2024Activity reportProject-TeamPICUBE

3.1.1 Towards the calculus of constructions

The $\lambda$-calculus, defined by Church  55, 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  37.

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  84.

In 1985, Coquand and Huet  58, 56 in the Formel team of INRIA-Rocquencourt explored an alternative approach based on Girard-Reynolds' system $F$  68, 90. 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.

3.1.2 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  57 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).

3.6.1 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.

3.6.2 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.

3.6.3 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.

3.6.4 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.

3.6.5 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.

3.6.6 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.

3.6.7 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.

3.7.1 Formalisation of mathematics.

We will work for this in a proactive way, in close collaboration with IRIF and IMJ-PRG gathering motivated mathematicians and computer scientists willing to formalise the mathematics they teach at the University, or on which they conduct their research work. Indeed, we aim at formalising classical mathematics curricula, pieces of contemporary mathematics as well as mathematical tools implemented in theoretical computer science. We will draw inspiration from the usually heuristic practice of mathematics, aiming to make the writing and reading of Coq documents altogether more intuitive, more straightforward and more flexible. This software development and mechanisation work will be combined with a study of the linguistic structure of mathematical documents, in collaboration with Philippe de Groote (Sémagramme): we think that there is a need to better understand the linguistic structures at work in the daily life of a mathematician and their formal nature.

More specifically, among the subjects of formalisation of mathematics we plan to carry in the team, three in particular should be mentioned:

• On the mathematical side, formalisation of parts of contemporary mathematics, in connection with members of the IMJ-PRG.
• On the CS side, we plan to formalise and certify algorithms on graphs (Jean-Jacques Lévy), with the aim of establishing links with other IRIF members working on graph algorithms, in particular with Laurent Viennot; we also intend to certify developments in the theory of automata and formal languages ​​or proof theory as used at IRIF, in particular with Thomas Colcombet.
• We also plan to contribute to an ambitious formalisation project aiming at designing a large Coq mathematical library covering undergraduate mathematics. This project is of course of a broader scope and will involve other research teams, within INRIA or elsewhere in France and abroad.

This research topic will therefore be based at the same time on a formalisation activity internal to the team, and on a long-term work of animation and construction of a scientific community involved in formalisation. We plan to contribute to the Coq training of our maths and CS colleagues (from PhD students to post-docs and those holding a permanent position), and not only grad students as it is more commonly the case, in particular through the organisation of regular sessions of working groups dedicated to helping colleagues in their formalisation tasks and by considering the opportunity to set up thematic schools in collaboration with the other teams of the institute contributing to Coq. A medium-term objective could be to achieve that some pure mathematics modules are based directly on formalised content and that some mathematics tutorials take the form of Coq exercise sessions, starting from existing works on Coq 41, 88, 54. This will require to develop close collaborations with the different communities of other proof assistants, especially those designed to be well-adapted to the formalisation of mathematics (Lean, Isabelle and Agda in particular). Understanding linguistic aspects of mathematical proofs will be a key to the success of our project.

3.7.2 Linguistics of proofs.

Formalising inevitably leads to a shift of perspective on what a mathematical proof is. From a mathematical standpoint, it is conventional, even if developments are emerging, to be satisfied of the mere possibility of a formalisation (which is never even sketched out) typically in the set-theoretical formalism, and to focus on the construction of a natural language discourse that is precise enough to convince the reader. From the proof assistant standpoint, the seminal vision which initially aimed only at making effective this virtuality, tends to evolve. New lines of communication and convergence are emerging: a proof is no longer strictly made up of logical inference rules, such as introducing or eliminating a connective, but it is made of more abstract entities such as the use of a lemma, the replacement of equals by equals, reasoning by induction, simplifying a polynomial expression, decomposing a formula in atoms, etc. With the arrival of a new generation of interactive proof engines 94, 38, a proof is no longer seen strictly as a tree derivation, but as a graph whose nodes can be refined with a reasonable degree of freedom; moreover, the order in which these transformations are applied interactively appears in the “incremental” format of the machine maths document. This is in line with the historical evolution of proof methods in order to escape from the “low level” of logic and get closer to more abstract conceptual levels used by humans. Our investigations will follow this line as we plan to analyse the linguistic structure of mathematical texts, with the ambition to develop the levels of abstraction which would eventually allow a direct formal understanding of a mathematical text at the level of mathematical discourse in which it is expressed. Examples of the sorts of linguistic structures we plan to analyse are “reasoning by analogy”, reasoning modulo isomorphism, or even modulo inclusion, and of course reasoning modulo general equational theories in general.

On this natural (mathematical) language processing part, we plan a collaboration with Philippe de Groote of the Sémagramme team (Inria Nancy & Loria), to identify the necessary structures for a flexible formalisation of vernacular mathematical proofs in order to implement this linguistic structure within Coq, procedural and discursive in nature rather than tree-like as described above. One of the objectives will be to be able to formalise in a more transparent and direct way mathematical texts, such Bourbaki's Éléments de mathématiques.

Regarding the design and development of a general mathematical library, it is undoubtedly too early to describe the directions we will take, but we have some ongoing discussions with Assia Mahboubi (Gallinette), Yves Bertot and Cyril Cohen (Scalp) around these essential questions. We also wish to develop the team's scientific interactions with Michael Soegtrop (Apple) and we closely follow his projects on proving algorithms of symbolic computing, around constructive reals and the formal integrator Rubi. We are also in contact with mathematicians from IMJ-PRG, and in particular with Antoine Chambert-Loir, who expressed a keen interest in formalisation.

We want to develop fruitful discussions with the communities of other proof assistants, such as Agda, Isabelle or Lean.

3.8.1 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 35) 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 60).

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 79, 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.

3.8.2 Higher-dimensional rewriting

Higher-dimensional categories are algebraic structures that contain, in essence, computational aspects. This has been recognised by Street 95, and independently by Burroni 48, 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 82, 83, 73, 74, and beyond: Petri nets 76 and formal proofs of classical and linear logic have been expressed in this framework 75. 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.

