Coercive left-composition.

>>> (All #. not) True
All {getAll = False}

The semantics with respect to bottoms are:

p #. ⊥ ≡ ⊥
p #. f ≡ p . f

Coercive right-composition.

>>> (stimes 2 .# Product) 3
Product {getProduct = 9}

The semantics with respect to bottoms are:

⊥ .# p ≡ ⊥
f .# p ≡ p . f

Reverse the type of a "packer".

>>> op All (All True)
True
>>> op (Identity . Sum) (Identity (Sum 3))
3

The workhorse of the package. Given a "packer" and a "higher order function" (hof), it handles the packing and unpacking, and just sends you back a regular old function, with the type varying based on the hof you passed.

The reason for the signature of the hof is due to ala not caring about structure. To illustrate why this is important, consider this alternative implementation of under2:

under2' :: (Coercible a b, Coercible a' b')
        => (a -> b) -> (b -> b -> b') -> (a -> a -> a')
under2' pa f o1 o2 = ala pa (\p -> uncurry f . bimap p p) (o1, o2)

Being handed the "packer", the hof may apply it in any structure of its choosing – in this case a tuple.

>>> ala Sum foldMap [1,2,3,4 :: Int] :: Int
10

The way it differs from the ala function in this package, is that it provides an extra hook into the "packer" passed to the hof.

However, this normally ends up being id, so ala wraps this function and passes id as the final parameter by default. If you want the convenience of being able to hook right into the hof, you may use this function.

>>> ala' Sum foldMap length ["hello", "world"] :: Int
10

>>> ala' First foldMap (readMaybe @Int) ["x", "42", "1"] :: Maybe Int
Just 42

A very simple operation involving running the function under the "packer".

>>> under Product (stimes 3) (3 :: Int) :: Int
27

The opposite of under. I.e., take a function which works on the underlying "unpacked" types, and switch it to a function that works on the "packer".

>>> over All not (All False) :: All
All {getAll = True}

Lower a binary function to operate on the underlying values.

>>> under2 Any (<>) True False :: Bool
True

The opposite of under2.

under lifted into a Functor.

over lifted into a Functor.