lens-5.3.3: Lenses, Folds and Traversals

Copyright | (C) 2012-16 Edward Kmett
License | BSD-style (see the file LICENSE)
Maintainer | Edward Kmett <ekmett@gmail.com>
Stability | provisional
Portability | Rank2Types
Safe Haskell | Trustworthy
Language | Haskell2010

Control.Lens.Type

Contents

- Other
- Lenses, Folds and Traversals
- Indexed
- Index-Preserving
- Common

Description

This module exports the majority of the types that need to appear in user signatures or in documentation when talking about lenses. The remaining types for consuming lenses are distributed across various modules in the hierarchy.

Synopsis

- type Equality (s :: k1) (t :: k2) (a :: k1) (b :: k2) = forall k3 (p :: k1 -> k3 -> Type) (f :: k2 -> k3). p a (f b) -> p s (f t)
- type Equality' (s :: k2) (a :: k2) = Equality s s a a
- type As (a :: k2) = Equality' a a
- type Iso s t a b = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Profunctor p, Functor f) => p a (f b) -> p s (f t)
- type Iso' s a = Iso s s a a
- type Prism s t a b = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Choice p, Applicative f) => p a (f b) -> p s (f t)
- type Prism' s a = Prism s s a a
- type Review t b = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Choice p, Bifunctor p, Settable f) => Optic' p f t b
- type AReview t b = Optic' (Tagged :: Type -> Type -> Type) Identity t b
- type Lens s t a b = forall (f :: Type -> Type). Functor f => (a -> f b) -> s -> f t
- type Lens' s a = Lens s s a a
- type Traversal s t a b = forall (f :: Type -> Type). Applicative f => (a -> f b) -> s -> f t
- type Traversal' s a = Traversal s s a a
- type Traversal1 s t a b = forall (f :: Type -> Type). Apply f => (a -> f b) -> s -> f t
- type Traversal1' s a = Traversal1 s s a a
- type Setter s t a b = forall (f :: Type -> Type). Settable f => (a -> f b) -> s -> f t
- type Setter' s a = Setter s s a a
- type Getter s a = forall (f :: Type -> Type). (Contravariant f, Functor f) => (a -> f a) -> s -> f s
- type Fold s a = forall (f :: Type -> Type). (Contravariant f, Applicative f) => (a -> f a) -> s -> f s
- type Fold1 s a = forall (f :: Type -> Type). (Contravariant f, Apply f) => (a -> f a) -> s -> f s
- type IndexedLens i s t a b = forall (f :: Type -> Type) (p :: Type -> Type -> Type). (Indexable i p, Functor f) => p a (f b) -> s -> f t
- type IndexedLens' i s a = IndexedLens i s s a a
- type IndexedTraversal i s t a b = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Indexable i p, Applicative f) => p a (f b) -> s -> f t
- type IndexedTraversal' i s a = IndexedTraversal i s s a a
- type IndexedTraversal1 i s t a b = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Indexable i p, Apply f) => p a (f b) -> s -> f t
- type IndexedTraversal1' i s a = IndexedTraversal1 i s s a a
- type IndexedSetter i s t a b = forall (f :: Type -> Type) (p :: Type -> Type -> Type). (Indexable i p, Settable f) => p a (f b) -> s -> f t
- type IndexedSetter' i s a = IndexedSetter i s s a a
- type IndexedGetter i s a = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Indexable i p, Contravariant f, Functor f) => p a (f a) -> s -> f s
- type IndexedFold i s a = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Indexable i p, Contravariant f, Applicative f) => p a (f a) -> s -> f s
- type IndexedFold1 i s a = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Indexable i p, Contravariant f, Apply f) => p a (f a) -> s -> f s
- type IndexPreservingLens s t a b = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Conjoined p, Functor f) => p a (f b) -> p s (f t)
- type IndexPreservingLens' s a = IndexPreservingLens s s a a
- type IndexPreservingTraversal s t a b = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Conjoined p, Applicative f) => p a (f b) -> p s (f t)
- type IndexPreservingTraversal' s a = IndexPreservingTraversal s s a a
- type IndexPreservingTraversal1 s t a b = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Conjoined p, Apply f) => p a (f b) -> p s (f t)
- type IndexPreservingTraversal1' s a = IndexPreservingTraversal1 s s a a
- type IndexPreservingSetter s t a b = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Conjoined p, Settable f) => p a (f b) -> p s (f t)
- type IndexPreservingSetter' s a = IndexPreservingSetter s s a a
- type IndexPreservingGetter s a = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Conjoined p, Contravariant f, Functor f) => p a (f a) -> p s (f s)
- type IndexPreservingFold s a = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Conjoined p, Contravariant f, Applicative f) => p a (f a) -> p s (f s)
- type IndexPreservingFold1 s a = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Conjoined p, Contravariant f, Apply f) => p a (f a) -> p s (f s)
- type Simple (f :: k1 -> k1 -> k2 -> k2 -> k) (s :: k1) (a :: k2) = f s s a a
- type LensLike (f :: k -> Type) s (t :: k) a (b :: k) = (a -> f b) -> s -> f t
- type LensLike' (f :: Type -> Type) s a = LensLike f s s a a
- type Over (p :: k -> Type -> Type) (f :: k1 -> Type) s (t :: k1) (a :: k) (b :: k1) = p a (f b) -> s -> f t
- type Over' (p :: Type -> Type -> Type) (f :: Type -> Type) s a = Over p f s s a a
- type IndexedLensLike i (f :: k -> Type) s (t :: k) a (b :: k) = forall (p :: Type -> Type -> Type). Indexable i p => p a (f b) -> s -> f t
- type IndexedLensLike' i (f :: Type -> Type) s a = IndexedLensLike i f s s a a
- type Optical (p :: k -> k1 -> Type) (q :: k2 -> k1 -> Type) (f :: k3 -> k1) (s :: k2) (t :: k3) (a :: k) (b :: k3) = p a (f b) -> q s (f t)
- type Optical' (p :: k -> k1 -> Type) (q :: k -> k1 -> Type) (f :: k -> k1) (s :: k) (a :: k) = Optical p q f s s a a
- type Optic (p :: k -> k1 -> Type) (f :: k2 -> k1) (s :: k) (t :: k2) (a :: k) (b :: k2) = p a (f b) -> p s (f t)
- type Optic' (p :: k -> k1 -> Type) (f :: k -> k1) (s :: k) (a :: k) = Optic p f s s a a

