This is a new version of an old article about lenses and prisms in Swift. This one contains many updates and more precisely reflects how I actually use these things in my work. This is also based on a talk I gave at iOSDevUK 2017 and Mobilization Conference 2017. I’m keeping the old article in draft as reference.

In date Sep 8 2018 I modified the article further, by including

`Affine`

in the discussion, again to reflect my current work better.

The concept of functional lens has become pretty popular in **functional programming** circles, and there are already good contributions for applying lenses to other, traditionally imperative/OO contexts, like Javascript. Brandon Williams has done an excellent job in introducing lenses to the Swift community, and in showing practical examples when working with something like `UIView`

, a radically **OOP construct** that an iOS developer has to work with on a daily basis.

I’d like to offer a more in depth view on why lenses can be useful in Swift, and also talk about a related concept called **Prism**: it’s likely that everyone that’s interested in lenses has heard about prisms but it seems like, while the basic intuition behind a lens can be grasped, the one behind a prism is kind of obscure. But before that, it might be important to discover again the concept of **Lens**: I’d like to do that because, by reading the comments for this video, I understand that **there’s some confusion**, mixed with the usual hostility towards functional programming concepts by some OOP people (luckily, a minority) who consider functional programming pointlessly complex: it’s really the opposite, that is, it’s simple but initially hard to grasp. And *simple vs. complex* is not the same thing as *easy vs. hard*.

So, let’s start from the beginning and talk about lenses, but in a more pragmatic way, that is, let’s consider some **practical problems** and see if a traditional, idiomatic (and imperative) Swift approach can be enough or not. Most of the code from this post is available in a Swift Playground, and I strongly recommend to follow the Playground while reading the post, even if I’m still going to recall most of the code here.

## Functional Getters and Setters

If you google “functional lens” you’re probably going to end up with a definition like “functional getter and setter”: in this context the word “functional” really means “immutable”. There are many advantages in using immutable data models - I’m not going into it for this article - but modifying immutable models (i.e. generating new models with something changed) can be a chore, because the whole model must be reconstructed by taking the previous values where they didn’t change, and setting the new values where they did. But Swift offers particular kinds of types, called “value types”, that have value semantics, which basically means that they have no identity and only represent a “value”, i.e., some kind of “information” that is implicitly immutable: a piece of information cannot change, but **new** information can be created, rendering the previous obsolete.

For example, a `Int`

is a value type: if I have a `2`

, and I’m adding `1`

, I’m not *mutating* the `2`

... I’m creating a new instance called `3`

. If it seems obvious it’s because *it is* obvious for a type like `Int`

, but in Swift we can also define complex types that have the exact same behavior.

Swift `struct`

and `enum`

are value types: if I give you an instance of a struct, it’s like I’m giving you a *copy* of my instance, which is not going to be mutated in any way, no matter what you do with your copy. This has many advantages and allows to safely reason about code. An interesting consequence is that one can safely define structs with `var`

properties, because if I set a new value for a property of a struct, what Swift really does *under the hood* is generating a whole new reference to that instance, such that others referring to it won’t see any change.

For this reason, if I define in my code a struct that have *memberwise* initializers - that is, they can be constructed with any possible value for the properties - I usually set all the properties as `var`

. There are cases where you still want the `let`

: maybe the struct **cannot** be constructed with any value for the properties (thus, some mutations wouldn’t make sense), or maybe you *really* need a class for a model. Anyway, in this article’s code all the model structs will have `var`

properties.

For all these reasons I don’t think that just the definition “functional getters and setters” is going to be useful for lenses and prisms, especially in languages that offer flexible value types like Swift. What makes these ideas interesting is their *composability*, that allows for building powerful abstractions, as we’ll see.

## A matter of relationship

Lenses and prisms (and programming with *optics* in general) allow us to establish **relationships between data structures**. In particular, a *lens* is called like that because it allows to *focus* on a specific part of a data structure, and consequently to act on that part, like retrieving it, or modifying it causing a modification of the whole data structure. A *prism* is a similar concept, but it works for sum types, that in Swift are represented via enum. Types like `class`

and `struct`

, that have *properties* (even a simple tuple), are called product types and are manipulated by lenses; while *sum* or **coproduct** types are types like `enum`

, that have *cases*, and are manipulated by prisms.

Let’s see it graphically:

In the image we can see a schematic representation of a generic data structure: ovals are struct or class types, thus, types with properties; rectangles are enum types; dotted lines are properties; dashed lines are cases for enums.

To define a `Lens`

from the type `A`

, that *contains* instances of the types `B`

, `C`

and `D`

, to one of the contained types, for example `B`

, I need two functions:

- a function called
`get`

that, given an`A`

, will return a`B`

; - a function called
`set`

that, given a`B`

, will return a*transforming function*for`A`

, i.e., a function of type`(A) -> A`

.

