Jason Shin

8 Jul 2019

•

18 min read

Github Repository: https://github.com/JasonShin/functional-programming-jargon.rs

Functional programming (FP) provides many advantages, and its popularity has been increasing as a result. However, each programming paradigm comes with its own unique jargon and FP is no exception. By providing a glossary, we hope to make learning FP easier.

Where applicable, this document uses terms defined in the Fantasy Land spec and Rust programming language to give code examples.

Also please be mindful that the project sometimes utilises FP related crates that are unfinished but contains a specific typeclass or data type that is appropriate to give an explanation. Many of them are experimental and are not suitable for production usage.

**Table of Contents**

- [Arity](# arity)
- [Higher-Order Functions (HOF)](# higher-order-functions-hof)
- [Closure](# closure)
- [Partial Application](# partial-application)
- [Currying](# currying)
- [Auto Currying](# auto-currying)
- [Referential Transparency](# referential-transparency)
- [Lambda](# lambda)
- [Lambda Calculus](# lambda-calculus)
- [Purity](# purity)
- [Side effects](# side-effects)
- [Idempotent](# idempotent)
- [Function Composition](# function-composition)
- [Point-Free Style](# point-free-style)
- [Predicate](# predicate)
- [Contracts](# contracts)
- [Category](# category)
- [Value](# value)
- [Constant](# constant)
- [Variance](# variance)
- [Higher Kinded Type](# higher-kinded-type-hkt)
- [Functor](# functor)
- [Pointed Functor](# pointed-functor)
- [Lifting](# lifting)
- [Equational Reasoning](# equational-reasoning)
- [Monoid](# monoid)
- [Monad](# monad)
- [Comonad](# comonad)
- [Applicative](# applicative)
- [Morphism](# morphism)
- [Endomorphism](# endomorphism)
- [Isomorphism](# isomorphism)
- [Homomorphism](# homomorphism)
- [Catamorphism](# catamorphism)
- [Hylomorphism](# hylomorphism)
- [Anamorphism](# anamorphism)

- [Setoid](# setoid)
- [Semigroup](# semigroup)
- [Foldable](# foldable)
- [Lens](# lens)
- [Type Signature](# type-signature)
- [Algebraic data type](# algebraic-data-type)
- [Sum Type](# sum-type)
- [Product Type](# product-type)

- [Option](# option)
- [Functional Programming References](# functional-programming-references)
- [Function Programming development in Rust Language](# functional-programming-development-in-rust-language)
- [My thought on this project](# my-thought-on-this-project)

The number of arguments a function takes. From words like unary, binary, ternary, etc. This word has the distinction of being composed of two suffixes, "-ary" and "-ity." Addition, for example, takes two arguments, and so it is defined as a binary function or a function with an arity of two. Such a function may sometimes be called "dyadic" by people who prefer Greek roots to Latin. Likewise, a function that takes a variable number of arguments is called "variadic," whereas a binary function must be given two and only two arguments, currying and partial application notwithstanding (see below).

```
let sum = |a: i32, b: i32| { a + b }; // The arity of sum is 2
```

A function which takes a function as an argument and/or returns a function.

```
let filter = | predicate: fn(&i32) -> bool, xs: Vec<i32> | {
return xs.into_iter().filter(predicate).collect::<Vec<i32>>();
};
```

```
let is_even = |x: &i32| { x % 2 == 0 };
```

```
filter(is_even, vec![1, 2, 3, 4, 5, 6]);
```

A closure is a scope which retains variables available to a function when it's created. This is important for [partial application](# partial-application) to work.

```
let add_to = | x: i32 | { move | y: i32 | { x + y } };
```

We can call `add_to`

with a number and get back a function with a baked-in `x`

. Notice that we also need to move the ownership of the y to the internal lambda.

```
let add_to_five = add_to(5);
```

In this case the `x`

is retained in `add_to_five`

's closure with the value `5`

. We can then call `add_to_five`

with the `y`

and get back the desired number.

```
add_to_five(3); // => 8
```

Closures are commonly used in event handlers so that they still have access to variables defined in their parents when they are eventually called.

**Further reading**

Partially applying a function means creating a new function by pre-filling some of the arguments to the original function.

To achieve this easily, we will be using a partial application crate

```
#[macro_use]
extern crate partial_application;
fn foo(a: i32, b: i32, c: i32, d: i32, mul: i32, off: i32) -> i32 {
(a + b*b + c.pow(3) + d.pow(4)) * mul - off
}
let bar = partial!( foo(_, _, 10, 42, 10, 10) );
assert_eq!(
foo(15, 15, 10, 42, 10, 10),
bar(15, 15)
); // passes
```

The partial application helps create simpler functions from more complex ones by baking in data when you have it. Curried functions are automatically partially applied.

