The telescoping languages approach achieves high performance from applications encoded as high-level scripts. The core idea is to pre-compile underlying libraries to generate multiple variants optimized for use in different possible contexts including different argument types. This paper proposes a type inference algorithm that enables this kind of specialization. The algorithm infers types over untyped library procedures before actual inputs are known. For our Matlab compiler, the notion of type captures matrix properties such as size, sparsity pattern, and data type. Type inference is necessary both to determine the minimum number of variants needed to handle all possible uses of the library procedure as well as to statically determine, for each variant, which optimized implementations should be dispatched at each call location. A key contribution that arose from this work is a notion of mutually exclusive types. To illustrate these types, we formalize the underlying type system, constraint-collection, and solution. Finally, we prove that the algorithm is polynomial under practical conditions.