3.2.1 A definitional proof-irrelevant version of Coq.

In the Coq proof assistant, the sort $\mathrm{𝐏𝐫𝐨𝐩}$ stands for the universe of types which are propositions. That is, when a term $P$ has type $\mathrm{𝐏𝐫𝐨𝐩}$, the only relevant fact is whether $P$ is inhabited (that is true) or not (that is false). This property, known as proof irrelevance, can be expressed formally as: $\forall xy:P,x=y$. Originally, the raison d'être of the sort $\mathrm{𝐏𝐫𝐨𝐩}$ was to characterise types with no computational meaning with the intention that terms of such types could be erased upon extraction. However, the assumption that every element of $\mathrm{𝐏𝐫𝐨𝐩}$ should be proof irrelevant has never been integrated to the system. Indeed, in Coq, proof irrelevance for the sort $\mathrm{𝐏𝐫𝐨𝐩}$ is not incorporated into the theory: it is only compatible with it, in the sense that its assumption does not give rise to an inconsistent theory. In fact, the exact status of the sort $\mathrm{𝐏𝐫𝐨𝐩}$ in Coq has never been entirely clarified, which explains in part this lack of integration. Homotopy type theory brings fresh thinking on this issue and suggests turning $\mathrm{𝐏𝐫𝐨𝐩}$ into the collection of terms that a certain static inference procedure tags as proof irrelevant. The goal of this task is to integrate this insight in the Coq system and to implement a definitional proof-irrelevant version of the sort $\mathrm{𝐏𝐫𝐨𝐩}$.

3.2.2 Extend the Coq proof assistant with a computational version of univalence

The univalence principle is becoming widely accepted as a very promising avenue to provide new foundations for mathematics and type theory. However, this principle has not yet been incorporated into a proof assistant. Indeed, the very mathematical structures (known as $\infty$-groupoids) motivating the theory remain to this day an active area of research. Moreover, a correct and decidable type checking procedure for the whole theory raises both computational complexity and logical coherence issues. Observational type theory 36, as implemented in Epigram, provides a first-stage approximation to homotopy type theory, but only deals with functional extensionality and does not capture univalence. Coquand and his collaborators have obtained significant results on the computational meaning of univalence using cubical sets 44, 49. Bickford has initiated a promising formalisation work 1 in the NuPRL system. However, a complete formalisation in intensional type theory remains an open problem.

Hence a major objective is to achieve a complete internalisation of univalence in intensional type theory, including an integration to a new version of Coq. We will strive to keep compatibility with previous versions, in particular from a performance point of view. Indeed, the additional complexity of homotopy type theory should not induce an overhead in the type checking procedure used by the software if we want our new framework to become rapidly adopted by the community. Concretely, we will make sure that the compilation time of Coq’s Standard Library will be of the same order of magnitude.

3.2.3 Extend the logical power of type theory without axioms in a modular way

Extending the power of a logic using model transformations (e.g., forcing transformation 76, 75 or the sheaf construction 106) is a classic topic of mathematical logic 50, 81. However, these ideas have not been much investigated in the setting of type theory, even though they may provide a useful framework for extending the logical power of proof assistant in a modular way. There is a good reason for this: with a syntactic notion of equality, the underlying structure of type theory does not conform to the structure of topos used in mathematical logic. A direct incorporation of the standard techniques is therefore not possible. However, a univalent notion of equality brings type theory closer to the required algebraic structure, as it corresponds to the notion of $\infty$-topos recently studied by Lurie 88. The goal of this task is to revisit model transformations in the light of the univalence principle, and to obtain in this way new internal transformations in type theory which can in turn be seen as compilation phases. The general notion of an internal syntactical translation has already been investigated in the team 45.

Multiple compilation phases

3.2.4 Methodology: Extending type theory with different compilation phases

The Gallinette project advocates the use of distinct compilation phases as a methodology for the design of a new generation of proof assistants featuring modular extensions of a core logic. The essence of a compiler is the separation of the complexity of a translation process into modular stages, and the organization of their re-composition. This idea finds a natural application in the design of complex proof assistants (Figure 1). For instance, the definition of type classes in Coq follows this pattern, and is morally given by the means of a translation into a type-class free kernel. More recently, a similar approach by compilation stages, using the forcing transformation, was used to relax the strict positivity condition guarding inductive types 76, 75. We believe that this flavour of compilation-based strategies offers a promising direction of investigation for the purpose of defining a decidable type checking algorithm for HoTT.

3.3.1 Models for integrating effects with dependent types

A crucial step is the integration of dependent types with effects, a topic which has remained “currently under investigation”  96 ever since the beginning. The difficulty resides in expressing the dependency of types on terms that can perform side-effects during the computation. On the side of denotational semantics, several extensions of categorical models for effects with dependent types have been proposed 33, 114 using axioms that should correspond to restrictions in terms of expressivity but whose practical implications, however, are not immediately transparent. On the side of logical approaches 71, 72, 82, 94, one first considers a drastic restriction to terms that do not compute, which is then relaxed by semantic means. On the side of systems for certified programming such as ${\mathrm{F}}^{☆}$, the type system ensures that types only depend on pure and terminating terms.

Thus, the recurring idea is to introduce restrictions on the dependency in order to establish an encapsulation of effects. In our approach, we seek a principled description of this idea by developing the concept of semantic value (thunkables, linears) which arose from foundational considerations 61, 109, 98 and whose relevance was highlighted in recent works  85, 102. The novel aspect of our approach is to seek a proper extension of type theory which would provide foundations for a classical type theory with axiom of choice in the style of Herbelin  71, but which moreover could be generalised to effects other than just control by exploiting an abstract and adaptable notion of semantic value.

3.3.2 Intuitionistic depolarisation

In our view, the common idea that evaluation order does not matter for pure and terminating computations should serve as a bridge between our proposals for dependent types in the presence of effects and traditional type theory. Building on the previous goal, we aim to study the relationship between semantic values, purity, and parametricity theorems 108, 63. Our goal is to characterise parametricity as a form of intuitionistic depolarisation following the method by which the first game model of full linear logic was given (Melliès 91, 92). We have two expected outcomes in mind: enriching type theory with intensional content without losing its properties, and giving an explanation of the dependent types in the style of Idris and ${\mathrm{F}}^{☆}$ where purity- and termination-checking play a role.

