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In this paper, we present sequent calculus for branching-time temporal logic with until operator. This sequent calculus uses efficient loop-check techinque. We prove that we can use not all but only several special sequents from the derivation tree for the loop-check. We use indexes to discover these special sequents in the sequent calculus. These restrictions let us to get efficient decision procedure based on introduced sequent calculus.

Restrictions for loop-check in sequent calculus for temporal logic with until operator