Deciding equivalence with sums and the empty type

The logical technique of focusing can be applied to the $\lambda$-calculus; in a simple type system with atomic types and negative type formers (functions, products, the unit type), its normal forms coincide with $\beta\eta$-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 $\beta\eta$-equivalence in presence of the empty type, the fact that it coincides with contextual equivalence, and a finite model property.