On the Generation of Quantified Lemmas

Abstract

In this paper we present an algorithmic method of lemma introduction. Given a proof in predicate logic with equality the algorithm is capable of introducing several universal lemmas. The method is based on an inversion of Gentzen's cut-elimination method for sequent calculus. The first step consists of the computation of a compact representation (a so-called decomposition) of Herbrand instances in a cut-free proof. Given a decomposition the problem of computing the corresponding lemmas is reduced to the solution of a secondorder unification problem (the solution conditions). It is shown that that there is always a solution of the solution conditions, the canonical solution. This solution yields a sequence of lemmas and, finally, a proof based on these lemmas. Various techniques are developed to simplify the canonical solution resulting in a reduction of proof complexity. Moreover, the paper contains a comprehensive empirical evaluation of the implemented method and gives an application to a mathematical proof.

Proof Technology and Learning in Mathematics: Common Issues and Perspectives

“…But such proofs could be much reduced if we could introduce cut rules, and therefore lemmas, by inverting cut-elimination. Recent results in this direction have been presented in Ebner, Hetzl, Leitsch, Reis, and Weller (2018).…”

Higher-order pattern generalization modulo equational theories

“…This related work does not consider anti-unification with higher-order terms in the presence of equational axioms. However, such a combination can be useful, for instance, for developing indexing techniques for higher-order theorem provers (Libal and Steen, 2016), in higher-order program manipulation tools, proof transformation (Ebner et al, 2019;Hetzl et al, 2014), and inductive theorem proving (Eberhard and Hetzl, 2015;Eberhard et al, 2017).…”

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].…”