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Strachey Parametricity and Game Semantics

We propose and analyse in this paper a criterion for Strachey parametricity, which is satisfied in particular by system F with its usual equality rules. The criterion is based on a simple restriction of the dinaturality diagram. Game semantics is used as an illustration of this criterion, providing both a parametric model and a non-parametric one. But it is also as a tool to explain how the criterion copes with second-order quantification, and to characterise dinaturality as a local property. We also see some limitations of the criterion, due to the requirement of totality we impose on strategies.