Introducing Quantified Cuts in Logic with Equality

Abstract

Abstract. Cut-introduction is a technique for structuring and compressing formal proofs. In this paper we generalize our cut-introduction method for the introduction of quantified lemmas of the form ∀x.A (for quantifier-free A) to a method generating lemmas of the form ∀x1 . . . ∀xn.A. Moreover, we extend the original method to predicate logic with equality. The new method was implemented and applied to the TSTP proof database. It is shown that the extension of the method to handle equality and quantifier-blocks leads to a substantial improvement of the old algorithm.

Herbrand's theorem revisited

“…For example, motivated by the aim to structure and compress automatically generated proofs, an algorithm for Π 1 cut-introduction based on proof grammars has been developed in [7]. This method has been implemented and empirically evaluated with good results in [8]. An extension of these techniques to the case of proofs with Π 1 -induction has led to a new technique for inductive theorem proving [4] which is currently being implemented.…”

Importing SMT and Connection proofs as expansion trees

“…It supports different proof formats, such as LK (with or without equality) for first and higher order logic, Robinson's resolution calculus [11], the schematic calculus LKS [4] and, more recently, expansion trees. It provides various algorithms for proofs, such as reductive cut-elimination [5], cut-elimination by resolution [2], cut-introduction [6], Skolemization, and translations between the proof formats. GAPT also comes with prooftool [3], an interactive proof visualization tool supporting all these formats.…”

“…Second of all, the use of a common representation facilitates the comparison of proofs and makes it possible to run and analyse algorithms developed for this representation without the need to adapt it to different formats. In particular, we have been using the imported proofs for experimenting proof compression via introduction of cuts [6]. Finally, it provides a simple sanity-check procedure and the possibility of building LK proofs.…”

Fast Cut-Elimination using Proof Terms: An Empirical Study

“…A computational implementation of Herbrand's theorem as provided by cut-elimination lies at the foundation of many applications in computational proof theory: if we can compress the Herbrand disjunction extracted from a proof using a special kind of tree grammar, then we can introduce a cut into the proof which reduces the number of quantifier inferences-in practice this method finds interesting non-analytic lemmas [25,23,22,10]. A similar approach can be used for automated inductive theorem proving, where the tree grammar generalizes a finite sequence of Herbrand disjunctions [9].…”