# Other

type Equality (s :: k1) (t :: k2) (a :: k1) (b :: k2) = forall k3 (p :: k1 -> k3 -> Type) (f :: k2 -> k3). p a (f b) -> p s (f t) #

A witness that `(a ~ s, b ~ t)`.

Note: Composition with an `Equality` is index-preserving.

type Equality' (s :: k2) (a :: k2) = Equality s s a a #

A `Simple` `Equality`.

type As (a :: k2) = Equality' a a #

Composable `asTypeOf`. Useful for constraining excess polymorphism, `foo . (id :: As Int) . bar`.

type Iso s t a b = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Profunctor p, Functor f) => p a (f b) -> p s (f t) #

Isomorphism families can be composed with another `Lens` using (`.`) and `id`.

Since every `Iso` is both a valid `Lens` and a valid `Prism`, the laws for those types imply the following laws for an `Iso` `f`:

    f . from f ≡ id
    from f . f ≡ id

Note: Composition with an `Iso` is index- and measure- preserving.

type Iso' s a = Iso s s a a #

    type Iso' = Simple Iso

type Prism s t a b = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Choice p, Applicative f) => p a (f b) -> p s (f t) #

A `Prism` `l` is a `Traversal` that can also be turned around with `re` to obtain a `Getter` in the opposite direction.

There are three laws that a `Prism` should satisfy:

First, if I `re` or `review` a value with a `Prism` and then `preview` or use (`^?`), I will get it back:

    preview l (review l b) ≡ Just b

Second, if you can extract a value `a` using a `Prism` `l` from a value `s`, then the value `s` is completely described by `l` and `a`:

    preview l s ≡ Just a ⟹ review l a ≡ s

Third, if you get non-match `t`, you can convert it result back to `s`:

    matching l s ≡ Left t ⟹ matching l t ≡ Left s

The first two laws imply that the `Traversal` laws hold for every `Prism` and that we `traverse` at most 1 element:

    lengthOf l x <= 1

It may help to think of this as an `Iso` that can be partial in one direction.

