The general orientation of our team is described by the short name given to it:
*Special Functions*, that is, particular mathematical functions that have
established names due to their importance in mathematical analysis, physics, and
other application domains. Indeed, we ambition to study special functions with
the computer, by combined means of computer algebra and formal methods.

Computer-algebra systems have been advertised for decades as software
for “doing mathematics by computer” . For
instance, computer-algebra libraries can uniformly generate a corpus
of mathematical properties about special functions, so as to display
them on an interactive website. This possibility was recently shown by the
computer-algebra component of the
team . Such
an automated generation significantly increases the reliability of the
mathematical corpus, in comparison to the content of existing static
authoritative handbooks. The importance of the validity of these
contents can be measured by the very wide audience that such handbooks
have had, to the point that a book
like remains one of the most cited
mathematical publications ever and has motivated the 10-year-long
project of writing its
successor .
However, can the mathematics produced “by computer” be considered as
*true* mathematics? More specifically, whereas it is nowadays
well established that the computer helps in discovering and observing
new mathematical phenomenons, can the mathematical statements produced
with the aid of the computer and the mathematical results computed by
it be accepted as valid mathematics, that is, as having the status of
mathematical *proofs*?
Beyond the reported weaknesses or
controversial design choices of mainstream computer-algebra systems,
the issue is more of an epistemological nature. It will not find its
solution even in the advent of the ultimate computer-algebra system:
the social process of peer-reviewing just falls short of evaluating
the results produced by computers, as reported by
Th. Hales after the publication of his proof of
the Kepler Conjecture about sphere packing.

A natural answer to this deadlock is to move to an alternative kind of
mathematical software and to use a proof assistant to check the
correctness of the desired properties or formulas. The recent success
of large-scale formalization projects, like the Four-Color Theorem of
graph theory , the above-mentioned Kepler
Conjecture , and, very recently, the Odd Order
Theorem of group theory

The Dynamic Dictionary of Mathematical Functions

The formal-proofs component of the team emanates from another project
of the MSR–Inria Joint Centre, namely the Mathematical Components
project (MathComp)

The present team takes benefit from these recent advances to explore the formal certification of the results collected in DDMF. The aim of this project is to concentrate the formalization effort on this delimited area, building on DDMF and the Algolib library, as well as on the Coq system and on the libraries developed by the MathComp project.

The following few opinions on computer algebra are, we believe, typical of computer-algebra users' doubts and difficulties when using computer-algebra systems:

Fredrik Johansson, expert in the multi-precision numerical evaluation
of special functions and in fast computer-algebra algorithms, writes
on his blog : “Mathematica is great for
cross-checking numerical values, but it's not unusual to run into
bugs, so *triple checking is a good habit*.” One answer in the
discussion is: “We can claim that Mathematica has [...] *an
impossible to understand semantics*: If Mathematica's output is
wrong then change the input. If you don't like the answer, change the
question. That seems to be the philosophy behind.”

Jacques Carette, former head of the maths group at Maplesoft, about a
bug when asking Maple to take the limit
`limit(f(n) * exp(-n), n = infinity)` for an undetermined
function `f`: “The problem is that there is an *implicit
assumption in the implementation* that unknown functions do not
`grow too fast'.”

As explained by the expert views above, complaints by computer-algebra users are often due to their misunderstanding of what a computer-algebra systems is, namely a purely syntactic tool for calculations, that the user must complement with a semantics. Still, robustness and consistency of computer-algebra systems are not ensured as of today, and, whatever Zeilberger may provocatively say in his Opinion 94 , a firmer logical foundation is necessary. Indeed, the fact is that many “bugs” in a computer-algebra system cannot be fixed by just the usual debugging method of tracking down the faulty lines in the code. It is sort of “by design”: assumptions that too often remain implicit are really needed by the design of symbolic algorithms and cannot easily be expressed in the programming languages used in computer algebra. A similar certification initiative has already been undertaken in the domain of numerical computing, in a successful manner , . It is natural to undertake a similar approach for computer algebra.

Some of the mathematical objects that interest our team are still totally untouched by formalization. When implementing them and their theory inside a proof assistant, we have to deal with the pervasive discrepancy between the published literature and the actual implementation of computer-algebra algorithms. Interestingly, this forces us to clarify our computer-algebraic view on them, and possibly make us discover holes lurking in published (human) proofs. We are therefore convinced that the close interaction of researchers from both fields, which is what we strive to maintain in this team, is a strong asset.

For a concrete example, the core of Zeilberger's creative telescoping manipulates rational functions up to simplifications. In summation applications, checking that these simplifications do not hide problematic divisions by 0 is most often left to the reader. In the same vein, in the case of integrals, the published algorithms do not check the convergence of all integrals, especially in intermediate calculations. Such checks are again left to the readers. In general, we expect to revisit the existing algorithms to ensure that they are meaningful for genuine mathematical sequences or functions, and not only for algebraic idealizations.

Another big challenge in this project originates in
the scientific difference between computer algebra and formal proofs.
Computer algebra seeks speed of calculation on *concrete
instances* of algebraic data structures (polynomials, matrices,
etc). For their part, formal proofs manipulate
symbolic expressions in terms of *abstract variables*
understood to represent generic elements of algebraic data
structures. In view of this, a continuous challenge is
to develop the right, hybrid thinking attitude that is able to
effectively manage concrete and abstract values simultaneously,
alternatively computing and proving with them.

