Copyright  (C) 20132016 University of Twente 2017 Myrtle Software Ltd 

License  BSD2 (see the file LICENSE) 
Maintainer  Christiaan Baaij <christiaan.baaij@gmail.com> 
Safe Haskell  Trustworthy 
Language  Haskell2010 
Extensions 

Synopsis
 data Vec :: Nat > Type > Type where
 length :: KnownNat n => Vec n a > Int
 lengthS :: KnownNat n => Vec n a > SNat n
 (!!) :: (KnownNat n, Enum i) => Vec n a > i > a
 head :: Vec (n + 1) a > a
 last :: Vec (n + 1) a > a
 at :: SNat m > Vec (m + (n + 1)) a > a
 indices :: KnownNat n => SNat n > Vec n (Index n)
 indicesI :: KnownNat n => Vec n (Index n)
 findIndex :: KnownNat n => (a > Bool) > Vec n a > Maybe (Index n)
 elemIndex :: (KnownNat n, Eq a) => a > Vec n a > Maybe (Index n)
 tail :: Vec (n + 1) a > Vec n a
 init :: Vec (n + 1) a > Vec n a
 take :: SNat m > Vec (m + n) a > Vec m a
 takeI :: KnownNat m => Vec (m + n) a > Vec m a
 drop :: SNat m > Vec (m + n) a > Vec n a
 dropI :: KnownNat m => Vec (m + n) a > Vec n a
 select :: CmpNat (i + s) (s * n) ~ 'GT => SNat f > SNat s > SNat n > Vec (f + i) a > Vec n a
 selectI :: (CmpNat (i + s) (s * n) ~ 'GT, KnownNat n) => SNat f > SNat s > Vec (f + i) a > Vec n a
 splitAt :: SNat m > Vec (m + n) a > (Vec m a, Vec n a)
 splitAtI :: KnownNat m => Vec (m + n) a > (Vec m a, Vec n a)
 unconcat :: KnownNat n => SNat m > Vec (n * m) a > Vec n (Vec m a)
 unconcatI :: (KnownNat n, KnownNat m) => Vec (n * m) a > Vec n (Vec m a)
 singleton :: a > Vec 1 a
 replicate :: SNat n > a > Vec n a
 repeat :: KnownNat n => a > Vec n a
 iterate :: SNat n > (a > a) > a > Vec n a
 iterateI :: KnownNat n => (a > a) > a > Vec n a
 generate :: SNat n > (a > a) > a > Vec n a
 generateI :: KnownNat n => (a > a) > a > Vec n a
 listToVecTH :: Lift a => [a] > ExpQ
 (++) :: Vec n a > Vec m a > Vec (n + m) a
 (+>>) :: KnownNat n => a > Vec n a > Vec n a
 (<<+) :: Vec n a > a > Vec n a
 concat :: Vec n (Vec m a) > Vec (n * m) a
 concatMap :: (a > Vec m b) > Vec n a > Vec (n * m) b
 shiftInAt0 :: KnownNat n => Vec n a > Vec m a > (Vec n a, Vec m a)
 shiftInAtN :: KnownNat m => Vec n a > Vec m a > (Vec n a, Vec m a)
 shiftOutFrom0 :: (Default a, KnownNat m) => SNat m > Vec (m + n) a > (Vec (m + n) a, Vec m a)
 shiftOutFromN :: (Default a, KnownNat n) => SNat m > Vec (m + n) a > (Vec (m + n) a, Vec m a)
 merge :: KnownNat n => Vec n a > Vec n a > Vec (2 * n) a
 replace :: (KnownNat n, Enum i) => i > a > Vec n a > Vec n a
 permute :: (Enum i, KnownNat n, KnownNat m) => (a > a > a) > Vec n a > Vec m i > Vec (m + k) a > Vec n a
 backpermute :: (Enum i, KnownNat n) => Vec n a > Vec m i > Vec m a
 scatter :: (Enum i, KnownNat n, KnownNat m) => Vec n a > Vec m i > Vec (m + k) a > Vec n a
 gather :: (Enum i, KnownNat n) => Vec n a > Vec m i > Vec m a
 reverse :: Vec n a > Vec n a
 transpose :: KnownNat n => Vec m (Vec n a) > Vec n (Vec m a)
 interleave :: (KnownNat n, KnownNat d) => SNat d > Vec (n * d) a > Vec (d * n) a
 rotateLeft :: (Enum i, KnownNat n) => Vec n a > i > Vec n a
 rotateRight :: (Enum i, KnownNat n) => Vec n a > i > Vec n a
 rotateLeftS :: KnownNat n => Vec n a > SNat d > Vec n a
 rotateRightS :: KnownNat n => Vec n a > SNat d > Vec n a
 map :: (a > b) > Vec n a > Vec n b
 imap :: forall n a b. KnownNat n => (Index n > a > b) > Vec n a > Vec n b
 smap :: forall k a b. KnownNat k => (forall l. SNat l > a > b) > Vec k a > Vec k b
