Newman's lemma
# 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.

