Fixed size vectors.

• Lists with their length encoded in their type
• Vector elements have an ASCENDING subscript starting from 0 and ending at length - 1.

The length of a Vector as an Int value.

>>> length (6 :> 7 :> 8 :> Nil)
3

Length of a Vector as an SNat value

"xs !! n" returns the n'th element of xs.

NB: vector elements have an ASCENDING subscript starting from 0 and ending at length - 1.

>>> (1:>2:>3:>4:>5:>Nil) !! 4
5
>>> (1:>2:>3:>4:>5:>Nil) !! (length (1:>2:>3:>4:>5:>Nil) - 1)
5
>>> (1:>2:>3:>4:>5:>Nil) !! 1
2
>>> (1:>2:>3:>4:>5:>Nil) !! 14
*** Exception: Clash.Sized.Vector.(!!): index 14 is larger than maximum index 4
...

Extract the first element of a vector

>>> head (1:>2:>3:>Nil)
1
>>> head Nil

<interactive>:...
    • Couldn't match type ‘1’ with ‘0’
      Expected type: Vec (0 + 1) a
        Actual type: Vec 0 a
    • In the first argument of ‘head’, namely ‘Nil’
      In the expression: head Nil
      In an equation for ‘it’: it = head Nil

Extract the last element of a vector

>>> last (1:>2:>3:>Nil)
3
>>> last Nil

<interactive>:...
    • Couldn't match type ‘1’ with ‘0’
      Expected type: Vec (0 + 1) a
        Actual type: Vec 0 a
    • In the first argument of ‘last’, namely ‘Nil’
      In the expression: last Nil
      In an equation for ‘it’: it = last Nil

"at n xs" returns n'th element of xs

NB: vector elements have an ASCENDING subscript starting from 0 and ending at length - 1.

>>> at (SNat :: SNat 1) (1:>2:>3:>4:>5:>Nil)
2
>>> at d1               (1:>2:>3:>4:>5:>Nil)
2

Generate a vector of indices.

>>> indices d4
<0,1,2,3>

Generate a vector of indices, where the length of the vector is determined by the context.

>>> indicesI :: Vec 4 (Index 4)
<0,1,2,3>

"findIndex p xs" returns the index of the first element of xs satisfying the predicate p, or Nothing if there is no such element.

>>> findIndex (> 3) (1:>3:>2:>4:>3:>5:>6:>Nil)
Just 3
>>> findIndex (> 8) (1:>3:>2:>4:>3:>5:>6:>Nil)
Nothing

"elemIndex a xs" returns the index of the first element which is equal (by ==) to the query element a, or Nothing if there is no such element.

>>> elemIndex 3 (1:>3:>2:>4:>3:>5:>6:>Nil)
Just 1
>>> elemIndex 8 (1:>3:>2:>4:>3:>5:>6:>Nil)
Nothing

Extract the elements after the head of a vector

>>> tail (1:>2:>3:>Nil)
<2,3>
>>> tail Nil

<interactive>:...
    • Couldn't match type ‘1’ with ‘0’
      Expected type: Vec (0 + 1) a
        Actual type: Vec 0 a
    • In the first argument of ‘tail’, namely ‘Nil’
      In the expression: tail Nil
      In an equation for ‘it’: it = tail Nil

Extract all the elements of a vector except the last element

>>> init (1:>2:>3:>Nil)
<1,2>
>>> init Nil

<interactive>:...
    • Couldn't match type ‘1’ with ‘0’
      Expected type: Vec (0 + 1) a
        Actual type: Vec 0 a
    • In the first argument of ‘init’, namely ‘Nil’
      In the expression: init Nil
      In an equation for ‘it’: it = init Nil

"take n xs" returns the n-length prefix of xs.

>>> take (SNat :: SNat 3) (1:>2:>3:>4:>5:>Nil)
<1,2,3>
>>> take d3               (1:>2:>3:>4:>5:>Nil)
<1,2,3>
>>> take d0               (1:>2:>Nil)
<>
>>> take d4               (1:>2:>Nil)

<interactive>:...
    • Couldn't match type ‘4 + n0’ with ‘2’
      Expected type: Vec (4 + n0) a
        Actual type: Vec (1 + 1) a
      The type variable ‘n0’ is ambiguous
    • In the second argument of ‘take’, namely ‘(1 :> 2 :> Nil)’
      In the expression: take d4 (1 :> 2 :> Nil)
      In an equation for ‘it’: it = take d4 (1 :> 2 :> Nil)

"takeI xs" returns the prefix of xs as demanded by the context.

>>> takeI (1:>2:>3:>4:>5:>Nil) :: Vec 2 Int
<1,2>

"drop n xs" returns the suffix of xs after the first n elements.

