On proof mining by cut-elimination

We present cut-elimination as a method of proof mining, in the sense that hidden mathematical information can be extracted by eliminating lemmas from proofs. We present reductive methods for cutelimination and the method ceres (cut-elimination by resolution). A comparison of ceres with reductive methods is given and it is shown that the asymptotic behavior of ceres is superior to that of reductive methods (nonelementary speed-up). It is illustrated, how ceres can be extended and applied in practice for analyzing mathematical proofs. Finally we we give an application of ceres to a well-known proof of the infinitude of primes by Fürstenberg; this proof uses topological lemmas based on arithmetic progressions. These topological lemmas of the proof are eliminated by ceres and Euclid’s construction of primes is extracted. We also touch the problem of cut-elimination by resolution on induction proofs and discuss the limits of the method.