## Section: New Results

### Metatheory and development of Coq

Participants : Pierre-Louis Curien, Hugo Herbelin, Pierre Letouzey, Yann Régis-Gianas, Matthieu Sozeau.

#### Models of type theory

Simplicial sets and their extensions as Kan complexes can serve as models of homotopy type theory. Hugo Herbelin extended his concrete type-theoretic formalisation of semi-simplicial sets [20] to simplicial sets.

#### Unification

Matthieu Sozeau is working in collaboration with Beta Ziliani (PhD at MPI-Saarbrücken, now assistant professor at Cordoba, Argentina) on formalising the unification algorithm used in Coq, which is central for working with advanced type inference features like Canonical Structures. This is the first precise formalisation of all the rules of unification including the ones used for canonical structure resolution and universes. The presentation includes a careful study of the heuristics used in the existing Coq algorithms, which can be added or removed from the new implementation modularly. This work has been presented at the ICFP'15 conference [31] .

#### Nominal techniques

Matthieu Sozeau cosupervised the internship of Gabriel Lewertowski with Nicolas Tabareau (Ascola team, Nantes), on the development of a library for nominal reasoning in Coq/Ssreflect. The goal of this internship was to study the use of nominal sets to ease the formalisation of programming language (meta-)theory. A library based on the Mathematical Components formalisation of finite sets and effective quotients was built, providing generic definitions of substitution and elimination operators for simple descriptions of programming language syntax as a grammar. This work was done in collaboration with Assia Mahboubi (Specfun) and Cyril Cohen (Marelle). It forms the basis for the formalisation of cubical type theory, a new type theory using name abstraction that implements an axiom-free version of Homotopy Type Theory.