## Séminaire Formath (Formalized Mathematics)

#### Jour, heure et lieu

Le lundi à 14h, 1007 ou 3052

Le calendrier des séances (format iCal).

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#### Contact(s)

Le séminaire Formath (Formalized mathematics) est le séminaire de l'équipe Picube.

Pour s'abonner à la liste de diffusion, consulter la page https://listes.irif.fr/sympa/info/formath ou écrire aux responsables du séminaire.

### Séances passées

#### Année 2024

Formath

Lundi 10 juin 2024, 14 heures, 3052 and bbb Link

**Sarah Reboullet** (Université Paris Cité, IRIF & INRIA Picube) *Fixing Internal Parametricity and Its Presheaf Model*

Formath

Lundi 27 mai 2024, 14 heures, 3052 and Zoom Link

**Laetitia Teodorescu** (IRIF, Université Paris Cité & INRIA FLOWERS & Picube) *Autotelic machine learning for programming and mathematics*

Formath

Lundi 29 avril 2024, 14 heures, 3052 and bbb Link

**Hugo Herbelin** (Picube) *Où se cache le calcul dans les topos élémentaires ?*

En tout cas, on commencera par voir le classificateur de sous-objet comme un hProp imprédicatif, par isoler le rôle de l'extensionnalité des propositions (= univalence pour hProp) dans la correspondance entre représentations fibrées et indexées des prédicats dans les topos, et par comprendre comment la somme disjointe peut se simuler en combinant le pouvoir de décision de la définition imprédicative de la disjonction avec le pouvoir de représentation des types par des parties singleton.

Formath

Mardi 26 mars 2024, 14 heures, 1020 and Renater

**Lasse Blaauwbroek** *A machine learning platform for proof synthesis in Coq*

In this talk, I will give a general tutorial on Tactician. We start with a usage demonstration from an end-user perspective. Then we'll take a look at Tactician's internal architecture. There are several data formats for machine learning, including a novel graph-based representation of the Calculus of Existential Inductive Constructions, where every node in the graph represents a unique term modulo alpha-equivalence. This data is made available both as an offline dataset and through interaction with Coq using a rich remote procedure calling (RPC) protocol. We'll see how we can easily implement our own external solving agent in Python.

Formath

Lundi 18 mars 2024, 14 heures, Grenoble

**Gdr Ifm** *Journées nationales du GDR (pas d'exposé)*

Formath

Lundi 11 mars 2024, 14 heures, 3052

**Gabriel Scherer** *Random generation of well-typed terms: a constraint-based approach*

Most existing generators integrate a large part of the implementation of a type-checker, “inverted” to assist in top-down random term generation by efficiently discarding choices that result in ill-typed programs. Scaling this approach to a full language would result in implementing two sophisticated type-checkers, once in the compiler and once in the program generator. This is one type-checker too many!

In this talk I will present preliminary work on reusing the constraint-based type inference approach [The Essence of ML-based type inference, Pottier and Rémy, 2004] to write a single type-checker that can be used for both purposes: to check types of user-provided programs, and to efficiently guide a random program generator. This only requires a fairly simple modification to constraint generators: parametrizing them over a search monad.

(Relevant previous work is [Making Random Judgments: Automatically Generating Well-Typed Terms from the Definition of a Type-System, Fetscher, Claessen, Pałka, Hughes, Findler]. They start from Redex, a Racket DSL to implement simple type systems, and derive a random program generator. The limitation of this approach is that Redex only works with fairly simple type system, in particular we do not expect it to scale to ML-style type inference with polymorphism, which is the part that hand-crafted generators also struggle on. Our work builds on top of constraint-solving approaches that are known to scale well to ML-style polymorphic; but it is preliminary and we have only implemented support for the simply-typed lambda-calculus for now.)

Formath

Lundi 4 mars 2024, 14 heures, 3052 and bbb Link

**Guillaume Baudart** *Schedule Agnostic Semantics for Reactive Probabilistic Programming*

In talk I will present two schedule agnostic semantics for a probabilistic synchronous language. The key idea is to interpret probabilistic expressions as a stream of un-normalized density functions which maps random variable values to a result and positive score. The co-iterative semantics extends the original semantics to interpret mutually recursive equations using a fixpoint operator. The relational semantics directly manipulates streams and is thus a better fit to reason about program equivalence. We use the relational semantics to prove the correctness of a program transformation required to run an optimized inference algorithm for state-space models with constant parameters.