 zipWith :: (a > b > c) > Vec n a > Vec n b > Vec n c
 zipWith3 :: (a > b > c > d) > Vec n a > Vec n b > Vec n c > Vec n d
 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)
 zip3 :: Vec n a > Vec n b > Vec n c > Vec n (a, b, c)
 zip4 :: Vec n a > Vec n b > Vec n c > Vec n d > Vec n (a, b, c, d)
 zip5 :: Vec n a > Vec n b > Vec n c > Vec n d > Vec n e > Vec n (a, b, c, d, e)
 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)
 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)
 izipWith :: KnownNat n => (Index n > a > b > c) > Vec n a > Vec n b > Vec n c
 unzip :: Vec n (a, b) > (Vec n a, Vec n b)
 unzip3 :: Vec n (a, b, c) > (Vec n a, Vec n b, Vec n c)
 unzip4 :: Vec n (a, b, c, d) > (Vec n a, Vec n b, Vec n c, Vec n d)
 unzip5 :: Vec n (a, b, c, d, e) > (Vec n a, Vec n b, Vec n c, Vec n d, Vec n e)
 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)
 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)
 foldr :: (a > b > b) > b > Vec n a > b
 foldl :: (b > a > b) > b > Vec n a > b
 foldr1 :: (a > a > a) > Vec (n + 1) a > a
 foldl1 :: (a > a > a) > Vec (n + 1) a > a
 fold :: (a > a > a) > Vec (n + 1) a > a
 ifoldr :: KnownNat n => (Index n > a > b > b) > b > Vec n a > b
 ifoldl :: KnownNat n => (a > Index n > b > a) > a > Vec n b > a
 dfold :: forall p k a. KnownNat k => Proxy (p :: TyFun Nat Type > Type) > (forall l. SNat l > a > (p @@ l) > p @@ (l + 1)) > (p @@ 0) > Vec k a > p @@ k
 dtfold :: forall p k a. KnownNat k => Proxy (p :: TyFun Nat Type > Type) > (a > p @@ 0) > (forall l. SNat l > (p @@ l) > (p @@ l) > p @@ (l + 1)) > Vec (2 ^ k) a > p @@ k
 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
 scanl :: (b > a > b) > b > Vec n a > Vec (n + 1) b
 scanr :: (a > b > b) > b > Vec n a > Vec (n + 1) b
 postscanl :: (b > a > b) > b > Vec n a > Vec n b
 postscanr :: (a > b > b) > b > Vec n a > Vec n b
 mapAccumL :: (acc > x > (acc, y)) > acc > Vec n x > (acc, Vec n y)
 mapAccumR :: (acc > x > (acc, y)) > acc > Vec n x > (acc, Vec n y)
 stencil1d :: KnownNat n => SNat (stX + 1) > (Vec (stX + 1) a > b) > Vec ((stX + n) + 1) a > Vec (n + 1) b
 stencil2d :: (KnownNat n, KnownNat m) => SNat (stY + 1) > SNat (stX + 1) > (Vec (stY + 1) (Vec (stX + 1) a) > b) > Vec ((stY + m) + 1) (Vec ((stX + n) + 1) a) > Vec (m + 1) (Vec (n + 1) b)
 windows1d :: KnownNat n => SNat (stX + 1) > Vec ((stX + n) + 1) a > Vec (n + 1) (Vec (stX + 1) a)
 windows2d :: (KnownNat n, KnownNat m) => SNat (stY + 1) > SNat (stX + 1) > Vec ((stY + m) + 1) (Vec ((stX + n) + 1) a) > Vec (m + 1) (Vec (n + 1) (Vec (stY + 1) (Vec (stX + 1) a)))
 toList :: Vec n a > [a]
 bv2v :: KnownNat n => BitVector n > Vec n Bit
 v2bv :: KnownNat n => Vec n Bit > BitVector n
 lazyV :: KnownNat n => Vec n a > Vec n a
 data VCons (a :: Type) (f :: TyFun Nat Type) :: Type
 asNatProxy :: Vec n a > Proxy n
 seqV :: KnownNat n => Vec n a > b > b
 forceV :: KnownNat n => Vec n a > Vec n a
 seqVX :: KnownNat n => Vec n a > b > b
 forceVX :: KnownNat n => Vec n a > Vec n a
 traverse# :: forall a f b n. Applicative f => (a > f b) > Vec n a > f (Vec n b)
 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)
Vec
tor data type
data Vec :: Nat > Type > Type where Source #
Fixed size vectors.
pattern (:>) :: a > Vec n a > Vec (n + 1) a infixr 5  Add an element to the head of a vector.
