We discuss forward and backward subsumption for the propositional case of the inverse method. Forward subsumption prevents the introduction of new sequents which could be obtained from present sequents by weakening. Backward subsumption deletes old sequent if they can be obtained from a new one by weakening. We explore ordering techniques for acceptable efficiency of forward subsumption and consider if more can be gained from backward subsumption than simply deleting the subsumed clauses.
We also consider proof term assignment techniques for the forward sequent calculus.