3.1.1 Proofs as programs

Proof theory is the branch of logic devoted to the study of the structure of proofs. An essential contributor to this field is Gentsen  62 who developed in 1935 two logical formalisms that are now central to the study of proofs. These are the so-called “natural deduction”, a syntax that is particularly well-suited to simulate the intuitive notion of reasoning, and the so-called “sequent calculus”, a syntax with deep geometric properties that is particularly well-suited for proof automation.

Proof theory gained a remarkable importance in computer science when it became clear, after genuine observations first by Curry in 1958  55, then by Howard and de Bruijn at the end of the 60's  74, 98, that proofs had the very same structure as programs: for instance, natural deduction proofs can be identified as typed programs of the ideal programming language known as $\lambda$-calculus.

This proofs-as-programs correspondence has been the starting point to a large spectrum of researches and results contributing to deeply connect logic and computer science. In particular, it is from this line of work that Coquand and Huet's Calculus of Constructions  54, 52 stemmed out – a formalism that is both a logic and a programming language and that is at the source of the Coq system   95.

3.1.2 Towards the calculus of constructions

The $\lambda$-calculus, defined by Church  51, is a remarkably succinct model of computation that is defined via only three constructions (abstraction of a program with respect to one of its parameters, reference to such a parameter, application of a program to an argument) and one reduction rule (substitution of the formal parameter of a program by its effective argument). The $\lambda$-calculus, which is Turing-complete, i.e. which has the same expressiveness as a Turing machine (there is for instance an encoding of numbers as functions in $\lambda$-calculus), comes with two possible semantics referred to as call-by-name and call-by-value evaluations. Of these two semantics, the first one, which is the simplest to characterise, has been deeply studied in the last decades  40.

To explain the Curry-Howard correspondence, it is important to distinguish between intuitionistic and classical logic: following Brouwer at the beginning of the 20th century, classical logic is a logic that accepts the use of reasoning by contradiction while intuitionistic logic proscribes it. Then, Howard's observation is that the proofs of the intuitionistic natural deduction formalism exactly coincide with programs in the (simply typed) $\lambda$-calculus.

A major achievement has been accomplished by Martin-Löf who designed in 1971 a formalism, referred to as modern type theory, that was both a logical system and a (typed) programming language  85.

In 1985, Coquand and Huet  54, 52 in the Formel team of INRIA-Rocquencourt explored an alternative approach based on Girard-Reynolds' system $F$  63, 91. This formalism, called the Calculus of Constructions, served as logical foundation of the first implementation of Coq in 1984. Coq was called CoC at this time.

3.1.3 The Calculus of Inductive Constructions

The first public release of CoC dates back to 1989. The same project-team developed the programming language Caml (nowadays called OCaml and coordinated by the Gallium team) that provided the expressive and powerful concept of algebraic data types (a paragon of it being the type of lists). In CoC, it was possible to simulate algebraic data types, but only through a not-so-natural not-so-convenient encoding.

In 1989, Coquand and Paulin  53 designed an extension of the Calculus of Constructions with a generalisation of algebraic types called inductive types, leading to the Calculus of Inductive Constructions (CIC) that started to serve as a new foundation for the Coq system. This new system, which got its current definitive name Coq, was released in 1991.

In practice, the Calculus of Inductive Constructions derives its strength from being both a logic powerful enough to formalise all common mathematics (as set theory is) and an expressive richly-typed functional programming language (like ML but with a richer type system, no effects and no non-terminating functions).

3.2.1 The underlying logic and the verification kernel

The architecture adopts the so-called de Bruijn principle: the well-delimited kernel of Coq ensures the correctness of the proofs validated by the system. The kernel is rather stable with modifications tied to the evolution of the underlying Calculus of Inductive Constructions formalism. The kernel includes an interpreter of the programs expressible in the CIC and this interpreter exists in two flavours: a customisable lazy evaluation machine written in OCaml and a call-by-value bytecode interpreter written in C dedicated to efficient computations. The kernel also provides a module system.

3.2.2 Programming and specification languages

The concrete user language of Coq, called Gallina, is a high-level language built on top of the CIC. It includes a type inference algorithm, definitions by complex pattern-matching, implicit arguments, mathematical notations and various other high-level language features. This high-level language serves both for the development of programs and for the formalisation of mathematical theories. Coq also provides a large set of commands. Gallina and the commands together forms the Vernacular language of Coq.

3.2.3 Standard library

The standard library is written in the vernacular language of Coq. There are libraries for various arithmetical structures and various implementations of numbers (Peano numbers, implementation of $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ with binary digits, implementation of $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ using machine words, axiomatisation of $ℝ$). There are libraries for lists, list of a specified length, sorts, and for various implementations of finite maps and finite sets. There are libraries on relations, sets, orders.

3.2.4 Tactics

The tactics are the methods available to conduct proofs. This includes the basic inference rules of the CIC, various advanced higher level inference rules and all the automation tactics. Regarding automation, there are tactics for solving systems of equations, for simplifying ring or field expressions, for arbitrary proof search, for semi-decidability of first-order logic and so on. There is also a powerful and popular untyped scripting language for combining tactics into more complex tactics.

Note that all tactics of Coq produce proof certificates that are checked by the kernel of Coq. As a consequence, possible bugs in proof methods do not hinder the confidence in the correctness of the Coq checker. Note also that the CIC being a programming language, tactics can have their core written (and certified) in the own language of Coq if needed.

3.2.5 Extraction

Extraction is a component of Coq that maps programs (or even computational proofs) of the CIC to functional programs (in OCaml, Scheme or Haskell). Especially, a program certified by Coq can further be extracted to a program of a full-fledged programming language then benefiting of the efficient compilation, linking tools, profiling tools, ... of the target language.

3.2.6 Documentation

Coq is a feature-rich system and requires extensive training in order to be used proficiently; current documentation includes the reference manual, the reference for the standard library, as well as tutorials, and related tooling [sphinx plugins, coqdoc]. The jsCoq tool allows writing interactive web pages were Coq programs can be embedded and executed.

3.2.7 Proof development infrastructure

Coq is used in large-scale proof developments, and provides users miscellaneous tooling to help with them: the coq_makefile and Dune build systems help with incremental proof-checking; the Coq OPAM repository contains a package index for most Coq developments; the CoqIDE, ProofGeneral, jsCoq, and VSCoq user interfaces are environments for proof writing; and the Coq's API does allow users to extend the system in many important ways. Among the current extensions we have QuickChik, a tool for property-based testing; STMCoq and CoqHammer integrating Coq with automated solvers; ParamCoq, providing automatic derivation of parametricity principles; MetaCoq for metaprogramming; Equations for dependently-typed programming; SerAPI, for data-centric applications; etc... This also includes the main open Coq repository living at Github.

3.3.1 Type-checking and proof automation

In between the decidable type systems of conventional data-types based programming languages and the full expressiveness of logically undecidable formulae, an active field of research explores a spectrum of decidable or semi-decidable type systems for possible use in dependently typed programming languages. At the beginning of the spectrum, this includes, for instance, the system F's extension ML${}_F$ of the ML type system or the generalisation of abstract data types with type constraints (G.A.D.T.) such as found in the Haskell programming language. At the other side of the spectrum, one finds arbitrary complex type specification languages (e.g. that a sorting function returns a list of type “sorted list”) for which more or less powerful proof automation tools exist – generally first-order ones.