3.3.3 Developing the rewriting theory of calculi with effects

An integrated type theory with effects requires an understanding of evaluation order from the point of view of rewriting. For instance, rewriting properties can entail the decidability of some conversions, allowing the automation of equational reasoning in types  31. They can also provide proofs of computational consistency (that terms are not all equivalent) by showing that extending calculi with new constructs is conservative  111. In our approach, the $\lambda$-calculus is replaced by a calculus modelling the evaluation in an abstract machine  56. We have shown how this approach generalises the previous semantic and proof-theoretic approaches 37, 84, 86, and overcomes their shortcomings 99.

One goal is to prove computational consistency or decidability of conversions purely using advanced rewriting techniques following a technique introduced in 111. Another goal is the characterisation of weak reductions: extensions of the operational semantics to terms with free variables that preserve termination, whose iteration is equivalent to strong reduction 32, 59. We aim to show that such properties derive from generic theorems of higher-order rewriting 107, so that weak reduction can easily be generalised to richer systems with effects.

3.3.4 Direct models and categorical coherence

Proof theory and rewriting are a source of coherence theorems in category theory, which show how calculations in a category can be simplified with an embedding into a structure with stronger properties 89, 80. We aim to explore such results for categorical models of effects 84, 55. Our key insight is to consider the reflection between indirect and direct models 61, 98 as a coherence theorem: it allows us to embed the traditional models of effects into structures for which the rewriting and proof-theoretic techniques from the previous section are effective.

Building on this, we are further interested in connecting operational semantics to 2-category theory, in which a second dimension is traditionally considered for modelling conversions of programs rather than equivalences. This idea has been successfully applied for the $\lambda$-calculus 77, 73 but does not scale yet to more realistic models of computation. In our approach, it has already been noticed that the expected symmetries coming from categorical dualities are better represented, motivating a new investigation into this long-standing question.

3.3.5 Models of effects and resources

The unified theory of effects and resources 55 prompts an investigation into the semantics of safe and automatic resource management, in the style of Modern C++ and Rust. Our goal is to show how advanced semantics of effects, resources, and their combination arise by assembling elementary blocks, pursuing the methodology applied by Melliès and Tabareau in the context of continuations 93. For instance, combining control flow (exceptions, return) with linearity allows us to describe in a precise way the “Resource Acquisition Is Initialisation” idiom in which the resource safety is ensured with scope-based destructors. A further step would be to reconstruct uniqueness types and borrowing using similar ideas.

3.4.1 Gradual Certified Programming

One of the main issues faced by a programmer starting to internalise in a proof assistant code written in a more permissive world is that type theory is constrained by a strict type discipline which lacks flexibility. Concretely, as soon as you start giving a more precise type/specification to a function, the rest of the code interacting with this function needs to be more precise too. To address this issue, the Gallinette team will put strong efforts into the development of gradual typing in type theory to allow progressive integration of code that comes from a more permissive world.

Indeed, on the way to full verification, programmers can take advantage of a gradual approach in which some properties are simply asserted instead of proven, subject to dynamic verification. Tabareau and Tanter have made preliminary progress in this direction 113. This work, however, suffers from a number of limitations, the most important being the lack of a mechanism for handling the possibility of runtime errors within Coq. Instead of relying on axioms, this project will explore the application of Section 3.3 to embed effects in Coq. This way, instead of postulating axioms for parts of the development that are too hard/marginal to be dealt with, the system adds dynamic checks. Then, after extraction, we get a program that corresponds to the initial program but with dynamic checks for parts that have not been proven, ensuring that the program will raise an error instead of going outside its specification.

This will yield new foundations of gradual certified programming, both more expressive and practical. We will also study how to integrate previous techniques with the extraction mechanism of Coq programs to OCaml, in order to exploit the exception mechanism of OCaml.

3.4.2 Imperative features and object polymorphism in the Coq proof assistant

3.4.3 Robust tactics for proof engineering for the scaling of formalised libraries

When developing certified software, a major part of the effort is spent not only on writing proof scripts, but on rewriting them, either for the purpose of code maintenance or because of more significant changes in the base definitions. Regrettably, proof scripts suffer more often than not from a bad programming style, and too many proof developers casually neglect the most elementary principles of well-behaved programmers. As a result, many proof scripts are very brittle, user-defined tactics are often difficult to extend, and sometimes even lack a clear specification. Formal libraries are thus generally very fragile pieces of software. One reason for this unfortunate situation is that proof engineering is very badly served by the tools currently available to the users of the Coq proof assistant, starting with its tactic language. One objective of the Gallinette team is to develop better tools to write proof scripts.

Completing and maintaining a large corpus of formalised mathematics requires a well-designed tactic language. This language should both accommodate the possible specific needs of the theories at stake, and help with diagnostics at refactoring time. Coq's tactic language is in fact two-leveled. First, it includes a basic tactic language, to organise the deductive steps in a proof script and to perform the elementary bureaucracy. Its second layer is a meta-programming language, which allows users to define their own new tactics at toplevel. Our first direction of work consists in the investigation of the appropriate features of the basic tactic language. For instance, the design of the Ssreflect tactic language, and its support for the small scale reflection methodology 66, has been a key ingredient in at least two large scale formalisation endeavours: the Four Colour Theorem 65 and of the Odd Order Theorem 64. Building on our experience with the Ssreflect tactic language, we will contribute to the ongoing work on the basic tactic language for Coq. The second objective of this task is to contribute to the design of a typed tactic language. In particular, we will build on the work of Ziliani and his collaborators 115, extending it with reasoning about the effects that tactics have on the “state of a proof” (e.g. number of sub-goals, metavariables in context). We will also develop a novel approach for incremental type checking of proof scripts, so that programmers gain access to a richer discovery—engineering interaction with the proof assistant.

3.5.1 Certified Code Refactoring

In the context of refactoring of C programs, we intend to formalise program transformations that are written in an imperative style to test the usability of our addition of effects in the proof assistant. This subject has been chosen based on the competence of members of the team.

