The S-calculus: Its Typed and Its Extended Versions

We present in this paper the simply typed version of the s-calculus (cf. KR95]) and prove the strong normalisation of the well typed terms. We also present an extension of the s-calculus: the s e-calculus and prove its local connuence on open terms and the weak normalisation of its corresponding calculus of substitutions s e. The strong normalisation of s e is still an open problem to challenge the rewriting community.