7.1.1 Coq

• Name:
  The Coq Proof Assistant
• Keyword:
  Proof assistant
• 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 integrated development environment (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:
  An overview of the new features and changes, along with the full list of contributors is available at https://coq.inria.fr/refman/changes.html#version-8-20 .
• News of the Year:
  Coq version 8.20 adds a new rewrite rule mechanism along with a few new features, a host of improvements to the virtual machine, the notation system, Ltac2 and the standard library.
• URL:
  http://­coq.­inria.­fr/
• Contact:
  Matthieu Sozeau
• Participants:
  Yves Bertot, Frédéric Besson, Tej Chajed, Cyril Cohen, Pierre Corbineau, Pierre Courtieu, Maxime Dénès, 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, Erik 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, Theo Zimmermann

7.1.2 coq-serapi

• Keywords:
  Interaction, Coq, Ocaml, Data centric, User Interfaces, GUI (Graphical User Interface), Toolkit
• Scientific 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).
• 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 for Coq 8.15 -
• URL:
  https://­github.­com/­ejgallego/­coq-serapi
• Publication:
  hal-01384408
• Contact:
  Emilio Jesus Gallego Arias
• Participants:
  Karl Palmskog, Theo Zimmermann, Shachar Itzhaky, Jason Gross
• Partner:
  KTH Royal Institute of Technology

7.1.3 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:
  - Port to Coq 8.14 - Improved packaging system for libraries - Improved display and interaction - Settings panel
• URL:
  https://­github.­com/­ejgallego/­jscoq
• Publication:
  hal-01425752
• Contact:
  Emilio Jesus Gallego Arias
• Participants:
  Emilio Jesus Gallego Arias, Shachar Itzhaky
• Partners:
  Mines ParisTech, Technion

7.1.4 pyCoq

• Keywords:
  Coq, Python
• Functional Description:
  PyCoq is a set of bindings and libraries allowing to interact with the Coq interactive proof assistant from inside Python 3.
• Release Contributions:
  Initial release
• URL:
  https://­github.­com/­ejgallego/­pycoq
• Contact:
  Emilio Jesus Gallego Arias
• Participant:
  Thierry Martinez

7.1.5 OCaml

• Keywords:
  Programming language, Functional programming, Compilers
• Functional Description:
  The OCaml language is a functional programming language that combines safety with expressiveness through the use of a precise and flexible type system with automatic type inference. The OCaml system is a comprehensive implementation of this language, featuring two compilers (a bytecode compiler, for fast prototyping and interactive use, and a native-code compiler producing efficient machine code for x86, ARM, PowerPC, RISC-V and System Z), a debugger, and a documentation generator. Many other tools and libraries are contributed by the user community and organized around the OPAM package manager.
• URL:
  https://­ocaml.­org/
• Publications:
  hal-04884634, hal-04681703, hal-04794404, hal-03917754, hal-03947986, hal-04407119, hal-03146495, hal-03510931, hal-03145030, hal-01929508, hal-03125031, hal-00772993, hal-00914493, hal-00914560, inria-00074804, hal-01499973, hal-01499946
• Contact:
  Florian Angeletti
• Participants:
  Florian Angeletti, Damien Doligez, Xavier Leroy, Luc Maranget, Gabriel Scherer, David Allsopp, Stephen Dolan, Alain Frisch, Jacques Garrigue, Anil Madhavapeddy, Kc Sivaramakrishnan, Nicolas Ojeda Bar, Leo White

7.1.6 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
• Publication:
  hal-03910313
• Contact:
  Guillaume Munch
• Participants:
  Guillaume Munch, Gabriel Scherer

7.1.7 ProbZelus

• Keywords:
  Probabilistic Programming, Synchronous Language
• Scientific Description:
  ProbZelus is a probabilistic reactive language which provides the facilities of a synchronous language to write control software, with probabilistic constructs to model uncertainties and perform inference-in-the-loop.
• Functional Description:
  ProbZelus is built on top of Zelus a dataflow language à la Scade/Lustre and offers several streaming inference techniques including classic Sequential Monte Carlo (SMC) algorithms and semi-symbolic inference algorithm based on delayed sampling.
• URL:
  https://­github.­com/­IBM/­probzelus
• Contact:
  Guillaume Baudart
• Partners:
  CSAIL, IBM

7.1.8 coq-lsp

• Keywords:
  IDE, Coq, HCI, Interactive Theorem Proving, Program verification
• Functional Description:
  coq-lsp is a Language Server and Visual Studio Code extension for the Coq Proof Assistant. Key features of coq-lsp are: continuous and incremental document checking, advanced error recovery, hybrid Coq/markdown document support, multiple workspace support, positional goals and information panel, performance data, extensible command-line compiler, plugin system, and more.
• URL:
  https://­github.­com/­ejgallego/­coq-lsp
• Contact:
  Emilio Jesus Gallego Arias
• Partner:
  Technion

8.1.1 Proof-theory of logics with induction and coinduction.

In the line of exploring inductive and coinductive reasoning based infinitary logics, akin to on infinite descent reasoning, Saurin and Bauer pursued the study of cut-elimination in the following directions:

• Bauer and Saurin finally completed the above result providing the first syntactic cut-elimination result for a (circular) proof system containing the full modal $\mu$-calculus: to achieve this, they design a linear version of the modal $\mu$-calculus and prove cut-elimination for a family of linear fixed-point logics with super-exponential, building on previous results by Bauer and Laurent. This result not-only allows to cover the case of the modal $\mu$-calculus but also treat most of the known light logics in the literature, extended with least and greatest fixed-points. A paper has been accepted for publication at FoSSaCS 2025 18.

• In the realm of light logics deriving from linear logic by tuning the exponential modalities in order to bound the complexity of cut-elimination, the profusion of proof systems sharing lots of common features induce the need for cut-elimination theorems for each logic whose proofs are often redundant. A number of approaches in proof theory have been adopted to cope with this need, making either the cut-elimination proofs more modular or deriving cut-elimination from the same property of another system by translation. Bauer and Saurin consider this issue from the point of view of enhancing linear logic with least and greatest fixed-points and considering such a variety of exponential connectives.

  They provided a uniform cut-elimination theorem for a parametrized system with fixed-points by combining two approaches: cut-elimination proofs by translation to another system and the identification of sufficient conditions for cut-elimination.

