F13.39> : The correctness of PTK is obvious, and the completeness follows from Theorem 1 below and the completeness of PTK. In the following, we show that PTK and PTK are polynomially equivalent, and that the mutual simulations also respect the depth of proofs. This was claimed without proof in [3], where PTK was rst dened. Theorem 1. If P is a proof in PTK, then there is a proof P 0 in PTK of the same end-sequent. The size of P 0 is linear in the size of P , and the formula depths of P and P 0 are the same. Proof. Each application of the rule T n k -right is replaced by a subproof that is built as follows: From the second premise we get by w