Automated Deduction - A Basis for Applications Volume I Foundations - Calculi and Methods Volume II Systems and Implementation Techniques Volume III Applications

1. BASIC CONCEPTS OF INTERACTIVE THEOREM PROVING Interactive Theorem Proving ultimately aims at the construction of powerful reasoning tools that let us (computer scientists) prove things we cannot prove without the tools, and the tools cannot prove without us. Interaction typi­ cally is needed, for example, to direct and control the reasoning, to speculate or generalize strategic lemmas, and sometimes simply because the conjec­ ture to be proved does not hold. In software verification, for example, correct versions of specifications and programs typically are obtained only after a number of failed proof attempts and subsequent error corrections. Different interactive theorem provers may actually look quite different: They may support different logics (first-or higher-order, logics of programs, type theory etc.), may be generic or special-purpose tools, or may be tar­ geted to different applications. Nevertheless, they share common concepts and paradigms (e.g. architectural design, tactics, tactical reasoning etc.). The aim of this chapter is to describe the common concepts, design principles, and basic requirements of interactive theorem provers, and to explore the band­ width of variations. Having a 'person in the loop', strongly influences the design of the proof tool: proofs must remain comprehensible, - proof rules must be high-level and human-oriented, - persistent proof presentation and visualization becomes very important.

One Interactive Theorem Proving.- 1. Structured Specifications and Interactive Proofs with KIV.- 2. Proof Theory at Work: Program Development in the Minlog System.- 3. Interactive and automated proof construction in type theory.- 4. Integrating Automated and Interactive Theorem Proving.- Two Representation and Optimization Techniques.- 5. Term Indexing.- 6. Developing Deduction Systems: The Toolbox Style.- 7. Specifications of Inference Rules: Extensions of the PTTP Technique.- 8. Proof Analysis, Generalization and Reuse.- Three Parallel Inference Systems.- 9. Parallel Term Rewriting with PaReDuX.- 10. Parallel Theorem Provers Based on SETHEO.- 11. Massively Parallel Reasoning.- Four Comparision and Cooperation of Theorem Provers.- 12. Extension Methods in Automated Deduction.- 13. A Comparison of Equality Reasoning Heuristics.- 14. Cooperating Theorem Provers.