Apologies for the repost. The post is a literate Haskell file and as my first attempt was formatted for 80 character lines, it no longer worked after the lines got mangled. Hopefully this time it'll come out right. --------------------------------------------------------------------- I've recently been playing with Sebastian Fischer's explicit sharing monad library (found at http://sebfisch.github.com/explicit-sharing) and after looking at the code noticed the implementation memoizes values into an IntMap of untyped values that are coerced into the correct type on retrieval, which is a sensible and practical way to implement this. Just for fun I thought I'd have a go at implementing a (very impractical) version without type casting but, disappointingly, I didn't quite get there. I ran out of steam by the time I got around to playing with parameterized monads by which stage the type signatures started looking a bit scary. That and I needed to get back to my projects. This has been sitting on my hard drive for a couple of weeks now and I've finally decided to clean it up and post it, maybe someone else will find it amusing. The post is about HList inspired type level implementation of Okasaki's skew binary random access lists indexed by Roman numerals, using ideas from Ralf Hinze's paper "Functional Pearl: Typed Quote/Antiquote - Or: Compile-time Parsing" and some fun tricks from the MLton wiki: http://www.cs.ox.ac.uk/ralf.hinze/publications/Quote.pdf http://mlton.org/Fold http://mlton.org/Fold01N http://mlton.org/VariableArityPolymorphism With that out of the way, we start of by declaring all the different extensions we'll need.
{-# LANGUAGE FlexibleInstances #-} {-# LANGUAGE FunctionalDependencies #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE ScopedTypeVariables #-} {-# LANGUAGE TypeFamilies #-} {-# LANGUAGE TypeSynonymInstances #-}
These two are ugly. We need undecidable instances for bunch of things and overlapping instances for the type equality trick borrowed from the HList paper: http://homepages.cwi.nl/~ralf/HList/paper.pdf
{-# LANGUAGE OverlappingInstances #-} {-# LANGUAGE UndecidableInstances #-}
For GHC 7 we also need to increase the context stack.
{-# OPTIONS_GHC -fcontext-stack=100 #-}
Define Boolean values at the type level.
data T = T data F = F
And now Peano numbers. We don't actually use the constructors for anything but if we omitted them we'd have to use another extension, EmptyDataDecls.
data Z = Z data S n = S n
Here we create some types for heterogeneous lists and trees. Note that Pair has an extra type parameter. We provide two different versions of the cons operation. One requires that extra parameter but does not require undecidable instances. Unfortunately we still need undecidable instances later on for indexing.
data Pair i x y = Pair x y deriving Show data Nil = Nil deriving Show data Tree l x r = Node l x r deriving Show data Leaf x = Leaf x deriving Show
type Pair' x y = Pair () x y
nil :: Nil nil = Nil
Equality, less than, addition, subtraction and multiplication by 2 type functions for Peano numbers.
type family Equals m n
type instance Equals Z Z = T type instance Equals Z (S n) = F type instance Equals (S n) Z = F type instance Equals (S m) (S n) = Equals m n
type family LessThan m n
type instance LessThan Z Z = F type instance LessThan Z (S n) = T type instance LessThan (S n) Z = F type instance LessThan (S m) (S n) = LessThan m n
type family Add m n
type instance Add Z n = n type instance Add n Z = n type instance Add (S m) (S n) = S (S (Add m n))
type family Sub m n
type instance Sub n Z = n type instance Sub (S m) (S n) = Sub m n
type family Mul2 n
type instance Mul2 Z = Z type instance Mul2 (S n) = S (S (Mul2 n))
And finally tree depth and weight/size.
