Relational Parametricity for a Polymorphic Linear Lambda Calculus

This paper presents a novel syntactic logical relation for a polymorphic linear λ-calculus that treats all types as linear and introduces the constructor ! to account for intuitionistic terms, and System F◦—an extension of System F that uses kinds to distinguish linear from intuitionistic types. We define a logical relation for open values under both open linear and intuitionistic contexts, then extend it for open terms with evaluation and open relation substitutions. Relations that instantiate type quantifiers are for open terms and types. We demonstrate the applicability of this logical relation through its soundness with respect to contextual equivalence, along with free theorems for linearity that are difficult to achieve by closed logical relations. When interpreting types on only closed terms, the model defaults to a closed logical relation that is both sound and complete with respect to contextual equivalence and is sufficient to reason about isomorphisms of type encodings. All of our results have been mechanically verified in Coq.