Using Extended Tactics to Do Proof Transformations

In this thesis we develop a comprehensive human-oriented theorem proving system that integrates several different proof systems. The main theorem proving environment centers around a natural Gentzen first-order logic system. This allows construction of natural proofs, encourages user involvement in the search for proofs, and facilitates understanding of the resulting proofs. We integrate more abstract automatically generated proofs such as resolution refutations by transforming them to proofs in the Gentzen system. Expansion trees are another proof system used as an intermediate stage in transformations between the abstract and natural systems. They are a compact representation useful for transformations and other computations. We develop a programming language approach to theorem proving based on tactics and tacticals. Our extended tactics provide a method for doing proof transformations, as well as facilitate interactive theorem proving, allowing full integration of interactive and automatic theorem proving. In the system, we explicitly represent proofs in each proof system and view expansion tree proofs as types for Gentzen proof terms. This explicit proof representation allows proofs to be manipulated as meaningful data objects and used in various computations. For example, the proof terms in the natural Gentzen system can be used to obtain natural language explanations of proofs. We foresee several applications for this kind of theorem proving system, such as use as a logic tutor, a tool for doing mathematics, or an enhanced reasoner and explanation facility for existing A1 systems. Comments University of Pennsylvania Department of Computer and Information Science Technical Report No. MSCIS-86-89. This technical report is available at ScholarlyCommons: https://repository.upenn.edu/cis_reports/666 Using Extended Tactics To Do Proof Transformations