type family Depth t
type instance Depth (Leaf x ) = S Z type instance Depth (Tree l x r) = S (Depth l)
type family Weight t
type instance Weight (Leaf x ) = S Z type instance Weight (Tree l x r) = S (Mul2 (Weight l))
For reference, here is normal implementation of random access lists newtype List a = List { unList :: [(Int, Tree a)] } deriving Show data Tree a = Leaf a | Node (Tree a) a (Tree a) deriving Show cons :: a -> List a -> List a cons x (List wts) = List $ case wts of (v, l) : (w, r) : wts' | v == w -> (1 + w + w, Node l x r) : wts' _ -> (1 , Leaf x ) : wts List here is a list of pairs of balanced binary trees and their weights. The cons function has exactly two cases. The second case is trivial, we are mostly interested in the first case. If the weights of the two trees at the front of the list are the same, we combine them into a new tree with the newly inserted element at the top. Now we are ready to implement cons. This first version is simpler but requires undecidable instances.
class Cons' x y z | x y -> z where cons' :: x -> y -> z
Here we cover the cases of inserting an element into an empty list, and inserting an element into a list with a single element. These cover two out of four different possibilities when inserting a new element.
instance Cons' x Nil (Pair' (Leaf x) Nil) where cons' = Pair . Leaf
instance Cons' x (Pair' y Nil) (Pair' (Leaf x) (Pair' y Nil)) where cons' = Pair . Leaf
The remaining two cases occur when we have at least two elements in the list. This is where we need to compare weights of trees. First observation is that we don't need weight of a tree, depth is sufficient. Now we have a problem. We'd like to be able to write something like instance (Depth l ~ u, Depth r ~ v, Equals u v ~ T) => Cons' x (Pair' l (Pair' r y)) (Pair' (Tree l x r) y) where ... instance (Depth l ~ u, Depth r ~ v, Equals u v ~ F) => Cons' x y (Pair' (Leaf x) y) where ... but we get a functional dependency conflict in the second instance as the type variable y might stand for Pair' l (Pair' r y) and the functional dependency already says the result should be Pair' (Tree l x r) y, not Pair' (Leaf x) (Pair' l (Pair' r y)). We need to merge these two instances into one and then branch on the result of the comparison of tree depths. To do this, we introduce another type class to do the branching.
class ConsIf c x y z | c x y -> z where consIf :: c -> x -> y -> z
And merge the instances.
instance ( Depth l ~ u , Depth r ~ v , Equals u v ~ c , ConsIf c x (Pair' l (Pair' r y)) z ) =>
Cons' x (Pair' l (Pair' r y)) z where
cons' = consIf (undefined :: c)
instance ConsIf F x y (Pair' (Leaf x) y) where consIf _ = Pair . Leaf
instance (Depth l ~ u, Depth r ~ u) =>
ConsIf T x (Pair' l (Pair' r y)) (Pair' (Tree l x r) y) where
consIf _ x (Pair l (Pair r y)) = Pair (Node l x r) y
We can now test our cons function.
t0' = cons' 'a' $ cons' (33 :: Int) $ cons' (4.4 :: Float) $ cons' (88 :: Integer) $ cons' ([1 .. 8] :: [Int]) $ cons' "moose" $ cons' ((6, 3) :: (Int, Int)) $ nil
And ghci tells us the type is *Main> :t t0' t0' :: Pair' (Tree (Tree (Leaf Float) Int (Leaf Integer)) Char (Tree (Leaf [Char]) [Int] (Leaf (Int, Int)))) Nil Now we get to the second version that doesn't need the undecidable instances extension. In this one we go a step back and attempt to keep track of depth information in the type of the list. When we again end up at the point where we need to compare if two trees at the front of the list are of the same depth, we run into a problem. instance Cons x (Pair i l (Pair i r y)) (Pair (S i) (Tree l x r) y) where ... instance Cons x (Pair i y (Pair j z w)) (Pair (S Z) x (Pair i y (Pair j z w))) where ... This clearly won't work as we again get a functional dependency conflict. The j type variable could be the same as i in which case we have two different rules for what the result of consing x with (Pair i l (Pair i r y)) should be. As we don't want to use undecidable instances, we are stuck. Unless we can find a way to disambiguate the two instances. Instead of keeping track of depth information, we could try to track how many elements we need to complete a new tree. That way instead of counting up, we are counting down towards zero and it becomes easy to differentiate the two instances as only the one where the depths of the two trees are equal will contain a Z. Keeping track of how many elements are missing turns out to be a problem, at least as long as we don't want to use undecidable instances. Fortunately we can replace the number of elements missing with the number of levels the tree at the front needs before its depth matches the tree that follows.
