We propose a framework for building Nakamoto-style proof-of-work blockchains where blocks are treated differently in the ``longest chain rule''. The crucial parameter is a weight function assigning different weights to blocks according to their hash value. Our framework enables the analysis of different weight functions while proving all statements at the appropriate level of abstraction. This allows us to quickly derive protocol guarantees for different weight functions. We exemplify the usefulness of our framework by capturing the classical Bitcoin protocol as well as exponentially growing functions as special cases. We show the typical properties---chain growth, chain quality and common prefix---for both, and further show that the latter provide an additional guarantee, namely a weak form of optimistic responsiveness. More precisely, we prove for a certain class of exponentially growing weight functions that in periods without corruption, the confirmation time only depends on the unknown actual network delay instead of the known upper bound.