Applications in combinatorics and mathematical physics frequently involve equations of so high orders and so large sizes, that computing or even storing all their coefficients is impossible on existing computers. Making this tractable is an extraordinary challenge. The approach we believe in is to design algorithms of good—ideally quasi-optimal—complexity in order to extract precisely the required data from the equations, while avoiding the computationally intractable task of completely expanding them into an explicit representation.

Typical applications with expected high impact are the automatic discovery and algorithmic proof of results in combinatorics and mathematical physics for which human proofs are currently unattainable.

The implementation of certified symbolic computations on special functions in the Coq proof assistant requires both investigating new formalization techniques and renewing the traditional computer-algebra viewpoint on these standard objects. Large mathematical objects typical of computer algebra occur during formalization, which also requires us to improve the efficiency and ergonomics of Coq. In order to feed this interdisciplinary activity with new motivating problems, we additionally pursue a research activity oriented towards experimental mathematics in application domains that involve special functions. We expect these applications to pose new algorithmic challenges to computer algebra, which in turn will deserve a formal-certification effort. Finally, DDMF is the motivation and the showcase of our progress on the certification of these computations. While striving to provide a formal guarantee of the correctness of the information it displays, we remain keen on enriching its mathematical content by developing new computer-algebra algorithms.

Our formalization effort consists in organizing a cooperation between a computer-algebra system and a proof assistant. The computer-algebra system is used to produce efficiently algebraic data, which are later processed by the proof assistant. The success of this cooperation relies on the design of appropriate libraries of formalized mathematics, including certified implementations of certain computer-algebra algorithms. On the other side, we expect that scrutinizing the implementation and the output of computer-algebra algorithms will shed a new light on their semantics and on their correctness proofs, and help clarifying their documentation.

The appropriate framework for the study of efficient algorithms for
special functions is *algebraic*.
Representing algebraic theories as Coq formal libraries
takes benefit from the methodology emerging from the success of
ambitious projects like the formal proof of a major classification
result in finite-group theory (the Odd Order
Theorem) .

Yet, a number of the objects we need to formalize in the present context has never been investigated using any interactive proof assistant, despite being considered as commonplaces in computer algebra. For instance there is up to our knowledge no available formalization of the theory of non-commutative rings, of the algorithmic theory of special-functions closures, or of the asymptotic study of special functions. We expect our future formal libraries to prove broadly reusable in later formalizations of seemingly unrelated theories.

Another peculiarity of the mathematical objects we are going to manipulate
with the Coq system is their size. In order to provide a formal guarantee
on the data displayed by DDMF, two related axes of research have to be
pursued.
First, efficient algorithms dealing with these large objects have
to be programmed and run in Coq.
Recent evolutions of the Coq system to improve the efficiency of
its internal computations , make this objective
reachable. Still, how to combine the aforementioned formalization
methodology with these cutting-edge evolutions of Coq remains
one of the prospective aspects of our project.
A second need is to help users *interactively*
manipulate large expressions occurring in their conjectures, an objective
for which little has been done so far. To address this need,
we work on improving the ergonomics of the system
in two ways: first, ameliorating the reactivity of Coq in its interaction
with the user; second, designing and implementing extensions of its
interface to ease our formalization activity. We expect the outcome of
these lines of research to be useful to a wider audience, interested in
manipulating large formulas on topics possibly unrelated to special functions.

Our algorithm certifications inside Coq intend to simulate
well-identified components of our Maple packages, possibly by
reproducing them in Coq. It would however not have been judicious to
re-implement them inside Coq in a systematic way. Indeed for a number of its
components, the output of the algorithm is more easily checked than
found, like for instance the solving of a linear system.
Rather, we delegate the discovery of the solutions to an
external, untrusted oracle like Maple. Trusted computations inside
Coq then formally validate the correctness of the a priori
untrusted output. More often than not, this validation consists in
implementing and executing normalization procedures *inside*
Coq. A challenge of this automation is to make sure they go to scale
while remaining efficient, which requires a Coq version of
non-trivial computer-algebra algorithms. A first, archetypal example we expect to
work on is a non-commutative generalization of the normalization
procedure for elements of rings .

Generally speaking, we design algorithms for manipulating special functions symbolically, whether univariate or with parameters, and for extracting algorithmically any kind of algebraic and analytic information from them, notably asymptotic properties. Beyond this, the heart of our research is concerned with parametrised definite summations and integrations. These very expressive operations have far-ranging applications, for instance, to the computation of integral transforms (Laplace, Fourier) or to the solution of combinatorial problems expressed via integrals (coefficient extractions, diagonals). The algorithms that we design for them need to really operate on the level of linear functional systems, differential and of recurrence. In all cases, we strive to design our algorithms with the constant goal of good theoretical complexity, and we observe that our algorithms are also fast in practice.

