I am curious about the possibility of developing Haskell programs spontaneously with proofs about their properties and have the type checker verify the proofs for us, in a way one would do in a dependently typed language.

In the exercise below, I tried to redo part of the merge-sort example in Altenkirch, McBride, and McKinna’s introduction to Epigram: deal the input list into a binary tree, and fold the tree by the function merging two sorted lists into one. The property I am going to show is merely that the length of the input list is preserved.

Given that dependent types and GADTs are such popular topics, I believe the same must have been done before, and there may be better ways to do it. If so, please give me some comments or references. Any comments are welcomed.

> {-# OPTIONS_GHC -fglasgow-exts #-}

To begin with, we define the usual type-level representation of natural numbers and lists indexed by their lengths.

> data Z = Z       deriving Show
> data S a = S a   deriving Show

> data List a n where
>   Nil :: List a Z
>   Cons :: a -> List a n -> List a (S n)

Append

To warm up, let us see the familiar “append” example. Unfortunately, unlike Omega, Haskell does not provide type functions. I am not sure which is the best way to emulate type functions. One possibility is to introduce the following GADT:

> data Plus m n k where    --- m + n = k
>   PlusZ :: Plus Z n n
>   PlusS :: Plus m n k -> Plus (S m) n (S k)

such that Plus m n k represents a proof that m + n = k.

Not having type functions, one of the possible ways to do append is to have the function, given two lists of lengths m and n, return a list of length k and a proof that m + n = k. Thus, the type of append would be:

   append :: List a m -> List a n
               -> exists k. (List a k, Plus m n k)

In Haskell, the existential quantifier is mimicked by forall. We define:

> data DepSum a p = forall i . DepSum (a i) (p i)

The term “dependent sum” is borrowed from the Omega tutorial of Sheard, Hook, and Linger. Derek Elkins and Dan Licata explained to me why they are called dependent sums, rather than products.

The function append can thus be defined as:

> append :: List a m -> List a n -> DepSum (List a) (Plus m n)
> append Nil ys = DepSum ys PlusZ
> append (Cons x xs) ys =
>    case (append xs ys) of
>      DepSum zs p -> DepSum (Cons x zs) (PlusS p)

Another possibility is to provide append a proof that m + n = k. The type and definition of append would be:

< append :: Plus m n k -> List a m -> List a n -> List a k
< append PlusZ Nil ys = ys
< append (PlusS pf) (Cons x xs) ys = Cons x (append pf xs ys)

I thought the second append would be more difficult to use: to append two lists, I have to provide a proof about their lengths! It turns out that this append actually composes easier with other parts of the program. We will come to this later.

Some Lemmas

Here are some lemmas represented as functions on terms. The function incAssocL, for example, converts a proof of m + (1+n) = k to a proof of (1+m) + n = k.

> incAssocL :: Plus m (S n) k -> Plus (S m) n k
> incAssocL PlusZ = PlusS PlusZ
> incAssocL (PlusS p) = PlusS (incAssocL p)

> incAssocR :: Plus (S m) n k -> Plus m (S n) k
> incAssocR (PlusS p) = plusMono p

> plusMono :: Plus m n k -> Plus m (S n) (S k)
> plusMono PlusZ = PlusZ
> plusMono (PlusS p) = PlusS (plusMono p)

For example, the following function revcat performs list reversal by an accumulating parameter. The invariant we maintain is m + n = k. To prove that the invariant holds, we have to use incAssocL.

