Type-level programming

Here is a list of exercises related to nested data types, GADTs, and generic programming.


Here is a datatype of contracts:

data Contract :: * -> * where
  Pred :: (a -> Bool) -> Contract a
  Fun  :: Contract a -> Contract b -> Contract (a -> b)

A contract can be a predicate for a value of arbitrary type. For functions, we offer contracts that contain a precondition on the arguments, and a postcondition on the results.

Contracts can be attached to values by means of assert. The idea is that assert will cause run-time failure if a contract is violated, and otherwise return the original result:

assert :: Contract a -> a -> a
assert (Pred p)       x = if p x then x else error "contract violation"
assert (Fun pre post) f = assert post . f . assert pre

For function contracts, we first check the precondition on the value, then apply the original function, and finally check the postcondition on the result. Note that the case for Fun makes use of the fact that the Fun constructor targets only function contracts. Because of this knowledge, GHC allows us to apply f as a function.

For example, the following contract states that a number is positive:

pos :: (Num a, Ord a) => Contract a
pos = Pred (> 0)

We have

assert pos 2 == 2
assert pos 0 ==  (contract violation error)

Exercise 1: Define a contract

true :: Contract a

such that for all values x, the equation

assert true x == x

holds. Prove this equation using equational reasoning.

Often, we want the postcondition of a function to be able to refer to the actual argument that has been passed to the function. Therefore, let us change the type of Fun:

DFun :: Contract a -> (a -> Contract b) -> Contract (a -> b)

The postcondition now depends on the function argument.

Exercise 2: Adapt the function assert to the new type of DFun.

Exercise 3: Define a combinator

(==>) :: Contract a -> Contract b -> Contract (a -> b)

that reexpresses the behaviour of the old Fun constructor in terms of the new and more general one.

Exercise 4: Define a contract suitable for the list index function (!!), i.e., a contract of type Contract ([a] -> Int -> a) that checks if the integer is a valid index for the given list.

Exercise 5: Define a contract

preserves :: Eq b => (a -> b) -> Contract (a -> a)

where assert (preserves p) f x fails if and only if the value of p x is different from the value of p (f x). Examples:

assert (preserves length) reverse  "Hello"       == "olleH"
assert (preserves length) (take 5) "Hello"       == "Hello"
assert (preserves length) (take 5) "Hello world" == 

Exercise 6: Consider

preservesPos  = preserves (>0)
preservesPos' = pos ==> pos

Is there a difference between assert preservesPos and assert preservesPos'? If yes, give an example where they show different behaviour. If not, try to prove their equality using equational reasoning.

We can add another contract constructor:

List :: Contract a -> Contract [a]

The corresponding case of assert is as follows:

assert (List c) xs = map (assert c) xs

Exercise 7: Consider

allPos  = List pos
allPos' = Pred (all (> 0))

Describe the differences between assert allPos and assert allPos', and more generally between using List versus using Pred to describe a predicate on lists. (Hint: Think carefully and consider different situations before giving your answer. What about using the allPos and allPos' contracts as parts of other contracts? What about lists of functions? What about infinite lists? What about strict and non-strict functions working on lists?)

Nested data types

Here is a nested data type for square matrices:

type Square      = Square' Nil  -- note that it is eta-reduced
data Square' t a = Zero (t (t a)) | Succ (Square' (Cons t) a)

data Nil    a = Nil
data Cons t a = Cons a (t a)

Exercise 1. Give Haskell code that represents the following two square matrices as elements of the Square data type:

\[\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \quad \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}\]

Let’s investigate how we can derive an equality function on square matrices. We do so very systematically by deriving an equality function for each of the four types. We follow a simple, yet powerful principle: type abstraction corresponds to term abstraction, and type application corresponds to term application.

What does this mean? If a type f is parameterized over an argument a, then in general, we have to know how equality is defined on a in order to define equality on f a. Therefore we define

eqNil :: (a -> a -> Bool) -> (Nil a -> Nil a -> Bool)
eqNil eqA Nil Nil = True

In this case, the a is not used in the definition of Nil , so it is not surprising that we do not use eqA in the definition of eqNil. But what about Cons? The data type Cons has two arguments t and a, so we expect two arguments to be passed to eqCons, something like

eqCons eqT eqA (Cons x xs) (Cons y ys) = eqA x y && ...

But what should the type of eqT be? The t is of kind * -> *, so it can’t be t -> t -> Bool. We can argue that we should use t a -> t a -> Bool, because we use t applied to a in the definition of Cons. However, a better solution is to recognise that, being a type constructor of kind * -> *, an equality function on t should take an equality function on its argument as a parameter. And, moreover, it does not matter what this parameter is! A function like eqNil is polymorphic in type a, so let us require that eqT is polymorphic in the argument type as well:

eqCons :: (forall b . (b -> b -> Bool) -> (t b -> t b -> Bool))
       -> (a -> a -> Bool)
       -> (Cons t a -> Cons t a -> Bool)
eqCons eqT eqA (Cons x xs) (Cons y ys) = eqA x y && eqT eqA xs ys

Now you can see how we apply eqT to eqA when we want equality at type t a – the type application corresponds to term application.

Exercise 2. A type with a forall on the inside requires the extension RankNTypes to be enabled. Try to understand what the difference is between a function of the type of eqCons and a function with the same type but the forall omitted. Can you omit the forall in the case of eqCons and does the function still work?

Now, on to Square'. The type of eqSquare' follows exactly the same idea as the type of eqCons:

eqSquare' :: (forall b . (b -> b -> Bool) -> (t b -> t b -> Bool))
          -> (a -> a -> Bool)
          -> (Square' t a -> Square' t a -> Bool)

We now for the first time have more than one constructor, so we actually have to give multiple cases. Let us first consider comparing two applications of Zero:

eqSquare' eqT eqA (Zero xs) (Zero ys) = eqT (eqT eqA) xs ys

Note how again the structure of the definition follows the structure of the type. We have a value of type t (t a). We compare it using eqT, passing it an equality function for values of type t a. How? By using eqT eqA. The remaining cases are as follows:

eqSquare' eqT eqA (Succ xs) (Succ ys) = eqSquare' (eqCons eqT) eqA xs ys
eqSquare' eqT eqA _         _         = False

The idea is the same – let the structure of the recursive calls follow the structure of the type.

Exercise 3. Again, try removing the forall from the type of eqSquare'. Does the function still type check? Try to explain!

Now we’re done:

eqSquare :: (a -> a -> Bool) -> Square a -> Square a -> Bool
eqSquare = eqSquare' eqNil

Test the function. We can now also give an Eq instance for Square – this requires the minor language extension TypeSynonymInstances, because Haskell 98 does not allow type synonyms like Square to be used in instance declarations:

instance Eq a => Eq (Square a) where
  (==) = eqSquare (==)

Exercise 4. Systematically follow the scheme just presented in order to define a Functor instance for square matrices. I.e., derive a function mapSquare such that you can define

instance Functor Square where
  fmap = mapSquare

This instance requires Square to be defined in eta-reduced form in the beginning, because Haskell does not allow partially applied type synonyms. If we had defined Square differently

type Square a = Square' Nil a

we cannot make Square an instance of the class Functor.

Exercise 5. Why is this restriction in place? Try to find problems arising from partially applied type synonyms, and describe them (as concisely as possible) with a few examples.