We describe an algorithmic method of proof compression based on the introduction of Π 2 -cuts into a cut-free LK-proof. The current approach is based on an inversion of Gentzen's cut-elimination method and extends former methods for introducing Π 1 -cuts. The Herbrand instances of a cut-free proof π of a sequent S are described by a grammar G which encodes substitutions defined in the elimination of quantified cuts. We present an algorithm which, given a grammar G, constructs a Π 2 -cut formula A and a proof π ′ of S with one cut on A. It is shown that, by this algorithm, we can achieve an exponential proof compression.

A tableau calculus is proposed, based on a compressed representation of clauses, where literals sharing a similar shape may be merged. The inferences applied on these literals are fused when possible, which reduces the size of the proof. It is shown that the obtained proof procedure is sound, refutationally complete and allows to reduce the size of the tableau by an exponential factor. The approach is compatible with all usual refinements of tableaux.

Abstract. Most interesting proofs in mathematics contain an inductive argument which requires an extension of the LK-calculus to formalize. The most commonly used calculi for induction contain a separate rule or axiom which reduces the valid proof theoretic properties of the calculus. To the best of our knowledge, there are no such calculi which allow cut-elimination to a normal form with the subformula property, i.e. every formula occurring in the proof is a subformula of the end sequent. Proof schemata are a variant of LK-proofs able to simulate induction by linking proofs together. There exists a schematic normal form which has comparable proof theoretic behaviour to normal forms with the subformula property. However, a calculus for the construction of proof schemata does not exist. In this paper, we introduce a calculus for proof schemata and prove soundness and completeness with respect to a fragment of the inductive arguments formalizable in Peano arithmetic.

Proof schemata are a variant of LK-proofs able to simulate various induction schemes in first-order logic by adding so called proof links to the standard first-order LK-calculus. Proof links allow proofs to reference proofs thus giving proof schemata a recursive structure. Unfortunately, applying reductive cutelimination is non-trivial in the presence of proof links. Borrowing the concept of lazy instantiation from functional programming, we evaluate proof links locally allowing reductive cut-elimination to proceed past them. Though, this method cannot be used to obtain cut-free proof schemata, we nonetheless obtain important results concerning the schematic CERES method, that is a method of cut-elimination for proof schemata based on resolution. In "Towards a clausal analysis of cut-elimination", it was shown that reductive cut-elimination transforms a given LK-proof in such a way that a subsumption relation holds between the pre-and post-transformation characteristic clause sets, i.e. the clause set representing the cut-structure of an LK-proof. Let CLpϕ 1 q be the characteristic clause set of a normal form ϕ 1 of an LK-proof ϕ that is reached by performing reductive cut-elimination on ϕ without atomic cut elimination. Then CLpϕ 1 q is subsumed by all characteristic clause sets extractable from any application of reductive cut-elimination to ϕ. Such a normal form is referred to as an ACNF top and plays an essential role in methods of cut-elimination by resolution. These results can be extended to proof schemata through our "lazy instantiation" of proof links, and provides an essential step toward a complete cut-elimination method for proof schemata.

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