Clash.Sized.Vector

Vector data type

data Vec :: Nat -> Type -> Type where

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.

Constructors

Nil :: Vec 0 a

Cons :: a -> Vec n a -> Vec (n + 1) a infixr 5

Bundled Patterns

pattern (:>) :: a -> Vec n a -> Vec (n + 1) a infixr 5

Add an element to the head of a vector.

>>> 3:>4:>5:>Nil
<3,4,5>
>>> let x = 3:>4:>5:>Nil
>>> :t x
x :: Num a => Vec 3 a

Can be used as a pattern:

>>> let f (x :> y :> _) = x + y
>>> :t f
f :: Num a => Vec ((n + 1) + 1) a -> a
>>> f (3:>4:>5:>6:>7:>Nil)
7

Also in conjunctions with (:<):

>>> let g (a :> b :> (_ :< y :< x)) = a + b +  x + y
>>> :t g
g :: Num a => Vec ((((n + 1) + 1) + 1) + 1) a -> a
>>> g (1:>2:>3:>4:>5:>Nil)
12

pattern (:<) :: Vec n a -> a -> Vec (n + 1) a infixl 5

Add an element to the tail of a vector.

>>> (3:>4:>5:>Nil) :< 1
<3,4,5,1>
>>> let x = (3:>4:>5:>Nil) :< 1
>>> :t x
x :: Num a => Vec 4 a

Can be used as a pattern:

>>> let f (_ :< y :< x) = y + x
>>> :t f
f :: Num a => Vec ((n + 1) + 1) a -> a
>>> f (3:>4:>5:>6:>7:>Nil)
13

Also in conjunctions with (:>):

>>> let g (a :> b :> (_ :< y :< x)) = a + b +  x + y
>>> :t g
g :: Num a => Vec ((((n + 1) + 1) + 1) + 1) a -> a
>>> g (1:>2:>3:>4:>5:>Nil)
12

Instances

Functor (Vec n)

Defined in Clash.Sized.Vector

Methods

fmap :: (a -> b) -> Vec n a -> Vec n b

(<$) :: a -> Vec n b -> Vec n a

KnownNat n => Applicative (Vec n)

Defined in Clash.Sized.Vector

Methods

pure :: a -> Vec n a

(<*>) :: Vec n (a -> b) -> Vec n a -> Vec n b

liftA2 :: (a -> b -> c) -> Vec n a -> Vec n b -> Vec n c

(*>) :: Vec n a -> Vec n b -> Vec n b

(<*) :: Vec n a -> Vec n b -> Vec n a

(KnownNat n, 1 <= n) => Foldable (Vec n)

Defined in Clash.Sized.Vector

Methods

fold :: Monoid m => Vec n m -> m

foldMap :: Monoid m => (a -> m) -> Vec n a -> m

foldMap' :: Monoid m => (a -> m) -> Vec n a -> m

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

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

foldl :: (b -> a -> b) -> b -> Vec n a -> b

foldl' :: (b -> a -> b) -> b -> Vec n a -> b

foldr1 :: (a -> a -> a) -> Vec n a -> a

foldl1 :: (a -> a -> a) -> Vec n a -> a

toList :: Vec n a -> [a]

null :: Vec n a -> Bool

length :: Vec n a -> Int

elem :: Eq a => a -> Vec n a -> Bool

maximum :: Ord a => Vec n a -> a

minimum :: Ord a => Vec n a -> a

sum :: Num a => Vec n a -> a

product :: Num a => Vec n a -> a

(KnownNat n, 1 <= n) => Traversable (Vec n)

Defined in Clash.Sized.Vector

Methods

traverse :: Applicative f => (a -> f b) -> Vec n a -> f (Vec n b)

sequenceA :: Applicative f => Vec n (f a) -> f (Vec n a)

mapM :: Monad m => (a -> m b) -> Vec n a -> m (Vec n b)

sequence :: Monad m => Vec n (m a) -> m (Vec n a)

(KnownNat n, Eq a) => Eq (Vec n a)

Defined in Clash.Sized.Vector

Methods

(==) :: Vec n a -> Vec n a -> Bool

(/=) :: Vec n a -> Vec n a -> Bool

(KnownNat n, Typeable a, Data a) => Data (Vec n a)

Defined in Clash.Sized.Vector

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Vec n a -> c (Vec n a)

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Vec n a)

toConstr :: Vec n a -> Constr

dataTypeOf :: Vec n a -> DataType

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Vec n a))

