Proof systems for Moss' coalgebraic logic

We study Gentzen-style proof theory of the finitary version of the coalgebraic logic introduced by L. Moss. The logic captures the behaviour of coalgebras for a large class of set functors. The syntax of the logic, defined uniformly with respect to a finitary coalgebraic type functor T  , uses a single modal operator ∇T∇T of arity given by the functor T   itself, and its semantics is defined in terms of a relation lifting functor T¯. An axiomatization of the logic, consisting of modal distributive laws, has been given together with an algebraic completeness proof in work of C. Kupke, A. Kurz and Y. Venema.In this paper, following our previous work on structural proof theory of the logic in the special case of the finitary powerset functor, we present cut-free, one- and two-sided sequent calculi for the finitary version of Moss' coalgebraic logic for a general finitary functor T in a uniform way. For the two-sided calculi to be cut-free we use a language extended with the boolean dual of the nabla modality.