>>> drop (SNat :: SNat 3) (1:>2:>3:>4:>5:>Nil)
<4,5>
>>> drop d3               (1:>2:>3:>4:>5:>Nil)
<4,5>
>>> drop d0               (1:>2:>Nil)
<1,2>
>>> drop d4               (1:>2:>Nil)

<interactive>:...
    • Couldn't match expected type ‘2’ with actual type ‘4 + n0’
      The type variable ‘n0’ is ambiguous
    • In the first argument of ‘print’, namely ‘it’
      In a stmt of an interactive GHCi command: print it

"dropI xs" returns the suffix of xs as demanded by the context.

>>> dropI (1:>2:>3:>4:>5:>Nil) :: Vec 2 Int
<4,5>

"select f s n xs" selects n elements with step-size s and offset f from xs.

>>> select (SNat :: SNat 1) (SNat :: SNat 2) (SNat :: SNat 3) (1:>2:>3:>4:>5:>6:>7:>8:>Nil)
<2,4,6>
>>> select d1 d2 d3 (1:>2:>3:>4:>5:>6:>7:>8:>Nil)
<2,4,6>

"selectI f s xs" selects as many elements as demanded by the context with step-size s and offset f from xs.

>>> selectI d1 d2 (1:>2:>3:>4:>5:>6:>7:>8:>Nil) :: Vec 2 Int
<2,4>

Split a vector into two vectors at the given point.

>>> splitAt (SNat :: SNat 3) (1:>2:>3:>7:>8:>Nil)
(<1,2,3>,<7,8>)
>>> splitAt d3 (1:>2:>3:>7:>8:>Nil)
(<1,2,3>,<7,8>)

Split a vector into two vectors where the length of the two is determined by the context.

>>> splitAtI (1:>2:>3:>7:>8:>Nil) :: (Vec 2 Int, Vec 3 Int)
(<1,2>,<3,7,8>)

Split a vector of (n * m) elements into a vector of "vectors of length m", where the length m is given.

>>> unconcat d4 (1:>2:>3:>4:>5:>6:>7:>8:>9:>10:>11:>12:>Nil)
<<1,2,3,4>,<5,6,7,8>,<9,10,11,12>>

Split a vector of (n * m) elements into a vector of "vectors of length m", where the length m is determined by the context.

>>> unconcatI (1:>2:>3:>4:>5:>6:>7:>8:>9:>10:>11:>12:>Nil) :: Vec 2 (Vec 6 Int)
<<1,2,3,4,5,6>,<7,8,9,10,11,12>>

Create a vector of one element

>>> singleton 5
<5>

"replicate n a" returns a vector that has n copies of a.

>>> replicate (SNat :: SNat 3) 6
<6,6,6>
>>> replicate d3 6
<6,6,6>

"repeat a" creates a vector with as many copies of a as demanded by the context.

>>> repeat 6 :: Vec 5 Int
<6,6,6,6,6>

"iterate n f x" returns a vector starting with x followed by n repeated applications of f to x.

iterate (SNat :: SNat 4) f x == (x :> f x :> f (f x) :> f (f (f x)) :> Nil)
iterate d4 f x               == (x :> f x :> f (f x) :> f (f (f x)) :> Nil)

>>> iterate d4 (+1) 1
<1,2,3,4>

"interate n f z" corresponds to the following circuit layout:

"iterate f x" returns a vector starting with x followed by n repeated applications of f to x, where n is determined by the context.

iterateI f x :: Vec 3 a == (x :> f x :> f (f x) :> Nil)

>>> iterateI (+1) 1 :: Vec 3 Int
<1,2,3>

"interateI f z" corresponds to the following circuit layout:

"generate n f x" returns a vector with n repeated applications of f to x.

generate (SNat :: SNat 4) f x == (f x :> f (f x) :> f (f (f x)) :> f (f (f (f x))) :> Nil)
generate d4 f x               == (f x :> f (f x) :> f (f (f x)) :> f (f (f (f x))) :> Nil)

>>> generate d4 (+1) 1
<2,3,4,5>

"generate n f z" corresponds to the following circuit layout:

"generateI f x" returns a vector with n repeated applications of f to x, where n is determined by the context.

generateI f x :: Vec 3 a == (f x :> f (f x) :> f (f (f x)) :> Nil)

>>> generateI (+1) 1 :: Vec 3 Int
<2,3,4>

"generateI f z" corresponds to the following circuit layout:

Create a vector literal from a list literal.

$(listToVecTH [1::Signed 8,2,3,4,5]) == (8:>2:>3:>4:>5:>Nil) :: Vec 5 (Signed 8)

>>> [1 :: Signed 8,2,3,4,5]
[1,2,3,4,5]
>>> $(listToVecTH [1::Signed 8,2,3,4,5])
<1,2,3,4,5>

Append two vectors.

>>> (1:>2:>3:>Nil) ++ (7:>8:>Nil)
<1,2,3,7,8>

Add an element to the head of a vector, and extract all but the last element.