This implies that `Lens`

is parametrized by two types, the type on which I’m applying the lens, usually referred as `Whole`

, and the type onto which the lens is focusing, called `Part`

. Thus, a full generic representation of a lens can be given like the following:

```
struct Lens<Whole,Part> {
let get: (Whole) -> Part
let set: (Part) -> (Whole) -> Whole
}
```

A `Prism`

is the same for coproducts. We can define a prism from `D`

(an enum) to `F`

(the associated type to one of its cases). What’s a `get`

in this case? It’s a function that takes a `D`

and *tries* to yield a `F`

: I said “tries” because a `F`

can only be retrieved if the `D`

instance *is in the right case*, otherwise we can simply return `nil`

. That’s why the `get`

for prisms is a `tryGet`

, that returns a `Part?`

. About the `set`

, given a `F`

we can completely reconstruct a `D`

via the `case`

function, so its more like an `inject`

, in which we produce a *new* `D`

without any knowledge of the previous one. That being said, a nice representation of `Prism`

is the following:

```
struct Prism<Whole,Part> {
let tryGet: (Whole) -> Part?
let inject: (Part) -> Whole
}
```

There are actually **many ways** to define lenses and prisms, and the code I showed defines a very simple representation that I find most useful in my Swift code. Maybe in future articles I’ll explore different representations, but my intention is to give a pragmatic approach, based on what seems useful in day-to-day Swift code, and what I actually use in my own code.

I’ll spend a few more words on `Prism<Whole,Part>`

, with a practical example. Suppose I have the following data structure:

```
enum ViewState<T> {
case empty
case processing(String)
case failed(Error)
case completed(T)
}
```

`ViewState<T>`

represents the model for a view that can be in many states, depending on the progress of a particular process: for example, the process could fail, yielding an `Error`

, or could complete successfully, with some generic `T`

value. Let’s define a `Prism`

on `ViewState<T>`

: because it has 4 cases, we can define 4 prisms. For example, for the `processing`

case we can define a prism where the `Whole`

is `ViewState<T>`

(as for any prism defined on this data structure) and the part is `String`

, i.e., the associated type to the `processing`

case. The final type of this prism will then be `Prism<ViewState<T>,String>`

. Notice that a prism defined on the `empty`

case would be of type `Prism<ViewState<T>,()>`

, because `()`

(or `Void`

in Swift) is the *implicit* associated type of that case.

## From *View* to *Model*

As many have probably noticed, the perfect use case for lenses and prisms is when we need to act on a tree-like data structure, where many parent-child relationships are established: for each parent-child pair we can define a lens or a prism, depending on the type of the parent (product or coproduct). As iOS developers (or mobile developers in general) we deal with one of these structures on a daily basis, and it’s the *view hierarchy*. While we could define lenses directly on `UIView`

and on its subclasses, I find it preferable to work with **view models**, that is, objects that represent views but are not view themselves. I’m not strictly talking about the Model-View-ViewModel architectural pattern, in which the ViewModel could be a mutable class with identity and logic: I’m simply referring to using structs and enums to mediate the content of views, something I could refer to as *functional ViewModel*. A ViewModel like this is dumb, has no logic, only data, and is acted upon *from the outside* with functions, functions like the ones defined in lenses and prisms.

Let’s see an example:

```
struct LoginPage {
var title: String
var credendials: CredentialBox
var buttonState: ViewState<Button>
}
struct CredentialBox {
var usernameField: TextField
var passwordField: TextField
}
struct TextField {
var text: String
var placeholder: String?
var secureText: Bool
}
struct Button {
var title: String
var enabled: Bool
}
```

`LoginPage`

is a model for a login page (which is probably going to be a `UIViewController`

in a iOS app), with all its parts: it includes a `title`

that’s simply a `String`

, a `CredentialBox`

struct represented by two `TextField`

that are themselves structs, and a `buttonState`

that’s represented by a `ViewState<Button>`

where `Button`

is another struct. Everything here is `var`

because any of these models can be constructed with any value, and any property can change in any way at any time. But, because we’re dealing with value types, mutating this `LoginPage`

actually means creating a new reference, and that’s only going to be possible if we assign an instance of `LoginPage`

to a variable, annotated with `var`

.

Adding lenses to these structs is simple, and there are many ways to do that. One possible way is to extend each type with a *nested* type called `lens`

that’s going to act as *namespace*, and then defining all the lenses as `static`

properties of this `lens`

, like in the following example:

```
extension CredentialBox {
enum lens {
static let usernameField = Lens<CredentialBox,TextField>.init(
get: { $0.usernameField },
set: { part in
{ whole in
var m = whole
m.usernameField = part
return m
}
})
}
}
```

As you can see, `CredentialBox.lens.usernameField`

is a lens from `CredentialBox`

to `TextField`

, and because it’s a static property, only an instance of this lens is going to be alive in the entire lifecycle of the application: it’s simply something always available, that can be referred when needed.