Every `Prism` is a valid `Traversal`.

Every `Iso` is a valid `Prism`.

For example, you might have a ``Prism'` `Integer` `Natural`` allows you to always go from a `Natural` to an `Integer`, and provide you with tools to check if an `Integer` is a `Natural` and/or to edit one if it is.

    nat :: Prism' Integer Natural
    nat = prism toInteger $ \ i ->
       if i < 0
       then Left i
       else Right (fromInteger i)

Now we can ask if an `Integer` is a `Natural`.

    >>> 5^?nat
    Just 5

    >>> (-5)^?nat
    Nothing

We can update the ones that are:

    >>> (-3,4) & both.nat *~ 2
    (-3,8)

And we can then convert from a `Natural` to an `Integer`.

    >>> 5 ^. re nat -- :: Natural
    5

Similarly we can use a `Prism` to `traverse` the `Left` half of an `Either`:

    >>> Left "hello" & _Left %~ length
    Left 5

or to construct an `Either`:

    >>> 5^.re _Left
    Left 5

such that if you query it with the `Prism`, you will get your original input back.

    >>> 5^.re _Left ^? _Left
    Just 5

Another interesting way to think of a `Prism` is as the categorical dual of a `Lens` -- a co-`Lens`, so to speak. This is what permits the construction of `outside`.

Note: Composition with a `Prism` is index-preserving.

type Prism' s a = Prism s s a a #

A `Simple` `Prism`.

type Review t b = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Choice p, Bifunctor p, Settable f) => Optic' p f t b #

This is a limited form of a `Prism` that can only be used for `re` operations.

Like with a `Getter`, there are no laws to state for a `Review`.

You can generate a `Review` by using `unto`. You can also use any `Prism` or `Iso` directly as a `Review`.

type AReview t b = Optic' (Tagged :: Type -> Type -> Type) Identity t b #

If you see this in a signature for a function, the function is expecting a `Review` (in practice, this usually means a `Prism`).

# Lenses, Folds and Traversals

type Lens s t a b = forall (f :: Type -> Type). Functor f => (a -> f b) -> s -> f t #

A `Lens` is actually a lens family as described in http://comonad.com/reader/2012/mirrored-lenses/.

With great power comes great responsibility and a `Lens` is subject to the three common sense `Lens` laws:

1) You get back what you put in:

    view l (set l v s)  ≡ v

2) Putting back what you got doesn't change anything:

    set l (view l s) s  ≡ s

3) Setting twice is the same as setting once:

    set l v' (set l v s) ≡ set l v' s

These laws are strong enough that the 4 type parameters of a `Lens` cannot vary fully independently. For more on how they interact, read the "Why is it a Lens Family?" section of http://comonad.com/reader/2012/mirrored-lenses/.

There are some emergent properties of these laws:

1) ``set` l s` must be injective for every `s` This is a consequence of law #1

2) ``set` l` must be surjective, because of law #2, which indicates that it is possible to obtain any `v` from some `s` such that ``set` s v = s`

3) Given just the first two laws you can prove a weaker form of law #3 where the values `v` that you are setting match:

    set l v (set l v s) ≡ set l v s

Every `Lens` can be used directly as a `Setter` or `Traversal`.

You can also use a `Lens` for `Getting` as if it were a `Fold` or `Getter`.

Since every `Lens` is a valid `Traversal`, the `Traversal` laws are required of any `Lens` you create:

    l pure ≡ pure
    fmap (l f) . l g ≡ getCompose . l (Compose . fmap f . g)

    type Lens s t a b = forall f. Functor f => LensLike f s t a b

type Lens' s a = Lens s s a a #

    type Lens' = Simple Lens

type Traversal s t a b = forall (f :: Type -> Type). Applicative f => (a -> f b) -> s -> f t #

A `Traversal` can be used directly as a `Setter` or a `Fold` (but not as a `Lens`) and provides the ability to both read and update multiple fields, subject to some relatively weak `Traversal` laws.

