Descendants and Origins in Term Rewriting

In this paper we treat various aspects of a notion that is central in term rewriting namely that of descendants or residuals We address both rst order term rewriting and calculus their nitary as well as their in nitary variants A recurrent theme is the Parallel Moves Lemma Next to the classical notion of descendant we introduce an extended version known as origin tracking Origin tracking has many applications Here it is employed to give new proofs of three classical theorems the Genericity Lemma in calculus the theorem of Huet and L evy on needed reduc tions in rst order term rewriting and Berry s Sequentiality Theorem in in nitary calculus Note This article is based on a lecture given by Jan Willem Klop at RTA held in Tsukuba Japan