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##### PI.R2 - 2013

Overall Objectives
Software and Platforms
Partnerships and Cooperations
Bibliography

## Section: New Results

### Type theory and the foundations of Coq

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

#### Substitutions and isomorphisms

Pierre-Louis Curien completed his joint work with Richard Garner and Martin Hofmann on relating syntax unstrictification through coercions with model strictification (cf. $\pi {r}^{2}$ report 2012), adding a careful treatment of identity types. The corresponding paper was accepted for publication in the TCS special issue for Glynn Winskel's anniversary.

#### Homotopy type theory

Hugo Herbelin, Matthieu Sozeau and Pierre-Louis Curien participated to the univalent foundations program. A collaborative book [18] on the results of this program has been published.

#### 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 has been accepted for publication in a special issue of MSCS on homotopy type theory  [22] .

The technique he used generalises to provide type-theoretic constructions for arbitrary presheaves on Reedy categories, thus including simplicial sets. In particular, this provides with a formulation of simplicial sets where degeneracies are decidable, which is not the case with the definition as a presheaf.

Hugo Herbelin also investigated hybrid constructive definitions of simplicial sets where face maps are axiomatised but degeneracies are built. Again, this provides with a formulation of simplicial sets where it is decidable whether a given simplex is degenerate or not.

#### Internalizing the setoid model of type theory

As an example use of the new polymorphic universe extension of Coq, Matthieu Sozeau developed together with Nicolas Tabareau (Inria Ascola team, École des mines Nantes) a complete groupoid model of type theory, following the seminal work of Hofmann and Streicher. A preliminary paper presenting a partial generalization of this model to 2-goupoids was written and will be resubmitted [23] .

A completed version of this model has since been formalized and will be submitted shortly. This model showcases the use of the polymorphic universes: in the course of its formalization we uncovered hidden assumptions in the interpretation of substitution and sigma types in the original presentation thanks to the universe system.

#### Proof irrelevance, eta-rules

Matthieu Sozeau finished his implementation of a proof-irrelevant system but did not publish it. Indeed, the homotopy type theory interpretation suggests new ways to introduce proof-irrelevance using bracket types that seem to significantly depart from the syntactic treatment developped by Werner and himself. An investigation of the relationship between the presentation of the calculus of inductive constructions given by Hugo Herbelin and Arnaud Spiwack in [44] which includes the bracket construction and the aforementioned syntactic version will be part of a master's internship supervised by Matthieu Sozeau in 2014.