (33 1) 39 63 53 30 Proving Correctness of the Translation from Mini-ML to the CAM with the Coq Proof Development System Abstract: In this report we show how we proved correctness of the translation from a small applicative language with rec...