) 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...

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...