A Formalization of the Smith Normal Form in Higher-Order Logic

  1. Divasón, Jose
  2. Thiemann, René
  1. 1 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

  2. 2 University of Innsbruck
    info

    University of Innsbruck

    Innsbruck, Austria

    ROR https://ror.org/054pv6659

Revista:
Journal of Automated Reasoning

ISSN: 0168-7433 1573-0670

Año de publicación: 2022

Tipo: Artículo

DOI: 10.1007/S10817-022-09631-5 GOOGLE SCHOLAR lock_openAcceso abierto editor

Otras publicaciones en: Journal of Automated Reasoning

Repositorio institucional: lock_openAcceso abierto Editor

Resumen

This work presents formal correctness proofs in Isabelle/HOL of algorithms to transform a matrix into Smith normal form, a canonical matrix form, in a general setting: the algorithms are written in an abstract form and parameterized by very few simple operations. We formally show their soundness provided the operations exist and satisfy some conditions, which always hold on Euclidean domains. We also provide a formal proof on some results about the generality of such algorithms as well as the uniqueness of the Smith normal form. Since Isabelle/HOL does not feature dependent types, the development is carried out by switching conveniently between two different existing libraries by means of the lifting and transfer package and the use of local type definitions, a sound extension to HOL.

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