  To be usable in practice, interactive theorem provers need to provide
convenient and efficient means of writing expressions, definitions, and proofs.
This involves inferring information that is often left implicit in an ordinary
mathematical text, and resolving ambiguities in mathematical expressions. We
refer to the process of passing from a quasi-formal and partially-specified
expression to a completely precise formal one as elaboration. We describe an
elaboration algorithm for dependent type theory that has been implemented in
the Lean theorem prover. Lean's elaborator supports higher-order unification,
type class inference, ad hoc overloading, insertion of coercions, the use of
tactics, and the computational reduction of terms. The interactions between
these components are subtle and complex, and the elaboration algorithm has been
carefully designed to balance efficiency and usability. We describe the central
design goals, and the means by which they are achieved.