> revcat :: List a m -> List a n -> DepSum (List a) (Plus m n)
> revcat Nil ys = DepSum ys PlusZ
> revcat (Cons x xs) ys =
>     case revcat xs (Cons x ys) of
>        DepSum zs p -> DepSum zs (incAssocL p)

Merge

Apart from the proof manipulations, the function merge is not very different from what one would expect:

> merge :: Ord a => List a m -> List a n -> DepSum (List a) (Plus m n)
> merge Nil ys = DepSum ys PlusZ
> merge (Cons x xs) Nil = append (Cons x xs) Nil
> merge (Cons x xs) (Cons y ys)
>   | x <= y = case merge xs (Cons y ys) of
>                  DepSum zs p -> DepSum (Cons x zs) (PlusS p)
>   | otherwise = case merge (Cons x xs) ys of
>                   DepSum zs p -> DepSum (Cons y zs) (plusMono p)

The lemma plusMono is used to convert a proof of m + n = k to a proof of m + (1+n) = 1+k.

Sized Trees

We also index binary trees by their sizes:

> data Tree a n where
>   Nul :: Tree a Z
>   Tip :: a -> Tree a (S Z)
>   Bin :: Tree a n1 -> Tree a n ->
>             (Plus p n n1, Plus n1 n k) -> Tree a k

The two trees given to the constructor Bin have sizes n1 and n respectively. The resulting tree, of size k, comes with a proof that n1 + n = k. Furthermore, we want to maintain an invariant that n1 either equals n, or is bigger than n by one. This is represented by the proof Plus p n n1. In the definition of insertT later, p is either PlusZ or PlusS PlusZ.

Lists to Trees

The function insertT inserts an element into a tree:

> insertT :: a -> Tree a n -> Tree a (S n)
> insertT x Nul = Tip x
> insertT x (Tip y) = Bin (Tip x) (Tip y) (PlusZ, PlusS PlusZ)
> insertT x (Bin t u (PlusZ, p)) =
>      Bin (insertT x t) u (PlusS PlusZ, PlusS p)
> insertT x (Bin t u (PlusS PlusZ, p)) =
>      Bin t (insertT x u) (PlusZ, PlusS (incAssocR p))

Note that whenever we construct a tree using Bin, the first proof, corresponding to the difference in size of the two subtrees, is either PlusZ or PlusS PlusZ.

The counterpart of foldr on indexed list is defined by:

> foldrd :: (forall k . (a -> b k -> b (S k))) -> b Z
>               -> List a n -> b n
> foldrd f e Nil = e
> foldrd f e (Cons x xs) = f x (foldrd f e xs)

The result is also an indexed type (b n).

The function deal :: List a n -> Tree a n, building a tree out of a list, can be defined as a fold:

> deal :: List a n -> Tree a n
> deal = foldrd insertT Nul

Trees to Lists, and Merge Sort

The next step is to fold through the tree by the function merge. The first two clauses are simple:

> mergeT :: Ord a => Tree a n -> List a n
> mergeT Nul = Nil
> mergeT (Tip x) = Cons x Nil

For the third clause, one would wish that we could write something as simple as:

   mergeT (Bin t u (_,p1)) =
      case merge (mergeT t) (mergeT u) of
        DepSum xs p -> xs

However, this does not type check. Assume that t has size n1, and u has size n. The DepSum returned by merge consists of a list of size i, and a proof p of type Plus m n i, for some i. The proof p1, on the other hand, is of type P m n k for some k. Haskell does not know that Plus m n is actually a function and cannot conclude that i=k.

To explicitly state the equality, we assume that there is a function plusFn which, given a proof of m + n = i and a proof of m + n = k, yields a function converting an i in any context to a k. That is:

   plusFn :: Plus m n i -> Plus m n k
               -> (forall f . f i -> f k)

The last clause of mergeT can be written as:

mergeT (Bin t u (_,p1)) =
   case merge (mergeT t) (mergeT u) of
     DepSum xs p -> plusFn p p1 xs

How do I define plusFn? On Haskell-Cafe, Jim Apple suggested this implementation (by the way, he maintains a very interesting blog on typed programming):

  plusFn :: Plus m n h -> Plus m n k -> f h -> f k
  plusFn PlusZ PlusZ x = x
  plusFn (PlusS p1) (PlusS p2) x =
      case plusFn p1 p2 Equal of
        Equal -> x

  data Equal a b where
      Equal :: Equal a a

Another implementation, which looks closer to the previous work on type equality [3, 4, 5], was suggested by apfelmus:

