Data.Sized

This module provides the functionality to make length-parametrized types from existing ListLike and Functor sequential types.

Most of the complexity of operations for Sized f n a are the same as original operations for f. For example, !! is O(1) for Sized Vector n a but O(i) for Sized [] n a.

This module also provides powerful view types and pattern synonyms to inspect the sized sequence. See Views and Patterns for more detail.

Main Data-types

data Sized f n a

Sized wraps a sequential type f and makes length-parametrized version. GHC's type natural is currently poor, so we adopt Peano numeral here.

Here, f must be the instance of Functor and ListLike (f a) a for all a. This constraint is expressed by ListLikeF. Folding and traversing function such as all and foldl' is available via Foldable or Traversable class, if f is the instance of them.

Since 0.1.0.0

Instances

Typeable ((k -> *) -> Nat -> k -> *) (Sized k)

Functor f => Functor (Sized * f n)

Foldable f => Foldable (Sized * f n)

Traversable f => Traversable (Sized * f n)

Eq (f a) => Eq (Sized k f n a)

Ord (f a) => Ord (Sized k f n a)

Show (f a) => Show (Sized k f n a)

ListLikeF f => FoldableLL (Sized * f n a) a

Since 0.1.0.0

type ListLikeF f = (Functor f, Forall (LLF f))

Functor f such that there is instance ListLike (f a) a for any a.

Since 0.1.0.0

data SomeSized f a where

Sized vector with the length is existentially quantified. This type is used mostly when the return type's length cannot be statically determined beforehand.

SomeSized sn xs :: SomeSized f a stands for the Sized sequence xs of element type a and length sn.

Since 0.1.0.0

Constructors

SomeSized :: (ListLikeF f, SingI n) => SNat n -> Sized f (n :: Nat) a -> SomeSized f a

Instances

Eq (f a) => Eq (SomeSized f a)

Show (f a) => Show (SomeSized f a)

Typeable ((* -> *) -> * -> *) SomeSized

Accessors

Length information

length :: ListLikeF f => Sized f n a -> Int

Returns the length of wrapped containers. If you use unsafeFromList or similar unsafe functions, this function may return different value from type-parameterized length.

Since 0.1.0.0

sLength :: SingI n => Sized f n a -> SNat n

SNat version of length.

Since 0.1.0.0

null :: ListLikeF f => Sized f n a -> Bool

Test if the sequence is empty or not.

Since 0.1.0.0

Indexing

(!!) :: ListLikeF f => Sized f (S m) a -> Int -> a

(Unsafe) indexing with Ints. If you want to check boundary statically, use %!! or sIndex.

Since 0.1.0.0

(%!!) :: ListLikeF f => Sized f n a -> Ordinal n -> a

Safe indexing with Ordinals.

Since 0.1.0.0

index :: ListLikeF f => Int -> Sized f (S m) c -> c

Flipped version of !!.

Since 0.1.0.0

sIndex :: ListLikeF f => Ordinal n -> Sized f n c -> c

Flipped version of %!!.

Since 0.1.0.0

head :: ListLikeF f => Sized f (S n) a -> a

Take the first element of non-empty sequence. If you want to make case-analysis for general sequence, see Views and Patterns section.

Since 0.1.0.0

last :: ListLikeF f => Sized f (S n) a -> a

Take the last element of non-empty sequence. If you want to make case-analysis for general sequence, see Views and Patterns section.

Since 0.1.0.0

uncons :: ListLikeF f => Sized f (S n) a -> (a, Sized f n a)

Take the head and tail of non-empty sequence. If you want to make case-analysis for general sequence, see Views and Patterns section.

Since 0.1.0.0

unsnoc :: ListLikeF f => Sized f (S n) a -> (Sized f n a, a)

Take the init and last of non-empty sequence. If you want to make case-analysis for general sequence, see Views and Patterns section.

