The Subject Reduction Property in the λΠ-calculus modulo

In type theory, the subject reduction (or type preservation) property states that the type of a λ-term is preserved under reduction. This article studies this property in the context of the λΠ-calculus modulo, a variant of the λ-calculus with dependent types (λΠ-calculus) where β-reduction is extended with user-defined object-level and type-level rewrite rules. We show that it is equivalent to the following property called Π-injectivity or product-compatibility: if product types are convertible then their components are pairwise convertible. We also show that subject reduction implies uniqueness of type and that both properties are undecidable. Finally we give a new decidable criterion ensuring subject reduction.