Jesús Aransay

A Mechanized Proof of the Basic Perturbation Lemma

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 also on the ongoing stages of our project to give a certified version of some of the algorithms present in the Kenzo symbolic computation system.

Generating certified code from formal proofs: a case study in homological algebra

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 algorithm in ML. In the formal proof, algebraic structures occurring in the Basic Perturbation Lemma are represented in a way, which is not directly amenable to code generation with the available tools. Interestingly, this representation is required in the proof, while for the algorithm simpler data structures are sufficient. Our approach is to establish a link between the non-executable setting of the proof and the executable representation in the algorithm, which is to be generated. This correspondence is established within the logical framework of Isabelle/HOL—that is, it is formally proved correct. The generated code is applied to and illustrated with a number of examples.

Formalization and Execution of Linear Algebra: From Theorems to Algorithms

In this work we present a formalization of the 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 standard distribution of the proof assistant). Hence, our proof shows the adequacy of such an infrastructure for the formalization of Linear Algebra. Moreover, we enrich the proof with an additional formalization of its computational meaning; to this purpose, we choose to implement the Gauss-Jordan elimination algorithm for matrices over fields, prove it correct, and then apply the Isabelle code generation facility that permits to execute the formalized algorithm. For the algorithm to be code generated, we use again the implementation of matrices available in the HOL Multivariate Analysis library, and enrich it with some necessary features. We report on the precise modifications that we introduce to get code execution from the original representation, and on the performance of the code obtained. We present an alternative verified type refinement of vectors that outperforms the original version. This refinement performs well enough as to be applied to the computation of the rank of some biomedical digital images. Our work proves itself as a suitable basis for the formalization of numerical Linear Algebra in HOL provers that can be successfully applied for computations of real case studies.

Four Approaches to Automated Reasoning with Differential Algebraic Structures

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 structures, partial functions in a logic of total functions, or the level of abstraction in formal proofs. Different approaches aiming at solving these problems will be evaluated and classified according to features such as the degree of mechanization obtained or the direct correspondence to the mathematical proofs. Prom this study, an environment for further developments in Homological Algebra will be proposed.

Formalisation in higher-order logic and code generation to functional languages of the Gauss-Jordan algorithm

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 verified algorithm. The aim of this research is not to compete with specialised numerical implementations of Gausslike algorithms, but to show that formal proofs in this area can be used to generate usable functional programs. The obtained programs show compelling performance in comparison to some other verified and functional versions, and accomplish some challenging tasks, such as the computation of determinants of matrices of big integers and the computation of the homology of matrices representing digital images.

Formalisation of the computation of the echelon form of a matrix in Isabelle/HOL

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 polynomials over a field. This is possible since the algorithm has been parameterised by a (possibly non-computable) operation that returns the Bézout coefficients of a pair of elements of a ring. The echelon form is also used to compute determinants and inverses of matrices. As a by-product, some algebraic structures have been implemented (principal ideal domains, Bézout domains, etc.). In order to improve performance, the algorithm has been refined to immutable arrays inside of Isabelle and code can be generated to functional languages as well.

Modelling Differential Structures in Proof Assistants: The Graded Case

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 is illustrated by developing a formal proof of the Trivial Perturbation Lemma in both systems.

Generalizing a Mathematical Analysis Library in Isabelle/HOL

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 it has not been tackled yet. Using that library, we were able to prove the Gauss-Jordan algorithm over real matrices, but our interest lied on generating verified code for matrices over arbitrary fields, greatly increasing the range of applications of such an algorithm. This short paper presents the steps that we did and the methodology that we devised to generalize such a library, which were successful to generalize the Gauss-Jordan algorithm to matrices over arbitrary fields.

Mechanized reasoning in homological algebra

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 knowledge in the algorithmic nature of this mathematical result and to test different possibilities offered by Isabelle to formally prove theorems in Homological Algebra. A detailed mathematical proof of the Perturbation Lemma, a methodology to obtain certified software in Homological Algebra, a suitable infrastructure to formalize the proof, a complete Isabelle formal proof, and a discussion on the constructiveness of the mathematical results introduced are presented.

A Formalisation in HOL of the Fundamental Theorem of Linear Algebra and Its Application to the Solution of the Least Squares Problem

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