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