Length-reducing rewriting system

Definition
A rewriting system is said to be length-reducing if for every rewrite, the length of the word on the right is strictly less than the length of the word on the left.

Note that while this term makes sense for a general rewriting system of any monoid, it can also be made sense of in the context of rewriting system for a group.

Stronger properties
Some stronger properties of rewriting systems are:


 * Special rewriting system
 * Monadic rewriting system

Weaker properties

 * Finitely terminating rewriting system

Metaproperties
To check whether a rewriting system is length-reducing, we simply need to check, for every rewrite, whether that is length-reducing. That is, the property is simply a certain property being true for each rewrite.

That a free product of length-reducing rewriting systems is length-reducing follows from its being locally testable.