**Further reading**

The process of converting a function that takes multiple arguments into a function that takes them one at a time.

Each time the function is called it only accepts one argument and returns a function that takes one argument until all arguments are passed.

```
fn add(x: i32) -> impl Fn(i32)-> i32 {
return move |y| x + y;
}
let add5 = add(5);
add5(10); // 15
```

**Further reading**

Transforming a function that takes multiple arguments into one that is given less than its correct number of arguments returns a function that takes the rest. When the function gets the correct number of arguments it is then evaluated.

Although Auto Currying is not possible in Rust right now, there is a debate on this issue on the Rust forum: https://internals.rust-lang.org/t/auto-currying-in-rust/149/22

An expression that can be replaced with its value without changing the behaviour of the program is said to be referentially transparent.

Say we have function greet:

```
let greet = || "Hello World!";
```

Any invocation of `greet()`

can be replaced with `Hello World!`

hence greet is referentially transparent.

An anonymous function that can be treated like a value.

```
fn increment(i: i32) -> i32 { i + 1 }
let closure_annotated = |i: i32| { i + 1 };
let closure_inferred = |i| i + 1;
```

Lambdas are often passed as arguments to Higher-Order Functions. You can assign a lambda to a variable, as shown above.

A branch of mathematics that uses functions to create a universal model of computation.

A function is pure if the return value is only determined by its input values, and does not produce side effects.

```
let greet = |name: &str| { format!("Hi! {}", name) };
greet("Jason"); // Hi! Jason
```

As opposed to each of the following:

```
let name = "Jason";
let greet = || -> String {
format!("Hi! {}", name)
};
greet(); // String = "Hi! Jason"
```

The above example's output is based on data stored outside of the function...

```
let mut greeting: String = "".to_string();
let mut greet = |name: &str| {
greeting = format!("Hi! {}", name);
};
greet("Jason");
assert_eq!("Hi! Jason", greeting); // Passes
```

... and this one modifies state outside of the function.

A function or expression is said to have a side effect if apart from returning a value, it interacts with (reads from or writes to) external mutable state.

```
use std::time::SystemTime;
let now = SystemTime::now();
```

```
println!("IO is a side effect!");
// IO is a side effect!
```

A function is idempotent if reapplying it to its result does not produce a different result.

```
// Custom immutable sort method
let sort = | x: Vec<i32> | -> Vec<i32> {
let mut cloned_x = x.clone();
cloned_x.sort();
return cloned_x;
};
```

Then we can use the sort method like

```
let x = vec![2 ,1];
let sorted_x = sort(sort(x.clone()));
let expected = vec![1, 2];
assert_eq!(sorted_x, expected); // passes
```

```
let abs = | x: i32 | -> i32 {
return x.abs();
};
let x: i32 = 10;
let result = abs(abs(x));
assert_eq!(result, x); // passes
```

The act of putting two functions together to form a third function where the output of one function is the input of the other. Below is an example of compose function is Rust.

```
macro_rules! compose {
( $last:expr ) => { $last };
( $head:expr, $($tail:expr), +) => {
compose_two($head, compose!($($tail),+))
};
}
fn compose_two<A, B, C, G, F>(f: F, g: G) -> impl Fn(A) -> C
where
F: Fn(A) -> B,
G: Fn(B) -> C,
{
move |x| g(f(x))
}
```

Then we can use it like

```
let add = | x: i32 | x + 2;
let multiply = | x: i32 | x * 2;
let divide = | x: i32 | x / 2;
let intermediate = compose!(add, multiply, divide);
let subtract = | x: i32 | x - 1;
let finally = compose!(intermediate, subtract);
let expected = 11;
let result = finally(10);
assert_eq!(result, expected); // passes
```

Writing functions where the definition does not explicitly identify the arguments used. This style usually requires currying or other Higher-Order functions. A.K.A Tacit programming.