We are currently working on the formalisation of refactoring tools in Coq 52. Automatic refactoring of programs in industrial languages is difficult because of the large number of potential interactions between language features that are difficult to predict and to test. Indeed, all available refactoring tools suffer from bugs : they fail to ensure that the generated program has the same behaviour as the input program. To cope with that difficulty, we have chosen to build a refactoring tool with Coq : a program transformation is written in the Coq programming language, then proven correct on all possible inputs, and then an OCaml executable program is generated by the platform. We rely on the CompCert C formalisation of the C language. CompCert is currently the most complete formalisation of an industrial language, which justifies that choice. We have three goals in that project :

• Build a refactoring tool that programmers can rely on and make it available in a popular platform (such as Eclipse, IntelliJ or Frama-C).
• Explore large, drastic program transformations such as replacing a design architecture for an other one, by applying a sequence of small refactoring operations (as we have done for Java and Haskell programs before 51, 48, 34), while ensuring behaviour preservation.
• Explore the use of enhancements of proof systems on large developments. For instance, refactoring tools are usually developed in the imperative/object paradigm, so the extension of Coq with side effects or with object features proposed in the team can find a direct use-case here.

3.5.2 Certified Constraint Programming

We plan to make use of the internalisation of the object-oriented paradigm in the context of constraint programming. Indeed, this domain is made of very complex algorithms that are often developed using object-oriented programming (as it is the case for instance for CHOCO, which is developed in the Tasc Group at IMT Atlantique, Nantes). We will in particular focus on filtering algorithms in constraint solvers, for which research publications currently propose new algorithms with manual proofs. Their formalisation in Coq is challenging. Another interesting part of constraint solving to formalise is the part that deals with program generation (as opposed to extraction). However, when there are numerous generated pieces of code, it is not realistic to prove their correctness manually, and it can be too difficult to prove the correctness of a generator. So we intend to explore a middle path that consists in generating a piece of code along with its corresponding proof (script or proof term). A target application could be interval constraints (for instance Allen interval algebra or region connection calculus) that can generate thousands of specialised filtering algorithms for a small number of variables 41.

Finally, Rémi Douence has already worked (articles publishing 68, 105, 58, 42, 39, 38, PhD Thesis advising 104, 40) with different members of the Tasc team. Currently, he supervises with Nicolas Beldiceanu the PhD Thesis of Jovial Cheukam Ngouonou in the Tasc team. He studies graph invariants to enhance learning algorithms. This work requires proofs, manually done for now, we would like to explore when these proofs could be mechanized.

3.5.3 Certified Symbolic Computation

We will investigate how the addition of effects in the Coq proof assistant can facilitate the marriage of computer algebra with formal proofs. Computer algebra systems on one hand, and proof assistants on the other hand, are both designed for doing mathematics with the help of a computer, by the means of symbolic computations. These two families of systems are however very different in nature: computer algebra systems allow for implementations faithful to the theoretical complexity of the algorithms, whereas proof assistants have the expressiveness to specify exactly the semantics of the data-structures and computations.

Experiments have been run that link computer algebra systems with Coq 57, 47. These bridges rely on the implementation of formal proof-producing core algorithms like normalisation procedures. Incidentally, they require non trivial maintenance work to survive the evolution of both systems. Other proof assistants like the Isabelle/HOL system make use of so-called reflection schemes: the proof assistant can produce code in an external programming language like SML, but also allows to import the values output by these extracted programs back inside the formal proofs. This feature extends the trusted base of code quite significantly but it has been used for major achievements like a certified symbolic/numeric ODE solver 74.

We would like to bring Coq closer to the efficiency and user-friendliness of computer algebra systems: for now it is difficult to use the Coq programming language so that certified implementations of computer algebra algorithms have the right, observable, complexity when they are executed inside Coq. We see the addition of effects to the proof assistant as an opportunity to ease these implementations, for instance by making use of caching mechanisms or of profiling facilities. Such enhancements should enable the verification of computation-intensive mathematical proofs that are currently beyond reach, like the validation of Helfgott's proof of the weak Goldbach conjecture 70.

6.1.1 Ltac2

• Keywords:
  Coq, Proof assistant
• Functional Description:
  A replacement for Ltac, the tactic language of Coq.
• Contact:
  Pierre-Marie Pedrot

6.1.2 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 8.13 to 8.15, 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 is a new major release of Equations, working with Coq 8.13 and 8.14. This version adds an improved syntax (less ,-separation), integration with the Coq-HoTT library and numerous bug fixes. See the reference manual for details.

  This version introduces minor breaking changes along with the following features:

  Enhancements of pattern interpretation

  No explicit shadowing of pattern variables is allowed anymore. This fixes numerous bugs where generated implicit names introduced by the elaboration of patterns could shadow user-given names, leading to incorrect names in right-hand sides and confusing environments.

  Improved syntax for "concise" clauses separated by |, at top-level or inside with subprograms. We no longer require to separate them by ,. For example, the following definition is now accepted:

  Equations foo : nat -> nat := | 0 => 1 | S n => S (foo n). The old syntax is however still supported for backwards compatibility.

  Multiple patterns can be separated by , in addition to |, as in:

  Equations trans {A} {x y z : A} (e : x = y) (e' : y = z) : x = z := | 1, 1 => 1. Require Import Equations.Equations. does not work anymore. One has to use Require Import Equations.Prop.Equations to load the plugin's default instance where equality is in Prop. From Equations Require Import Equations is unaffected.

  Use Require Import Equations.HoTT.All to use the HoTT variant of the library compatible with the Coq HoTT library The plugin then reuses the definition of paths from the HoTT library and all its constructions are universe polymorphic. As for the HoTT library alone, coq must be passed the arguments -noinit -indices-matter to use the library and plugin. The coq-equations opam package depends optionally on coq-hott, so if coq-hott is installed before it, coq-equations will automatically install the HoTT library variant in addition to the standard one. This variant of Equations allows to write very concise dependent pattern-matchings on equality:

  Require Import Equations.HoTT.All. Equations sym {A} {x y : A} (e : x = y) : y = x := | 1 => 1. New attribute #[tactic=tac] to set locally the default tactic to solve remaining holes. The goals on which the tactic applies are now always of the form Γ |- τ where Γ is the context where the hole was introduced and τ the expected type, even when using the Obligation machinery to solve them, resulting in a possible incompatibility if the obligation tactic treated the context differently than the conclusion. By default, the program_simpl tactic performs a simpl call before introducing the hypotheses, so you might need to add a simpl in * to your tactics.

  New attributes #[derive(equations=yes,no, eliminator=yes|no)] can be used in place of the (noeqns, noind) flags which are deprecated.

