Extracting Proofs from Branch-and-Prune

δ-Complete decision procedures can solve SMT problems over the reals with a wide range of nonlinear functions, allowing “δ-bounded errors”. The scalability of such procedures usually depends on efficient numerical procedures, whose implementation can be error-prone. It is important for δ-complete solvers to provide certificates to prove the correctness of their answers. We show how to do this for DPLL〈ICP〉, a general solving framework based on Interval Constraint Propagation. We focus on the construction of proof trees for the “unsat” answers and the proof-checking of their correctness. Besides certifying solvers, we find our approach a promising one for automated theorem proving over the reals, exploiting the power of numerical algorithms in a formal way. One direct application is to establish many nonlinear lemmas in the Flyspeck project, for the formal proof of the Kepler Conjecture. This research was sponsored by the National Science Foundation grants no. DMS1068829, no. CNS0926181 and no. CNS0931985, the GSRC under contract no. 1041377, the Semiconductor Research Corporation under contract no. 2005TJ1366, and the Office of Naval Research under award no. N000141010188.