A Verified Implementation of Algebraic Numbers in Isabelle/HOL

Sebastiaan J.C. Joosten, René Thiemann (Corresponding Author), Akihisa Yamada

Research output: Contribution to journalArticleAcademicpeer-review

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Abstract

We formalize algebraic numbers in Isabelle/HOL. Our development serves as a verified implementation of algebraic operations on real and complex numbers. We moreover provide algorithms that can identify all the real or complex roots of rational polynomials, and two implementations to display algebraic numbers, an approximative version and an injective precise one. We obtain verified Haskell code for these operations via Isabelle’s code generator. The development combines various existing formalizations such as matrices, Sturm’s theorem, and polynomial factorization, and it includes new formalizations about bivariate polynomials, unique factorization domains, resultants and subresultants.
Original languageEnglish
Number of pages27
JournalJournal of automated reasoning
DOIs
Publication statusE-pub ahead of print/First online - 9 Dec 2018
Externally publishedYes

Fingerprint

Polynomials
Factorization

Keywords

  • UT-Hybrid-D
  • Algebraic numbers
  • Real algebraic geometry
  • Resultants
  • deal
  • Theorem proving

Cite this

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title = "A Verified Implementation of Algebraic Numbers in Isabelle/HOL",
abstract = "We formalize algebraic numbers in Isabelle/HOL. Our development serves as a verified implementation of algebraic operations on real and complex numbers. We moreover provide algorithms that can identify all the real or complex roots of rational polynomials, and two implementations to display algebraic numbers, an approximative version and an injective precise one. We obtain verified Haskell code for these operations via Isabelle’s code generator. The development combines various existing formalizations such as matrices, Sturm’s theorem, and polynomial factorization, and it includes new formalizations about bivariate polynomials, unique factorization domains, resultants and subresultants.",
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A Verified Implementation of Algebraic Numbers in Isabelle/HOL. / Joosten, Sebastiaan J.C.; Thiemann, René (Corresponding Author); Yamada, Akihisa.

In: Journal of automated reasoning, 09.12.2018.

Research output: Contribution to journalArticleAcademicpeer-review

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T1 - A Verified Implementation of Algebraic Numbers in Isabelle/HOL

AU - Joosten, Sebastiaan J.C.

AU - Thiemann, René

AU - Yamada, Akihisa

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N2 - We formalize algebraic numbers in Isabelle/HOL. Our development serves as a verified implementation of algebraic operations on real and complex numbers. We moreover provide algorithms that can identify all the real or complex roots of rational polynomials, and two implementations to display algebraic numbers, an approximative version and an injective precise one. We obtain verified Haskell code for these operations via Isabelle’s code generator. The development combines various existing formalizations such as matrices, Sturm’s theorem, and polynomial factorization, and it includes new formalizations about bivariate polynomials, unique factorization domains, resultants and subresultants.

AB - We formalize algebraic numbers in Isabelle/HOL. Our development serves as a verified implementation of algebraic operations on real and complex numbers. We moreover provide algorithms that can identify all the real or complex roots of rational polynomials, and two implementations to display algebraic numbers, an approximative version and an injective precise one. We obtain verified Haskell code for these operations via Isabelle’s code generator. The development combines various existing formalizations such as matrices, Sturm’s theorem, and polynomial factorization, and it includes new formalizations about bivariate polynomials, unique factorization domains, resultants and subresultants.

KW - UT-Hybrid-D

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KW - Real algebraic geometry

KW - Resultants

KW - deal

KW - Theorem proving

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