The standard input format for Satisfiability Modulo Theories (SMT) solvers has now reached its second version and integrates many of the features useful for users to interact with their favourite SMT solver. However, although many SMT solvers do output proofs, no standardised proof format exists. We, here, propose for discussion at the PxTP Workshop a generic proof format in the SMT-LIB philosophy that is flexible enough to be easily recast for any SMT solver. The format is configurable so that the proof can be provided by the solver at the desired level of detail. 1

The problem of computing Craig Interpolants has recently received a lot of interest. In this paper, we address the problem of efficient generation of interpolants for some important fragments of first-order logic, which are amenable for effective decision procedures, called Satisfiability Modulo Theory solvers. We make the following contributions. First, we provide interpolation procedures for several basic theories of interest: the theories of linear arithmetic over the rationals, difference logic over rationals and integers, and UTVPI over rationals and integers. Second, we define a novel approach to interpolate combinations of theories, that applies to the Delayed Theory Combination approach. Efficiency is ensured by the fact that the proposed interpolation algorithms extend state-ofthe-art algorithms for Satisfiability Modulo Theories. Our experimental evaluation shows that the MathSAT SMT solver can produce interpolants with minor overhead in search, and much more efficiently than other competitor solvers.

SMT solvers are nowadays pervasive in verification tools. When the verification is about a critical system, the result of the SMT solver is also critical and cannot be trusted. The SMT-LIB 2.0 is a standard interface for SMT solvers but does not specify the output of the get-proof command. We present a proof system that is geared towards SMT solvers and follows their conceptually modular architecture. Our proof system makes a clear distinction between propositional and theory reasoning. Moreover, individual theories provide specific proof systems that are combined using the Nelson-Oppen proof scheme. We propose specific proof systems for linear real arithmetic (LRA) and uninterpreted functions (EUF) and discuss proof generation and proof checking. We have evaluated the cost of generating proofs in our proof system. Our experiments on benchmarks taken from the SMT-LIB library show that the simple mechanisms used in our approach suffice for a large majority of the selected benchmarks. 1