Logic of Proofs and Labels with a Complete Set of Operations

We develop a framework in which operations on proofs can be specified and studied. Proofs are treated as structures built from atomic proofs and references by means of computable operations.Our approach extends the ideas of Logic of Proofs (Artemov, 1995, Bull. Symb. Logic, 7, 1–36) in which the proof predicate ‘t is a proof of F ’ is incorporated into the propositional language. We introduce an additional storage predicate ‘x is a label for F ’ (Yavorskaya, 2005, J. Logic Comput., 15, 517–537).In this article which is essentially a continuation of Yavorskaya's work, we study a natural example of a logic with operations on proofs and labels. This logic $\cal {LPS^{*}}$ is decidable and complete with respect to its intended semantics. $\cal {LPS^{*}}$ is capable to internalize its own proofs, and operations of $\cal {LPS^{*}}$ suffice to realize all operations specified in the language with proofs and labels.