Refinement of bounds of the heights of terms in the most general unifer

We consider the most general unifer of a pair of terms. Our goal is to estimate the heights of terms in the unifer respect to the depth differences of variables in given terms. The depth difference of a variable is the maximum of the differences between the depths of occurrences of the variable in the terms. Such bounds are useful in finding upper bounds of the heights of terms in proofs in Gentzen-type sequential calculi. We construct a series of examples proving an exponential lower bound on the heights of terms in the most general unifer. We also improve the previously known upper bound of heights in the most general unifer.