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Vec n a))

gmapT :: (forall b. Data b => b -> b) -> Vec n a -> Vec n a

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Vec n a -> r

gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Vec n a -> r

gmapQ :: (forall d. Data d => d -> u) -> Vec n a -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> Vec n a -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Vec n a -> m (Vec n a)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Vec n a -> m (Vec n a)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Vec n a -> m (Vec n a)

(KnownNat n, Ord a) => Ord (Vec n a)

Defined in Clash.Sized.Vector

Methods

compare :: Vec n a -> Vec n a -> Ordering

(<) :: Vec n a -> Vec n a -> Bool

(<=) :: Vec n a -> Vec n a -> Bool

(>) :: Vec n a -> Vec n a -> Bool

(>=) :: Vec n a -> Vec n a -> Bool

max :: Vec n a -> Vec n a -> Vec n a

min :: Vec n a -> Vec n a -> Vec n a

Show a => Show (Vec n a)

Defined in Clash.Sized.Vector

Methods

showsPrec :: Int -> Vec n a -> ShowS

show :: Vec n a -> String

showList :: [Vec n a] -> ShowS

KnownNat n => Generic (Vec n a)

In many cases, this Generic instance only allows generic functions/instances over vectors of at least size 1, due to the n-1 in the Rep (Vec n a) definition.

We'll have to wait for things like https://ryanglscott.github.io/2018/02/11/how-to-derive-generic-for-some-gadts/ before we can work around this limitation

Defined in Clash.Sized.Vector

Associated Types

type Rep (Vec n a) :: Type -> Type

Methods

from :: Vec n a -> Rep (Vec n a) x

to :: Rep (Vec n a) x -> Vec n a

Lift a => Lift (Vec n a)

Defined in Clash.Sized.Vector

Methods

lift :: Vec n a -> Q Exp

(KnownNat n, Arbitrary a) => Arbitrary (Vec n a)

Defined in Clash.Sized.Vector

Methods

arbitrary :: Gen (Vec n a)

shrink :: Vec n a -> [Vec n a]

CoArbitrary a => CoArbitrary (Vec n a)

Defined in Clash.Sized.Vector

Methods

coarbitrary :: Vec n a -> Gen b -> Gen b

(Default a, KnownNat n) => Default (Vec n a)

Defined in Clash.Sized.Vector

Methods

def :: Vec n a

NFData a => NFData (Vec n a)

Defined in Clash.Sized.Vector

Methods

rnf :: Vec n a -> ()

KnownNat n => Ixed (Vec n a)

Defined in Clash.Sized.Vector

Methods

ix :: Index (Vec n a) -> Traversal' (Vec n a) (IxValue (Vec n a))

(NFDataX a, KnownNat n) => NFDataX (Vec n a)

Defined in Clash.Sized.Vector

Methods

deepErrorX :: String -> Vec n a

hasUndefined :: Vec n a -> Bool

ensureSpine :: Vec n a -> Vec n a

rnfX :: Vec n a -> ()

ShowX a => ShowX (Vec n a)

Defined in Clash.Sized.Vector

Methods

showsPrecX :: Int -> Vec n a -> ShowS

showX :: Vec n a -> String

showListX :: [Vec n a] -> ShowS

(KnownNat n, BitPack a) => BitPack (Vec n a)

Defined in Clash.Sized.Vector

Associated Types

type BitSize (Vec n a) :: Nat

Methods

pack :: Vec n a -> BitVector (BitSize (Vec n a))

unpack :: BitVector (BitSize (Vec n a)) -> Vec n a

KnownNat n => Bundle (Vec n a)

Defined in Clash.Signal.Bundle

Associated Types

type Unbundled dom (Vec n a) = (res :: Type)

Methods

bundle :: forall (dom :: Domain). Unbundled dom (Vec n a) -> Signal dom (Vec n a)

unbundle :: forall (dom :: Domain). Signal dom (Vec n a) -> Unbundled dom (Vec n a)

KnownNat n => Bundle (Vec n a)

Defined in Clash.Signal.Delayed.Bundle

Associated Types

type Unbundled dom d (Vec n a) = (res :: Type)

Methods

bundle :: forall (dom :: Domain) (d :: Nat). Unbundled dom d (Vec n a) -> DSignal dom d (Vec n a)

unbundle :: forall (dom :: Domain) (d :: Nat). DSignal dom d (Vec n a) -> Unbundled dom d (Vec n a)

