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##### PI.R2 - 2013

Overall Objectives
Software and Platforms
Partnerships and Cooperations
Bibliography

## Section: Software and Platforms

### COQ (http://coq.inria.fr )

Participants : Bruno Barras [Inria Saclay] , Yves Bertot [Marelle team, Sophia] , Pierre Boutillier, Xavier Clerc [SED team] , Pierre Courtieu [CNAM] , Maxime Dénès [Marelle team, Sophia] , Julien Forest [CNAM] , Stéphane Glondu [CARAMEL team, Nancy Grand Est] , Benjamin Grégoire [Marelle team, Sophia] , Vincent Gross [Consultant at NBS Systems] , Hugo Herbelin [correspondant] , Pierre Letouzey, Assia Mahboubi [SpecFun team, Saclay] , Julien Narboux [University of Strasbourg] , Jean-Marc Notin [Ecole Polytechnique] , Christine Paulin [Proval team, Saclay] , Pierre-Marie Pédrot, Loïc Pottier [Marelle team, Sophia] , Matthias Puech, Yann Régis-Gianas, François Ripault, Matthieu Sozeau, Arnaud Spiwack [Abstraction team, ENS] , Pierre-Yves Strub [IMDEA, Madrid] , Enrico Tassi [SpecFun team, Saclay] , Benjamin Werner [Ecole Polytechnique] .

#### Version 8.5

Version 8.5 is expected to be released after the summer of 2014. It will be a major release of the Coq proof assistant, including 6 major new features:

• Parallel development and compilation, inside files and across files, by Enrico Tassi (Inria SpecFun), a result of the Paral-ITP ANR project.

• New proof engine of Arnaud Spiwack (formerly pi.r2 postdoc), more expressive and with clearer semantics.

• Native compilation by Maxime Dénès and Benjamin Grégoire (Inria Marelle, M. Dénès is now at the University of Pennsylvania). A compilation scheme from Coq to OCaml to native code, considerably improving on the previous virtual machine implementation by B. Grégoire.

• A Universe Polymorphic extension by Matthieu Sozeau that allows universe-generic developments, as required by the Homotopy Type Theory library for example.

• Primitive projections for records by Matthieu Sozeau.

• A new document generation system by F. Ripault and Yann Régis-Gianas.

A more detailed description of all the new features will be made in next year's report, but some elements can already be found below.

#### Evaluation algorithms

Pierre Boutillier has worked on the practical implementation of the unfolding algorithm for global constants that he proposed last year, so that it could become the default process to simplify terms by tactics. A formal presentation has been written in his PhD, to be defended in February 2014.

#### Internal representation of projections

The change of representation for record projections implemented by Matthieu Sozeau will be part of the 8.5 release. It provides not only exponential gains in performance (type-checking and comparison time and space usage) but also a better basis to work with canonical structures in the unification algorithm, allowing to improve for example the ssreflect inference mechanism significantly. Benchmarks on the HoTT library and a groupoid model construction confirm the exponential gain in performance.

#### Universes

Matthieu Sozeau followed up his work on universe polymorphism and uncovered important theoretical problems regarding conversion and unification of universe polymorphic constants in the presence of cumulativity and the Prop $\le$ Type rule. After a careful study of the alternative solutions, he designed a practical correction for the issue and developed a paper proof of conservativity of the complete new system over the original theory of Coq. A paper describing this work has been submitted. The universe polymorphic system, already in use by the HoTT community, will be part of the upcoming 8.5 release.

#### The Equations plugin

Matthieu Sozeau continued work on the Equations plugin and fixed the remaining bugs preventing full automation of a middle-size example of formalization of the normalization proof of a simply-typed lambda calculus. Wojciech Jedynak, a student of Dariusz Biernacki and Małgorzata Biernacka at the University of Wroclaw, is working under his supervision to do the remaining work before an official release of the plugin can be done. Wojciech Jedynak applied for a 4-month internship supervised by Matthieu Sozeau on an extension of Equations, to start in March 2014.

#### Internal architecture of the Coq software

With the help of many others, Pierre Letouzey organized in November 2013 the migration of the official Coq source repository from subversion to git. The native use of this decentralized version control system eases the exchange of code amongst Coq developpers (either from the Coq dev team or from external contributors).

Pierre Letouzey, Pierre-Marie Pédrot and Xavier Clerc have continued to work at improving the quality of the OCaml code which composes Coq :

• Many modules have been revised, in particular with cleaner naming convention.

• Almost all uses of the generic OCaml comparison has been chased and transformed into specific code. This avoided many potential bugs with advanced structures, while improving performances at the same time.

• The codes handling OCaml exceptions have been reworked to avoid undue interceptions of critical exceptions.

• Issues involving exceptions are now quite simpler to debug, thanks to easy-to-obtain backtraces.

#### Efficiency

Pierre-Marie Pédrot has been working on the overall optimization of Coq, by tracking hotspots in the code. Coq trunk is currently much more efficient than its v8.4 counterpart, and is about as quick as v8.3, while having been expanded with a lot of additional features.

Pierre Letouzey has improved the representation of Coq binary files : these files are now smaller (thanks to more sharing), and are reloaded quickly by Coq (thanks to deferred loading of opaque proof terms, which are large and almost never accessed by the user).

#### Documentation generation

François Ripault and Yann Régis-Gianas developed a new version of coqdoc, the documentation generator of Coq. This new implementation is based on the interaction protocol with the Coq system and should be more robust with respect to the evolution of Coq.

#### General maintenance

Pierre Letouzey has been the main maintainer of Coq with extra contributions from Hugo Herbelin, Pierre Boutillier, Matthieu Sozeau, Pierre-Marie Pédrot, ...

#### Modules in Coq

In 2013, Pierre Letouzey has proposed an important rework of the code implementing the module system of Coq. This code was inherited from the successive works of several PhD students, and was in pretty poor shape. While being equivalent in terms of features for the user, the new code should be quite more readable and robust, as well as more efficient: the memory sharing of modular structures should be better, leading to reduced memory footprint for Coq as well as smaller Coq compiled files.

#### The Coq extraction

Pierre Letouzey has collaborated with colleagues with the aim of extending the extraction tool to additional target languages:

• C++ with Gabriel Dos Reis and his student Robert Schumacher

• F# with David Monniaux

These experiments have been quite promising. In the case of C++ , an article has been written, it should be re-submitted soon for publication.

#### Formalisation in Coq

Hugo Herbelin's type-theoretic construction of semi-simplicial sets [22] has been formalised in Coq.

Matthieu Sozeau and Nicolas Tabareau formalised a groupoid model in Coq http://github.com/mattam82/groupoid .

Jaime Gaspar has verified in Coq the correctness of Jean-Louis Krivine's proof that Zermelo-Fraenkel set theory without choice $\mathrm{ZF}$ is contained in a variant ${\mathrm{ZF}}_{\epsilon }$ (useful for Krivine's classical realisability).