>>> 1 +>> (3:>4:>5:>Nil)
<1,3,4>
>>> 1 +>> Nil
<>

Add an element to the tail of a vector, and extract all but the first element.

>>> (3:>4:>5:>Nil) <<+ 1
<4,5,1>
>>> Nil <<+ 1
<>

Concatenate a vector of vectors.

>>> concat ((1:>2:>3:>Nil) :> (4:>5:>6:>Nil) :> (7:>8:>9:>Nil) :> (10:>11:>12:>Nil) :> Nil)
<1,2,3,4,5,6,7,8,9,10,11,12>

Shift in elements to the head of a vector, bumping out elements at the tail. The result is a tuple containing:

• The new vector
• The shifted out elements

>>> shiftInAt0 (1 :> 2 :> 3 :> 4 :> Nil) ((-1) :> 0 :> Nil)
(<-1,0,1,2>,<3,4>)
>>> shiftInAt0 (1 :> Nil) ((-1) :> 0 :> Nil)
(<-1>,<0,1>)

Shift in element to the tail of a vector, bumping out elements at the head. The result is a tuple containing:

• The new vector
• The shifted out elements

>>> shiftInAtN (1 :> 2 :> 3 :> 4 :> Nil) (5 :> 6 :> Nil)
(<3,4,5,6>,<1,2>)
>>> shiftInAtN (1 :> Nil) (2 :> 3 :> Nil)
(<3>,<1,2>)

Shift m elements out from the head of a vector, filling up the tail with Default values. The result is a tuple containing:

• The new vector
• The shifted out values

>>> shiftOutFrom0 d2 ((1 :> 2 :> 3 :> 4 :> 5 :> Nil) :: Vec 5 Integer)
(<3,4,5,0,0>,<1,2>)

Shift m elements out from the tail of a vector, filling up the head with Default values. The result is a tuple containing:

• The new vector
• The shifted out values

>>> shiftOutFromN d2 ((1 :> 2 :> 3 :> 4 :> 5 :> Nil) :: Vec 5 Integer)
(<0,0,1,2,3>,<4,5>)

Merge two vectors, alternating their elements, i.e.,

>>> merge (1 :> 2 :> 3 :> 4 :> Nil) (5 :> 6 :> 7 :> 8 :> Nil)
<1,5,2,6,3,7,4,8>

"replace n a xs" returns the vector xs where the n'th element is replaced by a.

NB: vector elements have an ASCENDING subscript starting from 0 and ending at length - 1.

>>> replace 3 7 (1:>2:>3:>4:>5:>Nil)
<1,2,3,7,5>
>>> replace 0 7 (1:>2:>3:>4:>5:>Nil)
<7,2,3,4,5>
>>> replace 9 7 (1:>2:>3:>4:>5:>Nil)
<1,2,3,4,*** Exception: Clash.Sized.Vector.replace: index 9 is larger than maximum index 4
...

Forward permutation specified by an index mapping, ix. The result vector is initialised by the given defaults, def, and an further values that are permuted into the result are added to the current value using the given combination function, f.

The combination function must be associative and commutative.

Backwards permutation specified by an index mapping, is, from the destination vector specifying which element of the source vector xs to read.

"backpermute xs is" is equivalent to "map (xs !!) is".

For example:

>>> let input = 1:>9:>6:>4:>4:>2:>0:>1:>2:>Nil
>>> let from  = 1:>3:>7:>2:>5:>3:>Nil
>>> backpermute input from
<9,4,1,6,2,4>

Copy elements from the source vector, xs, to the destination vector according to an index mapping is. This is a forward permute operation where a to vector encodes an input to output index mapping. Output elements for indices that are not mapped assume the value in the default vector def.

For example:

>>> let defVec = 0:>0:>0:>0:>0:>0:>0:>0:>0:>Nil
>>> let to = 1:>3:>7:>2:>5:>8:>Nil
>>> let input = 1:>9:>6:>4:>4:>2:>5:>Nil
>>> scatter defVec to input
<0,1,4,9,0,4,0,6,2>

NB: If the same index appears in the index mapping more than once, the latest mapping is chosen.

Backwards permutation specified by an index mapping, is, from the destination vector specifying which element of the source vector xs to read.

"gather xs is" is equivalent to "map (xs !!) is".

For example:

>>> let input = 1:>9:>6:>4:>4:>2:>0:>1:>2:>Nil
>>> let from  = 1:>3:>7:>2:>5:>3:>Nil
>>> gather input from
<9,4,1,6,2,4>

The elements in a vector in reverse order.