Since 0.1.0.0

Slicing

tail :: ListLikeF f => Sized f (S n) a -> Sized f n a

Take the tail of non-empty sequence. If you want to make case-analysis for general sequence, see Views and Patterns section.

Since 0.1.0.0

init :: ListLikeF f => Sized f (S n) a -> Sized f n a

Take the initial segment of non-empty sequence. If you want to make case-analysis for general sequence, see Views and Patterns section.

Since 0.1.0.0

take :: (ListLikeF f, (n :<<= m) ~ True) => SNat n -> Sized f m a -> Sized f n a

take k xs takes first k element of xs where the length of xs should be larger than k. It is really sad, that this function takes at least O(k) regardless of base container.

Since 0.1.0.0

takeAtMost :: ListLikeF f => SNat n -> Sized f m a -> Sized f (Min n m) a

take k xs takes first k element of xs at most. It is really sad, that this function takes at least O(k) regardless of base container.

Since 0.1.0.0

drop :: (ListLikeF f, (n :<<= m) ~ True) => SNat n -> Sized f m a -> Sized f (m :-: n) a

drop k xs drops first k element of xs and returns the rest of sequence, where the length of xs should be larger than k. It is really sad, that this function takes at least O(k) regardless of base container.

Since 0.1.0.0

splitAt :: (ListLikeF f, (n :<<= m) ~ True) => SNat n -> Sized f m a -> (Sized f n a, Sized f (m :-: n) a)

splitAt k xs split xs at k, where the length of xs should be less than or equal to k. It is really sad, that this function takes at least O(k) regardless of base container.

Since 0.1.0.0

splitAtMost :: ListLikeF f => SNat n -> Sized f m a -> (Sized f (Min n m) a, Sized f (m :-: n) a)

splitAtMost k xs split xs at k. If k exceeds the length of xs, then the second result value become empty. It is really sad, that this function takes at least O(k) regardless of base container.

Since 0.1.0.0

Construction

Initialisation

empty :: forall f a. ListLikeF f => Sized f Z a

Empty sequence.

Since 0.1.0.0

singleton :: forall f a. ListLikeF f => a -> Sized f One a

Sequence with one element.

Since 0.1.0.0

toSomeSized :: forall f a. ListLikeF f => f a -> SomeSized f a

Consruct the Sized sequence from base type, but the length parameter is dynamically determined and existentially quantified; see also SomeSized.

Since 0.1.0.0

replicate :: forall f n a. ListLikeF f => SNat n -> a -> Sized f n a

Replicates the same value.

Since 0.1.0.0

replicate' :: (SingI (n :: Nat), ListLikeF f) => a -> Sized f n a

replicate with the length inferred.

Since 0.1.0.0

Concatenation

cons :: ListLikeF f => a -> Sized f n a -> Sized f (S n) a

Append an element to the head of sequence.

Since 0.1.0.0

(<|) :: ListLikeF f => a -> Sized f n a -> Sized f (S n) a infixr 5

Infix version of cons.

Since 0.1.0.0

snoc :: ListLikeF f => Sized f n a -> a -> Sized f (S n) a

Append an element to the tail of sequence.

Since 0.1.0.0

(|>) :: ListLikeF f => Sized f n a -> a -> Sized f (S n) a infixl 5

Infix version of snoc.

Since 0.1.0.0

append :: ListLikeF f => Sized f n a -> Sized f m a -> Sized f (n :+ m) a

Append two lists.

Since 0.1.0.0

(++) :: ListLikeF f => Sized f n a -> Sized f m a -> Sized f (n :+ m) a infixr 5

Infix version of append.

Since 0.1.0.0

concat :: forall f f' m n a. (ListLikeF f, ListLikeF f') => Sized f' m (Sized f n a) -> Sized f (m :* n) a

Concatenates multiple sequences into one.

Since 0.1.0.0

Zips

zip :: forall f a b n m. ListLikeF f => Sized f n a -> Sized f m b -> Sized f (Min n m) (a, b)

Zipping two sequences. Length is adjusted to shorter one.

