Metalogical Frameworks II: Developing a Reflected Decision Procedure
Journal of Automated Reasoning
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Proving theorems is a creative act demanding new combinations of ideas and on occasion new methods of argument. For this reason, theorem proving systems need to be extensible. The provers should also remain correct under extension, so there must be a secure mechanism for doing this. The tactic-style provers pioneered by Edinburgh LCF provide a very effective way to achieve secure extensions, but in such systems, all new methods must be reduced to tactics. This is a drawback because there are other useful proof generating tools such as decision procedures; these include, for example, algorithms which reduce a deduction problem, such as arithmetic provability, to a computation on graphs.
The Nuprl system pioneered the combination of fixed decision procedures with tactics, but the issue of securely adding new ones was not solved. In this paper we show how to safely include user- defined decision procedures in theorem provers. The idea is to prove properties of the procedure inside the prover''s logic and then invoke a reflection rule to connect the procedure to the system. We also show that using a rich underlying logic permits an abstract account of the approach so that the results carry over to different implementations and other logics.
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