Our long-term goal is to design fast algorithms for a general method
for special-function integration (*creative telescoping*), and
make them applicable to general special-function inputs. Still, our
strategy is to proceed with simpler, more specific classes first
(rational functions, then algebraic functions, hyperexponential
functions, D-finite functions, non-D-finite functions; two variables,
then many variables); as well, we isolate analytic questions by
first considering types of integration with a more purely algebraic
flavor (constant terms, algebraic residues, diagonals of
combinatorics). In particular, we expect to extend our recent
approach to more general classes
(algebraic with nested radicals, for example): the idea is to speed up
calculations by making use of an analogue of Hermite reduction that avoids
considering certificates.
Homologous problems for summation will be addressed as well.

As a consequence of our complexity-driven approach to algorithms design, the algorithms mentioned in the previous paragraph are of good complexity. Therefore, they naturally help us deal with applications that involve equations of high orders and large sizes.

With regard to combinatorics, we expect to advance the algorithmic classification of combinatorial classes like walks and urns. Here, the goal is to determine if enumerative generating functions are rational, algebraic, or D-finite, for example. Physical problems whose modelling involves special-function integrals comprise the study of models of statistical mechanics, like the Ising model for ferro-magnetism, or questions related to Hamiltonian systems.

Number theory is another promising domain of applications. Here, we attempt an experimental approach to the automated certification of integrality of the coefficients of mirror maps for Calabi–Yau manifolds. This could also involve the discovery of new Calabi–Yau operators and the certification of the existing ones. We also plan to algorithmically discover and certify new recurrences yielding good approximants needed in irrationality proofs.

It is to be noted that in all of these application domains, we would so far use general algorithms, as was done in earlier works of ours , , . To push the scale of applications further, we plan to consider in each case the specifics of the application domain to tailor our algorithms.

In continuation of our past project of an encyclopedia at
http://

the algorithmic discussion of equations with parameters, leading to certified automatic case analysis based on arithmetic properties of the parameters;

lists of summation and integral formulas involving special functions, including validity conditions on the parameters;

guaranteed large-precision numerical evaluations.

Computer algebra manipulates symbolic representations of exact mathematical objects in a computer, in order to perform computations and operations like simplifying expressions and solving equations for “closed-form expressions”. The manipulations are often fundamentally of algebraic nature, even when the ultimate goal is analytic. The issue of efficiency is a particular one in computer algebra, owing to the extreme swell of the intermediate values during calculations.

Our view on the domain is that research on the algorithmic manipulation of special functions is anchored between two paradigms:

adopting linear differential equations as the right data structure for special functions,

designing efficient algorithms in a complexity-driven way.

It aims at four kinds of algorithmic goals:

algorithms combining functions,

functional equations solving,

multi-precision numerical evaluations,

guessing heuristics.

This interacts with three domains of research:

computer algebra, meant as the search for quasi-optimal algorithms for exact algebraic objects,

symbolic analysis/algebraic analysis;

experimental mathematics (combinatorics, mathematical physics, ...).

This view is made explicit in the present section.

Numerous special functions satisfy linear differential and/or
recurrence equations. Under a mild technical condition, the existence
of such equations induces a finiteness property that makes the main
properties of the functions decidable. We thus speak of
*D-finite functions*. For example, 60 % of the chapters in the
handbook describe D-finite functions.
In addition, the class is closed under a rich set of algebraic operations.
This makes linear functional equations just the right data structure
to encode and manipulate special functions. The power of this
representation was observed in the early
1990s , leading to the design of many
algorithms in computer algebra.
Both on the theoretical and algorithmic sides, the study of D-finite
functions shares much with neighbouring mathematical domains:
differential algebra,
D-module theory,
differential Galois theory,
as well as their counterparts for recurrence equations.

Differential/recurrence equations that define special functions can be
recombined to define: additions and
products of special functions; compositions of special functions;
integrals and sums involving special functions. Zeilberger's fast
algorithm for obtaining recurrences satisfied by parametrised binomial
sums was developed in the early 1990s already .
It is the basis of all modern definite summation and integration
algorithms. The theory was made fully rigorous and algorithmic in
later works, mostly by a group in Risc (Linz, Austria) and by members
of the
team , , , , , .
The past ÉPI Algorithms contributed several implementations
(*gfun* ,
*Mgfun* ).

Encoding special functions as defining linear functional equations postpones some of the difficulty of the problems to a delayed solving of equations. But at the same time, solving (for special classes of functions) is a sub-task of many algorithms on special functions, especially so when solving in terms of polynomial or rational functions. A lot of work has been done in this direction in the 1990s; more intensively since the 2000s, solving differential and recurrence equations in terms of special functions has also been investigated.

A major conceptual and algorithmic difference exists for numerical
calculations between data structures that fit on a machine word and
data structures of arbitrary length, that is, *multi-precision*
arithmetic. When multi-precision floating-point numbers became
available, early works on the evaluation of special functions were
just promising that “most” digits in the output were correct, and
performed by heuristically increasing precision during intermediate
calculations, without intended rigour. The original theory
has evolved in a
twofold way since the 1990s:
by making computable all constants hidden in asymptotic
approximations, it became possible to guarantee a *prescribed*
absolute precision; by employing state-of-the-art algorithms on
polynomials, matrices, etc, it became possible to have evaluation
algorithms in a time complexity that is linear in the output size, with a
constant that is not more than a few units.
On the implementation side, several original works
exist, one of which (*NumGfun* ) is
used in our DDMF.

