Inconsistency proving of CSPs is typically achieved by a combination of systematic search and arc consistency, which can both be characterized as resolution. However, it is well-known that there are cases where resolution produces exponential contradiction proofs, although proofs of polynomial size exist. For this reason, we will use optimization methods to reduce the proof size globally by 1. decomposing the original unsatisfiability problem into a conjunction of satisfiable subproblems and by 2. finding an ordering that separates the solution spaces of the subproblems. This principle allows Operation Research methods to prove the inconsistency of overconstrained linear programs even if domains are infinite. We exploit the principle for testing the satisfiability of global user requirements in product configuration problems.