## Section: New Results

### Towards proof canonicity in presence of disjunction and induction

Participants : Hichem Chihani, Danko Ilik.

The previous work on type isomorphisms showed a way to treat the problem of identity/canonicity of proofs for intuitionistic logic with disjunction, or, equivalently, the problem of the (non-)existence of a canonical eta-long normal form for lambda calculus with if-expressions, which is a long standing open question.

One can see this from the perspective of focusing sequent calculi. The asynchronous phase of proof search is an oriented application of type isomorphisms (by the formulas-as-types correspondence). As we already know that, in the absence of disjunction (sum types), a cut-free focused derivation is eta-long and unique (when the data provided by the synchronous phase is the same), what is necessary in order to handle disjunction is to propagate isomorphisms further than what usual sequent calculus allows. This is related in spirit to deep inference, but more conservative. An implementation of a canonical normalizer and a paper on the topic is under way.

We also intend to use the method to give a proof of focused cut-elimination for the sequent calculi LJF and LKF (at least, for the Sigma-2 fragment) extended with induction. A formal proof in Agda is under development.