  More precisely, they captured a broad range of systems, taking inspiration from Nigam and Miller's subexponentials and from Bauer and Laurent's super exponentials and proved a uniform proof of cut-elimination for all the systems of this family, including the linear modal $\mu$-calculus that is used in a crucial point of the previous item or systems closed to Baillot’s work on light logics with recursive types. A pre-print has been written 17.

• While a limitation of previous proofs of cut-elimination for circular and non-wellfounded proofs for linear logics with fixed-points is that they rely on the notion of multi-cut, that abstracts a series of consecutive cut inferences, and therefore are less fine-grained than cut-reduction dynamics based on the usual cut-inference, Bauer achieved in his PhD manuscrit being soon finalized a proof of cut-elimination for $\mu {\mathrm{𝖫𝖫}}^{\infty }$that does not rely on multi- cut and that captures a wider class of productive cut-reduction sequences.

8.1.2 Reversible computation with inductive types

Chardonnet, Saurin and Valiron submitted an extended journal version 21 of their CSL 2023 paper where they developed a term calculus for a class of type isomorphisms for circular $\mu {\mathrm{𝖬𝖠𝖫𝖫}}^{\infty }$, from which they extract a language for reversible computation with inductive types 51, 52.

8.1.3 Proof relevant interpolation

Analyzing Maehara's method for proving Craig's interpolation theorem, Saurin extracted a “proof relevant” interpolation theorem for first-order LL in the sense that if $\pi$ is a cut-free sequent proof of $A⊢B$, one can find a formula $C$ in the common vocabulary of $A$ and $B$ and proofs ${\pi }_1,{\pi }_2$ of $A⊢C$ and $C⊢B$ respectively such that ${\pi }_1$ composed with ${\pi }_2$ cut-reduces to $\pi$. This ensures that interpolation can be achieved while preserving the denotational content of proofs.

As a direct corollary, he obtained similar proof relevant interpolation results for LJ and LK using linear translations. This refined interpolation was then rephrased in terms of a cut-introduction process synthetizing the interpolant.

Finally, he analyzed the computational content of interpolation by proving and interpolation result for Curien and Herbelin's Duality of Computation. These results were presented at CIBD workshop and at the Logic Colloquium 2024. A pre-print has been written and will soon be submitted.

In the setting of MELL, Saurin, in a collaboration with Fiorillo and Osorio, achieved a direct interpolation in proof-nets, exploiting an informative various of the parsing correctness criterion, they are currently investigating how this can be generalized with Danos' contractibility criterion.

Together with Osorio, he also analyzed how to extend our methodology beyond the wellfounded setting, investigating how to proof-relevantly interpolate $\mu {\mathrm{𝖫𝖫}}^{\infty }$circular strongly valid cut-free pre-proofs.

8.1.4 Coinductive presentation of infinitary rewriting

During the fall 2024, Saurin, together with Cerda, studied the coinductive presentation of infinitary rewriting. In particular they proved the famous Compression Lemma directly in this coinductive setting. Still in an early phase of this work, a preprint is being written.

8.1.5 An operadic account of context-free grammars and languages

Melliès and Zeilberger develop in 85, 86 a categorical framework based on non-symmetric operads (= multicategories) to describe context-free grammars and establish the Chomsky-Schützenberger representation theorem for context-free languages.

In this approach, a context-free grammar is defined as a functor from an operad freely generated by a species of production rules, to an operad of spliced words introduced for the first time in this paper. A notion of automaton on a category $𝒞$ generalising the usual notion of automaton on words is formulated there as a functor from $𝒬\to 𝒞$ satisfying a unique-factorisation-lifting as well as a fibrewise finiteness property, where $𝒬$ describes the category of runs over the automaton.

One main benefit of the approach is that it provides a more high-level and conceptual account of traditional aspects of automata-theory. In particular, the fact that the intersection of a context-free language with a regular language is context-free is derived from the fact that pullbacks computed in the category of operads preserves context-free grammars seen as functors. Similarly, the Chomsky-Schützenberger representation theorem is derived from the observation that the functor which turns a category into its spliced arrow operad has a left adjoint functor which turns any operad to its category of contours. In particular, the fact that the unit of the adjunction transports every operad to its spliced operad of contours plays a fundamental role in the argument.

This work is part of a long-term project of integrating refinement systems, environment machines and automata theory in a single and unified framework.

8.1.6 A description of the indexed-to-fibered translation in type theory

In the continuation of his internship, S. Arambillete described the traditional indexed-to-fibered translation used to interpret type theory in locally closed cartesian category directly within the language of type theory.

Moana Jubert is investigating the notion of indexed presentation of presheaves (such as semi-simplicial sets, simplicial sets, cubical sets) through various categorical constructions: profunctors, 2-categorical version of fibrations and of Bénabou-Roubaud theorem (work in progress).

8.1.7 Explicit universe subtyping in type theory

V. Blazy worked on a variant of type theory with explicit universe subtyping, under the supervision of H. Herbelin and P. Letouzey. V. Blazy gave a talk at the EuroProofNet WG6 in Leuven. V. Blazy and H. Herbelin and P. Letouzey gave talks at the "journées sous-typage" of the "axe transverse LOGIQUES du LIRMM".

8.1.8 Parametric type theory and modal type theory

S. Reboullet had discovered last year that the presheaf model of parametric type theory of Bernardy, Coquand, Moulin failed to justify the definitional isomorphism between $A\to \mathrm{𝖳𝗒𝗉𝖾}$ and $A{\in }_i\mathrm{𝖳𝗒𝗉𝖾}$ present in the syntax of the type theory. She showed that, instead, the model satisfies an isomorphism between $A\to \left(1{\in }_i\mathrm{𝖳𝗒𝗉𝖾}\right)$ and $A{\in }_i\mathrm{𝖳𝗒𝗉𝖾}$ where $1{\in }_i\mathrm{𝖳𝗒𝗉𝖾}$, compared to $\mathrm{𝖳𝗒𝗉𝖾}$, implicitly exhibits the group of permutations between the dimensions available at a given stage of the derivation of a judgement of the theory.