class Cons x y z | x y -> z where cons :: x -> y -> z
Inserting an element into an empty list involves adding the element to the front and setting its the depth counter to 1.
instance Cons x Nil (Pair (S Z) (Leaf x) Nil) where cons = Pair . Leaf
If we are inserting an element into a list that already has a single tree, we again add the new element to the front, set its counter to 1 and we decrement the counter of the tree that now follows the newly inserted element.
instance Cons x (Pair (S i) y Nil) (Pair (S Z) (Leaf x) (Pair i y Nil)) where
cons x (Pair y z) = Pair (Leaf x) (Pair y z)
Here we have the case where the list already contains at least two trees but their depths are not equal. Like with the previous instance we insert the new element to the front and decrement the counter of the tree that now follows it.
instance Cons x (Pair (S i) y (Pair (S j) z w)) (Pair (S Z) (Leaf x) (Pair i y (Pair (S j) z w))) where
cons x (Pair y z) = Pair (Leaf x) (Pair y z)
That means we are left with one more case to handle. The case when the depths of the two trees are the same, or rather when the counter reaches 0. As we merge the two trees and create a new tree, we must also decrement the counter of the tree that follows. What this means is that we really need two instances. One where there are only two trees of the same depth in the list, and one where those two trees are followed by at least one more tree.
instance (Depth l ~ i, Depth r ~ i) => Cons x (Pair i l (Pair Z r Nil)) (Pair (S i) (Tree l x r) Nil) where
cons x (Pair l (Pair r y)) = Pair (Node l x r) y
As we merge the trees into a new tree, we reset and increment the depth counter. Note that we are using the first tree's counter. Reason for that is that the counter must represent the tree's depth as the tree at the front was either just inserted and so its counter is 1, or it was created when two smaller trees merged and its counter was set to its depth. Finally we must also decrement the counter of the tree that follows.
instance (Depth l ~ i, Depth r ~ i) => Cons x (Pair i l (Pair Z r (Pair (S j) y z))) (Pair (S i) (Tree l x r) (Pair j y z)) where
cons x (Pair l (Pair r (Pair y z))) = Pair (Node l x r) (Pair y z)
Now we can test the new cons function.
t0 = cons 'a' $ cons (33 :: Int) $ cons (4.4 :: Float) $ cons (88 :: Integer) $ cons ([1 .. 8] :: [Int]) $ cons "moose" $ cons ((6, 3) :: (Int, Int)) $ nil
The type is slightly more verbose. *Main> :t t0 t0 :: Pair (S (S (S Z))) (Tree (Tree (Leaf Float) Int (Leaf Integer)) Char (Tree (Leaf [Char]) [Int] (Leaf (Int, Int)))) Nil Here we introduce our wrapper for Roman numerals though we won't get to play with those just yet.
newtype R n = R { unR :: Int } deriving Show
We want to get indexing and updates out of the way first. The reference implementation for the indexing function is a bit more complex than the cons function. Each of the cases we need to handle has been marked with a comment so we can see which instance matches which part of the code. index :: List a -> Int -> Maybe a index (List ((w, t) : wts)) i | i < w = index' w i t -- 1 | otherwise = index (List wts) (i - w) -- 2 where index' 1 0 (Leaf x ) = Just x -- 3 index' _ _ (Leaf _ ) = Nothing index' w i (Node l x r) | i == 0 = Just x -- 4 | i <= w' = index' w' (i - 1 ) l -- 5 | otherwise = index' w' (i - 1 - w') r -- 6 where w' = w `div` 2 index _ _ = Nothing We don't need to do anything when the index is out of range - the type system takes care of that for us.
class Index i x y | i y -> x where index :: y -> R i -> x class IndexIf c i x y | c i y -> x where indexIf :: c -> y -> R i -> x
First we handle the easy cases.
