We use a deep embedding of the display calculus for relation algebras RA in the logical framework Isabelle/HOL to formalise a machine-checked proof of cut-admissibility for RA. Unlike other "implementations ", we explicitly formalise the structural induction in Isabelle /HOL and believe this to be the first full formalisation of cutadmissibility in the presence of explicit structural rules.

We use a deep embedding of the display calculus for relation algebras #RA in the logical framework Isabelle/HOL to formalise a new, machine-checked, proof of strong normalisation and cut-elimination for #RA which does not use measures on the size of derivations. Our formalisation generalises easily to other display calculi and can serve as a basis for formalised proofs of strong normalisation for the classical and intuitionistic versions of a vast range of substructural logics like the Lambek calculus, linear logic, relevant logic, BCK-logic, and their modal extensions. We believe this is the first full formalisation of a strong normalisation result for a sequent system using a logical framework.

Whilst results from Structural Proof Theory can be couched in many formalisms, it is the sequent calculus which is the most amenable of the formalisms to metamathematical treatment. Constructive syntactic proofs are filled with bureaucratic details; rarely are all cases of a proof completed in the literature. Two intermediate results can be used to drastically reduce the amount of effort needed in proofs of Cut admissibility: Weakening and Invertibility. Indeed, whereas there are proofs of Cut admissibility which do not use Invertibility, Weakening is almost always necessary. Use of these results simply shifts the bureaucracy, however; Weakening and Invertibility, whilst more easy to prove, are still not trivial. We give a framework under which sequent calculi can be codified and analysed, which then allows us to prove various results: for a calculus to admit Weakening and for a rule to be invertible in a calculus. For the latter, even though many calculi are investigated, the general condition is simple and easily verified. The results have been applied to G3ip, G3cp, G3c, G3s, G3-LC and G4ip. Invertibility is important in another respect; that of proof-search. Should all rules in a calculus be invertible, then terminating root-first proof search gives a decision procedure