This is joint work with Louis Mandel and Christine Tasson.

Formath

Lundi 26 février 2024, 14 heures, 3052

**Julien Narboux** (Université de Strasbourg) *Formalization of geometry, and automated theorem proving using constraint solving*

Formath

Lundi 12 février 2024, 14 heures, 3052 and bbb Link

**Esaïe Bauer** (IRIF, Université Paris Cité & INRIA Picube) *Cut-elimination for the circular modal mu-calculus: linear logic and super exponentials to the rescue*

During this talk, I will present a strategy to prove cut-elimination theorem for the modal mu-calculus. This strategy is to prove cut-elimination for a “linear translation” of the modal mu-calculus (that is to define a translation of the statements and proofs of the modal mu-calculus into a linear sequent calculus) and to deduce the desired cut-elimination results as corollaries.

While designing this linear translation, we come up with a sequent calculus exhibiting several modalities (or exponentials). It happens that the literature of linear logic offers tools to manage such calculi, for instance subexponentials by Nigam and Miller and super exponentials which is a work I did with Olivier Laurent.

We actually extend super exponentials with fixed-points and non-wellfounded proofs (of which the linear decomposition of the modal mu-calculus is an instance) and prove a syntactic cut-elimination theorem for these sequent calculi in the framework of non-wellfounded proof theory.

Back to the classical modal mu-calculus, we deduce its cut-elimination theorem from the above results, investigating both non-wellfounded and regular proofs.

This is a joint work with Alexis Saurin.

Formath

Lundi 22 janvier 2024, 14 heures, 3052 and bbb Link

**Houda Mouhcine (Inria Saclay) & Micaela Mayero (Université Paris 13, Lipn, Love)** *Double exposé*

Abstract: Formalizing real numbers in a formal proof tool represents a particular challenge. It is not only a question of representing numbers in computers but also of preserving all mathematical properties needed to make proofs. We will review the history of real numbers formalization in different proof assistants and the different ways of formalizing them, before giving an overview of real numbers libraries that exist today in provers. As application, we will finish by presenting real analysis developments in Coq.

2. [Deuxième exposé : Houda] Title: Formal proofs in applied mathematics: Formalization of Tonelli Theorem and Lagrange Finite Elements

Abstract. Numerical analysis, as an essential tool for solving complex mathematical problems through the use of algorithms and numerical approximations, is further fortified in this work by rigorously formalizing Tonelli's Theorem and Lagrange Finite Elements. These basic concepts are defined and validated using formal proof techniques, enhancing their credibility in various mathematical applications. In the field of integration, important results such as Tonelli's Theorem and the Lebesgue Induction Principle are covered. Furthermore, in the field of numerical resolution of partial differential equations, we explore finite elements, covering the general definition of finite elements, and geometric transformations, up to the full definition of simplicial Lagrange finite elements, including the unisolvance property.

Formath

Lundi 8 janvier 2024, 14 heures, 3052 and Galène Link (password: tensor)

**Loïc Pujet** (University of Stockholm) *Choice Principles in Observational Type Theory*

Unfortunately, it is difficult to use these quotients without any choice principles to extract information from proof-irrelevant truncations. In this talk, I will discuss the various choice principles one could hope to have in OTT, and I will use ideas from Higher Observational Type Theory to sketch a version of OTT with unique choice.

#### Année 2023

Formath

Lundi 11 décembre 2023, 14 heures, 3052

**Meven Lennon-Bertrand** (Université de Cambridge) *Definitional Functoriality for Dependent (Sub)Types*

I will explain how to repair this and add these extra equations in, and the proof techniques based on logical relations we need to set up to establish the good theoretic properties of the resulting type system, that we have recently formalised in Coq.

