Create a synchronous function from a combinational function describing a mealy machine

import qualified Data.List as L

macT
  :: Int        -- Current state
  -> (Int,Int)  -- Input
  -> (Int,Int)  -- (Updated state, output)
macT s (x,y) = (s',s)
  where
    s' = x * y + s

mac
  :: KnownDomain dom
  => Clock dom
  -> Reset dom
  -> Enable dom
  -> Signal dom (Int, Int)
  -> Signal dom Int
mac clk rst en = mealy clk rst en macT 0

>>> simulate (mac systemClockGen systemResetGen enableGen) [(0,0),(1,1),(2,2),(3,3),(4,4)]
[0,0,1,5,14...
...

Synchronous sequential functions can be composed just like their combinational counterpart:

dualMac
  :: KnownDomain dom
  => Clock dom
  -> Reset dom
  -> Enable dom
  -> (Signal dom Int, Signal dom Int)
  -> (Signal dom Int, Signal dom Int)
  -> Signal dom Int
dualMac clk rst en (a,b) (x,y) = s1 + s2
  where
    s1 = mealy clk rst en macT 0 (bundle (a,x))
    s2 = mealy clk rst en macT 0 (bundle (b,y))

A version of mealy that does automatic Bundleing

Given a function f of type:

f :: Int -> (Bool,Int) -> (Int,(Int,Bool))

When we want to make compositions of f in g using mealy, we have to write:

g clk rst en a b c = (b1,b2,i2)
  where
    (i1,b1) = unbundle (mealy clk rst en f 0 (bundle (a,b)))
    (i2,b2) = unbundle (mealy clk rst en f 3 (bundle (c,i1)))

Using mealyB however we can write:

g clk rst en a b c = (b1,b2,i2)
  where
    (i1,b1) = mealyB clk rst en f 0 (a,b)
    (i2,b2) = mealyB clk rst en f 3 (c,i1)

Create a synchronous function from a combinational function describing a moore machine

macT
  :: Int        -- Current state
  -> (Int,Int)  -- Input
  -> (Int,Int)  -- Updated state
macT s (x,y) = x * y + s

mac
  :: KnownDomain dom
  => Clock dom
  -> Reset dom
  -> Enable dom
  -> Signal dom (Int, Int)
  -> Signal dom Int
mac clk rst en = moore clk rst en macT id 0

>>> simulate (mac systemClockGen systemResetGen enableGen) [(0,0),(1,1),(2,2),(3,3),(4,4)]
[0,0,1,5,14...
...

Synchronous sequential functions can be composed just like their combinational counterpart:

dualMac
  :: KnownDomain dom
  => Clock dom
  -> Reset dom
  -> Enable dom
  -> (Signal dom Int, Signal dom Int)
  -> (Signal dom Int, Signal dom Int)
  -> Signal dom Int
dualMac clk rst en (a,b) (x,y) = s1 + s2
  where
    s1 = moore clk rst en macT id 0 (bundle (a,x))
    s2 = moore clk rst en macT id 0 (bundle (b,y))

A version of moore that does automatic Bundleing

Given a functions t and o of types:

t :: Int -> (Bool, Int) -> Int
o :: Int -> (Int, Bool)

When we want to make compositions of t and o in g using moore, we have to write:

g clk rst en a b c = (b1,b2,i2)
  where
    (i1,b1) = unbundle (moore clk rst en t o 0 (bundle (a,b)))
    (i2,b2) = unbundle (moore clk rst en t o 3 (bundle (c,i1)))

Using mooreB however we can write:

g clk rst en a b c = (b1,b2,i2)
  where
    (i1,b1) = mooreB clk rst en t o 0 (a,b)
    (i2,b2) = mooreB clk rst en t o 3 (c,i1)

Create a register function for product-type like signals (e.g. (Signal a, Signal b))

rP :: Clock dom -> Reset dom -> Enable dom
   -> (Signal dom Int, Signal dom Int)
   -> (Signal dom Int, Signal dom Int)
rP clk rst en = registerB clk rst en (8,8)

>>> simulateB (rP systemClockGen systemResetGen enableGen) [(1,1),(1,1),(2,2),(3,3)] :: [(Int,Int)]
[(8,8),(8,8),(1,1),(2,2),(3,3)...
...

Synchronizer based on two sequentially connected flip-flops.

• NB: This synchronizer can be used for bit-synchronization.

• NB: Although this synchronizer does reduce metastability, it does not guarantee the proper synchronization of a whole word. For example, given that the output is sampled twice as fast as the input is running, and we have two samples in the input stream that look like:

  [0111,1000]

  But the circuit driving the input stream has a longer propagation delay on msb compared to the lsbs. What can happen is an output stream that looks like this:

  [0111,0111,0000,1000]

  Where the level-change of the msb was not captured, but the level change of the lsbs were.

