## Section: New Results

### Automated theorem proving

Guillaume Bury defined a sound and complete extension of the tableaux method to handle linear arithmetic. The rules are based on a variant of the simplex algorithm for rational and real linear arithmetic, and a Branch&Bound algorithm for integer arithmetic.

Guillaume Bury defined an encoding of analytical tableaux rules as a theory for smt solvers. The theory acts like a lazy cnf conversion during the proof search and allows to integrate the cnf conversion into the resolution proof for unsatisfiable formulas. This work was implemented in mSAT.

Simon Cruanes added many improvements to Logtk, in particular a better algorithm to reduce formulas to Clausal Normal Form. A presentation of its design and implementation has been made at PAAR 2014[16] . He also used Zipperposition as a testbed for integer linear arithmetic; a sophisticated inference system for this fragment of arithmetic was designed and implemented in Zipperposition, including many redundancy criteria and simplification rules that make it efficient in practice. The arithmetic-enabled Zipperposition version entered CASC-J7 , the annual competition of Automated Theorem Provers, in the first-order theorems with linear arithmetic division where it had very promising results (on integer problems only, since Zipperposition doesn't handle rationals).

Another extension of Zipperposition has been performed by Julien Rateau,
Simon Cruanes, and David Delahaye, in order to deal with a fragment of set
theory in the same vein as the *STR$\dot{+}$VE$\subseteq $* prover [40] . This
extension relies on a specific normal form of literal, which only involves
the $\subseteq $, $\cap $, $\cup $, and complement set operators. In the future,
the idea is to use this extension in the framework of the *BWare* project
to verify *B* proof obligations coming from industrial benchmarks.

The current effort of research on Zipperposition focuses on extending superposition to handle structural induction, following the work from [45] . The current prototype is able to prove simple properties on natural numbers, binary trees and lists.

Kailiang Ji defined a set of rewrite rules for the equivalence between CTL formulas (denote them as ${R}_{CTL}$), by taking them as terms of designed predicates. For a given transition system model, we transform it into a set of rewrite rules (denote them as ${R}_{m}$). Then any CTL property of the transition system can be proved in deduction modulo ${R}_{CTL}\cup {R}_{m}$, by specifying the model checking problems into designed first-order formulas. This method was implemented in iProver Modulo, and the experimental evaluation was reported in workshop of Locali 2014.