On Spector's bar recursion

We show that Spector’s “restricted” form of bar recursion is sufficient (over system T ) to define Spector’s search functional. This new result is then used to show that Spector’s restricted form of bar recursion is in fact as general as the supposedly more general form of bar recursion. Given that these two forms of bar recursion correspond to the (explicitly controlled) iterated products of selection function and quantifiers, it follows that this iterated product of selection functions is T -equivalent to the corresponding iterated product of quantifiers.