We study cut-elimination in first-order classical logic. We construct a sequence of polynomial-length proofs having a non-elementary number of different cut-free normal forms. These normal forms are different in a strong sense: they not only represent different Herbrand-disjunctions but also differ in their propositional structure.
This result illustrates that the constructive content of a proof in classical logic is not uniquely determined but rather depends on the chosen method for extracting it.
"On the non-confluence of cut-elimination." J. Symbolic Logic 76 (1) 313 - 340, March 2011. https://doi.org/10.2178/jsl/1294171002