Abstract. Modus Ponens says that if you know A and you know that A implies B, then you know B. This is a basic rule that we take for granted

Streams are acyclic directed subgraphs of the logical flow graph of a proof and represent bundles of paths with the same origin and the same end. Streams can be described with a natural algebraic formalism which allows to explain in algebraic terms the evolution of proofs during cutelimination. In our approach, "logic" is often forgotten and combinatorial properties of graphs are taken into account to explain logical phenomena.

Proof theory is the branch of mathematical logic that investigates mathematical reasoning and mathematical proofs. This area emanated from Hilbert’s program calling for consistency proofs of formal theories. An interesting aspect of many consistency proofs is that they have a largely constructive character, relying on a proof transformation (like cut-elimination) or on a proof translation (like the functional interpretation). It is therefore often possible to regard a consistency proof not only as a general justification of a formal system, but also as a method that can be used to analyze concrete proofs formalized in that system. This viewpoint turned out to be very productive and several interesting mathematical results have been obtained by manually applying these methods to existing proofs. But the manual application of such methods is not the only option: Being of an algorithmic nature they can – in principle – be automated. There are, however, different requirements on a method, depending on wether it shall actually be implemented or merely understood to work in principle. The

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Abstract. Hilbert’s ε-calculus is based on an extension of the language of predicate logic by a term-forming operator εx. Two fundamental results about the ε-calculus, the first and second epsilon theorem, play a rôle similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand’s Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential theorems obtained by this elimination procedure. Keywords: Hilbert’s ε-calculus, epsilon theorems, Herbrand’s theorem, proof complexity 1.