Can be used as a pattern:
Also in conjunctions with (

pattern (:<) :: Vec n a > a > Vec (n + 1) a infixl 5  Add an element to the tail of a vector.
Can be used as a pattern:
Also in conjunctions with (

Instances
Functor (Vec n) Source #  
KnownNat n => Applicative (Vec n) Source #  
(KnownNat n, 1 <= n) => Foldable (Vec n) Source #  
Defined in Clash.Sized.Vector 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 # elem :: Eq a => a > Vec n a > Bool # maximum :: Ord a => Vec n a > a # minimum :: Ord a => Vec n a > a #  
(KnownNat n, 1 <= n) => Traversable (Vec n) Source #  
(KnownNat n, Eq a) => Eq (Vec n a) Source #  
(KnownNat n, Typeable a, Data a) => Data (Vec n a) Source #  
Defined in Clash.Sized.Vector 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) Source #  
Show a => Show (Vec n a) Source #  
KnownNat n => Generic (Vec n a) Source #  In many cases, this Generic instance only allows generic functions/instances over vectors of at least size 1, due to the n1 in the Rep (Vec n a) definition. We'll have to wait for things like https://ryanglscott.github.io/2018/02/11/howtoderivegenericforsomegadts/ before we can work around this limitation 
Lift a => Lift (Vec n a) Source #  
(KnownNat n, Arbitrary a) => Arbitrary (Vec n a) Source #  
CoArbitrary a => CoArbitrary (Vec n a) Source #  
Defined in Clash.Sized.Vector coarbitrary :: Vec n a > Gen b > Gen b #  
(Default a, KnownNat n) => Default (Vec n a) Source #  
Defined in Clash.Sized.Vector  
NFData a => NFData (Vec n a) Source #  
Defined in Clash.Sized.Vector  
KnownNat n => Ixed (Vec n a) Source #  
Defined in Clash.Sized.Vector  
(NFDataX a, KnownNat n) => NFDataX (Vec n a) Source #  
Defined in Clash.Sized.Vector  
ShowX a => ShowX (Vec n a) Source #  
(KnownNat n, BitPack a) => BitPack (Vec n a) Source #  
KnownNat n => Bundle (Vec n a) Source #  
KnownNat n => Bundle (Vec n a) Source #  
Defined in Clash.Signal.Delayed.Bundle  
(KnownNat n, AutoReg a) => AutoReg (Vec n a) Source #  
Defined in Clash.Class.AutoReg.Internal  
(LockStep en a, KnownNat n) => LockStep (Vec n en) (Vec n a) Source #  
type Unbundled t d (Vec n a) Source #  
Defined in Clash.Signal.Delayed.Bundle  
type HasDomain dom (Vec n a) Source #  
Defined in Clash.Class.HasDomain.HasSpecificDomain  
type TryDomain t (Vec n a) Source #  
Defined in Clash.Class.HasDomain.HasSingleDomain  
type Unbundled t (Vec n a) Source #  
Defined in Clash.Signal.Bundle  
type Rep (Vec n a) Source #  
Defined in Clash.Sized.Vector type Rep (Vec n a) = D1 ('MetaData "Vec" "Clash.Data.Vector" "clashprelude" '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) Source #  
Defined in Clash.Sized.Vector  
type IxValue (Vec n a) Source #  
Defined in Clash.Sized.Vector  
type BitSize (Vec n a) Source #  
Defined in Clash.Sized.Vector 
Accessors
Length information
Indexing
(!!) :: (KnownNat n, Enum i) => Vec n a > i > a Source #
"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 Source #
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 Source #
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
indices :: KnownNat n => SNat n > Vec n (Index n) Source #
Generate a vector of indices.
>>>
indices d4
<0,1,2,3>
indicesI :: KnownNat n => Vec n (Index n) Source #
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>
Extracting subvectors (slicing)
tail :: Vec (n + 1) a > Vec n a Source #
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 Source #
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 Source #
"take
n xs
" returns the nlength 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 Source #
"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 Source #
"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 Source #
"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 Source #
"select
f s n xs
" selects n elements with stepsize 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 Source #
"selectI
f s xs
" selects as many elements as demanded by the context
with stepsize 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) Source #
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) Source #
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) Source #
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) Source #
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
replicate :: SNat n > a > Vec n a Source #
"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 Source #
"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 Source #
"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:
generate :: SNat n > (a > a) > a > Vec n a Source #
"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:
Initialisation from a list
listToVecTH :: Lift a => [a] > ExpQ Source #
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 Source #
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 Source #
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 Source #
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 Source #
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 Source #
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>
:: 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>)
:: 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>)
:: (Default a, KnownNat m)  
=> SNat m 

> 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>)
:: (Default a, KnownNat n)  
=> SNat m 

> 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 Source #
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 Source #
"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
:: (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.