These have also been known as multilenses, but they have the signature and spirit of

    traverse :: Traversable f => Traversal (f a) (f b) a b

and the more evocative name suggests their application.

Most of the time the `Traversal` you will want to use is just `traverse`, but you can also pass any `Lens` or `Iso` as a `Traversal`, and composition of a `Traversal` (or `Lens` or `Iso`) with a `Traversal` (or `Lens` or `Iso`) using (`.`) forms a valid `Traversal`.

The laws for a `Traversal` `t` follow from the laws for `Traversable` as stated in "The Essence of the Iterator Pattern".

    t pure ≡ pure
    fmap (t f) . t g ≡ getCompose . t (Compose . fmap f . g)

One consequence of this requirement is that a `Traversal` needs to leave the same number of elements as a candidate for subsequent `Traversal` that it started with. Another testament to the strength of these laws is that the caveat expressed in section 5.5 of the "Essence of the Iterator Pattern" about exotic `Traversable` instances that `traverse` the same entry multiple times was actually already ruled out by the second law in that same paper!

type Traversal' s a = Traversal s s a a #

    type Traversal' = Simple Traversal

type Traversal1 s t a b = forall (f :: Type -> Type). Apply f => (a -> f b) -> s -> f t #

A `Traversal` which targets at least one element.

Note that since `Apply` is not a superclass of `Applicative`, a `Traversal1` cannot always be used in place of a `Traversal`. In such circumstances `cloneTraversal` will convert a `Traversal1` into a `Traversal`.

type Traversal1' s a = Traversal1 s s a a #

type Setter s t a b = forall (f :: Type -> Type). Settable f => (a -> f b) -> s -> f t #

The only `LensLike` law that can apply to a `Setter` `l` is that

    set l y (set l x a) ≡ set l y a

You can't `view` a `Setter` in general, so the other two laws are irrelevant.

However, two `Functor` laws apply to a `Setter`:

    over l id ≡ id
    over l f . over l g ≡ over l (f . g)

These can be stated more directly:

    l pure ≡ pure
    l f . untainted . l g ≡ l (f . untainted . g)

You can compose a `Setter` with a `Lens` or a `Traversal` using (`.`) from the `Prelude` and the result is always only a `Setter` and nothing more.

    >>> over traverse f [a,b,c,d]
    [f a,f b,f c,f d]

    >>> over _1 f (a,b)
    (f a,b)

    >>> over (traverse._1) f [(a,b),(c,d)]
    [(f a,b),(f c,d)]

    >>> over both f (a,b)
    (f a,f b)

    >>> over (traverse.both) f [(a,b),(c,d)]
    [(f a,f b),(f c,f d)]

type Setter' s a = Setter s s a a #

A `Setter'` is just a `Setter` that doesn't change the types.

These are particularly common when talking about monomorphic containers. e.g.

    sets Data.Text.map :: Setter' Text Char

    type Setter' = Simple Setter

type Getter s a = forall (f :: Type -> Type). (Contravariant f, Functor f) => (a -> f a) -> s -> f s #

A `Getter` describes how to retrieve a single value in a way that can be composed with other `LensLike` constructions.

Unlike a `Lens` a `Getter` is read-only. Since a `Getter` cannot be used to write back there are no `Lens` laws that can be applied to it. In fact, it is isomorphic to an arbitrary function from `(s -> a)`.

Moreover, a `Getter` can be used directly as a `Fold`, since it just ignores the `Applicative`.

type Fold s a = forall (f :: Type -> Type). (Contravariant f, Applicative f) => (a -> f a) -> s -> f s #

A `Fold` describes how to retrieve multiple values in a way that can be composed with other `LensLike` constructions.

A ``Fold` s a` provides a structure with operations very similar to those of the `Foldable` typeclass, see `foldMapOf` and the other `Fold` combinators.

By convention, if there exists a `foo` method that expects a ``Foldable` (f a)`, then there should be a `fooOf` method that takes a ``Fold` s a` and a value of type `s`.

A `Getter` is a legal `Fold` that just ignores the supplied `Monoid`.

