# Data.Vect

(++) : (xs : Vect m elem) -> (ys : Vect n elem) -> Vect (m + n) elem

  Append two vectors

  Fixity
    Left associative, precedence 7
data Elem : a -> Vect k a -> Type

  A proof that some element is found in a vector

  Here : Elem x (x :: xs)
  There : (later : Elem x xs) -> Elem x (y :: xs)
data Vect : (len : Nat) -> (elem : Type) -> Type

  Vectors: Generic lists with explicit length in the type

  len

    the length of the list

  elem

    the type of elements

  Nil : Vect 0 elem

    Empty vector

  (::) : (x : elem) -> (xs : Vect len elem) -> Vect (S len) elem

    A non-empty vector of length `S len`, consisting of a head element and
    the rest of the list, of length `len`.

    Fixity
      Left associative, precedence 7
catMaybes : Vect n (Maybe elem) -> (p : Nat ** Vect p elem)
concat : (xss : Vect m (Vect n elem)) -> Vect (m * n) elem

  Flatten a vector of equal-length vectors

delete : Eq elem => elem -> Vect len elem -> (p : Nat ** Vect p elem)
deleteAt : Fin (S len) -> Vect (S len) elem -> Vect len elem

  Construct a new vector consisting of all but the indicated element

deleteBy : (elem -> elem -> Bool) -> elem -> Vect len elem -> (p : Nat ** Vect p elem)
diag : Vect len (Vect len elem) -> Vect len elem
drop : (n : Nat) -> Vect (n + m) elem -> Vect m elem

  Remove the first n elements of a Vect

  n

    the number of elements to remove

dropDrop : (n : Nat) -> (m : Nat) -> (xs : Vect (n + m + l) a) -> (ys : Vect (n + (m + l)) a) -> xs ~=~ ys -> drop (n + m) xs = drop m (drop n ys)

  `drop (n + m) = drop m . drop n`

dropElem : (xs : Vect (S k) t) -> (p : Elem x xs) -> Vect k t

  Remove the element at the given position.

  xs

    The vector to be removed from

  p

    A proof that the element to be removed is in the vector

dropPrefix : (ns : Vect n a) -> (ms : Vect m a) -> drop n (ns ++ ms) = ms

  Adding a prefix and then dropping the prefix does nothing. Or,
  adding a suffix and then taking the suffix gets the suffix.

dropTakeTakeDrop : (n : Nat) -> (m : Nat) -> (xs : Vect (n + m + l) a) -> (ys : Vect (n + (m + l)) a) -> xs ~=~ ys -> drop n (take (n + m) xs) = take m (drop n ys)

  `drop n . take (n + m) = take m . drop n`.

  Or: there are two ways to extract a subsequence.

dropWhile : (p : elem -> Bool) -> Vect len elem -> (q : Nat ** Vect q elem)

  Remove the longest prefix of a Vect such that all removed elements satisfy some
  Boolean predicate.

  p

    the predicate

elem : Eq elem => (x : elem) -> (xs : Vect len elem) -> Bool

  Use the default Boolean equality on elements to search for an item

  x

    what to search for

  xs

    where to search

elemBy : (p : elem -> elem -> Bool) -> (e : elem) -> (xs : Vect len elem) -> Bool

  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

elemIndex : Eq elem => elem -> Vect m elem -> Maybe (Fin m)
elemIndexBy : (elem -> elem -> Bool) -> elem -> Vect m elem -> Maybe (Fin m)
elemIndices : Eq elem => elem -> Vect m elem -> List (Fin m)
elemIndicesBy : (elem -> elem -> Bool) -> elem -> Vect m elem -> List (Fin m)
exactLength : (len : Nat) -> (xs : Vect m a) -> Maybe (Vect len a)

  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

filter : (elem -> Bool) -> Vect len elem -> (p : Nat ** Vect p elem)

  Find all elements of a vector that satisfy some test

find : (p : elem -> Bool) -> (xs : Vect len elem) -> Maybe elem

  Find the first element of the vector that satisfies some test

  p

    the test to satisfy

findIndex : (elem -> Bool) -> Vect len elem -> Maybe (Fin len)

  Find the index of the first element of the vector that satisfies some test

findIndices : (elem -> Bool) -> Vect m elem -> List (Fin m)

  Find the indices of all elements that satisfy some test

foldl1 : (t -> t -> t) -> Vect (S n) t -> t

  Foldl without seeding the accumulator

foldr1 : (t -> t -> t) -> Vect (S n) t -> t

  Foldr without seeding the accumulator

foldrImpl : (t -> acc -> acc) -> acc -> (acc -> acc) -> Vect n t -> acc
fromList : (l : List elem) -> Vect (length l) elem

  Convert a list to a vector.