> newtype Equal a b = Proof { coerce :: forall f . f a -> f b }

> newtype Succ f a  = InSucc { outSucc :: f (S a) }

> equalZ :: Equal Z Z
> equalZ = Proof id

> equalS :: Equal m n -> Equal (S m) (S n)
> equalS (Proof eq) = Proof (outSucc . eq . InSucc)

> plusFnEq :: Plus m n i -> Plus m n k -> Equal i k
> plusFnEq PlusZ     PlusZ     = Proof id
> plusFnEq (PlusS x) (PlusS y) = equalS (plusFn x y)

> plusFn :: Plus m n i -> Plus m n k -> f i -> f k
> plusFn p1 p2 = coerce (plusFnEq p1 p2)

Now that we have both deal and mergeT, merge sort is simply their composition:

> msort :: Ord a => List a n -> List a n
> msort = mergeT . deal

The function mergeT can be defined using a fold on trees as well. Such a fold might probably look like this:

> foldTd :: (forall m n k . Plus m n k -> b m -> b n -> b k)
>           -> (a -> b (S Z)) -> b Z
>           -> Tree a n -> b n
> foldTd f g e Nul = e
> foldTd f g e (Tip x) = g x
> foldTd f g e (Bin t u (_,p)) =
>    f p (foldTd f g e t) (foldTd f g e u)

   mergeT :: Ord a => Tree a n -> List a n
   mergeT = foldTd merge' (\x -> Cons x Nil) Nil
    where merge' p1 xs ys =
            case merge xs ys of
              DepSum xs p -> plusFn p p1 xs

I am not sure whether this is a "reasonable" type for foldTd.

Passing in the Proof as an Argument

Previously I thought the second definition of append would be more difficult to use, because we will have to construct a proof about the lengths before calling append. In the context above, however, it may actually be more appropriate to use this style of definitions.

An alternative definition of merge taking a proof as an argument can be defined by:

< merge :: Ord a => Plus m n k -> List a m ->
<                       List a n -> List a k
< merge PlusZ Nil ys = ys
< merge pf (Cons x xs) Nil = append pf (Cons x xs) Nil
< merge (PlusS p) (Cons x xs) (Cons y ys)
<   | x <= y    = Cons x (merge p xs (Cons y ys))
<   | otherwise = Cons y (merge (incAssocL p) (Cons x xs) ys)

A definition of mergeT using this definition of merge follows immediately because we can simply use the proof coming with the tree:

< mergeT :: Ord a => Tree a n -> List a n
< mergeT Nul = Nil
< mergeT (Tip x) = Cons x Nil
< mergeT (Bin t u (_,p1)) =
<   merge p1 (mergeT t) (mergeT u)

I don't know which approach can be called more "natural". However, both Jim Apple and apfelmus pointed out that type families, a planned new feature of Haskell, may serve as a kind of type functions. With type families, the append example would look like ( code from apfelmus):

  type family   Plus :: * -> * -> *
  type instance Plus Z n     = n
  type instance Plus (S m) n = S (Plus m n)

  append :: (Plus m n ~ k) => List a m -> List a n -> List a k
  append Nil         ys = ys
  append (Cons x xs) ys = Cons x (append xs ys)

apfelmus commented that "viewed with the dictionary translation for type classes in mind, this is probably exactly the alternative type of append you propose: append :: Plus m n k -> List a m -> List a n -> List a k".