“Differential approximation”, or “Guessing”, is an operation to get an ODE likely to be satisfied by a given approximate series expansion of an unknown function. This has been used at least since the 1970s and is a key stone in spectacular applications in experimental mathematics . All this is based on subtle algorithms for Hermite–Padé approximants . Moreover, guessing can at times be complemented by proven quantitative results that turn the heuristics into an algorithm . This is a promising algorithmic approach that deserves more attention than it has received so far.

The main concern of computer algebra has long been to prove the feasibility of a given problem, that is, to show the existence of an algorithmic solution for it. However, with the advent of faster and faster computers, complexity results have ceased to be of theoretical interest only. Nowadays, a large track of works in computer algebra is interested in developing fast algorithms, with time complexity as close as possible to linear in their output size. After most of the more pervasive objects like integers, polynomials, and matrices have been endowed with fast algorithms for the main operations on them , the community, including ourselves, started to turn its attention to differential and recurrence objects in the 2000s. The subject is still not as developed as in the commutative case, and a major challenge remains to understand the combinatorics behind summation and integration. On the methodological side, several paradigms occur repeatedly in fast algorithms: “divide and conquer” to balance calculations, “evaluation and interpolation” to avoid intermediate swell of data, etc. .

Handbooks collecting mathematical properties aim at serving as
reference, therefore trusted, documents. The decision of
several authors or maintainers of such knowledge bases to move from paper
books , , to websites and wikis

Several attempts have been made in order to extend existing computer-algebra systems with symbolic manipulations of logical formulas. Yet, these works are more about extending the expressivity of computer-algebra systems than about improving the standards of correctness and semantics of the systems. Conversely, several projects have addressed the communication of a proof system with a computer-algebra system, resulting in an increased automation available in the proof system, to the price of the uncertainty of the computations performed by this oracle.

More ambitious projects have tried to design a new computer-algebra system providing an environment where the user could both program efficiently and elaborate formal and machine-checked proofs of correctness, by calling a general-purpose proof assistant like the Coq system. This approach requires a huge manpower and a daunting effort in order to re-implement a complete computer-algebra system, as well as the libraries of formal mathematics required by such formal proofs.

The move to machine-checked proofs of the mathematical correctness of the output of computer-algebra implementations demands a prior clarification about the often implicit assumptions on which the presumably correctly implemented algorithms rely. Interestingly, this preliminary work, which could be considered as independent from a formal certification project, is seldom precise or even available in the literature.

A number of authors have investigated ways to organize the communication of a chosen computer-algebra system with a chosen proof assistant in order to certify specific components of the computer-algebra systems, experimenting various combinations of systems and various formats for mathematical exchanges. Another line of research consists in the implementation and certification of computer-algebra algorithms inside the logic , , or as a proof-automation strategy. Normalization algorithms are of special interest when they allow to check results possibly obtained by an external computer-algebra oracle . A discussion about the systematic separation of the search for a solution and the checking of the solution is already clearly outlined in .

Significant progress has been made in the certification of numerical applications by formal proofs. Libraries formalizing and implementing floating-point arithmetic as well as large numbers and arbitrary-precision arithmetic are available. These libraries are used to certify floating-point programs, implementations of mathematical functions and for applications like hybrid systems.

To be checked by a machine, a proof needs to be expressed in a constrained, relatively simple formal language. Proof assistants provide facilities to write proofs in such languages. But, as merely writing, even in a formal language, does not constitute a formal proof just per se, proof assistants also provide a proof checker: a small and well-understood piece of software in charge of verifying the correctness of arbitrarily large proofs. The gap between the low-level formal language a machine can check and the sophistication of an average page of mathematics is conspicuous and unavoidable. Proof assistants try to bridge this gap by offering facilities, like notations or automation, to support convenient formalization methodologies. Indeed, many aspects, from the logical foundation to the user interface, play an important role in the feasibility of formalized mathematics inside a proof assistant.

While many logical foundations for mathematics have been proposed, studied, and implemented, type theory is the one that has been more successfully employed to formalize mathematics, to the notable exception of the Mizar system , which is based on set theory. In particular, the calculus of construction (CoC) and its extension with inductive types (CIC) , have been studied for more than 20 years and been implemented by several independent tools (like Lego, Matita, and Agda). Its reference implementation, Coq , has been used for several large-scale formalizations projects (formal certification of a compiler back-end; four-color theorem). Improving the type theory underlying the Coq system remains an active area of research. Other systems based on different type theories do exist and, whilst being more oriented toward software verification, have been also used to verify results of mainstream mathematics (prime-number theorem; Kepler conjecture).

The most distinguishing feature of CoC is that computation is promoted to the status of rigorous logical argument. Moreover, in its extension CIC, we can recognize the key ingredients of a functional programming language like inductive types, pattern matching, and recursive functions. Indeed, one can program effectively inside tools based on CIC like Coq. This possibility has paved the way to many effective formalization techniques that were essential to the most impressive formalizations made in CIC.

Another milestone in the promotion of the computations-as-proofs feature of Coq has been the integration of compilation techniques in the system to speed up evaluation. Coq can now run realistic programs in the logic, and hence easily incorporates calculations into proofs that demand heavy computational steps.

