Copyright  (c) Ross Paterson 2002 

License  (c) Ross Paterson 2002 
Maintainer  libraries@haskell.org 
Stability  provisional 
Portability  portable 
Safe Haskell  Trustworthy 
Basic arrow definitions, based on
* Generalising Monads to Arrows, by John Hughes,
Science of Computer Programming 37, pp67111, May 2000.
plus a couple of definitions (returnA
and loop
) from
* A New Notation for Arrows, by Ross Paterson, in ICFP 2001,
Firenze, Italy, pp229240.
These papers and more information on arrows can be found at
http://www.haskell.org/arrows/.
 class Category a => Arrow a where
 newtype Kleisli m a b = Kleisli {
 runKleisli :: a > m b
 returnA :: Arrow a => a b b
 (^>>) :: Arrow a => (b > c) > a c d > a b d
 (>>^) :: Arrow a => a b c > (c > d) > a b d
 (>>>) :: Category cat => cat a b > cat b c > cat a c
 (<<<) :: Category cat => cat b c > cat a b > cat a c
 (<<^) :: Arrow a => a c d > (b > c) > a b d
 (^<<) :: Arrow a => (c > d) > a b c > a b d
 class Arrow a => ArrowZero a where
 zeroArrow :: a b c
 class ArrowZero a => ArrowPlus a where
 (<+>) :: a b c > a b c > a b c
 class Arrow a => ArrowChoice a where
 class Arrow a => ArrowApply a where
 app :: a (a b c, b) c
 newtype ArrowMonad a b = ArrowMonad (a () b)
 leftApp :: ArrowApply a => a b c > a (Either b d) (Either c d)
 class Arrow a => ArrowLoop a where
 loop :: a (b, d) (c, d) > a b c
Arrows
class Category a => Arrow a whereSource
The basic arrow class.
Minimal complete definition: arr
and first
, satisfying the laws
arr
id =id
arr
(f >>> g) =arr
f >>>arr
gfirst
(arr
f) =arr
(first
f)first
(f >>> g) =first
f >>>first
gfirst
f >>>arr
fst
=arr
fst
>>> ffirst
f >>>arr
(id
*** g) =arr
(id
*** g) >>>first
ffirst
(first
f) >>>arr
assoc
=arr
assoc
>>>first
f
where
assoc ((a,b),c) = (a,(b,c))
The other combinators have sensible default definitions, which may be overridden for efficiency.
arr :: (b > c) > a b cSource
Lift a function to an arrow.
first :: a b c > a (b, d) (c, d)Source
Send the first component of the input through the argument arrow, and copy the rest unchanged to the output.
second :: a b c > a (d, b) (d, c)Source
A mirror image of first
.
The default definition may be overridden with a more efficient version if desired.
(***) :: a b c > a b' c' > a (b, b') (c, c')Source
Split the input between the two argument arrows and combine their output. Note that this is in general not a functor.
The default definition may be overridden with a more efficient version if desired.
(&&&) :: a b c > a b c' > a b (c, c')Source
Fanout: send the input to both argument arrows and combine their output.
The default definition may be overridden with a more efficient version if desired.
Kleisli arrows of a monad.
Kleisli  

Monad m => Category * (Kleisli m)  
MonadFix m => ArrowLoop (Kleisli m)  Beware that for many monads (those for which the 
Monad m => ArrowApply (Kleisli m)  
Monad m => ArrowChoice (Kleisli m)  
MonadPlus m => ArrowPlus (Kleisli m)  
MonadPlus m => ArrowZero (Kleisli m)  
Monad m => Arrow (Kleisli m) 
Derived combinators
returnA :: Arrow a => a b bSource
The identity arrow, which plays the role of return
in arrow notation.
Righttoleft variants
(<<^) :: Arrow a => a c d > (b > c) > a b dSource
Precomposition with a pure function (righttoleft variant).
(^<<) :: Arrow a => (c > d) > a b c > a b dSource
Postcomposition with a pure function (righttoleft variant).
Monoid operations
Conditionals
class Arrow a => ArrowChoice a whereSource
Choice, for arrows that support it. This class underlies the
if
and case
constructs in arrow notation.
Minimal complete definition: left
, satisfying the laws
left
(arr
f) =arr
(left
f)left
(f >>> g) =left
f >>>left
gf >>>
arr
Left
=arr
Left
>>>left
fleft
f >>>arr
(id
+++ g) =arr
(id
+++ g) >>>left
fleft
(left
f) >>>arr
assocsum
=arr
assocsum
>>>left
f
where
assocsum (Left (Left x)) = Left x assocsum (Left (Right y)) = Right (Left y) assocsum (Right z) = Right (Right z)
The other combinators have sensible default definitions, which may be overridden for efficiency.
left :: a b c > a (Either b d) (Either c d)Source
Feed marked inputs through the argument arrow, passing the rest through unchanged to the output.
right :: a b c > a (Either d b) (Either d c)Source
A mirror image of left
.
The default definition may be overridden with a more efficient version if desired.
(+++) :: a b c > a b' c' > a (Either b b') (Either c c')Source
Split the input between the two argument arrows, retagging and merging their outputs. Note that this is in general not a functor.
The default definition may be overridden with a more efficient version if desired.
() :: a b d > a c d > a (Either b c) dSource
Fanin: Split the input between the two argument arrows and merge their outputs.
The default definition may be overridden with a more efficient version if desired.
ArrowChoice (>)  
Monad m => ArrowChoice (Kleisli m) 
Arrow application
class Arrow a => ArrowApply a whereSource
Some arrows allow application of arrow inputs to other inputs. Instances should satisfy the following laws:
first
(arr
(\x >arr
(\y > (x,y)))) >>>app
=id
first
(arr
(g >>>)) >>>app
=second
g >>>app
first
(arr
(>>> h)) >>>app
=app
>>> h
Such arrows are equivalent to monads (see ArrowMonad
).
ArrowApply (>)  
Monad m => ArrowApply (Kleisli m) 
newtype ArrowMonad a b Source
The ArrowApply
class is equivalent to Monad
: any monad gives rise
to a Kleisli
arrow, and any instance of ArrowApply
defines a monad.
ArrowMonad (a () b) 
ArrowApply a => Monad (ArrowMonad a)  
Arrow a => Functor (ArrowMonad a)  
(ArrowApply a, ArrowPlus a) => MonadPlus (ArrowMonad a)  
Arrow a => Applicative (ArrowMonad a)  
ArrowPlus a => Alternative (ArrowMonad a) 
leftApp :: ArrowApply a => a b c > a (Either b d) (Either c d)Source
Any instance of ArrowApply
can be made into an instance of
ArrowChoice
by defining left
= leftApp
.
Feedback
class Arrow a => ArrowLoop a whereSource
The loop
operator expresses computations in which an output value
is fed back as input, although the computation occurs only once.
It underlies the rec
value recursion construct in arrow notation.
loop
should satisfy the following laws:
 extension

loop
(arr
f) =arr
(\ b >fst
(fix
(\ (c,d) > f (b,d))))  left tightening

loop
(first
h >>> f) = h >>>loop
f  right tightening

loop
(f >>>first
h) =loop
f >>> h  sliding

loop
(f >>>arr
(id
*** k)) =loop
(arr
(id
*** k) >>> f)  vanishing

loop
(loop
f) =loop
(arr
unassoc >>> f >>>arr
assoc)  superposing

second
(loop
f) =loop
(arr
assoc >>>second
f >>>arr
unassoc)
where
assoc ((a,b),c) = (a,(b,c)) unassoc (a,(b,c)) = ((a,b),c)