User blog comment:Plain'N'Simple/A question for proof-theory experts/@comment-35470197-20191029224813/@comment-35470197-20191030124008

> are you using "N" here as "the set of natural numbers" or as a set of ordinals?

Oops. Sorry for the inconcsistency of the notation. (I first denoted by T the ordinal notation but I found that T already standed for the theory.) I editted the answer.

> If you can answer an analogous question to a theory you are more familiar with

I do not know a single example of T for which the proof length of the well-foundedness of t_n is explicitly given. However, I guess that the result for PA and NT (the number theory in Pohlers 7.2) is well-known to experts. If we carefully read Gentzen's proof, we will obtain the relation of the proof length under PA of the well-foundedness of ω↑↑n and that of ω↑↑(n+1) with respect to the ordinal notation associated to Cantor normal form. The relation for NT is essentially written in Pohlers 7.4, although I have never computed the exact growth rate.

If the growth rate of the proof length of the well-foundedness of ω↑↑n is significantly smaller than E, then we obtain a googological approximation between X and E. I heard that the cut elimination contributes to the proof length as at most tetration or pentation levels, the assumption might be true for PA.