LGtk/ADT lenses

Problem description

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

For example, one can view and edit pairs with fstLens and sndLens. These two lenses form a toolbox for editing pairs in the sense that given pairs p :: (a, b) and q :: (a, b), by succesive get and set operations p can be changed to be equivalent to q:

q == 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?

Existing solutions

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:)

Unfortunately headLens and tailLens does not provide a complete toolbox, one cannot change an empty list to a non-empty list with them, for example.

ADT lenses

The proposed solution, summarized:

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

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))

Usage

Suppose that we have a state s of type

type S = (Bool, (Int, [Int]))

If we view the state through listLens :: Lens S [Int], we see a list of Ints. So we can view the list.

We can edit the list with the following lenses:

With fstLens :: Lens S Bool the top level constructor of the list can be viewed and edited: False corresponds to [] and True corresponds to (:).
With headLens = fstLens . sndLens :: Lens S Int the first element of the list can be viewed and edited. Note that if the top level constructor of the list is [], the first element can still be edited; the change will only be visible through listLens when the constructor is changed back to (:). (This may seem to be odd, but for certain applications this is the right behaviour.)
With tailLens = sndLens . sndLens :: Lens S Int the tail of the list can be viewed and edited.

Note that for editing the tail of the list, we need an s' :: S such that s viewed through tailLens is the same as s' viewed through listLens. Explained on a figure:

How s' is created is a related but different question. See LGtk/Semantics#Dependent_reference_creation.

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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LGtk/ADT lenses

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