Strong normalisation in two Pure Pattern Type Systems

Abstract

Pure Pattern Type Systems (P 2 T S) combine the frameworks and capabilities of rewriting and λ-calculus within a unified setting. Their type systems, which are adapted from Barendregt's λ-cube, are especially interesting from a logical point of view. Until now, strong normalisation, which is an essential property for logical soundness, has only been conjectured: in this paper, we give a positive answer for the simply-typed system and the dependently-typed system. The proof is based on a translation of terms and types from P 2 T S 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 T S into the System Fω. 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 T S. We prove that the encoding is correct with respect to reductions and typing, and we conclude with the strong normalisation of simply-typed P 2 T S terms. The strong normalisation with dependent types is in turn obtained by an intermediate translation into simply-typed terms.