3.4.1 Control operators and classical logic

Indeed, a significant extension of the Curry-Howard correspondence has been obtained at the beginning of the 90's thanks to the seminal observation by Griffin  64 that some operators known as control operators were typable by the principle of double negation elimination ($\neg \neg A\Rightarrow A$), a principle that enables classical reasoning.

Control operators are used to jump from one location of a program to another. They were first considered in the 60's by Landin  81 and Reynolds  90 and started to be studied in an abstract way in the 80's by Felleisen et al  59, leading to Parigot's $\lambda \mu$-calculus  88, a reference calculus that is in close Curry-Howard correspondence with classical natural deduction. In this respect, control operators are fundamental pieces to establish a full connection between proofs and programs.

3.4.2 Sequent calculus

The Curry-Howard interpretation of sequent calculus started to be investigated at the beginning of the 90's. The main technicality of sequent calculus is the presence of left introduction inference rules, for which two kinds of interpretations are applicable. The first approach interprets left introduction rules as construction rules for a language of patterns but it does not really address the problem of the interpretation of the implication connective. The second approach, started in 1994, interprets left introduction rules as evaluation context formation rules. This line of work led in 2000 to the design by Hugo Herbelin and Pierre-Louis Curien of a symmetric calculus exhibiting deep dualities between the notion of programs and evaluation contexts and between the standard notions of call-by-name and call-by-value evaluation semantics.

3.4.3 Abstract machines

Abstract machines came as an intermediate evaluation device, between high-level programming languages and the computer microprocessor. The typical reference for call-by-value evaluation of $\lambda$-calculus is Landin's SECD machine  82 and Krivine's abstract machine for call-by-name evaluation  78, 77. A typical abstract machine manipulates a state that consists of a program in some environment of bindings and some evaluation context traditionally encoded into a “stack”.

3.4.4 Delimited control

Delimited control extends the expressiveness of control operators with effects: the fundamental result here is a completeness result by Filinski  60: any side-effect expressible in monadic style (and this covers references, exceptions, states, dynamic bindings, ...) can be simulated in $\lambda$-calculus equipped with delimited control.

3.5.1 Higher-dimensional algebra

Like ordinary categories, higher-dimensional categorical structures originate in algebraic topology. Indeed, $\infty$-groupoids have been initially considered as a unified point of view for all the information contained in the homotopy groups of a topological space $X$: the fundamental $\infty$-groupoid$\Pi \left(X\right)$ of $X$ contains the elements of $X$ as 0-dimensional cells, continuous paths in $X$ as 1-cells, homotopies between continuous paths as 2-cells, and so on. This point of view translates a topological problem (to determine if two given spaces $X$ and $Y$ are homotopically equivalent) into an algebraic problem (to determine if the fundamental groupoids $\Pi \left(X\right)$ and $\Pi \left(Y\right)$ are equivalent).

In the last decades, the importance of higher-dimensional categories has grown fast, mainly with the new trend of categorification that currently touches algebra and the surrounding fields of mathematics. Categorification is an informal process that consists in the study of higher-dimensional versions of known algebraic objects (such as higher Lie algebras in mathematical physics 39) and/or of “weakened” versions of those objects, where equations hold only up to suitable equivalences (such as weak actions of monoids and groups in representation theory 58).

The categorification process has also reached logic, with the introduction of homotopy type theory. After a preliminary result that had identified categorical structures in type theory 73, it has been observed recently that the so-called “identity types” are naturally equiped with a structure of $\infty$-groupoid: the 1-cells are the proofs of equality, the 2-cells are the proofs of equality between proofs of equality, and so on. The striking resemblance with the fundamental $\infty$-groupoid of a topological space led to the conjecture that homotopy type theory could serve as a replacement of set theory as a foundational language for different fields of mathematics, and homotopical algebra in particular.

3.5.2 Higher-dimensional rewriting

Higher-dimensional categories are algebraic structures that contain, in essence, computational aspects. This has been recognised by Street 94, and independently by Burroni 47, when they have introduced the concept of computad or polygraph as combinatorial descriptions of higher categories. Those are directed presentations of higher-dimensional categories, generalising word and term rewriting systems.

In the recent years, the algebraic structure of polygraph has led to a new theory of rewriting, called higher-dimensional rewriting, as a unifying point of view for usual rewriting paradigms, namely abstract, word and term rewriting 80, 84, 69, 70, and beyond: Petri nets 72 and formal proofs of classical and linear logic have been expressed in this framework 71. Higher-dimensional rewriting has developed its own methods to analyse computational properties of polygraphs, using in particular algebraic tools such as derivations to prove termination, which in turn led to new tools for complexity analysis 44.

3.5.3 Squier theory

The homotopical properties of higher categories, as studied in mathematics, are in fact deeply related to the computational properties of their polygraphic presentations. This connection has its roots in a tradition of using rewriting-like methods in algebra, and more specifically in the works of Anick 36 and Squier 93, 92: Squier has proved that, if a monoid $M$ can be presented by a finite, terminating and confluent rewriting system, then its third integral homology group $H_3(M,\mathbb{Z})$ is finitely generated and the monoid $M$ has finite derivation type (a property of homotopical nature). This allowed him to conclude that finite convergent rewriting systems were not a universal solution to decide the word problem of finitely generated monoids. Since then, Yves Guiraud and Philippe Malbos have shown that this connection was part of a deeper unified theory when formulated in the higher-dimensional setting 11, 12, 67, 68, 66.

In particular, the computational content of Squier's proof has led to a constructive methodology to produce, from a convergent presentation, coherent presentations and polygraphic resolutions of algebraic structures, such as monoids 11 and algebras 10. A coherent presentation of a monoid $M$ is a 3-dimensional combinatorial object that contains not only a presentation of $M$ (generators and relations), but also higher-dimensional cells, corresponding each to two fundamentally different proofs of the same equality: this is, in essence, the same as the proofs of equality of proofs of equality in homotopy type theory. When this process of “unfolding” proofs of equalities is pursued in every dimension, one gets a polygraphic resolution of the starting monoid $M$. This object has the following desirable qualities: it is free and homotopically equivalent to $M$ (in the canonical model structure of higher categories 79, 37). A polygraphic resolution of an algebraic object $X$ is a faithful formalisation of $X$ on which one can perform computations, such as homotopical or homological invariants of $X$. In particular, this has led to new algorithms and proofs in representation theory 8, and in homological algebra 6510. See 13 for a summary of these results.

7.1.1 Coq

• Name: The Coq Proof Assistant
• Keywords: Proof, Certification, Formalisation
• Scientific Description: Coq is an interactive proof assistant based on the Calculus of (Co-)Inductive Constructions, extended with universe polymorphism. This type theory features inductive and co-inductive families, an impredicative sort and a hierarchy of predicative universes, making it a very expressive logic. The calculus allows to formalize both general mathematics and computer programs, ranging from theories of finite structures to abstract algebra and categories to programming language metatheory and compiler verification. Coq is organised as a (relatively small) kernel including efficient conversion tests on which are built a set of higher-level layers: a powerful proof engine and unification algorithm, various tactics/decision procedures, a transactional document model and, at the very top an IDE.
• Functional Description: Coq provides both a dependently-typed functional programming language and a logical formalism, which, altogether, support the formalisation of mathematical theories and the specification and certification of properties of programs. Coq also provides a large and extensible set of automatic or semi-automatic proof methods. Coq's programs are extractible to OCaml, Haskell, Scheme, ...
• Release Contributions:

  Some highlights from this release are:

  - Introduction of primitive persistent arrays in the core language, implemented using imperative persistent arrays. - Introduction of definitional proof irrelevance for the equality type defined in the SProp sort. - Many improvements to the handling of notations, including number notations, recursive notations and notations with bindings. A new algorithm chooses the most precise notation available to print an expression, which might introduce changes in printing behavior.

