Control.Arrow

The basic arrow class.

Instances should satisfy the following laws:

• arr id = id

• arr (f >>> g) = arr f >>> arr g

• first (arr f) = arr (first f)

• first (f >>> g) = first f >>> first g

• first f >>> arr fst = arr fst >>> f

• first f >>> arr (id *** g) = arr (id *** g) >>> first f

• first (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.

Precomposition with a pure function (right-to-left variant).

Postcomposition with a pure function.

Postcomposition with a pure function (right-to-left variant).

Precomposition with a pure function.

Any instance of ArrowApply can be made into an instance of ArrowChoice by defining left = leftApp.

The identity arrow, which plays the role of return in arrow notation.

Right-to-left composition

Left-to-right composition

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).

Choice, for arrows that support it. This class underlies the if and case constructs in arrow notation.

Instances should satisfy the following laws:

• left (arr f) = arr (left f)

• left (f >>> g) = left f >>> left g

• f >>> arr Left = arr Left >>> left f

• left f >>> arr (id +++ g) = arr (id +++ g) >>> left f

• left (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.

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)

The ArrowApply class is equivalent to Monad: any monad gives rise to a Kleisli arrow, and any instance of ArrowApply defines a monad.

A monoid on arrows.

Kleisli arrows of a monad.