## Section: New Results

### Proof normalization in sequent calculus

Participant : Stéphane Lengrand.

One way of obtaining cut-free proofs is to normalize a proof that contains cuts. Stéphane Lengrand has studied the termination of normalization procedures in sequent calculi.

The methodology uses proof-terms and rewrite systems on these objects, which have to be proved terminating.

Lengrand and Kikuchi in [19] have designed such a cut-elimination system for one of the simplest, and non-focused, sequent calculus for intuitionistic logic. Yet it provides a computation model that simulates the -calculus in a strong sense. Thus, termination of the system is at least as strong as that of the (simply-typed) -calculus.

Lengrand in [37] has shown the termination of a cut-elimination system presented in [3] for a focused sequent calculus in relation to call-by-value semantics.

The proof term approach could provide a systematic method to obtain cut-elimination results in focused sequent calculi such as LJF and LKF [6] .