Formath

Lundi 20 novembre 2023, 14 heures, 3052 and bbb Link

**Paul Laforgue** (IRIF, Université de Paris, Nomadic Labs) *Characterisations of the must-preorder for asynchronous processes, mechanised in Coq.*

The must-preorder, introduced by De Nicola and Hennessy in 1984, considers one process to be a refinement of another if every test satisfied by the former is also satisfied by the latter. In practice, this corresponds to updating the software running on a server without clients noticing any change.

In this talk I will present a new characterisation of the must-preorder for asynchronous processes which is simpler than the existing ones and generalize the standard characterization defined for synchronous processes.

Our work led to the implementation of a Coq framework. I shall discuss how we leveraged Coq typeclasses to make our theorems applicable to any Labelled Transition System (LTS) that respects Selinger's axioms for output-buffered agents.

Formath

Lundi 13 novembre 2023, 14 heures, 3052 SG

**Antoine Chambert-Loir** (IMJ-PRG, Université Paris Cité) *Simplicité des groupes (une expérience incomplète de formalisation)*

Je présenterai mon travail sur la formalisation en Lean de la simplicité du groupe alterné sur au moins 5 lettres qui repose sur un critère d'Iwasawa.

Ce théorème, souvent attribué à Galois lui-même, possède de nombreuses preuves, certaines assez courtes, et celle que j'ai choisie est en fin de compte assez longue.

Je tenterai de justifier pourquoi son schéma, les concepts qu'elle met en œuvre et les ramifications qu'elle offre la rendent, à mon sens, appropriée pour figurer dans une librairie de mathématiques formelles.

Formath

Lundi 6 novembre 2023, 14 heures, 3052 SG

**Alexis Saurin** (IRIF) *When interpolation is proof-relevant: Craig-Lyndon interpolation as cut-introduction*

This interpolation is proof-relevant in the following sense: if π proves A ⊢ B, then we can find (i) a formula C in the common vocabulary of A and B and (ii) two proofs π1 and π2 of A ⊢ C and C ⊢ B respectively, such that π1 composed with π2 cut-reduces to π. Similar proof-relevant interpolation results for LJ and LK are obtained as immediate corollaries via usual linear translations.

I will explain how this result follows straightforwardly from the usual proof-theoretic method for proving interpolation and how the construction of a proof-relevant interpolant can be viewed as a cut-introduction procedure.

Finally, I shall discuss how to extend the above result to (a fragment of) circular proofs of linear logic with least and greatest fixed-points.

Formath

Lundi 23 octobre 2023, 14 heures, 3052

**Félix Castro** (IRIF - Université Paris CIté) *Revisiting the model of Hereditary Recursive Operations*

Formath

Lundi 16 octobre 2023, 14 heures, 3052

**Thiago Felicissimo** (Deducteam, INRIA & LFM, Paris-Saclay) *Generic bidirectional typing for dependent type theories*

Formath

Lundi 9 octobre 2023, 14 heures, 3052

**Pierre Letouzey** *A propos d'une curieuse famille de fonctions récursives imbriquées due à Hofstadter*

G(0)=0 puis G(n)=n-G(G(n-1)) pour tout entier n>0

Il définit également une fonction H selon le même modèle, mais avec trois appels récursifs imbriqués au lieu de deux pour G, et mentionne que cela continue ensuite, pour n'importe quel degré d'imbrication. Cet exposé est consacré à l'étude de la famille de fonctions que l'on obtient ainsi. Je parlerai de leurs liens intimes avec (au moins!) des arbres rationnels infinis, des variantes des nombres de Fibonacci et leur systèmes de numération associés, des mots morphiques, des nombres de Pisot, des fractales de Rauzy. Je présenterai également le développement Coq correspondant, certifiant presque toutes les propriétés en question, y compris une conjecture OEIS concernant H. Cela m'a amené à considérer en Coq des réels, complexes, polynômes, matrices, etc, ce qui est encore loin d'être une sinécure. Il sera enfin question d'une conjecture personnelle récalcitrante : cette famille de fonctions semble croissante point-à-point.

NB: Il y aura un recouvrement certain mais loin d'être total avec un exposé précédent aux journées PPS 2018, consacré à l'époque à G et à un exercice au lecteur posé par Hostadter à propos de “l'arbre miroir” de G. Pas besoin ici d'avoir vu cet exposé 2018 !

