Methods of Lemma Extraction in Natural Deduction Proofs

Abstract

The existing examples of natural deduction proofs, either declarative or procedural, indicate that often the legibility of proof scripts is of secondary importance to the authors. As soon as the computer accepts the proof script, many authors do not work on improving the parts that could be shortened and do not avoid repetitions of technical sub-deductions, which often could be replaced by a single lemma. This article presents selected properties of reasoning passages that may be used to determine if a reasoning passage can be extracted from a proof script, transformed into a lemma and replaced by a reference to the newly created lemma. Additionally, we present methods for improving the legibility of the reasoning that remains after the extraction of the lemmas.

Combining the Syntactic and Semantic Representations of Mizar Proofs

“…Let us fix such a let-assume-thus step. We encapsulate a part of deduction beginning from the step using the created list of rules as a sub-deduction that proves the reconstructed thesis (the correctness condition of such cut introduction have been developed in [25]). Then we replace this step by a conclusion where the original thesis is given as the proposition and we refer to the sub-deduction and have ∀ y : object HIDDENM1 .…”

Four Decades of Mizar

“…Proof graphs measuring information flow available in the MML can be useful for that purpose as well [7,19]. Such enhancements are part of routine maintenance of Mizar Mathematical Library that includes generalizing notions and theorems (and this new representation can help to obtain interesting mathematical results [5]), elimination of redundancy [9], preserving integrity [23], etc.…”

Definitional Expansions in Mizar

“…MIZAR is also a common name for the components of the system: a formal language derived from the mathematical vernacular and dozens of computer programs responsible for formal verification, optimization of proofs [24,25]. An integrant part of the system is the Mizar Mathematical Library (MML), a collection of articles written in the MIZAR language.…”