Since 0.1.0.0

zipSame :: forall f n a b. ListLikeF f => Sized f n a -> Sized f n b -> Sized f n (a, b)

zip for the sequences of the same length.

Since 0.1.0.0

zipWith :: forall f a b c m n. ListLikeF f => (a -> b -> c) -> Sized f n a -> Sized f m b -> Sized f (Min n m) c

Zipping two sequences with funtion. Length is adjusted to shorter one.

Since 0.1.0.0

zipWithSame :: forall f a b c n. ListLikeF f => (a -> b -> c) -> Sized f n a -> Sized f n b -> Sized f n c

zipWith for the sequences of the same length.

Since 0.1.0.0

unzip :: forall f n a b. ListLikeF f => Sized f n (a, b) -> (Sized f n a, Sized f n b)

Unzipping the sequence of tuples.

Since 0.1.0.0

Transformation

map :: Functor f => (a -> b) -> Sized f n a -> Sized f n b

Map function.

Since 0.1.0.0

reverse :: ListLikeF f => Sized f n a -> Sized f n a

Reverse function.

Since 0.1.0.0

intersperse :: ListLikeF f => a -> Sized f n a -> Sized f ((Two :* n) :-: One) a

Intersperces.

Since 0.1.0.0

nub :: (ListLikeF f, Eq a) => Sized f n a -> SomeSized f a

Remove all duplicates.

Since 0.1.0.0

sort :: (ListLikeF f, Ord a) => Sized f n a -> Sized f n a

Sorting sequence by ascending order.

Since 0.1.0.0

sortBy :: ListLikeF f => (a -> a -> Ordering) -> Sized f n a -> Sized f n a

Generalized version of sort.

Since 0.1.0.0

insert :: (ListLikeF f, Ord a) => a -> Sized f n a -> Sized f (S n) a

Insert new element into the presorted sequence.

Since 0.1.0.0

insertBy :: ListLikeF f => (a -> a -> Ordering) -> a -> Sized f n a -> Sized f (S n) a

Generalized version of insert.

Since 0.1.0.0

Conversion

List

toList :: ListLikeF f => Sized f n a -> [a]

Convert to list.

Since 0.1.0.0

fromList :: forall f n a. ListLikeF f => SNat n -> [a] -> Maybe (Sized f n a)

If the given list is shorter than n, then returns Nothing Otherwise returns Sized f n a consisting of initial n element of given list.

Since 0.1.0.0

fromList' :: (ListLikeF f, SingI (n :: Nat)) => [a] -> Maybe (Sized f n a)

fromList with the result length inferred.

Since 0.1.0.0

unsafeFromList :: forall f n a. ListLikeF f => SNat n -> [a] -> Sized f n a

Unsafe version of fromList. If the length of the given list does not equal to n, then something unusual happens.

Since 0.1.0.0

unsafeFromList' :: (SingI (n :: Nat), ListLikeF f) => [a] -> Sized f n a

unsafeFromList with the result length inferred.

Since 0.1.0.0

fromListWithDefault :: forall f n a. ListLikeF f => SNat n -> a -> [a] -> Sized f n a

Construct a Sized f n a by padding default value if the given list is short.

Since 0.1.0.0

fromListWithDefault' :: (SingI (n :: Nat), ListLikeF f) => a -> [a] -> Sized f n a

fromListWithDefault with the result length inferred.

Since 0.1.0.0

Base container

unsized :: Sized f n a -> f a

Forget the length and obtain the wrapped base container.

Since 0.1.0.0

toSized :: ListLikeF f => SNat n -> f a -> Maybe (Sized f n a)

If the length of the input is shorter than n, then returns Nothing. Otherwise returns Sized f n a consisting of initial n element of the input.

