6 research outputs found
Computing Persistent Homology within Coq/SSReflect
Persistent homology is one of the most active branches of Computational
Algebraic Topology with applications in several contexts such as optical
character recognition or analysis of point cloud data. In this paper, we report
on the formal development of certified programs to compute persistent Betti
numbers, an instrumental tool of persistent homology, using the Coq proof
assistant together with the SSReflect extension. To this aim it has been
necessary to formalize the underlying mathematical theory of these algorithms.
This is another example showing that interactive theorem provers have reached a
point where they are mature enough to tackle the formalization of nontrivial
mathematical theories
Formalized linear algebra over Elementary Divisor Rings in Coq
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 uniqueness of the Smith
normal form up to multiplication by units. We present formally verified
algorithms computing this normal form on a variety of coefficient structures
including Euclidean domains and constructive principal ideal domains. We also
study different ways to extend B\'ezout domains in order to be able to compute
the Smith normal form of matrices. The extensions we consider are: adequacy
(i.e. the existence of a gdco operation), Krull dimension and
well-founded strict divisibility