On Zucker’s isomorphism for LJ and its extension to pure type systems (submitted

Abstract

Abstract. It is shown how the sequent calculus LJ can be embedded into a simple extension of the λ-calculus by generalized applications, called ΛJ. The reduction rules of cut elimination and normalization can be precisely correlated, if explicit substitutions are added to ΛJ. The resulting system ΛJ ✷ is proved strongly normalizing, thus showing strong normalization for Gentzen’s cut elimination steps. This refines previous results by Zucker, Pottinger and Herbelin on the isomorphism between natural deduction and sequent calculus. The concept of generalized applications extends to Pure Type Systems, so that in particular sequent calculus analogues for all systems of the λ-cube arise. Cut elimination and strong β-normalization are shown to be equivalent for all Pure Type Systems