• News of the Year:
  Equations 1.3, released first in September 2021 brings bugfixes, an improved grammar and more robust proof automation tactics.
• URL:
  http://­mattam82.­github.­io/­Coq-Equations/
• Publications:
  hal-01671777, hal-01248807, inria-00628862
• Contact:
  Matthieu Sozeau
• Participants:
  Matthieu Sozeau, Cyprien Mangin

6.1.3 Math-Components

• Name:
  Mathematical Components library
• Keyword:
  Proof assistant
• Functional Description:
  The Mathematical Components library is a set of Coq libraries that cover the prerequiste for the mechanization of the proof of the Odd Order Theorem.
• Release Contributions:

  This release is compatible with Coq 8.10, 8.11 and Coq 8.12. The main changes are:

  - support for Coq 8.7, 8.8 and 8.9 have been dropped,

  - a change of implementation of intervals and the updated theory,

  - the addition of kernel lemmas for matrices,

  - generalized many lemmas for path and sorted,

  - several lemma additions, name changes and bug fixes.

• URL:
  http://­math-comp.­github.­io/­math-comp/
• Contact:
  Assia Mahboubi
• Participants:
  Alexey Solovyev, Andrea Asperti, Assia Mahboubi, Cyril Cohen, Enrico Tassi, François Garillot, Georges Gonthier, Ioana Pasca, Jeremy Avigad, Laurence Rideau, Laurent Théry, Russell O'Connor, Sidi Ould Biha, Stéphane Le Roux, Yves Bertot

6.1.4 Math-comp-analysis

• Name:
  Mathematical Components Analysis
• Keyword:
  Proof assistant
• Functional Description:
  This library adds definitions and theorems for real numbers and their mathematical structures
• Release Contributions:
  Compatible with MathComp 1.12.0, and Coq 8.11, 8.12 and 8.13.
• News of the Year:
  In 2020, we unified norms and absolute values, added a topology and pseudo-metric on the extended reals, and provided a more complete theory about sequences and measure theory.
• URL:
  https://­github.­com/­math-comp/­analysis
• Publication:
  hal-01719918
• Contact:
  Cyril Cohen
• Participants:
  Cyril Cohen, Georges Gonthier, Marie Kerjean, Assia Mahboubi, Damien Rouhling, Laurence Rideau, Pierre-Yves Strub, Reynald Affeldt
• Partners:
  Ecole Polytechnique, AIST Tsukuba

6.1.5 MetaCoq

• Keyword:
  Coq
• Scientific Description:

  The MetaCoq project aims to provide a certified meta-programming environment in Coq. It builds on Template-Coq, a plugin for Coq originally implemented by Malecha (Extensible proof engineering in intensional type theory, Harvard University, 2014), which provided a reifier for Coq terms and global declarations, as represented in the Coq kernel, as well as a denotation command. Recently, it was used in the CertiCoq certified compiler project (Anand et al., in: CoqPL, Paris, France, 2017), as its front-end language, to derive parametricity properties (Anand and Morrisett, in: CoqPL’18, Los Angeles, CA, USA, 2018). However, the syntax lacked semantics, be it typing semantics or operational semantics, which should reflect, as formal specifications in Coq, the semantics of Coq ’s type theory itself. The tool was also rather bare bones, providing only rudimentary quoting and unquoting commands. MetaCoq generalizes it to handle the entire polymorphic calculus of cumulative inductive constructions, as implemented by Coq, including the kernel’s declaration structures for definitions and inductives, and implement a monad for general manipulation of Coq’s logical environment. The MetaCoq framework allows Coq users to define many kinds of general purpose plugins, whose correctness can be readily proved in the system itself, and that can be run efficiently after extraction. Examples of implemented plugins include a parametricity translation and a certified extraction to call-by-value lambda-calculus. The meta-theory of Coq itself is verified in MetaCoq along with verified conversion, type-checking and erasure procedures providing higly trustable alternatives to the procedures in Coq's OCaml kernel. MetaCoq is hence a foundation for the development of higher-level certified tools on top of Coq's kernel. A meta-programming and proving framework for Coq.

  MetaCoq is made of 4 main components: - The entry point of the project is the Template-Coq quoting and unquoting library for Coq which allows quotation and denotation of terms between three variants of the Coq AST: the OCaml one used by Coq's kernel, the Coq one defined in MetaCoq and the one defined by the extraction of the MetaCoq AST, allowing to extract OCaml plugins from Coq implementations. - The PCUIC component is a full formalization of Coq's typing and reduction rules, along with proofs of important metatheoretic properties: weakening, substitution, validity, subject reduction and principality. The PCUIC calculus differs slightly from the Template-Coq one and verified translations between the two are provided. - The checker component contains verified implementations of weak-head reduction, conversion and type inference for the PCUIC calculus, along with a verified checker for Coq theories. - The erasure compoment contains a verified implementation of erasure/extraction from PCUIC to untyped (call-by-value) lambda calculus extended with a dummy value for erased terms.

• Functional Description:
  MetaCoq is a framework containing a formalization and verified implementation of Coq's kernel in Coq along with a verified erasure procedure. It provides tools for manipulating Coq terms and developing certified plugins (i.e. translations, compilers or tactics) in Coq.
• Release Contributions:
  This version is a beta-release including a fully-functional reification and denotation support, and the verified type-checking and erasure procedures. The metatheory proofs are not entirely completed.
• News of the Year:
  The verification of Coq's typechecking and conversion algorithm was completed, resulting in a publication at POPL'20. During this year we improved the erasure procedure, verified completeness in addition to soundness of the conversion algorithm and completed the subject reduction and principality proofs for the PCUIC calculus. MetaCoq was used to show the confluence and subject reduction of an extension of Coq with rewrite rules, presented in an article at POPL'21.
• URL:
  https://­metacoq.­github.­io
• Publications:
  hal-02901011, hal-02380196, hal-02167423, hal-01809681
• Contact:
  Matthieu Sozeau
• Participants:
  Abhishek Anand, Danil Annenkov, Meven Bertrand, Jakob Botsch Nielsen, Simon Boulier, Cyril Cohen, Yannick Forster, Kenji Maillard, Gregory Malecha, Matthieu Sozeau, Nicolas Tabareau, Theo Winterhalter
• Partners:
  Concordium Blockchain Research Center, Aarhus University, Denmark, Saarland University

6.1.6 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:

  Coq version 8.14 integrates many usability improvements, as well as an important change in the core language. The main changes include:

  - The internal representation of match has changed to a more space-efficient and cleaner structure, allowing the fix of a completeness issue with cumulative inductive types in the type-checker. The internal representation is now closer to the user-level view of match, where the argument context of branches and the inductive binders in and as do not carry type annotations.