Unlike a `Traversal` a `Fold` is read-only. Since a `Fold` cannot be used to write back there are no `Lens` laws that apply.

type Fold1 s a = forall (f :: Type -> Type). (Contravariant f, Apply f) => (a -> f a) -> s -> f s #

A relevant Fold (aka `Fold1`) has one or more targets.

# Indexed

type IndexedLens i s t a b = forall (f :: Type -> Type) (p :: Type -> Type -> Type). (Indexable i p, Functor f) => p a (f b) -> s -> f t #

Every `IndexedLens` is a valid `Lens` and a valid `IndexedTraversal`.

type IndexedLens' i s a = IndexedLens i s s a a #

    type IndexedLens' i = Simple (IndexedLens i)

type IndexedTraversal i s t a b = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Indexable i p, Applicative f) => p a (f b) -> s -> f t #

Every `IndexedTraversal` is a valid `Traversal` or `IndexedFold`.

The `Indexed` constraint is used to allow an `IndexedTraversal` to be used directly as a `Traversal`.

The `Traversal` laws are still required to hold.

In addition, the index `i` should satisfy the requirement that it stays unchanged even when modifying the value `a`, otherwise traversals like `indices` break the `Traversal` laws.

type IndexedTraversal' i s a = IndexedTraversal i s s a a #

    type IndexedTraversal' i = Simple (IndexedTraversal i)

type IndexedTraversal1 i s t a b = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Indexable i p, Apply f) => p a (f b) -> s -> f t #

type IndexedTraversal1' i s a = IndexedTraversal1 i s s a a #

type IndexedSetter i s t a b = forall (f :: Type -> Type) (p :: Type -> Type -> Type). (Indexable i p, Settable f) => p a (f b) -> s -> f t #

Every `IndexedSetter` is a valid `Setter`.

The `Setter` laws are still required to hold.

type IndexedSetter' i s a = IndexedSetter i s s a a #

    type IndexedSetter' i = Simple (IndexedSetter i)

type IndexedGetter i s a = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Indexable i p, Contravariant f, Functor f) => p a (f a) -> s -> f s #

Every `IndexedGetter` is a valid `IndexedFold` and can be used for `Getting` like a `Getter`.

type IndexedFold i s a = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Indexable i p, Contravariant f, Applicative f) => p a (f a) -> s -> f s #

Every `IndexedFold` is a valid `Fold` and can be used for `Getting`.

type IndexedFold1 i s a = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Indexable i p, Contravariant f, Apply f) => p a (f a) -> s -> f s #

# Index-Preserving

type IndexPreservingLens s t a b = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Conjoined p, Functor f) => p a (f b) -> p s (f t) #

An `IndexPreservingLens` leaves any index it is composed with alone.

type IndexPreservingLens' s a = IndexPreservingLens s s a a #

    type IndexPreservingLens' = Simple IndexPreservingLens

type IndexPreservingTraversal s t a b = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Conjoined p, Applicative f) => p a (f b) -> p s (f t) #

An `IndexPreservingTraversal` leaves any index it is composed with alone.

type IndexPreservingTraversal' s a = IndexPreservingTraversal s s a a #

    type IndexPreservingTraversal' = Simple IndexPreservingTraversal

type IndexPreservingTraversal1 s t a b = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Conjoined p, Apply f) => p a (f b) -> p s (f t) #

type IndexPreservingTraversal1' s a = IndexPreservingTraversal1 s s a a #

type IndexPreservingSetter s t a b = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Conjoined p, Settable f) => p a (f b) -> p s (f t) #

An `IndexPreservingSetter` can be composed with a `IndexedSetter`, `IndexedTraversal` or `IndexedLens` and leaves the index intact, yielding an `IndexedSetter`.

type IndexPreservingSetter' s a = IndexPreservingSetter s s a a #

    type IndexedPreservingSetter' i = Simple IndexedPreservingSetter

type IndexPreservingGetter s a = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Conjoined p, Contravariant f, Functor f) => p a (f a) -> p s (f s) #

An `IndexPreservingGetter` can be used as a `Getter`, but when composed with an `IndexedTraversal`, `IndexedFold`, or `IndexedLens` yields an `IndexedFold`, `IndexedFold` or `IndexedGetter` respectively.