instance Index Z x (Leaf x ) where index (Leaf x ) _ = x -- 3 instance Index Z x (Tree l x r) where index (Node _ x _) _ = x -- 4
Cases 5 and 6 require the use of a second type class so we can branch on the result of the comparison i <= w, though we rewrite it to w < i' + 1 to make use of the LessThan type family and to differentiate instance 6 from instance 4. Instead of halving the weight of the tree we are currently working on, we get the weight of one of its children.
instance ( Weight l ~ w , LessThan w (S i) ~ c , IndexIf c i x (Tree l y r) ) =>
Index (S i) x (Tree l y r) where
index t _ = indexIf (undefined :: c) t (undefined :: R i) -- 5, 6
instance Index i x l => IndexIf F i x (Tree l y r) where indexIf _ (Node l _ _) = index l -- 5
instance ( Weight r ~ w , Sub i w ~ j , Index j x r ) => IndexIf T i x (Tree l y r) where
indexIf _ (Node _ _ r) _ = index r (undefined :: R j) -- 6
Cases 1 and 2 are handled in a similar way.
instance ( Weight t ~ w , LessThan i w ~ c , IndexIf c i x (Pair j t y) ) => Index i x (Pair j t y) where
index = indexIf (undefined :: c) -- 1, 2
instance Index i x t => IndexIf T i x (Pair j t y) where indexIf _ (Pair t _) = index t -- 1
instance ( Weight t ~ w , Sub i w ~ j , Index j x y ) => IndexIf F i x (Pair k t y) where
indexIf _ (Pair _ y) _ = index y (undefined :: R j) -- 2
Update is similar. update :: Int -> a -> List a -> List a update i x (List ((w, t) : wts)) = List $ if i < w then (w, update' w i t) : wts -- 1 else (w, t) : unList (update (i - w) x (List wts)) -- 2 where update' 1 0 (Leaf _ ) = Leaf x -- 3 update' _ _ (Leaf _ ) = error "update" update' w i (Node l x' r) | i == 0 = Node l x r -- 4 | i <= w' = Node l' x' r -- 5 | otherwise = Node l x' r' -- 6 where w' = w `div` 2 l' = update' w' (i - 1 ) l r' = update' w' (i - 1 - w') r update _ _ _ = error "update"
class Index i x z => Update i x y z | i x y -> z where update :: y -> R i -> x -> z
class UpdateIf c i x y z | c i x y -> z where updateIf :: c -> y -> R i -> x -> z
instance Update Z x (Leaf y) (Leaf x) where update _ _ x = Leaf x -- 3
instance Update Z x (Tree l y r) (Tree l x r) where update (Node l _ r) _ x = Node l x r -- 4
instance ( Weight l ~ w , LessThan w (S i) ~ c , Index (S i) x t , UpdateIf c i x (Tree l y r) t) =>
Update (S i) x (Tree l y r) t where
update t _ = updateIf (undefined :: c) t (undefined :: R i) -- 5, 6
instance Update i x l l' => UpdateIf F i x (Tree l y r) (Tree l' y r) where
updateIf _ (Node l y r) i x = Node (update l i x) y r -- 5
instance ( Weight r ~ w , Sub i w ~ j , Update j x r r' ) =>
UpdateIf T i x (Tree l y r) (Tree l y r') where
updateIf _ (Node l y r) _ x = -- 6 Node l y (update r (undefined :: R j) x)
instance ( Weight t ~ w , LessThan i w ~ c , Index i x p , UpdateIf c i x (Pair j t y) p ) =>
Update i x (Pair j t y) p where
update = updateIf (undefined :: c) -- 1, 2
instance Update i x t t' => UpdateIf T i x (Pair j t y) (Pair j t' y) where
updateIf _ (Pair t y) i x = Pair (update t i x) y -- 1
instance ( Weight t ~ w , Sub i w ~ j , Update j x y z ) =>
UpdateIf F i x (Pair k t y) (Pair k t z) where
updateIf _ (Pair t y) _ x = -- 2 Pair t (update y (undefined :: R j) x)
Maybe we could combine Update and Index type classes into a single class? But we won't attempt that here. Now that we have indexing and updates, we need values we can use to index the list. Typing Z, S Z, S (S Z) and so on is too tedious and not very readable. We could use templates to predefine a whole bunch of type level numbers. That's kind of ugly though. And we can do it without using any extensions, in plain Haskell 98. We start off by making I a type synonym for S. Then we define types II to X, XX to C in steps of 10, CC to M in steps of 100 and finally we define MM and MMM. I don't know how Roman numerals are supposed to work above 3999 so that's the largest number we will be able to encode. Probably for the best as the type system doesn't seem too happy about having to deal with types of this sort.
