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

We devise a calculus based on the resolution and paramodulation rules and operating on schemata of formulæ. These schemata are de ned inductively, using convergent rewrite systems encoding primitive recursive de nitions. The main original feature of this calculus is that the rules operate on formulæ or terms occurring at arbitrary deep positions inside the considered schemata, thus a ecting the corresponding rewrite system. Each inference step in the new calculus corresponds to several applications of the usual resolution or paramodulation rules over the considered instances. The calculus has been implemented in the proof editor Shred (available on the web). 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 Fürtsenberg's proof of the in nite of prime numbers [9].