Sequent Calculi with procedure calls

In this paper, we introduce two focussed sequent calculi, LK(T ) and LK(T ), that are based on Miller-Liang’s LKF system [LM09] for polarised classical logic. The novelty is that those sequent calculi integrate the possibility to call a decision procedure for some background theory T , and the possibility to polarise literals "on the fly" during proof-search. These features are used in other works [FLM12, FGLM13] to simulate the DPLL(T ) procedure [NOT06] as proof-search in the extension of LK(T ) with a cut-rule. In this report we therefore prove cut-elimination in LK(T ). Contrary to what happens in the empty theory, the polarity of literals affects the provability of formulae in presence of a theory T . On the other hand, changing the polarities of connectives does not change the provability of formulae, only the shape of proofs. In order to prove this, we introduce a second sequent calculus, LK(T ) that extends LK(T ) with a relaxed focussing discipline, but we then show an encoding of LK(T ) back into the more restrictive system LK(T ). We then prove completeness of LK(T ) (and therefore of LK(T )) with respect to firstorder reasoning modulo the ground propositional lemmas of the background theory T .