如何将Lens定义为State值本身？

Question

While trying to understand State monad and use of Lens with it, I arrived at a surprisingly trivial definition of a lens for a simple counter:

self :: ASetter s s s s
self = ($)

incrementUsingLens :: State Int ()
incrementUsingLens = self %= (+1)

Since

type ASetter s t a b = (a -> Identity b) -> s -> Identity t

In my case is just

type ASetter s s s s = (s -> Identity s) -> s -> Identity s

Is this is indeed a correct definition for a lens into a state variable? I'm worried that I might be missing some laws or other assumptions.

Answer 1

lens calls this do-nothing optic simple . Note that simple is an Equality , and Equality is at the very bottom of the optics hierarchy, which means you can use it not only as a do-nothing setter, but also as a do-nothing lens, prism, etc.

I'm worried that I might be missing some laws or other assumptions.

The setter laws say that, for a foo setter, over foo should follow the functor laws:

over foo id = id
over foo (g . f) = over foo g . over foo f

If you try this with simple / id , you will find the laws hold trivially. The same goes for the other optic laws.

Answer 2

Yes, ($) , also called id , is also a lens in the representation of the lens package. Though here we have get , put , modify that work just as well, the identity lens can still be useful sometimes.