A Constructive Proof of Cut Elimination for a System of Full Second   Order Logic
  In this paper we present a constructive proof of cut elimination for a system of full second order logic with the structural rules absorbed and using sets instead of sequences. The standard problem of the cutrank growth is avoided by using a new parameter for the induction, the cutweight. This technique can also be applied to first order logic. 
Sandro Skansi
math.LO cs.LO