Normalisation is Insensible to λ-term Identity or Difference

This paper analyses the computational behaviour of λterm applications. The properties we are interested in are weak normalisation (i.e. there is a terminating reduction) and strong normalisation (i.e. all reductions are terminating). One can prove that the application of a λ-term M to a fixed number n of copies of the same arbitrary strongly normalising λ-term is strongly normalising if and only if the application of M to n different arbitrary strongly normalising λ-terms is strongly normalising. I.e. one has that M X . . .X } {{ } n is strongly normalising, for an arbitrary strongly normalising X , if and only if MX1 . . .Xn is strongly normalising for arbitrary strongly normalising X1, . . . , Xn. The analogous property holds when replacing strongly normalising by weakly normalising. As an application of the result on strong normalisation the λ-terms whose interpretation is the top element (in the environment which associates the top element to all variables) of the Honsell-Lenisa model turn out to be exactly the λ-terms which, applied to an arbitrary number of strongly normalising λ-terms, always produces strongly normalising λ-terms. This proof uses a finitary logical description of the model by means of intersection types. This answers an open question stated by Dezani, Honsell and Motohama.