Strong normalisation in two Pure Pattern Type Systems
Pure Pattern Type Systems (P 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 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 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 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 TS terms.