Because of their different choice for the underlying logic, other proof assistants have to simulate computations outside the formal system, and indeed fewer attempts to formalize mathematical proofs involving heavy calculations have been made in these tools. The only notable exception, which was finished in 2014, the Kepler conjecture, required a significant work to optimize the rewriting engine that simulates evaluation in Isabelle/HOL.

Programs run and proved correct inside the logic are especially useful for the conception of automated decision procedures. To this end, inductive types are used as an internal language for the description of mathematical objects by their syntax, thus enabling programs to reason and compute by case analysis and recursion on symbolic expressions.

The output of complex and optimized programs external
to the proof assistant can also be stamped with a formal proof of
correctness when their result is easier to *check* than to
*find*. In that case one can benefit from their efficiency
without compromising the level of confidence on their output at the
price of writing and certify a
checker inside the logic. This approach, which has been successfully
used in various contexts,
is very relevant to the present research project.

Representing abstract algebra in a proof assistant has been studied for long. The libraries developed by the MathComp project for the proof of the Odd Order Theorem provide a rather comprehensive hierarchy of structures; however, they originally feature a large number of instances of structures that they need to organize. On the methodological side, this hierarchy is an incarnation of an original work based on various mechanisms, primarily type inference, typically employed in the area of programming languages. A large amount of information that is implicit in handwritten proofs, and that must become explicit at formalization time, can be systematically recovered following this methodology.

The MathComp library was consistently designed after uniform principles of software engineering. These principles range from simple ones, like naming conventions, to more advanced ones, like generic programming, resulting in a robust and reusable collection of formal mathematical components. This large body of formalized mathematics covers a broad panel of algebraic theories, including of course advanced topics of finite group theory, but also linear algebra, commutative algebra, Galois theory, and representation theory. We refer the interested reader to the online documentation of these libraries , which represent about 150,000 lines of code and include roughly 4,000 definitions and 13,000 theorems.

Topics not addressed by these libraries and that might be relevant to the present project include real analysis and differential equations. The most advanced work of formalization on these domains is available in the HOL-Light system , , , although some existing developments of interest , are also available for Coq. Another aspect of the MathComp libraries that needs improvement, owing to the size of the data we manipulate, is the connection with efficient data structures and implementations, which only starts to be explored.

The user of a proof assistant describes the proof he wants to formalize in the system using a textual language. Depending on the peculiarities of the formal system and the applicative domain, different proof languages have been developed. Some proof assistants promote the use of a declarative language, when the Coq and Matita systems are more oriented toward a procedural style.

The development of the large, consistent body of MathComp libraries has prompted the need to design an alternative and coherent language extension for the Coq proof assistant , , enforcing the robustness of proof scripts to the numerous changes induced by code refactoring and enhancing the support for the methodology of small-scale reflection.

The development of large libraries is quite a novelty for the Coq system. In particular any long-term development process requires the iteration of many refactoring steps and very little support is provided by most proof assistants, with the notable exception of Mizar . For the Coq system, this is an active area of research.

Pierre Lairez has been awarded this year the “Ecole Polytechnique thesis prize”, for his PhD thesis defended in 2014 .

Keywords: Proof - Certification - Formalisation

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, ...

Participants: Benjamin Grégoire, Enrico Tassi, Bruno Barras, Yves Bertot, Pierre Boutillier, Xavier Clerc, Pierre Courtieu, Maxime Denes, Stéphane Glondu, Vincent Gross, Hugo Herbelin, Pierre Letouzey, Assia Mahboubi, Julien Narboux, Jean-Marc Notin, Christine Paulin-Mohring, Pierre-Marie Pédrot, Loïc Pottier, Matthias Puech, Yann Régis-Gianas, François Ripault, Matthieu Sozeau, Arnaud Spiwack, Pierre-Yves Strub, Benjamin Werner, Guillaume Melquiond and Jean-Christophe Filliâtre

Partners: CNRS - Université Paris-Sud - ENS Lyon - Université Paris-Diderot

Contact: Hugo Herbelin

URL: http://

Dynamic Mathematics on the Web

Functional Description

Programming tool for controlling the generation of mathematical websites that embed dynamical mathematical contents generated by computer-algebra calculations. Implemented in OCaml.

Participants: Frédéric Chyzak, Alexis Darrasse and Maxence Guesdon

Contact: Frédéric Chyzak

Encyclopedia of Combinatorial Structures

Functional Description

On-line mathematical encyclopedia with an emphasis on sequences that arise in the context of decomposable combinatorial structures, with the possibility to search by the first terms in the sequence, keyword, generating function, or closed form.

Participants: Stéphanie Petit, Alexis Darrasse, Frédéric Chyzak and Maxence Guesdon

Contact: Frédéric Chyzak

Mathematical Components library

Functional Description

The Mathematical Components library is a set of Coq libraries that cover the mechanization of the proof of the Odd Order Theorem.

