## Section: New Results

### Metatheory and development of Coq

Participants : Hugo Herbelin, Pierre Letouzey, Yann Régis-Gianas, Matthieu Sozeau, Cyprien Mangin, Théo Zimmermann.

#### Homotopy type theory

Hugo Herbelin worked on the computational contents of extensional equality in type theory. Exploiting the idea introduced in Cohen, Coquand, Huber and Mörtberg's Cubical Type Theory of equality as abstraction over a geometrical dimension, he developed a direct-style system of notations for a scoped iterated parametricity semantics. The resulting logic respects equivalence of types by construction, thus providing a simple computational content to the key axiom of Homotopy Type Theory, namely the axiom of univalence.

#### Proof irrelevance and Homotopy Type Theory

Gaëtan Gilbert (PhD student of N. Tabareau, Gallinette and M. Sozeau, started in 2016) is studying the integration of a new notion of propositions, called *strict* propositions, in the calculus of inductive constructions. This new sort dubbed sProp supports definitional proof-irrelevance (two proofs of a strict proposition are always convertible), while maintaining compatibility with Univalence or Uniqueness of Identity Proofs. The goal of this work is to provide a more comfortable programming experience in the system by allowing more proofs to be identified “for free” during conversion. This should have an impact both on programming with dependent types (avoiding issues with coercions during typechecking) and for the development of homotopy type theory (avoiding “trivial” transports of equality proofs on natural numbers for example). Gaëtan Gilbert has developed a prototype version integrating this extension in Coq.

#### Extensionality and Intensionality in Type Theory

Théo Winterhalter (internship co-advised by Matthieu Sozeau and Andrej Bauer in 2017, now PhD student at Inria Nantes, co-advised by Nicolas Tabareau and Matthieu Sozeau) studied a translation from extensional to intensional type theory during his internship with Matthieu Sozeau and a general framework for formalising variants of type theory previously with Andrej Bauer at the University of Ljubljana in Slovenia. They developed a revised version of the translation by Nicolas Oury which doesn't require the use of John Major equality nor suspicious axioms associated to it. It results in a mixed translation that can transport derivations of extensional type theory into intensional 2-level type theory (with an original, syntactic presentation of the latter). This allows in principle to use the convenience of the reflection rule of equality in proofs while being able to derive decorated terms checkable by the kernel of a 2-level variant of Coq: one where both a univalent equality and a strict equality with uniqueness of identity proofs can cohabit. They are working on a Coq formalisation of this result using the Template-Coq framework, which will be extracted to a translation plugin to provide this facility in Coq itself.

#### Dependent pattern-matching

Cyprien Mangin developed a new simplification engine on top of the Equations plugin. This simplification engine is similar to the one of Cockx [73], allowing an interpretation of dependent pattern-matching that is independent of axioms like UIP or Univalence. While refining the implementation, he also designed a few optimisations allowing for a smarter compilation scheme, in terms of the required properties of the objects and the size of the generated proofs. Matthieu Sozeau concentrated on making the treatment of recursive functions more robust and complete, leading to the first tool of this kind for Coq that can handle both mutual and nested structurally recursive functions along with nested well-founded definitions. The elimination principle generation part of the system was adapted accordingly, putting the tool in good position to replace the previous Function tool of Coq that supports neither dependent pattern-matching nor nested fixpoints. Matthieu Sozeau developed a number of examples showcasing the tool, the largest one having actually been first developed by a student of the MPRI 2.7.2 course. An article presenting this tool and the smart case analysis method is in revision [53]. Version 1.0 of the system was released in December 2017. Cyprien Mangin gave a demo / presentation of the tool at the EUTypes Type Theory Tools workshop in January 2017 and will present a poster and demonstration of the new version at PEPM 2018 in Los Angeles.

Thierry Martinez started the implementation of a dependent pattern-matching compilation algorithm in Coq based on the PhD thesis work of Pierre Boutillier and on the internship work of Meven Bertrand. The algorithm based on small inversion and generalisation is the object of a paper to be submitted to the TYPES post-proceedings.

#### Transferring theorems along isomorphisms

Following his work on theorem transfer along (iso)morphisms, Théo Zimmermann has started to explore more fundamental aspects that are connected to it: the concept of logical relation, which was originally invented to prove behavioral equivalence of programs and served to formalise parametricity, seems, following Hermida, Reddy and Robinson, to correspond to a very generic relational notion of morphism that was precisely the one needed for transfer lemmas.

#### Unification

Matthieu Sozeau has developped a complete reimplementation of the basic tactics of Coq in terms of the type-inference unification algorithm of Coq. This work is scheduled to be integrated in part in the 8.8 version of Coq due next year. It should provide a clean slate for development of the 9 series of Coq relying solely on an algorithm close to the one studied with Beta Ziliani in [22].

#### Cumulativity for Inductive Types

Together with Amin Timany (PhD student of Bart Jacobs at KU Leuven), Matthieu Sozeau developed an extension of the Calculus of Inductive Constructions featuring cumulativity for inductive types [44]. This extension is useful for developments using universe polymorphism like Category Theory formalisations and the HoTT library [36] but also crucial to develop syntactic program translations that add structures to types, as advocated by Boulier et al [66], requiring to validate the cumulativity rule on sigma types. They showed the relative consistency of this extension of the calculus using a set-theoretic model, inspired by the one of Lee and Werner [102] for proof-irrelevance. This extension is integrated in the 8.7 release of Coq and involved a large amount of design and implementation work in particular in relation with the unification strategy used in presence of subtyping and delta reduction, extending the framework studied in [34]. An article describing this work is in revision.