A weakened version of the Cutting Plane (CP) proof system with a restriction on the degree of falsity of intermediate inequalities was introduced by Goerdt. He proved an exponential lower bound for CP proofs with degree of falsity bounded by n/(log~2 n+1), where n is the number of variables. Hirsch and Nikolenko strengthened this result by establishing a direct connection between CP and Resolution proofs. This result implies an exponential lower bound on the proof length of the Tseitin-Urquhart tautologies, when the degree of falsity is bounded by cn for some constant c. In this paper we generalize this result for extensions of Lovasz-Schrijver calculi (LS), namely for LS~k+CP~k proof systems introduced by Grigoriev et al. We show that any LS~k+CP~k proof with bounded degree of falsity can be transformed into a Res(k) proof. We also prove lower and upper bounds for the new system.
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