type I n = S n
type II n = I (I n) type III n = I (II n) type IV n = I (III n) type V n = I (IV n) type VI n = I (V n) type VII n = I (VI n) type VIII n = I (VII n) type IX n = I (VIII n) type X n = I (IX n)
type XX n = X (X n) type XXX n = X (XX n) type XL n = X (XXX n) type L n = X (XL n) type LX n = X (L n) type LXX n = X (LX n) type LXXX n = X (LXX n) type XC n = X (LXXX n) type C n = X (XC n)
type CC n = C (C n) type CCC n = C (CC n) type CD n = C (CCC n) type D n = C (CD n) type DC n = C (D n) type DCC n = C (DC n) type DCCC n = C (DCC n) type CM n = C (DCCC n) type M n = C (CM n)
type MM n = M (M n) type MMM n = M (MM n)
Now for the fun part. We'd like to be able to construct only valid Roman numerals and that the type signatures match the values. We start of by defining the return and lift functions as described in Ralf Hinze's paper. We don't go all the way to creating a monad as these two functions are the only thing we need.
ret a f = f a lift f = ret . f
Now before we continue, we need a way to ensure that only valid numbers are constructed. We can group numbers in the range I to X into one group, let's call it ones, X to C into group tens, C to M into group hundreds and M, MM and MMM are in group thousands. A valid number can have at most 1 digit from each group. Digits also have to be ordered from largest to smallest, left to right. To do that, we could try and reuse the LessThen type family from before and add constraints so that a digit from the tens group can only be appended to a number less than ten. And a digit from the hundreds group can only be applied to a number less than a hundred. This approach works but it makes the GHC's type system choke and evaluating types of expressions takes minutes. We could also create a class for each of the groups and a set of combinators that can only be applied to a group of a lower rank. This works better. But we can go a step further than that - we don't actually need any type classes. All we need is one more phantom type for the rank of the number. So we introduce a second type that resembles R, only has one extra type variable.
newtype R' n s = R' { unR' :: Int } deriving Show
Here we make use of ret and lift to create polyvariadic functions for constructing Roman numerals. rn starts the construction and nr finalizes it.
rn :: ((R' n s -> R n) -> r) -> r rn f = ret (R . unR') f
nr :: (R' Z (IV Z) -> r) -> r nr f = f (R' 0)
Trying them in ghci we get *Main> rn nr R {unR = 0} *Main> :t rn nr rn nr :: R Z We have a zero. Notice in the type signature for nr, rank of zero is 4. Now we need to generate the digits. We can use a single function to generate all of them.
mkR n = lift (\f -> f . R' . (+ n) . unR')
And we encode the constraints in the type. Digits from the ones group can only be applied to a number of rank 4. There is only one number of rank 4 and that is zero. Applying a digit from the ones group to a zero gives us a number of rank 3. Digits from the tens group can be applied to numbers of ranks 3 or 4 and return a number of rank 2. And so on.
