The Presentation of Proofs at the Assertion Level

Abstract

Most automated theorem provers suffer from the problem that they can produce proofs only in formalisms difficult to understand even for experienced mathematicians. Efforts have been made to transform such machine generated proofs into natural deduction (ND) proofs. Although the single steps are now easy to understand, the entire proof is usually at a low level of abstraction, containing too many tedious steps. Therefore, it is not adequate as input to natural language generation systems. To overcome these problems, we propose a new intermediate representation, called ND style proofs at the assertion level. After illustrating the notion intuitively, we show that the assertion level steps can be justified by domain-specific inference rules, and that these rules can be represented compactly in a tree structure. Finally, we describe a procedure which substantially shortens ND proofs by abstracting them to the assertion level, and report our experience with further transformation into natural language.