On the discriminating power of tests in resource lambda-calculus

Since its discovery, differential linear logic (DLL) inspired numerous domains. In denotational semantics, categorical models of DLL are now commune, and the simplest one is Rel, the category of sets and relations. In proof theory this naturally gave birth to differential proof nets that are full and complete for DLL. In turn, these tools can naturally be translated to their intuitionistic counterpart. By taking the co-Kleisly category associated to the ! comonad, Rel becomes MRel, a model of the λ-calculus that contains a notion of differentiation. Proof nets can be used naturally to extend the λ-calculus into the lambda calculus with resources, a calculus that contains notions of linearity and differentiations. Of course MRel is a model of the λ-calculus with resources, and it has been proved adequate, but is it fully abstract? That was a strong conjecture of Bucciarelli, Carraro, Ehrhard and Manzonetto in [4]. However, in this paper we exhibit a counterexample. Moreover, to give more intuition on the essence of the counterexample and to look for more generality, we will use an extension of the resource λ-calculus also introduced by Bucciarelli et al in [4] for which M∞ is fully abstract, the tests.