Invariants , patterns and weights for ordering

We prove that any simpliication order over arbtrary terms is an extension of an order by weight, by consideringa 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.