  If you want to have safe word-synchronization use asyncFIFOSynchronizer.

Synchronizer implemented as a FIFO around a synchronous RAM. Based on the design described in Clash.Tutorial, which is itself based on the design described in http://www.sunburst-design.com/papers/CummingsSNUG2002SJ_FIFO1.pdf. However, this FIFO uses a synchronous dual-ported RAM which, unlike those designs using RAM with an asynchronous read port, is nearly guaranteed to actually synthesize into one of the dual-ported RAMs found on most FPGAs.

NB: This synchronizer can be used for word-synchronization. NB: This synchronizer will only work safely when you set up the proper bus skew and maximum delay constraints inside your synthesis tool for the clock domain crossings of the gray pointers.

An asynchronous/combinational ROM with space for n elements

See also:

• See Clash.Sized.Fixed and Clash.Prelude.BlockRam for ideas on how to use ROMs and RAMs.
• A large Vec for the content may be too inefficient, depending on how it is constructed. See asyncRomFile and asyncRomBlob for different approaches that scale well.

An asynchronous/combinational ROM with space for 2^n elements

See also:

• See Clash.Sized.Fixed and Clash.Prelude.BlockRam for ideas on how to use ROMs and RAMs.
• A large Vec for the content may be too inefficient, depending on how it is constructed. See asyncRomFilePow2 and asyncRomBlobPow2 for different approaches that scale well.

A ROM with a synchronous read port, with space for n elements

• NB: Read value is delayed by 1 cycle
• NB: Initial output value is undefined, reading it will throw an XException

See also:

• See Clash.Sized.Fixed and Clash.Explicit.BlockRam for ideas on how to use ROMs and RAMs.
• A large Vec for the content may be too inefficient, depending on how it is constructed. See romFile and romBlob for different approaches that scale well.

A ROM with a synchronous read port, with space for 2^n elements

• NB: Read value is delayed by 1 cycle
• NB: Initial output value is undefined, reading it will throw an XException

See also:

• See Clash.Sized.Fixed and Clash.Explicit.BlockRam for ideas on how to use ROMs and RAMs.
• A large Vec for the content may be too inefficient, depending on how it is constructed. See romFilePow2 and romBlobPow2 for different approaches that scale well.

An asynchronous/combinational ROM with space for n elements

See also:

• See Clash.Sized.Fixed and Clash.Prelude.BlockRam for ideas on how to use ROMs and RAMs.

An asynchronous/combinational ROM with space for 2^n elements

See also:

• See Clash.Sized.Fixed and Clash.Prelude.BlockRam for ideas on how to use ROMs and RAMs.

A ROM with a synchronous read port, with space for n elements

• NB: Read value is delayed by 1 cycle
• NB: Initial output value is undefined, reading it will throw an XException

See also:

• See Clash.Sized.Fixed and Clash.Explicit.BlockRam for ideas on how to use ROMs and RAMs.

A ROM with a synchronous read port, with space for 2^n elements

• NB: Read value is delayed by 1 cycle
• NB: Initial output value is undefined, reading it will throw an XException

See also:

• See Clash.Sized.Fixed and Clash.Explicit.BlockRam for ideas on how to use ROMs and RAMs.

Create a RAM with space for n elements

• NB: Initial content of the RAM is undefined, reading it will throw an XException

See also:

• See Clash.Explicit.BlockRam for more information on how to use a RAM.

Create a RAM with space for 2^n elements

• NB: Initial content of the RAM is undefined, reading it will throw an XException

See also:

• See Clash.Prelude.BlockRam for more information on how to use a RAM.

Create a block RAM with space for n elements

• NB: Read value is delayed by 1 cycle
• NB: Initial output value is undefined, reading it will throw an XException

See also:

• See Clash.Explicit.BlockRam for more information on how to use a block RAM.
• Use the adapter readNew for obtaining write-before-read semantics like this: readNew clk rst en (blockRam clk inits) rd wrM.
• A large Vec for the initial content may be too inefficient, depending on how it is constructed. See blockRamFile and blockRamBlob for different approaches that scale well.

Example

bram40
  :: Clock  dom
  -> Enable  dom
  -> Signal dom (Unsigned 6)
  -> Signal dom (Maybe (Unsigned 6, Bit))
  -> Signal dom Bit
bram40 clk en = blockRam clk en (replicate d40 1)

Create a block RAM with space for 2^n elements

• NB: Read value is delayed by 1 cycle
• NB: Initial output value is undefined, reading it will throw an XException

See also:

• See Clash.Prelude.BlockRam for more information on how to use a block RAM.
• Use the adapter readNew for obtaining write-before-read semantics like this: readNew clk rst en (blockRamPow2 clk inits) rd wrM.
• A large Vec for the initial content may be too inefficient, depending on how it is constructed. See blockRamFilePow2 and blockRamBlobPow2 for different approaches that scale well.

