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### Introduction

Todayâ€™s topic is a big one, and itâ€™s only the beginning of a long, deep journey. We want to consider the seemingly humble `map`

function in all of its glory. Many people point to the presence of `map`

in a language as being a strong indicator of the language having â€śfunctionalâ€ť tendencies. Youâ€™ll often hear â€ślanguage ABC has support for functional concepts like `map`

, `filter`

, `reduce`

, etcâ€ť, and the mention of `map`

is always first! Why is that?!

Swift must be doubly functional because it comes with two `map`

s! One on arrays and one on optionals!

We want to build intuition for why `map`

seems to be so ubiquitous in â€śfunctional-leaningâ€ť languages, and in fact there are lots of other `map`

s lurking in the shadows of our everyday Swift code that we have yet to explore.

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### Exercises

Implement a

`map`

function on dictionary values, i.e.`map: ((V) -> W) -> ([K: V]) -> [K: W]`

Does it satisfy

`map(id) == id`

?Implement the following function:

`transformSet: ((A) -> B) -> (Set<A>) -> Set<B>`

We do not call this

`map`

because it turns out to not satisfy the properties of`map`

that we saw in this episode. What is it about the`Set`

type that makes it subtly different from`Array`

, and how does that affect the genericity of the`map`

function?Recall that one of the most useful properties of

`map`

is the fact that it distributes over compositions, i.e.`map(f >>> g) == map(f) >>> map(g)`

for any functions`f`

and`g`

. Using the`transformSet`

function you defined in a previous example, find an example of functions`f`

and`g`

such that`transformSet(f >>> g) != transformSet(f) >>> transformSet(g)`

This is why we do not call this function

`map`

.There is another way of modeling sets that is different from

`Set<A>`

in the Swift standard library. It can also be defined as function`(A) -> Bool`

that answers the question â€śis`a: A`

contained in the set.â€ť Define a type`struct PredicateSet<A>`

that wraps this function. Can you define the following?`map: ((A) -> B) -> (PredicateSet<A>) -> PredicateSet<B>`

What goes wrong?

Try flipping the direction of the arrow in the previous exercise. Can you define the following function?

`fakeMap: ((B) -> A) -> (PredicateSet<A>) -> PredicateSet<B>`

What kind of laws do you think

`fakeMap`

should satisfy?Sometimes we deal with types that have multiple type parameters, like

`Either`

and`Result`

. For those types you can have multiple`map`

s, one for each generic, and no one version is â€śmoreâ€ť correct than the other. Instead, you can define a`bimap`

function that takes care of transforming both type parameters at once. Do this for`Result`

and`Either`

.Write a few implementations of the following function:

`func r<A>(_ xs: [A]) -> A? { }`

Continuing the previous exercise, can you generalize your implementations of

`r`

to a function`[A] -> B?`

if you had a function`f: (A) -> B`

?`func s<A, B>(_ f: (A) -> B, _ xs: [A]) -> B? { }`

What features of arrays and optionals do you need to implement this?

Derive a relationship between

`r`

, any function`f: (A) -> B`

, and the`map`

on arrays and optionals.This relationship is the â€śfree theoremâ€ť for

`r`

â€™s signature.

### References

#### Theorems for Free

**Philip Wadler â€˘ Saturday Jul 1, 1989**

This famous paper describes â€śtheorems for freeâ€ť, in which if you write down a generic function signature, you can derive theorems that the function satisfies. This works in any language that has parametric polymorphism, as Swift does.