No Speedup for Geometric Theories

Geometric theories based on classical logic are conservative over their intuitionistic counterparts for geometric implications. The latter result (sometimes referred to as Barr's theorem) is squarely a consequence of Gentzen's Hauptsatz. Prima facie though, cut elimination can result in superexponentially longer proofs. In this chapter, it is shown that the transformation of a classical proof of a geometric implication in a geometric theory into an intuitionistic proof can be achieved in a feasible way.