not true in the model where , and the interpretation where ). Consequently, is satisfiable (it is satisfied by the same model and interpretation). However, a closed tableau could be generated with and , and substituting with would generate a closure. A correct procedure is to first make universal quantifiers explicit, thus generating . The following two variants are also correct. Applying to the whole tableau a substitution to the free variables of the tableau is a correct rule, provided that this substitution is free for the formula representing the tableau. In other worlds, applying such a substitution leads to a tableau whose formula is still a consequence of the input set. Using most general unifiers automatically ensures that the condition of freeness for the tableau is met. While in general every variable has to be replaced with the same term in the whole tableau, there are some special cases in which this is not necessary. Tableaux with unification can be proved complete: if a set of formulae is unsatisfiable, it has a tableau-with-unification proof. However, actually finding such a proof may be a difficult problem. Contrarily to the case without unification, applying a substitution can modify the existing part of a tableau; while applying a substitution closes at least a branch, it may make other branches impossible to close (even if the set is unsatisfiable). A solution to this problem is delayed instantiation: no substitution is applied until one that closes all branches at the same time is found. With this variant, a proof for an unsatisfiable set can always be found by a suitable policy of application of the other rules. This method however requires the whole tableau to be kept in memory: the general method closes branches, which can be then discarded, while this variant does not close any branch until the end. The problem that some tableaux that can be generated are impossible to close even if the set is unsatisfiable is common to other sets of tableau expansion rules: even if some specific sequences of application of these rules allow constructing a closed tableau (if the set is unsatisfiable), some other sequences lead to tableaux that cannot be closed. General solutions for these cases are outlined in the "Searching for a tableau" section. Tableau calculi and their properties A tableau calculus is a set of rules that allows building and modification of a tableau. Propositional