The implementation is trivial, there’s really only one **sane** way to write this: the `get`

function simply gets the value from the `usernameField`

property, while the `set`

grabs a `var`

reference to the `whole`

that’s passed in, mutates it and then returns it: we can do this because `usernameField`

is defined as `var`

in `CredentialBox`

. Notice that the `set`

function is a *higher-order function*, because it takes a value and returns *a new function*: we generally want this kind of thing for the `set`

part of a lens, so that we can *prepare* a function that will transform the `Whole`

with a new `Part`

.

The story is similar for `Prism`

. For example, we might want to the define the `Prism`

for the `processing`

case in `ViewState<T>`

:

```
extension ViewState {
enum prism {
static var processing: Prism<ViewState,String> {
return Prism<ViewState,String>.init(
tryGet: {
guard case .processing(let message) = $0 else {
return nil
}
return message
},
inject: { .processing($0) })
}
}
}
```

Unfortunately, with this strategy, we cannot assign the `Prism`

to a *stored* static property (because `ViewState`

is a generic type): not a big deal, we’ll use a computed property (still static). Again, the implementation is trivial, there’s only one sane way to write this.

## Let’s do this

Suppose we want to modify a current instance of `LoginPage`

. A new set of values is represented by the following tuple:

```
let initialState = (
title: "Welcome back!",
username: savedUsername,
buttonState: ViewState<Button>.completed(Button.init(
title: "Login",
enabled: false)))
```

`initialState`

contains a bunch of initial settings for the `LoginPage`

: `savedUsername`

might be the username string stored in the `UserDefaults`

.

Because our `LoginPage`

is mutable - in the sense we discussed earlier for value types, that is, not really mutable but it can be manipulated like it was - we can simply do this:

```
var m_newModel = oldModel
m_newModel.title = initialState.title
m_newModel.credendials.usernameField.text = initialState.username
m_newModel.buttonState = initialState.buttonState
```

The problem with this approach, which is of course the most basic one for `var`

instances, is that we’re not clearly representing the relationship between the `LoginPage`

and its parts: we’re directly accessing the properties of the model, and in the case of the username text we need to call 3-levels deep of properties (that is, `.credendials.usernameField.text`

). It would be really useful if we had objects that *encapsulate* the access to properties at any level, for example if we were able to encapsulate the access to the username field text, that is, the *relationship* with the `LoginPage`

, such that if something changes in the future for this chain of relationships, nothing is going to break in the place where we use the object to manipulate the property itself.

A `Lens`

does exactly this: *encapsulates* the relationship between a data structure and one of its parts. But for now we only defined lenses from a type to its direct properties. To represent with a lens something like `.credendials.usernameField.text`

we need to **compose** lenses with matching `Part/Whole`

parameters. What I mean is something like this:

```
Lens<A,B> + Lens<B,C> = Lens<A,C>
```

If a lens’ `Part`

is the same as another lens’ `Whole`

we can *chain* them together, generating a single lens that goes *one level deeper* in the data structure represented by the first lens’ `Whole`

. It turns out that we can always do this in a completely generic way, we only need the types to match correctly. In fact, we can extend both `Lens`

and `Prism`

with a `then`

function that will take another lens/prism for which the `Whole`

is the same as the first lens/prism `Part`

, and return a new lens/prism that will cover the extended range of types:

```
extension Lens {
func then<Subpart>(_ other: Lens<Part,Subpart>) -> Lens<Whole,Subpart> {
return Lens<Whole,Subpart>(
get: { other.get(self.get($0)) },
set: { (subpart: Subpart) in
{ (whole: Whole) -> Whole in
self.set(other.set(subpart)(self.get(whole)))(whole)
}
})
}
}
```

The implementation seems complex, in particular the line `self.set(other.set(subpart)(self.get(whole)))(whole)`

, but actually there’s really only one sane way to write this: the types really guide the implementation here, and if it compiles it’s almost certainly going to work. Actually, the only other option for implementing the `set`

function so that Swift compiles is by naively returning the `whole`

input unchanged, which is of course wrong.

We can do the same for `Prism`

:

```
Prism<A,B> + Prism<B,C> = Prism<A,C>
extension Prism {
func then<Subpart>(_ other: Prism<Part,Subpart>) -> Prism<Whole,Subpart> {
return Prism<Whole,Subpart>(
tryGet: { self.tryGet($0).flatMap(other.tryGet) },
inject: { self.inject(other.inject($0)) })
}
}
```

