## Section: New Results

### Sequent Calculus with Calls to a Decision Procedure

Participants : Mahfuza Farooque, Stéphane Lengrand.

In the PSI project, we have designed a version of the focussed sequent calculus (for first-order classical logic) that can call external decision procedures. Since the last Activity Report, we have finished proving the essential meta-theory for it: soundness, invertibility of asynchronous rules, cut-elimination, the fact that polarities do not affect provability but only the shape of proofs, and finally completeness.

The first properties are the object of [27] , while the latter ones have been obtained later in 2012.

A side-product of this meta-theory is a technical device that could be
used to encode other techniques from automated reasoning like
*connection tableaux*.

Secondly, we have encoded the SMT-solving algorithm DPLL(T) as the incremental construction of proof-trees in that sequent calculus [29] , [28] . A very interesting aspect of the encodings is that the basic rules of DPLL(T) makes use of cuts on atoms in sequent calculus, while the advanced jrules (e.g. backjumping) makes use of general cuts. This sheds a new light on the computational speed-ups that those advanced rules provide.

We have done the encoding for two distinct presentations of DPLL(T) in the literature, and we have formalised the connection between those two descriptions [29] .