Editing LGtk/ADT lenses (section)

=== Example: List lens ===

The lens for lists which forms a complete toolbox:

<haskell>
import Data.Lens.Common
</haskell>

<haskell>
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))
</haskell>

Here <hask>Bool</hask> is used as the constructor tag, and <hask>a</hask> and <hask>[a]</hask> are the components of the list (head and tail). Intead of a triple we use two pairs, so we reach the parts with <hask>fstLens</hask> and <hask>sndLens</hask>.

==== List lens usage ====

Suppose that we have a state <hask>s</hask> of type

<haskell>
type S = (Bool, (Int, [Int]))
</haskell>


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

* <hask>listLens :: Lens S [Int]</hask> edits the '''complete list'''.
* <hask>fstLens :: Lens S Bool</hask> edits the '''top level constructor''' of the list: <hask>False</hask> corresponds to <hask>[]</hask> and <hask>True</hask> corresponds to <hask>(:)</hask>.
* <hask>headLens = fstLens . sndLens :: Lens S Int</hask> edits the '''head''' of the list.
* <hask>tailLens = sndLens . sndLens :: Lens S [Int]</hask> edits the '''tail''' of the list.

Remarks:

* If the top level constructor of the list is <hask>[]</hask>, the head and the tail of the list can still be edited; the change will only be visible through <hask>listLens</hask> when the constructor is changed back to <hask>(:)</hask>. This may seem to be odd, but for many applications this is the right behaviour. See the interpretation subsection below.
* For editing the tail of the tail of the list, we need an <hask>s' :: S</hask> such that <hask>s</hask> viewed through <hask>tailLens</hask> is the same as <hask>s'</hask> viewed through <hask>listLens</hask>. Explained on a figure:

[[Image:ADT.png]]

On the figure, edges are lenses and nodes are ''references''. One possible definition of references is given in [[LGtk/Semantics#References]]. How <hask>s'</hask> can be created and how <hask>s</hask> and <hask>s'</hask> can be kept in sync is an important but separate question. [[LGtk/Semantics#Dependent_reference_creation]] describes a possible solution.