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 lambda-abstraction at the level of families. We abandon the use of beta-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.

44 pages

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