Proof Transformation and Expansion with a Parameterizable Inference Machine a 2 U U F a 2 F Def a 6 2 F U F a 6 2 U Def

1 Motivation Tactical theorem provers or proof planners, such as Clam 4], HOL3], or mega 2] provide proof tactics and/or proof methods to support abstract level proof construction and to improve the communication with the user. Tactics and methods typically abbreviate frequently recurring low level proof patterns. A system like mega 2] is capable of an explicit maintenance of abstract level proof objects and support a stepwise expansion of them in pure calculus level proofs while in other systems, e.g. HOL, the application of a tactic immediately introduces a calculus level derivation. The common problem is to describe and operationalize the mapping of the abstract level inferences in terms of a sequence of less abstract inferences. A similar problem arises when diierent deduction systems are integrated and complex inference steps of one system have to be expressed in terms of inferences of the other system. In mega this expansion or transformation knowledge is currently encoded in form of precise programs associated with each tactic/method. This approach clearly suuers on the problem that the programs have to be adapted whenever one of the inference systems is modiied { even in case of minor modiications like the permutation of the premise order in one single inference rule. In order to overcome these problems we propose to use a parameterizable inference machine as proof expansion and transformation mechanism. The parameters are the set of target inference rules whose composition suuces to justify the high-level derivation. Thus the execution of the transformation/expansion program is replaced by controlled proof search over a restricted set of target inferences. Thereby we gain a more exible and stable transformation/expansion mechanism as well as shorter and better maintainable speciications. As a concrete pa-rameterizable inference machine for mega we suggest to employ the agent based inference mechanism-ants. 2 Expansion/Transformation in mega We brieey sketch some examples for proof transformation and expansion in the context of mega: E1: The mega tactic modus tollens (mt): The expansion of mt is precisely deened in program P employing the tactic contrapos (cp) and the rule modus ponens (mp) in the following way: a) b :b :a mt P ?! a) b :b) :a cp :b :a mp Tactic cp is associated with further expansion information (where rule mp, for instance, is employed again) such that in a stepwise fashion megas pure natural deduction calculus level is reached in the end. Modifying the order of …