A predicate is a function that returns true or false for a given value. A common use of a predicate is as the callback for array filter.

```
let predicate = | a: &i32 | a.clone() > 2;
let result = (vec![1, 2, 3, 4]).into_iter().filter(predicate).collect::<Vec<i32>>();
assert_eq!(result, vec![3, 4]); // passes
```

A contract specifies the obligations and guarantees of the behavior from a function or expression at runtime. This acts as a set of rules that are expected from the input and output of a function or expression, and errors are generally reported whenever a contract is violated.

```
let contract = | x: &i32 | -> bool {
return x > &10;
};
let add_one = | x: &i32 | -> Result<i32, String> {
if contract(x) {
return Ok(x + 1);
}
return Err("Cannot add one".to_string());
};
```

Then you can use `add_one`

like

```
let expected = 12;
match add_one(&11) {
Ok(x) => assert_eq!(x, expected),
_ => panic!("Failed!")
}
```

A category in category theory is a collection of objects and morphisms between them. In programming, typically types act as the objects and functions as morphisms.

To be a valid category 3 rules must be met:

- There must be an identity morphism that maps an object to itself.
Where
`a`

is an object in some category, there must be a function from`a -> a`

. - Morphisms must compose.
Where
`a`

,`b`

, and`c`

are objects in some category, and`f`

is a morphism from`a -> b`

, and`g`

is a morphism from`b -> c`

;`g(f(x))`

must be equivalent to`(g • f)(x)`

. - Composition must be associative
`f • (g • h)`

is the same as`(f • g) • h`

Since these rules govern composition at very abstract level, category theory is great at uncovering new ways of composing things.

**Further reading**

Anything that can be assigned to a variable.

```
let a = 5;
let b = vec![1, 2, 3];
let c = "test";
```

A variable that cannot be reassigned once defined.

```
let a = 5;
a = 3; // error!
```

Constants are [referentially transparent](# referential-transparency). That is, they can be replaced with the values that they represent without affecting the result.

Variance in functional programming refers to subtyping between more complex types related to subtyping between their components.

Unlike other usage of variance in Object Oriented Programming like Typescript or C# or functional programming language like Scala or Haskell

Variance in Rust is used during the type checking against type and lifetime parameters. Here are examples:

- By default, all lifetimes are co-variant except for
`'static`

because it outlives all others `'static`

is always contra-variant to others regardless of where it appears or used- It is
`in-variant`

if you use`Cell<T>`

or`UnsafeCell<T>`

in`PhatomData`

**Further Reading**

- https://github.com/rust-lang/rustc-guide/blob/master/src/variance.md
- https://nearprotocol.com/blog/understanding-rust-lifetimes/

Rust does not support Higher Kinded Types yet. First of all, HKT is a
type with a "hole" in it, so you can declare a type signature such as `trait Functor<F<A>>`

.

Although Rust lacks in a native support for HKT, we always have a walk around called Lightweight Higher Kinded Type

An implementation example of above theory in Rust would look like below:

```
pub trait HKT<A, B> {
type URI;
type Target;
}
// Lifted Option
impl<A, B> HKT<A, B> for Option<A> {
type URI = Self;
type Target = Option<B>;
}
```

Higher Kinded Type is crucial for functional programming in general.

**Further Reading**

- https://gist.github.com/CMCDragonkai/a5638f50c87d49f815b8
- https://www.youtube.com/watch?v=ERM0mBPNLHc

An object that implements a map function which, while running over each value in the object to produce a new functor of the same type, adheres to two rules:

**Preserves identity**

```
object.map(x => x) ≍ object
```

**Composable**

```
object.map(compose(f, g)) ≍ object.map(g).map(f)
```

(`f`

, `g`

are arbitrary functions)

For example, below can be considered as a functor-like operation

```
let v: Vec<i32> = vec![1, 2, 3].into_iter().map(| x | x + 1).collect();
assert_eq!(v, vec![2, 3, 4]); // passes while mapping the original vector and returns a new vector
```

While leveraging the [HKT implementation](# higher-kinded-type-hkt), You can define a trait that represents Functor like below

```
pub trait Functor<A, B>: HKT<A, B> {
fn fmap<F>(self, f: F) -> <Self as HKT<A, B>>::Target
where F: FnOnce(A) -> B;
}
```

Then use it against a type such as [Option](# https://doc.rust-lang.org/std/option/index.html) like

```
impl<A, B> Functor<A, B> for Option<A> {
fn fmap<F>(self, f: F) -> Self::Target
where
F: FnOnce(A) -> B
{
self.map(f)
}
}
#[test]
fn test_functor() {
let z = Option::fmap(Some(1), |x| x + 1).fmap(|x| x + 1); // Return Option<B>
assert_eq!(z, Some(3)); // passes
}
```

An object with an of function that puts any single value into it.

```
#[derive(Debug, PartialEq, Eq)]
enum Maybe<T> {
Nothing,
Just(T),
}
impl<T> Maybe<T> {
fn of(x: T) -> Self {
return Maybe::Just(x);
}
}
```

Then use it like

```
let pointed_functor = Maybe::of(1);
assert_eq!(pointed_functor, Maybe::Just(1));
```

Lifting in functional programming typically means to lift a function into a context (a Functor or Monad).
For example, give a function `a -> b`

and lift it into a `List`

then the signature would
look like `List[a] -> List[b]`

.