Studying multimodal type theory, she also proposed a new left-Kan-extension kind of presheaf model alternative to the right-Kan-extension model of e.g. Gratzer, Kavvos, Nuyts and Birkedal.

8.1.9 Oriented simplices and oriented cubes

In collaboration with F. Métayer, Herbelin supervised the internship of Manuel Catz on a combinatorial description of Street's orientals that simultaneously builds by induction both oriented simplices and oriented cubes, respectively following a unary and binary parametricity recipe.

8.1.10 A parametricity-based construction of simplicial and cubical sets

R. Ramachandra and Herbelin extended their parametricity-based construction in Coq of semi-simplicial and semi-cubical sets in indexed form 77 to the case of cubical sets and to a variant of simplicial sets with non-standard degeneracies. Indeed, the construction showed that, unexpectedly, what is called degeneracies in simplicial sets is not the equivalent of degeneracies in cubical sets but the equivalent of (so-called) connections. What is equivalent in simplicial sets to degeneracies from cubical sets is instead a form of "induction principle", that is, that any simplex in any dimension satisfies an induction principle stated in the next dimension.

Thanks to a study of the exact dependencies between the different components of the construction, they also simplified the first construction, replacing the need for equational reasoning by computations. This will allow to add further structures to the construction (connections, symmetries, sigma-types, pi-types and universes) more easily.

8.1.11 Proving Gödel's completeness theorem using forcing in direct style

H. Herbelin supervised the internship of Rui Li on designing a formal logic with a mutable reference in which Gödel's completeness theorem can be proved by forcing in "direct style", that by interpreting a proof of the forcing translation of the statement of completeness as if it were a proof of the original statement, but using an explicit memory.

8.1.12 Rigidifying $\infty$-categories into models of homotopy type theory

E. M. Cherradi has carried on his work on the interpretation of homotopy type theory in $\infty$-categories. To that purpose, he developed a construction based on the Yoneda embedding whose aim is to rigidify quasicategories and turn them into models of homotopy type theory. He is currently working on a conjecture by Kapulkin and Lumsdaine stated 81 which expresses that the relative category of locally cartesian closed quasicategories is equivalent up to a Dwyer-Kan equivalence to the relative category of comprehension categories (models of dependent types) with dependent sums, extensional dependent products, and identity types.

8.1.13 Realisability and non-realisability of choice and collection axioms

F. Castro investigated the strength of different variants of choice and collection axioms in intuitionistic and classical second order logic. In particular, using a clever argument, he showed that while the axiom of collection with domain is realisable in intuitionistic second order logic, the corresponding axiom of choice with domain is not.

8.1.14 Crossing the languages of proof theory, programming and topos theory: the case of higher-order logic

Observing that Church's Higher-Order Logic, Girard's System $F_{\omega }$, Zermelo set theory, and elementary topos all (morally) have the logical strength of higher-order logic (when natural numbers are added), Herbelin started to explore how to formulate all of them in a common language. He gave a talk about it in June 2024 at the working group "Logique, Homotopie et Catégories" of the Groupe de Recherche Informatique Fondamentale et Mathématiques of CNRS.

8.1.15 Geometric coherence theorems

In joint work with G. Laplante-Anfossi  8, P.-L. Curien investigated geometric coherence theorems on regular CW-complexes in general, and on polytopes in particular. There, coherence means that any two parallel cellular paths (sequences of consecutive edges) are combinatorially homotopic, meaning that one can move from one to the other by steps consisting in changing a portion of a path fitting on the boundary of a 2-face of the complex, replacing it with its complement (the rest of the boundary). It was Kapranov's somewhat elliptic slogan that Mac Lane's coherence theorem had an instant proof after “translating” it into the “language” of associahedra (a family of polytopes which he described combinatorially and was later realised geometrically). In further work, Laplante-Anfossi and Curien studied coherence in a special family of polytopes, hypergraph polytopes a.k.a. nestohedra, and showed how coherence there can be proved via the methods of term rewriting systems (as was observed for Mac Lane's theorem in G. Huet's 1984 DEA course notes on category theory) 24.

8.1.16 Higher-dimensional propositional calculus

In joint work with A. Bucciarelli, A. Ledda, F. Paoli and A. Salibra, P.-L. Curien investigated a sequent calculus for a presentation of classical logic based on a famlily of $n$-valued versions of the if then else construct. A simple proof of completeness is obtained by noticing that the introduction rules for these connectives are reversible. The work will appear in the Journal of Logic of IGPL (5).

8.1.17 Universal algebra

In joint work with A. Bucciarelli, A. De Faveri and A. Salibra (IRIF), P.-L. Curien revisited Birkhoff's theorem (characterizing the classes of all models of some given equational theory) under the lenses of infinitary algebras (i.e. admitting operations of infinite arity), resulting in a new proof of this old theorem and in some new generalisations of it 19.

8.1.18 Presentations of monoids in Coq

During his short four months postdoc funded by FSMP and Inria (from November 2023 to February 2024), A. Duric made initial steps towards a formalisation of some of the background and results of his PhD thesis on Garside theory, which was considerably developed by the late mathematician Patrick Dehornoy (Université de Caen and associate member of IRIF), and in which presentations enjoy a good notion of normal form generalising the setting of Coxeter groups. Duric is now a postdoc in Ljubljana (working on ring theory).

8.1.19 Linguistic of mathematics

S. Arambillete explored the capabilities of ForThel and Naproche for interpreting mathematics written in natural language. He started to explore the use of plural in the mathematical natural language, as, e.g., in "the vectors are linearly independent" (a property applying to a plural collection) vs "the family of vectors is free" (a property applying to a singular object).

