In [1,2] Zhang shows how the complexity of cut elimination depends on the nesting of quantifiers in cut formulas. By studying the role of contractions we can refine that analysis and show how the complexity depends on a combination of contractions and quantifier nesting. With the refined analysis the upper bound on cut elimination coincides with Statman's lower bound. Every non-elementary growth example must display a combination of nesting of quantifiers and contractions similar to Statman's lower bound example. The upper and lower bounds on cut elimination immediately translate into bounds on Herbrand's theorem. Finally we discuss the role of quantifier alternations and show an elementary upper bound for the -Λ-case (resp. -Λ-case).
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