Keeping sums under control

In a recent paper [31], I presented with Marcelo Fiore and Roberto Di Cosmo a new normalisation tool for the -calculus with sum types, based on the technique of normalisation by evaluation, and more precisely on techniques developped by Olivier Danvy for partial evaluation, using control operators. The main characteristic of this tool is that it produces a result in a canonical form we introduced. That is to say: two -equivalent terms will be normalised into (almost) identical terms. It was not the case with the traditional algorithm, which could even lead to an explosion of the size of code. This canonical form is an -long -normal form with constraints, which capture the denition of -long normal form for the -calculus without sums, and reduces drastically the -conversion possibilities for sums. The present paper recall the denition of these normal forms and the normalisation algorithm, and shows how it is possible to use these tools to solve a problem of characterization of type isomorphisms. Indeed, the canonical form allowed to nd the complicated counterexamples we exhibited in another work [6], that proves that type isomorphisms in the -calculus with sums are not nitely axiomatisable. What’s more, when proving that these terms are isomorphisms, the new partial evaluation algorithm avoids an explosion of the size of the term that arises with the old one.