Formalized linear algebra over Elementary Divisor Rings in Coq

This paper presents a Coq formalization of linear algebra over elementary divisor rings, that is, rings where every matrix is equivalent to a matrix in Smith normal form. The main results are the formalization that these rings support essential operations of linear algebra, the classification theorem of finitely presented modules over such rings and the uniqueness of the Smith normal form up to multiplication by units. We present formally verified algorithms computing this normal form on a variety of coefficient structures including Euclidean domains and constructive principal ideal domains. We also study different ways to extend B\'ezout domains in order to be able to compute the Smith normal form of matrices. The extensions we consider are: adequacy (i.e. the existence of a gdco operation), Krull dimension $\leq 1$ and well-founded strict divisibility

Gradual Certified Programming in Coq

Expressive static typing disciplines are a powerful way to achieve high-quality software. However, the adoption cost of such techniques should not be under-estimated. Just like gradual typing allows for a smooth transition from dynamically-typed to statically-typed programs, it seems desirable to support a gradual path to certified programming. We explore gradual certified programming in Coq, providing the possibility to postpone the proofs of selected properties, and to check "at runtime" whether the properties actually hold. Casts can be integrated with the implicit coercion mechanism of Coq to support implicit cast insertion a la gradual typing. Additionally, when extracting Coq functions to mainstream languages, our encoding of casts supports lifting assumed properties into runtime checks. Much to our surprise, it is not necessary to extend Coq in any way to support gradual certified programming. A simple mix of type classes and axioms makes it possible to bring gradual certified programming to Coq in a straightforward manner.Comment: DLS'15 final version, Proceedings of the ACM Dynamic Languages Symposium (DLS 2015

Matrices à blocs et en forme canonique

International audienceNous présentons une formalisation réalisée avec Coq visant essentiellement à prouver l'existence des formes matricielles canoniques de Frobenius et de Jordan, ainsi que leurs propriétés. Nous définissons formellement des notions importantes, comme les matrices diagonales par blocs ou les matrices compagnes, et prouvons des résultats intermédiaires originaux, comme le théorème fondamental de similitude sur un corps, ou encore l'unicité de la forme normale de Smith. Outre la formalisation de la théorie de la réduction des endormorphismes des espaces vectoriels de dimension finie, ce travail ouvre la voie à la certification d'algorithmes efficaces de calcul du polynôme caractéristique ou de la forme de Frobenius

A refinement-based approach to computational algebra in COQ

International audienceWe describe a step-by-step approach to the implementation and formal verification of efficient algebraic algorithms. Formal specifications are expressed on rich data types which are suitable for deriving essential theoretical properties. These specifications are then refined to concrete implementations on more efficient data structures and linked to their abstract counterparts. We illustrate this methodology on key applications: matrix rank computation, Winograd's fast matrix product, Karatsuba's polynomial multiplication, and the gcd of multivariate polynomials

Refinements for Free!

International audienceFormal verification of algorithms often requires a choice be-tween definitions that are easy to reason about and definitions that are computationally efficient. One way to reconcile both consists in adopt-ing a high-level view when proving correctness and then refining stepwise down to an efficient low-level implementation. Some refinement steps are interesting, in the sense that they improve the algorithms involved, while others only express a switch from data representations geared towards proofs to more efficient ones geared towards computations. We relieve the user of these tedious refinements by introducing a framework where correctness is established in a proof-oriented context and automatically transported to computation-oriented data structures. Our design is gen-eral enough to encompass a variety of mathematical objects, such as rational numbers, polynomials and matrices over refinable structures. Moreover, the rich formalism of the Coq proof assistant enables us to develop this within Coq, without having to maintain an external tool

Full reduction at full throttle

International audienceEmerging trends in proof styles and new applications of interactive proof assistants exploit the computational facilities of the provided proof language, reaping enormous benefits in proof size and convenience to the user. However, the resulting proof objects really put the proof assistant to the test in terms of computational time required to check them. We present a novel translation of the terms of the full Calculus of (Co)Inductive Constructions to OCAML programs. Building on this translation, we further present a new fully featured version of COQ that offloads much of the computation required during proof checking to a vanilla, state of the art and fine tuned compiler. This modular scheme yields substantial performance improvements over existing systems at a reduced implementation cost. The work presented here builds on previous work described in [GL02], but we place particular emphasis in this paper on the fact that this scheme is in fact an instance of untyped normalization by evaluation [FR04, Lin05, AHN08, Boe10]

Tropane and related alkaloid skeletons via a radical [3+3]-annulation process.

Tropanes and related bicyclic alkaloids are highly attractive compounds possessing a broad biological activity. Here we report a mild and simple protocol for the synthesis of N-arylated 8-azabicyclo[3.2.1]octane and 9-azabicyclo[3.3.1]nonane derivatives. It provides these valuable bicyclic alkaloid skeletons in good yields and high levels of diastereoselectivity from simple and readily available starting materials using visible-light photoredox catalysis. These bicyclic aniline derivatives are hardly accessible via the classical Robinson tropane synthesis and represent a particularly attractive scaffold for medicinal chemistry. This unprecedented annulation process takes advantage of the unique reactivity of ethyl 2-(acetoxymethyl)acrylate as a 1,3-bis radical acceptor and of cyclic N,N-dialkylanilines as radical 1,3-bis radical donors. The success of this process relies on efficient electron transfer processes and highly selective deprotonation of aminium radical cations leading to the key α-amino radical intermediates

Formal Foundations of 3D Geometry to Model Robot Manipulators

International audienceWe are interested in the formal specification of safety properties of robot manipulators down to the mathematical physics. To this end, we have been developing a formalization of the mathematics of rigid body transformations in the COQ proof-assistant. It can be used to address the forward kinematics problem, i.e., the computation of the position and orientation of the end-effector of a robot manipulator in terms of the link and joint parameters. Our formalization starts by extending the Mathematical Components library with a new theory for angles and by developing three-dimensional geometry. We use these theories to formalize the foundations of robotics. First, we formalize a comprehensive theory of three-dimensional rotations, including exponentials of skew-symmetric matrices and quaternions. Then, we provide a formalization of the various representations of rigid body transformations: isometries, homogeneous representation, the Denavit-Hartenberg convention, and screw motions. These ingredients make it possible to formalize robot manipulators: we illustrate this aspect by an application to the SCARA robot manipulator

Foundational Property-Based Testing

International audienceIntegrating property-based testing with a proof assistant creates an interesting opportunity: reusable or tricky testing code can be formally verified using the proof assistant itself. In this work we introduce a novel methodology for formally verified property-based testing and implement it as a foundational verification framework for QuickChick, a port of QuickCheck to Coq. Our framework enables one to verify that the executable testing code is testing the right Coq property. To make verification tractable, we provide a systematic way for reasoning about the set of outcomes a random data generator can produce with non-zero probability, while abstracting away from the actual probabilities. Our framework is firmly grounded in a fully verified implementation of QuickChick itself, using the same underlying verification methodology. We also apply this methodology to a complex case study on testing an information-flow control abstract machine, demonstrating that our verification methodology is modular and scalable and that it requires minimal changes to existing code

Towards a certified computation of homology groups for digital images

International audienceIn this paper we report on a project to obtain a verified computation of homology groups of digital images. The methodology is based on program- ming and executing inside the COQ proof assistant. Though more research is needed to integrate and make efficient more processing tools, we present some examples partially computed in COQ from real biomedical images