Bridging the -and S-styles of Explicit Substitutions ?

We present the ! and !e calculi, the two-sorted (term and substitution) versions of the s (cf. KR95a]) and se (cf. KR96a]) cal-culi, respectively. We establish an isomorphism between the s-calculus and the term restriction of the !-calculus, which extends to an isomor-phism between se and the term restriction of !e. Since the ! and !e calculi are given in the style of the-calculus (cf. ACCL91]) they bridge calculi between s and and between se and and thus we are able to better understand one calculus in terms of the other. We improve our knowledge on the open problem of strong normalisation (SN) of the associated calculus of substitutions se by showing SN for two sub-calculi (we use the isomorphism with !e for the proof of SN of one of them). Finally, we present typed versions of all the calculi and check that the above mentioned isomorphism preserves types. As a consequence, the !-calculus is a calculus in the-style that simulates one step-reduction, is connuent (on closed terms), preserves strong normalisation, its associated calculus of substitutions is SN, and its simply typed version is also SN. Moreover, the !-calculus possesses an extension !e that is connuent on open terms (terms with eventual metavariables of sort term only) and whose simply typed version is weakly normalising (on open term). As far as we know, !-calculus is the rst calculus in the-style that has all those properties. However, the two open problems of the SN of the associated calculus of substitution of !e and of the preservation of strong normalisation of !e remain unsolved.