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Algebraic Numbers in Isabelle/HOL

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Interactive Theorem Proving (ITP 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9807))

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Abstract

We formalize algebraic numbers in Isabelle/HOL, based on existing libraries for matrices and Sturm’s theorem. Our development serves as a verified implementation for real and complex numbers, and it admits to compute roots and completely factor real and complex polynomials, provided that all coefficients are rational numbers. Moreover, we provide two implementations to display algebraic numbers, an injective and expensive one, and a faster but approximative version.

To this end, we mechanize several results on resultants, which also required us to prove that polynomials over a unique factorization domain form again a unique factorization domain. We moreover formalize algorithms for factorization of integer polynomials: Newton interpolation, factorization over the integers, and Kronecker’s factorization algorithm, as well as a factorization oracle via Berlekamp’s algorithm with the Hensel lifting.

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Notes

  1. 1.

    See http://math-comp.github.io/math-comp.

  2. 2.

    Here one cannot just evaluate the polynomial on the algebraic point and test the result is 0; we are defining the basic arithmetic operations needed for this evaluation.

  3. 3.

    However, we use a faster computer with 3.5 GHz instead of 2.66 GHz.

  4. 4.

    We contributed our formalization to the development version of Isabelle (May 2016). There one will find the general “”.

  5. 5.

    As for the division algorithm, we have not been able to work with Isabelle’s existing type class for GCDs, as the GCD on polynomials is only available for fields.

  6. 6.

    These optimizations became part of the development version of Isabelle (May 2016).

  7. 7.

    We thank one of the anonymous reviewers for pointing us to this equality test.

  8. 8.

    Note that the quotient type can be in principle defined also directly on top of , such that the quotient and invariant construction is done in one step, but then code generator will fail in Isabelle 2016.

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Acknowledgments

We thank the anonymous reviewers for their helpful comments. The early abortion in our divisibility test for integer polynomials is due to Sebastiaan Joosten. This research was supported by the Austrian Science Fund (FWF) project Y757.

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Authors and Affiliations

  1. University of Innsbruck, Innsbruck, Austria

    René Thiemann & Akihisa Yamada

Authors
  1. René Thiemann
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  2. Akihisa Yamada
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Correspondence to René Thiemann .

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Editors and Affiliations

  1. Inria Nancy – Grand Est, Villers-lès-Nancy, France

    Jasmin Christian Blanchette

  2. Inria Nancy – Grand Est, Villers-lès-Nancy, France

    Stephan Merz

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Thiemann, R., Yamada, A. (2016). Algebraic Numbers in Isabelle/HOL. In: Blanchette, J., Merz, S. (eds) Interactive Theorem Proving. ITP 2016. Lecture Notes in Computer Science(), vol 9807. Springer, Cham. https://doi.org/10.1007/978-3-319-43144-4_24

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  • DOI: https://doi.org/10.1007/978-3-319-43144-4_24

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-43143-7

  • Online ISBN: 978-3-319-43144-4

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