**Further Reading**

- https://wiki.haskell.org/Lifting
- https://stackoverflow.com/questions/43482772/difference-between-lifting-and-higher-order-functions
- https://stackoverflow.com/questions/2395697/what-is-lifting-in-haskell/2395956

When an application is composed of expressions and devoid of side effects, truths about the system can be derived from the parts.

An object with a function that "combines" that object with another of the same type.

One simple monoid is the addition of numbers:

```
1 + 1
// i32: 2
```

In this case number is the object and `+`

is the function.

An "identity" value must also exist that when combined with a value doesn't change it.

The identity value for addition is `0`

.

```
1 + 0
// i32: 1
```

It's also required that the grouping of operations will not affect the result (associativity):

```
1 + (2 + 3) == (1 + 2) + 3
// bool: true
```

Array concatenation also forms a monoid:

```
[vec![1, 2, 3], vec![4, 5, 6]].concat();
// Vec<i32>: vec![1, 2, 3, 4, 5, 6]
```

The identity value is empty array `[]`

```
[vec![1, 2], vec![]].concat();
// Vec<i32>: vec![1, 2]
```

If identity and compose functions are provided, functions themselves form a monoid:

```
fn identity<A>(a: A) -> A {
return a;
}
```

`foo`

is any function that takes one argument.

```
compose(foo, identity) ≍ compose(identity, foo) ≍ foo
```

A [Monad](https://github.com/fantasyland/fantasy-land# monad) is a trait that implements `Applicative`

and `Chain`

specifications. `chain`

is
like `map`

except it un-nests the resulting nested object.

First, `Chain`

type can be implemented like below:

```
pub trait Chain<A, B>: HKT<A, B> {
fn chain<F>(self, f: F) -> <Self as HKT<A, B>>::Target
where F: FnOnce(A) -> <Self as HKT<A, B>>::Target;
}
impl<A, B> Chain<A, B> for Option<A> {
fn chain<F>(self, f: F) -> Self::Target
where F: FnOnce(A) -> <Self as HKT<A, B>>::Target {
self.and_then(f)
}
}
```

Then `Monad`

itself can simply derive `Chain`

and `Applicative`

```
pub trait Monad<A, F, B>: Chain<A, B> + Applicative<A, F, B>
where F: FnOnce(A) -> B {}
impl<A, F, B> Monad<A, F, B> for Option<A>
where F: FnOnce(A) -> B {}
#[test]
fn monad_example() {
let x = Option::of(Some(1)).chain(|x| Some(x + 1));
assert_eq!(x, Some(2)); // passes
}
```

`pure`

is also known as `return`

in other functional languages. `flat_map`

is also known as `bind`

in other languages.

An object that has `extract`

and `extend`

functions.

```
trait Extend<A, B>: Functor<A, B> + Sized {
fn extend<W>(self, f: W) -> <Self as HKT<A, B>>::Target
where
W: FnOnce(Self) -> B;
}
trait Extract<A> {
fn extract(self) -> A;
}
trait Comonad<A, B>: Extend<A, B> + Extract<A> {}
```

Then we can implement these types for Option

```
impl<A, B> Extend<A, B> for Option<A> {
fn extend<W>(self, f: W) -> Self::Target
where
W: FnOnce(Self) -> B,
{
self.map(|x| f(Some(x)))
}
}
impl<A> Extract<A> for Option<A> {
fn extract(self) -> A {
self.unwrap() // is there a better way to achieve this?
}
}
```

Extract takes a value out of a functor.

```
Some(1).extract(); // 1
```

Extend runs a function on the Comonad.

```
Some(1).extend(|co| co.extract() + 1); // Some(2)
```

An applicative functor is an object with an `ap`

function. `ap`

applies a function in the object to a value in another
object of the same type. Given a pure program `g: (b: A) -> B`

, we must lift it to `g: (fb: F<A>) -> F<B>`

. In order to achieve
this, we will introduce another [higher kinded type](# higher-kinded-type-hkt), called `HKT3`

that is capable of doing this.

