Linear Higher-Order Pre-Unification

We develop a pre-unification algorithm in the style of Huet for the linear lambda-calculus lambda^{-> -o & T} which includes intuitionistic functions (->), linear functions (-o), additive pairing (&), and additive unit (T). This procedure conveniently operates on an efficient representation of lambda^{-> -o & T}, the spine calculus S^{-> -o & T} for which we define the concept of weak-head normal form. We prove the soundness and completeness of our algorithm with respect to the proper notion of definitional equality for S^{-> -o & T}, and illustrate the distinctive aspects of linear higher-order unification by means of examples. We also show that, surprisingly, a similar pre-unification algorithm does not exist for certain sublanguages. Applications lie in proof search, logic programming, and logical frameworks based on linear type theories.