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EMV is the international protocol standard for smartcard payment and is used in over 9 billion cards worldwide. Despite the standard's advertised security, various issues have been previously uncovered, deriving from logical flaws that are hard to spot in EMV's lengthy and complex specification, running over 2,000 pages. We formalize a comprehensive symbolic model of EMV in Tamarin, a state-of-the-art protocol verifier. Our model is the first that supports a fine-grained analysis of all relevant security guarantees that EMV is intended to offer. We use our model to automatically identify flaws that lead to two critical attacks: one that defrauds the cardholder and a second that defrauds the merchant. First, criminals can use a victim's Visa contactless card to make payments for amounts that require cardholder verification, without knowledge of the card's PIN. We built a proof-of-concept Android application and successfully demonstrated this attack on real-world payment terminals. Second, criminals can trick the terminal into accepting an unauthentic offline transaction, which the issuing bank should later decline, after the criminal has walked away with the goods. This attack is possible for implementations following the standard, although we did not test it on actual terminals for ethical reasons. Finally, we propose and verify improvements to the standard that prevent these attacks, as well as any other attacks that violate the considered security properties. The proposed improvements can be easily implemented in the terminals and do not affect the cards in circulation.