Minimal from Classical Proofs

Let A be a formula without implications, and Γ consist of formulas containing disjunction and falsity only negatively and implication only positively. Orevkov (1968) and Nadathur (2000) proved that classical derivability of A from Γ implies intuitionistic derivability, by a transformation of derivations in sequent calculi. We give a new proof of this result (for minimal rather than intuitionistic logic), where the input data are natural deduction proofs in long normal form (given as proof terms via the Curry-Howard correspondence) involving stability axioms for relations; the proof gives a quadratic algorithm to remove the stability axioms. This can be of interest for computational uses of classical proofs.