Formalization and Execution of Linear Algebra: From Theorems to Algorithms

Aransay, Jesús; Divasón, Jose

doi:10.1007/978-3-319-14125-1_1

Cited by 9 publications

(9 citation statements)

References 16 publications

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“…As discussed in Section 7.3, this development improves over the work presented in [22] by providing computable definitions of determinants and the Gauss-Jordan elimination algorithm. This work in PVS complements existing work on computable formalizations of linear algebra in other theorem provers [2,8,17,27].…”

Section: Related Workmentioning

confidence: 52%

Formally-Verified Decision Procedures for Univariate Polynomial Computation Based on Sturm’s and Tarski’s Theorems

2015

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Sturm's theorem is a well-known result in real algebraic geometry that provides a function that computes the number of roots of a univariate polynomial in a semi-open interval, not counting multiplicity. A generalization of Sturm's theorem is known as Tarski's theorem, which provides a linear relationship between functions known as Tarski queries and cardinalities of certain sets. The linear system that results from this relationship is in fact invertible and can be used to explicitly count the number of roots of a univariate polynomial on a set defined by a system of polynomial relations. This paper presents a formalization of these results in the PVS theorem prover, including formal proofs of Sturm's and Tarski's theorems. These theorems are at the basis of two decision procedures, which are implemented as computable functions in PVS. The first, based on Sturm's theorem, determines satisfiability of a single polynomial relation over an interval. The second, based on Tarski's theorem, determines the satisfiability of a system of polynomial relations over the real line. The soundness and completeness properties of these decision procedures are formally verified in PVS. The procedures and their correctness properties enable the implementation of PVS strategies for automatically proving existential and universal statements on polynomial systems. Since the decision procedures are formally verified in PVS, the soundness of the strategies depends solely on the internal logic of PVS rather than on an external oracle.

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Section: Related Workmentioning

confidence: 52%

Formally-Verified Decision Procedures for Univariate Polynomial Computation Based on Sturm’s and Tarski’s Theorems

2015

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“…(defconst * XxY * (create−setoid 'X−Y−inv 'X−Y−eq)) (5) Finally, the user can certify that *XxY* is a setoid using the macro call:…”

Section: Cartesian Product Of Setoidsmentioning

confidence: 99%

Modelling algebraic structures and morphisms in ACL2

Heras

Martín-Mateos

Pascual

2015

AAECC

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In this paper, we present how algebraic structures and morphisms can be modelled in the ACL2 theorem prover. Namely, we illustrate a methodology for implementing a set of tools that facilitates the formalisations related to algebraic structuresas a result, an algebraic hierarchy ranging from setoids to vector spaces has been developed. The resultant tools can be used to simplify the development of generic theories about algebraic structures. In particular, the benefits of using the tools presented in this paper, compared to a from-scratch approach, are especially relevant when working with complex mathematical structures; for example, the structures employed in Algebraic Topology. This work shows that ACL2 can be a suitable tool for formalising algebraic concepts coming, for instance, from computer algebra systems.

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“…Our formalization is certainly not unique: we have been heavily inspired by existing linear algebra developments, particularly in Isabelle/HOL [3,21,43] and Coq [26].…”

Section: Linear Algebramentioning

confidence: 99%

“…More casual conversations occur in a Zulip chat room. 3 Eleven community members have been designated as maintainers of the library. These people are responsible for approving pull requests in their areas of expertise.…”

Section: Github and Zulipmentioning

confidence: 99%

The lean mathematical library

Community

2020

Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs

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This paper describes mathlib, a community-driven effort to build a unified library of mathematics formalized in the Lean proof assistant. Among proof assistant libraries, it is distinguished by its dependently typed foundations, focus on classical mathematics, extensive hierarchy of structures, use of large-and small-scale automation, and distributed organization. We explain the architecture and design decisions of the library and the social organization that has led to its development.

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Formalization and Execution of Linear Algebra: From Theorems to Algorithms

Cited by 9 publications

References 16 publications

Formally-Verified Decision Procedures for Univariate Polynomial Computation Based on Sturm’s and Tarski’s Theorems

Formally-Verified Decision Procedures for Univariate Polynomial Computation Based on Sturm’s and Tarski’s Theorems

Modelling algebraic structures and morphisms in ACL2

The lean mathematical library

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