(KnownNat n, AutoReg a) => AutoReg (Vec n a)

Defined in Clash.Class.AutoReg.Internal

Methods

autoReg :: forall (dom :: Domain). (HasCallStack, KnownDomain dom) => Clock dom -> Reset dom -> Enable dom -> Vec n a -> Signal dom (Vec n a) -> Signal dom (Vec n a)

(LockStep en a, KnownNat n) => LockStep (Vec n en) (Vec n a)

Defined in Clash.Prelude.DataFlow

Methods

lockStep :: forall (dom :: Domain). DataFlow dom (Vec n en) Bool (Vec n a) (Vec n a)

stepLock :: forall (dom :: Domain). DataFlow dom Bool (Vec n en) (Vec n a) (Vec n a)

type Unbundled t d (Vec n a)

Defined in Clash.Signal.Delayed.Bundle

type Unbundled t d (Vec n a) = Vec n (DSignal t d a)

type HasDomain dom (Vec n a)

Defined in Clash.Class.HasDomain.HasSpecificDomain

type HasDomain dom (Vec n a) = HasDomain dom a

type TryDomain t (Vec n a)

Defined in Clash.Class.HasDomain.HasSingleDomain

type TryDomain t (Vec n a) = TryDomain t a

type Unbundled t (Vec n a)

Defined in Clash.Signal.Bundle

type Unbundled t (Vec n a) = Vec n (Signal t a)

type Rep (Vec n a)

Defined in Clash.Sized.Vector

type Rep (Vec n a) = D1 ('MetaData "Vec" "Clash.Data.Vector" "clash-prelude" 'False) (C1 ('MetaCons "Nil" 'PrefixI 'False) (U1 :: Type -> Type) :+: C1 ('MetaCons "Cons" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Vec (n - 1) a))))

type Index (Vec n a)

Defined in Clash.Sized.Vector

type Index (Vec n a) = Index n

type IxValue (Vec n a)

Defined in Clash.Sized.Vector

type IxValue (Vec n a) = a

type BitSize (Vec n a)

Defined in Clash.Sized.Vector

type BitSize (Vec n a) = n * BitSize a

Accessors

Length information

length :: KnownNat n => Vec n a -> Int

The length of a Vector as an Int value.

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

lengthS :: KnownNat n => Vec n a -> SNat n

Length of a Vector as an SNat value

Indexing

(!!) :: (KnownNat n, Enum i) => Vec n a -> i -> a

"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
...

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

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

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

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 :: SNat m -> Vec (m + (n + 1)) a -> a

"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

indices :: KnownNat n => SNat n -> Vec n (Index n)

Generate a vector of indices.

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

indicesI :: KnownNat n => Vec n (Index n)

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 :: KnownNat n => (a -> Bool) -> Vec n a -> Maybe (Index n)

"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 :: (KnownNat n, Eq a) => a -> Vec n a -> Maybe (Index n)

"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

Extracting sub-vectors (slicing)

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

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

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

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 :: SNat m -> Vec (m + n) a -> Vec m a

"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 :: KnownNat m => Vec (m + n) a -> Vec m a

"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 :: SNat m -> Vec (m + n) a -> Vec n a

"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>:...: error:
    • Couldn't match...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 :: KnownNat m => Vec (m + n) a -> Vec n a

"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 :: CmpNat (i + s) (s * n) ~ 'GT => SNat f -> SNat s -> SNat n -> Vec (f + i) a -> Vec n a

"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 :: (CmpNat (i + s) (s * n) ~ 'GT, KnownNat n) => SNat f -> SNat s -> Vec (f + i) a -> Vec n a

"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>

Splitting

splitAt :: SNat m -> Vec (m + n) a -> (Vec m a, Vec n a)

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>)

splitAtI :: KnownNat m => Vec (m + n) a -> (Vec m a, Vec n a)

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>)

unconcat :: KnownNat n => SNat m -> Vec (n * m) a -> Vec n (Vec m a)

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>>

unconcatI :: (KnownNat n, KnownNat m) => Vec (n * m) a -> Vec n (Vec m a)

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>>

Construction

Initialisation

singleton :: a -> Vec 1 a

Create a vector of one element

>>> singleton 5
<5>

replicate :: SNat n -> a -> Vec n a

"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 :: KnownNat n => a -> Vec n a

"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 :: SNat n -> (a -> a) -> a -> Vec n a

"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>

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

iterateI :: KnownNat n => (a -> a) -> a -> Vec n a

"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>

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

generate :: SNat n -> (a -> a) -> a -> Vec n a

"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 :: KnownNat n => (a -> a) -> a -> Vec n a

"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:

Initialisation from a list

listToVecTH :: Lift a => [a] -> ExpQ

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>

Concatenation

(++) :: Vec n a -> Vec m a -> Vec (n + m) a infixr 5

Append two vectors.

