Proofs, rewriting systems, reachability

[void4]

Given a rewriting system, in which circumstances can properties of derivations/proofs be estimated in advance, before knowing anything else about the derivation?

Derivations are of a specific length, some derivations may have the minimum number of steps to construct the string, tree or other object (which can correspond to a proof, for example). Having a measure of distance provokes thinking about proof space. Does it have any known properties?

https://math.stackexchange.com/questions/3477810/estimating-meta-mathematical-properties-of-conjectures

https://twitter.com/dd4ta/status/1153091658331934721

Fixed growth rates for https://en.wikipedia.org/wiki/Look-and-say_sequence:
https://www.youtube.com/watch?v=ea7lJkEhytA