Abstract
Pure Pattern Type Systems (P 2 TS) combine in a unified setting the capabilities of rewriting and λ-calculus. Their type systems, adapted from Barendregt’s λ-cube are especially interesting from a logical point of view.Strong normalization, an essential property for logical soundness, had only been conjectured so far: in this paper, we give a positive answer for the simply-typed system.
The proof is based on a translation of terms and types from (P 2 TS) into the λ-calculus. First, we deal with untyped terms, ensuring that reductions are faithfully mimicked in the λ-calculus. For this, we rely on an original encoding of the pattern matching capability of (P 2 TS) into the λ-calculus.
Then we show how to translate types: the expressive power of System Fω is needed in order to fully reproduce the original typing judgments of (P 2 TS) We prove that the encoding is correct with respect to reductions and typing, and we conclude with the strong normalization of simply-typed (P 2 TS) terms.
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Wack, B. (2004). The Simply-Typed Pure Pattern Type System Ensures Strong Normalization. In: Levy, JJ., Mayr, E.W., Mitchell, J.C. (eds) Exploring New Frontiers of Theoretical Informatics. IFIP International Federation for Information Processing, vol 155. Springer, Boston, MA. https://doi.org/10.1007/1-4020-8141-3_48
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DOI: https://doi.org/10.1007/1-4020-8141-3_48
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