Lemma Extraction Criteria Based on Properties of Theorem Statements

Both easily readable and obscure proof scripts can be found in the Mizar Mathematical Library (MML). Many authors do not want to invest additional e orts in improving readability of their deductions, once the computer accepts their scripts. In their opinion, text of the proofs are ignored by most of the readers and the readability is essential only at the top level of scripts where statements of theorems are formulated. However, the analysis of such scripts is unavoidable if we have to rebuild some theorems to make them stronger or more easily applicable. Therefore, it is important to develop tools that can improve legibility of proofs, in particular those that shorten reasoning by removing technical sub-deductions, and this requires development of criteria that lead to extraction of statements which, in the opinion of human readers, describe well the extracted reasoning. We propose characteristics of formula complexity that can be applied to determine which sub-deductions should be extracted so that resulting lemmas are more comprehensible. To better understand their signi cance we study the distribution of these characteristics on statements of theorems that are collected in the MML.