>>> reverse (1:>2:>3:>4:>Nil)
<4,3,2,1>

Transpose a matrix: go from row-major to column-major

>>> let xss = (1:>2:>Nil):>(3:>4:>Nil):>(5:>6:>Nil):>Nil
>>> xss
<<1,2>,<3,4>,<5,6>>
>>> transpose xss
<<1,3,5>,<2,4,6>>

"interleave d xs" creates a vector:

<x_0,x_d,x_(2d),...,x_1,x_(d+1),x_(2d+1),...,x_(d-1),x_(2d-1),x_(3d-1)>

>>> let xs = 1 :> 2 :> 3 :> 4 :> 5 :> 6 :> 7 :> 8 :> 9 :> Nil
>>> interleave d3 xs
<1,4,7,2,5,8,3,6,9>

Dynamically rotate a Vector to the left:

>>> let xs = 1 :> 2 :> 3 :> 4 :> Nil
>>> rotateLeft xs 1
<2,3,4,1>
>>> rotateLeft xs 2
<3,4,1,2>
>>> rotateLeft xs (-1)
<4,1,2,3>

NB: use rotateLeftS if you want to rotate left by a static amount.

Dynamically rotate a Vector to the right:

>>> let xs = 1 :> 2 :> 3 :> 4 :> Nil
>>> rotateRight xs 1
<4,1,2,3>
>>> rotateRight xs 2
<3,4,1,2>
>>> rotateRight xs (-1)
<2,3,4,1>

NB: use rotateRightS if you want to rotate right by a static amount.

Statically rotate a Vector to the left:

>>> let xs = 1 :> 2 :> 3 :> 4 :> Nil
>>> rotateLeftS xs d1
<2,3,4,1>

NB: use rotateLeft if you want to rotate left by a dynamic amount.

Statically rotate a Vector to the right:

>>> let xs = 1 :> 2 :> 3 :> 4 :> Nil
>>> rotateRightS xs d1
<4,1,2,3>

NB: use rotateRight if you want to rotate right by a dynamic amount.

"map f xs" is the vector obtained by applying f to each element of xs, i.e.,

map f (x1 :> x2 :>  ... :> xn :> Nil) == (f x1 :> f x2 :> ... :> f xn :> Nil)

and corresponds to the following circuit layout:

Apply a function of every element of a vector and its index.

>>> :t imap (+) (2 :> 2 :> 2 :> 2 :> Nil)
imap (+) (2 :> 2 :> 2 :> 2 :> Nil) :: Vec 4 (Index 4)
>>> imap (+) (2 :> 2 :> 2 :> 2 :> Nil)
<2,3,*** Exception: Clash.Sized.Index: result 4 is out of bounds: [0..3]
...
>>> imap (\i a -> fromIntegral i + a) (2 :> 2 :> 2 :> 2 :> Nil) :: Vec 4 (Unsigned 8)
<2,3,4,5>

"imap f xs" corresponds to the following circuit layout:

Apply a function to every element of a vector and the element's position (as an SNat value) in the vector.

>>> let rotateMatrix = smap (flip rotateRightS)
>>> let xss = (1:>2:>3:>Nil):>(1:>2:>3:>Nil):>(1:>2:>3:>Nil):>Nil
>>> xss
<<1,2,3>,<1,2,3>,<1,2,3>>
>>> rotateMatrix xss
<<1,2,3>,<3,1,2>,<2,3,1>>

zipWith generalises zip by zipping with the function given as the first argument, instead of a tupling function. For example, "zipWith (+)" applied to two vectors produces the vector of corresponding sums.

zipWith f (x1 :> x2 :> ... xn :> Nil) (y1 :> y2 :> ... :> yn :> Nil) == (f x1 y1 :> f x2 y2 :> ... :> f xn yn :> Nil)

"zipWith f xs ys" corresponds to the following circuit layout:

NB: zipWith is strict in its second argument, and lazy in its third. This matters when zipWith is used in a recursive setting. See lazyV for more information.

zipWith3 generalises zip3 by zipping with the function given as the first argument, instead of a tupling function.

zipWith3 f (x1 :> x2 :> ... xn :> Nil) (y1 :> y2 :> ... :> yn :> Nil) (z1 :> z2 :> ... :> zn :> Nil) == (f x1 y1 z1 :> f x2 y2 z2 :> ... :> f xn yn zn :> Nil)

"zipWith3 f xs ys zs" corresponds to the following circuit layout:

NB: zipWith3 is strict in its second argument, and lazy in its third and fourth. This matters when zipWith3 is used in a recursive setting. See lazyV for more information.

zip takes two vectors and returns a vector of corresponding pairs.

>>> zip (1:>2:>3:>4:>Nil) (4:>3:>2:>1:>Nil)
<(1,4),(2,3),(3,2),(4,1)>

zip takes three vectors and returns a vector of corresponding triplets.

>>> zip3 (1:>2:>3:>4:>Nil) (4:>3:>2:>1:>Nil) (5:>6:>7:>8:>Nil)
<(1,4,5),(2,3,6),(3,2,7),(4,1,8)>

Zip two vectors with a functions that also takes the elements' indices.

