Moessner's Theorem: An Exercise in Coinductive Reasoning in Coq

Moessner's Theorem describes a construction of the sequence of powers $$1^n, 2^n, 3^n, \ldots $$1n,2n,3n,', 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 Salie's generalizations could also be proved using almost the same bisimulation.