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

(+>>) :: KnownNat n => a -> Vec n a -> Vec n a infixr 4

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
<>

(<<+) :: Vec n a -> a -> Vec n a infixl 4

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
<>

concat :: Vec n (Vec m a) -> Vec (n * m) a

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>

concatMap :: (a -> Vec m b) -> Vec n a -> Vec (n * m) b

Map a function over all the elements of a vector and concatentate the resulting vectors.

>>> concatMap (replicate d3) (1:>2:>3:>Nil)
<1,1,1,2,2,2,3,3,3>

shiftInAt0

Arguments

:: KnownNat n

=> Vec n a

The old vector

-> Vec m a

The elements to shift in at the head

-> (Vec n a, Vec m a)

(The new vector, shifted out elements)

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>)

shiftInAtN

Arguments

:: KnownNat m

=> Vec n a

The old vector

-> Vec m a

The elements to shift in at the tail

-> (Vec n a, Vec m a)

(The new vector, shifted out elements)

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>)

shiftOutFrom0

Arguments

:: (Default a, KnownNat m)

=> SNat m

m, the number of elements to shift out

-> Vec (m + n) a

The old vector

-> (Vec (m + n) a, Vec m a)

(The new vector, shifted out elements)

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>)

shiftOutFromN

Arguments

:: (Default a, KnownNat n)

=> SNat m

m, the number of elements to shift out

-> Vec (m + n) a

The old vector

-> (Vec (m + n) a, Vec m a)

(The new vector, shifted out elements)

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 :: KnownNat n => Vec n a -> Vec n a -> Vec (2 * n) a

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>

Modifying vectors

replace :: (KnownNat n, Enum i) => i -> a -> Vec n a -> Vec n a

"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
...

Permutations

permute

Arguments

:: (Enum i, KnownNat n, KnownNat m)

=> (a -> a -> a)

Combination function, f

-> Vec n a

Default values, def

-> Vec m i

Index mapping, is

-> Vec (m + k) a

Vector to be permuted, xs

-> Vec n a

Forward permutation specified by an index mapping, ix. The result vector is initialized 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.

backpermute

Arguments

:: (Enum i, KnownNat n)

=> Vec n a

Source vector, xs

-> Vec m i

Index mapping, is

-> Vec m a

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>

scatter

Arguments

:: (Enum i, KnownNat n, KnownNat m)

=> Vec n a

Default values, def

-> Vec m i

Index mapping, is

-> Vec (m + k) a

Vector to be scattered, xs

-> Vec n a

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.

gather

Arguments

:: (Enum i, KnownNat n)

=> Vec n a

Source vector, xs

-> Vec m i

Index mapping, is

-> Vec m a

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>

Specialised permutations

reverse :: Vec n a -> Vec n a

The elements in a vector in reverse order.

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

transpose :: KnownNat n => Vec m (Vec n a) -> Vec n (Vec m a)

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

Arguments

:: (KnownNat n, KnownNat d)

=> SNat d

Interleave step, d

-> Vec (n * d) a

-> Vec (d * n) a

"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>

rotateLeft :: (Enum i, KnownNat n) => Vec n a -> i -> Vec n a

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.

rotateRight :: (Enum i, KnownNat n) => Vec n a -> i -> Vec n a

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.

rotateLeftS :: KnownNat n => Vec n a -> SNat d -> Vec n a

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.

rotateRightS :: KnownNat n => Vec n a -> SNat d -> Vec n a

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.

