Certified Kruskal’s Tree Theorem

Abstract

This article presents the first formalization of Kurskal's tree theorem in a proof assistant. The Isabelle/HOL development is along the lines of Nash-Williams' original minimal bad sequence argument for proving the tree theorem. Along the way, proofs of Dickson's lemma and Higman's lemma, as well as some technical details of the formalization are discussed.

Certified ACKBO

“…Concerning well-foundedness, we actually have formalized two different proofs with slightly different implications. Our first proof follows Yamada et al in establishing that ACKBO is a simplification order (which implies wellfoundedness) and relies on an earlier formalization of Kruskal's tree theorem [7]. However, this result only holds for TRSs over a finite set of function symbols.…”

Well quasi-orders and the functional interpretation

“…The full formalisation of Nash-Williams' minimal bad sequence argument has already been studied in e.g. [25] (in Minlog) and [27] (in Isabelle/HOL), and we hope that our formal proof sketched in Chapter 3 may prove informative to those working in a more hands-on manner on the formalisation of WQO theory in proof assistants. In Chapters 4-5 we took the opportunity to present G ödel's functional interpretation in a way that would appeal to readers not already familiar with it.…”

“…Then since n = Ψ N ( ū) from the left hand side we have P(v, Ψ N (v)). Now, since v = {u k } φ for some element in the learning procedure LC φ ξ,C 0 [ ū], by (27) we have…”