Proof Theory at Work: Complexity Analysis of Term Rewrite Systems

In this article we introduce the notion of a generalized system of fundamental sequences and we define its associated slow-growing hierarchy. We claim that these concepts are genuinely related to the classification of the complexity—the derivation length— of rewrite systems for which termination is provable by a standard termination ordering. To substantiate this claim, we re-obtain multiple recursive bounds on the the derivation length for rewrite systems terminating under lexicographic path ordering, originally established by the second author.