  Motivated by recent results of Kapron and Steinberg (LICS 2018) we introduce
new forms of iteration on length in the setting of applied lambda-calculi for
higher-type poly-time computability. In particular, in a type-two setting, we
consider functionals which capture iteration on input length which bound
interaction with the type-one input parameter, by restricting to a constant
either the number of times the function parameter may return a value of
increasing size, or the number of times the function parameter may be applied
to an argument of increasing size. We prove that for any constant bound, the
iterators obtained are equivalent, with respect to lambda-definability over
type-one poly-time functions, to the recursor of Cook and Urquhart which
captures Cobham's notion of limited recursion on notation in this setting.
