Control.LensFunction

This module provides an "applicative" (functional) way of composing lenses through the data type L. For example, this module enables us to define a "lens" version of unlines as follows.

unlinesF :: [L s String] -> L s String
unlinesF []     = new ""
unlinesF (x:xs) = catLineF x (unlinesF xs)
  where catLineF = lift2 catLineL

catLineL :: Lens' (String, String) String
catLineL = ...

To make a lens from such "lens functions", one can use unlifting functions (unlift, unlift2, unliftT) as follows.

unlinesL :: Lens' [String] String
unlinesL = unliftT unlinesF

The obtained lens works as expected (here ^., & and .~ are taken from Control.Lens).

>>> ["banana", "orange", "apple"] ^. unlinesL
"banana\norange\napple\n"
>>> ["banana", "orange", "apple"] & unlinesL .~ "Banana\nOrange\nApple\n"
["Banana","Orange","Apple"]

One can understand that L s a is an updatable a. The type [L s String] -> L s String of unlinesF tells us that we can update only the list elements. Actually, insertion or deletion of lines to the view will fail, as below.

>>> ["banana", "orange", "apple"] & unlinesL .~ "Banana\nOrange\nApple"
*** Exception: ...
>>> ["banana", "orange", "apple"] & unlinesL .~ "Banana\nOrange\nApple\n\n"
*** Exception: ...

If you want to reflect insertions and deletions, one have to write a function of type L s [String] -> L s String, which says that the list structure itself would be updatable. To write a function of this type, liftC and liftC2 functions would be sometimes useful.

unlinesF' :: L s [String] -> L s String
unlinesF' = liftC (foldWithDefault "" "n") (lift catLineL')

catLineL' :: Lens' (Either () (String,String)) String
catLineL' = ...

foldWithDefault :: a -> (Lens' (Either () (a,b)) b) -> Lens' [a] b
foldWithDefault d f = ...

Core Datatype

data L s a

An abstract type for "updatable" data. Bidirectional programming through our module is to write manipulation of this datatype.

Categorical Notes

The type constructor L s together with lift, unit and pair defines a lax monoidal functor from the category of lenses to that of Haskell functions. The lift function does transfor a lens to a function. The unit and pair functions are the core operations on this lax monoidal functor. Any lifting functions defined in this module can be defined by these three functions.

Other constructors for Lens'

lens' :: (s -> (v, v -> s)) -> Lens' s v

A variant of lens. Sometimes, this function would be easier to use because one can re-use a source information to define a "put".

Functions handling pairs and containers

unit :: L s ()

The unit element in the lifted world.

Let elimUnitL and elimUnitR are lenses defined as follows.

elimUnitL = lens ((x,()) -> x) (_ x -> (x,()))
elimUnitR = lens (((),x) -> x) (_ x -> ((),x))

Then, we have the following laws.

lift2 elimUnitL x unit = x

lift2 elimUnitR unit x = x

pair :: L s a -> L s b -> L s (a, b)

A paring function of L s a-typed values. This function can be defined by lift2 as below.

pair = lift2 (lens id (const id))

list :: [L s a] -> L s [a]

Similar to pair, but this function is for lists. This is a derived function, because this can be defined by using lift and pair.

sequenceL :: (Eq (t ()), Traversable t) => t (L s a) -> L s (t a)

A data-type generic version of list. The contraint Eq (t ()) says that we can check the equivalence of shapes of containers t.

Lifting Functions

new :: Eq a => a -> L s a

The nullary version of a lifting function. Since there is no source, every view generated by new is not updatable.

The function will throw ConstantUpdateException, if its view is updated.

lift :: Lens' a b -> forall s. L s a -> L s b

The lifting function. Note that it forms a functor from the category of lenses to the category of sets and functions.

unlift is a left-inverse of this function.

unlift (lift x) = x

lift2 :: Lens' (a, b) c -> L s a -> L s b -> L s c

The lifting function for binary lenses. unlift2 is a left inverse of this function.

unlift2 (lift2 l) = l

This function can be defined from lift and pair as below.

lift2 l x y = lift l (pair x y)

NB: This is not a right inverse of unlift2.