References

• [1] Thorsten Altenkirch, Conor McBride, and James McKinna. Why Dependent Types Matter.
• [2] Tim Sheard, James Hook, and Nathan Linger. GADTs + Extensible Kinds = Dependent Programming.
• [3] James Cheney, Ralf Hinze. A lightweight implementation of generics and dynamics.
• [4] Stephanie Weirich, Type-safe cast: (functional pearl), ICFP 2000.
• [5]Arthur I. Baars , S. Doaitse Swierstra, Typing dynamictyping, ACM SIGPLAN Notices, v.37 n.9, p.157-166, September 2002

Appendix. Auxiliary Functions

> instance Show a => Show (List a n) where
>  showsPrec _ Nil = ("[]"++)
>  showsPrec _ (Cons x xs) = shows x . (':':) . shows xs

> instance Show (Plus m n k) where
>   showsPrec _ PlusZ = ("pZ"++)
>   showsPrec p (PlusS pf) = showParen (p>=10) (("pS " ++) .
>                              showsPrec 10 pf)

> instance Show a => Show (Tree a n) where
>   showsPrec _ Nul = ("Nul"++)
>   showsPrec p (Tip x) = showParen (p >= 10) (("Tip " ++) . shows x)
>   showsPrec p (Bin t u pf) =
>     showParen (p>=10)
>      (("Bin "++) . showsPrec 10 t . (' ':) . showsPrec 10 u .
>       (' ':) . showsPrec 10 pf)

Inductive types

Inductive types are least fixed-points. An indcutive type μF is simulated by

 forall x . (F x -> x) -> x

In Haskell, we need a constructor to wrap a “forall” quantifier.

> newtype LFix f = LF (forall x. (f x -> x) -> x)
> instance Show (LFix f) where
>   show _ = ""

> foldL :: Functor f => (f x -> x) -> LFix f -> x
> foldL f (LF x) = x f

> inL :: Functor f => f (LFix f) -> LFix f
> inL fx = LF (\\f -> f . fmap (foldL f) $ fx)

> outL :: Functor f => LFix f -> f (LFix f)
> outL = foldL (fmap inL)

Example: lists.

> data F a x = L | R a x deriving Show
> instance Functor (F a) where
>   fmap f L = L
>   fmap f (R a x) = R a (f x)

> type ListL a = LFix (F a)

A ListL can only be constructed out of nil and cons.

    Main> consL 1 (consL 2 (consL 3 nilL))
   

> nilL = inL L
> consL a x = inL (R a x)

We can convert a ListL to a ‘real’ Haskell list.

   Main> lList $ consL 1 (consL 2 (consL 3 nilL))
   [1,2,3]

> lList :: LFix (F a) -> [a]
> lList = foldL ll
>   where ll L = []
>         ll (R a x) = a : x

However, a ListL has to be explicitly built, so there is no infinite ListL.

Coinductive types

A coinductive type νF is simulatd by

 exists x . (x -> F x, x)

We represent it in Haskell using the property exists x . F x == forall y . (forall x . F x -> y) -> y:

> data GFix f = forall x . GF (x -> f x, x)
> instance Show (GFix f) where
>   show _ = ""

> unfoldG :: Functor f => (x -> f x) -> x -> GFix f
> unfoldG g x = GF (g,x)

> outG :: Functor f => GFix f -> f (GFix f)
> outG (GF (g,x)) = fmap (unfoldG g) . g $ x

> inG :: Functor f => f (GFix f) -> GFix f
> inG = unfoldG (fmap outG)

Example:

> type ListG a = GFix (F a)

ListG can be constructed out of nil and cons, as well as an unfold.

> nilG = inG L
> consG a x = inG (R a x)

> fromG :: Int -> ListG Int
> fromG m = unfoldG ft m
>    where ft m = R m (m+1)

However, one can perform only a finite number of outG.

  Main> fmap outG . outG . fromG $ 1
  R 1 (R 2 )

Isomorphism

The function force witnesses the isomorphism between LFix and GFix.

> force :: Functor f => GFix f -> LFix f
> force = inL . fmap force . outG     -- recursion!

