A paramodulation-based calculus for refuting schemata of clause sets defined by rewrite rules

Abstract

International audienceWe devise a calculus based on the resolution and paramodulation rules and operating on schemata of formulæ. These schemata are defined inductively, using convergent rewrite systems encoding primitive recursive definitions. The main original feature of this calculus is that the rules operate on formulæ or terms occurring at arbitrarily deep positions inside the considered schemata, possibly by applying transformations on the corresponding rewrite system. Each inference step in the new calculus corresponds to several applications of the usual resolution or paramodulation rules over the instances of the considered schemata. The calculus has been implemented in the proof editor Shred. As an example of application we provide a formal refutation of a schema of clause sets generated by applying the CERES cut-elimination method on Fürstenberg's proof of the infinity of prime numbers