### 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 second-order 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.

Original language | English |
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Pages (from-to) | 95-126 |

Number of pages | 32 |

Journal | Journal of Automated Reasoning |

Volume | 63 |

Issue number | 1 |

DOIs | |

Publication status | Published - 15 Jun 2019 |

Externally published | Yes |

### Keywords

- Cut-introduction
- Herbrand’s theorem
- Lemma generation
- Proof theory
- The resolution calculus

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## Cite this

*Journal of Automated Reasoning*,

*63*(1), 95-126. https://doi.org/10.1007/s10817-018-9462-8