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) Maria C. F. Ferreira 1 and Delia Kesner 2 and Laurence Puel 2 1 Dep. de Inform'atica, Fac. de Ciencias e Tecnologia, Univ. Nova de Lisboa, Quinta da Torre, 2825 Monte de Caparica, Portugal, cf@fct.unl.pt. 2 CNRS & Lab. de Rech. en Informatique, Bat 490, Univ. de Paris-Sud, 91405 Orsay Cedex, France, fkesner,puelg@lri.fr. Abstract. We study preservation of fi-strong normalization by d and dn , two confluent -calculi with explicit substitutions defined in [10]; the particularity of these calculi is that both have a composition operator for substitutions. We develop an abstract simulation technique allowing to reduce preservation of fi-strong normalization of one calculus to that of another one, and apply said technique to reduce preservation of fi-strong normalization of d and dn to that of f , another calculus having no composition operator. Then, preservation of fi-strong normalization of f is shown using the same technique as in [2]. As a consequence, d and dn become the fir...

We present typed pattern calculus with explicit pattern matching and explicit substitutions, where both the typing rules and the reduction rules are modeled on the same logical proof system, namely Gentzen sequent calculus for minimal logic. Our calculus is inspired by the Curry-Howard Isomorphism, in the sense that types, both for patterns and terms, correspond to propositions, terms correspond to proofs, and term reduction corresponds to sequent proof normalization performed by cut elimination. The calculus enjoys subject reduction, confluence, preservation of strong normalization w.r.t a system with meta-level substitutions, and strong normalization for well-typed terms, and, as a consequence, can be seen as an implementation calculus for functional formalisms using meta-level operations for pattern matching and substitutions.

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.

In the frame of intuitionistic logic and type theory, it is well known that there is an isomorphism between types and propositions; the Curry-Howard Isomorphism. However, it is less clear the relation between terms construction and proofs development. The main difficulty arises when we try to represent incomplete proofs as terms describing a state of knowledge where some part of the proof is built, but another part remains to be built. The pieces of proof terms that are unknown are called places-holders. We present a theoretical approach to place-holders in type theory. In this approach place-holders are represented by metavariables and terms are built incrementally by instantiation of metavariables. We show how an appropriate extension to typed -calculus with explicit substitutions and explicit typing of metavariables allows to identify terms construction and proofs development activities.

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.

. The \Delta-calculus is a -calculus with a local operator closely related to normalisation procedures in classical logic and control operators in functional programming. We introduce \Deltaexp, an explicit substitution calculus for \Delta, show it preserves strong normalisation and that its simply typed version is strongly normalising. Interestingly, \Deltaexp is the first example for which the decency method of showing preservation of strong normalisation (PSN) works whereas the structure preserving method which is based on the decency method does not. In particular, \Deltaexp is a very simple calculus yet is not structure preserving. This shows that the structure preserving notion intended to give a general description of calculi of explicit substitution that satisfy PSN, is restrictive. To our knowledge, \Deltaexp is the first calculus of explicit substitution that is not structure preserving. 5 1 Introduction Explicit substitutions were introduced in [1] as a bridge between -cal...

This paper proposes an implementation of objects and functions via a calculus with explicit substitutions which is confluent and preserves strong normalization. The source calculus corresponds to the combination of the \sigma-calculus of Abadi and Cardelli [AC96] and the \lambda-calculus, and the target calculus corresponds to an extension of the former calculus with explicit substitutions. The interesting feature of our calculus is that substitutions are separated -- and treated accordingly -- in two different kinds: those used to encode ordinary substitutions and those encoding invoke substitutions. When working with explicit substitutions, this differentiation is essential to encode \lambda-calculus into \sigma-calculus in a conservative way, following the style proposed in [AC96].

In the frame of intuitionistic logic and type theory, it is well known that there is an isomorphism between types and propositions; the Curry-Howard Isomorphism. However, it is less clear the relation between terms construction and proofs development. The main difficulty arises when we try to represent incomplete proofs as terms describing a state of knowledge where some part of the proof is built, but another part remains to be built. The pieces of proof terms that are unknown are called places-holders. We present a theoretical approach to place-holders in type theory. In this approach place-holders are represented by metavariables and terms are built incrementally by instantiation of metavariables. We show how an appropriate extension to typed -calculus with explicit substitutions and explicit typing of metavariables allows to identify terms construction and proofs development activities. Representaci'on de pruebas en la teor'ia de tipos: Estado del arte Resumen En el marco de la l...