  The length of the list should be statically known.

fromList' : Vect len elem -> (l : List elem) -> Vect (length l + len) elem
hasAny : Eq elem => Vect m elem -> Vect len elem -> Bool

  Check if any element of xs is found in elems using the default Boolean equality test

hasAnyBy : (p : elem -> elem -> Bool) -> (elems : Vect m elem) -> (xs : Vect len elem) -> Bool

  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

head : Vect (S len) elem -> elem

  Only the first element of the vector

index : Fin len -> Vect len elem -> elem

  Extract a particular element from a vector

indexReplicate : (x : a) -> (n : Nat) -> (i : Fin n) -> index i (replicate n x) = x
indexTranspose : (x : Fin o) -> (y : Fin i) -> (xss : Vect o (Vect i a)) -> index y (index x xss) = index x (index y (transpose xss))
indexZipWith : (f : a -> b -> c) -> (as : Vect n a) -> (bs : Vect n b) -> (i : Fin n) -> index i (zipWith f as bs) = f (index i as) (index i bs)
init : Vect (S len) elem -> Vect len elem

  All but the last element of the vector

insertAt : Fin (S len) -> elem -> Vect len elem -> Vect (S len) elem

  Insert an element at a particular index

intersperse : (sep : elem) -> (xs : Vect len elem) -> Vect (len + pred len) elem

  Alternate an element between the other elements of a vector

  sep

    the element to intersperse

  xs

    the vector to separate with `sep`

isElem : DecEq a => (x : a) -> (xs : Vect n a) -> Dec (Elem x xs)

  A decision procedure for Elem

isPrefixOf : Eq elem => Vect m elem -> Vect len elem -> Bool
isPrefixOfBy : (elem -> elem -> Bool) -> Vect m elem -> Vect len elem -> Bool
isSuffixOf : Eq elem => Vect m elem -> Vect len elem -> Bool
isSuffixOfBy : (elem -> elem -> Bool) -> Vect m elem -> Vect len elem -> Bool
last : Vect (S len) elem -> elem

  The last element of the vector

length : (xs : Vect len elem) -> Nat

  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.

  len

    the length (provably equal to the return value)

  xs

    the vector

lookup : Eq key => key -> Vect n (key, val) -> Maybe val

  Find the assocation of some key using the default Boolean equality test

lookupBy : (p : key -> key -> Bool) -> (e : key) -> (xs : Vect n (key, val)) -> Maybe val

  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

mapElem : Elem x xs -> Elem (f x) (map f xs)
mapMaybe : (f : a -> Maybe b) -> (xs : Vect len a) -> (m : Nat ** Vect m b)

  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

maybeToVect : Maybe elem -> (p : Nat ** Vect p elem)
merge : Ord elem => Vect n elem -> Vect m elem -> Vect (n + m) elem
mergeBy : (elem -> elem -> Ordering) -> (xs : Vect n elem) -> (ys : Vect m elem) -> Vect (n + m) elem

  Merge two ordered vectors

neitherHereNorThere : Not (x = y) -> Not (Elem x xs) -> Not (Elem x (y :: xs))

  An item not in the head and not in the tail is not in the Vect at all

noEmptyElem : Elem x [] -> Void

  Nothing can be in an empty Vect

nub : Eq elem => Vect len elem -> (p : Nat ** Vect p elem)

  Make the elements of some vector unique by the default Boolean equality

nubBy : (elem -> elem -> Bool) -> Vect len elem -> (p : Nat ** Vect p elem)

  Make the elements of some vector unique by some test

overLength : (len : Nat) -> (xs : Vect m a) -> Maybe (p : Nat ** Vect (plus p len) a)

  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

partition : (elem -> Bool) -> Vect len elem -> ((p : Nat ** Vect p elem), (q : Nat ** Vect q elem))
range : Vect len (Fin len)
replaceAt : Fin len -> elem -> Vect len elem -> Vect len elem

  Replace an element at a particlar index with another

replaceByElem : (xs : Vect k t) -> Elem x xs -> t -> Vect k t
replaceElem : (xs : Vect k t) -> Elem x xs -> (y : t) -> (ys : Vect k t ** Elem y ys)
replicate : (len : Nat) -> (x : elem) -> Vect len elem

  Repeate some value some number of times.

  len

    the number of times to repeat it

  x

    the value to repeat

reverse : Vect len elem -> Vect len elem

  Reverse the order of the elements of a vector

scanl : (res -> elem -> res) -> res -> Vect len elem -> Vect (S len) res

  The scanl function is similar to foldl, but returns all the intermediate
  accumulator states in the form of a vector.

scanl1 : (elem -> elem -> elem) -> Vect len elem -> Vect len elem

  The scanl1 function is a variant of scanl that doesn't require an explicit
  starting value.
  It assumes the first element of the vector to be the starting value and then
  starts the fold with the element following it.

