Which simple types have a unique inhabitant?

We study the question of whether a given type has a unique in-habitant modulo program equivalence. In the setting of simply-typed lambda-calculus with sums, equipped with the strong βη-equivalence, we show that uniqueness is decidable. We present a saturating focused logic that introduces irreducible cuts on positive types “as soon as possible”. Backward search in this logic gives an effective algorithm that returns either zero, one or two distinct in-habitants for any given type. Preliminary application studies show that such a feature can be useful in strongly-typed programs, in-ferring the code of highly-polymorphic library functions, or “glue code ” inside more complex terms. Categories and Subject Descriptors F.3.3 [Studies of Program Constructs]: Type structur