CERES: An analysis of Fürstenberg’s proof of the infinity of primes

Understanding Resolution Proofs through Herbrand’s Theorem

“…As Skolem functions often possess a natural mathematical interpretation (see e.g. [8]), we opt for displaying expansion trees that still include the Skolem functions in order to increase their readability.…”

“…It plays the key role in Luckhardt's (manual) analysis [5] of Roth's theorem where it has been used to obtain polynomial bounds (which were obtained independently and by purely mathematical as opposed to logical methods by Bombieri and van der Poorten in [6]). The extraction and analysis of Herbrandsequents as described by Hetzl et al [7] has also been used in the computerassisted analysis of Fürstenberg's topological proof of the infinity of primes by Baaz et al [8] which yielded Euclid's original argument via cut-elimination. Herbrand's theorem and methods based on it have furthermore also been used in a number of smaller case studies such as [9] by Baaz et al or [10] by Urban.…”

Cut Elimination for First Order Gödel Logic by Hyperclause Resolution

“…3 The end-hypersequent H σ of the HG-proof σ that forms the input of hyperCERES can be of two forms: either it contains only weak quantifier occurrences or it consists of prenex formulas only. 4 In the latter case we have to Skolemize the proof first (step 1) and de-Skolemize it after cut elimination (step 7): 5. apply θ to the reduced proofs R 1 (σ ), .…”

“…This method, called CERES 1 is based on the resolution calculus and has been successfully employed for the in depth analysis of proofs in number theory (e.g., [5]). It is moreover also of theoretical interest due to its global nature and other essential differences, compared to the traditional, local Gentzen-and Schütte-Tait-style cut elimination methods [18,20].…”

“…It might surprise the reader that we rely on the cut-free completeness of HG in a paper dealing with cut elimination. However, this just emphasizes the fact that we are interested in a particular transformation of proofs with cuts (i.e., 'lemmas') into cut-free proofs, that is adequate for automatization and proof analysis (compare [9,5]). …”