splitAt : (n : Nat) -> (xs : Vect (n + m) elem) -> (Vect n elem, Vect m elem)

  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.
  It is equivalent to `(take n xs, drop n xs)` (`splitAtTakeDrop`),
  but is more efficient.

  n

    the index to split at

  xs

    the `Vect` to split in two

splitAtTakeDrop : (n : Nat) -> (xs : Vect (n + m) a) -> splitAt n xs = (take n xs, drop n xs)
tail : Vect (S len) elem -> Vect len elem

  All but the first element of the vector

take : (n : Nat) -> Vect (n + m) elem -> Vect n elem

  Get the first n elements of a Vect

  n

    the number of elements to take

takeDropConcat : (n : Nat) -> (xs : Vect (n + m) a) -> take n xs ++ drop n xs = xs

  A `Vect` may be restored from its components.

takePrefix : (ns : Vect n a) -> (ms : Vect m a) -> take n (ns ++ ms) = ns

  Adding a prefix and then taking the prefix gets the prefix. Or,
  adding a suffix and then dropping the suffix does nothing.

takeTake : (n : Nat) -> (m : Nat) -> (xs : Vect (n + m + l) a) -> (ys : Vect (n + (m + l)) a) -> xs ~=~ ys -> take n (take (n + m) xs) = take n ys

  `take n . take (n + m) = take n`

takeWhile : (p : elem -> Bool) -> Vect len elem -> (q : Nat ** Vect q elem)

  Take the longest prefix of a Vect such that all elements satisfy some
  Boolean predicate.

  p

    the predicate

transpose : Vect m (Vect n elem) -> Vect n (Vect m elem)

  Transpose a `Vect` of `Vect`s, turning rows into columns and vice versa.

  This is like zipping all the inner `Vect`s together and is equivalent to `traverse id` (`transposeTraverse`).

  As the types ensure rectangularity, this is an involution, unlike `Prelude.List.transpose`.

transposeCons : (xs : Vect o a) -> (xss : Vect o (Vect i a)) -> transpose (zipWith (::) xs xss) = xs :: transpose xss
transposeTraverse : (xss : Vect o (Vect i a)) -> transpose xss = traverse id xss
traverseIdCons : (xs : Vect o a) -> (xss : Vect o (Vect i a)) -> traverse id (pure (::) <*> xs <*> xss) = xs :: traverse id xss
unzip : (xs : Vect n (a, b)) -> (Vect n a, Vect n b)

  Convert a vector of pairs to a pair of vectors

unzip3 : (xs : Vect n (a, b, c)) -> (Vect n a, Vect n b, Vect n c)

  Convert a vector of three-tuples to a triplet of vectors

updateAt : (i : Fin len) -> (f : elem -> elem) -> (xs : Vect len elem) -> Vect len elem

  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

vectAppendAssociative : (xs : Vect xLen elem) -> (ys : Vect yLen elem) -> (zs : Vect zLen elem) -> xs ++ ys ++ zs = (xs ++ ys) ++ zs
vectConsCong : (x : elem) -> (xs : Vect len elem) -> (ys : Vect m elem) -> (xs = ys) -> x :: xs = x :: ys
vectInjective1 : x :: xs ~=~ y :: ys -> x ~=~ y
vectInjective2 : x :: xs ~=~ y :: ys -> xs ~=~ ys
vectMustBeNil : (xs : Vect 0 a) -> xs = []
vectNilRightNeutral : (xs : Vect n a) -> xs ++ [] = xs
vectToMaybe : Vect len elem -> Maybe elem
zip : (xs : Vect n a) -> (ys : Vect n b) -> Vect n (a, b)

  Combine two equal-length vectors pairwise

zip3 : (xs : Vect n a) -> (ys : Vect n b) -> (zs : Vect n c) -> Vect n (a, b, c)

  Combine three equal-length vectors elementwise into a vector of tuples

zipWith : (f : a -> b -> c) -> (xs : Vect n a) -> (ys : Vect n b) -> Vect n c

  Combine two equal-length vectors pairwise with some function.

  f

    the function to combine elements with

  xs

    the first vector of elements

  ys

    the second vector of elements

zipWith3 : (a -> b -> c -> d) -> (xs : Vect n a) -> (ys : Vect n b) -> (zs : Vect n c) -> Vect n d

  Combine three equal-length vectors into a vector with some function

zipWithIsLiftA2 : (f : a -> b -> c) -> (as : Vect n a) -> (bs : Vect n b) -> zipWith f as bs = pure f <*> as <*> bs
zipWithIsLiftA3 : (f : a -> b -> c -> d) -> (as : Vect n a) -> (bs : Vect n b) -> (cs : Vect n c) -> zipWith3 f as bs cs = pure f <*> as <*> bs <*> cs