A lambda-calculus with explicit weakening and explicit substitution

Since Mellies showed that λσ (a calculus of explicit substitutions) does not preserve the strong normalization of the β-reduction, it has become a challenge to find a calculus satisfying the following properties: step-by-step simulation of the β-reduction, confluence on terms with metavariables, strong normalization of the calculus of substitutions and preservation of the strong normalization of the λ-calculus. We present here such a calculus. The main novelty of this 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.

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