8.2.1 Coherent differentiation

Differential Linear Logic (DiLL) has been introduced by Ehrhard and Regnier in the mid 2000's. This extension of linear logic provides a new interpretation of the exponentials, turning them into a modality related to communication rather than to the sole resource replicability and erasing. DiLL also provided a new understanding of resource calculi and of intersection types, allowing to apply to programs and proofs an operation of approximation which is a syntactic version of the standard Taylor expansion of functions. Until 2021, it seemed that DiLL was doomed to feature a strong form of nondeterminism due to the interaction between differentiation and the structural rule of contraction (Leibniz rule), making it incompatible with stable or sequential denotational interpretations of programs. The article 63 presents Coherent Differentiation, a new denotational setting discovered in 2021, featuring differentiation operations while being compatible with stable and sequential interpretations. A syntactic account of this new approach to differentiation has also been proposed by the author in paper 61. This paper presents a differential extension of the standard programming language PCF (a simply typed, Turing complete, purely functional and call-by-name programming language). It is shown that, contrarily to the differential lambda-calculus considered earlier, this coherent differential PCF has an deterministic operational semantics which can be expressed by means of a Krivine abstract machine. Together with his PhD student Walch, Ehrhard has shown in 22 how this coherent differential approach can be slightly modified to take into account the whole Taylor expansion of terms by means of a simple type constructs which implements the Faà di Bruno formula.

8.2.2 Continuous probabilities and Linear Logic

Probabilistic coherence spaces are a model of Linear Logic where programs are interpreted as "analytic functions" acting on spaces whose elements are (generalized) subprobability distributions. In spite of its good properties, this model does not take into account probabilities on continuous spaces such as the real line. This problem has been partially solved 6 years ago by Ehrhard, together with Michele Pagani and Christine Tasson. The model of measurable cones they introduced was however missing a general theory of integration, mainly required for interpreting sampling in call-by-value and call-by-push-value functional programming languages. In a joint work with Guillaume Geoffroy, Ehrhard has developed a model of integrable cones 64 which solves this issue and enjoys much better properties than the initial one. For instance, the category of integrable cones contains the category of measurable spaces and sub-stochastic kernels as a full subcategory.

8.2.3 Causality structures in probabilistic rewriting

The question of describing the causal structure of symbolic rewriting systems such as the lambda-calculus is at the heart of an old and vibrant connection established in the 1980s between proof theory and concurrency theory. The idea is that the process of reducing beta-redexes generates causal structures similar to what one finds in Petri nets and process calculi such as CCS or the $\pi$-calculus. Typically, the reduction of one beta-redex $u$ can create a beta-redex $v$ whose reduction can in turn create a third beta-redex $w$, and so on. These causal structures give rise to the notion of family designed by Jean-Jacques Lévy in his theory of optimality. They also appear in term rewriting and graph rewriting and play a fundamental role in the description of the stochastic behaviour of probabilistic rewriting. Together with Nicolas Behr (IRIF) and Noam Zeilberger (LIX), Melliès has developed in 40 a framework where these causality structures can be described in a unified way in the language of double categories. The idea is to compose rewriting rules seen as representable presheaves under the action of context extension. Composition is defined using a generalisation to double categories of the usual Day convolution tensor product of presheaves on a monoidal category. One main novelty of the work is to integrate for the first time unification and rewriting in the same categorical framework.

In joint work with van Gool, Melliès and Moreau introduced in 72 a notion of profinite $\lambda$-calculus which provides a new topological account of $\lambda$-terms, different from the usual Scott domain topology and based on new insights on the connection between typed $\lambda$-calculus and automata theory. The construction provides to every type $A$ a compactification of the set of $\lambda$-terms of type $A$ defined as a profinite limit of the interpretations of the type $A$ in the category of finite sets. The profinite limit may be also understood as providing the Stone dual of the boolean algebra of regular languages of $\lambda$-terms of type $A$ introduced by Salvati 92. The categorical definition of profinite $\lambda$-terms is identified in 72 with a definition based on parametricity. This establishes an important and unexpected connection between polymorphism and automata theory. The profinite completion of the Church encoding of finite words in a given alphabet $\Sigma$ is shown to coincide with the usual notion of profinite monoid. Then, in the continuation of this work, Moreau establishes in joint work with Nguyen 11 that the definition of regular language of simply typed $\lambda$-terms does not depend on the choice of (locally finite) cartesian closed category where the $\lambda$-calculus is interpreted, as long as the category of interpretation has enough points. They also establish the important unifying result that the semantic definition by Salvati 92 of regular languages of simply typed $\lambda$-terms coincides with the purely syntactic definition by Hillebrand and Kanellakis 78.

8.3.1 Survey on the Coq community

The goal of the article 32 is to better understand Coq users and community, and to make informed decisions about the research and development challenges around a system such as Coq. A key point on the paper is the multidisciplinary research team, bringing together researchers from several areas in Computer and Social sciences. Inspired by previous Surveys in Coq, OCaml, and open source world, a survey was designed and ran for the Coq community totaling 109 questions, and obtaining 466 answers, to this date, the largest survey done on users of Interactive Theorem Provers. The data were analyzed using rigorous regression methods common on the social sciences.

8.3.2 Reducing the technical debts of the Coq proof assistant

Herbelin simplified and extended the component of the Coq proof assistant in charge of declaring new constants. This resulting in several Pull Requests fixing bugs and factorising code, guided by the objectives of the Coq Enhancement Proposal #42.

Herbelin fixed several bugs of the main proof-level simplification algorithm of Coq (tactic simpl), also investigating how to stop unnecessarily unfolding fixpoint expressions that have a global names, for the daily satisfaction of users.

8.3.3 Implicit arguments with default instance

S. Arambillete implemented an extension of Coq allowing to specify a default value or default algorithm to infer some implicit arguments of a function. This is particularly useful for granting side conditions supposed to be proved from information present in the current context.

8.3.4 Relaxing some constraints of the guard condition

Herbelin presented at TYPES 2024 two improvements of the guard condition (that is of the algorithm checking the termination of recursive functions in Coq). These improvements are needed to implement Agda-style dependent pattern-matching using Monin-Boutillier technique of "small inversion" (see Coq PR #16097).

8.3.5 Automatic generation of logical schemes

Herbelin supervised the internship of Félix Loyau-Kahn on simplifying and generalising the component of Coq in charge of supporting the declaration of logical schemes.

8.4.1 Snapshottable stores

Participants: Clément Allain [Cambium], Basile Clément [OCamlPro], Alexandre Moine [Cambium], Gabriel Scherer.

