Call-by-value, call-by-name, and strong normalization for the classical sequent calculus

Abstract : We present a typed calculus LambdaXi isomorphic to the implicational fragment of the classical sequent calculus LK. Reductions in LK eliminate the cut-rule by local rewriting steps, which correspond to the evaluation of explicit substitutions in the calculus. This bridges the gap between Curien and Herbelin's LambdaBarMuMu~-calculus and Urban's rewriting system for proofs. Encodings of one into the other are defined, and from one of them we derive the strong normalization of LambdaBarMuMu~. Identifying two reduction strategies CBV and CBN in Urban's rewriting system enables us to derive two corresponding semantics of continuations from those of LambdaBarMuMu~, via the other encoding.