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A Formal Proof of the Computation of Hermite Normal Form in a General Setting

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Artificial Intelligence and Symbolic Computation (AISC 2018)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11110))

Abstract

In this work, we present a formal proof of an algorithm to compute the Hermite normal form of a matrix based on our existing framework for the formalisation, execution, and refinement of linear algebra algorithms in Isabelle/HOL. The Hermite normal form is a well-known canonical matrix analogue of reduced echelon form of matrices over fields, but involving matrices over more general rings, such as Bézout domains. We prove the correctness of this algorithm and formalise the uniqueness of the Hermite normal form of a matrix. The succinctness and clarity of the formalisation validate the usability of the framework.

This work has been supported by the projects MTM2014-54151-P and MTM2017-88804-P from Ministerio de Economía y Competitividad (Gobierno de España).

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Notes

  1. 1.

    The Isabelle file that serialises to UArrays is available from our website [16].

  2. 2.

    Neither records nor locales [9] are used for this task, although they are a valid alternative.

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Authors and Affiliations

  1. Universidad de La Rioja, Logroño, La Rioja, Spain

    Jose Divasón & Jesús Aransay

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  1. Jose Divasón
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  2. Jesús Aransay
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Correspondence to Jose Divasón .

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Editors and Affiliations

  1. University of Edinburgh, Edinburgh, United Kingdom

    Jacques Fleuriot

  2. Beihang University, Beijing, China

    Dongming Wang

  3. Karlsruhe Institute of Technology, Karlsruhe, Germany

    Jacques Calmet

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Divasón, J., Aransay, J. (2018). A Formal Proof of the Computation of Hermite Normal Form in a General Setting. In: Fleuriot, J., Wang, D., Calmet, J. (eds) Artificial Intelligence and Symbolic Computation. AISC 2018. Lecture Notes in Computer Science(), vol 11110. Springer, Cham. https://doi.org/10.1007/978-3-319-99957-9_3

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  • DOI: https://doi.org/10.1007/978-3-319-99957-9_3

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