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The Weighted Path Order of Yamada is a powerful technique for provingtermination. It is also supported by CeTA, a certifier for checking untrustedtermination proofs. To be more precise, CeTA contains a verified function thatcomputes for two terms whether one of them is larger than the other for a givenWPO, i.e., where all parameters of the WPO have been fixed. The problem of thisverified function is its exponential runtime in the worst case.  Therefore, in this work we develop a polynomial time implementation of WPOthat is based on memoization. It also improves upon an earlier verifiedimplementation of the Recursive Path Order: the RPO-implementation uses fullterms as keys for the memory, a design which simplified the soundness proofs,but has some runtime overhead. In this work, keys are just numbers, so that thelookup in the memory is faster. Although trivial on paper, this changeintroduces some challenges for the verification task.

A Verified Efficient Implementation of the Weighted Path Order