Type-checkers and SMT solvers need efficient support for backtracking their data structures. In 4 we designed and implemented an OCaml library, Store, that provides such backtracking support in a generic way, and verified a core subset of the library in the Coq/Rocq proof assistant, using the Iris separation logic.

In a follow-up work 10 we design and implement better support for adding backtracking support to existing data structures through an abstract interface, without modifying their implementation.

8.4.2 Tail Modulo Cons, OCaml, and Relational Separation Logic

Participants: Clément Allain [Cambium], Frédéric Bour [Cambium], Basile Clément [Cambium], Francois Pottier [Cambium], Gabriel Scherer.

In 3 we implemented a new program transformation in the OCaml compiler, formalized it on an idealized functional language, and verified its correctness using the Coq/Rocq proof assistant and its Iris separation logic.

A novel technical aspect of our work is to use a simulation relation within Iris to justify a (relational) program logic that is expressive enough to reason about a compiler transformation. We had to extend the previous work on simulations in Iris (SimulIris) to support abstract "protocols", used in our case to support a change of calling convention.

8.4.3 Type inference for OCaml

Participants: Olivier Martinot [Partout], Alistair O'Brien [Cambridge University], Francois Pottier [Cambium], Gabriel Scherer.

We are starting an active collaboration with Alistair O'Brien, from Cambridge university, on constraint-based type inference for OCaml. Particular topics of interest include "frozen constraints" and inference of GADTs. We have not yet published this work besides Olivier Martinot's PhD manuscrit, the undergraduate thesis of Alistair O'Brien, and a small technical report 28.

8.4.4 Static Analysis for Probabilistic Programming

Participants: Ellie Y. Cheng [MIT], Eric Atkinson [Binghampton University], Guillaume Baudart, Louis Mandel [IBM Research], Michael Carbin.

Advanced probabilistic programming languages using hybrid particle filtering combine symbolic exact inference and Monte Carlo methods to improve inference performance. These systems use heuristics to partition random variables within the program into variables that are encoded symbolically and variables that are encoded with sampled values, and the heuristics are not necessarily aligned with the developer's performance evaluation metrics. In 6, we introduced inference plans, a programming interface that enables developers to control the partitioning of random variables during hybrid particle filtering. We developped Siren, a new PPL that enables developers to use annotations to specify inference plans, and an abstract-interpretation-based static analysis for Siren for determining inference plan satisfiability.

8.6.1 NLIR: Natural Language Intermediate Representation for Mechanized Theorem Proving

Participants: Laetitia Teodorescu [AdaptiveML], Guillaume Baudart, Emilio Jesus Gallego Arias, Marc Lelarge [Argo], Jules Viennot.

In 12, we developed an LLM-based agent that can interact with the Rocq proof assistant. Our key idea is to rely on natural language as much as possible when generating proofs. Using natural language leverages the strength of LLMs, and allows us to use chain-of-thought by asking for an informal mathematical proof before generating the formal proof, making it more intuitive and comprehensible compared to purely automatic formal techniques. Additionally, partial proofs expressed in natural language are easier for humans to understand, adapt, or reuse, allowing for greater flexibility and collaboration between machine-generated suggestions and human mathematicians. We proposed two interactive proof protocols both leveraging natural language reasoning: tactic-by-tactic proof construction, and hierarchical proof templating. We then coupled both protocols with standard search algorithms leveraging feedback from the ITP and using natural language to rerank proof candidates. Using our method with GPT-4o we can successfully synthesize proofs for 58% of the first 100/260 lemmas from the newly published Busy Beaver proofs.

10.1.1 Visits of international scientists

10.2.1 Horizon Europe

MeReMath Marie Curie fellowship on cordis.europa.eu

• Title:
  Mechanised Reverse Mathematics in the Calculus of Inductive Constructions
• Duration:
  From June 1, 2024 to May 31, 2026
• Beneficiary:
  Dominik Kirst
• Inria contact:
  Hugo Herbelin
• Summary:

  “Foundations of mathematics” labels the centuries-old interdisciplinary vision to secure the logical basis for mathematics and its applications. Gödel's celebrated completeness theorem for first-order logic, identifying semantic truth with syntactic deduction, is a key result heralding the formal phase of that vision. Yet, this and other foundational results have not been fully characterised regarding their logical strength and computational content, limiting their understanding and applicability.

  The MeReMath project aims at closing this gap by systematically employing computer mechanisation to the programme of reverse mathematics, the ongoing effort to identify the exact logical principles underlying completeness and other results. For this analysis, the project will use the calculus of inductive constructions (CIC) as a logical base system, embodying an agnostic intuitionistic base system unveiling fine logical structure, and the Coq proof assistant, an interactive software tool for modelling logical reasoning and its computational content.

  Specifically, continuing previous research of the experienced researcher (ER) and the supervisor, the MeReMath project will contribute the first comprehensive constructive and computational analysis of the completeness theorem, taking into account all dimensions relevant to its logical strength and implementing modularly mechanised proofs as executable Coq code. By further accommodating a similar analysis of the related Löwenheim-Skolem theorems and other results in the canon, the main outcome of the MeReMath project will be a well-designed, collaboratively developed Coq library for (constructive) reverse mathematics, with novel logical observations and mechanisation techniques developed on the way.

  Hosted at the IRIF lab in Paris, the project will be developed in the centre of the original creation of CIC and Coq, providing a world-class environment for the project’s aims and the ER’s academic career prospects.

GDR and associated working groups (GT).

Our team participates to the GDR Informatique Mathématique, in the LHC (Logique, Homotopie, Catégories) and Scalp (Structures formelles pour le calcul et les preuves) working groups. Alexis Saurin is coordinator of the Scalp working group (see website here).

Some members also participate to the GDR Homotopie, federating French researchers working on classical topics of algebraic topology and homological algebra, such as homotopy theory, group homology, K-theory, deformation theory, and on more recent interactions of topology with other themes, such as higher categories and theoretical computer science.