Element-wise operations

Mapping

map :: (a -> b) -> Vec n a -> Vec n b

"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:

imap :: forall n a b. KnownNat n => (Index n -> a -> b) -> Vec n a -> Vec n b

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: X: 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:

smap :: forall k a b. KnownNat k => (forall l. SNat l -> a -> b) -> Vec k a -> Vec k b

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>>

Zipping

zipWith :: (a -> b -> c) -> Vec n a -> Vec n b -> Vec n c

zipWith generalizes 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 :: (a -> b -> c -> d) -> Vec n a -> Vec n b -> Vec n c -> Vec n d

zipWith3 generalizes 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.

zipWith4 :: (a -> b -> c -> d -> e) -> Vec n a -> Vec n b -> Vec n c -> Vec n d -> Vec n e

zipWith5 :: (a -> b -> c -> d -> e -> f) -> Vec n a -> Vec n b -> Vec n c -> Vec n d -> Vec n e -> Vec n f

zipWith6 :: (a -> b -> c -> d -> e -> f -> g) -> Vec n a -> Vec n b -> Vec n c -> Vec n d -> Vec n e -> Vec n f -> Vec n g

zipWith7 :: (a -> b -> c -> d -> e -> f -> g -> h) -> Vec n a -> Vec n b -> Vec n c -> Vec n d -> Vec n e -> Vec n f -> Vec n g -> Vec n h

zip :: Vec n a -> Vec n b -> Vec n (a, b)

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)>

zip3 :: Vec n a -> Vec n b -> Vec n c -> Vec n (a, b, c)

zip3 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)>

zip4 :: Vec n a -> Vec n b -> Vec n c -> Vec n d -> Vec n (a, b, c, d)

zip4 takes four vectors and returns a list of quadruples, analogous to zip.

zip5 :: Vec n a -> Vec n b -> Vec n c -> Vec n d -> Vec n e -> Vec n (a, b, c, d, e)

zip5 takes five vectors and returns a list of five-tuples, analogous to zip.

zip6 :: Vec n a -> Vec n b -> Vec n c -> Vec n d -> Vec n e -> Vec n f -> Vec n (a, b, c, d, e, f)

zip6 takes six vectors and returns a list of six-tuples, analogous to zip.

zip7 :: Vec n a -> Vec n b -> Vec n c -> Vec n d -> Vec n e -> Vec n f -> Vec n g -> Vec n (a, b, c, d, e, f, g)

zip7 takes seven vectors and returns a list of seven-tuples, analogous to zip.

izipWith :: KnownNat n => (Index n -> a -> b -> c) -> Vec n a -> Vec n b -> Vec n c

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: X: 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.

Unzipping

unzip :: Vec n (a, b) -> (Vec n a, Vec n b)

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 :: Vec n (a, b, c) -> (Vec n a, Vec n b, Vec n c)

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>)

unzip4 :: Vec n (a, b, c, d) -> (Vec n a, Vec n b, Vec n c, Vec n d)

unzip4 takes a vector of quadruples and returns four vectors, analogous to unzip.

unzip5 :: Vec n (a, b, c, d, e) -> (Vec n a, Vec n b, Vec n c, Vec n d, Vec n e)

unzip5 takes a vector of five-tuples and returns five vectors, analogous to unzip.

unzip6 :: Vec n (a, b, c, d, e, f) -> (Vec n a, Vec n b, Vec n c, Vec n d, Vec n e, Vec n f)

unzip6 takes a vector of six-tuples and returns six vectors, analogous to unzip.

unzip7 :: Vec n (a, b, c, d, e, f, g) -> (Vec n a, Vec n b, Vec n c, Vec n d, Vec n e, Vec n f, Vec n g)

unzip7 takes a vector of seven-tuples and returns seven vectors, analogous to unzip.

Folding

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

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

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

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

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

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:

ifoldr :: KnownNat n => (Index n -> a -> b -> b) -> b -> Vec n a -> b

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:

ifoldl :: KnownNat n => (a -> Index n -> b -> a) -> a -> Vec n b -> a

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:

Specialised folds

dfold

Arguments

:: forall p k a. KnownNat k

=> Proxy (p :: TyFun Nat Type -> Type)

The motive

-> (forall l. SNat l -> a -> (p @@ l) -> p @@ (l + 1))

Function to fold.

NB: The SNat l is not the index (see (!!)) to the element a. SNat l is the number of elements that occur to the right of a.

-> (p @@ 0)

Initial element

-> Vec k a

Vector to fold over

-> p @@ k

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)).

dtfold

Arguments

:: forall p k a. KnownNat k

=> Proxy (p :: TyFun Nat Type -> Type)

The motive

-> (a -> p @@ 0)

Function to apply to every element

-> (forall l. SNat l -> (p @@ l) -> (p @@ l) -> p @@ (l + 1))

Function to combine results.