  See the Zenodo citation for more information on this release: https://zenodo.org/record/4501022#.YB00r5NKjlw

• News of the Year:

  Coq version 8.13 integrates many usability improvements, as well as extensions of the core language. The main changes include: - Introduction of primitive persistent arrays in the core language, implemented using imperative persistent arrays. - Introduction of definitional proof irrelevance for the equality type defined in the SProp sort. - Cumulative record and inductive type declarations can now specify the variance of their universes. - Various bugfixes and uniformization of behavior with respect to the use of implicit arguments and the handling of existential variables in declarations, unification and tactics. - New warning for unused variables in catch-all match branches that match multiple distinct patterns. - New warning for Hint commands outside sections without a locality attribute, whose goal is to eventually remove the fragile default behavior of importing hints only when using Require. The recommended fix is to declare hints as export, instead of the current default global, meaning that they are imported through Require Import only, not Require. See the following rationale and guidelines for details. - General support for boolean attributes. - Many improvements to the handling of notations, including number notations, recursive notations and notations with bindings. A new algorithm chooses the most precise notation available to print an expression, which might introduce changes in printing behavior. - Tactic improvements in lia and its zify preprocessing step, now supporting reasoning on boolean operators such as Z.leb and supporting primitive integers Int63. - Typing flags can now be specified per-constant / inductive. - Improvements to the reference manual including updated syntax descriptions that match Coq's grammar in several chapters, and splitting parts of the tactics chapter to independent sections.

  See the changelog for an overview of the new features and changes, along with the full list of contributors. https://coq.github.io/doc/v8.13/refman/changes.html#version-8-13

• URL: http://coq.inria.fr/
• Authors: Bruno Barras, Yves Bertot, Frédéric Besson, Pierre Corbineau, Cristina Cornes, Judicaël Courant, Pierre Courtieu, Pierre Crégut, David Delahaye, Maxime Denes, Jean-Christophe Filliâtre, Julien Forest, Emilio Jesus Gallego Arias, Gaëtan Gilbert, Georges Gonthier, Benjamin Grégoire, Hugo Herbelin, Gérard Huet, Vincent Laporte, Pierre Letouzey, Assia Mahboubi, Pascal Manoury, Guillaume Melquiond, César Munoz, Chetan Murthy, Amokrane Saibi, Catherine Parent, Christine Paulin Mohring, Pierre-Marie Pédrot, Loïc Pottier, Matthieu Sozeau, Arnaud Spiwack, Enrico Tassi, Laurent Théry, Benjamin Werner, Théo Zimmermann
• Contacts: Hugo Herbelin, Matthieu Sozeau
• Participants: Yves Bertot, Frédéric Besson, Tej Chajed, Cyril Cohen, Pierre Corbineau, Pierre Courtieu, Maxime Denes, Jim Fehrle, Julien Forest, Emilio Jesús Gallego Arias, Gaëtan Gilbert, Georges Gonthier, Benjamin Grégoire, Jason Gross, Hugo Herbelin, Vincent Laporte, Olivier Laurent, Assia Mahboubi, Kenji Maillard, Érik Martin-Dorel, Guillaume Melquiond, Pierre-Marie Pédrot, ClÉment Pit-Claudel, Kazuhiko Sakaguchi, Vincent Semeria, Michael Soegtrop, Arnaud Spiwack, Matthieu Sozeau, Enrico Tassi, Laurent Théry, Anton Trunov, Li-Yao Xia, Théo Zimmermann
• Partners: CNRS, Université Paris-Sud, ENS Lyon, Université Paris-Diderot

7.1.2 Equations

• Keywords: Coq, Dependent Pattern-Matching, Proof assistant, Functional programming
• Scientific Description:

  Equations is a tool designed to help with the definition of programs in the setting of dependent type theory, as implemented in the Coq proof assistant. Equations provides a syntax for defining programs by dependent pattern-matching and well-founded recursion and compiles them down to the core type theory of Coq, using the primitive eliminators for inductive types, accessibility and equality. In addition to the definitions of programs, it also automatically derives useful reasoning principles in the form of propositional equations describing the functions, and an elimination principle for calls to this function. It realizes this using a purely definitional translation of high-level definitions to core terms, without changing the core calculus in any way, or using axioms.

  The main features of Equations include:

  Dependent pattern-matching in the style of Agda/Epigram, with inaccessible patterns, with and where clauses. The use of the K axiom or a proof of K is configurable, and it is able to solve unification problems without resorting to the K rule if not necessary.

  Support for well-founded and mutual recursion using measure/well-foundedness annotations, even on indexed inductive types, using an automatic derivation of the subterm relation for inductive families.

  Support for mutual and nested structural recursion using with and where auxilliary definitions, allowing to factor multiple uses of the same nested fixpoint definition. It proves the expected elimination principles for mutual and nested definitions.

  Automatic generation of the defining equations as rewrite rules for every definition.

  Automatic generation of the unfolding lemma for well-founded definitions (requiring only functional extensionality).

  Automatic derivation of the graph of the function and its elimination principle. In case the automation fails to prove these principles, the user is asked to provide a proof.

  A new dependent elimination tactic based on the same splitting tree compilation scheme that can advantageously replace dependent destruction and sometimes inversion as well. The as clause of dependent elimination allows to specify exactly the patterns and naming of new variables needed for an elimination.

  A set of Derive commands for automatic derivation of constructions from an inductive type: its signature, no-confusion property, well-founded subterm relation and decidable equality proof, if applicable.

• Functional Description:

  Equations is a function definition plugin for Coq (supporting Coq 8.11 to 8.13, with special support for the Coq-HoTT library), that allows the definition of functions by dependent pattern-matching and well-founded, mutual or nested structural recursion and compiles them into core terms. It automatically derives the clauses equations, the graph of the function and its associated elimination principle.

  Equations is based on a simplification engine for the dependent equalities appearing in dependent eliminations that is also usable as a separate tactic, providing an axiom-free variant of dependent destruction.

• Release Contributions:

  This version of Equations is based on an improved simplification engine for the dependent equalities appearing during dependent eliminations that is also usable as a separate dependent elimination tactic, providing an axiom-free variant of dependent destruction and a more powerful form of inversion.

  This is a bugfix release of version 1.2 working with Coq 8.11 to Coq 8.13 See https://mattam82.github.io/Coq-Equations/equations/2019/05/17/1.2.html for the 1.2 release notes. New in this version:

  Fixed issue #297: dependent elimination simplification mistreated let-bindings Fixed issue #295: discrepancy between the syntax in the manual and the implementation Ensure that NoConfusion is derived before EqDec as it is necessary to solve the corresponding obligation.

