idris-0.10: libs/base/Data/Vect.idr
module Data.Vect
import public Data.Fin
import Language.Reflection
%access public
%default total
infixr 7 ::
||| Vectors: Generic lists with explicit length in the type
%elim
data Vect : Nat -> Type -> Type where
||| Empty vector
Nil : Vect Z a
||| A non-empty vector of length `S k`, consisting of a head element and
||| the rest of the list, of length `k`.
(::) : (x : a) -> (xs : Vect k a) -> Vect (S k) a
-- Hints for interactive editing
%name Vect xs,ys,zs,ws
--------------------------------------------------------------------------------
-- Length
--------------------------------------------------------------------------------
||| Calculate the length of a `Vect`.
|||
||| **Note**: this is only useful if you don't already statically know the length
||| and you want to avoid matching the implicit argument for erasure reasons.
||| @ n the length (provably equal to the return value)
||| @ xs the vector
length : (xs : Vect n a) -> Nat
length [] = 0
length (x::xs) = 1 + length xs
||| Show that the length function on vectors in fact calculates the length
private lengthCorrect : (n : Nat) -> (xs : Vect n a) -> length xs = n
lengthCorrect Z [] = Refl
lengthCorrect (S n) (x :: xs) = rewrite lengthCorrect n xs in Refl
--------------------------------------------------------------------------------
-- Indexing into vectors
--------------------------------------------------------------------------------
||| All but the first element of the vector
tail : Vect (S n) a -> Vect n a
tail (x::xs) = xs
||| Only the first element of the vector
head : Vect (S n) a -> a
head (x::xs) = x
||| The last element of the vector
last : Vect (S n) a -> a
last (x::[]) = x
last (x::y::ys) = last $ y::ys
||| All but the last element of the vector
init : Vect (S n) a -> Vect n a
init (x::[]) = []
init (x::y::ys) = x :: init (y::ys)
||| Extract a particular element from a vector
index : Fin n -> Vect n a -> a
index FZ (x::xs) = x
index (FS k) (x::xs) = index k xs
||| Insert an element at a particular index
insertAt : Fin (S n) -> a -> Vect n a -> Vect (S n) a
insertAt FZ y xs = y :: xs
insertAt (FS k) y (x::xs) = x :: insertAt k y xs
insertAt (FS k) y [] = absurd k
||| Construct a new vector consisting of all but the indicated element
deleteAt : Fin (S n) -> Vect (S n) a -> Vect n a
deleteAt FZ (x::xs) = xs
deleteAt {n = S m} (FS k) (x::xs) = x :: deleteAt k xs
deleteAt {n = Z} (FS k) (x::xs) = absurd k
deleteAt _ [] impossible
||| Replace an element at a particlar index with another
replaceAt : Fin n -> t -> Vect n t -> Vect n t
replaceAt FZ y (x::xs) = y :: xs
replaceAt (FS k) y (x::xs) = x :: replaceAt k y xs
||| Replace the element at a particular index with the result of applying a function to it
||| @ i the index to replace at
||| @ f the update function
||| @ xs the vector to replace in
updateAt : (i : Fin n) -> (f : t -> t) -> (xs : Vect n t) -> Vect n t
updateAt FZ f (x::xs) = f x :: xs
updateAt (FS k) f (x::xs) = x :: updateAt k f xs
--------------------------------------------------------------------------------
-- Subvectors
--------------------------------------------------------------------------------
||| Get the first n elements of a Vect
||| @ n the number of elements to take
take : (n : Nat) -> Vect (n + m) a -> Vect n a
take Z xs = []
take (S k) (x :: xs) = x :: take k xs
||| Remove the first n elements of a Vect
||| @ n the number of elements to remove
drop : (n : Nat) -> Vect (n + m) a -> Vect m a
drop Z xs = xs
drop (S k) (x :: xs) = drop k xs
||| Take the longest prefix of a Vect such that all elements satisfy some
||| Boolean predicate.
|||
||| @ p the predicate
takeWhile : (p : a -> Bool) -> Vect n a -> (q ** Vect q a)
takeWhile p [] = (_ ** [])
takeWhile p (x::xs) =
let (len ** ys) = takeWhile p xs
in if p x then
(S len ** x :: ys)
else
(_ ** [])
||| Remove the longest prefix of a Vect such that all removed elements satisfy some
||| Boolean predicate.