NB: The SNat l indicates the depth/height of the node in the tree that is created by applying this function. The leafs of the tree have depth/height 0, and the root of the tree has height k.

-> Vec (2 ^ k) a

Vector to fold over.

NB: Must have a length that is a power of 2.

-> p @@ k

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 add function, as defined in the instance ExtendingNum instance of Index. However, we cannot simply use fold to create a tree-structure of addes:

>>> :{
let populationCount' :: (KnownNat (n+1), KnownNat (n+2))
                     => BitVector (n+1) -> Index (n+2)
    populationCount' = fold add . 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 ‘add’
      In the first argument of ‘(.)’, namely ‘fold add’
      In the expression: fold add . 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 add, where the result is larger than the arguments, we must use a dependently typed fold in 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 -> add 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)).

vfold :: forall k a b. KnownNat k => (forall l. SNat l -> a -> Vec l b -> Vec (l + 1) b) -> Vec k a -> Vec k b

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:

Prefix sums (scans)

scanl :: (b -> a -> b) -> b -> Vec n a -> Vec (n + 1) b

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

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

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

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:

mapAccumL :: (acc -> x -> (acc, y)) -> acc -> Vec n x -> (acc, Vec n y)

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:

mapAccumR :: (acc -> x -> (acc, y)) -> acc -> Vec n x -> (acc, Vec n y)

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:

Stencil computations

stencil1d

Arguments

:: KnownNat n

=> SNat (stX + 1)

Windows length stX, at least size 1

-> (Vec (stX + 1) a -> b)

The stencil (function)

-> Vec ((stX + n) + 1) a

-> Vec (n + 1) b

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>

stencil2d

Arguments

:: (KnownNat n, KnownNat m)

=> SNat (stY + 1)

Window hight stY, at least size 1

-> SNat (stX + 1)

Window width stX, at least size 1

-> (Vec (stY + 1) (Vec (stX + 1) a) -> b)

The stencil (function)

-> Vec ((stY + m) + 1) (Vec ((stX + n) + 1) a)

-> Vec (m + 1) (Vec (n + 1) b)

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

Arguments

:: KnownNat n

=> SNat (stX + 1)

Length of the window, at least size 1

-> Vec ((stX + n) + 1) a

-> Vec (n + 1) (Vec (stX + 1) a)

"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

Arguments

:: (KnownNat n, KnownNat m)

=> SNat (stY + 1)

Window hight stY, at least size 1

-> SNat (stX + 1)

Window width stX, at least size 1

-> Vec ((stY + m) + 1) (Vec ((stX + n) + 1) a)

-> Vec (m + 1) (Vec (n + 1) (Vec (stY + 1) (Vec (stX + 1) a)))

"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>>>>

Conversions

toList :: Vec n a -> [a]

Convert a vector to a list.

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

bv2v :: KnownNat n => BitVector n -> Vec n Bit

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>

v2bv :: KnownNat n => Vec n Bit -> BitVector n

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

Misc

lazyV :: KnownNat n => Vec n a -> Vec n a

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>

data VCons (a :: Type) (f :: TyFun Nat Type) :: Type

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)

Instances

type Apply (VCons a :: TyFun Nat Type -> Type) (l :: Nat)

Defined in Clash.Sized.Vector

type Apply (VCons a :: TyFun Nat Type -> Type) (l :: Nat) = Vec l a

asNatProxy :: Vec n a -> Proxy n

Vector as a Proxy for Nat

seqV :: KnownNat n => Vec n a -> b -> b infixr 0

Evaluate all elements of a vector to WHNF, returning the second argument

forceV :: KnownNat n => Vec n a -> Vec n a

Evaluate all elements of a vector to WHNF

seqVX :: KnownNat n => Vec n a -> b -> b infixr 0

Evaluate all elements of a vector to WHNF, returning the second argument. Does not propagate XExceptions.

forceVX :: KnownNat n => Vec n a -> Vec n a

Evaluate all elements of a vector to WHNF. Does not propagate XExceptions.

Primitives

Traversable instance

traverse# :: forall a f b n. Applicative f => (a -> f b) -> Vec n a -> f (Vec n b)

BitPack instance

concatBitVector# :: (KnownNat n, KnownNat m) => Vec n (BitVector m) -> BitVector (n * m)

unconcatBitVector# :: forall n m. (KnownNat n, KnownNat m) => BitVector (n * m) -> Vec n (BitVector m)