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>
:: (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.
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 Source #
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) Source #
Transpose a matrix: go from rowmajor to columnmajor
>>>
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_(d1),x_(2d1),x_(3d1)>
>>>
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 Source #
Dynamically rotate a Vec
tor 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 Source #
Dynamically rotate a Vec
tor 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 Source #
Statically rotate a Vec
tor 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 Source #
Statically rotate a Vec
tor 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.
Elementwise operations
Mapping
map :: (a > b) > Vec n a > Vec n b Source #
"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 Source #
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 Source #
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 Source #
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 Source #
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.
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 Source #
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 Source #
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 Source #
zip :: Vec n a > Vec n b > Vec n (a, b) Source #
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) Source #
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)>
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) Source #
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) Source #
izipWith :: KnownNat n => (Index n > a > b > c) > Vec n a > Vec n b > Vec n c Source #
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) Source #
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) Source #
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>)
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) Source #
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) Source #
Folding
foldr :: (a > b > b) > b > Vec n a > b Source #
foldr
, applied to a binary operator, a starting value (typically
the rightidentity 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: "
produces a linear structure, which has a depth, or
delay, of O(foldr
f z xs"
). Use length
xsfold
if your binary operator f
is
associative, as "
produces a structure with a depth of
O(log_2(fold
f xs"
)).length
xs
foldl :: (b > a > b) > b > Vec n a > b Source #
foldl
, applied to a binary operator, a starting value (typically
the leftidentity 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)
8.333333333333333e3
"foldl
f z xs
" corresponds to the following circuit layout:
NB: "
produces a linear structure, which has a depth, or
delay, of O(foldl
f z xs"
). Use length
xsfold
if your binary operator f
is
associative, as "
produces a structure with a depth of
O(log_2(fold
f xs"
)).length
xs
foldr1 :: (a > a > a) > Vec (n + 1) a > a Source #
foldr1
is a variant of foldr
that has no starting value argument,
and thus must be applied to nonempty 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: "
produces a linear structure, which has a depth,
or delay, of O(foldr1
f z xs"
). Use length
xsfold
if your binary operator f
is
associative, as "
produces a structure with a depth of
O(log_2(fold
f xs"
)).length
xs
foldl1 :: (a > a > a) > Vec (n + 1) a > a Source #
foldl1
is a variant of foldl
that has no starting value argument,
and thus must be applied to nonempty 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)
8.333333333333333e3
"foldl1
f xs
" corresponds to the following circuit layout:
NB: "
produces a linear structure, which has a depth,
or delay, of O(foldl1
f z xs"
). Use length
xsfold
if your binary operator f
is
associative, as "
produces a structure with a depth of
O(log_2(fold
f xs"
)).length
xs
fold :: (a > a > a) > Vec (n + 1) a > a Source #
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 treelike
structure. The depth, or delay, of the structure produced by
"
", is hence fold
f xsO(log_2(
, and not
length
xs))O(
.length
xs)
NB: The binary operator "f
" in "
" must be associative.fold
f xs
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 Source #
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 Source #
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
:: 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 
> (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 instanceApply
(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: "
" creates a linear structure, which has a depth,
or delay, of O(dfold
m f z xs
). Look at length
xsdtfold
for a dependently typed
fold that produces a structure with a depth of O(log_2(
)).length
xs
:: 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 
> 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 treelike 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 .. k1]
(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)
)
treestructure 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 bitwidth, 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 treestructure of
add
es:
>>>
:{
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 instanceApply
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 offold
, we had to restrict ourBitVector
argument to have bitwidth 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 bitwidth for every layer, from the VHDL/(System)Verilog produced by the Clash compiler.
NB: The depth, or delay, of the structure produced by
"
" is O(log_2(dtfold
m f g xs
)).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 Source #
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 Source #
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 Source #
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
mapAccumL :: (acc > x > (acc, y)) > acc > Vec n x > (acc, Vec n y) Source #
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) Source #
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
:: 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 
1dimensional 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>
:: (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) 
2dimensional 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>>
:: 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>>
:: (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
Misc
lazyV :: KnownNat n => Vec n a > Vec n a Source #
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 zipWith
s 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 Source #
seqV :: KnownNat n => Vec n a > b > b infixr 0 Source #
Evaluate all elements of a vector to WHNF, returning the second argument
seqVX :: KnownNat n => Vec n a > b > b infixr 0 Source #
Evaluate all elements of a vector to WHNF, returning the second argument.
Does not propagate XException
s.
forceVX :: KnownNat n => Vec n a > Vec n a Source #
Evaluate all elements of a vector to WHNF. Does not propagate XException
s.