On the proof theory of type two functionals based on primitive recursive operations. Steiner, D. & Strahm, T. Mathematical Logic Quarterly, 52(3):237–252, 2006.
On the proof theory of type two functionals based on primitive recursive operations [pdf]Paper  bibtex   
@Article{ss06,
  Author         = {David Steiner and Thomas Strahm},
  Title          = {On the proof theory of type two functionals based on
                   primitive recursive operations},
  Journal        = {Mathematical Logic Quarterly},
  Volume         = {52},
  Number         = {3},
  Pages          = {237{--}252},
  url            = {2006/ss06.pdf},
  year           = 2006
}
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