Section: New Software and Platforms

Super Zenon and Zenon Modulo

Several extensions of the Zenon automated theorem prover (developed by Damien Doligez at Inria in the Gallium team) to Deduction modulo have been studied. These extensions intend to be applied in the context of the automatic verification of proof rules and obligations coming from industrial applications formalized using the B method.

The first extension, developed by Mélanie Jacquel and David Delahaye, is called Super Zenon and is an extension of Zenon to superdeduction, which can be seen as a variant of Deduction modulo. This extension is a generalization of previous experiments  [42] together with Catherine Dubois and Karim Berkani (Siemens), where Zenon has been used and extended to superdeduction to deal with the B set theory and automatically prove proof rules of Atelier B. This generalization consists in allowing us to apply the extension of Zenon to superdeduction to any first order theory by means of a heuristic that automatically transforms axioms of the theory into rewrite rules. This work is described in [13] [35] , which also proposes a study of the possibility of recovering intuition from automated proofs using superdeduction.

The second extension, developed by Pierre Halmagrand, David Delahaye, Damien Doligez, and Olivier Hermant, is called Zenon Modulo and is an extension of Zenon to Deduction modulo. Compared to Super Zenon, this extension allows us to deal with rewrite rules both over propositions and terms. Like Super Zenon, Zenon Modulo is able to deal with any first order theory by means of a similar heuristic. To assess the approach of Zenon Modulo, we have applied this extension to the first order problems coming from the TPTP library. An increase of the number of proved problems has been observed, with in particular a significant increase in the category of set theory. Over these problems of the TPTP library, we have also observed a significant proof size reduction, which confirms this aspect of Deduction modulo. These results are gathered into two publications  [33] , [34] .

The third extension, developed by Guillaume Bury and David Delahaye, is an extension of Zenon to (rational and integer) linear arithmetic (using the simplex algorithm), that has been integrated to Zenon Modulo by Guillaume Bury and Pierre Halmagrand, in order to be applied in the framework of the B set theory to the verification of proof obligations of Atelier B [17] . Experiments have been conducted over the benchmarks of the BWare project, and it turns out that more than 95% of the proof obligations are proved thanks to this extension.