Since 0.1.0.0

toSized' :: (ListLikeF f, SingI (n :: Nat)) => f a -> Maybe (Sized f n a)

toSized with the result length inferred.

Since 0.1.0.0

unsafeToSized :: SNat n -> f a -> Sized f n a

Unsafe version of toSized. If the length of the given list does not equal to n, then something unusual happens.

Since 0.1.0.0

unsafeToSized' :: (SingI (n :: Nat), ListLikeF f) => f a -> Sized f n a

unsafeToSized with the result length inferred.

Since 0.1.0.0

toSizedWithDefault :: ListLikeF f => SNat n -> a -> f a -> Sized f n a

Construct a Sized f n a by padding default value if the given list is short.

Since 0.1.0.0

toSizedWithDefault' :: (SingI (n :: Nat), ListLikeF f) => a -> f a -> Sized f n a

toSizedWithDefault with the result length inferred.

Since 0.1.0.0

Querying

Partitioning

data Partitioned f n a where

The type Partitioned f n a represents partitioned sequence of length n. Value Partitioned lenL ls lenR rs stands for:

• Entire sequence is divided into ls and rs, and their length are lenL and lenR resp.

• lenL + lenR = n

Since 0.1.0.0

Constructors

Partitioned :: (ListLikeF f, SingI n, SingI m) => SNat n -> Sized f (n :: Nat) a -> SNat m -> Sized f (m :: Nat) a -> Partitioned f (n :+ m) a

takeWhile :: ListLikeF f => (a -> Bool) -> Sized f n a -> SomeSized f a

Take the initial segment as long as elements satisfys the predicate.

Since 0.1.0.0

dropWhile :: ListLikeF f => (a -> Bool) -> Sized f n a -> SomeSized f a

Drop the initial segment as long as elements satisfys the predicate.

Since 0.1.0.0

span :: ListLikeF f => (a -> Bool) -> Sized f n a -> Partitioned f n a

Invariant: ListLike (f a) a instance must be implemented to satisfy the following property: length (fst (span p xs)) + length (snd (span p xs)) == length xs Otherwise, this function introduces severe contradiction.

Since 0.1.0.0

break :: ListLikeF f => (a -> Bool) -> Sized f n a -> Partitioned f n a

Invariant: ListLike (f a) a instance must be implemented to satisfy the following property: length (fst (break p xs)) + length (snd (break p xs)) == length xs Otherwise, this function introduces severe contradiction.

Since 0.1.0.0

partition :: ListLikeF f => (a -> Bool) -> Sized f n a -> Partitioned f n a

Invariant: ListLike (f a) a instance must be implemented to satisfy the following property: length (fst (partition p xs)) + length (snd (partition p xs)) == length xs Otherwise, this function introduces severe contradiction.

Since 0.1.0.0

Searching

elem :: (ListLikeF f, Eq a) => a -> Sized f n a -> Bool

Membership test; see also notElem.

Since 0.1.0.0

notElem :: (ListLikeF f, Eq a) => a -> Sized f n a -> Bool

Negation of elem.

Since 0.1.0.0

find :: ListLikeF f => (a -> Bool) -> Sized f n a -> Maybe a

Find the element satisfying the predicate.

Since 0.1.0.0

findIndex :: ListLikeF f => (a -> Bool) -> Sized f n a -> Maybe Int

findIndex p xs find the element satisfying p and returns its index if exists.

Since 0.1.0.0

sFindIndex :: (SingI n, ListLikeF f) => (a -> Bool) -> Sized f n a -> Maybe (Ordinal n)

Ordinal version of findIndex.

Since 0.1.0.0

findIndices :: ListLikeF f => (a -> Bool) -> Sized f n a -> [Int]

findIndices p xs find all elements satisfying p and returns their indices.

Since 0.1.0.0

sFindIndices :: (SingI n, ListLikeF f) => (a -> Bool) -> Sized f n a -> [Ordinal n]

Ordinal version of findIndices.

