Version 3 (modified by simonpj, 4 months ago)

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# Newtype wrappers

This page proposes newtype wrappers, a new feature for Haskell indended to make newtypes more flexible and useful. It tackles head-on the problem underlying #7542 and #2110.

## The problem

Suppose we have

```newtype Age = MkAge Int
```

Then if n :: Int, we can convert n to an Age thus: MkAge n :: Age. Moreover, this conversion is a type conversion only, and involves no runtime instructions whatsoever. This cost model -- that newtypes are free -- is important to Haskell programmers, and encourages them to use newtypes freely to express type distinctions without introducing runtime overhead.

Alas, the newtype cost model breaks down when we involve other data structures. Suppose we have these declarations

```data T a   = TLeaf a     | TNode (Tree a) (Tree a)
data S m a = SLeaf (m a) | SNode (S m a) (S m a)
```

and we have these variables in scope

```x1 :: [Int]
x2 :: Char -> Int
x3 :: T Int
x4 :: S IO Int
```

Can we convert these into the corresponding forms where the Int is replaced by Age? Alas, not easily, and certainly not without overhead.

• For x1 we can write map MkAge x1 :: [Age]. But this does not follow the newtype cost model: there will be runtime overhead from executing the map at runtime, and sharing will be lost too. Could GHC optimise the map somehow? This is hard; apart from anything else, how would GHC know that map was special? And it it gets worse.
• For x2 we'd have to eta-expand: (\y -> MkAge (x2 y)) :: Char -> Age. But this isn't good either, because eta exapansion isn't semantically valid (if x2 was bottom, seq could distinguish the two). See #7542 for a real life example.
• For x3, we'd have to map over T, thus mapT MkAge x3. But what if mapT didn't exist? We'd have to make it. And not all data types have maps. S is a harder one: you could only map over S-values if m was a functor. There's a lot of discussion abou this on #2110.

## The proposal

Clearly what we want is a way to "lift" newtype constructors (and dually deconstructors) over arbitrary types, so that whenever we have some type blah Int blah we can convert it to the type blah Age blah, and vice versa.

Tantalisingly,  System FC, GHC's internal Core language, already has exactly this!

• A newtype constructor turns into an FC cast:
```  MkAge x    turns into       x |> AgeNTCo
where
AgeNTCo :: Int ~ Age
```

The |> is a cast, and the AgeNTCo is a coercion axiom witnessng the equality of Int and Age.

• Coercions can be lifted, so that
```  [AgeNTCo]       :: [Int] -> [Age]
Char -> AgeNTCo :: (Char -> Int) ~ (Char -> Age)
T AgeNTCo       :: T Int ~ T Age
S IO AgeNTCo    :: S IO Int ~ S IO Age
```

So all we need is concrete syntax to allow you to ask for these lifed coercions in Haskell. I couldn't think of a good way to do this, but now I have a proposal:

```  newtype wrap w1 :: [Int] -> [Age])
newtype wrap w2 :: (Char -> Int) -> (Char -> Age)
newtype wrap w3 :: T Int -> T Age
..etc..
```

and dually

```  newtype unwrap u1 :: [Age] -> [Int])
newtype unwrap u2 :: (Char -> Age) -> (Char -> Int)
..etc...
```

This brings into scope the variables w1, w2, etc, with the declared types. Applications of these wrappers and unwrappers have the same cost model as newtype constructors themselves: they are free.

More precisely

• The type specified in a newtype wrap/unwrap declaration must be of the form type1 -> type2.
• wrap and unwrap are keywords, but only when they occur right after the keyword newtype.
• Wrappers/unwrappers can be polymorphic
```  newtype wrap foo :: [(Int,b)] -> [(Age,b)]
```
• Multiple "layers" of newtype can be wrapped at once (just as in foreign declarations). For example:
```  newtype Fun a = MkFun (a -> a)
newtype Age = MkAge Int
newtype wrap foo :: (Int -> Int) -> Fun Age
```

## Abstraction

Suppose we have

```	module Map( ... ) where
data Map a b = ...blah blah...

module Age( ... ) where
newtype Age = MkAge Int
```

Now suppose we want a newtype wrapper like this

```	import Map
import Age

newtype wrap foo :: Map Int Bool -> Map Age Bool
```

Could we write foo by hand? (This is a good criterion, I think.) Only if we could see the data constructors of both Map and Age.

• If we can't see the data constructor of Age we might miss an invariant that Ages are supposed to have. For example, they might be guaranteed positive.
• If we can't see the data constructors of Map, we might miss an invariant of Maps. For example, maybe Map is represented as a list of pairs, ordered by the keys. Then, if Age orders in the reverse way to Int, it would obviously be bad to substitute.

Invariants like these are difficult to encode in the type system, so we use "exporting the constructors" as a proxy for "I trust the importer to maintain invariants". The "Internals" module name convention is a signal that you must be particularly careful when importing this module; runtime errors may result if you screw up.

One possible conclusion: if we have them at all, newtype wrappers should only work if you can see the constructors of both the newtype, and the type you are lifting over.

But that's not very satisfactory either.

• There are some times (like IO) where it *does* make perfect sense to lift newtypes, but where we really don't want to expose the representation.
• Actually Map is also a good example: while Map Age Bool should not be converted to Map Int Bool, it'd be perfectly fine to convert Map Int Age to Map Int Int.
• The criterion must be recursive. For example if we had
```	 data Map a b = MkMap (InternalMap a b)
```
it's no good being able to see the data constructor MkMap; you need to see the constructors of InternalMap too.

The right thing is probably to use kinds, and all this is tantalisingly close to the system suggested in  Generative type abstraction and type-level computation. Maybe we should be able to declare Map to be indexed (rather than parametric) in its first parameter.

More thought required.

## Type soundness

This proposal rests on very similar foundations to the "newtype deriving" feature, and suffers from precisely the same issue with type soundness; see #1496, #5498, #4846, #7148. For example, consider

```  type family F a
type instance F Int = Int
type instance F Age = Char

data T a = MkT (F a)

newtype wrap bad :: T Int -> T Age

bogus :: Int -> Char
bogus n = case (bad (MkT n)) of
MkT c -> c
```

The problem is, as usual, the type function hiding inside T's definition. The solution is described in [ http://research.microsoft.com/en-us/um/people/simonpj/papers/ext-f/ Generative type abstraction and type-level computation]. It is still not implemented, alas, but adding the newtype wrappers introduces no problems that we do not already have.