Strong Normalization of Explicit Substitutions via Cut Elimination in Proof Nets (Extended Abstract)

In this paper, we show the correspondence existing between normalization in calculi with explicit substitution and cut elimination in sequent calculus for Linear Logic, via Proof Nets. This correspondence allows us to prove that a typed version of the λx-calculus [30, 29] is strongly normalizing, as well as of all the calculi isomorphic to it such as λυ [24], λs [19], λd [21], and λf [11]. In order to achieve this result, we introduce a new notion of reduction in Proof Nets: this extended reduction is still confluent and strongly normalizing, and is of interest of its own, as it correspond to more identifications of proofs in Linear Logic that differ by inessential details. These results show that calculi with explicit substitutions are really an intermediate formalism between lambda calculus and proof nets, and suggest a completely new way to look at the problems still open in the field of explicit substitutions.