Let $\mathcal{R} = (R, \Sigma)$ be a term rewrite system over a signature $\Sigma$ with set of basic rewrite rules $R$. It is known that $\mathcal{R}$ is terminating IF every basic rewrite rule $l \to r \in R$ has the following two properties:
The length of $l$ is strictly greater than the length of $r$.
For any variable $x$, the number of occurrences of $x$ in $l$ is $\geq$ the number of occurrences of $x$ in $r$.
I don't think that the converse is true (e.g. the term rewriting system for the theory of groups is terminating, but the first condition fails for the associativity rule). But if we know that $\mathcal{R}$ is terminating, can we perhaps say something similar but slightly weaker about the basic rewrite rules of $\mathcal{R}$? For example, if $\mathcal{R}$ is terminating, then must it be the case that every basic rewrite rule satisfies (2), and is such that the length of the left term is equal to or greater than the length of the right term (rather than strictly greater)?
In sum, I'm wondering if we can conclude anything about the syntactic properties of the basic rewrite rules of $\mathcal{R}$, if we know that $R$ is terminating.