(\x y -> x) /= lift2 (unlift2 (\x y -> x))

>>> set (unlift (\z -> (\x y -> x) z z)) "A" "B"
"B"
>>> set (unlift (\z -> lift2 (unlift2 (\x y -> x)) (z,z))) "A" "B"
Error: ...

liftT :: (Eq (t ()), Traversable t) => Lens' (t a) b -> forall s. t (L s a) -> L s b

A datatype-generic version of lift2

liftLens :: (a -> b) -> (a -> b -> a) -> forall s. L s a -> L s b

Just a composition of lift and lens. Sometimes, this function would be more efficient than the composition due to eliminated conversion from the lens to the internal representation.

Since both of the internal and the external representations are functions (= normal forms), we have to pay the conversion cost for each time when the lifted lens function is evaluated, even in the lazy evaluation.

We actually has the RULE to make the composition of lift and lens to liftLens. However, the rule may not be fired especially when profiling codes are inserted by GHC.

liftLens' :: (a -> (b, b -> a)) -> forall s. L s a -> L s b

Just a composition of lift and lens'. This function has the similar role to liftLens.

Unlifting Functions

unlift :: Eq a => (forall s. L s a -> L s b) -> Lens' a b

The unlifting function, satisfying unlift (lift x) = x.

unlift2 :: (Eq a, Eq b) => (forall s. L s a -> L s b -> L s c) -> Lens' (a, b) c

The unlifting function for binary functions, satisfying unlift2 (lift2 x) = x.

unliftT :: (Eq a, Eq (t ()), Traversable t) => (forall s. t (L s a) -> L s b) -> Lens' (t a) b

The unlifting function for functions that manipulate data structures, satisfying unliftT (liftT x) = x if x keeps the shape. The constraint Eq (t ()) says that we can compare the shapes of given two containers.

Functions for Handling Observations

Core Monad

data R s a

An abstract monad used to keep track of observations. By this monad, we can inspect the value of 'L s a'-data.

It is worth noting that we cannot change the inspection result to ensure the consistency property (aka PutGet in some context).

Instances

Monad (R s)

Functor (R s)

Applicative (R s)

Lifting Observations

observe :: Eq w => L s w -> R s w

A primitive used to define liftO and liftO2. With observe, one can inspect the current value of a lifted '(L s a)'-value as below.

f x :: L s A -> R s (L s B)
f x = do viewX <- observe x
         ... computation depending on viewX ...

Once the observe function is used in a lens function, the lens function becomes not able to change change the "observed" value to ensure the correctness.

liftO :: Eq w => (a -> w) -> L s a -> R s w

Lifting of observations. A typical use of this function would be as follows.

f x :: L s Int -> R s (L s B)
f x = do b liftO ( 0) x 
         if b then ... else ...           

liftO2 :: Eq w => (a -> b -> w) -> L s a -> L s b -> R s w

Lifting of binary observations

Unlifting Functions

unliftM :: Eq a => (forall s. L s a -> R s (L s b)) -> Lens' a b

A monadic version of unlift.

unliftM2 :: (Eq a, Eq b) => (forall s. L s a -> L s b -> R s (L s c)) -> Lens' (a, b) c

A monadic version of unlift2.

unliftMT :: (Eq a, Eq (t ()), Traversable t) => (forall s. t (L s a) -> R s (L s b)) -> Lens' (t a) b

A monadic version of unliftT.

Lifting Functions for Combinators

liftC :: Eq a => (Lens' a b -> Lens' c d) -> (forall s. L s a -> L s b) -> forall s. L s c -> L s d

A lifting function for lens combinators. One can understand that the universal quantification for the second argument as closedness restriction.

liftC2 :: (Eq a, Eq c) => (Lens' a b -> Lens' c d -> Lens' e f) -> (forall s. L s a -> L s b) -> (forall s. L s c -> L s d) -> forall s. L s e -> L s f

Similar to liftC, but this function is for binary lens combinators.

module Control.LensFunction.Exception