Jesús Aransay

Universidad de La Rioja (Spain) | UNIRIOJA · Mathematics and Computation

In recent decades, the use of information systems has become widespread, thanks to greater data availability, as well as the improvement of the power and capacity of computer systems and programs. Specifically, the management of geographical data through the so-called Geographic Information Systems (GIS) has brought about a revolution in the abilit...

La utilización de datos geográficos y programas informáticos específicos como los Sistemas de Información Geográfica (SIG) es muy común en disiciplinas de muy diversa índole dentro de la planificación de grados y masters universitarios. De hecho, en el caso de la Universidad de La Rioja, se utilizan en diez titulaciones de las 26 ofertadas, pertene...

In this paper we show how a thoughtful reusing of libraries can provide concise proofs of non-trivial mathematical results. Concretely, we formalise in Isabelle/HOL a proof of the Fundamental Theorem of Linear Algebra for vector spaces over inner product spaces, the Gram–Schmidt process of orthogonalising vectors over \(\mathbb {R}\), its applicati...

In this contribution we present a formalised algorithm in the Isabelle/HOL proof assistant to compute echelon forms, and, as a consequence, characteristic polynomials of matrices. We have proved its correctness over Bézout domains, but its executability is only guaranteed over Euclidean domains, such as the integer ring and the univariate polynomia...

We present the execution tests and benchmarks of some Linear Algebra programs generated from their veri fied formalisation in Isabelle/HOL; more concretely, the Gauss-Jordan algorithm and the QR decomposition, together with the techniques used to improve the performance of the extracted code, are described.

We present the execution tests and benchmarks of some Linear Algebra programs generated from their veri fied formalisation in Isabelle/HOL; more concretely, the Gauss-Jordan algorithm and the QR decomposition, together with the techniques used to improve the performance of the extracted code, are described.

The HOL Multivariate Analysis Library (HMA) of Isabelle/HOL is focused on concrete types such as $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{R}^n$ and on algebraic structures such as real vector spaces and Euclidean spaces, represented by means of type classes. The generalization of HMA to more abstract algebraic structures is something desirable but i...

In this paper, we present a formalisation in a proof assistant, Isabelle/HOL, of a naive version of the Gauss-Jordan algorithm, with explicit proofs of some of its applications; and, additionally, a process to obtain versions of this algorithm in two different functional languages (SML and Haskell) by means of code generation techniques from the ve...

In this work, we present an interoperability framework that enables the translation of specifications (signature of functions and lemma statements) among different theorem provers. This translation is based on a new intermediate XML language, called XLL, and is performed almost automatically. As a case study, we focus on porting developments from I...

Computer Algebra systems are widely spread because of some of their remarkable features such as their ease of use and performance. Nonetheless, this focus on performance sometimes leads to unwanted consequences: algorithms and computations are implemented and carried out in a way which is sometimes not transparent to the users, and that can lead to...

In this work we present a formalization of the \emph{Rank Nullity} theorem of Linear Algebra in Isabelle/HOL. The formalization is of interest because of various reasons. First, it has been carried out based on the representation of mathematical structures proposed in the HOL Multivariate Analysis library of Isabelle/HOL (which is part of the stand...

In this paper we describe the performance results that have been obtained after executing a verified Linear Algebra SML program automatically generated from an Isabelle/HOL formalization. This SML program computes the reduced row echelon form of a matrix, using the well-known Gauss-Jordan algorithm. We explain how the code generator of Isabelle has...

In this work we present the formalization of some well-known results in Abstract Algebra with the proof assistant Isabelle/HOL. Our interest focuses in finite-dimensional vector spaces, and our proposal is to closely follow the presentation made in a popular book of the field by P. R. Halmos. The main result in this work proves that every finite-di...

In this article, two different mechanized reasoning tools (Coq and Isabelle/HOL) are used in order to prove some basic but significant properties extracted from a symbolic computation system called Kenzo. In particular, we focused on a property called ‘cancellation theorem’, which can be applied to the proof of the idempotence property of a differe...

We apply current theorem proving technology to certified code in the domain of abstract algebra. More concretely, based on a formal proof of the Basic Perturbation Lemma (a central result in homological algebra) in the prover Isabelle/HOL, we apply various code generation techniques, which lead to certified implementations of the associated algorit...

In this work we propose a representation of graded algebraic structures and morphisms over them appearing in the field of Homological Algebra in the proof assistants Isabelle and Coq. We provide particular instances of these representations in both systems showing the correctness of the representation. Moreover the adequacy of such representations...

We present a complete mechanized proof of the result in homological algebra known as basic perturbation lemma. The proof has been carried out in the proof assistant Isabelle, more concretely, in the implementation of higher-order logic (HOL) available in the system. We report on the difficulties found when dealing with abstract algebra in HOL, and...

In this work we face the problem of obtaining a certified version of a crucial algorithm in Homological Algebra, known as Perturbation Lemma. This lemma is intensively used in the software system Kenzo, devoted to symbolic computation in Homological Algebra. To this end we use the proof assistant Isabelle. Our motivations are to increase the knowle...

En la tesis se aborda el problema de obtener una versión certificada de un resultado fundamental en Álgebra Homológica, conocido como "Lema de Perturbación Básico". Dicho resultado es una pieza central del sistema "Kenzo", software dedicado al cálculo simbólico en Álgebra Homológica. Para tal fin, se utiliza el asistente de demostración "Isabelle"....

In this paper, an approach to synthesize correct programs from specifications is presented. The idea is to extract code from def- initions appearing in statements which have been mechanically proved with the help of a proof assistant. This approach has been found when proving the correctness of certain Computer Algebra programs (for Al- gebraic Top...

While implementing a proof for the Basic Perturbation Lemma (a central result in Homological Algebra) in the theorem prover Isabelle one faces problems such as the implementation of algebraic struc-tures, partial functions in a logic of total functions, or the level of ab-straction in formal proofs. Different approaches aiming at solving these prob...

We present a possible solution to some problems to mecha-nize proofs in Homological Algebra: how to deal with partial functions in a logic of total functions and how to get a level of abstraction that allows the prover to work with morphisms in an equational way.

In this paper, a project to develop a computer-aided proof of the Basic Perturbation Lemma is presented. This Perturbation Lemma is one of the central results in algorithmic algebraic topology and to obtain a mechanised proof of it, would be a first step to increase the reliability of several symbolic computation systems in this area. Techniques to...

While implementing a proof for the Basic Perturbation Lemma (a central result in Homological Algebra) in the theorem prover Isabelle one faces problems such as the implementation of algebraic struc- tures, partial functions in a logic of total functions, or the level of ab- straction in formal proofs. Dierent approaches aiming at solving these prob...

We study one of Mirian’s last works, entitled “Formalizing simplicial topology in ACL2” [M. Andrés et al., in: Seventh international workshop on the ACL2 theorem prover and its applications, held in Austin (Texas), 34–39 (2007)] which was presented and defended by her in the main meeting of that workshop. We explain the main results included in tha...