A Formalisation in HOL of the Fundamental Theorem of Linear Algebra and Its Application to the Solution of the Least Squares Problem

Aransay, Jesús; Divasón, Jose

doi:10.1007/s10817-016-9379-z

Cited by 4 publications

(2 citation statements)

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“…Despite this limitation, the HA library has been shown to be very useful for formalizing linear algebra algorithms. Indeed, based on the HA library, we successfully complete several linear algebra developments [ 2 – 4 , 20 ].…”

Section: Preliminariesmentioning

confidence: 99%

“…In our previous works using the HA library [ 2 – 4 , 20 ], we do not only impose finite types to model the rows and columns of matrices, but we also require more conditions (an explicit enumeration of its universe and some basic arithmetical properties) which were encoded by means of a type class named . The class is designed to facilitate the proofs and to be later instantiated by executable types.…”

Section: Preliminariesmentioning

confidence: 99%

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A Formalization of the Smith Normal Form in Higher-Order Logic

Divasón

Thiemann

2022

J Autom Reasoning

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This work presents formal correctness proofs in Isabelle/HOL of algorithms to transform a matrix into Smith normal form, a canonical matrix form, in a general setting: the algorithms are written in an abstract form and parameterized by very few simple operations. We formally show their soundness provided the operations exist and satisfy some conditions, which always hold on Euclidean domains. We also provide a formal proof on some results about the generality of such algorithms as well as the uniqueness of the Smith normal form. Since Isabelle/HOL does not feature dependent types, the development is carried out by switching conveniently between two different existing libraries by means of the lifting and transfer package and the use of local type definitions, a sound extension to HOL.

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Section: Preliminariesmentioning

confidence: 99%