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Bitcoin is a peer-to-peer payment system proposed by Nakamoto in 2008. Based on the Nakamoto consensus, Bagaria, Kannan, Tse, Fanti, and Viswanath proposed the Prism protocol in 2018 and showed that it achieves near-optimal blockchain throughput while maintaining a similar level of security as bitcoin. Previous probabilistic security guarantees for the bitcoin and Prism backbone protocols were either established under a simplified discrete-time model or expressed in terms of exponential order results. This paper presents a streamlined and strengthened analysis under a more realistic continuous-time model. A fully rigorous model for blockchains is developed with no restrictions on adversarial miners except for an upper bound on their aggregate mining rate. The only assumption on the peer-to-peer network is that all block propagation delays are upper bounded by a constant. A new notion of "t-credible blockchains" is introduced, which, together with some carefully defined "typical" events concerning block production over time intervals, is crucial to establish probabilisitic security guarantees in continuous time. A blockchain growth theorem, a blockchain quality theorem, and a common prefix theorem are established with explicit probability bounds. Moreover, under a certain typical event which occurs with probability close to $1$, a valid transaction that is deep enough in one credible blockchain is shown to be permanent in the sense that it must be found in} in all future credible blockchains.