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.

66 pages

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