type IndexPreservingFold s a = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Conjoined p, Contravariant f, Applicative f) => p a (f a) -> p s (f s) #

An `IndexPreservingFold` can be used as a `Fold`, but when composed with an `IndexedTraversal`, `IndexedFold`, or `IndexedLens` yields an `IndexedFold` respectively.

type IndexPreservingFold1 s a = forall (p :: Type -> Type -> Type) (f :: Type -> Type). (Conjoined p, Contravariant f, Apply f) => p a (f a) -> p s (f s) #

# Common

type Simple (f :: k1 -> k1 -> k2 -> k2 -> k) (s :: k1) (a :: k2) = f s s a a #

A `Simple` `Lens`, `Simple` `Traversal`, ... can be used instead of a `Lens`,`Traversal`, ... whenever the type variables don't change upon setting a value.

    _imagPart :: Simple Lens (Complex a) a
    traversed :: Simple (IndexedTraversal Int) [a] a

Note: To use this alias in your own code with ``LensLike` f` or `Setter`, you may have to turn on `LiberalTypeSynonyms`.

This is commonly abbreviated as a "prime" marker, e.g. `Lens'` = `Simple` `Lens`.

type LensLike (f :: k -> Type) s (t :: k) a (b :: k) = (a -> f b) -> s -> f t #

Many combinators that accept a `Lens` can also accept a `Traversal` in limited situations.

They do so by specializing the type of `Functor` that they require of the caller.

If a function accepts a ``LensLike` f s t a b` for some `Functor` `f`, then they may be passed a `Lens`.

Further, if `f` is an `Applicative`, they may also be passed a `Traversal`.

type LensLike' (f :: Type -> Type) s a = LensLike f s s a a #

    type LensLike' f = Simple (LensLike f)

type Over (p :: k -> Type -> Type) (f :: k1 -> Type) s (t :: k1) (a :: k) (b :: k1) = p a (f b) -> s -> f t #

This is a convenient alias for use when you need to consume either indexed or non-indexed lens-likes based on context.

type Over' (p :: Type -> Type -> Type) (f :: Type -> Type) s a = Over p f s s a a #

This is a convenient alias for use when you need to consume either indexed or non-indexed lens-likes based on context.

    type Over' p f = Simple (Over p f)

type IndexedLensLike i (f :: k -> Type) s (t :: k) a (b :: k) = forall (p :: Type -> Type -> Type). Indexable i p => p a (f b) -> s -> f t #

Convenient alias for constructing indexed lenses and their ilk.

type IndexedLensLike' i (f :: Type -> Type) s a = IndexedLensLike i f s s a a #

Convenient alias for constructing simple indexed lenses and their ilk.

type Optical (p :: k -> k1 -> Type) (q :: k2 -> k1 -> Type) (f :: k3 -> k1) (s :: k2) (t :: k3) (a :: k) (b :: k3) = p a (f b) -> q s (f t) #

    type LensLike f s t a b = Optical (->) (->) f s t a b

    type Over p f s t a b = Optical p (->) f s t a b

    type Optic p f s t a b = Optical p p f s t a b

type Optical' (p :: k -> k1 -> Type) (q :: k -> k1 -> Type) (f :: k -> k1) (s :: k) (a :: k) = Optical p q f s s a a #

    type Optical' p q f s a = Simple (Optical p q f) s a

type Optic (p :: k -> k1 -> Type) (f :: k2 -> k1) (s :: k) (t :: k2) (a :: k) (b :: k2) = p a (f b) -> p s (f t) #

A valid `Optic` `l` should satisfy the laws:

    l pure ≡ pure
    l (Procompose f g) = Procompose (l f) (l g)

This gives rise to the laws for `Equality`, `Iso`, `Prism`, `Lens`, `Traversal`, `Traversal1`, `Setter`, `Fold`, `Fold1`, and `Getter` as well along with their index-preserving variants.

    type LensLike f s t a b = Optic (->) f s t a b

type Optic' (p :: k -> k1 -> Type) (f :: k -> k1) (s :: k) (a :: k) = Optic p f s s a a #

    type Optic' p f s a = Simple (Optic p f) s a