ANR RECIPROG:

Kostia Chardonnet, Abhishek De, Thomas Ehrhard, Farzad Jafarrahmani, Hugo Herbelin, Paul-André Melliès, Daniela Petrisan and Alexis Saurin (coordinator) are members of the four year RECIPROG project. RECIPROG is an ANR collaborative project (a.k.a. PRC) started in the fall 2021-2022 and running till the end of 2025. ReCiProg aims at extending the proofs-as-programs correspondence to recursive programs and circular proofs for logic and type systems using induction and coinduction. The project aims at contributing 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.

The project is coordinated by Alexis Saurin and has four sites: IRIF in Paris where the team Picube is located, LIP in Lyon, LIS in Marseille and LS2N in Nantes.

INRIA Challenge LiberAbaci:

Hugo Herbelin participates in the Inria Challenge LiberAbaci. LiberAbaci is a collaborative project aimed at improving the accessibility of the Coq interactive proof system for an audience of mathematics students in the early academic years. The lead is Yves Bertot and the involved teams are: Cambium (Paris), Camus (Strasbourg), Gallinette (Nantes) PiCube (Paris), Spades (Grenoble), Stamp (Sophia Antipolis), Toccata (Saclay), LIPN (Laboratoire d'Informatique de Paris Nord).

INRIA Challenge LLM4Code:

Emilio J. Gallego Arias, Guillaume Baudart, Paul-André Melliès, and Jules Viennot participates in the Inria Challenge LLM4Code. The goal of the LLM4Code défi is to leverage Large Language Models (LLM) capabilities to build code assistants that can enhance both reliability and productivity. In particular, one axis of the défi focuses on the interaction between proof assistant and LLMs.

The défi is co-led by M. Acher and G. Baudart and involves 11 inria teams, the LaBRI, Software Heritage and Sopra Steria.

Chocola seminar

Our team is active in the organization and attendance of the Chocola seminar, being held monthly in Lyon. (Alexis Saurin and Gabriel Scherer are among the 12 people in the organizing committee.)

11.1.1 Scientific events: organisation

11.1.2 Scientific events: selection

11.1.3 Journal

11.1.4 Invited talks

• G. Baudart and C. Tasson gave an invited course on reactive probabilistic programming at the journées Francophones des Langages Applicatifs (JFLA) 2024.
• G. Baudart and C. Tasson gave a course on probabilistic programming at École des Jeunes Chercheurs et Chercheuses en Informatique Fondamentale et ses Mathématiques (EJCIM) 2024 14.
• G. Baudart was an invited speaker at the Probabilistic Programming workshop of the Modelling and inference for pandemic preparedness programme at the Isaac Newton Institute in 2024.
• G. Baudart was an invited speaker at the INRIA/KAIST workshop 2024.
• P.-L. Curien was an invited speaker at SLALM 2024 (Latin American conference in logic) in Montevideo (Uruguay) in July 2024
• T. Ehrhard was an invited speaker at the CIRM conference Differential $\lambda$-calculus and Differential Linear Logic, 20 years later in 2024.
• T. Ehrhard was invited to give a talk at ICALP'24 in Tallinn (Alonzo Church Award ceremony)
• T. Ehrhard was invited speaker at the meeting of the Associazione Italiana di Logica e sue Applicazioni in Udine, in 2024.
• H. Herbelin was an invited speaker at the workshop PACMAN 2024 in Verona.
• H. Herbelin was an invited speaker at the EPFL SYSTEMF seminar in March 2024 in Lausanne.
• H. Herbelin was an invited speaker at the mini-workshop Structures in Foundations of Mathematics in Septembre 2024 in Padova.
• J-J. Lévy talked about the 1996 Ariane 501 bug, talk entitled 'A Small bug, A Big bang' at Huzhou College, Zhejiang China, on November 15.
• J-J. Lévy talked about a revised work of a chapter in the 2024 book entitled 'The French School of Programming' (ed. B. Meyer, Springer) at the PPS seminar, Irif on October 24; at ECNU Shanghai on November 20; at Nanjing University on November 22. This talk is entitled 'Tracking Redexes in the Lambda-Calculus - revisited) [see http://jeanjacqueslevy.net/talks]
• P-A. Melliès was an invited speaker at the Logic Colloquium 2025 at Goteborg, Sweden.

11.1.5 Scientific expertise

• H. Herbelin evaluated a project for the Austrian Science Fund (FWF).

11.2.1 Teaching

• Master: Guillaume Baudart and Gabriel Scherer: “Synchronous Programming” (M2), TDs, Université Paris Cité
• Master: Guillaume Baudart : “Probabilistic Programming Languages” (M2), Lectures and TDs, MPRI
• Master: Thomas Ehrhard : “Modèles des langages de programmation” (M2), Lecture and TDs, MPRI
• Master: Paul-André Melliès : “Modèles des langages de programmation” (M2), Lecture and TDs, MPRI
• Master: Alexis Saurin : “Functional programming and formal proofs in Coq” (M2), Lectures and TPs, LMFI
• Master: Alexis Saurin : “Second-order quantification and fixed-points in logic” (M2), Lectures, LMFI
• Aggregation: Guillaume Baudart : “Introduction to Software Engineering” (préparation à l'aggrégation d'informatique), Lectures and TDs
• Master: Hugo Herbelin: "Preuves assistées par ordinateur" (M1), Lectures, Université Paris Cité
• Master: Gabriel Scherer: "Programmation fonctionnelle et systèmes de types" (M2), Lectures, MPRI.

• Bachelor: "Projets de programmation OCaml" (L3), TP (practical sessions), Université Paris Cité

11.2.2 Supervision

11.2.3 Juries

• T. Ehrhard was a reviewer of the PhD thesis of Louis Lemonnier
• H. Herbelin was a reviewer of the PhD thesis of Pietro Sabelli. He was an examiner of the PhD committee of Clément Blaudeau and Jui-Hsuan Wu.
• P-A. Melliès was a jury member for the HDR defense of Sam van Gool.
• G. Scherer was a jury member for the PhD defense of Colin Gonzalez.
• A. Saurin was a jury member for the PhD defence of Guillermo Menéndez Turata (Amsterdam).

11.3.1 Productions (articles, videos, podcasts, serious games, ...)

• G. Baudart, J. Narboux and G. Scherer supervised activities for schoolchildren for the Fête de la science 2024 at Université Paris Cité.