Formath

Lundi 2 octobre 2023, 14 heures 15, 3052 SG

**Emilio Gallego Arias** (IRIF & PICUBE, INRIA) *Flèche: Incremental Validation for Hybrid Formal Documents*

Flèche provides a formal document model, based on a notion of functional interpretation for a family of languages plus algebraic effects capturing document dynamics.

Flèche targets existing systems with non-trivial interfaces and documents in their daily use.

Some features of Flèche are: a) hybrid formal documents, where natural language is intermixed with its formal counterpart; b) modular error recovery and handling of incomplete documents; c) incremental, full-project model; d) standard document edition workflow.

We have released a Flèche implementation for the Coq Proof Assistant, `coq-lsp`, that uses the Language Server Protocol (LSP) as an editor API. Together with LSP-capable editors like Visual Studio Code, `coq-lsp` provides a new user interface for the Coq system with unique characteristics.

Formath

Mardi 12 septembre 2023, 10 heures 15, Salle 146, Bâtiment Olympe de Gouges

**Ian Shillito And Iris Van Der Giessen** *Interpolation(s) via proof theory: sequents, Coq, and beyond sequents.*

This will be a joint talk of a duration of about 1H45. Il s'agira d'un exposé commun, d'une durée d'environ 1H45

Formath

Lundi 5 juin 2023, 14 heures, 146 (Olympe de Gouges)

**Wendlasida Ouedraogo** (INRIA (Paris Saclay)) *Source code optimization for safety critical systems*

This research work is situated in the context of one such system, the communication-based train control (CBTC) system of Siemens Mobility France which operates a number of driverless subway systems around the World, including Paris lines 1, 4, and 14. That system is certified according to the industrial norm EN-50128 and up to the highest Safety Integrity Level 4, required for safety-critical systems with potentially catastrophic consequences. In this context, the thesis looks for an answer to the question of how to automatically optimize the execution time performance of such systems without losing the previously obtained safety guarantees.

The answer provided is provably correct optimization of source code. A first contribution is a source-to-source compiler for VCP Ada (a subset of Ada) programs, that optimizes source code while preserving the formal semantics of the programs. The compiler has been proven correct in the Coq proof assistant and guarantees the equivalence of execution between the original and the optimized program. The compiler copes with the complexities of the platform, due to hardware safety measures, which is important, since real-world safety-critical systems are often susceptible to having such measures, potentially conflicting with existing formally proven optimizing compilers. Moreover, choosing the approach of a source-to-source compilation shows to have methodological advantages over an approach to proven optimizations having a number of intermediate languages, allowing to simplify and diminishing the needed proof effort.

A second contribution is to the problem of provably correct lexical analysis of compilers, which has previously not received a lot of research attention, a weak link in any compilation chain using a proven or qualified compiler. We develop CoqLex, the first Coq-formalized lexer generator, based on a modification of an existing Coq implementation of regular expression matching via Brzozowski derivatives.

The developed theory and tools have been applied to optimize real-world VCP Ada programs of CBTC systems, consisting of thousands of source files, with promising results.

Formath

Lundi 22 mai 2023, 14 heures, Olympe de Gouges, 146

**Pablo Donato** (Partout Team, LIX laboratory) *Deep Inference for Graphical Theorem Proving*

In the first part of the talk, I will present our so-called proof-by-action paradigm, where the user builds proofs by executing gestural actions directly on the proof state. Typically, this corresponds to the possibility of clicking, dragging and dropping items which represent the conclusion and hypotheses of the current goal. This will be illustrated through a demonstration of Actema, a prototype of graphical interface implementing these ideas in the setting of intuitionistic first-order logic. The main feature of Actema is its drag-and-drop proof tactic, which can be seen as a powerful generalization of the standard “apply” and “rewrite” tactics found in most proof assistants. I will discuss its proof-theoretical foundation, a technique called “subformula linking”, as well as how we were able to make it available in the Coq proof assistant by building the coq-actema plugin.