I actually prefer to use an operator for composing lenses and prisms this way, the `..`

operator, which can be read as *and then*: `A..B`

would be read as *A, and then, B*.

```
precedencegroup LeftCompositionPrecedence {
associativity: left
}
infix operator .. : LeftCompositionPrecedence
extension Lens {
static func .. <Subpart> (lhs: Lens<Whole,Part>, rhs: Lens<Part,Subpart>) -> Lens<Whole,Subpart> {
return lhs.then(rhs)
}
}
extension Prism {
static func .. <Subpart> (lhs: Prism<Whole,Part>, rhs: Prism<Part,Subpart>) -> Prism<Whole,Subpart> {
return lhs.then(rhs)
}
}
```

Now we can simply grab some lenses:

```
let titleLens = LoginPage.lens.title
let usernameTextLens = LoginPage.lens.credentials..CredentialBox.lens.usernameField..TextField.lens.text
let buttonStateLens = LoginPage.lens.buttonState
```

A very ugly way to use these lenses to modify our original model would be to apply their `set`

function over and over again on `oldModel`

: because `set`

, applied to a `Part`

, returns a function that transforms the `Whole`

, we can generate these `(Whole) -> Whole`

functions and apply them one after the other to the `oldModel`

. The resulting code is nothing to be proud of:

```
let newModel = titleLens.set(initialState.title)(usernameTextLens.set(initialState.username)(buttonStateLens.set(initialState.buttonState)(oldModel)))
```

It would useful if we were able to combine again these lenses, but in a different way: these lenses all share the same `Whole`

, but focus on different `Part`

s. The composition we need is something like this:

```
Lens<A,B1> + Lens<A,B2> = Lens<A,(B1,B2)>
Prism<A,B1> + Prism<A,B2> = Prism<A,Either<B1,B2>>
```

The resulting lens will focus on **both** `B1`

and `B2`

, while the resulting prism will focus on **either** `B1`

or `B2`

; to represent this *exclusive or* relationship for `Prism`

I used a very simple `Either`

enum:

```
enum Either<A,B> {
case left(A)
case right(B)
}
```

Can we combine lenses and prims generically in this way? Of course we can. I like to call this function `zip`

for both `Lens`

and `Prism`

:

```
extension Lens {
static func zip<Part1,Part2>(
_ a: Lens<Whole,Part1>,
_ b: Lens<Whole,Part2>)
-> Lens<Whole,(Part1,Part2)>
where Part == (Part1,Part2)
{
return Lens<Whole,(Part1,Part2)>(
get: { (a.get($0),b.get($0)) },
set: { parts in { whole in b.set(parts.1)(a.set(parts.0)(whole)) } })
}
static func zip<A,B,C>(_ a: Lens<Whole,A>, _ b: Lens<Whole,B>, _ c: Lens<Whole,C>) -> Lens<Whole,(A,B,C)> where Part == (A,B,C) {
return Lens<Whole,(A,B,C)>(
get: { (a.get($0),b.get($0),c.get($0)) },
set: { parts in { whole in c.set(parts.2)(b.set(parts.1)(a.set(parts.0)(whole))) } })
}
}
extension Prism {
static func zip<Part1,Part2>(
_ a: Prism<Whole,Part1>,
_ b: Prism<Whole,Part2>)
-> Prism<Whole,Either<Part1,Part2>>
where Part == Either<Part1,Part2>
{
return Prism<Whole,Either<Part1,Part2>>(
tryGet: { a.tryGet($0).map(Either.left) ?? b.tryGet($0).map(Either.right) },
inject: { part in
switch part {
case .left(let value):
return a.inject(value)
case .right(let value):
return b.inject(value)
}
})
}
}
```

I also gave a definition for `zip`

and 3 lenses (of course it can be done for any number of lenses), because this is the one we’re going to use in our example. We can finally define our `initialStateLens`

:

```
let initialStateLens = Lens.zip(
titleLens,
usernameTextLens,
buttonStateLens)
let newModel = initialStateLens.set(initialState)(oldModel)
```

`initialStateLens`

encapsulates the entirety of the relationships between data types that we need to apply to the `oldModel`

to get the new one.

What about prisms? Prisms are for enums, so in our example we’re probably going to need them when working with the button state. Suppose for example that we’re in the `processing`

state, so we’re currently showing a message to the user (like “Please wait”). We’d like to modify the `LoginPage`

by setting a new message as time passes, considering the logic expressed by the following function:

```
func advanceProcessingMessage(_ previous: String) -> String {
switch previous {
case "":
return "Please wait"
case "Please wait":
return "Almost there"
case "Almost there":
return "ALMOST THERE"
default:
return previous + "!"
}
}
```

This is just a pure function, expressing a simple, isolated piece of logic. But this is not a pure `set`