Example

bram32
  :: Clock dom
  -> Enable dom
  -> Signal dom (Unsigned 5)
  -> Signal dom (Maybe (Unsigned 5, Bit))
  -> Signal dom Bit
bram32 clk en = blockRamPow2 clk en (replicate d32 1)

Create a block RAM with space for n elements

• NB: Read value is delayed by 1 cycle
• NB: Initial output value is undefined, reading it will throw an XException

See also:

• See Clash.Prelude.BlockRam for more information on how to use a block RAM.
• Use the adapter readNew for obtaining write-before-read semantics like this: readNew clk rst en (blockRamBlob clk en content) rd wrM.

Create a block RAM with space for 2^n elements

• NB: Read value is delayed by 1 cycle
• NB: Initial output value is undefined, reading it will throw an XException

See also:

• See Clash.Prelude.BlockRam for more information on how to use a block RAM.
• Use the adapter readNew for obtaining write-before-read semantics like this: readNew clk rst en (blockRamBlobPow2 clk en content) rd wrM.

Efficient storage of memory content

It holds n words of BitVector m.

Create a MemBlob binding from a list of values

Since this uses Template Haskell, nothing in the arguments given to createMemBlob can refer to something defined in the same module.

Example

createMemBlob "content" Nothing [15 :: Unsigned 8 .. 17]

ram clk en = blockRamBlob clk en content

>>> import qualified Prelude as P
>>> let es = [ Nothing, Just (7 :: Unsigned 8), Just 8 ]
>>> :{
createMemBlob "content0" (Just 0) es
createMemBlob "content1" (Just 1) es
x = 1
:}

>>> let pr = mapM_ (putStrLn . show)
>>> pr $ P.map pack es
0b0_...._....
0b1_0000_0111
0b1_0000_1000
>>> pr $ unpackMemBlob content0
0b0_0000_0000
0b1_0000_0111
0b1_0000_1000
>>> pr $ unpackMemBlob content1
0b0_1111_1111
0b1_0000_0111
0b1_0000_1000

>>> :{
createMemBlob "contentN" Nothing es
x = 1
:}

<interactive>:...: error:...
    packBVs: cannot convert don't care values. Please specify a mapping to a definite value.

Create a MemBlob from a list of values

Since this uses Template Haskell, nothing in the arguments given to memBlobTH can refer to something defined in the same module.

Example

ram clk en = blockRamBlob clk en $(memBlobTH Nothing [15 :: Unsigned 8 .. 17])

>>> import qualified Prelude as P
>>> let es = [ Nothing, Just (7 :: Unsigned 8), Just 8 ]
>>> content0 = $(memBlobTH (Just 0) es)
>>> content1 = $(memBlobTH (Just 1) es)
>>> let pr = mapM_ (putStrLn . show)
>>> pr $ P.map pack es
0b0_...._....
0b1_0000_0111
0b1_0000_1000
>>> pr $ unpackMemBlob content0
0b0_0000_0000
0b1_0000_0111
0b1_0000_1000
>>> pr $ unpackMemBlob content1
0b0_1111_1111
0b1_0000_0111
0b1_0000_1000

>>> $(memBlobTH Nothing es)

<interactive>:...: error:...
    • packBVs: cannot convert don't care values. Please specify a mapping to a definite value.
    • In the untyped splice: $(memBlobTH Nothing es)

Convert a MemBlob back to a list

NB: Not synthesizable

Create a read-after-write block RAM from a read-before-write one

Produces vendor-agnostic HDL that will be inferred as a true dual-port block RAM

Any value that is being written on a particular port is also the value that will be read on that port, i.e. the same-port read/write behavior is: WriteFirst. For mixed-port read/write, when port A writes to the address port B reads from, the output of port B is undefined, and vice versa.

Port operation

Give a pulse when the Signal goes from minBound to maxBound

Give a pulse when the Signal goes from maxBound to minBound

Give a pulse every n clock cycles. This is a useful helper function when combined with functions like regEn or mux, in order to delay a register by a known amount.

Oscillate a Bool for a given number of cycles, given the starting state.

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.

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)

Convert a vector to a list.

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

NB: This function is not synthesizable

"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 :> Nil

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

Append two vectors.

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

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)).

"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:

"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
...

"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 :> Nil
>>> drop d3               (1:>2:>3:>4:>5:>Nil)
4 :> 5 :> Nil
>>> drop d0               (1:>2:>Nil)
1 :> 2 :> Nil

>>> 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

"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

"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

Extract the first element of a vector

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

>>> head Nil

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

"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 :> Nil

"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 :> Nil

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

"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 :> Nil

Split a vector into two vectors at the given point.

