Rule-Elimination Theorems

Cut-elimination theorems constitute one of the most important classes of theorems of proof theory. Since Gentzen's proof of the cut-elimination theorem for the system $\mathbf{LK}$, several other proofs have been proposed. Even though the techniques of these proofs can be modified to sequent systems other than $\mathbf{LK}$, they are essentially of a very particular nature; each of them describes an algorithm to transform a given proof to a cut-free proof. However, due to its reliance on heavy syntactic arguments and case distinctions, such an algorithm makes the fundamental structure of the argument rather opaque. We, therefore, consider rules abstractly, within the framework of logical structures familiar from universal logic \`a la Jean-Yves B\'eziau, and aim to clarify the essence of the so-called ``elimination theorems''. To do this, we first give a non-algorithmic proof of the cut-elimination theorem for the propositional fragment of $\mathbf{LK}$. From this proof, we abstract the essential features of the argument and define something called ``normal sequent structures'' relative to a particular rule. We then prove a version of the rule-elimination theorem for these. Abstracting even more, we define ``abstract sequent structures'' and show that for these structures, the corresponding version of the ``rule''-elimination theorem has a converse as well.