|||
||| @ p the predicate
dropWhile : (p : a -> Bool) -> Vect n a -> (q ** Vect q a)
dropWhile p [] = (_ ** [])
dropWhile p (x::xs) =
if p x then
dropWhile p xs
else
(_ ** x::xs)
--------------------------------------------------------------------------------
-- Transformations
--------------------------------------------------------------------------------
||| Reverse the order of the elements of a vector
reverse : Vect n a -> Vect n a
reverse xs = go [] xs
where go : Vect n a -> Vect m a -> Vect (n+m) a
go {n} acc [] = rewrite plusZeroRightNeutral n in acc
go {n} {m=S m} acc (x :: xs) = rewrite sym $ plusSuccRightSucc n m
in go (x::acc) xs
||| Alternate an element between the other elements of a vector
||| @ sep the element to intersperse
||| @ xs the vector to separate with `sep`
intersperse : (sep : a) -> (xs : Vect n a) -> Vect (n + pred n) a
intersperse sep [] = []
intersperse sep (x::xs) = x :: intersperse' sep xs
where
intersperse' : a -> Vect n a -> Vect (n + n) a
intersperse' sep [] = []
intersperse' {n=S n} sep (x::xs) = rewrite sym $ plusSuccRightSucc n n
in sep :: x :: intersperse' sep xs
--------------------------------------------------------------------------------
-- Conversion from list (toList is provided by Foldable)
--------------------------------------------------------------------------------
fromList' : Vect n a -> (l : List a) -> Vect (length l + n) a
fromList' ys [] = ys
fromList' {n} ys (x::xs) =
rewrite (plusSuccRightSucc (length xs) n) ==>
Vect (plus (length xs) (S n)) a in
fromList' (x::ys) xs
||| Convert a list to a vector.
|||
||| The length of the list should be statically known.
fromList : (l : List a) -> Vect (length l) a
fromList l =
rewrite (sym $ plusZeroRightNeutral (length l)) in
reverse $ fromList' [] l
--------------------------------------------------------------------------------
-- Building (bigger) vectors
--------------------------------------------------------------------------------
||| Append two vectors
(++) : Vect m a -> Vect n a -> Vect (m + n) a
(++) [] ys = ys
(++) (x::xs) ys = x :: xs ++ ys
||| Repeate some value n times
||| @ n the number of times to repeat it
||| @ x the value to repeat
replicate : (n : Nat) -> (x : a) -> Vect n a
replicate Z x = []
replicate (S k) x = x :: replicate k x
--------------------------------------------------------------------------------
-- Zips and unzips
--------------------------------------------------------------------------------
||| Combine two equal-length vectors pairwise with some function
zipWith : (a -> b -> c) -> Vect n a -> Vect n b -> Vect n c
zipWith f [] [] = []
zipWith f (x::xs) (y::ys) = f x y :: zipWith f xs ys
||| Combine three equal-length vectors into a vector with some function
zipWith3 : (a -> b -> c -> d) -> Vect n a -> Vect n b -> Vect n c -> Vect n d
zipWith3 f [] [] [] = []
zipWith3 f (x::xs) (y::ys) (z::zs) = f x y z :: zipWith3 f xs ys zs
||| Combine two equal-length vectors pairwise
zip : Vect n a -> Vect n b -> Vect n (a, b)
zip = zipWith (\x,y => (x,y))
||| Combine three equal-length vectors elementwise into a vector of tuples
zip3 : Vect n a -> Vect n b -> Vect n c -> Vect n (a, b, c)
zip3 = zipWith3 (\x,y,z => (x,y,z))
||| Convert a vector of pairs to a pair of vectors
unzip : Vect n (a, b) -> (Vect n a, Vect n b)
unzip [] = ([], [])
unzip ((l, r)::xs) with (unzip xs)
| (lefts, rights) = (l::lefts, r::rights)
||| Convert a vector of three-tuples to a triplet of vectors
unzip3 : Vect n (a, b, c) -> (Vect n a, Vect n b, Vect n c)
unzip3 [] = ([], [], [])
unzip3 ((l,c,r)::xs) with (unzip3 xs)
| (lefts, centers, rights) = (l::lefts, c::centers, r::rights)
--------------------------------------------------------------------------------
-- Equality
--------------------------------------------------------------------------------
implementation (Eq a) => Eq (Vect n a) where
(==) [] [] = True
(==) (x::xs) (y::ys) = x == y && xs == ys
--------------------------------------------------------------------------------
-- Order
--------------------------------------------------------------------------------
implementation Ord a => Ord (Vect n a) where
compare [] [] = EQ
compare (x::xs) (y::ys) = compare x y `thenCompare` compare xs ys
--------------------------------------------------------------------------------
-- Maps
--------------------------------------------------------------------------------
implementation Functor (Vect n) where
map f [] = []
map f (x::xs) = f x :: map f xs
||| Map a partial function across a vector, returning those elements for which
||| the function had a value.
