The Newman lemma is a useful result to prove good properties of a rewriting system. At the best, a rewriting system is strongly normalizing and confluent. The lemma allows to verify only the weaker property of local confluence, once we have first proved the strong normalization, in the case where the rewriting system possesses these two good properties.

Let $(X,\rightarrow)$ be an abstract rewriting system ie. a set equipped with a binary relation $\rightarrow$. The Newman lemma asserts that:

If $\rightarrow$ is strongly normalizing and locally confluent then, it is confluent.

The original article:

- Maxwell Herman Alexander Newman,
*On theories with a combinatorial definition of “equivalence”*, Annals of Mathematics**43**2 (1942) 223–243 [doi:10.2307/2269299]

See also:

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