The Simply-typed Pure Pattern Type System Ensures Strong Normalization

Pure Pattern Type Systems (P2 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 P2 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 P2 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 P2 TS. We prove that the encoding is correct with respect to reductions and typing, and we conclude with the strong normalization of simply-typed P2 TS terms.

• Type: Conference
• Year: 2004
• Where: IFIP
• Authors: Benjamin Wack
• Keywords: Communications, IFIPTCS 2004, P2 TS, Strong Normalization, Type Systems