Using Reeection to Build Eecient and Certiied Decision Procedures

In this paper we explain how computational reeection can help build eecient certiied decision procedure in reduction systems. We have developped a decision procedure on abelian rings in the Coq system but the approach we describe applies to all reduction systems that allow the deenition of concrete types (or datatypes). We show that computational reeection is more eecient than an LCF-like approach to implement decision procedures in a reduction system. We discuss the concept of total reeection, which we have investigated in Coq using two facts: the extraction process available in Coq and the fact that the implementation language of the Coq system can be considered as a sublanguage of Coq. Total reeection is not yet implemented in Coq but we can test its performance as the extraction process is eeective. Both reeection and total reeection are conservative extensions of the reduction system in which they are used. We also discuss performance and related approaches. In the paper,we assume basic knowledges of ML and proof-checkers.