# LGtk/ADT lenses

### From HaskellWiki

## Contents |

## 1 Problem description

Lenses provide uniform and compositional way to view and edit data structures.

For example, one can view and edit pairs withq == setL fstLens (getL fstLens q) (setL sndLens (getL sndLens q) p)

Similarly, there is a toolbox of lenses for records which toolbox contains one lens for each record field.

Are there a toolbox of lenses for algebraic data types with multiple constructors?

## 2 Existing solutions

### 2.1 Partial lenses

The data-lens library provides partial lenses which are isomorphic to

type PartialLens a b = (a -> Maybe b, a -> Maybe (b -> a))

The flollowing partial lenses are defined for lists:

headLens :: PartialLens [a] a headLens = (get, set) where get [] = Nothing get (h:t) = Just h set [] = Nothing set (h:t) = Just (:t)

tailLens :: PartialLens [a] [a] tailLens = (get, set) where get [] = Nothing get (h:t) = Just t set [] = Nothing set (h:t) = Just (h:)

### 2.2 Other solutions

*Please help to extend the list of known solutions.*

## 3 ADT lenses

The proposed solution, summarized:

**As a lens toolbox for an ADT, use a lens whose codomain is the ADT and whose domain is tuple of the constructor tag and the ADT components.**

Let's see specific examples before the generic descripton of the proposed lens.

### 3.1 Example: List lens

The lens for lists which forms a complete toolbox:

`import Data.Lens.Common`

listLens :: Lens (Bool, (a, [a])) [a] listLens = lens get set where get (False, _) = [] get (True, (l, r)) = l: r set [] (_, x) = (False, x) set (l: r) _ = (True, (l, r))

### 3.2 Usage

Suppose that we have a statetype S = (Bool, (Int, [Int]))

We can view and edit the list through the following lenses:

- edits thelistLens :: Lens S [Int]
**complete list**. - edits thefstLens :: Lens S Bool
**top level constructor**of the list:corresponds toFalseand[]corresponds toTrue.(:) - edits theheadLens = fstLens . sndLens :: Lens S Int
**head**of the list. - edits thetailLens = sndLens . sndLens :: Lens S [Int]
**tail**of the list.

Remarks:

- If the top level constructor of the list is , the head and the tail of the list can still be edited; the change will only be visible through[]when the constructor is changed back tolistLens. This may seem to be odd, but for many applications this is the right behaviour.(:)
- For editing the tail of the tail of the list, we need an such thats' :: Sviewed throughsis the same astailLensviewed throughs'. Explained on a figure:listLens

## 4 Links and references

I have not seen this technique described before. Please help to extend the list of papers / blog entries, where this or similar technique is used.

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