Copyright  (C) 20132015, University of Twente 

License  BSD2 (see the file LICENSE) 
Maintainer  Christiaan Baaij <christiaan.baaij@gmail.com> 
Safe Haskell  None 
Language  Haskell2010 
Extensions 

 data Vec :: Nat > * > * where
 (<:) :: Vec n a > a > Vec (n + 1) a
 singleton :: a > Vec 1 a
 head :: Vec (n + 1) a > a
 tail :: Vec (n + 1) a > Vec n a
 last :: Vec (n + 1) a > 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
 at :: SNat m > Vec (m + (n + 1)) a > 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
 (++) :: Vec n a > Vec m a > Vec (n + m) a
 (+>>) :: KnownNat n => a > Vec n a > Vec n a
 (<<+) :: KnownNat n => Vec n a > a > Vec n a
 concat :: Vec n (Vec m a) > Vec (n * m) a
 zip :: Vec n a > Vec n b > Vec n (a, b)
 unzip :: Vec n (a, b) > (Vec n a, Vec n b)
 zip3 :: Vec n a > Vec n b > Vec n c > Vec n (a, b, c)
 unzip3 :: Vec n (a, b, c) > (Vec n a, Vec n b, Vec n c)
 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 (m + n)) => SNat m > Vec (m + n) a > (Vec (m + n) a, Vec m 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)
 merge :: Vec n a > Vec n a > Vec (n + n) a
 map :: (a > b) > Vec n a > Vec n 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
 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
 scanl :: (b > a > b) > b > Vec n a > Vec (n + 1) b
 scanr :: (a > b > b) > b > Vec n a > Vec (n + 1) b
 sscanl :: (b > a > b) > b > Vec n a > Vec n b
 sscanr :: (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)
 dfold :: Proxy (p :: TyFun Nat * > *) > (forall l. Proxy l > a > (p $ l) > p $ (l + 1)) > (p $ 0) > Vec k a > p $ k
 vfold :: (forall l. a > Vec l b > Vec (l + 1) b) > Vec k a > Vec k b
 (!!) :: (KnownNat n, Integral i) => Vec n a > i > a
 replace :: (KnownNat n, Integral i) => Vec n a > i > a > Vec n a
 maxIndex :: KnownNat n => Vec n a > Integer
 length :: KnownNat n => Vec n a > Integer
 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
 reverse :: Vec n a > Vec n a
 toList :: Vec n a > [a]
 v :: Lift a => [a] > ExpQ
 lazyV :: KnownNat n => Vec n a > Vec n a
 asNatProxy :: Vec n a > Proxy n
 concatBitVector# :: KnownNat m => Vec n (BitVector m) > BitVector (n * m)
 unconcatBitVector# :: (KnownNat n, KnownNat m) => BitVector (n * m) > Vec n (BitVector m)
Vec
tor constructors
data Vec :: Nat > * > * where Source
Fixed size vectors
 Lists with their length encoded in their type
Vec
tor elements have an ASCENDING subscript starting from 0 and ending atmaxIndex
(==length
 1).
>>>
(3:>4:>5:>Nil)
<3,4,5>>>>
:t (3:>4:>5:>Nil)
(3:>4:>5:>Nil) :: Num a => Vec (2 + 1) a
Functor (Vec n) Source  
KnownNat n => Applicative (Vec n) Source  NB: Not synthesisable 
Foldable (Vec n) Source  
Traversable (Vec n) Source  NB: Not synthesisable 
Eq a => Eq (Vec n a) Source  
Ord a => Ord (Vec n a) Source  
Show a => Show (Vec n a) Source  
(Default a, KnownNat n) => Default (Vec n a) Source  
Lift a => Lift (Vec n a) Source  
(KnownNat n, KnownNat (BitSize a), BitPack a) => BitPack (Vec n a) Source  
KnownNat n => Bundle (Vec n a) Source  
type Unbundled' t (Vec n a) = Vec n (Signal' t a) Source  
type BitSize (Vec n a) = * n (BitSize a) Source 
(<:) :: Vec n a > a > Vec (n + 1) a infixl 5 Source
Add an element to the tail of a vector.
>>>
(3:>4:>5:>Nil) <: 1
<3,4,5,1>>>>
:t (3:>4:>5:>Nil) <: 1
(3:>4:>5:>Nil) <: 1 :: Num a => Vec (3 + 1) a
Standard Vec
tor functions
Extracting subVec
tors
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
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
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
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
take :: SNat m > Vec (m + n) a > Vec m a Source
take
n
, applied to a vector 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’ The type variable ‘n0’ is ambiguous Expected type: Vec (4 + n0) a Actual type: Vec (1 + 1) a 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>:... Couldn't match expected type ‘2’ with actual type ‘4 + n0’ The type variable ‘n0’ is ambiguous In the first argument of ‘print’, namely ‘it’ In a stmt of an interactive GHCi command: print it
dropI :: 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>
Combining Vec
tors
(++) :: 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
<>
(<<+) :: KnownNat n => 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>
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)>
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>)
zip3 :: Vec n a > Vec n b > Vec n c > Vec n (a, b, c) Source
zip
takes three vectors and returns a vector of corresponding triplets.