• News of the Year: Equations 1.2.3, released first in June 2020 brings bugfixes, a fixed grammar and more robust proof automation tactics.
• URL: http://mattam82.github.io/Coq-Equations/
• Publications: hal-01671777, hal-01248807, inria-00628862
• Authors: Cyprien Mangin, Matthieu Sozeau, Matthieu Sozeau
• Contact: Matthieu Sozeau
• Participants: Matthieu Sozeau, Cyprien Mangin

7.1.3 Rewr

• Name: Rewriting methods in algebra
• Keywords: Computer algebra system (CAS), Rewriting systems, Algebra
• Functional Description: Rewr is a prototype of computer algebra system, using rewriting methods to compute resolutions and homotopical invariants of monoids. The library implements various classical constructions of rewriting theory (such as completion), improved by experimental features coming from Garside theory, and allows homotopical algebra computations based on Squier theory. Specific functionalities have been developed for usual classes of monoids, such as Artin monoids and plactic monoids.
• URL: http://www.lix.polytechnique.fr/Labo/Samuel.Mimram/rewr
• Publications: hal-00326974, hal-00531242, hal-00682233, hal-00818253, hal-00932845, hal-01141226
• Authors: Yves Guiraud, Samuel Mimram
• Contact: Yves Guiraud
• Participants: Yves Guiraud, Samuel Mimram

7.1.4 Catex

• Keywords: LaTeX, String diagram, Algebra
• Functional Description: Catex is a Latex package and an external tool to typeset string diagrams easily from their algebraic expression. Catex works similarly to Bibtex.
• URL: https://www.irif.fr/~guiraud/catex/catex.zip
• Author: Yves Guiraud
• Contact: Yves Guiraud
• Participant: Yves Guiraud

7.1.5 Cox

• Keywords: Computer algebra system (CAS), Rewriting systems, Algebra
• Functional Description: Cox is a Python library for the computation of coherent presentations of Artin monoids, with experimental features to compute the lower dimensions of the Salvetti complex.
• URL: https://www.irif.fr/~guiraud/cox/cox.zip
• Publications: hal-00682233, hal-00818253
• Author: Yves Guiraud
• Contact: Yves Guiraud
• Participant: Yves Guiraud

7.1.6 jsCoq

• Keywords: Coq, Program verification, Interactive, Formal concept analysis, Proof assistant, Ocaml, Education, JavaScript
• Functional Description: jsCoq is an Online Integrated Development Environment for the Coq proof assistant and runs in your browser! It aims to enable new UI/interaction possibilities and to improve the accessibility of the Coq platform itself.
• Release Contributions: - Coq 8.10 support - Much improved interaction and general experience - Open / Save dialogs - AST and full serialization of Coq's datatypes - NPM packaging - Timeout support
• URL: https://github.com/ejgallego/jscoq
• Publication: hal-01425752
• Contact: Emilio Jesus Gallego Arias
• Participant: Emilio Jesus Gallego Arias
• Partners: Mines ParisTech, Technion, Israel Institute of Technology

7.1.7 coq-serapi

• Keywords: Interaction, Coq, Ocaml, Data centric, User Interfaces, GUI (Graphical User Interface), Toolkit
• Functional Description: SerAPI is a library for machine-to-machine interaction with the Coq proof assistant, with particular emphasis on applications in IDEs, code analysis tools, and machine learning. SerAPI provides automatic serialization of Coq's internal OCaml datatypes from/to JSON or S-expressions (sexps).
• Release Contributions: - Support Coq 8.10 - Serialization of extensive AST - Serialization of kernel structures - Support for kernel traces [dumping and replay] - Tokenization of Coq documents - Serialization to JSON - Improved protocol and printing - Bug fixes
• URL: https://github.com/ejgallego/coq-serapi
• Publication: hal-01384408
• Contact: Emilio Jesus Gallego Arias
• Participant: Karl Palmskog
• Partner: KTH Royal Institute of Technology

7.1.8 coqbot

• Keywords: Web API, Automation, Software engineering
• Functional Description:

  This software is a bot to help and automatize the development of the Coq proof assistant on the GitHub platform. It is written in OCaml and provides numerous features: synchronization between GitHub and GitLab to allow the use of GitLab for automatic testing (continuous integration), management of milestones on issues, management of the backporting process, merging of pull request upon request by maintainers, etc.

  Most of the features are used only for the development of Coq, but the synchronization with GitLab feature is also used in dozens of independent projects.

• Release Contributions:

  The Julien Coolen's internship final release.

  Added

  Integrate with Jason Gross' coq-bug-minimizer tool. Merge a branch in the coq repository if some conditions are met, by writing @coqbot: merge now in a comment. Parameterize the bot with a configuration file. Installation as a GitHub App is supported. Report CI status checks with the Checks API when using the GitHub app. Report errors of jobs in allow failure mode when the Checks API is used.

  Changed

  Refactored the architecture of the application and of the bot-components library Always create a merge commit when pushing to GitLab. More informative bot merge commit title for GitLab CI.

• URL: https://github.com/coq/bot/
• Contact: Théo Zimmermann

8.1.1 A theory of effects and resources

In collaboration with Thomas Letan (ANSSI), Yann Régis-Gianas developed and proved several properties of a simple web server implemented in Coq using FreeSpec, a compositional framework to verify effectful software components in Coq. This work has been presented at CPP 2020 28, 22.

8.1.2 Explicit syntax for stores

Hugo Herbelin et Étienne Miquey developed a calculus of explicit stores describing the operations and rewriting rules needed to explicitly manage and type extensible environments as part of a syntactic calculus 27. This is similar to calculi with explicit substitutions turning the meta-operation of substitution into a syntactic one, but adding here the possibility to dynamically expand or modify the substitution.

8.1.3 Computational contents of the axiom of choice

Félix Castro started his PhD under the joint supervision of Hugo Herbelin (INRIA) and Alexandre Miquel (IMERL, Facultad de Ingeniería, Universidad de la República, Uruguay) in September 2019. His PhD work focuses on the computational contents of Gödel's constructible universe, which, in particular, justifies the Axiom of Choice. (Previously, he worked on the formalisation of the ramified analytical hierarchy in classical second-order arithmetic.)

During his first year, he worked on the computational contents of logical translations by relativization (such as the syntactical translation implementing Gödel's constructible universe). His approach aims at establishing that, as the double-negation-translation justifies the addition of continuations in a programming language (representing proofs of an intuitionistic logic via the Curry-Howard correspondence), this kind of logical translation (by relativization) justifies the addition of a "quote" instruction in the programming counter-part of the logical system which is the source of the translation.

Hugo Herbelin developed in collaboration with Nuria Brede (U. Potsdam) a unified approach of the underlying logical structure of choice and bar induction principles 32.

8.1.4 Proof-search in proof nets

Alexis Saurin worked on proof search in a proof-net scenario, that is proof-net search. A key aspect of proof construction is a management of non-determinism in bottom-up sequent-proof construction, be it when the search succeeds or when facing a failure and the need for backtracking. This is partially dealt with by focussing proof-construction, which reduces drastically the search space while retaining completeness of the resulting proof space (both at the provability level and at the denotational level).

