Related papers: Formalized linear algebra over Elementary Divisor …
This is the second installment of an exposition of an ACL2 formalization of elementary linear algebra. It extends the results of Part I, which covers the algebra of matrices over a commutative ring, but focuses on aspects of the theory that…
Given an associative, not necessarily commutative, ring R with identity, a formal matrix calculus is introduced and developed for pairs of matrices over R. This calculus subsumes the theory of homogeneous systems of linear equations with…
This is the first installment of an exposition of an ACL2 formalization of elementary linear algebra, focusing on aspects of the subject that apply to matrices over an arbitrary commutative ring with identity, in anticipation of a future…
This paper describes a formalization of discrete real closed fields in the Coq proof assistant. This abstract structure captures for instance the theory of real algebraic numbers, a decidable subset of real numbers with good algorithmic…
In this paper we give a complete axiomatisation of qubit ZX-calculus via elementary transformations which are basic operations in linear algebra. This formalism has two main advantages. First, all the operations of the phases are algebraic…
A number of models of linear logic are based on or closely related to linear algebra, in the sense that morphisms are "matrices" over appropriate coefficient sets. Examples include models based on coherence spaces, finiteness spaces and…
This paper studies the unitary diagonalization of matrices over formal power series rings. Our main result shows that a normal matrix is unitarily diagonalizable if and only if its minimal polynomial completely splits over the ring and the…
The ever-growing complexity of mathematical proofs makes their manual verification by mathematicians very cognitively demanding. Autoformalization seeks to address this by translating proofs written in natural language into a formal…
This extended abstract is about an effort to build a formal description of a triangulation algorithm starting with a naive description of the algorithm where triangles, edges, and triangulations are simply given as sets and the most complex…
In this article we present a method for formally proving the correctness of the lazy algorithms for computing homographic and quadratic transformations -- of which field operations are special cases-- on a representation of real numbers by…
Matching logic is a formalism for specifying, and reasoning about, mathematical structures, using patterns and pattern matching. Growing in popularity, it has been used to define many logical systems such as separation logic with recursive…
We consider generalized $\Lambda$-structures on algebras and schemes over the ring of integers $\mathit{O}_K$ of a number field $K$. When $K=\mathbb{Q}$, these agree with the $\lambda$-ring structures of algebraic K-theory. We then study…
This notes explains how standard algorithms that construct sorting networks have been formalised and proved correct in the Coq proof assistant using the SSReflect extension.
For performance and verification in machine learning, new methods have recently been proposed that optimise learning systems to satisfy formally expressed logical properties. Among these methods, differentiable logics (DLs) are used to…
One can perform equational reasoning about computational effects with a purely functional programming language thanks to monads. Even though equational reasoning for effectful programs is desirable, it is not yet mainstream. This is partly…
While every matrix algebra over a field $K$ can be realized as a Leavitt path algebra, this is not the case for every graded matrix algebra over a graded field. We provide a complete description of graded matrix algebras over a field,…
The book is devoted to investigation of arithmetic of the matrix rings over certain classes of commutative finitely generated principal ideals domains. We mainly concentrate on constructing of the matrix factorization theory. We reveal a…
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…
Quantum superalgebras $su_{q}(m\mid n)$ are studied in the framework of $R$-matrix formalism. Explicit parametrization of $L^{(+)}$ and $L^{(-)}$ matrices in terms of $su_{q}(m\mid n)$ generators are presented. We also show that quantum…
This paper consists of three parts: (I) To develop general theory of a (large) class of central simple finite dimensional algebras and answering some natural questions about them (that in general situation it is not even clear how to…