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We present a new method for proving rank lower bounds for Cutting Planes (CP) and several procedures based on lifting due to Lovász and Schrijver (LS), when viewed as proof systems for unsatisfiability. We apply this method to obtain the following new results: First, we prove near-optimal rank bounds for Cutting Planes and Lovász-Schrijver proofs for several prominent unsatisfiable CNF examples, including random kCNF formulas and the Tseitin graph formulas. It follows from these lower boundsdoi:10.1109/sfcs.2003.1238206 dblp:conf/focs/Buresh-OppenheimGHMP03 fatcat:g773zgvpwbandjhsw6opzplfae