His approach consists in considering the correctness problem of proof-structure for the larger class of para-proof structures (ie proof-structure admitting a generalized axiom deriving any sequent) viewing proof-search and sequentialisation as dual aspects of partial proof structures (that is proof nets with open premisses). Revisiting in particular two related correctness criteria in the framework of para-proof-structures (contractibility and Lafont's parsing criterion), he obtains a proof-construction algorithm in which the proof space is not a search tree, as in sequent-calculus, but a dag allowing to share proof-construction paths, for which proof-construction corresponds to "unparsing". A paper is currently being written.

8.1.5 Algebraic Logic Programming

Emilio Jesús Gallego Arias collaborated with Jim Lipton from Wesleyan University on the development of algebraic models for proof search in the context of logic programming. In particular, we have extended the semantics of  61 to better account for the recursive definition of relations, using techniques inspired in step-indexing  43. A key point of the extension is that the index category does account for the particular proof search strategy as understood in logic programming. For example, the category $\langle \mathbb{N},<\rangle$ provides the categorical setting for breadth-first search.

8.2.1 Proof theory of non-wellfounded and circular proofs

8.2.2 On the denotational semantics of finitary and non-wellfounded proofs

Farzad Jafar-Rahmani started his PhD under the supervision of Thomas Ehrhard and Alexis Saurin in October 2019. His PhD work will focus on the denotational semantics of finitary and circular proofs of linear logic with fixed points.

Together with Ehrhard, he developed a categorical semantics of $\mu LL$ based on Seely categories and on strong functors acting on them, exhibiting two instances of this categorical setting: a relational semantics in which least and greatest fixed points are interpreted in the same way and a more refined interpretation in which sets are equipped with a notion of totality (non-uniform totality spaces), the relations preserving the totality. This later interpretation distinguishes least and greatest fixedpoints and shows that $\mu LL$ enjoys a denotational form of normalization of proofs.

The three of them are currently working on the interpretation of circular proofs and of the denotational counterpart of the validity condition of circular proofs. Moreover, they developed a system L for a polarized calculus and its categorical semantics.

8.2.3 Quantum programming languages with inductive and coinductive types

Kostia Chardonnet started his PhD under the supervision of Alexis Saurin and Benoît Valiron in November 2019. Chardonnet's PhD research focuses on extending quantum programming languages with inductive and coinductive types, under the hypothesis of quantum control (as in QML  35 compared to classical control). Chardonnet, Saurin and Valiron published a first work 26 presenting a language of type isomorphisms with inductive and coinductive types and understanding the connections of those reversible programs with $\mu MALL$ type isomorphisms and more specifically with $\mu MALL$ focused circular proof isomorphisms.

In a collaboration with Valiron and Vilmart, Chardonnet investigated an asynchronous model of Geometry of Interaction for the pure ZX-Calculus, a graphical language for quantum computation, and its extension to ground-processes. This GoI semantics takes the form of a Token Machine. They showed how to connect this new semantics to the usual standard interpretation of the ZX-diagrams.

8.2.4 Simplifying smartcontract programming via stream programming and spreadsheets

Colin Gonzalez is preparing a CIFFRE PhD under the joint supervision of Yann Régis-Gianas, Adrien Guatto and Ralf Treinen. His work aim at programming smartcontracts more easily by taking inspiration from spreadsheets and stream programming. To do this, his goal is to build a certified spreadsheet compiler to Michelson's smartcontracts.

Building on the similarity between the type a smartcontract executions and the type of streams, they first defined a programming language (Lisa) allowing structured programming of spreadsheets to be translated to a stream algebra. First results are the design of a Lisa interpreter, a propotype compiler from a toy spreadsheet language to Lisa and a compiler from a fragment of Lustre to Lisa.

8.3.1 Rewriting methods in higher algebra

Yves Guiraud works with Dimitri Ara (Univ. Aix-Marseille), Albert Burroni, Philippe Malbos (Univ. Lyon 1), François Métayer (Univ. Nanterre) and Samuel Mimram (École Polytechnique) on a reference book on the theory of polygraphs and higher-dimensional categories, and their applications in rewriting theory and homotopical algebra.

Yves Guiraud works with Marcelo Fiore (Univ. Cambridge) on the theoretical foundations of higher-dimensional algebra, in order to develop a common setting to develop rewriting methods for various algebraic structures at the same time. Practically, they aim at a definition of polygraphic resolutions of monoids in monoidal categories, based on the recent notion of $n$-oid in an $n$-oidal category. This theory will subsume the known cases of monoids and associative algebras, and encompass a wide range of objects, such as Lawvere theories (for term rewriting), operads (for Gröbner bases) or higher-order theories (for the $\lambda$-calculus).

Building on 6, Yves Guiraud is currently finishing with Matthieu Picantin (Univ. Paris Diderot) a work that generalises already known constructions such as the bar resolution, several resolutions defined by Dehornoy and Lafont  57, and the main results of Gaussent, Guiraud and Malbos on coherent presentations of Artin monoids 8, to monoids with a Garside family. This allows an extension of the field of application of the rewriting methods to other geometrically interesting classes of monoids, such as the dual braid monoids.

Still with Matthieu Picantin, Yves Guiraud develops an improvement of the classical Knuth-Bendix completion procedure, called the KGB (for Knuth-Bendix-Garside) completion procedure. The original algorithm tries to compute, from an arbitrary terminating rewriting system, a finite convergent presentation, by adding relations to solve confluence issues. Unfortunately, this algorithm fails on standard examples, like most Artin monoids with their usual presentations. The KGB procedure uses the theory of Tietze transformations, together with Garside theory, to also add new generators to the presentation, trying to reach the convergent Garside presentation identified in 6. The KGB completion procedure is partially implemented in the prototype Rewr, developed by Yves Guiraud and Samuel Mimram.

Yves Guiraud has started a collaboration with Najib Idrissi and Muriel Livernet (IMJ-PRG, Univ. Paris Diderot) whose aim is to understand the relation between several different methods known to compute small resolutions of algebras and operads: those based on rewriting methods (Anick, Squier) and those that stem from Koszul duality theory.

8.3.2 Coherent presentations of monoids

Alen Đurić, in collaboration with Pierre-Louis Curien and Yves Guiraud, has found a common generalisation of two distinct generalisations of a result originally due to Deligne, who had shown how to give coherent presentations of aspherical Artin monoids. Here, coherent means, given a presentation by generators and relations, that a number of generating "2-relations" between the relations is provided that suffices to show that "all diagrams involving the relations" can be paved by these generating 2-relations (this is the beginning of a higher-dimensional story). The two distinct generalisations have been given by Gaussent, Guiraud and Malbos and concern Artin groups without the aspherical restriction on the one hand, and so-called Garside monoids on the other hand. The common generalisation concerns a large family of monoids admitting a Garside family. Beyond the technical details, what is important in this new development is to have worked out general conditions that make the coherence argument work. This work is on the edge of being submitted to a journal.

8.3.3 Topological aspects of polygraphs

Amar Hadzihasanović has been a postdoc of the team (funded by FSMP) from end of November 2019 to end of September 2020. He has been working intensively on the study of shapes appropriate for the description of higher cells as needed in various approaches to higher categories and higher structures. During his stay, despite of the sanitary problems, Amar managed to considerably revise his last work "Diagrammatic sets and rewriting in weak higher categories". He also started a collaboration with Pierre-Louis Curien on a variation of the category of opetopes (see next section).

