Differential privacy (DP) has established itself as one of the standards for ensuring privacy of individual data. However, reasoning about DP is a challenging and error-prone task, hence methods for formal verification and refutation of DP properties have received significant interest in recent years. In this work, we present a novel method for automated formal refutation of $\epsilon$-DP. Our method refutes $\epsilon$-DP by searching for a pair of inputs together with a non-negative function over outputs whose expected value on these two inputs differs by a significant amount. The two inputs and the non-negative function over outputs are computed simultaneously, by utilizing upper expectation supermartingales and lower expectation submartingales from probabilistic program analysis, which we leverage to introduce a sound and complete proof rule for $\epsilon$-DP refutation. To the best of our knowledge, our method is the first method for $\epsilon$-DP refutation to offer the following four desirable features: (1)~it is fully automated, (2)~it is applicable to stochastic mechanisms with sampling instructions from both discrete and continuous distributions, (3)~it provides soundness guarantees, and (4)~it provides semi-completeness guarantees. Our experiments show that our prototype tool SuperDP achieves superior performance compared to the state of the art and manages to refute $\epsilon$-DP for a number of challenging examples collected from the literature, including ones that were out of the reach of prior methods.