>>>
zip3 (1:>2:>3:>4:>Nil) (4:>3:>2:>1:>Nil) (5:>6:>7:>8:>Nil)
<(1,4,5),(2,3,6),(3,2,7),(4,1,8)>
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>)
:: 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 (m + 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>)
Splitting Vec
tors
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 with length m, where 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 with length m, where 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>>
merge :: Vec n a > Vec n a > Vec (n + 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>
Applying functions to Vec
tor elements
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)
zipWith :: (a > b > c) > Vec n a > Vec n b > Vec n c Source
zipWith
generalises zip
by zipping with the function given
as the first argument, instead of a tupling function.
For example,
is applied to two vectors to produce the
vector of corresponding sums.zipWith
(+)
zipWith f (x1 :> x2 :> ... xn :> Nil) (y1 :> y2 :> ... :> yn :> Nil) == (f x1 y1 :> f x2 y2 :> ... :> f xn yn :> Nil)
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
generalises zip3
by zipping with the function given
as the first argument, instead of a tupling function.
zipWith3 f (x1 :> x2 :> ... xn :> Nil) (y1 :> y2 :> ... :> yn :> Nil) (z1 :> z2 :> ... :> zn :> Nil) == (f x1 y1 z1 :> f x2 y2 z2 :> ... :> f xn yn zn :> Nil)
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.
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
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
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
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
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.
NB: Not synthesisablefold
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
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>)
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>)
Special folds
:: Proxy (p :: TyFun Nat * > *)  The motive 
> (forall l. Proxy l > a > (p $ l) > p $ (l + 1))  Function to fold 
> (p $ 0)  Initial element 
> Vec k a  Vector to fold over 
> p $ k 
A dependently typed fold.
NB: Not synthesisable
Using lists, we can define append
(++
) using foldr
:
>>>
import qualified Prelude
>>>
let append xs ys = Prelude.foldr (:) ys xs
>>>
append [1,2] [3,4]
[1,2,3,4]
However, when we try to do the same for Vec
:
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: n1 ~ n1 + 1 Expected type: a > Vec n1 a > Vec n1 a Actual type: a > Vec n1 a > Vec (n1 + 1) a Relevant bindings include ys :: Vec n1 a (bound at ...) append' :: Vec n a > Vec n1 a > Vec n1 a (bound at ...) In the first argument of ‘foldr’, namely ‘(:>)’ In the expression: foldr (:>) ys xs
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 index into the vector. Using dfold
, we can
now correctly define (++
):
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' :: 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>
vfold :: (forall 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.
NB: Not synthesisable
Example:
cs a b = if a > b then (a,b) else (b,a) csRow y xs = let (y',xs') =mapAccumL
cs y xs in xs'<:
y' csSort =vfold
csRow
Builds a triangular structure of compare and swaps to sort a row.
>>>
csSort (7 :> 3 :> 9 :> 1 :> Nil)
<1,3,7,9>
Indexing Vec
tors
(!!) :: (KnownNat n, Integral i) => Vec n a > i > a Source
Vector index (subscript) operator.
NB: vector elements have an ASCENDING subscript starting from 0 and
ending at maxIndex
.
>>>
(1:>2:>3:>4:>5:>Nil) !! 4
5>>>
(1:>2:>3:>4:>5:>Nil) !! maxIndex (1:>2:>3:>4:>5:>Nil)
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
replace :: (KnownNat n, Integral i) => Vec n a > i > a > Vec n a Source
Replace an element of a vector at the given index (subscript).
NB: vector elements have an ASCENDING subscript starting from 0 and
ending at maxIndex
.
>>>
replace (1:>2:>3:>4:>5:>Nil) 3 7
<1,2,3,7,5>>>>
replace (1:>2:>3:>4:>5:>Nil) 0 7
<7,2,3,4,5>>>>
replace (1:>2:>3:>4:>5:>Nil) 9 7
<1,2,3,4,*** Exception: CLaSH.Sized.Vector.replace: index 9 is larger than maximum index 4
maxIndex :: KnownNat n => Vec n a > Integer Source
Index (subscript) of the last element in a Vec
tor
>>>
maxIndex (6 :> 7 :> 8 :> Nil)
2
length :: KnownNat n => Vec n a > Integer Source
Length of a Vec
tor as an Integer
>>>
length (6 :> 7 :> 8 :> Nil)
3
Generating Vec
tors
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>
iterateI :: KnownNat n => (a > a) > a > Vec n a Source
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>
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>
generateI :: KnownNat n => (a > a) > a > Vec n a Source
generate
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>
Misc
reverse :: Vec n a > Vec n a Source
Returns the elements in a vector in reverse order
>>>
reverse (1:>2:>3:>4:>Nil)
<4,3,2,1>
toList :: Vec n a > [a] Source
Convert a vector to a list
>>>
toList (1:>2:>3:>Nil)
[1,2,3]
NB: Not synthesisable
v :: Lift a => [a] > ExpQ Source
Create a vector literal from a list literal
$(v [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]>>>
$(v [1::Signed 8,2,3,4,5])
<1,2,3,4,5>
lazyV :: KnownNat n => Vec n a > Vec n a Source
For 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:
>>>
sortV (4 :> 1 :> 2 :> 3 :> Nil)
<*** Exception: <<loop>>
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>