It is known that LK, a system of propositional sequent calculus, without a cut rule (written in tree form) does not p-simulate LK with a full cut rule (written in tree form) [1, 6]. It is also known that LK with a full cut rule in tree-form p-simulates LK in DAG form. In this paper, we show that LK (in tree form) with a cut rule, where the complexity of cut formulas is bounded by (k + 1) has an exponential speed-up over the one bounded by k. We also show that LK (in tree form) with a bounded complexity cut rule does not polynomially simulate cut-free LK in DAG form.
