Hyperproperties relate multiple executions of a program and are commonly used to specify security and information-flow policies. Most existing work has focused on the verification of k-safety properties, i.e., properties that state that all k-tuples of execution traces satisfy a given property. In this paper, we study the automated verification of richer properties that combine universal and existential quantification over executions. Concretely, we consider ∀k∃l properties, which state that for all k executions, there exist l executions that, together, satisfy a property. This captures important non-k-safety requirements, including hyperliveness properties such as generalized non-interference, opacity, refinement, and robustness. We design an automated constraint-based algorithm for the verification of ∀k∃l properties. Our algorithm leverages a sound-and-complete program logic and a (parameterized) strongest postcondition computation. We implement our algorithm in a tool called ForEx and report on encouraging experimental results.