type RF n m s t k r = (R' n s -> k) -> ((R' m t -> k) -> r) -> r
i :: RF (I n) n (III Z) (IV s) k r ii :: RF (II n) n (III Z) (IV s) k r iii :: RF (III n) n (III Z) (IV s) k r iv :: RF (IV n) n (III Z) (IV s) k r v :: RF (V n) n (III Z) (IV s) k r vi :: RF (VI n) n (III Z) (IV s) k r vii :: RF (VII n) n (III Z) (IV s) k r viii :: RF (VIII n) n (III Z) (IV s) k r ix :: RF (IX n) n (III Z) (IV s) k r
x :: RF (X n) n (II Z) (III s) k r xx :: RF (XX n) n (II Z) (III s) k r xxx :: RF (XXX n) n (II Z) (III s) k r xl :: RF (XL n) n (II Z) (III s) k r l :: RF (L n) n (II Z) (III s) k r lx :: RF (LX n) n (II Z) (III s) k r lxx :: RF (LXX n) n (II Z) (III s) k r lxxx :: RF (LXXX n) n (II Z) (III s) k r xc :: RF (XC n) n (II Z) (III s) k r
c :: RF (C n) n (I Z) (II s) k r cc :: RF (CC n) n (I Z) (II s) k r ccc :: RF (CCC n) n (I Z) (II s) k r cd :: RF (CD n) n (I Z) (II s) k r d :: RF (D n) n (I Z) (II s) k r dc :: RF (DC n) n (I Z) (II s) k r dcc :: RF (DCC n) n (I Z) (II s) k r dccc :: RF (DCCC n) n (I Z) (II s) k r cm :: RF (CM n) n (I Z) (II s) k r
m :: RF (M n) n Z (I s) k r mm :: RF (MM n) n Z (I s) k r mmm :: RF (MMM n) n Z (I s) k r
i = mkR 1 ii = mkR 2 iii = mkR 3 iv = mkR 4 v = mkR 5 vi = mkR 6 vii = mkR 7 viii = mkR 8 ix = mkR 9
x = mkR 10 xx = mkR 20 xxx = mkR 30 xl = mkR 40 l = mkR 50 lx = mkR 60 lxx = mkR 70 lxxx = mkR 80 xc = mkR 90
c = mkR 100 cc = mkR 200 ccc = mkR 300 cd = mkR 400 d = mkR 500 dc = mkR 600 dcc = mkR 700 dccc = mkR 800 cm = mkR 900
m = mkR 1000 mm = mkR 2000 mmm = mkR 3000
We can now try to construct a number such as 2011.
r2011 = rn mm x i nr
*Main> r2011 R {unR = 2011} *Main> :t r2011 r2011 :: R (MM (X (I Z))) The type reflects the value we entered. Attempting to construct an invalid number gives us an error *Main> rn v vi nr <interactive>:1:0: Couldn't match expected type `Z' against inferred type `S s' Probable cause: `rn' is applied to too many arguments In the expression: rn v vi nr In the definition of `it': it = rn v vi nr I'll finish off this post with a generic map function. Here we use the type equality trick from the HList paper.
class GMap t p q | t p -> q where gmap :: t -> p -> q class GMapIf c t p q | c t p -> q where gmapIf :: c -> t -> p -> q
instance GMap (a -> b) Nil Nil where gmap _ = id
instance ( TypeEq a x c , GMapIf c (a -> b) (Leaf x) (Leaf y) ) =>
GMap (a -> b) (Leaf x) (Leaf y) where
gmap = gmapIf (undefined :: c)
instance ( TypeEq a x c , GMapIf c (a -> b) (Tree l x r) (Tree l' y r') ) =>
GMap (a -> b) (Tree l x r) (Tree l' y r') where
gmap = gmapIf (undefined :: c)
instance ( GMap (a -> b) x y , GMap (a -> b) p q ) =>
GMap (a -> b) (Pair i x p) (Pair i y q) where
gmap f (Pair x p) = Pair (gmap f x) (gmap f p)
instance GMapIf F (a -> b) (Leaf x) (Leaf x) where gmapIf _ _ = id
instance GMapIf T (a -> b) (Leaf a) (Leaf b) where gmapIf _ f (Leaf a) = Leaf (f a)
instance ( GMap (a -> b) l l' , GMap (a -> b) r r' ) =>
GMapIf F (a -> b) (Tree l x r) (Tree l' x r') where
gmapIf _ f (Node l x r) = Node (gmap f l) x (gmap f r)
instance ( GMap (a -> b) l l' , GMap (a -> b) r r' ) =>
GMapIf T (a -> b) (Tree l a r) (Tree l' b r') where
gmapIf _ f (Node l a r) = Node (gmap f l) (f a) (gmap f r)
type family Cast a type family Cast' t a type family Cast'' t a
type instance Cast a = Cast' () a type instance Cast' t a = Cast'' t a type instance Cast'' () a = a
class TypeEq x y c | x y -> c
instance Cast c ~ F => TypeEq x y c
instance TypeEq Char Char T instance TypeEq Int Int T instance TypeEq Float Float T instance TypeEq a a T => TypeEq [a] [a] T
And a few tests to see if it all works.
