Proof planning is an application of AI-planning in mathematical domains. As opposed to planning for domains such as blocks world or transportation, the domain knowledge for mathematical domains is dicult to extract. Hence proof planning requires clever knowledge engineering and representation of the domain knowledge. We think that on the one hand, the resulting domain denitions that include operators, supermethods, control-rules, and constraint solver are interesting in itself. On the other hand, they can provide ideas for modeling other realistic domains and for means of search reduction in planning. Therefore, we present proof planning and an exemplary domain used for planning proofs of so-called limit theorems that lead to proofs that were beyond the capabilities of other current proof planners and theorem provers. 1 Introduction While humans can cope with long and complex proofs and have strategies to avoid less promising proof paths, classical automated theore...

This paper reports on the integration of the higher-order theorem proving environment Tps [Andrews et al., 1996] into the mathematical assistant Ωmega [Benzmuller et al., 1997]. Tps can be called from mega either as a black box or as an interactive system. In black box mode, the user has control over the parameters which control proof search in Tps; in interactive mode, all features of the Tps-system are available to the user. If the subproblem which is passed to Tps contains concepts defined in Ωmega's database of mathematical theories, these definitions are not instantiated but are also passed to Tps. Using a special theory which contains proof tactics that model the ND-calculus variant of Tps within mega, any complete or partial proof generated in Tps can be translated one to one into an mega proof plan. Proof transformation is realised by proof plan expansion in Ωmega's 3-dimensional proof data structure, and remains transparent to the user.

Proof planning is an alternative methodology to classical automated theorem proving based on exhaustive search that was first introduced by Bundy [8]. The goal of this paper is to extend the current realm of proof planning to cope with genuinely mathematical problems such as the well-known limit theorems first investigated for automated theorem proving by Bledsoe. The report presents a general methodology and contains ideas that are new for proof planning and theorem proving, most importantly ideas for search control and for the integration of domain knowledge into a general proof planning framework. We extend proof planning by employing explicit control-rules and supermethods. We combine proof planning with constraint solving.

A lot of the human ability to prove hard mathematical theorems can be ascribed to a problem-specific problem solving know-how. Such knowledge is intrinsically incomplete. In order to prove related problems human mathematicians, however, can go beyond the acquired knowledge by adapting their know-how to new related problems. These two aspects, having rich experience and extending it by need, can be simulated in a proof planning framework: the problem-specific reasoning knowledge is represented in form of declarative planning operators, called methods; since these are declarative, they can be mechanically adapted to new situations by so-called metamethods. In this contribution we apply this framework to two prominent proofs in theorem proving, first, we present methods for proving the ground completeness of binary resolution, which essentially correspond to key lemmata, and then, we show how these methods can be reused for the proof of the ground completeness of lock resolution.

Introduction In the following sections I am going to describe why it might be a good idea to search for proofs via a proof plan rather than directly on calculus level (section 2). Also I am going to describe what a proof planning system needs to provide in order to instantiate these hopes (section 3). Then I am going to argue briefly that methods as described in [Bun88, MB96, HKRS94, Seh95] can be used to implement the relevant features. 2 Motivation Traditional Automated Theorem Proving (ATP) suffers from search in exponential search trees. In case one uses higher order object languages -- which seems to be appropriate for most interesting tasks -- there are further complications which are not easy to overcome. Compared to this traditional ATP approach, human mathematicians work differently: They plan their proofs on a very high level and fill in the details later. In addition they do not stick to one direction of argument. Interesting aspects