Linearity and iterator types for Gödel's System

System LI is a linear λ -calculus with numbers and an iterator, which, although imposing linearity restrictions on terms, has all the computational power of Gödel’s System T . System LI owes its power to two features: the use of a closed reduction strategy (which permits the construction of an iterator on an open function, but only iterates the function after it becomes closed), and the use of a liberal typing rule for iterators based on iterative types. In this paper, we study these new types, and show how they relate to intersection types. We also give a sound and complete type reconstruction algorithm for System LI .