>>> izipWith (\i a b -> i + a + b) (2 :> 2 :> Nil)  (3 :> 3:> Nil)
<*** Exception: Clash.Sized.Index: result 3 is out of bounds: [0..1]
...
>>> izipWith (\i a b -> fromIntegral i + a + b) (2 :> 2 :> Nil) (3 :> 3 :> Nil) :: Vec 2 (Unsigned 8)
<5,6>

"imap f xs" corresponds to the following circuit layout:

NB: izipWith is strict in its second argument, and lazy in its third. This matters when izipWith is used in a recursive setting. See lazyV for more information.

unzip transforms a vector of pairs into a vector of first components and a vector of second components.

>>> unzip ((1,4):>(2,3):>(3,2):>(4,1):>Nil)
(<1,2,3,4>,<4,3,2,1>)

unzip3 transforms a vector of triplets into a vector of first components, a vector of second components, and a vector of third components.

>>> unzip3 ((1,4,5):>(2,3,6):>(3,2,7):>(4,1,8):>Nil)
(<1,2,3,4>,<4,3,2,1>,<5,6,7,8>)

foldr, applied to a binary operator, a starting value (typically the right-identity of the operator), and a vector, reduces the vector using the binary operator, from right to left:

foldr f z (x1 :> ... :> xn1 :> xn :> Nil) == x1 `f` (... (xn1 `f` (xn `f` z))...)
foldr r z Nil                             == z

>>> foldr (/) 1 (5 :> 4 :> 3 :> 2 :> Nil)
1.875

"foldr f z xs" corresponds to the following circuit layout:

NB: "foldr f z xs" produces a linear structure, which has a depth, or delay, of O(length xs). Use fold if your binary operator f is associative, as "fold f xs" produces a structure with a depth of O(log_2(length xs)).

foldl, applied to a binary operator, a starting value (typically the left-identity of the operator), and a vector, reduces the vector using the binary operator, from left to right:

foldl f z (x1 :> x2 :> ... :> xn :> Nil) == (...((z `f` x1) `f` x2) `f`...) `f` xn
foldl f z Nil                            == z

>>> foldl (/) 1 (5 :> 4 :> 3 :> 2 :> Nil)
8.333333333333333e-3

"foldl f z xs" corresponds to the following circuit layout:

NB: "foldl f z xs" produces a linear structure, which has a depth, or delay, of O(length xs). Use fold if your binary operator f is associative, as "fold f xs" produces a structure with a depth of O(log_2(length xs)).

foldr1 is a variant of foldr that has no starting value argument, and thus must be applied to non-empty vectors.

foldr1 f (x1 :> ... :> xn2 :> xn1 :> xn :> Nil) == x1 `f` (... (xn2 `f` (xn1 `f` xn))...)
foldr1 f (x1 :> Nil)                            == x1
foldr1 f Nil                                    == TYPE ERROR

>>> foldr1 (/) (5 :> 4 :> 3 :> 2 :> 1 :> Nil)
1.875

"foldr1 f xs" corresponds to the following circuit layout:

NB: "foldr1 f z xs" produces a linear structure, which has a depth, or delay, of O(length xs). Use fold if your binary operator f is associative, as "fold f xs" produces a structure with a depth of O(log_2(length xs)).

foldl1 is a variant of foldl that has no starting value argument, and thus must be applied to non-empty vectors.

foldl1 f (x1 :> x2 :> x3 :> ... :> xn :> Nil) == (...((x1 `f` x2) `f` x3) `f`...) `f` xn
foldl1 f (x1 :> Nil)                          == x1
foldl1 f Nil                                  == TYPE ERROR

>>> foldl1 (/) (1 :> 5 :> 4 :> 3 :> 2 :> Nil)
8.333333333333333e-3

"foldl1 f xs" corresponds to the following circuit layout:

NB: "foldl1 f z xs" produces a linear structure, which has a depth, or delay, of O(length xs). Use fold if your binary operator f is associative, as "fold f xs" produces a structure with a depth of O(log_2(length xs)).

fold is a variant of foldr1 and foldl1, but instead of reducing from right to left, or left to right, it reduces a vector using a tree-like structure. The depth, or delay, of the structure produced by "fold f xs", is hence O(log_2(length xs)), and not O(length xs).

NB: The binary operator "f" in "fold f xs" must be associative.

fold f (x1 :> x2 :> ... :> xn1 :> xn :> Nil) == ((x1 `f` x2) `f` ...) `f` (... `f` (xn1 `f` xn))
fold f (x1 :> Nil)                           == x1
fold f Nil                                   == TYPE ERROR

>>> fold (+) (5 :> 4 :> 3 :> 2 :> 1 :> Nil)
15

"fold f xs" corresponds to the following circuit layout:

Right fold (function applied to each element and its index)

>>> let findLeftmost x xs = ifoldr (\i a b -> if a == x then Just i else b) Nothing xs
>>> findLeftmost 3 (1:>3:>2:>4:>3:>5:>6:>Nil)
Just 1
>>> findLeftmost 8 (1:>3:>2:>4:>3:>5:>6:>Nil)
Nothing

"ifoldr f z xs" corresponds to the following circuit layout:

Left fold (function applied to each element and its index)

>>> let findRightmost x xs = ifoldl (\a i b -> if b == x then Just i else a) Nothing xs
>>> findRightmost 3 (1:>3:>2:>4:>3:>5:>6:>Nil)
Just 4
>>> findRightmost 8 (1:>3:>2:>4:>3:>5:>6:>Nil)
Nothing

"ifoldl f z xs" corresponds to the following circuit layout:

A dependently typed fold.