: it actually *depends on the previous value*. Both lens’ `set`

and prism’s `inject`

don’t give access to the previous value, but only allow to modify a data structure with a brand new value. We need some kind of `modify`

function (sometimes called `over`

) that instead of taking a new `Part`

to yield a transformation of the `Whole`

, takes a *transformation of the Part*, i.e., a function of type

`(Part) -> Part`

. Can we write this `modify`

function in a completely generic way? Yep:```
extension Lens {
func modify(_ transform: @escaping (Part) -> Part) -> (Whole) -> Whole {
return { whole in self.set(transform(self.get(whole)))(whole) }
}
}
extension Prism {
func tryModify(_ transform: @escaping (Part) -> Part) -> (Whole) -> Whole {
return { whole in self.tryGet(whole).map { self.inject(transform($0)) } ?? whole }
}
}
```

Notice that, for `Prism`

is `tryModify`

: this is because it will only modify an instance of an enum if it’s in the right case; if it’s not, the resulting `(Whole) -> Whole`

function will be an *identity*, i.e., a function that does nothing on the input, and simply returns it.

Now, we have a lens from `LoginPage`

to `ViewState<Button>`

, and a prism from `ViewState<Button>`

to `String`

: can we compose them? Because composition has been the main point here from the very start of the article: combining small, simple things to form more complex but more useful things. Unfortunately there’s really no easy way to combine a lens and a prism, at least considering the representation we gave for them. Luckily, we really don’t care, because under the hood we’re simply dealing with *pure functions*, that compose if the types they act upon compose. By composing functions I mean something like this:

```
(A -> B) + (B -> C) = (A -> C)
```

This kind of composition can always be done, no matter what `A`

, `B`

and `C`

are, and will always produce something meaningful if the functions involved are *pure*, which means that they have no side-effects, and that for the same input they’ll always return the same output. A classic way of composing functions is by using a *dot* operator `•`

, defined like the following:

```
infix operator •
func • <A,B,C> (
_ left: @escaping (B) -> C,
_ right: @escaping (A) -> B)
-> (A) -> C
{
return { left(right($0)) }
}
```

The operator will simply apply both functions to a certain argument, one after the other. Now that we have a way to easily compose functions, we can generate a function that will modify the entire login page by applying a modification of the message for the processing state:

```
let buttonStateLens = LoginPage.lens.buttonState
let processingPrism = ViewState<Button>.prism.processing
let onProcessing = buttonStateLens.modify • processingPrism.tryModify
```

From a completely generic standpoint, we’re obtaining the `onProcessing`

function with a composition like the following:

```
(B -> B) -> (A -> A) • (C -> C) -> (B -> B) = (C -> C) -> (A -> A)
```

Or, by using the actual types:

```
(ViewState<Button> -> ViewState<Button>) -> (LoginPage -> LoginPage) • (String -> String) -> (ViewState<Button> -> ViewState<Button>) = (String -> String) -> (LoginPage -> LoginPage)
```

Thus, the result is a function of type `(String -> String) -> (LoginPage -> LoginPage)`

, that is, a higher-order function that takes a function that modifies a `String`

(our processing message) and returns a function that modifies the entire `LoginPage`

: in a way, this function takes a transformation of a small subpart of the data structure, and *lifts* it into a transformation of the whole structure.

Finally, we can apply our `onProcessing`

function like the following:

```
let newModel = onProcessing(advanceProcessingMessage)(oldModel)
```

## It’s all nice and well, but...

Up to this point we’ve been working mostly with *types*: words like *lens*, *prism*, *zip*, *tryModify* et cetera, are really just *conventions* for expressing what is really expressed by the types involved, in particular by the types of the functions that we composed and applied to our data structures. Types here are the only interface that we used to combine together the various pieces, and if types don’t change, nothing *should* break. Unfortunately, types might be not enough.

If I give you a lens, what I’m really giving you is functions based on a pair of types, a `Whole`

and a `Part`

: but just the interface doesn’t make sure that the lens is going to work as intended (for example, that it’s actually going to modify the data structure as we’d expect). How can we be sure that a lens or a prism behave properly? The answer is **LAWS**.

As programmers we don’t frequently talk about what *laws* or *axioms* our code is supposed to respect. **But we really should**. In fact, a discussion about laws is always needed when we’re building a large system from small pieces, because we need to **trust** the pieces: we wouldn’t be able to reason about code at all if we weren’t able to assert some *properties* on our pieces. And actually, when we write **tests** we’re really asserting that certain invariants are going to hold: for example, that a function called with a specific input will always return a specific output. But the problem with tests is that, usually, the underlying axioms that we’re asserting are **implicit** in the tests, meaning that we’re not really defining them. A nasty consequence of this is that we could end up writing tests that are ineffective (because we’re not really testing against all the laws) or redundant (because a certain law is already made sure by another test). Let’s define this axioms explicitly for lenses and prisms.

For a very well-behaved `Lens`

we need to prove 3 laws:

`getSet`

: if I`get`

a`B`

from a data structure`A`

, and then I`set`

it back,`A`

is not going to change;`setGet`

: if I`set`

a`B`

into a data structure`A`

and then I`get`

it back, the retrieved`B`

must be equal to the initial one;`setSet`

:`set`

is*idempotent*, meaning that if I`set`

a`B`

into an`A`

multiple times, it’s going to be exactly as if I`set`

it just once.

A well-behaved `Prism`

has similar laws, that basically state that a full instance of the `Whole`

is *completely defined* by each and everyone of its `Part`

s:

`tryGetInject`

: if I’m able to`tryGet`

a`B`

from an`A`

,`inject`

ing it back yields*exactly*the same`A`

;`injectTryGet`

: if I construct an`A`

by injecting a`B`

, I must be able to`tryGet`

exactly the same`B`

.

Once we have these laws defined, we can test lenses and prisms by asserting that the laws hold *for any possible* `Whole`

and `Part`

. I like to implement these kind of assertions as functions that return `true`

or `false`

, defined in a `Law`

namespace, like the following:

```
struct LensLaw {
static func setGet<Whole, Part>(
_ lens: Lens<Whole,Part>,
_ whole: Whole,
_ part: Part)
-> Bool where Part: Equatable
{
return lens.get(lens.set(part)(whole)) == part
}
}
```

This `LensLaw.setGet`

function takes a `Lens<Whole,Part>`

as argument, together with a `Whole`

and a `Part`

, and returns a `Bool`

simply by equating the retrieved part to the initial part (in fact, `Part`

must be `Equatable`

for the law to be asserted). Calling this function with a certain `Whole`

and `Part`

will **only** prove that the lens behaves for those specific instances. What we want to do instead is to call this function with **many random instances** of `Whole`

and `Part`

, and then check if the function always returns `true`

. This procedure is usually called *property-based testing*, and can be performed with libraries like SwiftCheck.

For example, using SwiftCheck we could write a test case like the following:

```
func testLensBehaves() {
let lens = Whole.lens.someProperty
property("Whole.lens.someProperty respects the setGet law") <- forAll { (whole: Whole, part: Part) in
LensLaw.setGet(lens,whole,part)
}
}
```

SwiftCheck is going to call the closure passed to `forAll`

many times, with many different *random* values of `whole`

and `part`

(the types must conform to certain protocols, please refer to SwiftCheck documentation for the details), and the test will pass only if the closure returns `true`

at each call.

## As always, boilerplate

Many of you might have noticed that the code for implementing lenses and prisms for our types is basically *boilerplate*: it has always the same structure, with only the names of properties and types changing. This has two implications:

- we don’t really need to test the code for
*trivial*lenses and prisms, that is, lenses and prisms that are defined for properties and cases; - we can automatically
**generate code**with tools like Sourcery.

For lenses in particular, starting from Swift 4, we don’t even need code generation thanks to the new `KeyPath`

type. From a `WritableKeyPath<Root,Value>`

we can always generate a `Lens<Root,Value>`

with code like the following:

```
extension WritableKeyPath {
var lens: Lens<Root,Value> {
return Lens<Root,Value>.init(
get: { whole in whole[keyPath: self] },
set: { part in
{ whole in
var m = whole
m[keyPath: self] = part
return m
}
})
}
}
let passwordLens = (\LoginPage.credentials.passwordField.text).lens
```

Unfortunately, this is not going to work if the `Value`

property onto which the lens is focusing is `let`

, thus immutable - we won’t get a `WritableKeyPath`

for it - and also `KeyPath`

is not available for enums, so code generation is still our friend here.

For example, the following template will generate the lenses for all the properties of structs, equipped with a memberwise initializer, annotated with the comment `//sourcery: lens`

(please refer to Sourcery documentation for more info):

```
{% for type in types.structs|annotated:"lens" %}
extension {{ type.name }} {
enum lens {
{% for variable in type.variables|!static|!computed %}
static let {{ variable.name }} = Lens<{{ type.name }}, {{ variable.typeName }}>(
get: { $0.{{ variable.name }} },
set: { part in
{ whole in
{{ type.name }}.init({% for argument in type.variables|!static|!computed %}{{ argument.name }}: {% if variable.name == argument.name %}part{% else %}whole.{{ argument.name }}{% endif %}{% if not forloop.last %}, {% endif %}{% endfor %})
}
}
){% endfor %}
}
}
{% endfor %}
```

A slightly different template is going to be needed if the type is generic (because we won’t be able to add static stored properties to it), but that’s the general strategy.

Tests are still going to be important, though, when we need to prove that functions like `then`

and `zip`

work as intended. Also, in some cases we really want to write some lenses and prisms manually, and consequently we’re really interested in testing them. For example we can define *prepared* lenses for particular data structures, like a `Dictionary`

. I frequently use the following `Lens`

, that focuses on a particular key of a dictionary:

```
extension Dictionary {
static func lens(at key: Key) -> Lens<Dictionary,Value?> {
return Lens<Dictionary,Value?>(
get: { $0[key] },
set: { part in
{ whole in
var m_dict = whole
m_dict[key] = part
return m_dict
}
})
}
}
```

This seems perfectly fine, but I want to be able to test this, and laws are the answer.

Notice that the returned lens here is of type `Lens<Dictionary,Value?>`

: the `Part`

is `Optional`

and this is going to give us trouble in composition. What I mean is, we were able to generically compose lenses like this:

```
Lens<A,B> + Lens<B,C> = Lens<A,C>
```

But here, we could deal with something like this:

```
Lens<A,B?> + Lens<B,C> = ?
```

`B?`

and `B`

are not the same, so the regular composition doesn’t work. A `Lens`

is simply not enough for this, we cannot make a lens *failable* while keeping it a proper lens that follows the laws. But it turns out that there’s *another* optic that works exactly like a failable lens, and it’s called `Affine`

:

```
struct Affine<Whole,Part> {
let tryGet: (Whole) -> Part?
let trySet: (Part) -> (Whole) -> Whole?
}
```

An `Affine<Whole,Part>`

looks exactly like the mix of a lens and prism, for it has `tryGet`

, like a prism, while the `set`

is actually a `trySet`

that *tries* to set a value, but can fail, and that’s reflected but the returned type, that is, `Whole?`

.

As an example, we can define an `Affine`

on the element at a certain index in an array:

```
extension Array {
static func affine(at index: Int) -> Affine<Array,Element> {
return Affine<Array,Element>.init(
tryGet: { array in
guard array.indices.contains(index) else { return nil }
return array[index]
},
trySet: { element in
{ array in
guard array.indices.contains(index) else { return nil }
var m = array
_ = m.remove(at: index)
m.insert(element, at: index)
return m
}
})
}
}
```

Affines compose like lenses and prisms, with the `then`

method:

```
extension Affine {
func then <Subpart> (_ other: Affine<Part,Subpart>) -> Affine<Whole,Subpart> {
return Affine<Whole,Subpart>.init(
tryGet: { s in self.tryGet(s).flatMap(other.tryGet) },
trySet: { bp in
{ s in
self.tryGet(s)
.flatMap { a in other.trySet(bp)(a) }
.flatMap { b in self.trySet(b)(s) }
}
})
}
}
```

`Affine`

must follow laws very similar to those of `Lens`

, that is, `trySetTryGet`

, `tryGetTrySet`

and `trySetTrySet`

. Interestingly, we can turn any lens or prism into an affine:

```
extension Lens {
func toAffine() -> Affine<Whole,Part> {
return Affine<Whole,Part>.init(
tryGet: self.get,
trySet: self.set)
}
}
extension Prism {
func toAffine() -> Affine<Whole,Part> {
return Affine<Whole,Part>.init(
tryGet: self.tryGet,
trySet: { part in self.tryModify { _ in part } })
}
}
```

Now we can solve the problem related to composing `Lens<A,B?>`

with `Lens<B,C>`

. We’ll leverage a very specific prism, that is, *the prism on an Optional<Wrapped>*:

```
extension Optional {
static var prism: Prism<Optional,Wrapped> {
return Prism<Optional,Wrapped>.init(
tryGet: { $0 },
inject: Optional.some)
}
}
```

Now we simply need a little algebra:

```
/// we cannot do this
Lens<A,B?> + Lens<B,C> = ?
/// but we can stick a prism in the middle
Lens<A,B?> + Prism<B?,B> + Lens<B,C>
/// we then turn everything into an affine
Affine<A,B?> + Affine<B?,B> + Affine<B,C>
/// finally, we compose the affines
Affine<A,B?> + Affine<B?,B> + Affine<B,C>
--> Affine<A,B> + Affine<B,C>
--> Affine<A,C>
/// the final result goes from A to C, with no optionals
Affine<A,C>
```

Notice that, because we can always turn a lens or a prism into an affine, it means that we can compose lenses with affines and prisms with affines, which means that, finally, **the affine is the result of the composition between a lens and prism**, which is maybe the big piece that was still missing:

```
Lens<A,B> + Prism<B,C> = Affine<A,C>
Prism<A,B> + Lens<B,C> = Affine<A,C>
```

`Affine`

solves our *nullability* problem by employing a special focusing strategy: it either focuses on *something* or on *nothing*, and this idea is encoded in its very semantics. Notice that, once you’re in `Affine`

, you simply can’t get out, because *everything* will be considered nullable from that on, and there’s no alternative in terms of optics: a weak link makes for a weak chain. To avoid using `Affine`

we need to avoid `Optional`

altogether, but usually `Optional`

is exactly what we want, because it better expresses our logic. There are some specific cases, though, where we could cook-up a data structure
where, in some place, instead of `Optional`

we use a *default value*, and that would remove the need for `Affine`

(for example, instead of `Dictionary<Key,Value>`

we could imagine a `DefaultedDictionary<Key,Value>`

): it’s up to you to choose the right solution for each specific problem.

## Conclusion

`Lens`

, `Prism`

and `Affine`

are abstractions over the relationships between a data structure and its parts. They allow to encapsulate such relationships with any level of composition, both *vertical* (like from a parent to a child of a child) and *horizontal* (like from a parent to different children in parallel). This relationship can be extended in many different ways: we can *induce structure* from a child to a parent, like for example in giving *ordering logic* to a data structure starting from a comparator in one of its parts (it was shown by Brandon Williams in this talk).

The main use case is representing the modifications induced into a data structure by modifying its parts, and composition plays a huge role. A nice, simple use case for prisms is also to assert that an enum is in a particular case, like you would do with `guard case`

but with a single expression, for example with a function like this:

```
extension Prism {
func isCase(_ whole: Whole) -> Bool {
return tryGet(whole) != nil
}
}
```

In fact, once you get the basic definition laid out for `Lens`

and `Prism`

, a universe of possibilities opens: we just need to explore it.

## Bonus Theory: the Lens/Prism duality

Another piece of trivia that you get when googling about functional lenses and prisms is that “prism is the *dual* of lens”. The mathematical concept of *duality* is pretty complex in general, but in this particular case it means that from a lens we can get something else (with different properties) by simply “reversing the arrows”, or in other words, by reversing all the (directed) relationships. For example if we have a function of type:

```
(A) -> B
```

The dual of this function is going to be:

```
(B) -> A
```

This is **not** the *inverse function* of `(A) -> B`

: the inverse function is the function that, applied after the first, gives us back exactly the value with which we started. This is simply a function with switched input and output, and the “dual” procedure simply allows us to discover that such a thing exists (pretty obvious, in this case).

A little more interesting is the dual of a *product type*: we defined `struct`

and `class`

as product types, in the sense that they’re constructed with a set of properties. The quintessential product type is a *tuple*, which **only** has properties (optionally tagged with names in Swift) and no other methods or behavior. A more rigorous way to see this is in term of *projections*: for every property, we have a function (the *getter*) from the type of the struct or class to the type of the property. So for example the following type:

```
struct Product<A,B> {
let a: A
let b: B
}
```

can be defined completely by the functions:

```
let projectA: (Product<A,B>) -> A = { $0.a }
let projectB: (Product<A,B>) -> B = { $0.b }
```

To define a type that’s dual to `Product<A,B>`

we need to turn the projections into *injections*, like the following:

```
let injectA: (A) -> ???<A,B> = { ??? }
let injectB: (B) -> ???<A,B> = { ??? }
```

As many have probably already imagined, the `???`

type that actually works in this case is an enum:

```
enum Coproduct<A,B> {
case a(A)
case b(B)
}
```

with the injections becoming:

```
let injectA: (A) -> Coproduct<A,B> = { .a($0) }
let injectB: (B) -> Coproduct<A,B> = { .b($0) }
```

An enum is thus a *coproduct* type (*co* stands for *dual to*).

Now, we defined lenses for product types, let’s then *discover* prisms for coproducts as *colenses* with the same procedure.

First, we need a different representation for `Lens<Whole,Part>`

that’s a little more basic, less convenient but more explicit:

```
struct Lens<Whole,Part> {
let get: (Whole) -> Part
let set: (Product<Whole,Part>) -> Whole
}
```

Notice that the `set`

function is not *partially applied*, and `Product<Whole,Part>`

is just a more explicit way of representing the simple tuple `(Whole,Part)`

.

Let’s reverse everything: functions will reverse the arrows, and products become coproducts:

```
struct ???<Whole,Part> {
let ???: (Part) -> Whole
let ???: (Whole) -> Coproduct<Whole,Part>
}
```

We’re going to call this `Prism`

. The first function is an injection (it’s the dual of the `get`

projection) so let’s call it `inject`

.

```
struct Prism<Whole,Part> {
let inject: (Part) -> Whole
let ???: (Whole) -> Coproduct<Whole,Part>
}
```

The second function is a little more complex: it takes a `Whole`

and returns an enum with two cases:

- the first case is the
`Whole`

again, which means that we really didn’t get anything out of it; - the second case is
`Part`

, which means that we were able to get what we were looking for.

Thus, let’s call this function `tryGet`

. To achieve the prism representation we used in the article, we can observe that in `Coproduct<Whole,Part>`

the `Part`

case is the only one that’s really interesting, so for what matters to us it could as well be `Coproduct<(),Part>`

; but the latter is basically `Optional<Part>`

(in fact, the `none`

case in `Optional`

implicitly has `()`

as associated type). Finally, by switching the order of the properties we get:

```
struct Prism<Whole,Part> {
let tryGet: (Whole) -> Part?
let inject: (Part) -> Whole
}
```

which is the representation we’re been using the whole time. Notice that we could have *discovered* this simply by starting with `Lens`

and being instructed about how to *reverse the arrows*.

Until next time.