8.3.4 Opetopes

Cédric Ho Thanh defended his PhD thesis on October 15, 2020 31. Since November 2020, he is a postdoc in Ichiro Hasuo's lab, National Institute for Informatics, Tokyo. During the last year of his PhD, he pursued his collaboration with Chaitanya Leena Subramaniam on "opetopic algebras". Their most important result is to have found a common generalisation of the model category constructions of Joyal on simplicial sets on one hand, and of Moerdijk and coauthors on dendroidal sets. This work is part of his PhD thesis, and also submitted to journals33.

The on-going collaboration of Pierre-Louis Curien and Amar Hadzihasanović mentioned above concerns a treatment of opetopes and opetopic sets by means of techniques from poset topology: those techniques are well-adapted for the description of an alternative notion of category of opetopes, where the objects are restricted to be positive opetopes, but the morphisms are more liberal: in this category, the treatment of degeneracies migrates from objects to morphisms. The current focus of the joint research is to find a good presentation of this alternative category by generators and relations.

8.3.5 Foundations and formalisation of higher algebra

Antoine Allioux (PhD started in February 2018), Eric Finster, Yves Guiraud and Matthieu Sozeau are exploring the development of higher algebra in type theory. To formalise higher algebra, one needs a new source of coherent structures in type theory. During the first year of Allioux's PhD, they studied an internalisation of polynomial monads (of which opetopes and $\infty$-categories are instances) in type theory, which ought to provide such a coherent algebraic structure, inspired by the work of Kock et al 76. They later realised that this internalisation is however incoherent as presented in pure type theory, essentially because of its reliance on equality types.

Since then, they switched to a different view, instead extending type theory with a universe of strict polynomial monads. These strict structures allow, in turn, to present internally weak structures such as $(\infty ,1)$-categories. In particular, they were able to internalize the known result that types are $\infty$-groupoids  96, 83. All attempts to formalize these weak structures in pure type theory had failed so far and it was conjectured that it was impossible to formalize them without modifying type theory. A paper has been submitted early 2021  34. They are now concentrating on developing the theory of $(\infty ,1)$-categories in this new framework.

8.4.1 Meta-programming and metatheory of Coq

Vincent Blazy, Hugo Herbelin and Pierre Letouzey started a work aiming at making explicit the universe subtyping in the Calculus of Constructions (master internship of Vincent Blazy, then start of his PhD thesis). The first goal is to detect more easily each use of the Prop-Type cumulativity in Coq, with potential application to Coq extraction and also to the mathematical foundations.

8.4.2 Homotopy type theory and synthetic homotopy

Hugo Herbelin and Hugo Moeneclaey worked on the explicit construction of a model for internal iterated parametricity using cubes. The progresses on this very technical task have been presented at the HoTT-UF'20 workshop.

Hugo Moeneclaey gave an alternative approach to building models for external iterated parametricity, sidestepping most technical problems using the theory of locally presentable categories. This method is applicable to build models for other interpretations of type theory.

Hugo Moeneclaey worked on a presentation of identity types in an arbitrary non-recursive higher inductive type as higher inductive types themselves. As an example the loop space of the circle is the type freely generated by a point and an auto-equivalence. This was done using the definition of higher inductive types by Kaposi and Kovacs. This work is still in progress.

8.4.3 Computational characterizations of relevant subsystems of second order arithmetic

Hugo Herbelin and Firmin Martin developed a variant of Girard-Reynolds' System F based on the axiom of separability rather than the axiom of comprehension. They derive from it a realisability semantics for the systems WKL${}_0$ and ATR${}_0$, two systems of the reverse mathematics of second order arithmetic which are precisely characterized by such a separability axiom (existence of a set separating any two disjoint formulas from a given class of arithmetic formulas).

8.4.4 Dependent pattern-matching

From August 2020, Thierry Martinez resumed full time 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. Significantly enough, he exploited OCaml's GADT to ensure strong static invariants of his implementation.

8.4.5 Software engineering aspects of the development of Coq

In January 2020, Théo Zimmermann was recruited on a three-year fixed term position to contribute both to the collaborative maintenance and evolution effort around Coq and its community, and to further investigate these software engineering aspects through empirical methods. His practical contributions for 2020 include:

• a restructuration of Coq's reference manual and the review of many documentation pull requests (in collaboration with the external contributor Jim Fehrle) that led to significant improvements of the documentation of Coq (see the details in Coq's changelog for version 8.12);
• participation to the effort (led by the external contributor Michael Soegtrop) to create a "Coq platform", that will be the new official distribution method for both the Coq software and a selection of important Coq packages;
• diverse community management and tooling contributions with various community members, including the management of the coq-community organization (which he founded during his PhD thesis in 2018) and which now hosts over 50 projects by over 30 maintainers.

During the summer, Théo Zimmermann supervised the internship of Julien Coolen on the maintenance and evolution of coqbot. this bot, which was created by Théo Zimmermann during his PhD thesis in 2018, is relied on every day by the Coq developers for the management of the development activity on GitHub. This internship led to significant additions to the bot. One of the most impactful addition is a new feature allowing maintainers to merge pull requests with a comment. Previously, the maintainers had to use a script and set up a PGP key, and this had turned out to be a significant barrier of entry to the role of maintainer.

Théo Zimmermann's research focused on the model of community organizations for the collaborative maintenance of packages which he discovered during his PhD thesis and applied to create the coq-community organization mentioned above. He published and presented a paper about it in the ICSE workshop SoHEAL 2020 29. Since then, he has pursued this investigation further in collaboration with Jean-Rémy Falleri from LaBRI (Bordeaux).

Emilio J. Gallego Arias continued work on improving the incremental proof checking infrastructure for Coq, with a particular focus on the integration of Coq with the industrial-scale Dune build system. Jointly with the rest of the Dune development team, support for composition of Coq “theories” was added and released, together with many other fixes and improvements, including extraction and native compute support, and the full porting of the Coq codebase to Dune. This new tooling, despite its experimental nature, has seen a good adoption by users and researchers. Emilio J. Gallego Arias was invited to a one week “Dune Retreat” — funded by Jane Street — where extended discussion and work about this project was pursued with the rest of the Dune team.

Emilio J. Gallego Arias and Karl Palmskog released a new version of the Coq SerAPI tool, with improvements and new functionality to enable new applications such as  86. Significant feedback from researchers has been gathered as to improve the tool, with a particular focus on mechanized-treatment of proofs by diverse tooling such as machine learning  49 or software engineering tooling.

Emilio J. Gallego Arias and Shachar Itzhaky released a new version of the Coq educational frontend jsCoq 7, with significant improvements both in the frontend, the backend, and the package system. In particular, jsCoq has reached the milestone where we consider that standard Coq textbooks such as Software Foundations run in a stable fashion. jsCoq has been used in several Coq courses in the 2020 year (we estimate at least in the dozens).

Emilio J. Gallego Arias maintains an ongoing collaboration with the Deducteam group at INRIA Saclay on the topic of interactive proof methods and standards; work in 2020 has included the co-supervision of two interns and improvements towards the specification and use of the Language Server Protocol in the context of interactive proof assistants.

8.4.6 Maintenance, development and coordination of the development of Coq

Hugo Herbelin, Emilio J. Gallego Arias and Théo Zimmermann, helped by members from Gallinette (Nantes) and Stamp (ex-Marelle, Sophia-Antipolis), devoted an important part of their time to coordinate the development, to review propositions of extensions of Coq from external and/or young contributors, and to propose themselves extensions.