|||
||| The first projection of the resulting pair (ie the length) will always be
||| at most the length of the input vector. This is not, however, guaranteed
||| by the type.
|||
||| @ f the partial function (expressed by returning `Maybe`)
||| @ xs the vector to check for results
mapMaybe : (f : a -> Maybe b) -> (xs : Vect n a) -> (m : Nat ** Vect m b)
mapMaybe f [] = (_ ** [])
mapMaybe f (x::xs) =
let (len ** ys) = mapMaybe f xs
in case f x of
Just y => (S len ** y :: ys)
Nothing => ( len ** ys)
--------------------------------------------------------------------------------
-- Folds
--------------------------------------------------------------------------------
foldrImpl : (t -> acc -> acc) -> acc -> (acc -> acc) -> Vect n t -> acc
foldrImpl f e go [] = go e
foldrImpl f e go (x::xs) = foldrImpl f e (go . (f x)) xs
implementation Foldable (Vect n) where
foldr f e xs = foldrImpl f e id xs
--------------------------------------------------------------------------------
-- Special folds
--------------------------------------------------------------------------------
||| Flatten a vector of equal-length vectors
concat : Vect m (Vect n a) -> Vect (m * n) a
concat [] = []
concat (v::vs) = v ++ concat vs
||| Foldr without seeding the accumulator
foldr1 : (t -> t -> t) -> Vect (S n) t -> t
foldr1 f (x::xs) = foldr f x xs
||| Foldl without seeding the accumulator
foldl1 : (t -> t -> t) -> Vect (S n) t -> t
foldl1 f (x::xs) = foldl f x xs
--------------------------------------------------------------------------------
-- Scans
--------------------------------------------------------------------------------
scanl : (b -> a -> b) -> b -> Vect n a -> Vect (S n) b
scanl f q [] = [q]
scanl f q (x::xs) = q :: scanl f (f q x) xs
--------------------------------------------------------------------------------
-- Membership tests
--------------------------------------------------------------------------------
||| Search for an item using a user-provided test
||| @ p the equality test
||| @ e the item to search for
||| @ xs the vector to search in
elemBy : (p : a -> a -> Bool) -> (e : a) -> (xs : Vect n a) -> Bool
elemBy p e [] = False
elemBy p e (x::xs) = p e x || elemBy p e xs
||| Use the default Boolean equality on elements to search for an item
||| @ x what to search for
||| @ xs where to search
elem : Eq a => (x : a) -> (xs : Vect n a) -> Bool
elem = elemBy (==)
||| Find the association of some key with a user-provided comparison
||| @ p the comparison operator for keys (True if they match)
||| @ e the key to look for
lookupBy : (p : a -> a -> Bool) -> (e : a) -> (xs : Vect n (a, b)) -> Maybe b
lookupBy p e [] = Nothing
lookupBy p e ((l, r)::xs) = if p e l then Just r else lookupBy p e xs
||| Find the assocation of some key using the default Boolean equality test
lookup : Eq a => a -> Vect n (a, b) -> Maybe b
lookup = lookupBy (==)
||| Check if any element of xs is found in elems by a user-provided comparison
||| @ p the comparison operator
||| @ elems the vector to search
||| @ xs what to search for
hasAnyBy : (p : a -> a -> Bool) -> (elems : Vect m a) -> (xs : Vect n a) -> Bool
hasAnyBy p elems [] = False
hasAnyBy p elems (x::xs) = elemBy p x elems || hasAnyBy p elems xs
||| Check if any element of xs is found in elems using the default Boolean equality test
hasAny : Eq a => Vect m a -> Vect n a -> Bool
hasAny = hasAnyBy (==)
--------------------------------------------------------------------------------
-- Searching with a predicate
--------------------------------------------------------------------------------
||| Find the first element of the vector that satisfies some test
||| @ p the test to satisfy
