26 research outputs found
A formally verified proof of the prime number theorem
The prime number theorem, established by Hadamard and de la Vall'ee Poussin
independently in 1896, asserts that the density of primes in the positive
integers is asymptotic to 1 / ln x. Whereas their proofs made serious use of
the methods of complex analysis, elementary proofs were provided by Selberg and
Erd"os in 1948. We describe a formally verified version of Selberg's proof,
obtained using the Isabelle proof assistant.Comment: 23 page
Introduction to Milestones in Interactive Theorem Proving
On March 8, 2018, Tobias Nipkow celebrated his sixtieth birthday. In anticipation of the occasion, in January 2016, two of his former students, Gerwin Klein and Jasmin Blanchette, and one of his former postdocs, Andrei Popescu, approached the editorial board of the Journal of Automated Reasoning with a proposal to publish a surprise Festschrift issue in his honor. The e-mail was sent to twenty-six members of the board, leaving out one, for reasons that will become clear in a moment. It is a sign of the love and respect that Tobias commands from his colleagues that within two days every recipient of the e-mail had responded favorably and enthusiastically to the proposal
Elaboration in Dependent Type Theory
To be usable in practice, interactive theorem provers need to provide
convenient and efficient means of writing expressions, definitions, and proofs.
This involves inferring information that is often left implicit in an ordinary
mathematical text, and resolving ambiguities in mathematical expressions. We
refer to the process of passing from a quasi-formal and partially-specified
expression to a completely precise formal one as elaboration. We describe an
elaboration algorithm for dependent type theory that has been implemented in
the Lean theorem prover. Lean's elaborator supports higher-order unification,
type class inference, ad hoc overloading, insertion of coercions, the use of
tactics, and the computational reduction of terms. The interactions between
these components are subtle and complex, and the elaboration algorithm has been
carefully designed to balance efficiency and usability. We describe the central
design goals, and the means by which they are achieved
Formalising Fisher's Inequality: Formal Linear Algebraic Proof Techniques in Combinatorics
The formalisation of mathematics is continuing rapidly, however combinatorics
continues to present challenges to formalisation efforts, such as its reliance
on techniques from a wide range of other fields in mathematics. This paper
presents formal linear algebraic techniques for proofs on incidence structures
in Isabelle/HOL, and their application to the first formalisation of Fisher's
inequality. In addition to formalising incidence matrices and simple techniques
for reasoning on linear algebraic representations, the formalisation focuses on
the linear algebra bound and rank arguments. These techniques can easily be
adapted for future formalisations in combinatorics, as we demonstrate through
further application to proofs of variations on Fisher's inequality.Comment: Accepted to ITP 2022, to be published in conference proceeding
A Mechanized Proof of the Max-Flow Min-Cut Theorem for Countable Networks
Aharoni et al. [Ron Aharoni et al., 2010] proved the max-flow min-cut theorem for countable networks, namely that in every countable network with finite edge capacities, there exists a flow and a cut such that the flow saturates all outgoing edges of the cut and is zero on all incoming edges. In this paper, we formalize their proof in Isabelle/HOL and thereby identify and fix several problems with their proof. We also provide a simpler proof for networks where the total outgoing capacity of all vertices other than the source is finite. This proof is based on the max-flow min-cut theorem for finite networks
Verification of the (1–δ)-Correctness Proof of CRYSTALS-KYBER with Number Theoretic Transform
This paper describes a formalization of the specification and the algorithm of the cryptographic scheme CRYSTALS-KYBER as well as the verification of its (1 − δ)-correctness proof. During the formalization, a problem in the correctness proof was uncovered. In order to amend this
issue, a necessary property on the modulus parameter of the CRYSTALS-KYBER algorithm was introduced. This property is already implicitly fulfilled by the structure of the modulus prime used in the number theoretic transform (NTT). The NTT and its convolution theorem
in the case of CRYSTALS-KYBER was formalized as well. The formalization was realized in the theorem prover Isabelle