This paper is concerned with strong normalisation of cut-elimination for a standard intuitionistic sequent calculus. The cut-elimination procedure is based on a rewrite system for proof-terms with cut-permutation rules allowing the simulation of β-reduction. Strong normalisation of the typed terms is inferred from that of the simply-typed λ-calculus, using the notions of safe and minimal reductions as well as a simulation in Nederpelt-Klop’s λI-calculus. It is also shown that the type-free terms enjoy the preservation of strong normalisation (PSN) property with respect to β-reduction in an isomorphic image of the type-free λ-calculus.

We present a typing system for the λ-calculus, with non-idempotent intersection types. As it is the case in (some) systems with idempotent intersections, a λ-term is typable if and only if it is strongly normalising. Nonidempotency brings some further information into typing trees, such as a bound on the longest β-reduction sequence reducing a term to its normal form. We actually present these results in Klop’s extension of λ-calculus, where the bound that is read in the typing tree of a term is refined into an exact measure of the longest reduction sequence. This complexity result is, for longest reduction sequences, the counterpart of de Carvalho’s result for linear head-reduction sequences.

A proposition in the author’s “Delayed substitutions”, RTA’07, concerned with preservation, by certain permutative reductions, of the length of the longest β-reduction sequence from a λ-term, has an incomplete proof in that paper. The purpose of this note is to fix that proof. This is achieved through a new, general result, from which one can prove preservation of the length of the longest β-reduction by other notions of reduction.