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We present a simple theory explaining the construction and the correctness ofan incremental and worst-case optimal decision procedure for modal logic witheventualities. The procedure gives an abstract account of important aspects ofGor\'e and Widmann's PDL prover. Starting from an input formula, the proceduregrows a Pratt-style graph tableau until the tableau proves or disproves thesatisfiability of the formula. The procedure provides a basis for practicalprovers since satisfiability and unsatisfiability of formulas can often bedetermined with small tableaux.

Correctness of an Incremental and Worst-Case Optimal Decision Procedure for Modal Logic with Eventualities