find : (p : a -> Bool) -> (xs : Vect n a) -> Maybe a
find p [] = Nothing
find p (x::xs) = if p x then Just x else find p xs
||| Find the index of the first element of the vector that satisfies some test
findIndex : (a -> Bool) -> Vect n a -> Maybe (Fin n)
findIndex p [] = Nothing
findIndex p (x :: xs) = if p x then Just 0 else map FS (findIndex p xs)
||| Find the indices of all elements that satisfy some test
findIndices : (a -> Bool) -> Vect m a -> List (Fin m)
findIndices p [] = []
findIndices p (x :: xs) = let is = map FS $ findIndices p xs
in if p x then 0 :: is else is
elemIndexBy : (a -> a -> Bool) -> a -> Vect m a -> Maybe (Fin m)
elemIndexBy p e = findIndex $ p e
elemIndex : Eq a => a -> Vect m a -> Maybe (Fin m)
elemIndex = elemIndexBy (==)
elemIndicesBy : (a -> a -> Bool) -> a -> Vect m a -> List (Fin m)
elemIndicesBy p e = findIndices $ p e
elemIndices : Eq a => a -> Vect m a -> List (Fin m)
elemIndices = elemIndicesBy (==)
--------------------------------------------------------------------------------
-- Filters
--------------------------------------------------------------------------------
||| Find all elements of a vector that satisfy some test
filter : (a -> Bool) -> Vect n a -> (p ** Vect p a)
filter p [] = ( _ ** [] )
filter p (x::xs) =
let (_ ** tail) = filter p xs
in if p x then
(_ ** x::tail)
else
(_ ** tail)
||| Make the elements of some vector unique by some test
nubBy : (a -> a -> Bool) -> Vect n a -> (p ** Vect p a)
nubBy = nubBy' []
where
nubBy' : Vect m a -> (a -> a -> Bool) -> Vect n a -> (p ** Vect p a)
nubBy' acc p [] = (_ ** [])
nubBy' acc p (x::xs) with (elemBy p x acc)
| True = nubBy' acc p xs
| False with (nubBy' (x::acc) p xs)
| (_ ** tail) = (_ ** x::tail)
||| Make the elements of some vector unique by the default Boolean equality
nub : Eq a => Vect n a -> (p ** Vect p a)
nub = nubBy (==)
deleteBy : (a -> a -> Bool) -> a -> Vect n a -> (p ** Vect p a)
deleteBy _ _ [] = (_ ** [])
deleteBy eq x (y::ys) =
let (len ** zs) = deleteBy eq x ys
in if x `eq` y then (_ ** ys) else (S len ** y ::zs)
delete : (Eq a) => a -> Vect n a -> (p ** Vect p a)
delete = deleteBy (==)
--------------------------------------------------------------------------------
-- Splitting and breaking lists
--------------------------------------------------------------------------------
||| A tuple where the first element is a Vect of the n first elements and
||| the second element is a Vect of the remaining elements of the original Vect
||| It is equivalent to (take n xs, drop n xs)
||| @ n the index to split at
||| @ xs the Vect to split in two
splitAt : (n : Nat) -> (xs : Vect (n + m) a) -> (Vect n a, Vect m a)
splitAt n xs = (take n xs, drop n xs)
partition : (a -> Bool) -> Vect n a -> ((p ** Vect p a), (q ** Vect q a))
partition p [] = ((_ ** []), (_ ** []))
partition p (x::xs) =
let ((leftLen ** lefts), (rightLen ** rights)) = partition p xs in
if p x then
((S leftLen ** x::lefts), (rightLen ** rights))
else
((leftLen ** lefts), (S rightLen ** x::rights))
--------------------------------------------------------------------------------
-- Predicates
--------------------------------------------------------------------------------
isPrefixOfBy : (a -> a -> Bool) -> Vect m a -> Vect n a -> Bool
isPrefixOfBy p [] right = True
isPrefixOfBy p left [] = False
isPrefixOfBy p (x::xs) (y::ys) = p x y && isPrefixOfBy p xs ys
isPrefixOf : Eq a => Vect m a -> Vect n a -> Bool
isPrefixOf = isPrefixOfBy (==)
isSuffixOfBy : (a -> a -> Bool) -> Vect m a -> Vect n a -> Bool
isSuffixOfBy p left right = isPrefixOfBy p (reverse left) (reverse right)
isSuffixOf : Eq a => Vect m a -> Vect n a -> Bool
