Extracted from #13140 (closed).

Today, there is a strange asymmetry between data types, for which the decomposition rule holds (if T A ~R T B then A ~ρ B, where ρ is the role of the type), and newtypes, for which the decomposition rule is unsound.

I believe the root cause of this problem is the fact that we only maintain a single role per type parameter, while in fact what we need is a role *range* (i.e., and lower and upper role bound) to specify what inferences can be made about a type. Here's how it works.

Every type parameter is ascribed a role range, specifying the possible roles by which the type parameter might possibly be used. For example, if I write data T a = MkT a, a is used exactly at representational role, but we could also say that a *might* be used nominally, giving the role range nominal-representational.

The lower bound (nominal is lowest in subroling) specifies at what role the application rule is valid: if I say that the role is at least nominal, then I must provide a ~N b evidence to show that T a ~R T b. The upper bound (phantom is highest) specifies at what role the decomposition rule is valid. If I say that the role is at most phantom, I learn nothing from decomposition; but if I say the role is at most representational, when T A ~R T B, I learn A ~R B. Clearly, the role range nominal-phantom permits the most implementations, but gives the client the least information about equalities.

How do we tell if a role range is compatible with a type? The lower bound (what we call a role today) is computed by propagating roles through, but allowing substitution of roles as per the subroling relationship N <= R <= P. To compute the upper bound, we do exactly the same rules, but with the opposite subroling relation: P <= R <= N.

Some examples:

type role T representational..representational
newtype T a = MkT a
-- T a ~R T b implies a ~R b

type role T nominal..representational -- NB: nominal..nominal illegal!
newtype T a = MkT a
-- T a ~R T b implies a ~R b, BUT
-- a ~R b is insufficient to prove T a ~R T b (you need a ~N b)

type role T nominal..phantom -- NB: nominal..representational illegal!
newtype T a = MkT Int
-- T a ~R T b implies a ~P b (i.e. we don't learn anything)
-- a ~N b implies T a ~R T b

Richard wonders if we could use this to solve the "recursive newtype unwrapping" problem. Unfortunately, because our solver is guess-free, we must also maintain the most precise role for every type constructor. See #13140 (closed)##13358