  - A new coqnative binary performs separate native compilation of libraries, starting from a .vo file. It is supported by coq_makefile.

  - Improvements to typeclasses and canonical structure resolution, allowing more terms to be considered as classes or keys.

  - More control over notations declarations and support for primitive types in string and number notations.

  - Removal of deprecated tactics, notably omega, which has been replaced by a greatly improved lia, along with many bug fixes.

  - New Ltac2 APIs for interaction with Ltac1, manipulation of inductive types and printing.

  Many changes and additions to the standard library in the numbers, vectors and lists libraries. A new signed primitive integers library Sint63 is available in addition to the unsigned Uint63 library.

• News of the Year:
  Coq version 8.14 integrates many usability improvements, as well as an important change in the core language. See the changelog at https://coq.inria.fr/refman/changes.html#version-8-14 for an overview of the new features and changes, along with the full list of contributors.
• URL:
  http://­coq.­inria.­fr/
• Contact:
  Matthieu Sozeau
• Participants:
  Yves Bertot, Frederic Besson, Tej Chajed, Cyril Cohen, Pierre Corbineau, Pierre Courtieu, Maxime Denes, Jim Fehrle, Julien Forest, Emilio Jesús Gallego Arias, Gaetan Gilbert, Georges Gonthier, Benjamin Grégoire, Jason Gross, Hugo Herbelin, Vincent Laporte, Olivier Laurent, Assia Mahboubi, Kenji Maillard, Érik Martin-Dorel, Guillaume Melquiond, Pierre-Marie Pedrot, 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

6.1.7 memprof-limits

• Keyword:
  Library
• Scientific Description:

  Memprof-limits is an implementation of per-thread global memory limits, and per-thread allocation limits à la Haskell, and CPU-bound thread cancellation, for OCaml, compatible with multiple threads.

  Memprof-limits interrupts the execution by raising an asynchronous exception: an exception that can arise at almost any location in the program. It is provided with a guide on how to recover from asynchronous exceptions and other unexpected exceptions, summarising for the first time practical knowledge acquired in OCaml by the Coq proof assistant as well as in other programming languages.

  Memprof-limits is probabilistic, as it is based on the statistical memory accountant memprof. It is provided with a statistical analysis that the user can rely on to have guarantees about the enforcement of limits.

• Functional Description:

  Memprof-limits is an implementation of (per-thread) global memory limits, (per-thread) allocation limits, and cancellation of CPU-bound threads, for OCaml. Memprof-limits interrupts a computation by raising an exception asynchronously and offers features to recover from them such as interrupt-safe resources.

  It is provided with an extensive documentation with examples which explains what must be done to ensure one recovers from an interrupt. This documentation summarises for the first time the experience acquired in OCaml in the Coq proof assistant, as well as in other situations in other programming languages.

• Release Contributions:
  Initial version.
• News of the Year:
  Version 0.2.0, first official and supported (non-prototype) release.
• URL:
  https://­gitlab.­com/­gadmm/­memprof-limits
• Publication:
  hal-03517592
• Author:
  Guillaume Munch
• Contact:
  Guillaume Munch

6.1.8 ocaml-boxroot

• Keywords:
  Interoperability, Library, Ocaml, Rust
• Scientific Description:
  Boxroot is an implementation of roots for the OCaml GC based on concurrent allocation techniques. These roots are designed to support a calling convention to interface between Rust and OCaml code that reconciles the latter's foreign function interface with the idioms from the former.
• Functional Description:
  Boxroot implements fast movable roots for OCaml in C. A root is a data type which contains an OCaml value, and interfaces with the OCaml GC to ensure that this value and its transitive children are kept alive while the root exists. This can be used to write programs in other languages that interface with programs written in OCaml.
• URL:
  https://­gitlab.­com/­ocaml-rust/­ocaml-boxroot
• Contact:
  Guillaume Munch
• Participants:
  Guillaume Munch, Gabriel Scherer

7.1.1 Type Theory

7.1.2 Proof Assistants

Many of our work on proof assistants are based on the MetaCoq project23.

Games, mobile processes, and functions.

In 21, we establish a tight connection between two models of the λ-calculus, namely Milner's encoding into the π-calculus (precisely, the Internal π-calculus), and operational game semantics (OGS). We first investigate the operational correspondence between the behaviours of the encoding provided by π and OGS. We do so for various LTSs: the standard LTS for π and a new 'concurrent' LTS for OGS; an 'output-prioritised' LTS for π and the standard alternating LTS for OGS. We then show that the equivalences induced on λ-terms by all these LTSs (for π and OGS) coincide. These connections allow us to transfer results and techniques between π and OGS. In particular we import up-to techniques from π onto OGS and we derive congruence and compositionality results for OGS from those of π. The study is illustrated for call-by-value; similar results hold for call-by-name.

Complete trace models of state and control.

In 19, we consider a hierarchy of four typed call-by-value languages with either higher-order or ground-type references and with either callcc or no control operator. Our first result is a fully abstract trace model for the most expressive setting, featuring both higher-order references and callcc, constructed in the spirit of operational game semantics. Next we examine the impact of suppressing higher-order references and callcc in contexts and provide an operational explanation for the game-semantic conditions known as visibility and bracketing respectively. This allows us to refine the original model to provide fully abstract trace models of interaction with contexts that need not use higher-order references or callcc. Along the way, we discuss the relationship between error- and termination-based contextual testing in each case, and relate the two to trace and complete trace equivalence respectively. Overall, the paper provides a systematic development of operational game semantics for all four cases, which represent the state-based face of the so-called semantic cube.

Compositional relational reasoning via operational game semantics.

In 20, 27, we show how to use operational game semantics as a guide to develop relational techniques for establishing contextual equivalences with respect to contexts drawn from a hierarchy of four call-by-value higher-order languages: with either general or ground-type references and with either call/cc or no control operator. In game semantics, differences between the contexts can be captured by the absence or presence of the O-visibility and O-bracketing conditions.The proposed technique, which we call Kripke normal-form bisimulations, combines insights from normal-form bisimulation and Kripke logical relations with game semantics. In particular, the role of the heap and the name history is abstracted away using Kripke-style world transition systems. The differences between the four kinds of contexts manifest themselves through simple local conditions that can be shown to correspond to O-visibility and O-bracketing, as applicable.The technique is sound and complete by virtue of correspondence with operational game semantics. Moreover, it sheds a new light on other related developments, such as backtracking and private transitions in Kripke logical relations, which can be related to specific phenomena in game models.

