## Section: New Results

### Proof theory

G. Burel developed a general framework, focusing with selection, of which various logical systems are instances: ordinary focusing, refinements of resolution, deduction modulo theory, superdeduction and beyond [20]. This strengthens links between sequent calculi and resolution methods.

F. Gilbert developed a constructivization algorithm, taking as input the classical proof of some formula and generating as output, whenever possible, a constructive proof of the same formula. This result has been published and presented in [14].

F. Gilbert submitted his PhD dissertation (work document [25]), centered on the extension of higher-order logic with predicate subtyping. Predicate subtyping is a key feature of the proof assistant PVS, allowing to define types from predicates – for instance, using this feature, the type of even numbers can be defined from the corresponding predicate. The core of this work is the definition of a language of verifiable certificates for predicate subtyping, as well as the proof of two properties of this language: a cut-elimination theorem, a theorem of conservativity over higher-order logic. F. Gilbert presented this language of certificates as well as the cut-elimination theorem at the workshop TYPES 2017.