Haskell Wiki (opens new window) has an example of a non-parametric parameter of a type function:

type family Inspect x
type instance Inspect Age = Int    
type instance Inspect Int = Bool

Here x is non-parametric because to determine the outcome of applying Inspect to a type argument, the type function must inspect x.

In this case, the role of x is nominal. We can declare the role explicitly with the RoleAnnotations extension:

type role Inspect nominal

An example of a parametric parameter of a type function:

data List a = Nil | Cons a (List a)

type family DoNotInspect x
type instance DoNotInspect x = List x

Here x is parametric because to determine the outcome of applying DoNotInspect to a type argument, the type function do not need to inspect x.

In this case, the role of x is representational. We can declare the role explicitly with the RoleAnnotations extension:

type role DoNotInspect representational

A phantom type parameter (opens new window) has a phantom role. Phantom roles cannot be declared explicitly.

Remarks

See also SafeNewtypeDeriving (opens new window).