Strong Normalization of the Typed lambda_ws-calculus

International audienceThe lambda_ws-calculus is a lambda-calculus with explicit substitutions that satisfies the desired properties of such a calculus: step by step simulation of beta, confluence on terms with meta-variables and preservation of the strong normalization. It was conjectured that simply typed terms of lambda_ws are strongly normalizable. This was proved in by Di Cosmo & al. by using a translation of lambda_ws into the proof nets of linear logic. We give here a direct and elementary proof of this result. The strong normalization is also proved for terms typable with second order types (the extension of Girard's system~F). This is a new result