Section: New Results

Type theory and the foundations of Coq

Participants : Pierre Boutillier, Pierre-Louis Curien, Hugo Herbelin, Pierre-Marie Pédrot, Yann Régis-Gianas, Matthieu Sozeau, Arnaud Spiwack.

Description of type theory

Hugo Herbelin and Arnaud Spiwack completed and published their characterisation of the type constructions of Coq in terms of atomic constructions rather than their usual description as a monolithic scheme [23] . This work permitted both a more pedagogical presentation of Coq's type system, and a more tractable and composable mathematical model of Coq on which meta-properties can be stated and proved.

Models of type theory

Simplicial sets and their extensions as Kan complexes can serve as models of homotopy type theory. Hugo Herbelin developed a concrete type-theoretic formalisation of semi-simplicial sets following ideas from Steve Awodey, Peter LeFanu Lumsdaine and other researchers both at Carnegie-Mellon University and at the Institute of Advanced Study. This is in the process of being published in a special issue of MSCS on homotopy type theory [9] .

The technique scales to provide type-theoretic constructions for arbitrary presheaves on Reedy categories, thus including simplicial sets.

Proof irrelevance, eta-rules

During his master's internship supervised by Matthieu Sozeau, Philipp Haselwarter studied a formulation of proof-irrelevance based on the rooster and the syntactic bracket presentation by Spiwack and Herbelin [23] . This resulted in a decomposition of the calculus cleanly showing the use of smashing and a better understanding of the restricted elimination rules of propositions. It also clearly shows that the inductive type for accessibility, used to justify general wellfounded definitions, can not be interpreted as a proof-irrelevant proposition in this calculus.

Unification

Matthieu Sozeau is continuing work in collaboration with Beta Ziliani (PhD at MPI-Saarbrücken) 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. The presentation currently excludes some heuristics that were added on top of the core algorithm in Coq, until they can be studied more carefully. This work, part of B. Ziliani's thesis, was presented at the UNIF'14 workshop [28] and the Coq workshop in Vienna. A submission is in preparation.

Foundations and paradoxes

Arnaud Spiwack generalised previous works by Herman Geuvers and Hugo Herbelin to implement Hurkens's paradox of the impredicative system $U^{-}$. The resulting Coq implementation, which is completely independent from the impredicative features of Coq, generalises the two special cases which were previously used to prove negative results about impredicativity in Coq.