Since 0.1.0.0

elemIndex :: (Eq a, ListLikeF f) => a -> Sized f n a -> Maybe Int

Returns the index of the given element in the list, if exists.

Since 0.1.0.0

sElemIndex :: (SingI n, ListLikeF f, Eq a) => a -> Sized f n a -> Maybe (Ordinal n)

Ordinal version of elemIndex

Since 0.1.0.0

elemIndices :: (ListLikeF f, Eq a) => a -> Sized f n a -> [Int]

Returns all indices of the given element in the list.

Since 0.1.0.0

sElemIndices :: (SingI n, ListLikeF f, Eq a) => a -> Sized f n a -> [Ordinal n]

Ordinal version of elemIndices

Since 0.1.0.0

Views and Patterns

With GHC's ViewPatterns and PatternSynonym extensions, we can pattern-match on arbitrary Sized f n a if f is list-like functor. Curretnly, there are two direction view and patterns: Cons and Snoc. Assuming underlying sequence type f has O(1) implementation for null, head (resp. last) and tail (resp. init), We can view and pattern-match on cons (resp. snoc) of Sized f n a in O(1).

Views

With ViewPatterns extension, we can pattern-match on Sized value as follows:

slen :: (SingI n, ListLikeF f) => Sized f n a -> SNat n
slen (viewCons -> NilCV)    = SZ
slen (viewCons -> _ ::- as) = SS (slen as)
slen _                          = error "impossible"

The constraint (SingI n, ListLikeF f) is needed for view function. In the above, we have extra wildcard pattern (_) at the last. Code compiles if we removed it, but current GHC warns for incomplete pattern, although we know first two patterns exhausts all the case.

Equivalently, we can use snoc-style pattern-matching:

slen :: (SingI n, ListLikeF f) => Sized f n a -> SNat n
slen (viewSnoc -> NilSV)     = SZ
slen (viewSnoc -> as :-:: _) = SS (slen as)

Patterns

So we can pattern match on both end of sequence via views, but it is rather clumsy to nest it. For example:

nextToHead :: (ListLikeF f, SingI n) => Sized f (S (S n)) a -> a
nextToHead (viewCons -> _ ::- (viewCons -> a ::- _)) = a

In such a case, with PatternSynonyms extension we can write as follows:

nextToHead :: (ListLikeF f, SingI n) => Sized f (S (S n)) a -> a
nextToHead (_ :< a :< _) = a

Of course, we can also rewrite above slen example using PatternSynonyms:

slen :: (SingI n, ListLikeF f) => Sized f n a -> SNat n
slen NilL      = SZ
slen (_ :< as) = SS (slen as)
slen _           = error "impossible"

So, we can use :< and NilL (resp. :> and NilR) to pattern-match directly on cons-side (resp. snoc-side) as we usually do for lists. :<, NilL, :> and NilR are neither functions nor data constructors, but pattern synonyms so we cannot use them in expression contexts. For more detail on pattern synonyms, see GHC Users Guide and HaskellWiki.

Definitions

viewCons :: forall f a n. (SingI n, ListLikeF f) => Sized f n a -> ConsView f n a

Case analysis for the cons-side of sequence.

Since 0.1.0.0

data ConsView f n a where

View of the left end of sequence (cons-side).

Since 0.1.0.0

Constructors

NilCV :: ConsView f Z a

(::-) :: SingI n => a -> Sized f n a -> ConsView f (S n) a infixr 5

viewSnoc :: forall f n a. (SingI n, ListLikeF f) => Sized f n a -> SnocView f n a

Case analysis for the snoc-side of sequence.

Since 0.1.0.0

data SnocView f n a where

View of the left end of sequence (snoc-side).

Since 0.1.0.0

Constructors

NilSV :: SnocView f Z a

(:-::) :: SingI n => Sized f n a -> a -> SnocView f (S n) a infixl 5

(:<)

NilL

(:>)

NilR