Aim of this work is to investigate from a proof-theoretic viewpoint a propositional and a predicate sequent calculus with an omega-type schema of inference that naturally interpret the propositional and the predicate until-free fragments of Linear Time Logic LTL respectively. The two calculi are based on a natural extension of ordinary sequents and of standard modal rules. We examine the pure propositional case (no extralogical axioms), the propositional and the first order predicate cases (both with a possibly infinite set of extralogical axioms). For each system we provide a syntactic proof of cut elimination and a proof of completeness.
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