A lambda-calculus with explicit weakening and explicit substitution

International audienceSince Melliès has shown that $\lambda\sigma$ (a calculus of explicit substitutions) does not preserve the strong normalization of the $\beta$-reduction, it became a challenge to find a calculus satisfying the following properties: step by step simulation of the beta-reduction, confluence on terms with metavariables, strong normalization of the calculus of substitutions and preservation of the strong normalization of the $\lambda$-calculus. We present here such a calculus. The main novelty of the calculus (given with de Bruijn indices) is the use of labels that represent updating functions and correspond to explicit weakening. A typed version is also presented