isSuffixOf = isSuffixOfBy (==)
--------------------------------------------------------------------------------
-- Conversions
--------------------------------------------------------------------------------
maybeToVect : Maybe a -> (p ** Vect p a)
maybeToVect Nothing = (_ ** [])
maybeToVect (Just j) = (_ ** [j])
vectToMaybe : Vect n a -> Maybe a
vectToMaybe [] = Nothing
vectToMaybe (x::xs) = Just x
--------------------------------------------------------------------------------
-- Misc
--------------------------------------------------------------------------------
catMaybes : Vect n (Maybe a) -> (p ** Vect p a)
catMaybes [] = (_ ** [])
catMaybes (Nothing::xs) = catMaybes xs
catMaybes ((Just j)::xs) =
let (_ ** tail) = catMaybes xs
in (_ ** j::tail)
diag : Vect n (Vect n a) -> Vect n a
diag [] = []
diag ((x::xs)::xss) = x :: diag (map tail xss)
range : {n : Nat} -> Vect n (Fin n)
range {n=Z} = []
range {n=S _} = FZ :: map FS range
||| Transpose a Vect of Vects, turning rows into columns and vice versa.
|||
||| As the types ensure rectangularity, this is an involution, unlike `Prelude.List.transpose`.
transpose : {n : Nat} -> Vect m (Vect n a) -> Vect n (Vect m a)
transpose [] = replicate _ []
transpose (x :: xs) = zipWith (::) x (transpose xs)
--------------------------------------------------------------------------------
-- Applicative/Monad/Traversable
--------------------------------------------------------------------------------
implementation Applicative (Vect k) where
pure = replicate _
fs <*> vs = zipWith apply fs vs
||| This monad is different from the List monad, (>>=)
||| uses the diagonal.
implementation Monad (Vect n) where
m >>= f = diag (map f m)
implementation Traversable (Vect n) where
traverse f [] = pure Vect.Nil
traverse f (x::xs) = [| Vect.(::) (f x) (traverse f xs) |]
--------------------------------------------------------------------------------
-- Show
--------------------------------------------------------------------------------
implementation Show a => Show (Vect n a) where
show = show . toList
--------------------------------------------------------------------------------
-- Properties
--------------------------------------------------------------------------------
vectConsCong : (x : a) -> (xs : Vect n a) -> (ys : Vect m a) -> (xs = ys) -> (x :: xs = x :: ys)
vectConsCong x xs xs Refl = Refl
vectNilRightNeutral : (xs : Vect n a) -> xs ++ [] = xs
vectNilRightNeutral [] = Refl
vectNilRightNeutral (x :: xs) =
vectConsCong _ _ _ (vectNilRightNeutral xs)
vectAppendAssociative : (x : Vect xLen a) -> (y : Vect yLen a) -> (z : Vect zLen a) -> x ++ (y ++ z) = (x ++ y) ++ z
vectAppendAssociative [] y z = Refl
vectAppendAssociative (x :: xs) ys zs =
vectConsCong _ _ _ (vectAppendAssociative xs ys zs)
--------------------------------------------------------------------------------
-- DecEq
--------------------------------------------------------------------------------
vectInjective1 : {xs, ys : Vect n a} -> {x, y : a} -> x :: xs = y :: ys -> x = y
vectInjective1 {x=x} {y=x} {xs=xs} {ys=xs} Refl = Refl
vectInjective2 : {xs, ys : Vect n a} -> {x, y : a} -> x :: xs = y :: ys -> xs = ys
vectInjective2 {x=x} {y=x} {xs=xs} {ys=xs} Refl = Refl
implementation DecEq a => DecEq (Vect n a) where
decEq [] [] = Yes Refl
decEq (x :: xs) (y :: ys) with (decEq x y)
decEq (x :: xs) (x :: ys) | Yes Refl with (decEq xs ys)
decEq (x :: xs) (x :: xs) | Yes Refl | Yes Refl = Yes Refl
decEq (x :: xs) (x :: ys) | Yes Refl | No neq = No (neq . vectInjective2)
decEq (x :: xs) (y :: ys) | No neq = No (neq . vectInjective1)
{- The following definition is elaborated in a wrong case-tree. Examination pending.
