This paper presents a type theory in which it is possible to directly manipulate n-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system. Further,… (More)
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In this paper we report on a project to obtain a verified computation of homology groups of digital images. The methodology is based on programming and executing inside the COQ proof assistant. Though more research is needed to integrate and make efficient more processing tools, we present some examples partially computed in COQ from real biomedical images.
<i>Persistent homology</i> is one of the most active branches of <i>computational algebraic topology</i> with applications in several contexts such as optical character recognition or analysis of point cloud data. In this article, we report on the formal development of certified programs to compute <i>persistent Betti numbers</i>, an instrumental tool of… (More)
We present a formalization of coherent and strongly discrete rings in type theory. This is a fundamental structure in constructive algebra that represents rings in which it is possible to solve linear systems of equations. These structures have been instantiated with Bézout domains (for instance Z and k[x]) and Prüfer domains (generalization of Dedekind… (More)
This paper presents a Coq formalization of linear algebra over elementary divisor rings, that is, rings where every matrix is equivalent to a matrix in Smith normal form. The main results are the formalization that these rings support essential operations of linear algebra, the classification theorem of finitely presented modules over such rings and the… (More)