We consider feasibility of linear integer problems in the context of verification systems such as SMT solvers or theorem provers. Although satisfiability of linear integer problems is decidable, many state-of-the-art implementations neglect termination in favor of efficiency. We present the calculus CutSat++ that is sound, terminating, complete, and leaves enough space for model assumptions and simplification rules in order to be efficient in practice. CutSat++ combines model-driven reasoning and quantifier elimination to the feasibility of linear integer problems.