Emilio J. Gallego Arias and Théo Zimmerman took the roles of release managers for the Coq 8.12 release cycle. Coq 8.12.0 was released in July, Coq 8.12.1 was released in November and Coq 8.12.2 was released in December.

Cyril Cohen (Stamp) and Théo Zimmermann replaced the Coq Gitter communication channels by a Zulip server supporting multiple topics. Hundreds of messages are posted every week.

In 2020, Hugo Herbelin contributed about 200 pull requests to Coq, covering maintenance, bug fixes, extensions of the support for notations, refinements of the phase of elaboration of user syntax to checked expresions, code cleanup (especially regarding implicit arguments and notations).

Emilio J. Gallego Arias contributed over 300 changes to Coq of diverse nature, most of them to improve the Coq codebase in preparation for further work on incremental and multi-core aware type checking. He also reviewed and merged over 200 pull requests, with around  2000 total interactions [comments, issues, etc...].

8.5.1 Proofs and surfaces

The joint work of Pierre-Louis Curien with Jovana Obradović (former PhD student of the team and now researcher at the Institute of Mathematics of the Serbian Academy of Sciences), Zoran Petrić and other Serbian colleagues on designing a formal deductive system capturing the proofs of incidence theorems has been published in the journal Annals of Pure and Applied Logic 23.

8.5.2 A Coq formalisation of the first-order predicate calculus

In relation with a logic course for master students, Pierre Letouzey continued a Coq formalisation of the first-order predicate calculus, with the help of Samuel Ben Hamou (internship, 1st year ENS Saclay). The logical rules are expressed in a natural deduction style (with explicit contexts), with two possible low-level representations of formulas (quantifiers with names or “locally nameless”). After a completeness theorem last year, this time the use of specific theories like Peano and ZF have been investigated. This development is available at https://gitlab.math.univ-paris-diderot.fr/letouzey/natded

8.5.3 Lexing and regular expressions in Coq

Pierre Letouzey and Yann Régis-Gianas continued working on a Coq formalisation of regular expressions (with complement and conjunction) and their Brzozowski derivatives. This year, Antimorov derivatives have also been studied in this Coq formalization. Many techniques have also been attempted to prove correct the exact details used in a real-world implementation (ml-ulex), but a complete proof of this implementation is still elusive.

8.5.4 Sensitivity Conjecture in Coq

Daniel de Rauglaudre started a formalization in Coq the Sensitivity Conjecture, which became a Theorem in 2019 thanks to Hao Huang  75. The sensitivity conjecture remained an open-problem for more than thirty years, aiming to relate the sensitivity of a Boolean function results to its input values to other complexity measures of Boolean functions, such as block sensitivity. De Rauglaudre started to formalize Huang's very succinct proof of the conjecture.

For proving some lemmas in this theorem, numerous formalizations in Linear Algebra (matrices, determinants, eigenvalues, permutations, etc.) had to be implemented. In this context, a study of algebra of ring-like structures has been started, and some syntax of iterators have been studied and added. This development is available at at https://github.com/roglo/coq_sensitivity.

8.5.5 Proofs of algorithms on graphs

Jean-Jacques Lévy and Chen Ran (a PhD student at the Institute of Software, Beijing) pursued their work about formal proofs of graph algorithms. Their goal is to provide computer-checked proofs of algorithms that remain the human-readable. In 2019, they presented at ITP 2019 a joint paper with Cyril Cohen, Stephan Merz and Laurent Théry on this work  48. This article compared formal proofs in three different systems (Why3, Coq, Isabelle/HOL) of Tarjan's linear-time algorithm (1972) computing the strongly connected components in directed graphs.

In July 2020, Chen Ran passed her PhD degree at Iscas, Beijing (adviser Zhang Wenhui). Jean-Jacques Lévy currently works on a proof of the implementation of this algorithm with imperative programming and memory pointers. He also plans to produce formal proofs of other abstract algorithms such as the Hopcroft-Tarjan (1972) linear-time algorithm for planarity testing in undirected graphs.

8.5.6 Iterated parametricity and semi-cubical sets

Hugo Herbelin and Ramkumar Ramachandra started the formalization in Coq of an original dependently-typed construction of semi-cubical sets inspired by the parametricity translation (see Section 8.4.2). This highly stressed the limits of Coq.

8.5.7 Datalog and low-level binary analysis

Emilio J. Gallego Arias collaborated with Stefania Dumbrava and Cody Roux on the use of our verified Datalog engine  45 for the analysis of low-level binary code; we have made significant progress towards the verification in Coq of the alias analysis proposed in  46.

8.5.8 Mechanism design

Emilio J. Gallego Arias collaborated with Pierre Jouvelot and Lucas Sguerra on the formalized verification of the general Vickrey-Clarke-Groves (see for example  87) mechanism using Coq. The work has resulted in a paper submission early 2021.

10.1.1 Inria associate team not involved in an IIL

Pierre-Louis Curien is a member of the Equipe associée CRECOGI (Concurrent, Resourceful and Effectful COmputation, by Geometry of Interaction) coordinated by Ugo dal Lago (Focus team, Bologna), with the National Institute of Informatics, Tokyo (Ichiro Hasuo) which planned to have its final meeting as a Shonan meeting, Japan, in November 2020. This event has been postponed to November 2021.

Pierre-Louis Curien is coordinator on the French side of the project VIP (Verification, Interaction, and Proofs) (2017-2020) of the Inria - Chinese Academy of Sciences (CAS) program, with the Institute of Sotware of the CAS. The final meeting of the project, to be held in Beijing, has been postponed to 2021. The Chinese coordinator Ying Jiang has retired and been replaced as coordinator by Peng Wu.

10.2.1 Visits to international teams

11.1.1 Scientific events: organisation

11.1.2 Scientific events: selection

11.1.3 Journal

11.1.4 Invited talks

Théo Zimmermann gave an invited talk at IRIT on the challenges in the collaborative evolution of Coq and its ecosystem.

Pierre-Louis Curien is an invited speaker at the Conference “Braids and beyond”, in memory of Patrick Dehornoy, which was planned in September 2020 in Caen and has been postponed to 2021.

Emilio J. Gallego Arias gave an invited talk at SystemX at the invitation of Kalpana Singh on the subject of applications to Coq to SystemX projects.

Hugo Herbelin gave an invited talk at the JFLA conference on cubical type theory.

11.1.5 Research administration

Yves Guiraud is an elected member of the “Conseil scientifique” of the “UFR de mathématiques de l'Université de Paris”

Teaching

• Master: Alexis Saurin, Outils classiques pour la correspondance de Curry-Howard, 24H, LMFI, Université de Paris. (spring 2020)
• Master: Alexis Saurin, Cours fondamental de théorie de la démonstration, 36H, LMFI, Université de Paris. (fall 2020)
• Master: Alexis Saurin, Programmation synchrone, 24H, M2 informatique, Université de Paris. (fall 2020)
• Master: Hugo Herbelin and Hugo Moeneclaey taught a class on Homotopy Type Theory, respectively 30H and 18H, LMFI, University de Paris. (winter 2020)
• Master: Pierre Letouzey teaches two short courses to the LMFI Master 2 students : “Programming in Coq” and “Introduction to computed-aided formal proofs” (2*24H). These two courses come in addition to Pierre Letouzey’s regular duty as teacher in the Computer Science department of Université de Paris (including a course on Compilation to M2-Pro students and a course on computed-aided formal proofs to M1 students).
• Group tutorial: in his visit to Wesleyan university, Emilio J. Gallego Arias helped Jim Lipton on a “group tutorial” on Blockchain Technologies with 18 participating students. A group tutorial is a student-driven teaching format. The idea was to supervise a student on the use of Coq in the Smart Contract context but plans were altered by the sanitary crisis.

