A Simplified Account of the Metatheory of Linear LF

We present a variant of the linear logical framework LLF that avoids the restriction that well-typed forms be in pre-canonical form and adds λ-abstraction at the level of families. We abandon the use of β-conversion as definitional equality in favor of a set of typed definitional equality judgments that include rules for parallel conversion and extensionality. We show type-checking is decidable by giving an algorithm to decide definitional equality for well-typed terms and showing the algorithm is sound and complete. The algorithm and the proof of its correctness are simplified by the fact that they apply only to well-typed terms and may therefore ignore the distinction between intuitionistic and linear hypotheses. This material is based on work supported in part by NSF grants CCR-9984812 and CCR-0121633. Any opinions, findings, and conclusions or recommendations in this publication are those of the authors and do not reflect the views of this agency.