## Section: New Results

### Metatheory and development of Coq

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

#### Dependent pattern-matching

Hugo Herbelin supervised the internship of Meven Bertrand on compiling dependent pattern-matching using a combination of techniques known as small inversion and generalization, as a following of Pierre Boutillier's PhD.

#### Transferring theorems along isomorphisms

Théo Zimmermann has developed a tool for transferring theorems along isomorphic structures. The long-term objective is to provide a language of proof methods matching the level of abstraction common in mathematics. Théo Zimmermann is applying his tool to introduce higher "mathematical" levels of abstraction to the basic Coq method for applying theorems. The proof of concept of this idea will be presented at the TTT POPL workshop in January.

#### Unification

Matthieu Sozeau worked in collaboration with Beta Ziliani (assistant professor at Córdoba, Argentina) on a journal version of the formalisation of the unification algorithm used in Coq, which is central for working with advanced type inference features like Canonical Structures. The presentation of this journal version is incremental (it is presented feature by feature), with an aim of easing the understanding of how the algorithm actually works for users who want to take advantage of it. It has been accepted for publication in the Journal of Functional Programming.

#### Explicit Cumulativity

Pierre Letouzey started exploring with the help of Matthieu Sozeau a version of Coq's logic (CIC) where the cumulativity rule would be explicit. This cumulativity rule is a form of coercion between Coq universes, and is done silently in Coq up to now. Having a version of CIC where the use of the cumulativity bewteen Prop and Type is traceable woud be of great interest. In particular this would lead to a solid ground for the Coq extraction tool and solve some of its current limitations. Moreover, an explicit cumulativity would also help significantly the studies of Coq theoretical models. Preliminary results are encouraging, but this work has not been finalized yet. This work is related to the studies of Ali Assaf (Google Zurich, formerly PhD student in the team Deducteam), but uses different technical choices for different goals. This work is now pursued by Gaëtan Gilbert (PhD student of Nicolas Tabareau and Matthieu Sozeau at the École des Mines in Nantes), with the goal of providing a version of the calculus of constructions with definitional proof-irrelevance. The absence of explicit cumulativity between Prop and Type was identified in earlier work by Benjamin Werner and Giesik Lee as an important obstacle to building models of the theory, we hence expect this work to simplify the (relative) consistency proof of the theory.