Invariants, Patterns and Weights for Ordering Terms

We prove that any simpliication order over arbtrary terms is an extension of an order by weight, by considering a related monadic term algebra called the spine. We show that any simpliication order on the spine lifts to an order on the full term algebra. Conversely a simpliication ordering on the term algebra deenes a weight function on the spine, which in turn can be lifted to a weight order on the original ground terms which contains the original order. We investigate the Knuth Bendix and polynomial orders in this light. We provide a general framework for ordering terms by counting embedded patterns, which gives rise to many new orderings. We examine the recursive path order in this context.