The Logic of Reusable Propositional Output with the Fulfilment Constraint

This paper shows the equivalence of three ways of expressing a certain strong consistency constraint – called the fulfilment constraint – on proofs of the logic of reusable propositional output: as a global requirement on proofs, as a local requirement on labels of formulas, and by phasing of proof rules. More specifically, we first show that the fulfilment constraint may be expressed either as a requirement on the historical structure of the proof tree or as a requirement on the contents of labels attached to its nodes. Second, we show that labelled proofs may be rewritten into a tightly phased form in which rules are applied in a fixed order. Third, we show that when a proof is in such a phased form, the consistency check on labels becomes redundant.