# 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 withSimilarly, there is a complete toolbox of lenses for records, the toolbox contains one lens for each record field.

Now, the problem is, 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:

**Use one lens for each ADT type, with reversed direction.**

### 3.1 Example: List lens

The ADT lens for lists:

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 edit the list with the following lenses:

- With the top level constructor of the list can be viewed and edited:fstLens :: Lens S Boolcorresponds toFalseand[]corresponds toTrue.(:)
- With the first element of the list can be viewed and edited. Note that if the top level constructor of the list isheadLens = fstLens . sndLens :: Lens S Int, the first element 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 certain applications this is the right behaviour.)(:)
- With the tail of the list can be viewed and edited.tailLens = sndLens . sndLens :: Lens S Int

## 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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