< LGtk

## 1 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?

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

### 2.2 Other solutions

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

#### 3.1.1 List lens usage

Suppose that we have a state
s
of type
`type S = (Bool, (Int, [Int]))`

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

• listLens :: Lens S [Int]
edits the complete list.
• fstLens :: Lens S Bool
edits the top level constructor of the list:
False
corresponds to
[]
and
True
corresponds to
(:)
.
• headLens = fstLens . sndLens :: Lens S Int
edits the head of the list.
• tailLens = sndLens . sndLens :: Lens S [Int]
edits the 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
listLens
when the constructor is changed back to
(:)
. 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
s' :: S
such that
s
viewed through
tailLens
is the same as
s'
viewed through
listLens
. Explained on a figure:

On the figure, edges are lenses and nodes are references. One possible definition of references is described in LGtk/Semantics#References. How
s'
can be created and how
s
and
s'
can be kept in sync is a related but separate question. LGtk/Semantics#Dependent_reference_creation describes a possible solution.

TODO