## Section: New Results

### Realisability models for cut-elimination in focused systems

Participant : Stéphane Graham-Lengrand.

This result is part of the effort to build meaningful semantics for classical proofs, here based on a polarisation of logical formulae: positive or negative.

Following work by Zeilberger [80] , a computational interpretation of cut-elimination in the focused systems LJF and LKF can be given: proofs of positive formulae provide structured data, while proofs of negative formulae consume such data; focusing allows the description of the interaction between the two kinds of proofs as pure pattern-matching.

First, we showed this at a level of abstraction where formulae are no longer made of syntax, yet we also extended the approach so that it could treat quantifiers.

Second, we connected this interpretation to realisability semantics, more precisely orthogonality models, where positive formulae are interpreted as sets of data, and negative formulae are interpreted as their orthogonal sets.

Our construction of orthogonality models for the focused systems LKF and LJF describe the pattern-matching process of cut-elimination in terms of orthogonality. This result has been proved in the Coq proof assistant and forms the second part of [11] .