Participants: Andrea Asperti, Jeremy Avigad, Yves Bertot, Cyril Cohen, François Garillot, Georges Gonthier, Stéphane Le Roux, Assia Mahboubi, Sidi Ould Biha, Ioana Pasca, Laurence Rideau, Alexey Solovyev, Enrico Tassi and Russell O'connor

Contact: Assia Mahboubi

URL: http://

Functional Description

Coq normalization tool and decision procedure for expressions in commutative ring theories. Implemented in Coq and OCaml. Integrated in the standard distribution of the Coq proof assistant since 2005.

Contact: Assia Mahboubi

Functional Description

Ssreflect is a tactic language extension to the Coq system, developed by the Mathematical Components team.

Participants: Cyril Cohen, Yves Bertot, Laurence Rideau, Enrico Tassi, Laurent Théry, Assia Mahboubi and Georges Gonthier

Contact: Yves Bertot

Periods of rational integrals are specific integrals, with respect to one or
several variables, whose integrand is a rational function and whose domain of
integration is closed. This particular class of integrals contains large
families of functions naturally occurring in combinatorics and statistical
physics, such as diagonals, constant terms and positive part of rational
functions. Periods involving one parameter are classically known to satisfy
*Picard-Fuchs equations*, a special type of linear differential equations
with a very rich analytic and arithmetic structure. As for other
special-function manipulations, handling periods through those differential
equations is a good way to actually compute them, and this was the topic of
Pierre Lairez' PhD thesis defended in 2014 and
awarded the “Ecole Polytechnique thesis prize” in 2015.

Computing multivariate integrals is one speciality of the team and our algorithms are known to treat much more general integrals than just periods of rational integrals. However, integration is still slow in practice when the number of variables goes increasing. By looking at periods of rational functions, the hope is to obtain relevant complexity bounds and faster algorithms.

The goal of reaching relevant theoretical complexity bounds had been reached in 2013 but a practically fast algorithm was still missing. This year, we described a new algorithm which is efficient in practice , though its complexity is not known. This algorithm allows to compute quickly integrals that are too big to be computed with previous algorithms. As a challenging benchmark, we computed 210 integrals given by Batyrev and Kreuzer in their work on Calabi–Yau varieties. This achievement gave strong visibility to the paper and allowed a quick dissemination of the implementation, which is provided in Magma under a CeCILL B license. The algorithm is now used on a regular basis by several teams. We know of:

Tom Coates' team (Dpt. of Mathematics, Imperial College, London, UK), which uses the software in their work about mirror symmetry and classification of Fano varieties;

Duco van Straten (Institute of Mathematics, University of Mainz, Germany), who uses the software in his work in algebraic geometry;

Gert Alkmvist (Dpt. of Mathematics, University of Lund, Sweden), who uses the software in his work of enumerating the Calabi–Yau differential equations.

Multiple binomial sums form a large class of multi-indexed sequences, closed under partial summation, which contains most of the sequences obtained by multiple summation of binomial coefficients and also all the sequences with algebraic generating function. We study in the representation of the generating functions of binomial sums by integrals of rational functions. The outcome is twofold. Firstly, we show that a univariate sequence is a multiple binomial sum if and only if its generating function is the diagonal of a rational function. Secondly we propose algorithms that decide the equality of multiple binomial sums and that compute recurrence relations for them. In conjunction with geometric simplifications of the integral representations, this approach behaves well in practice. The process avoids the computation of certificates and the problem of accurate summation that afflicts discrete creative telescoping, both in theory and in practice.

Diagonals of rational functions naturally occur in lattice statistical mechanics and enumerative combinatorics. In all the examples emerging from physics, the minimal linear differential operators annihilating these diagonals of rational functions have been shown to actually possess orthogonal or symplectic differential Galois groups. In order to understand the emergence of such orthogonal or symplectic groups, we exhaustively analyze in three (constrained) sets of diagonals of rational functions, corresponding respectively to rational functions of three variables, four variables and six variables. The conclusion is that, even for these sets of examples which, at first sight, have no relation with physics, their differential Galois groups are always orthogonal or symplectic groups. We also discuss conditions on the rational functions such that the operators annihilating their diagonals do not correspond to orthogonal or symplectic differential Galois groups, but rather to generic special linear groups.

The diagonal of a multivariate power series

Counting lattice paths obeying various geometric constraints is a classical
topic in combinatorics and probability theory. Many recent works deal with the
enumeration of 2-dimensional walks with prescribed steps confined to the
positive quadrant. A notoriously difficult case concerns the so-called
*Gessel walks*: they are planar walks confined to the positive quarter
plane, that move by unit steps in any of the following directions: West,
North-East, East and South-West. In 2001, Ira Gessel conjectured a closed-form
expression for the number of such walks of a given length starting and ending
at the origin. In 2008, Kauers, Koutschan and Zeilberger gave a computer-aided
proof of this conjecture. The same year, Bostan and Kauers showed, using again
computer algebra tools, that the trivariate generating function of Gessel
walks is algebraic. We propose in the first “human
proofs” of these results. They are derived from a new expression for the
generating function of Gessel walks in terms of special functions. This work
has been published in the prestigious journal *Transactions of the AMS*.

