## Section: Research Program

### Automated theorem proving

Deduction modulo has originally been proposed to solve a problem in
automated theorem proving and some of the early work in this area
focused on the design of an automated theorem proving method called
*Resolution modulo*, but this method was so complex that it was
never implemented. This method was simplified in 2010 [5]
and it could
then be implemented. This implementation that builds on the
iProver effort [46] is called iProver modulo.

iProver modulo gave surprisingly good results [4] , so
that we use it now to search for proofs in many areas: in the theory
of classes—also known as *B* set theory—, on finite
structures, etc. Similar ideas have also been implemented for the
tableau method with in particular several extensions of the *Zenon*
automated theorem prover. More precisely, two extensions have been
realized: the first one is called *Super Zenon* [13] [35] and is an
extension to superdeduction (which is a variant of Deduction modulo), and
the second one is called
*Zenon Modulo* [33] , [34] and is an
extension to Deduction modulo. Both extensions have been extensively
tested over first order problems (of the TPTP library), and also
provide good results in terms of number of proved problems. In
particular, these tools provide good performances in set theory, so
that *Super Zenon* has been successfully applied to verify *B* proof
rules of *Atelier B* (work in collaboration with
*Siemens*). Similarly, we plan to apply *Zenon Modulo* in the framework of
the *BWare* project to verify *B* proof obligations coming from
the modeling of industrial applications.

More generally, we believe that proof-checking and automated theorem proving have a lot to learn from each other, because a proof is both a static linguistic object justifying the truth of a proposition and a dynamic process of proving this proposition.