Theorems for free from separation logic specifications.

Separation logic specifications with abstract predicates intuitively enforce a discipline that constrains when and how calls may be made between a client and a library. Thus a separation logic specification of a library intuitively enforces a protocol on the trace of interactions between a client and the library. In 10, we show how to formalize this intuition and demonstrate how to derive free theorems about such interaction traces from abstract separation logic specifications. We present several examples of free theorems. In particular, we prove that a so-called logically atomic concurrent separation logic specification of a concurrent module operation implies that the operation is linearizable. All the results presented in this paper have been mechanized and formally proved in the Coq proof assistant using the Iris higher-order concurrent separation logic framework.

Temporal Refinements for Guarded Recursive Types.

In 24, 28, we propose a logic for temporal properties of higher-order programs that handle infinite objects like streams or infinite trees, represented via coinductive types. Specifications of programs use safety and liveness properties. Programs can then be proven to satisfy their specification in a compositional way, our logic being based on a type system. The logic is presented as a refinement type system over the guarded λ-calculus, a λ-calculus with guarded recursive types. The refinements are formulae of a modal μ-calculus which embeds usual temporal modal logics. The semantics of our system is given within a rich structure, the topos of trees, in which we build a realizability model of the temporal refinement type system.

Models of programming languages mixing effects and resources: Resource management in OCaml.

One goal of resource management is to ensure the correctness of programs in the face of failures and interruptions. We have proposed a model for asynchronous interrupt-safety in OCaml and developed a library based on this model that implements resource limits and thread cancellation for OCaml 25.

A Formal Proof of the Irrationality of $\zeta \left(3\right)$.

In 13, we present a complete formal verification of a proof that the evaluation of the Riemann zeta function at 3 is irrational, using the Coq proof assistant. This result was first presented by Apéry in 1978, and the proof we have formalized essentially follows the path of his original presentation. The crux of this proof is to establish that some sequences satisfy a common recurrence. We formally prove this result by an a posteriori verification of calculations performed by computer algebra algorithms in a Maple session. The rest of the proof combines arithmetical ingredients and asymptotic analysis, which we conduct by extending the Mathematical Components libraries.

Unsolvability of the Quintic Formalized in Dependent Type Theory.

In 17, we describe an axiom-free Coq formalization that there does not exist a general method for solving by radicals polynomial equations of degree greater than 4. This development includes a proof of Galois’ Theorem of the equivalence between solvable extensions and extensions solvable by radicals. The unsolvability of the general quintic follows from applying this theorem to a well chosen polynomial with unsolvable Galois group.

Machine-checked computer-aided mathematics.

Proof assistants are pieces of software designed for the realization of digital libraries of formalized mathematics. The latter libraries contain definitions, statements, and proofs, all formalized in a fixed variant of logic. In particular, the verification of the well-formedness of statements, and of the correctness of proofs, boils down to a mechanical process, associated with the underlying logical formalism. The kernel of a proof assistant is the software component which performs this verification, while the actual proof assistant implements a collection of automation techniques, which allow users to conduct in practice the formalization of arbitrarily sophisticated mathematical definitions and theories. The memoir 26 presents an overview of three main contributions to the formal verification of mathematical theories in dependent type theory. The first of these contributions deals with the realization of a library of digitized mathematics covering the standard undergraduate background in algebra, as well as some more advanced chapters in finite group theory. The two other contributions are related to the issues pertaining to the formal verification of computational mathematical proofs, by the means of symbolic algorithms and of rigorous numerical methods respectively.

9.1.1 Associate Teams in the framework of an Inria International Lab or in the framework of an Inria International Program

9.1.2 Participation in other International Programs

A. Mahboubi holds a part-time endowed professor position in the Department of Mathematics at the Vrije Universiteit Amsterdam (the Netherlands).

9.2.1 Visits of international scientists

9.3.1 FP7 & H2020 projects

NUSCAP

• Title:
  Numerical Safety for Computer-Aided Proofs
• Program:
  ANR AAPG2020,
• Type:
  PRC, CES 48
• Duration:
  Feb 2021 - Jan 2024
• Coordinator:
  UMR CNRS - ENS Lyon - UCB Lyon 1 - INRIA 5668
• Local Contact:
  Assia Mahboubi
• Summary:
  The last twenty years have seen the advent of computer-aided proofs in mathematics and this trend is getting more and more important. They request various levels of numerical safety, from fast and stable computations to formal proofs of the computations. Hovewer, the necessary tools and routines are usually ad hoc, sometimes unavailable, or inexistent. On a complementary perspective, numerical safety is also critical for complex guidance and control algorithms, in the context of increased satellite autonomy. We plan to design a whole set of theorems, algorithms and software developments, that will allow one to study a computational problem on all (or any) of the desired levels of numerical rigor. Key developments include fast and certified spectral methods and polynomial arithmetic, with subsequent formal verifications. There will be a strong feedback between the development of our tools and the applications that motivate it.

ReCiProg

• Title:
  Reasoning on Circular proofs for Programming
• Program:
  ANR AAPG2021,
• Type:
  PRC, CES 48
• Duration:
  Jan 2022 - Jan 2025
• Coordinator:
  UMR CNRS - IRIF - Université de Paris
• Local Contact:
  Guilhem Jaber
• Summary:
  ReCiProg is a collaborative project (Lyon-Marseille-Nantes-Paris) aiming at extending the proofs-as-programs correspondence (also known as Curry-Howard correspondence) to recursive programs and circular proofs for logic and type systems using induction and coinduction. The project will contribute both to the necessary theoretical foundations of circular proofs and to the software development allowing to enhance the use of coinductive types and coinductive reasoning in the Coq proof assistant: such coinductive types present, in the current state of the art serious defects that the project will aim at solving.

