Abstract
Moessner’s Theorem describes a construction of the sequence of powers \((1^n, 2^n, 3^n, \ldots )\), by repeatedly dropping and summing elements from the sequence of positive natural numbers. The theorem was presented by Moessner in 1951 without a proof and later proved and generalized in several directions. More recently, a coinductive proof of the original theorem was given by Niqui and Rutten. We present a formalization of their proof in the Coq proof assistant. This formalization serves as a non-trivial illustration of the use of coinduction in Coq. During the formalization, we discovered that Long and Salié’s generalizations could also be proved using (almost) the same bisimulation.
The work in this paper was developed when all authors were at the Radboud University, The Netherlands.
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Notes
- 1.
We use Leibniz equality for the heads because we only deal with streams of integers. In general, for example to consider streams of streams, this is still too restrictive.
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Acknowledgments
The authors thank Dexter Kozen, Jan Rutten, Olivier Danvy, and the anonymous referees for useful comments and discussions. The last author learned from Frank many years ago that a good steak, a glass of excellent wine, and fantastic company can make the hardest of days seem distant and irrelevant in the grand scheme of things! Frank, we wish you the very best for the years to come!
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Dedicated to Frank de Boer on the occasion of his 60 $$^{th}$$ t h birthday.
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Krebbers, R., Parlant, L., Silva, A. (2016). Moessner’s Theorem: An Exercise in Coinductive Reasoning in Coq . In: Ábrahám, E., Bonsangue, M., Johnsen, E. (eds) Theory and Practice of Formal Methods. Lecture Notes in Computer Science(), vol 9660. Springer, Cham. https://doi.org/10.1007/978-3-319-30734-3_21
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