t1 = gmap (length :: String -> Int) t0 t2 = gmap (length :: [Int] -> Int) t1 t3 = gmap ((+ 1000) :: Int -> Int) t2
t4 = cons (t0 `index` rn iv nr) t3
t5 = update t4 (rn iv nr) "asdf"
t1' = gmap (length :: String -> Int) t0' t2' = gmap (length :: [Int] -> Int) t1' t3' = gmap ((+ 1000) :: Int -> Int) t2'
t4' = cons' (t0' `index` rn iv nr) t3'
t5' = update t4' (rn iv nr) "asdf"
As a bonus the implementation of type-level Roman numerals in Standard ML. SML/NJ has a fit trying to load the code. It seems to cope with it if constructors for values above 100 are commented out. And it gives really nice type signatures. MLton has no problems compiling the code, but fully expands all type signatures. (**********************************************************************) signature ROMAN = sig type z type 'n i type 'n ii = 'n i i type 'n iii = 'n i ii type 'n iv = 'n i iii type 'n v = 'n i iv type 'n vi = 'n i v type 'n vii = 'n i vi type 'n viii = 'n i vii type 'n ix = 'n i viii type 'n x = 'n i ix type 'n xx = 'n x x type 'n xxx = 'n x xx type 'n xl = 'n x xxx type 'n l = 'n x xl type 'n lx = 'n x l type 'n lxx = 'n x lx type 'n lxxx = 'n x lxx type 'n xc = 'n x lxxx type 'n c = 'n x xc type 'n cc = 'n c c type 'n ccc = 'n c cc type 'n cd = 'n c ccc type 'n d = 'n c cd type 'n dc = 'n c d type 'n dcc = 'n c dc type 'n dccc = 'n c dcc type 'n cm = 'n c dccc type 'n m = 'n c cm type 'n mm = 'n m m type 'n mmm = 'n m mm type 'n r type ('n, 's) r' val toInt : 'n r -> int val <| : ((('n, 's) r' -> 'n r) -> 'r) -> 'r val |> : ((z, z iv) r' -> 'r) -> 'r type ('n, 'm, 's, 't, 'k, 'r) rf = (('n, 's) r' -> 'k) -> ((('m, 't) r' -> 'k) -> 'r) -> 'r val i : ('n i , 'n, z iii, 's iv , 'k, 'r) rf val ii : ('n ii , 'n, z iii, 's iv , 'k, 'r) rf val iii : ('n iii , 'n, z iii, 's iv , 'k, 'r) rf val iv : ('n iv , 'n, z iii, 's iv , 'k, 'r) rf val v : ('n v , 'n, z iii, 's iv , 'k, 'r) rf val vi : ('n vi , 'n, z iii, 's iv , 'k, 'r) rf val vii : ('n vii , 'n, z iii, 's iv , 'k, 'r) rf val viii : ('n viii, 'n, z iii, 's iv , 'k, 'r) rf val ix : ('n ix , 'n, z iii, 's iv , 'k, 'r) rf val x : ('n x , 'n, z ii , 's iii, 'k, 'r) rf val xx : ('n xx , 'n, z ii , 's iii, 'k, 'r) rf val xxx : ('n xxx , 'n, z ii , 's iii, 'k, 'r) rf val xl : ('n xl , 'n, z ii , 's iii, 'k, 'r) rf val l : ('n l , 'n, z ii , 's iii, 'k, 'r) rf val lx : ('n lx , 'n, z ii , 's