PhD Supervision

• PhD started: Vincent Blazy, Fine-grain structure of universe subtyping in Coq and applications to program certification and mathematical foundations, Université de Paris, started in September 2020, supervised by Hugo Herbelin and Pierre Letouzey.
• PhD started: Colin Gonzalez, Certified compilation of spreadsheets as Tezos smart contract, Université de Paris, CIFRE contract started in September 2020, supervised by Yann Régis-Gianas, Benjamin Canou (Nomadic Labs), Adrien guatto and Ralf Treinen.
• PhD in progress: Félix Castro, Computational contents of the model of constructible sets, cotutelle between Université de Paris and Universidad de la República (Montevideo, Uruguay), started in September 2019, supervised by Hugo Herbelin and Alexandre Miquel.
• PhD in progress: Kostia Chardonnet, Inductive and coinductive types in quantum programming languages, Université Paris Saclay, started in November 2019, supervised by Alexis Saurin and Benoît Valiron.
• PhD in progress: Alen Đurić, Normalisation for monoids and higher categories, Université de Paris, started in October 2O19, supervised by Yves Guiraud and Pierre-Louis Curien.
• PhD in progress: Farzad Jafar-Rahmani, Denotational semantics of circular and non-wellfounded proofs, Université de Paris, started in October 2019, supervised by Thomas Ehrhard and Alexis Saurin.
• PhD inprogress: Hugo Moeneclaey, Syntax of spheres in homotopy type theory, Université de Paris, started in September 2019, supervised by Hugo Herbelin.
• PhD in progress: Antoine Allioux, Opetopes in Type Theory, Université de Paris, since March 2018, supervised by Yves Guiraud and Matthieu Soseau.
• PhD in progress: Abhishek De, Proof-nets for fixed-point logics and non-well-founded proofs, Université de Paris, since October 2018, supervised by Alexis Saurin.
• PhD in progress: Lucas Massoni Sguerra, Formal verification of mechanism design, Université PSL, since January 2018, supervised by Gérard Memmi, Pierre Jouvelot, and Emilio J. Gallego Arias.
• PhD in progress: Rémi Nollet, Validity conditions for circular proofs, Université de Paris, since October 2016, supervised by Alexis Saurin and Christine Tasson.
• PhD ended: Cédric Ho Thanh, Opetopes for higher-dimensional rewriting and koszulity, Université de Paris, defended October 15 2020, supervised by Pierre-Louis Curien and Samuel Mimram.
• PhD ended: Simon Forest, Computational descriptions of higher categories, École Polytechnique, defended on 12 January 2021, supervised by Samuel Mimram (École Polytechnique) and Yves Guiraud

Internships

• Hugo Herbelin and Pierre Letouzey supervised the master internship of Vincent Blazy on an extension of the Calculus of Constructions by an explicit subtyping of Prop into an infinite universe hierarchy.
• Pierre Letouzey supervised the internship of Samuel Ben Hamou (1st year ENS Saclay) on the extensions of a 1st-order logic course certified in Coq.
• Théo Zimmermann supervised the internship of Julien Coolen (june - august 2020) on the maintenance and evolution of coqbot.
• Hugo Herbelin supervised the master internship of Firmin Martin (master 1) on a realizability semantics of the axiom of separability using a refined version of System F (see Section 8.1.3).
• Emilio J. Gallego co-supervised with Fréderic Blanqui two interns on the subject of user interfaces for the lambdapi theorem prover: François Lefoulon (1A, June-July 2020) and Ashish Kumar Barnawal (Dec-Jan 2020).

Juries

• Hugo Herbelin was a member of the PhD committee of Théo Winterhalter.

International journals

• 22 article ThomasT. Letan, YannY. Régis-Gianas, PierreP. Chifflier and GuillaumeG. Hiet. Modular verification of programs with effects and effects handlers Formal Aspects of Computing December 2020
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• 23 article MarinaM. Milicevic, DjordjeD. Baralic, JovanaJ. Obradovic, ZoranZ. Petric, MladenM. Zekic, RadeR. Zivaljevic and Pierre-LouisP.-L. Curien. Proofs and surfaces Annals of Pure and Applied Logic 171 9 2020
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• 24 article MatthieuM. Sozeau, AbhishekA. Anand, SimonS. Boulier, CyrilC. Cohen, YannickY. Forster, FabianF. Kunze, GregoryG. Malecha, NicolasN. Tabareau and ThéoT. Winterhalter. The MetaCoq Project Journal of Automated Reasoning February 2020
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• 25 articleMatthieuM. Sozeau, SimonS. Boulier, YannickY. Forster, NicolasN. Tabareau and ThéoT. Winterhalter. Coq Coq Correct! Verification of Type Checking and Erasure for Coq, in CoqProceedings of the ACM on Programming LanguagesJanuary 2020, 1-28
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International peer-reviewed conferences

• 26 inproceedingsKostiaK. Chardonnet, AlexisA. Saurin and BenoîtB. Valiron. Toward a Curry-Howard Equivalence for Linear, Reversible ComputationRC 2020 - 12th international conference on Reversible ComputationLNCSReversible Computation12227Oslo / Virtual, NorwayJuly 2020, 144-152
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• 27 inproceedingsHugoH. Herbelin and ÉtienneÉ. Miquey. A calculus of expandable stores: Continuation-and-environment-passing style translationsProceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS '20), July 8--11, 2020, Saarbrücken, GermanyLICS 2020 - 35th ACM/IEEE Symposium on Logic in Computer ScienceSaarbrücken / Virtual, GermanyJuly 2020, 564-577
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• 28 inproceedingsThomasT. Letan and YannY. Régis-Gianas. FreeSpec: Specifying, Verifying and Executing Impure Computations in CoqCPP 2020 - 9th ACM SIGPLAN International Conference on Certified Programs and ProofsNouvelle-Orléans, United StatesJanuary 2020, 1-15
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• 29 inproceedings ThéoT. Zimmermann. A first look at an emerging model of community organizations for the long-term maintenance of ecosystems' packages SoHeal 2020 - 3rd International Workshop on Software Health Seoul / Virtual, South Korea May 2020
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Edition (books, proceedings, special issue of a journal)

• 30 book Zaynah LeaZ. Dargaye and YannY. Regis-Gianas. 31ème Journées Francophones des Langages Applicatifs January 2020
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Doctoral dissertations and habilitation theses

• 31 thesis CédricC. Ho Thanh. Opetopes: Syntactic and Algebraic Aspects Université de Paris October 2020
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Reports & preprints

• 32 misc NuriaN. Brede and HugoH. Herbelin. On the logical structure of choice and bar induction principles March 2021
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• 33 misc CédricC. Ho Thanh and ChaitanyaC. Leena Subramaniam. Opetopic algebras III: Presheaf models of homotopy-coherent opetopic algebras January 2020
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