Using lists, we can define append (a.k.a. Data.List.++) in terms of Data.List.foldr:

>>> import qualified Data.List
>>> let append xs ys = Data.List.foldr (:) ys xs
>>> append [1,2] [3,4]
[1,2,3,4]

However, when we try to do the same for Vec, by defining append' in terms of Clash.Sized.Vector.foldr:

append' xs ys = foldr (:>) ys xs

we get a type error:

>>> let append' xs ys = foldr (:>) ys xs

<interactive>:...
    • Occurs check: cannot construct the infinite type: ... ~ ... + 1
      Expected type: a -> Vec ... a -> Vec ... a
        Actual type: a -> Vec ... a -> Vec (... + 1) a
    • In the first argument of ‘foldr’, namely ‘(:>)’
      In the expression: foldr (:>) ys xs
      In an equation for ‘append'’: append' xs ys = foldr (:>) ys xs
    • Relevant bindings include
        ys :: Vec ... a (bound at ...)
        append' :: Vec n a -> Vec ... a -> Vec ... a
          (bound at ...)

The reason is that the type of foldr is:

>>> :t foldr
foldr :: (a -> b -> b) -> b -> Vec n a -> b

While the type of (:>) is:

>>> :t (:>)
(:>) :: a -> Vec n a -> Vec (n + 1) a

We thus need a fold function that can handle the growing vector type: dfold. Compared to foldr, dfold takes an extra parameter, called the motive, that allows the folded function to have an argument and result type that depends on the current length of the vector. Using dfold, we can now correctly define append':

import Data.Singletons.Prelude
import Data.Proxy

data Append (m :: Nat) (a :: *) (f :: TyFun Nat *) :: *
type instance Apply (Append m a) l = Vec (l + m) a

append' xs ys = dfold (Proxy :: Proxy (Append m a)) (const (:>)) ys xs

We now see that append' has the appropriate type:

>>> :t append'
append' :: KnownNat k => Vec k a -> Vec m a -> Vec (k + m) a

And that it works:

>>> append' (1 :> 2 :> Nil) (3 :> 4 :> Nil)
<1,2,3,4>

NB: "dfold m f z xs" creates a linear structure, which has a depth, or delay, of O(length xs). Look at dtfold for a dependently typed fold that produces a structure with a depth of O(log_2(length xs)).

A combination of dfold and fold: a dependently typed fold that reduces a vector in a tree-like structure.

As an example of when you might want to use dtfold we will build a population counter: a circuit that counts the number of bits set to '1' in a BitVector. Given a vector of n bits, we only need we need a data type that can represent the number n: Index (n+1). Index k has a range of [0 .. k-1] (using ceil(log2(k)) bits), hence we need Index n+1. As an initial attempt we will use sum, because it gives a nice (log2(n)) tree-structure of adders:

populationCount :: (KnownNat (n+1), KnownNat (n+2))
                => BitVector (n+1) -> Index (n+2)
populationCount = sum . map fromIntegral . bv2v

The "problem" with this description is that all adders have the same bit-width, i.e. all adders are of the type:

(+) :: Index (n+2) -> Index (n+2) -> Index (n+2).

This is a "problem" because we could have a more efficient structure: one where each layer of adders is precisely wide enough to count the number of bits at that layer. That is, at height d we want the adder to be of type:

Index ((2^d)+1) -> Index ((2^d)+1) -> Index ((2^(d+1))+1)

We have such an adder in the form of the plus function, as defined in the instance ExtendingNum instance of Index. However, we cannot simply use fold to create a tree-structure of pluses:

>>> :{
let populationCount' :: (KnownNat (n+1), KnownNat (n+2))
                     => BitVector (n+1) -> Index (n+2)
    populationCount' = fold plus . map fromIntegral . bv2v
:}

<interactive>:...
    • Couldn't match type ‘((n + 2) + (n + 2)) - 1’ with ‘n + 2’
      Expected type: Index (n + 2) -> Index (n + 2) -> Index (n + 2)
        Actual type: Index (n + 2)
                     -> Index (n + 2) -> AResult (Index (n + 2)) (Index (n + 2))
    • In the first argument of ‘fold’, namely ‘plus’
      In the first argument of ‘(.)’, namely ‘fold plus’
      In the expression: fold plus . map fromIntegral . bv2v
    • Relevant bindings include
        populationCount' :: BitVector (n + 1) -> Index (n + 2)
          (bound at ...)

because fold expects a function of type "a -> a -> a", i.e. a function where the arguments and result all have exactly the same type.

