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Comparing cubes of typed and type assignment systems

Abstract : We study the cube of type assignment systems, as introduced in [10], and confront it with Barendregt's typed λ-cube [3]. The first is obtained from the latter through applying a natural type erasing function E to derivation rules, that erases type information from terms. In particular , we address the question whether a judgement, derivable in a type assignment system, is always an erasure of a derivable judgement in a corresponding typed system; we show that this property holds only for the systems without polymorphism. The type assignment systems we consider satisfy the properties 'subject reduction' and 'strong normalization'. Moreover, we define a new type assignment cube that is isomorphic to the typed one.
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Steffen van Bakel, Luigi Liquori, Simona Ronchi Della Rocca, Pawel Urzyczyn. Comparing cubes of typed and type assignment systems. Annals of Pure and Applied Logic, Elsevier Masson, 1997, 86 Issue 3, pp.267-303. ⟨10.1016/S0168-0072(96)00036-X⟩. ⟨hal-01154638⟩

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