Why there is no AGetter?

Posted on 2017-03-13 by Oleg Grenrus lens

This question was recently asked on lens issue tracker. The short answer, by Edward Kmett is

Mainly because we have functions, which perfectly characterize getters.

Let's explore that in more detail:

This post is a literate Haskell file, and as we work with lens, we need to import them.

{-# LANGUAGE RankNTypes, TypeOperators #-}
module NoAGetter where
import Control.Lens
import Data.Type.Equality  -- for a small proof

#Existential answer

Keep in mind AGetter, if it existed, wouldn't work with views or any of the more general combinators other than view. So it'd be a thing that only existed so we could turn it into a function.

Suppose, there is AGetter:

-- | When you see this as an argument to a function, it expects a Getter.
type AGetter s a = LensLike' (Retext a) s a

then we could put AGetters into a container. Similarly as we have a version of (^.), the (^#) :: s -> ALens s t a b -> a that works on ALens, we'll new combinator, say (^-) :: s -> AGetter s a -> b.

To put it differently: The ALens is isomorphic to Lens, but we cannot use it directly as Lens:

Control.Lens> :t view (_1 :: ALens' (Int, Bool) Int)

<interactive>:... error:
Couldn't match typeControl.Lens.Internal.Context.Pretext
                             (->) Int Int (Int, Bool)’
                     with ‘Const Int (Int, Bool)’
      Expected type: Getting Int (Int, Bool) Int
        Actual type: ALens' (Int, Bool) Int
In the first argument of ‘view’, namely
        ‘(_1 :: ALens' (Int, Bool) Int)’
      In the expression: view (_1 :: ALens' (Int, Bool) Int)

we need to clone the ALens, or use specialized combinators:

Control.Lens> :t view (cloneLens (_1 :: ALens' (Int, Bool) Int))
view (cloneLens (_1 :: ALens' (Int, Bool) Int))
  :: Control.Monad.Reader.Class.MonadReader (Int, Bool) m => m Int

#Constructive answer

The question is, what's Retext? If you explore the lens documentation, you'll notice that ALens uses Pretext, which is better Context, which is isomorphic to exists s. (s, Lens s t a b). By analogy, we need something isomorphic to exists s. (s, Getter s a). Getter s a is isomorphic to s -> a, so

exists s. (s, Getter s a) ~ exists s. (s, s -> a) ~ a

As Retext needs to be a Functor, we'll just add a phantom type-variable. So at the end, the Retext is an old friend of us:

type Retext = Const

The Context has more type arguments, and we cannot define a Sellable instance, but it would be sell = Const. Using this observation we can define cloneGetter and (^-), by adopting definitions of the cloneLens and the (^#):

cloneGetter :: AGetter s a -> Getter s a
cloneGetter g = to $ \s -> getConst (g Const s)

infixl 8 ^-
(^-) :: s -> AGetter s a -> a
s ^- g = getConst (g Const s)

And we can use these definitions:

*Main> ('a', 'b') ^- _1

*Main> let getters = [_1, _2] :: [AGetter (Int, Int) Int]
*Main> map (\g -> (42, 7) ^- g) getters

*Main> map (\g -> (42, 7) ^. cloneGetter g) getters


If we inline the type-aliases of AGetter, we'll get a Getting a s a:

AGetter s a
    = LensLike' (Const a) s a
    = LensLike (Const a) s s a a
    = (a -> Const a a) -> s -> Const a s
    = Getting a s a

If you really want to put Getters into containers, simplest solution seems to use Getting a s a: It's already in Control.Lens.

type AGetter2 s a = Getting a s a

cloneGetter2 :: AGetter2 s a -> Getter s a
cloneGetter2 g = to $ \s -> s ^. g

One should note, tat the AGetter and the AGetter2 are exactly the same:

proof :: AGetter s a :~: AGetter2 s a
proof = Refl

The corollary of above, is that need (^-), (^.) works well:

*Main> let getters2 = [_1, _2] :: [AGetter2 (Int, Int) Int]
*Main> map (\g -> (42, 7) ^. g) getters2

because of the type of (^.) :: s -> Getting a s a -> a.

Yet, we still need to clone the AGetter when the viewed type is changed (similarly as with ALens):

*Main> (42, True) ^. _1 . re _Show

*Main> (42, True) ^# (_1 :: ALens' (Int, Bool) Int) . re _Show
<interactive>:... error: ...
*Main> (42, True) ^. (_1 :: AGetter (Int, Bool) Int) . re _Show
<interactive>:... error:

*Main> (42, True) ^. cloneLens (_1 :: ALens' (Int, Bool) Int) . re _Show
*Main> (42, True) ^. cloneGetter (_1 :: AGetter (Int, Bool) Int) . re _Show

#Plain old function

On the other hand, just using getter functions is even simpler, though then we "escape" the lens eco-system:

type AGetter3 s a = s -> a

cloneGetter3 :: AGetter3 s a -> Getter s a
cloneGetter3 = to

To put AGetter3s into container, we need explicitly view them:

*Main> let getters3 = [view _1, view _2] :: [AGetter3 (Int, Int) Int]
*Main> map (\g -> (42, 7) ^. cloneGetter3 g) getters3


So, why there is no AGetter? In this post I tried to clarify, that having one doesn't add any(?) benefit over plain getter functions. Yet, you are free to play with the definitions here. Please tell me (e.g. via twitter), if you find some applications where AGetter is more elegant then a plain old function.

You can run this file with

stack --resolver=nightly-2017-03-01 ghci --ghci-options='-pgmL markdown-unlit' agetter.lhs

fetch the source from https://gist.github.com/phadej/fb02aea7de32fba94bc3de33de6ee2e6

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