11.3.2 Others science outreach relevant activities

• H. Herbelin organised the 2024 edition of the FSMP Horizons Maths event on "Preuve mathématique et sûreté logicielle" (helped with X. Leroy and C. Dubois).

International journals

• 3 articleC.Clément Allain, F.Frédéric Bour, B.Basile Clément, F.François Pottier and G.Gabriel Scherer. Tail Modulo Cons, OCaml, and Relational Separation Logic.Proceedings of the ACM on Programming Languages9POPLJanuary 2025, 2337-2363HALDOIback to text
• 4 articleC.Clément Allain, B.Basile Clément, A.Alexandre Moine and G.Gabriel Scherer. Snapshottable Stores.Proceedings of the ACM on Programming Languages8ICFPAugust 2024, 338-369HALDOIback to text
• 5 articleA.Antonio Bucciarelli, A.Antonio Ledda, F.Francesco Paoli, A.Antonino Salibra and P.-L.Pierre-Louis Curien. The higher dimensional propositional calculus.Logic Journal of the IGPL2025. In press. HALDOIback to text
• 6 articleE. Y.Ellie Y Cheng, E.Eric Atkinson, G.Guillaume Baudart, L.Louis Mandel and M.Michael Carbin. Inference Plans for Hybrid Particle Filtering.Proceedings of the ACM on Programming Languages9POPL2025, 10HALDOIback to text
• 7 articleP.-L.Pierre-Louis Curien, B.Bérénice Delcroix-Oger and J.Jovana Obradović. Tridendriform algebras on hypergraph polytopes.Algebraic Combinatorics2024. In press. HAL
• 8 articleG.Guillaume Laplante-Anfossi and P.-L.Pierre-Louis Curien. Topological proofs of categorical coherence.Cahiers de topologie et géométrie différentielle catégoriquesLXV4December 2024, 357-389HALback to text
• 9 articleI.Ingkarat Rak‐amnouykit, A.Ana Milanova, G.Guillaume Baudart, M.Martin Hirzel and J.Julian Dolby. Principled and practical static analysis for Python: Weakest precondition inference of hyperparameter constraints.Software: Practice and Experience5432024, 363-393HALDOI

International peer-reviewed conferences

• 10 inproceedingsB.Basile Clément, C.Camille Noûs and G.Gabriel Scherer. Storable types: free, absorbing, custom.36es Journées Francophones des Langages Applicatifs (JFLA 2025)Roiffé, FranceJanuary 2025HALback to text
• 11 inproceedingsV.Vincent Moreau and L. T.Lê Thành Dũng Nguyễn. Syntactically and Semantically Regular Languages of λ-Terms Coincide Through Logical Relations.32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)288Leibniz International Proceedings in Informatics (LIPIcs)Naples, ItalySchloss Dagstuhl – Leibniz-Zentrum für Informatik2024, 40:1-40:22HALDOIback to text
• 12 inproceedingsL.Laetitia Teodorescu, G.Guillaume Baudart, E. J.Emilio Jesús Gallego Arias and M.Marc Lelarge. NLIR: Natural Language Intermediate Representation for Mechanized Theorem Proving.MathAI@NeuRIPS 2024 - 4th Workshop on Mathematical Reasoning and AIVancouver, CanadaDecember 2024HALback to text

Conferences without proceedings

• 13 inproceedingsA.Alexis Saurin. Craig-Lyndon interpolation as cut-introduction.Logic Colloquium 2024 - European Summer Meeting of the Association for Symbolic LogicGothenburg, SwedenJune 2024HAL

Scientific book chapters

• 14 inbookG.Guillaume Baudart and C.Christine Tasson. Introduction to probabilistic programming.Informatique Fondamentale et ses Mathématiques Une photographie en 2024CNRS ÉditionsJune 2024, 117-156HALback to text

Reports & preprints

• 15 miscQ.Quentin Aristote. Monotone weak distributive laws over the lifted powerset monad in categories of algebras.November 2024HAL
• 16 miscE.Esaïe Bauer and A.Alexis Saurin. Cut-elimination for the circular modal mu-calculus: linear logic and super exponentials to the rescue.March 2024HAL
• 17 miscE.Esaïe Bauer and A.Alexis Saurin. Light logics with inductive and coinductive types: a uniform account of cut-elimination.September 2024HALDOIback to text
• 18 miscE.Esaïe Bauer and A.Alexis Saurin. On the cut-elimination of the modal µ-calculus: Linear Logic to the rescue.July 2024HALback to text
• 19 miscA.Antonio Bucciarelli, P.-L.Pierre-Louis Curien, A.Arturo de Faveri and A.Antonino Salibra. Birkhoff-style Theorems Through Infinitary Clone Algebras.November 2024HALback to text
• 20 miscA.Antonio Bucciarelli, A.Antonino Salibra and P.-L.Pierre-Louis Curien. Abstract clones as noncommutative monoids I.2024HAL
• 21 miscK.Kostia Chardonnet, A.Alexis Saurin and B.Benoît Valiron. A Curry-Howard Correspondence for Linear, Reversible Computation.February 2025HALback to text
• 22 miscT.Thomas Ehrhard and A.Aymeric Walch. Coherent Taylor expansion as a bimonad.December 2024HALback to text
• 23 miscH.Hugo Herbelin and D.Danko Ilik. An analysis of the constructive content of Henkin's proof of Gödel's completeness theorem.January 2024HAL
• 24 miscG.Guillaume Laplante-Anfossi and P.-L.Pierre-Louis Curien. Term rewriting on nestohedra.March 2024HALback to text
• 25 miscP.Pierre Letouzey. Generalized Hofstadter functions G, H and beyond: numeration systems and discrepancy.February 2025HAL
• 26 miscP.Pierre Letouzey, S.Shuo Li and W.Wolfgang Steiner. Pointwise order of generalized Hofstadter functions G, H and beyond.September 2024HALback to text
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• 29 miscH.Haoyi Zeng, Y.Yannick Forster, D.Dominik Kirst and T.Takako Nemoto. Post's Problem in Constructive Mathematics.January 2025HAL