2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/227456We obtain two results about the proof complexity of deep inference: 1) deep-inference proof systems are as powerful as Frege ones, even when both are extended with the Tseitin extension rule or with the substitution rule; 2) there are analytic deep-inference proof systems that exhibit an exponential speed-up over analytic Gentzen proof systems that they polynomially simulate.Minor improvements over the published version. Always updated version at <http://cs.bath.ac.uk/ag/p/PrComplDI.pdf>Computational ComplexityLogic in Computer ScienceLogicF.2.2; F.4.1text