Reconstruction Proofs at the Assertion Level

Most automated theorem provers suffer from the problem that they can produce proofs only in formalisms difficult to understand even for experienced mathematicians. Effort has been made to reconstruct natural deduction (ND) proofs from such machine generated proofs. Although the single steps in ND proofs are easy to understand, the entire proof is usually at a low level of abstraction, containing too many tedious steps. To obtain proofs similar to those found in mathematical textbooks, we propose a new formalism, called ND style proofs at the assertion level, where derivations are mostly justified by the application of a definition or a theorem. After characterizing the structure of compound ND proof segments allowing assertion level justification, we show that the same derivations can be achieved by domain-specific inference rules as well. Furthermore, these rules can be represented compactly in a tree structure. Finally, we describe a system called PROVERB, which substantially shortens ND proofs by abstracting them to the assertion level and then transforms them into natural language.