In order to accommodate the type of our plus, where the result is larger than the arguments, we must use a dependently typed fold in the the form of dtfold:

{-# LANGUAGE UndecidableInstances #-}
import Data.Singletons.Prelude
import Data.Proxy

data IIndex (f :: TyFun Nat *) :: *
type instance Apply IIndex l = Index ((2^l)+1)

populationCount' :: (KnownNat k, KnownNat (2^k))
                 => BitVector (2^k) -> Index ((2^k)+1)
populationCount' bv = dtfold (Proxy @IIndex)
                             fromIntegral
                             (\_ x y -> plus x y)
                             (bv2v bv)

And we can test that it works:

>>> :t populationCount' (7 :: BitVector 16)
populationCount' (7 :: BitVector 16) :: Index 17
>>> populationCount' (7 :: BitVector 16)
3

Some final remarks:

• By using dtfold instead of fold, we had to restrict our BitVector argument to have bit-width that is a power of 2.
• Even though our original populationCount function specified a structure where all adders had the same width. Most VHDL/(System)Verilog synthesis tools will create a more efficient circuit, i.e. one where the adders have an increasing bit-width for every layer, from the VHDL/(System)Verilog produced by the Clash compiler.

NB: The depth, or delay, of the structure produced by "dtfold m f g xs" is O(log_2(length xs)).

Specialised version of dfold that builds a triangular computational structure.

Example:

compareSwap a b = if a > b then (a,b) else (b,a)
insert y xs     = let (y',xs') = mapAccumL compareSwap y xs in xs' :< y'
insertionSort   = vfold (const insert)

Builds a triangular structure of compare and swaps to sort a row.

>>> insertionSort (7 :> 3 :> 9 :> 1 :> Nil)
<1,3,7,9>

The circuit layout of insertionSort, build using vfold, is:

scanl is similar to foldl, but returns a vector of successive reduced values from the left:

scanl f z (x1 :> x2 :> ... :> Nil) == z :> (z `f` x1) :> ((z `f` x1) `f` x2) :> ... :> Nil

>>> scanl (+) 0 (5 :> 4 :> 3 :> 2 :> Nil)
<0,5,9,12,14>

"scanl f z xs" corresponds to the following circuit layout:

NB:

last (scanl f z xs) == foldl f z xs

scanr is similar to foldr, but returns a vector of successive reduced values from the right:

scanr f z (... :> xn1 :> xn :> Nil) == ... :> (xn1 `f` (xn `f` z)) :> (xn `f` z) :> z :> Nil

>>> scanr (+) 0 (5 :> 4 :> 3 :> 2 :> Nil)
<14,9,5,2,0>

"scanr f z xs" corresponds to the following circuit layout:

NB:

head (scanr f z xs) == foldr f z xs

postscanl is a variant of scanl where the first result is dropped:

postscanl f z (x1 :> x2 :> ... :> Nil) == (z `f` x1) :> ((z `f` x1) `f` x2) :> ... :> Nil

>>> postscanl (+) 0 (5 :> 4 :> 3 :> 2 :> Nil)
<5,9,12,14>

"postscanl f z xs" corresponds to the following circuit layout:

postscanr is a variant of scanr that where the last result is dropped:

postscanr f z (... :> xn1 :> xn :> Nil) == ... :> (xn1 `f` (xn `f` z)) :> (xn `f` z) :> Nil

>>> postscanr (+) 0 (5 :> 4 :> 3 :> 2 :> Nil)
<14,9,5,2>

"postscanr f z xs" corresponds to the following circuit layout:

The mapAccumL function behaves like a combination of map and foldl; it applies a function to each element of a vector, passing an accumulating parameter from left to right, and returning a final value of this accumulator together with the new vector.

>>> mapAccumL (\acc x -> (acc + x,acc + 1)) 0 (1 :> 2 :> 3 :> 4 :> Nil)
(10,<1,2,4,7>)

"mapAccumL f acc xs" corresponds to the following circuit layout:

The mapAccumR function behaves like a combination of map and foldr; it applies a function to each element of a vector, passing an accumulating parameter from right to left, and returning a final value of this accumulator together with the new vector.

>>> mapAccumR (\acc x -> (acc + x,acc + 1)) 0 (1 :> 2 :> 3 :> 4 :> Nil)
(10,<10,8,5,1>)

"mapAccumR f acc xs" corresponds to the following circuit layout:

1-dimensional stencil computations

"stencil1d stX f xs", where xs has stX + n elements, applies the stencil computation f on: n + 1 overlapping (1D) windows of length stX, drawn from xs. The resulting vector has n + 1 elements.

