Deciding equivalence with sums and the empty type

The logical technique of focusing can be applied to the λ-calculus; in a simple type system with atomic types and negative type formers (functions, products, the unit type), its normal forms coincide with βη-normal forms. Introducing a saturation phase gives a notion of quasi-normal forms in presence of positive types (sum types and the empty type). This rich structure let us prove the decidability of βη-equivalence in presence of the empty type, the fact that it coincides with contextual equivalence, and with set-theoretic equality in all finite models.