Abstract
Pure Type Systems with universes (γPTS) provide a right frame to model programming languages features and are the core of Logical Frameworks, widely used in theorem proof systems. For these systems some authors propose a single rule, which includes both typing under β-conversion and the relation γ between universes. Our proposal adds an independent rule parameterized over a relation between sorts. Non trivial properties of the PTS like the weak strengthening lemma can be obtained in PTS by extending a method proposed by van Benthem Jutting and using weak-closure for γ-reduction. This lemma is important due to two main reasons: (1) it provides a condensing lemma that determines in the underlying logic system a cut rule that simplifies the task in proof assistant systems; (2) the proof of type checking decidability can be eased in some normalizing systems.
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Jiménez, B.C.R. (1998). Condensing Lemmas for Pure Type Systems with Universes. In: Haeberer, A.M. (eds) Algebraic Methodology and Software Technology. AMAST 1999. Lecture Notes in Computer Science, vol 1548. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49253-4_30
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DOI: https://doi.org/10.1007/3-540-49253-4_30
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