In this paper we relate different formulations of the DPLL(T ) procedure. The first formulation is based on a system of rewrite rules, which we denote DPLL(T ). The second formulation is an inference system of, which we denote LKDPLL(T ). The third formulation is the application of a standard proof-search mechanism in a sequent calculus LKp(T ) introduced here. We formalise an encoding from DPLL(T ) to LKDPLL(T ) that was, to our knowledge, never explicitly given and, in the case where DPLL(T ) is extended with backjumping and Lemma learning, never even implicitly given. We also formalise an encoding from LKDPLL(T ) to LKp(T ), building on Ivan Gazeau's previous work: we extend his work in that we handle the "-modulo-Theory" aspect of SAT-modulo-theory, by extending the sequent calculus to allow calls to a theory solver (seen as a blackbox). We also extend his work in that we handle advanced features of DPLL such as backjumping and Lemma learning, etc. Finally, we re fine the approach by starting to formalise quantitative aspects of the simulations: the complexity is preserved (number of steps to build complete proofs). Other aspects remain to be formalised (non-determinism of the search / width of search space).