DyVerSe

• Title:
  Dynamic Versatile Semantics
• Program:
  ANR AAPG2019,
• Type:
  PRC, CES 48
• Duration:
  Jan 2020 - Dec 2023
• Coordinator:
  Pierre Clairambault (CR CNRS, LIP, UMR 5668)
• Local Contact:
  Guillaume Munch-Maccagnoni
• Summary:
  DyVerSe aims to develop a theoretical framework for dynamic/game semantics for programming languages, capturing in one versatile setting a spectrum of computational features, representative of the heterogeneity of software (e.g. higher-order functions, concurrency, probabilities or other quantitative aspects). Our ambition is (1) to help unify denotational semantics by providing the missing link between various incompatible models focusing on specific aspects, and (2) to provide a toolbox to reason compositionally about the dynamic behaviour of programs, with an eye towards specification and verification.

Vercoma

• Atlanstic 2020/Attractivity grant
• Goal:
  Verified computer mathematics
• Coordinator:
  Assia Mahboubi.
• Duration:
  08/2018 - 12/2021.

ASCOC

• Atlanstic 2020/Amorçage grant
• Goal:
  Compositional Analysis of OCaml code
• Coordinator:
  G. Jaber.
• Duration:
  09/2020 - 12/2022

10.1.1 Scientific events: organisation

10.1.2 Scientific events: selection

10.1.3 Journal

10.1.4 Invited talks

• Matthieu Sozeau gave invited talks at the LFMTP'21 workshop and the HoTTest seminar on MetaCoq.
• Assia Mahboubi gave invited talks at the CSL'21 conference, at the FM'21 conference, at the Journées du GDR Informatique Mathématique, at the colloquium of mathematics of the university of Ljubjlana (Slovenia), at the chocola seminar (Lyon, France), at the colloquium of the School of Computer and Cyber Sciences at university of Alberta (USA), at the OWLS on-line seminar.
• Pierre-Marie Pédrot gave an invited talk at TYPES 2021.

10.1.5 Leadership within the scientific community

• Matthieu Sozeau has been elected as a member of the TYPES conference Steering Committee.
• Assia Mahboubi is a member of the steering committee of the CPP conference.

10.2.1 Teaching

• Licence : Julien Cohen, Discrete Mathematics, 48h, L1 (IUT), IUT Nantes, France
• Licence : Julien Cohen, Introduction to proof assistants (Coq), 8h, L2 (PEIP : IUT/Engineering school), Polytech Nantes, France
• Licence : Julien Cohen, Functional Programming (Scala), 22h, L2 (IUT), IUT Nantes, France
• Master : Julien Cohen, Object oriented programming (Java), 32h, M1 (Engineering school), Polytech Nantes, France
• Master : Julien Cohen, Functional programming (OCaml), 18h, M1 (Engineering school), Polytech Nantes, France
• Master : Julien Cohen, Tools for software engineering (proof with Frama-C, test, code management), 20h, M1 (Engineering school), Polytech Nantes, France
• Licence : Rémi Douence, Object Oriented Design and Programming, 45h, L1 (engineers), IMT-Atlantique, Nantes, France
• Licence : Rémi Douence, Object Oriented Design and Programming Project, 30h, L1 (apprenticeship), IMT-Atlantique, Nantes, France
• Master : Rémi Douence, Functional Programming with Haskell, 45h, M1 (engineers), IMT-Atlantique, Nantes, France
• Master : Rémi Douence, Functional Programming with Haskell, 20h, M1 (apprenticeship), IMT-Atlantique, Nantes, France
• Master : Rémi Douence, Formal Methods: Model checking with Alloy and from Haskell to Coq, 11h, M1 (apprenticeship), IMT-Atlantique, Nantes, France
• Master : Rémi Douence, Introduction to scientific research in computer science (Project: compilation in Java of Haskell Class Types), 45h, M2 (apprenticeship), IMT-Atlantique, Nantes, France
• Licence : Hervé Grall, Algorithms and Discrete Mathematics, 25h , L3 (engineers), IMT-Atlantique, Nantes, France
• Licence : Hervé Grall, Object Oriented Design and Programming, 25h , L3 (engineers), IMT-Atlantique, Nantes, France
• Licence, Master : Hervé Grall, Modularity and Typing, 40h, L3 and M1, IMT-Atlantique, Nantes, France
• Master : Hervé Grall, Service-oriented Computing, 40h, M1 and M2, IMT-Atlantique, Nantes, France
• Master : Hervé Grall, Research Project - (Linear) Logic Programming in Coq, 90h (1/3 supervised), M1 and M2, IMT-Atlantique, Nantes, France
• Licence : Guilhem Jaber, Computer Tools for Science, 18h, L1, Université de Nantes France
• Licence : Guilhem Jaber, Foundations of Computer Science, 54h, L3, Université de Nantes France
• Licence : Guilhem Jaber, Logic in Computer Science, 48h, L2, Université de Nantes France
• Licence : Guilhem Jaber, Functional Programming, 36h, L3, Université de Nantes France
• Master : Guilhem Jaber, Verification and Formal Proofs, 18h, M1, Université de Nantes, France
• Master : Nicolas Tabareau, Homotopy Type Theory, 24h, M2 LMFI, Université Paris Diderot, France
• Master: Assia Mahboubi, Machine-Checked Mathematics, 22.5h, M2, Vrije Universiteit Amsterdam, the Netherlands
• Master : Matthieu Sozeau, Proof Assistants, 24h, M2 MPRI, Université Paris Diderot, France

10.2.2 Supervision

• PhD in progress: Xavier Montillet, Rewriting and solvability for Call-by-push-value, Univ Nantes, advisors: Guillaume Munch-Maccagnoni and Nicolas Tabareau
• PhD in progress: Joachim Hotonnier, Deep Specification for Domain-Specific Modelling, advisors: Gerson Sunye (Naomod team), Massimo Tisi (Naomod team), Hervé Grall.
• PhD in progress: Loïc Pujet, Giving meaning to cubical type theory using forcing, Univ Nantes, advisors: Nicolas Tabareau
• PhD in progress: Meven Bertrand, Gradualizing the calculus of constructions, Univ Nantes, advisors: Nicolas Tabareau
• PhD in progress: Martin Baillon, Syntactic Models of Type Theory and Continuity Principles, Univ Nantes, advisors: Assia Mahboubi and Pierre-Marie Pédrot
• PhD in progress: Pierre Benjamin Giraud, Formalizing extraction of Coq to OCaml, Univ Nantes, advisors: Pierre-Marie Pédrot, Matthieu Sozeau and Nicolas Tabareau
• PhD in progress: Enzo Crance, Automated theorem proving and dependent types: automated reasoning for interactive proof assistants, Univ Nantes, advisors: Denis Cousineau and Assia Mahboubi
• PhD in progress: Antoine Allioux, Coherent Higher Structures in Homotopy Type Theory, Univ Paris Diderot, advisors: Pierre-Louis Curien (Univ. Paris Diderot), Eric Finster (Univ. Birmingham) and Matthieu Sozeau
• PhD in progress: Christopher Hughes, Transport of mathematical properties in type theory: structures, morphisms, refinements, Univ Nantes, advisors: Cyril Cohen (Inria Sophia Antipolis Méditerranée) and Assia Mahboubi
• PhD in progress: Hamza Jaafar, Operational Game Semantics for OCaml, Univ. Nantes, advisor: Guilhem Jaber and Nicolas Tabareau
• PhD in progress: Peio Borthelle, Formalized Game Semantics for Interoperability, Univ. Savoie Mont-Blanc, advisor: Tom Hirschowitz and Guilhem Jaber

