Experience Implementing a Performant Category-Theory Library in Coq

We describe our experience implementing a broad category-theory library in Coq. Category theory and computational performance are not usually mentioned in the same breath, but we have needed substantial engineering effort to teach Coq to cope with large categorical constructions without slowing proof script processing unacceptably. In this paper, we share the lessons we have learned about how to represent very abstract mathematical objects and arguments in Coq and how future proof assistants might be designed to better support such reasoning. One particular encoding trick to which we draw attention allows category-theoretic arguments involving duality to be internalized in Coq's logic with definitional equality. Ours may be the largest Coq development to date that uses the relatively new Coq version developed by homotopy type theorists, and we reflect on which new features were especially helpful.Comment: The final publication will be available at link.springer.com. This version includes a full bibliography which does not fit in the Springer version; other than the more complete references, this is the version submitted as a final copy to ITP 201

Point-Free, Set-Free Concrete Linear Algebra

International audienceWe show how a simple variant of Gaussian elimination can be used to model abstract linear algebra directly, using matrices only to represent all categories of objects, with operations such as subspace intersection and sum. We can even provide effective support for direct sums and subalgebras. We have formalized this work in Coq, and used it to develop all of the group representation theory required for the proof of the Odd Order Theorem, including results such as the Jacobson Density Theorem, Clifford's Theorem, the Jordan-Holder Theorem for modules, the Wedderburn Structure Theorem for semisimple rings (the basis for character theory).On présente une formalisation en Coq de l'algèbre linéaire où tous les objets sont représentés par des matrices, y compris les sous-espaces. Ce développement a été utilisé pour élaborer la formalisation des éléments de théorie de la représentation nécessaires à la prévue du théorème de Feit-Thompson

Construction of real algebraic numbers in Coq

This paper shows a construction in Coq of the set of real algebraic numbers, together with a formal proof that this set has a structure of discrete archimedian real closed field. This construction hence implements an interface of real closed field. Instances of such an interface immediately enjoy quantifier elimination thanks to a previous work. This work also intends to be a basis for the construction of complex algebraic numbers and to be a reference implementation for the certification of numerous algorithms relying on algebraic numbers in computer algebra

Particle identification using Boosted Decision Trees in the semi-digital hadronic calorimeter

International audienceThe CALICE Semi-Digital Hadronic CALorimeter (SDHCAL) prototype using Glass Resistive Plate Chambers as a sensitive medium is the first technological prototype in a family of high-granularity calorimeters developed by the CALICE collaboration to equip the experiments of future leptonic colliders. It was exposed to beams of hadrons, electrons and muons several times on the CERN PS and SPS beamlines in 2012, 2015 and 2016. We present here a new method of particle identification within the SDHCAL using the Boosted Decision Tree (BDT) method applied to the data collected in 2015. The performance of the method is tested first with GEANT4-based simulated events and then on the data collected in the SDHCAL in the energy range between 10 and 80 GeV with 10 GeV energy steps. The BDT method is then used to reject the electrons and muons which contaminate the SPS hadron beams

A Coq Formalization of Finitely Presented Modules

International audienceThis paper presents a formalization of constructive module theory in the intuitionistic type theory of Coq. We build an abstraction layer on top of matrix encodings, in order to represent finitely presented modules, and obtain clean definitions with short proofs justifying that it forms an abelian category. The goal is to use it as a first step to get certified programs for computing topological invariants, like homology groups and Betti numbers

Energy reconstruction for a hadronic calorimeter using multivariate data analysis methods

International audienceThe CALICE highly granular Semi-Digital Hadronic CALorimeter (SDHCAL) technological prototype provides rich information on the shape and structure of the hadronic showers. To exploit this information and to improve on the standard energy reconstruction method where only the total number of hits is used, we propose to use two methods based on MultiVariate data Analysis (MVA) techniques: the Multi-Layer Perceptron (MLP) and the Boosted Decision Trees with Gradient Boost (BDTG) . The two new methods achieve better energy linearity (Δ E/Ebeam ≤ 2%) with respect to the classic method ( Δ E/Ebeam ≤ 5%) and improve on the relative energy resolution. For instance, the MLP method achieves 6–7% relative improvement on the whole energy range when applied on samples of simulated π− events in the SDHCAL

A Language of Patterns for Subterm Selection

International audienceThis paper describes the language of patterns that equips the SSReflect proof shell extension for the Coq system. Patterns are used to focus proof commands on sub expressions of the conjecture under analysis in a declarative manner. They are designed to ease the writing of proof scripts and to increase their readability and maintainability. A pattern can identify the sub expression of interest approximating the sub expression itself, or its enclosing context or both. The user is free to choose the most convenient option. Patterns are matched following an extremely precise and predictable discipline, that is carefully designed to admit an efficient implementation. In this paper we report on the language of patterns, its matching algorithm and its usage in the formal library developed by the Mathematical Components team to support the verification of the Odd Order theorem

Description and stability of a RPC-based calorimeter in electromagnetic and hadronic shower environments

The CALICE Semi-Digital Hadron Calorimeter technological prototype completed in 2011 is a sampling calorimeter using Glass Resistive Plate Chamber (GRPC) detectors as the active medium. This technology is one of the two options proposed for the hadron calorimeter of the International Large Detector for the International Linear Collider. The prototype was exposed in 2015 to beams of muons, electrons, and pions of different energies at the CERN Super Proton Synchrotron. The use of this technology for future experiments requires a reliable simulation of its response that can predict its performance. GEANT4 combined with a digitization algorithm was used to simulate the prototype. It describes the full path of the signal: showering, gas avalanches, charge induction, and hit triggering. The simulation was tuned using muon tracks and electromagnetic showers for accounting for detector inhomogeneity and tested on hadronic showers collected in the test beam. This publication describes developments of the digitization algorithm. It is used to predict the stability of the detector performance against various changes in the data-taking conditions, including temperature, pressure, magnetic field, GRPC width variations, and gas mixture variations. These predictions are confronted with test beam data and provide an attempt to explain the detector properties. The data-taking conditions such as temperature and potential detector inhomogeneities affect energy density measurements but have a small impact on detector efficiency