Syntactic Conditions for Invertibility in Sequent Calculi

Formalised proofs of Cut admissibility often rely on the invertibility of the rules of a sequent calculus. We will present some sufficient conditions for when a rule is invertible with respect to a calculus, which we illustrate with many examples. Appropriate definitions are given for rarely defined intuitive notions, such as a formula being principal for a rule. It must be noted we give purely syntactic criteria; no guarantees are given as to the suitability of the rules. We also formalise some of the results in the proof assistant Isabelle, as a means to automating Cut admissibility proofs.