10.3.1 Interventions

• Guilhem Jaber and Assia Mahboubi participated to an artists residence at the Athenor theater (Saint-Nazaire, France). They participated to three performances in the frame of the Instants Fertiles $♯9$ festival.

International journals

• 10 articleL.Lars Birkedal, T.Thomas Dinsdale-Young, A.Armaël Guéneau, G.Guilhem Jaber, K.Kasper Svendsen and N.Nikos Tzevelekos. Theorems for free from separation logic specifications.Proceedings of the ACM on Programming Languages5ICFPAugust 2021, 1-29
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• 11 articleJ.Jesper Cockx, N.Nicolas Tabareau and T.Théo Winterhalter. The Taming of the Rew: A Type Theory with Computational Assumptions.Proceedings of the ACM on Programming Languages2021
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• 12 articleM.Meven Lennon-Bertrand, K.Kenji Maillard, N.Nicolas Tabareau and É.Éric Tanter. Gradualizing the Calculus of Inductive Constructions.ACM Transactions on Programming Languages and Systems (TOPLAS)2022
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• 13 articleA.Assia Mahboubi and T.Thomas Sibut-Pinote. A formal proof of the irrationality of ζ(3).Logical Methods in Computer ScienceFebruary 2021
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• 14 articleL.Loïc Pujet and N.Nicolas Tabareau. Observational Equality: Now For Good.Proceedings of the ACM on Programming Languages6POPLJanuary 2022
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• 15 articleN.Nicolas Tabareau, É.Éric Tanter and M.Matthieu Sozeau. The Marriage of Univalence and Parametricity.Journal of the ACM (JACM)681January 2021, 1-44
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International peer-reviewed conferences

• 16 inproceedingsM.Martin Baillon, A.Assia Mahboubi and P.-M.Pierre-Marie Pédrot. Gardening with the PythiaA model of continuity in a dependent setting.Computer Science LogicGöttingen, GermanyFebruary 2022
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• 17 inproceedingsS.Sophie Bernard, C.Cyril Cohen, A.Assia Mahboubi and P.-Y.Pierre-Yves Strub. Unsolvability of the Quintic Formalized in Dependent Type Theory.ITP 2021 - 12th International Conference on Interactive Theorem ProvingRome / Virtual, FranceJune 2021
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• 18 inproceedingsE.Eric Finster, A.Antoine Allioux and M.Matthieu Sozeau. Types are internal infinity-groupoids.LICS 2021Rome, ItalyJune 2021
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• 19 inproceedingsG.Guilhem Jaber and A. S.Andrzej S Murawski. Complete trace models of state and control.ESOP 2021 - 30th European Symposium on Programming12648Lecture Notes in Computer ScienceLuxembourg, LuxembourgSpringer International PublishingMarch 2021, 348-374
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• 20 inproceedingsG.Guilhem Jaber and A.Andrzej Murawski. Compositional relational reasoning via operational game semantics.LICS 2021 - 36th Annual ACM/IEEE Symposium on Logic in Computer ScienceProceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science23Rome, ItalyIEEEJune 2021, 1-13
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• 21 inproceedingsG.Guilhem Jaber and D.Davide Sangiorgi. Games, mobile processes, and functions.30th EACSL Annual Conference on Computer Science Logic (CSL 2022).Göttingen, GermanyOctober 2021
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• 22 inproceedingsM.Meven Lennon-Bertrand. Complete Bidirectional Typing for the Calculus of Inductive Constructions.12th International Conference on Interactive Theorem Proving (ITP 2021)193Leibniz International Proceedings in Informatics (LIPIcs)24Rome, ItalyJune 2021, 24:1--24:19
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• 23 inproceedingsM.Matthieu Sozeau. Touring the MetaCoq Project (Invited Paper).LFMTP 2021 - Logical Frameworks and Meta-Languages: Theory and PracticePittsburg, United StatesJuly 2021
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Conferences without proceedings

• 24 inproceedingsG.Guilhem Jaber and C.Colin Riba. Temporal Refinements for Guarded Recursive Types.ESOP 2021 - 30th European Symposium on Programming12648Lecture Notes in Computer Science2021-04-01, LuxembourgSpringer International PublishingMarch 2021, 548-578
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• 25 inproceedingsG.Guillaume Munch-Maccagnoni. Probabilistic resource limits using StatMemprof.OCaml Workshop 2021Online, FranceAugust 2021, 2
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Doctoral dissertations and habilitation theses

• 26 thesisA.Assia Mahboubi. Machine-checked computer-aided mathematics.Université de Nantes (UN), Nantes, FRA.January 2021
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Reports & preprints

• 27 miscG.Guilhem Jaber and A. S.Andrzej S. Murawski. Complete trace models of state and control (full version).January 2021
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• 28 miscG.Guilhem Jaber and C.Colin Riba. Temporal Refinements for Guarded Recursive Types (full version).January 2021
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• 29 miscK.Kenji Maillard, N.Nicolas Margulies, M.Matthieu Sozeau, N.Nicolas Tabareau and É.Éric Tanter. The Multiverse: Logical Modularity for Proof Assistants.August 2021
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• 30 miscT.Théo Zimmermann, J.Julien Coolen, J.Jason Gross, P.-M.Pierre-Marie Pédrot and G.Gaëtan Gilbert. Extending the team with a project-specific bot.December 2021
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