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Parametric Domain-theoretic models of Linear Abadi & Plotkin Logic

We present a formalization of a linear version of Abadi and Plotkin’s logic for parametricity for a polymorphic dual intuitionistic / linear type theory with fixed points, and show, following Plotkin’s suggestions, that it can be used to define a wide collection of types, including solutions to recursive domain equations. We further define a notion of parametric LAPL-structure and prove that it provides a sound and complete class of models for the logic. Finally, we present a concrete parametric parametric LAPL-structure based on suitable categories of partial equivalence relations over a universal model of the untyped lambda calculus.