Newman's lemma

In mathematics, in the theory of rewriting systems, Newman's lemma, also commonly called the diamond lemma, states that a terminating (or strongly normalizing) abstract rewriting system (ARS), that is, one in which there are no infinite reduction sequences, is confluent if it is locally confluent. In fact a terminating ARS is confluent precisely when it is locally confluent.

Equivalently, for every binary relation with no decreasing infinite chains and satisfying a weak version of the diamond property, there is a unique minimal element in every connected component of the relation considered as a graph.

Today, this is seen as a purely combinatorial result based on well-foundedness due to a proof of Gérard Huet in 1980. Newman's original proof was considerably more complicated.

Diamond lemma


In general, Newman's lemma can be seen as a combinatorial result about binary relations → on a set A (written backwards, so that a → b means that b is below a) with the following two properties:


 * → is a well-founded relation: every non-empty subset X of A has a minimal element (an element a of X such that a → b for no b in X). Equivalently, there is no infinite chain $a_{0} → a_{1} → a_{2} → a_{3} → ...$. In the terminology of rewriting systems, → is terminating.
 * Every covering is bounded below. That is, if an element a in A covers elements b and c in A in the sense that $a → b$ and $a → c$, then there is an element d in A such that $b ∗ → d$ and $c ∗ → d$, where $∗ →$ denotes the reflexive transitive closure of →.  In the terminology of rewriting systems, → is locally confluent.

The lemma states that if the above two conditions hold, then → is confluent: whenever $a ∗ → b$ and $a ∗ → c$, there is an element d such that $b ∗ → d$ and $c ∗ → d$. In view of the termination of →, this implies that every connected component of → as a graph contains a unique minimal element a, moreover $b ∗ → a$ for every element b of the component.

Textbooks

 * Term Rewriting Systems, Terese, Cambridge Tracts in Theoretical Computer Science, 2003. (book weblink)
 * Term Rewriting and All That, Franz Baader and Tobias Nipkow, Cambridge University Press, 1998 (book weblink)
 * John Harrison, Handbook of Practical Logic and Automated Reasoning, Cambridge University Press, 2009, ISBN 978-0-521-89957-4, chapter 4 "Equality".