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We present an implemented method for encoding reasoning problems of a discrete version of the classical logic event calculus in propositional conjunctive normal form, enabling the problems to be solved efficiently by off-the-shelf complete satisfiability (SAT) solvers. We build on the previous encoding method of Shanahan and Witkowski, extending it to support causal constraints, concurrent events, determining fluents, effect axioms with conditions, events triggered by conditions, gradual change, incompletely specified initial situations, state constraints, and release from the commonsense law of inertia. We present an alternative classical logic axiomatization of the event calculus and prove its equivalence to a standard axiomatization for integer time. We describe our encoding method based on the alternative axiomatization and prove its correctness. We evaluate the method on 14 benchmark reasoning problems for the event calculus and compare performance with the causal calculator on eight problems in the zoo world domain.