Data families vs Injective type families

Question

Now that we have injective type families, is there any remaining use case for using data families over type families?

Looking at past StackOverflow questions about d

Answer 1

type family T a = r | r -> a
data family D a

An injective type family T satisfies the injectivity axiom

if T a ~ T b then a ~ b

But a data family satisfies the much stronger generativity axiom

if D a ~ g b then D ~ g and a ~ b

(If you like: Because the instances of D define new types that are different from any existing types.)

In fact D itself is a legitimate type in the type system, unlike a type family like T, which can only ever appear in a fully saturated application like T a. This means

• D can be the argument to another type constructor, like MaybeT D. (MaybeT T is illegal.)

• You can define instances for D, like instance Functor D. (You can't define instances for a type family Functor T, and it would be unusable anyway because instance selection for, e.g., map :: Functor f => (a -> b) -> f a -> f b relies on the fact that from the type f a you can determine both f and a; for this to work f cannot be allowed to vary over type families, even injective ones.)