Characterization of strong normalizability for a sequent lambda calculus with co-control

Abstract

We study strong normalization in a lambda calculus of proof-terms with co-control for the intuitionistic sequent calculus. In this sequent lambda calculus, the management of formulas on the left hand side of typing judgements is "dual" to the management of formulas on the right hand side of the typing judgements in Parigot's lambdamu calculus - that is why our system has first-class "co-control". The characterization of strong normalization is by means of intersection types, and is obtained by analyzing the relationship with another sequent lambda calculus, without co-control, for which a characterization of strong normalizability has been obtained before. The comparison of the two formulations of the sequent calculus, with or without co-control, is of independent interest. Finally, since it is known how to obtain bidirectional natural deduction systems isomorphic to these sequent calculi, characterizations are obtained of the strongly normalizing proof-terms of such natural deduction systems.