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

Extract the elements after the head of a vector

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

>>> tail Nil

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

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

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

>>> take d4               (1:>2:>Nil)

<interactive>:...
    • Couldn't match type ‘4 + n0’ with ‘2’
      Expected: Vec (4 + n0) a
        Actual: 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)

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 :> Nil) :> (3 :> 4 :> Nil) :> (5 :> 6 :> Nil) :> Nil
>>> transpose xss
(1 :> 3 :> 5 :> Nil) :> (2 :> 4 :> 6 :> Nil) :> Nil

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 :> Nil,4 :> 3 :> 2 :> 1 :> Nil)

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

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) :> Nil

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) :> Nil

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 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.

Generate a vector of indices.

>>> indices d4
0 :> 1 :> 2 :> 3 :> Nil

The length of a Vector as an Int value.

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

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)).

"unfoldr n f s" builds a vector of length n from a seed value s, where every element a is created by successive calls of f on s. Unlike unfoldr from Data.List the generating function f cannot dictate the length of the resulting vector, it must be statically known.

a simple use of unfoldr:

>>> unfoldr d10 (\s -> (s,s-1)) 10
10 :> 9 :> 8 :> 7 :> 6 :> 5 :> 4 :> 3 :> 2 :> 1 :> Nil

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 :> Nil

Extract the last element of a vector

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

>>> last Nil

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

Extract all the elements of a vector except the last element

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

>>> init Nil

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

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)).

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 :> Nil

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

• NB:

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

• For a different trade-off between circuit size and logic depth for associative operators, see Clash.Sized.RTree

scanl with no seed value

>>> scanl1 (-) (1 :> 2 :> 3 :> 4 :> Nil)
1 :> -1 :> -4 :> -8 :> Nil

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)).

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 :> Nil

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

• NB:

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

• For a different trade-off between circuit size and logic depth for associative operators, see Clash.Sized.RTree

scanr with no seed value

>>> scanr1 (-) (1 :> 2 :> 3 :> 4 :> Nil)
-2 :> 3 :> -1 :> 4 :> Nil

The largest element of a non-empty vector

The least element of a non-empty vector

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

>>> replicate (SNat :: SNat 3) 6
6 :> 6 :> 6 :> Nil
>>> replicate d3 6
6 :> 6 :> 6 :> Nil

The elements in a vector in reverse order.

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

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 :> Nil

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 :> Nil

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:

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 :> Nil)

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

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

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

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

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

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

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

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

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

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

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

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

Create a vector of one element

>>> singleton 5
5 :> Nil

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 :> Nil

"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 :> Nil
>>> replace 0 7 (1:>2:>3:>4:>5:>Nil)
7 :> 2 :> 3 :> 4 :> 5 :> Nil
>>> replace 9 7 (1:>2:>3:>4:>5:>Nil)
1 :> 2 :> 3 :> 4 :> 5 :> *** Exception: Clash.Sized.Vector.replace: index 9 is out of bounds: [0..4]
...

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:

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:

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 -> extend (bitCoerce i) + a) (2 :> 2 :> 2 :> 2 :> Nil) :: Vec 4 (Unsigned 8)
2 :> 3 :> 4 :> 5 :> Nil

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

"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

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

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 :> Nil

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

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 :> Nil

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 2 is out of bounds: [0..1]
...
>>> izipWith (\i a b -> extend (bitCoerce i) + a + b) (2 :> 2 :> Nil) (3 :> 3 :> Nil) :: Vec 2 (Unsigned 8)
5 :> 6 :> Nil

"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.

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 :> Nil

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

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

doctests setup

>>> let compareSwapL a b = if a < b then (a,b) else (b,a)
>>> :{
let sortVL :: (Ord a, KnownNat (n + 1)) => Vec ((n + 1) + 1) a -> Vec ((n + 1) + 1) a
    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
:}

>>> :{
let 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
:}

Example usage

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 :> Nil

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 :> Nil

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 :> Nil) :> (1 :> 2 :> 3 :> Nil) :> (1 :> 2 :> 3 :> Nil) :> Nil
>>> rotateMatrix xss
(1 :> 2 :> 3 :> Nil) :> (3 :> 1 :> 2 :> Nil) :> (2 :> 3 :> 1 :> Nil) :> Nil

"iterateI 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 :> Nil

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

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

doctests setup

>>> :seti -XUndecidableInstances
>>> import Data.Singletons (Apply, Proxy (..), TyFun)
>>> data IIndex (f :: TyFun Nat Type) :: Type
>>> type instance Apply IIndex l = Index ((2^l)+1)
>>> :{
let 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)
:}

Example usage

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: Index (n + 2) -> Index (n + 2) -> Index (n + 2)
        Actual: 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
import Data.Proxy

data IIndex (f :: TyFun Nat Type) :: Type
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)).

Length of a Vector as an SNat value

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 :> Nil