Extension without cut

In proof theory one distinguishes sequent proofs with cut and cut-free sequent proofs, while for proof complexity one distinguishes Frege-systems and extended Frege-systems. In this paper we show how deep inference can provide a uniform treatment for both classifications, such that we can define cut-free systems with extension, which is neither possible with Frege-systems, nor with the sequent calculus. We show that the propositional pidgeon-hole principle admits polynomial-size proofs in a cut-free system with extension. We also define cut-free systems with substitution and show that the system with extension p-simulates the system with substitution. This yields a new (and simpler) proof that extended Frege-systems p-simulate Frege-systems with substitution. Finally, we propose a new class of tautologies that have short proofs in extended systems, but might not in Frege systems without extension.