Strong Normalization in two Pure Pattern Type Systems
Résumé
Pure Pattern Type Systems (P 2 T S ) combine in a unified setting the frameworks and 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 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 normalization of simply-typed P 2 T S terms. The strong normalization with dependent types is in turn obtained by an intermediate translation into simply-typed terms.
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