Formal Proofs of Hypergeometric Sums

Abstract

Algorithmic methods can successfully automate the proof, and even the discovery, of a large class of identities involving sums of hypergeometric terms. In particular, the Wilf-Zeilberger (WZ) algorithm is a uniform framework for a substantial class of hypergeometric summation problems. This algorithm can produce a rational function certificate that can, on the face of it, be used to verify the result by routine algebraic manipulations, independently of the working of the algorithm that discovered it. It is therefore very natural to consider using this certificate to produce, by automated means, a rigorous deductive proof in an interactive theorem prover. However, naive presentations of the WZ method tend to gloss over trivial-looking but rather knotty questions about zero denominators, which makes their rigorous formalization tricky and their ultimate logical justification somewhat obscure. We describe how we have handled these difficulties to produce rigorous WZ proofs inside the HOL Light theorem prover.

Revisiting the combinatorics of lifting line and 2D vortex lattice theory

“…The lifting line model proves to have a hidden combinatorial structure involving the triangular number, with the structure giving a clean form of model convergence on the number of segments n. The combinatorial expressions related to the lifting line and 2D vortex lattice theory can be simplified in presentation, rewriting in terms of central binomial coefficients that also allows for simple approximation. The matrix Q also serves as an excellent real world pedagogical case study (not too simple -not too complex) for more recent developments in applied math such as automated theorem proving (20)(21)(22)(23) , properties of M-Matrices and inverse M-Matrices (14,24,25) , along with ongoing research in the eigenstructure of infinite Toeplitz matrices (12,13) . The treatment shown here complements the traditional approach using Fourier analysis in demonstrating the special properties of the elliptical wing.…”

A bi-directional extensible interface between Lean and Mathematica

“…This skeptical approach is also taken in some projects that require one specific type of computer algebra computation instead of a generic link. Harrison [22] computes Wilf-Zeilberger certificates in Maxima and verifies them in HOL Light. While this work does establish an interface between the two systems, Harrison notes the convenience of a "manual" version, where users generate certificates in Maxima and transfer them to HOL Light by hand.…”