In the second part of the talk, I will present a more experimental line of research where I aim to get rid of the traditional symbolic formulas of logic, in favor of a more structured and iconic representation of the proof state. By “iconic”, I mean that the visual representation of logical statements should hint at their meaning, and provide affordances to the ways in which they can be manipulated. It turns out that this has already been investigated by the visionary logician C. S. Peirce at the end of the 19th century, who devised arguably the first diagrammatic proof calculus in history: the existential graphs. I will present my own take on the existential graphs, which has crystallized into a proof system I call the “flower calculus”. With only 7 rules of (deep) inference, the flower calculus is sound and complete for intuitionistic first-order logic. I will conclude with a demonstration of the Flower Prover, another prototype of GUI for theorem proving in the proof-by-action paradigm, whose actions map directly to the rules of the flower calculus.

Formath

Lundi 15 mai 2023, 14 heures, Salle 146, Bâtiment Olympe de Gouges

**Etienne Miquey** (I2M) *Evidenced frames: a unifying framework broadening realizability models (and that is formalised in Coq!)*

In this work, we (Liron Cohen, Ross Tate & myself) alternatively address these limitations by introducing evidenced frames: a general-purpose framework for building realizability models that support diverse effectful computations. We demonstrate that they define a very flexible notion of realizability models, allowing in particular models wherein the realizers themselves can be effectful, such as λ-terms that can manipulate state, reduce non-deterministically, or fail entirely. Beyond the broader notions of computation, we demonstrate that evidenced frames form a unifying framework for (realizability) models of higher-order dependent predicate logic. In particular, we prove that evidenced frames are complete with respect to these models, and that the existing completeness construction for implicative algebras—another foundational framework for realizability—factors through our simpler construction. As such, we conclude that evidenced frames offer an ideal domain for unifying and broadening realizability models.

Most of our results have been formalised in the Coq proof assistant, and I plan to devote some part of the talk to present this formalisation.

Formath

Mardi 9 mai 2023, 11 heures, 147 (Olympe de Gouges)

**Louise Dubois De Prisque** (LSV, CNRS & ENS Paris-Saclay) *Compositional pre-processing for Coq’s tactics*

Formath

Lundi 17 avril 2023, 14 heures, Olympe de Gouges (room 146)

**Arthur Charguéraud** (Camus team, ICPS, iCube, INRIA) *A Modern Eye on Weakest Preconditions*

Besides, I will present “characteristic formulae in weakest-precondition style”. These formulae also correspond to the weakest precondition of a unannotated program, that is, a program without specifications nor invariants. Characteristic formulae provide a practical way of embedding Separation Logic reasoning in a proof assistant such as Coq. This approach is implemented in my program verification tool CFML, and is described in an all-in-Coq course, Volume 6 of the Software Foundations series.

Furthermore, I will advertise for two cousins of the omni-big-step semantics: the coinductive-omni-big-step semantics, which captures partial correctness; and the omni-small-step semantics, which relates one configuration to its set of successor configurations. I will explain how omni-semantics may be used to: (1) carry out type-soundness proofs in a style that avoids quadratic case inspections, and (2) carry out compiler correctness proofs using forward simulation proofs, even in the case of nondeterministic semantics.

Formath

Lundi 3 avril 2023, 14 heures, 1007

**Olivier Laurent** (Équipe Plume, Laboratoire de l'Informatique du Parallélisme (LIP), ENS Lyon) *1, 2, 3, Yalla: Linear Logic in Coq*

https://perso.ens-lyon.fr/olivier.laurent/yalla/

The initial specification was to get a library which:

- would provide standard results on the proof theory of Linear Logic;
- could be reused, thus in particular able to deal with some variants of the system;
- should be compatible with the Curry-Howard interpretation of proofs, thus faithful to their computational content.

We will describe the evolution of the project and how it matches this specification through the different versions of the library: Yalla 1, Yalla 2 (current version), Yalla 3 (future directions).

Following all previous known works, Yalla 1 defined proofs in Prop. It relies on an explicit exchange rule for dealing with the computational content of proofs. Yalla 2 corresponds to a move to Type to be able to extract computational informations from proofs. Yalla 3 will rely on a different way of approaching exchange to be able to deal with local transformations of proofs more easily.

Formath

Lundi 20 mars 2023, 15 heures, 1007

**Riccardo Brasca** (IMJ-PRG, Université Paris-Cité) *Le dernier théorème de Fermat pour les premiers réguliers en Lean*