In this paper we present the first formal proof of an implementation of Koda and Ruskey's algorithm, an algorithm for generating all ideals of a forest poset as a Gray code. One contribution of this work is to exhibit the invariants of this algorithm, which proved to be challenging. We implemented, specified, and proved this algorithm using the Why3 tool. This allowed us to employ a combination of several automated theorem provers to discharge most of the verification conditions, and the Coq proof assistant for the remaining two.