iii, 'k, 'r) rf val lxx : ('n lxx , 'n, z ii , 's iii, 'k, 'r) rf val lxxx : ('n lxxx, 'n, z ii , 's iii, 'k, 'r) rf val xc : ('n xc , 'n, z ii , 's iii, 'k, 'r) rf val c : ('n c , 'n, z i , 's ii , 'k, 'r) rf val cc : ('n cc , 'n, z i , 's ii , 'k, 'r) rf val ccc : ('n ccc , 'n, z i , 's ii , 'k, 'r) rf val cd : ('n cd , 'n, z i , 's ii , 'k, 'r) rf val d : ('n d , 'n, z i , 's ii , 'k, 'r) rf val dc : ('n dc , 'n, z i , 's ii , 'k, 'r) rf val dcc : ('n dcc , 'n, z i , 's ii , 'k, 'r) rf val dccc : ('n dccc, 'n, z i , 's ii , 'k, 'r) rf val cm : ('n cm , 'n, z i , 's ii , 'k, 'r) rf val m : ('n m , 'n, z , 's i , 'k, 'r) rf val mm : ('n mm , 'n, z , 's i , 'k, 'r) rf val mmm : ('n mmm , 'n, z , 's i , 'k, 'r) rf end structure Roman :> ROMAN = struct datatype z = Z datatype 'n i = I of 'n type 'n ii = 'n i i type 'n iii = 'n i ii type 'n iv = 'n i iii type 'n v = 'n i iv type 'n vi = 'n i v type 'n vii = 'n i vi type 'n viii = 'n i vii type 'n ix = 'n i viii type 'n x = 'n i ix type 'n xx = 'n x x type 'n xxx = 'n x xx type 'n xl = 'n x xxx type 'n l = 'n x xl type 'n lx = 'n x l type 'n lxx = 'n x lx type 'n lxxx = 'n x lxx type 'n xc = 'n x lxxx type 'n c = 'n x xc type 'n cc = 'n c c type 'n ccc = 'n c cc type 'n cd = 'n c ccc type 'n d = 'n c cd type 'n dc = 'n c d type 'n dcc = 'n c dc type 'n dccc = 'n c dcc type 'n cm = 'n c dccc type 'n m = 'n c cm type 'n mm = 'n m m type 'n mmm = 'n m mm type 'n r = int datatype ('n, 's) r' = R of int type ('n, 'm, 's, 't, 'k, 'r) rf = (('n, 's) r' -> 'k) -> ((('m, 't) r' -> 'k) -> 'r) -> 'r fun toInt n = n local fun unR (R n) = n fun ret a f = f a fun lift f = ret o f fun mkR n k = lift (fn g => g o R o (fn x => x + n) o unR) k in fun <| f = ret unR f fun |> f = f (R 0) fun i k = mkR 1 k fun ii k = mkR 2 k fun iii k = mkR 3 k fun iv k = mkR 4 k fun v k = mkR 5 k fun vi k = mkR 6 k fun vii k = mkR 7 k fun viii k = mkR 8 k fun ix k = mkR 9 k fun x k = mkR 10 k fun xx k = mkR 20 k fun xxx k = mkR 30 k fun xl k = mkR 40 k fun l k = mkR 50 k fun lx k = mkR 60 k fun lxx k = mkR 70 k fun lxxx k = mkR 80 k fun xc k = mkR 90 k fun c k = mkR 100 k fun cc k = mkR 200 k fun ccc k = mkR 300 k fun cd k = mkR 400 k fun d k = mkR 500 k fun dc k = mkR 600 k fun dcc k = mkR 700 k fun dccc k = mkR 800 k fun cm k = mkR 900 k fun m k = mkR 1000 k fun mm k = mkR 2000 k fun mmm k = mkR 3000 k end end