Mechanizing the Metatheory of LF

TL;DR

This paper formalizes the metatheoretic properties of LF within Isabelle/HOL, verifying correctness proofs and deriving executable algorithms, thereby strengthening the foundation for LF-based systems.

Contribution

It provides the first formal verification of LF's metatheoretic properties in a theorem prover, resolving gaps and deriving executable type checking algorithms.

Findings

Formal verification of LF properties in Isabelle/HOL

Identification and resolution of proof gaps

Generation of executable type checking code

Abstract

LF is a dependent type theory in which many other formal systems can be conveniently embedded. However, correct use of LF relies on nontrivial metatheoretic developments such as proofs of correctness of decision procedures for LF's judgments. Although detailed informal proofs of these properties have been published, they have not been formally verified in a theorem prover. We have formalized these properties within Isabelle/HOL using the Nominal Datatype Package, closely following a recent article by Harper and Pfenning. In the process, we identified and resolved a gap in one of the proofs and a small number of minor lacunae in others. We also formally derive a version of the type checking algorithm from which Isabelle/HOL can generate executable code. Besides its intrinsic interest, our formalization provides a foundation for studying the adequacy of LF encodings, the correctness of Twelf-style metatheoretic reasoning, and the metatheory of extensions to LF.