Small step walks in 2D are by now quite well understood, but almost everything remains to be done in higher dimensions. We explored in the classification problem for 3-dimensional walks with unit steps confined to the positive octant. The first difficulty is their number: there are 11 074 225 cases (instead of 79 in dimension 2). In our work, we focused on the 35 548 that have at most six steps. We applied to them a combined approach, first experimental and then rigorous. Among the 35 548 cases, we first found 170 cases with a finite group; in the remaining cases, our experiments suggest that the group is infinite. We then rigorously proved D-finiteness of the generating series in all the 170 cases, with the exception of 19 intriguing step sets for which the nature of the generating function still remains unclear. In two challenging cases, no human proof is currently known, and we derived computer-algebra proofs, thus constituting the first proofs for those two step sets.

The

Goal-directed proof search in first-order logic uses meta-variables to delay the choice of witnesses; substitutions for such variables are produced when closing proof-tree branches, using first-order unification or a theory-specific background reasoner. We have investigated a generalization of such mechanisms whereby theory-specific constraints are produced instead of substitutions. In order to design modular proof-search procedures over such mechanisms, we provide a sequent calculus with meta-variables, which manipulates such constraints abstractly. Proving soundness and completeness of the calculus leads to an acclimatization that identifies the conditions under which abstract constraints can be generated and propagated in the same way unifiers usually are. We then extract from our abstract framework a component interface and a specification for concrete implementations of background reasoners. This is a common work with Damien Rouhling (ENS Lyon), Stéphane Lengrand (CNRS, LIX) and Jean-Marc Notin (CNRS, LIX), based on the PhD contributions of Mahfuza Farooque (unaffiliated). It is described in .

The interactivity needed by our on-line encyclopedia DDMF is made possible
by implementing it over our tool DynaMoW
(http://

The Encyclopedia of Combinatorial Structures (ECS,
http://

We have released a new version of the Mathematical Components
Library
(http://

*Mathematical Components* (project of the MSR–Inria
Joint Centre).

Goal: Investigate the design of large-scale, modular and reusable libraries of formalized mathematics, using the Coq proof assistant. This project successfully formalized the proof of the Odd Order Theorem, resulting in a corpus of libraries related to various areas of algebra.

Leader: G. Gonthier (MSR Cambridge). Participants: F. Chyzak, A. Mahboubi.

Website:
http://

**ParalITP** (ANR-11-INSE-001).

Goal: Improve the performances and the ergonomics of interactive provers by taking advantage of modern, parallel hardware.

Leader: B. Wolff (University of Orsay, Paris Paris-Sud). Participants: A. Mahboubi, C. Tankink.

Website: http://

**FastRelax** (ANR-14-CE25-0018).

Goal: Develop computer-aided proofs of numerical values, with certified and reasonably tight error bounds, without sacrificing efficiency.

Leader: B. Salvy (Inria, ÉNS Lyon). Participants: A. Mahboubi, Th. Sibut-Pinote.

Website: http://

Program: COST

Project acronym: EUTYPES (CA15123)

Project title: The European research network on types for programming and verification

Duration: October 2015 - October 2019

Coordinator: Herman Geuvers (Radboud University, Nijmegen, the Netherlands)

Other partners: Czech Republic, Estonia, Macedonia, Germany, Greece, the Netherlands, Norway, Poland, Serbia, Slovenia, United Kingdom.

Abstract: Types are pervasive in programming and information technology. A type defines a formal interface between software components, allowing the automatic verification of their connections, and greatly enhancing the robustness and reliability of computations and communications. In rich dependent type theories, the full functional specification of a program can be expressed as a type. Type systems have rapidly evolved over the past years, becoming more sophisticated, capturing new aspects of the behaviour of programs and the dynamics of their execution. This COST Action will give a strong impetus to research on type theory and its many applications in computer science, by promoting: (1) the synergy between theoretical computer scientists, logicians and mathematicians to develop new foundations for type theory, for example as based on the recent development of “homotopy type theory”, (2) the joint development of type theoretic tools as proof assistants and integrated programming environments, (3) the study of dependent types for programming and its deployment in software development, (4) the study of dependent types for verification and its deployment in software analysis and verification. The action will also tie together these different areas and promote cross-fertilisation. Europe has a strong type theory community, ranging from foundational research to applications in programming languages, verification and theorem proving, which is in urgent need of better networking. A COST Action that crosses the borders will support the collaboration between groups and complementary expertise, and mobilise a critical mass of existing type theory research.

A. Bostan has served in the organizing committee of the
*Journées Nationales de Calcul Formel* (JNCF 2015), the
annual meeting of the French computer algebra community.

A. Mahboubi has served in the organizing and scientific
committees of *Mathematics, Algorithms and Proofs*
(MAP 2016).

A. Bostan is part of the Scientific advisory board of the
conference series *Conference on Effective Methods in
Algebraic Geometry* (MEGA).

F. Chyzak has served as a conference program committee member
for the *Conference on Intelligent Computer Mathematics*
(CICM 2015).

A. Mahboubi has served as a program committee member
for the *25th International Conference on Automated
Deduction* (CADE 25).

A. Mahboubi has served as a program committee member
for the *21st International Conference on Types for
Proofs and Programs* (TYPES 2015).

A. Mahboubi has served as a program committee member
for the *Workshop on Logical Frameworks and Meta-Languages: Theory
and Practice* (LFMTP 2015).