>>> let xs = (1:>2:>3:>4:>5:>6:>Nil)
>>> :t xs
xs :: Num a => Vec 6 a
>>> :t stencil1d d2 sum xs
stencil1d d2 sum xs :: Num b => Vec 5 b
>>> stencil1d d2 sum xs
<3,5,7,9,11>

2-dimensional stencil computations

"stencil2d stY stX f xss", where xss is a matrix of stY + m rows of stX + n elements, applies the stencil computation f on: (m + 1) * (n + 1) overlapping (2D) windows of stY rows of stX elements, drawn from xss. The result matrix has m + 1 rows of n + 1 elements.

>>> let xss = ((1:>2:>3:>4:>Nil):>(5:>6:>7:>8:>Nil):>(9:>10:>11:>12:>Nil):>(13:>14:>15:>16:>Nil):>Nil)
>>> :t xss
xss :: Num a => Vec 4 (Vec 4 a)
>>> :t stencil2d d2 d2 (sum . map sum) xss
stencil2d d2 d2 (sum . map sum) xss :: Num b => Vec 3 (Vec 3 b)
>>> stencil2d d2 d2 (sum . map sum) xss
<<14,18,22>,<30,34,38>,<46,50,54>>

"windows1d stX xs", where the vector xs has stX + n elements, returns a vector of n + 1 overlapping (1D) windows of xs of length stX.

>>> let xs = (1:>2:>3:>4:>5:>6:>Nil)
>>> :t xs
xs :: Num a => Vec 6 a
>>> :t windows1d d2 xs
windows1d d2 xs :: Num a => Vec 5 (Vec 2 a)
>>> windows1d d2 xs
<<1,2>,<2,3>,<3,4>,<4,5>,<5,6>>

"windows2d stY stX xss", where matrix xss has stY + m rows of stX + n, returns a matrix of m+1 rows of n+1 elements. The elements of this new matrix are the overlapping (2D) windows of xss, where every window has stY rows of stX elements.

>>> let xss = ((1:>2:>3:>4:>Nil):>(5:>6:>7:>8:>Nil):>(9:>10:>11:>12:>Nil):>(13:>14:>15:>16:>Nil):>Nil)
>>> :t xss
xss :: Num a => Vec 4 (Vec 4 a)
>>> :t windows2d d2 d2 xss
windows2d d2 d2 xss :: Num a => Vec 3 (Vec 3 (Vec 2 (Vec 2 a)))
>>> windows2d d2 d2 xss
<<<<1,2>,<5,6>>,<<2,3>,<6,7>>,<<3,4>,<7,8>>>,<<<5,6>,<9,10>>,<<6,7>,<10,11>>,<<7,8>,<11,12>>>,<<<9,10>,<13,14>>,<<10,11>,<14,15>>,<<11,12>,<15,16>>>>

Convert a vector to a list.

>>> toList (1:>2:>3:>Nil)
[1,2,3]

Convert a BitVector to a Vec of Bits.

>>> let x = 6 :: BitVector 8
>>> x
0000_0110
>>> bv2v x
<0,0,0,0,0,1,1,0>

Convert a Vec of Bits to a BitVector.

>>> let x = (0:>0:>0:>1:>0:>0:>1:>0:>Nil) :: Vec 8 Bit
>>> x
<0,0,0,1,0,0,1,0>
>>> v2bv x
0001_0010

What you should use when your vector functions are too strict in their arguments.

For example:

-- Bubble sort for 1 iteration
sortV xs = map fst sorted :< (snd (last sorted))
 where
   lefts  = head xs :> map snd (init sorted)
   rights = tail xs
   sorted = zipWith compareSwapL lefts rights

-- Compare and swap
compareSwapL a b = if a < b then (a,b)
                            else (b,a)

Will not terminate because zipWith is too strict in its second argument.

In this case, adding lazyV on zipWiths second argument:

sortVL xs = map fst sorted :< (snd (last sorted))
 where
   lefts  = head xs :> map snd (init sorted)
   rights = tail xs
   sorted = zipWith compareSwapL (lazyV lefts) rights

Results in a successful computation:

>>> sortVL (4 :> 1 :> 2 :> 3 :> Nil)
<1,2,3,4>

NB: There is also a solution using flip, but it slightly obfuscates the meaning of the code:

sortV_flip xs = map fst sorted :< (snd (last sorted))
 where
   lefts  = head xs :> map snd (init sorted)
   rights = tail xs
   sorted = zipWith (flip compareSwapL) rights lefts

>>> sortV_flip (4 :> 1 :> 2 :> 3 :> Nil)
<1,2,3,4>

To be used as the motive p for dfold, when the f in "dfold p f" is a variation on (:>), e.g.:

map' :: forall n a b . KnownNat n => (a -> b) -> Vec n a -> Vec n b
map' f = dfold (Proxy @(VCons b)) (_ x xs -> f x :> xs)

Vector as a Proxy for Nat