Modelling algebraic structures and morphisms in ACL2

Heras, Jónathan; Martín-Mateos, Francisco-Jesús; Pascual, Vico

doi:10.1007/s00200-015-0252-9

Cited by 9 publications

(4 citation statements)

References 39 publications

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“…In Nuprl and Mizar, there are proofs of the Binomial Theorem for rings in [19] and [28], respectively, and a Mizar formalization of the First Isomorphism Theorem for rings [20]. ACL2 has a hierarchy of algebraic structures ranging from setoids to vector spaces that aims the formalization of computer algebra systems [17].…”

Section: Related Work and Work In Progress 41 Related Workmentioning

confidence: 99%

Formalizing Factorization on Euclidean Domains and Abstract Euclidean Algorithms

de Lima,

Borges Avelar,

Galdino

et al. 2024

Electron. Proc. Theor. Comput. Sci.

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This paper discusses the extension of the Prototype Verification System (PVS) sub-theory for rings, part of the PVS algebra theory, with theorems related to the division algorithm for Euclidean rings and Unique Factorization Domains that are general structures where an analog of the Fundamental Theorem of Arithmetic holds. First, we formalize the general abstract notions of divisibility, prime, and irreducible elements in commutative rings, essential to deal with unique factorization domains. Then, we formalize the landmark theorem, establishing that every principal ideal domain is a unique factorization domain. Finally, we specify the theory of Euclidean domains and formally verify that the rings of integers, the Gaussian integers, and arbitrary fields are Euclidean domains. To highlight the benefits of such a general abstract discipline of formalization, we specify a Euclidean gcd algorithm for Euclidean domains and formalize its correctness. Also, we show how this correctness is inherited under adequate parameterizations for the structures of integers and Gaussian integers.

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Section: Related Work and Work In Progress 41 Related Workmentioning

confidence: 99%

Formalizing Factorization on Euclidean Domains and Abstract Euclidean Algorithms

de Lima,

Borges Avelar,

Galdino

et al. 2024

Electron. Proc. Theor. Comput. Sci.

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“…Formalizations of the Binomial Theorem for rings are available in Nuprl [23] and Mizar [28]. In ACL2 a hierarchy of algebraic structures ranging from setoids to vector spaces is built focusing on the formalization of computer algebra systems [20]. The Algebra Library of Isabelle/HOL [3] provides a wide range of theorems on mathematical structures, including results on rings, groups, factorization over ideals, rings of integers and polynomial rings.…”

Section: Information About Formal Developmentsmentioning

confidence: 99%

Teaching Interactive Proofs to Mathematicians

Ayala-Rincón

Lima

2020

Electron. Proc. Theor. Comput. Sci.

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This work discusses an approach to teach to mathematicians the importance and effectiveness of the application of Interactive Theorem Proving tools in their specific fields of interest. The approach aims to motivate the use of such tools through short courses. In particular, it is discussed how, using as case-of-study algebraic notions and properties, the use of the proof assistant Prototype Verification System PVS is promoted to interest mathematicians in the development of their mechanized proofs.

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“…In the HOL/Isabelle Archive of Formal Proofs [22] one finds a number of proof libraries devoted to algebraic domains. Lately in ACL2 [1] an algebraic hierarchy has been built in order to support reasoning about Common Lisp programs [21].…”

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confidence: 99%

On algebraic hierarchies in mathematical repository of Mizar

Grabowski

Korniłowicz

Schwarzweller

2016

Annals of Computer Science and Information Systems

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Abstract-Mathematics, especially algebra, uses plenty of structures: groups, rings, integral domains, fields, vector spaces to name a few of the most basic ones. Classes of structures are closely connected -usually by inclusion -naturally leading to hierarchies that has been reproduced in different forms in different mathematical repositories. In this paper we give a brief overview of some existing algebraic hierarchies and report on the latest developments in the Mizar computerized proof assistant system. In particular we present a detailed algebraic hierarchy that has been defined in Mizar and discuss extensions of the hierarchy towards more involved domains. Taking fully formal approach into account we meet new difficulties comparing with its informal mathematical framework.

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Modelling algebraic structures and morphisms in ACL2

Cited by 9 publications

References 39 publications

Formalizing Factorization on Euclidean Domains and Abstract Euclidean Algorithms

Formalizing Factorization on Euclidean Domains and Abstract Euclidean Algorithms

Teaching Interactive Proofs to Mathematicians

On algebraic hierarchies in mathematical repository of Mizar

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