A. Bostan has served as reviewer for the *International
Symposium on Symbolic and Algebraic Computation* (ISSAC 2015).

F. Chyzak has served as reviewer for the *International
Symposium on Symbolic and Algebraic Computation* (ISSAC 2015).

A. Mahboubi has served as reviewer for the international
conferences *NASA Formal Methods Symposium* (NFM 2015),
*Certified Programs and Proofs* (CPP 2015), *Typed
Lambda Calculi and Applications* (TLCA 2015),
*Conference on Intelligent Computer Mathematics* (CICM
2015) and for the national conference *Journées Nationales
des Langages Applicatifs* (JFLA 2015).

Th. Sibut-Pinote has served as reviewer for the
*International Conference on Automated Deduction* (CADE
2015).

A. Bostan has served as reviewer for the Journal of Symbolic Computation, the Journal of Complexity and Applicable Algebra in Engineering, Communication and Computing.

F. Chyzak has served several times as a reviewer for the Journal of Symbolic Computation.

A. Mahboubi has served as a reviewer for the Journal of Formalized Reasoning and several times for Journal of Automated Reasoning.

A. Bostan has given an invited 3-hours lecture at the
Séminaire Lotharingien de Combinatoire in Ellwangen, Germany (March 2015)
and another 3-hours lecture at the *SFB-Workshop on Restricted Lattice Walks* in RISC, Hagenberg, Austria (May 2015).

A. Bostan has given an invited talk in the conference *Automatic Sequences, Number Theory, and Aperiodic Order*, held at the Technical University of Delft, Netherlands (Oct 2015).

A. Bostan has given a talk during the *Thematic Program
on Computer Algebra* (Fields Institute, Toronto, Canada,
September 2015).

F. Chyzak has given a number of talks on his ongoing work
(joint with A. Bostan of the team) on obtaining hypergeometric
closed-form expressions in the enumerative combinatorics of
walks: *Functional Equations in Limoges* (Limoges, March
2015), *Thematic Program on Computer Algebra* (Fields
Institute, Toronto, Canada, September 2015), *Séminaire
Philippe Flajolet* (Institut Henri Poincaré, Paris, October
2015).

A. Mahboubi has given an invited talk common to the conferences
*14th Asian Logic Conference* (ALC 15) and
*6th Indian Conference on Logic and its Application*
(ICLA 15), in Mumbai, India (January 2015).

A. Mahboubi has given an invited talk to the Workshop on Homotopy Type Theory / Univalent Foundations, satellite of the International Conference on Rewriting, Deduction, and Programming, in Warsaw, Poland (July 2015).

A. Mahboubi has given a talk during the
*Thematic Program on Computer Algebra* (Fields
Institute, Toronto, Canada, December 2015).

F. Chyzak is member of the steering committee of the
*Journées Nationales de Calcul Formel* (JNCF 2015), the
annual meeting of the French computer algebra community.

A. Mahboubi has been nominated as a member of the managment
committee the COST action EUTYPES (CA15123) *The European
research network on types for programming and
verification*, coordinated by Herman Geuvers.

A. Mahboubi is a member of the *Commission
Scientifique* of the Inria–Saclay-Île-de-France center.

Master: A. Bostan, *Algorithmes efficaces en calcul formel*, 18h, M2, MPRI, France

Master: F. Chyzak, *Algorithmes efficaces en calcul
formel*, 4.5h, M2, MPRI, France

Master: A. Mahboubi, *Assistants de preuve*, 18h, M2, MPRI,
France

License: L. Dumont, various courses, 64h, Université Paris-Sud, France.

License: Th. Sibut-Pinote, various courses, 64h, École Polytechnique, France.

PhD in progress: L. Dumont, *Algorithmique
efficace pour les diagonales, applications en
combinatoire, physique et théorie des nombres*, École
Polytechnique, started in September 2013, supervised by
A. Bostan and B. Salvy.

PhD in progress: Th. Sibut-Pinote, *Calcul numérique
et démonstrations mathématiques, de la rigueur à la preuve
formelle*, École Polytechnique, started in September 2014,
supervised by A. Mahboubi.

Master intership in progress (M1): G. Boisseau and
Th. Huffschmitt,
*Combination of decision procedures in presence of
meta-variables*, École
Polytechnique, supervised by A. Mahboubi (jointly with
S. Graham-Lengrand from LIX).

A. Bostan has served as a jury member of the French *Agrégation de Mathématiques – épreuve de modélisation, option C*.

A. Bostan has served as an examiner in the PhD jury of Cuang Tran, *Calcul formel dans la base des polynômes unitaires de Chebyshev*, Université Paris 6, October 9, 2015.

F. Chyzak has served as an examiner in the PhD jury of Suzy Maddah,
*Formal Reduction of Differential Systems*, Université de Limoges,
September 25, 2015.

A. Mahboubi has served as an examiner in the half-way
PhD defense of Pierre Boutry, *Learning environment for interactive
proof in geometry*, University of Strasbourg, June 15th, 2015.

A. Bostan has published, together with Kilian Raschel, a
popularization article titled *Compter les excursions sur un
échiquier* in the popular science magazine *Pour la Science*, the
French version of the *Scientific American*.