implementation DecEq a => DecEq (Vect n a) where
decEq [] [] = Yes Refl
decEq (x :: xs) (y :: ys) with (decEq x y, decEq xs ys)
decEq (x :: xs) (x :: xs) | (Yes Refl, Yes Refl) = Yes Refl
decEq (x :: xs) (y :: ys) | (_, No nEqTl) = No (\p => nEqTl (vectInjective2 p))
decEq (x :: xs) (y :: ys) | (No nEqHd, _) = No (\p => nEqHd (vectInjective1 p))
-}
--------------------------------------------------------------------------------
-- Elem
--------------------------------------------------------------------------------
||| A proof that some element is found in a vector
data Elem : a -> Vect k a -> Type where
Here : Elem x (x::xs)
There : Elem x xs -> Elem x (y::xs)
||| Nothing can be in an empty Vect
noEmptyElem : {x : a} -> Elem x [] -> Void
noEmptyElem Here impossible
||| An item not in the head and not in the tail is not in the Vect at all
neitherHereNorThere : {x, y : a} -> {xs : Vect n a} -> Not (x = y) -> Not (Elem x xs) -> Not (Elem x (y :: xs))
neitherHereNorThere xneqy xninxs Here = xneqy Refl
neitherHereNorThere xneqy xninxs (There xinxs) = xninxs xinxs
||| A decision procedure for Elem
isElem : DecEq a => (x : a) -> (xs : Vect n a) -> Dec (Elem x xs)
isElem x [] = No noEmptyElem
isElem x (y :: xs) with (decEq x y)
isElem x (x :: xs) | (Yes Refl) = Yes Here
isElem x (y :: xs) | (No xneqy) with (isElem x xs)
isElem x (y :: xs) | (No xneqy) | (Yes xinxs) = Yes (There xinxs)
isElem x (y :: xs) | (No xneqy) | (No xninxs) = No (neitherHereNorThere xneqy xninxs)
replaceElem : (xs : Vect k t) -> Elem x xs -> (y : t) -> (ys : Vect k t ** Elem y ys)
replaceElem (x::xs) Here y = (y :: xs ** Here)
replaceElem (x::xs) (There xinxs) y with (replaceElem xs xinxs y)
| (ys ** yinys) = (x :: ys ** There yinys)
replaceByElem : (xs : Vect k t) -> Elem x xs -> t -> Vect k t
replaceByElem (x::xs) Here y = y :: xs
replaceByElem (x::xs) (There xinxs) y = x :: replaceByElem xs xinxs y
mapElem : {xs : Vect k t} -> {f : t -> u} -> Elem x xs -> Elem (f x) (map f xs)
mapElem Here = Here
mapElem (There e) = There (mapElem e)
-- Some convenience functions for testing lengths
||| If the given Vect is the required length, return a Vect with that
||| length in its type, otherwise return Nothing
||| @len the required length
||| @xs the vector with the desired length
-- Needs to be Maybe rather than Dec, because if 'n' is unequal to m, we
-- only know we don't know how to make a Vect n a, not that one can't exist.
exactLength : {m : Nat} -> -- expected at run-time
(len : Nat) -> (xs : Vect m a) -> Maybe (Vect len a)
exactLength {m} len xs with (decEq m len)
exactLength {m = m} m xs | (Yes Refl) = Just xs
exactLength {m = m} len xs | (No contra) = Nothing
||| If the given Vect is at least the required length, return a Vect with
||| at least that length in its type, otherwise return Nothing
||| @len the required length
||| @xs the vector with the desired length
overLength : {m : Nat} -> -- expected at run-time
(len : Nat) -> (xs : Vect m a) -> Maybe (p ** Vect (plus p len) a)
overLength {m} n xs with (cmp m n)
overLength {m = m} (plus m (S y)) xs | (CmpLT y) = Nothing
overLength {m = m} m xs | CmpEQ
= Just (0 ** xs)
overLength {m = plus n (S x)} n xs | (CmpGT x)
= Just (S x ** rewrite plusCommutative (S x) n in xs)