If LFix and GFix are isomorphic, we are allowed to build hylomorphisms:

> fromTo :: Int -> Int -> LFix (F Int)
> fromTo m n = takeLessThan n . force . fromG $ m

> takeLessThan :: Int -> LFix (F Int) -> LFix (F Int)
> takeLessThan n = foldL (tlt n)
>   where tlt n L = nilL
>         tlt n (R m x) | n <= m = nilL
>                       | otherwise = consL m x

  Main> lList (fromTo 1 10)
  [1,2,3,4,5,6,7,8,9]

However, force has to be defined using general recursion. The price is that it is now possible to write programs that do not terminate.

I was trying to simulate church numerals and primitive recursion in second rank polymorphism of Haskell, and I thought I could do it without introducing a constructor by newtype. Church’s encoding of natural numbers can be expressed by the type:

> type N = forall a . (a -> a) -> a -> a

Basics

We first define zero, succ, and some constants:

> zero :: N
> zero = \f -> \a ->  a

> succ :: N -> N
> succ n = \f -> \a -> f (n f a)

> toInt :: N -> Int
> toInt n = n (1+) 0

> one, two, three :: N
> one = succ zero
> two = succ one
> three = succ two

Addition and multiplication can be defined as usual:

> add, mul :: N -> N -> N
> add m n = \f -> \a -> m f (n f a)
> mul m n = \f -> \a -> m (n f) a

Predecessor

Even pred typechecks — at least this particular version of pred:

> pred :: N -> N
> pred n f x = n (\g h -> h (g f)) (const x) id

I am more familiar with another way to implement pred, which uses pairs. This pred, on the other hand, can be understood by observing that the “fold” part returns a function that abstracts away the outermost f:

  pred n f x = dec n id
    where dec 0 = \h -> x
          dec (1+n) = \h -> h (f^n x)

Exponential

I know of two ways to define exp. However, only this one typechecks:

> exp :: N -> N -> N
> exp m n = n m

Another definition, making use of mul, does not typecheck!

  exp :: N -> N -> N
  exp m n = n (mul m) one

GHC returns a rather cryptic error message “Inferred type is less polymorphic than expected. Quantified type variable `a’ escapes.” The real reason is that polymorphic types in Haskell can only be instantiated with monomorphic types. If we spell out the explicit type applications, we get:

   exp  (m::N) (n::N) = n [N] ((mul m)::N->N) (one::N)

which is not allowed. On the other hand, the previous definition of exp does not instantiate type variables to polymorphic types:

   exp (m::N) (n::N) = /\b -> \(f::(b->b)) -> \(x::b) -> n[b->b] (m[b]) f x

For the same reason, primitive recursion does not type check:

 primrec :: (N -> a -> a) -> a -> N -> a
 primrec f c n =
   snd (n (\z -> pair (succ (fst z))
                      (f (fst z) (snd z)))
          (pair zero c))

nor does the pair-based definition of pred.

To overcome the problem we have to wrap N in newtype. The solution was presented by Oleg Kiselyov.

What happens if we have only monomorphic type when we define church numerals? Let us represent church numerals by the type (a -> a) -> a -> a for some monomorphic a:

> import Prelude hiding (succ, pred)

> type N a = (a -> a) -> a -> a

Zero and successor can still be defined. Their values do not matter.

> zero :: N a
> zero = error ""

> succ :: N a -> N a
> succ = error ""

Also assume we have pairs predefined:

> pair a b = (a,b)

We can even define primitive recursion, only that it does not have the desired type:

  primrec :: (N -> b -> b) -> b -> N -> b

Instead we get the type below:

> primrec :: (N a -> b -> b) -> b -> N (N a, b) -> b
> primrec f c n =
>   snd (n (\z -> pair (succ (fst z))
>                      (f (fst z) (snd z)))
>          (pair zero c))

Therefore, predecessor does not get the type pred :: N a -> N a
we want. Instead we have:

> pred :: N (N a, N a) -> N a
> pred = primrec (\n m -> n) zero

Ref.

Oleg Kiselyov, Predecessor and lists are not representable in simply typed lambda-calculus.