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We present a method for certifying that the values computed by an imperative program will be bounded by polynomials in the program's inputs. To this end, we introduce mwp-matrices and define a semantic relation ⊧ C : M, where C is a program and M is an mwp-matrix. It follows straightforwardly from our definitions that there exists M such that ⊧ C : M holds iff every value computed by C is bounded by a polynomial in the inputs. Furthermore, we provide a syntactical proof calculus and define the relation ⊢ C : M to hold iff there exists a derivation in the calculus where C : M is the bottom line. We prove that ⊢ C : M implies ⊧ C : M. By means of exhaustive proof search, an algorithm can decide if there exists M such that the relation ⊢ C : M holds, and thus, our results yield a computational method.