For this example, we will use Option datatype.

```
trait HKT3<A, B, C> {
type Target2;
}
impl<A, B, C> HKT3<A, B, C> for Option<A> {
type Target2 = Option<B>;
}
```

Since Applicative implements Apply for `ap`

and `Pure`

for `of`

according to [Fantasy Land specification](https://github.com/fantasyland/fantasy-land# applicative)
we must implement the types like below:

```
// Apply
trait Apply<A, F, B> : Functor<A, B> + HKT3<A, F, B>
where F: FnOnce(A) -> B,
{
fn ap(self, f: <Self as HKT3<A, F, B>>::Target2) -> <Self as HKT<A, B>>::Target;
}
impl<A, F, B> Apply<A, F, B> for Option<A>
where F: FnOnce(A) -> B,
{
fn ap(self, f: Self::Target2) -> Self::Target {
self.and_then(|v| f.map(|z| z(v)))
}
}
// Pure
trait Pure<A>: HKT<A, A> {
fn of(self) -> <Self as HKT<A, A>>::Target;
}
impl<A> Pure<A> for Option<A> {
fn of(self) -> Self::Target {
self
}
}
// Applicative
trait Applicative<A, F, B> : Apply<A, F, B> + Pure<A>
where F: FnOnce(A) -> B,
{} // Simply derives Apply and Pure
impl<A, F, B> Applicative<A, F, B> for Option<A>
where F: FnOnce(A) -> B,
{}
```

Then we can use Option Applicative like this:

```
let x = Option::of(Some(1)).ap(Some(|x| x + 1));
assert_eq!(x, Some(2));
```

A transformation function.

A function where the input type is same as the output.

```
// uppercase :: &str -> String
let uppercase = |x: &str| x.to_uppercase();
// decrement :: i32 -> i32
let decrement = |x: i32| x - 1;
```

A pair of transformations between 2 types of objects that is structural in nature and no data is lost.

For example, 2D coordinates could be stored as a i32 vector [2,3] or a struct {x: 2, y: 3}.

```
#[derive(PartialEq, Debug)]
struct Coords {
x: i32,
y: i32,
}
let pair_to_coords = | pair: (i32, i32) | Coords { x: pair.0, y: pair.1 };
let coords_to_pair = | coords: Coords | (coords.x, coords.y);
assert_eq!(
pair_to_coords((1, 2)),
Coords { x: 1, y: 2 },
); // passes
assert_eq!(
coords_to_pair(Coords { x: 1, y: 2 }),
(1, 2),
); // passes
```

A homomorphism is just a structure preserving map. In fact, a functor is just a homomorphism between categories as it preserves the original category's structure under the mapping.

```
assert_eq!(A::of(f).ap(A::of(x)), A::of(f(x))); // passes
assert_eq!(
Either::of(|x: &str| x.to_uppercase(x)).ap(Either::of("oreos")),
Either::of("oreos".to_uppercase),
); // passes
```

A `reduceRight`

function that applies a function against an accumulator and each value of the array (from right-to-left)
to reduce it to a single value.

```
let sum = |xs: Vec<i32>| xs.iter().fold(0, |mut sum, &val| { sum += val; sum });
assert_eq!(sum(vec![1, 2, 3, 4, 5]), 15);
```

An `unfold`

function. An `unfold`

is the opposite of `fold`

(`reduce`

). It generates a list from a single value.

```
let count_down = unfold((8_u32, 1_u32), |state| {
let (ref mut x1, ref mut x2) = *state;
if *x1 == 0 {
println!("stopping!");
return None;
}
let next = *x1 - *x2;
let ret = *x1;
*x1 = next;
Some(ret)
});
assert_eq!(
count_down.collect::<Vec<u32>>(),
vec![8, 7, 6, 5, 4, 3, 2, 1],
);
```

The combination of anamorphism and catamorphism.

It's the opposite of paramorphism, just as anamorphism is the opposite of catamorphism. Whereas with paramorphism, you combine with access to the accumulator and what has been accumulated, apomorphism lets you unfold with the potential to return early.

An object that has an `equals`

function which can be used to compare other objects of the same type.

It must obey following rules to be `Setoid`

`a.equals(a) == true`

(reflexivity)`a.equals(b) == b.equals(a)`

(symmetry)`a.equals(b)`

and`b.equals(c)`

then`a.equals(c)`

(transitivity)

Make a Vector a setoid:

Note that I am treating `Self`

/ `self`

like `a`

.

```
trait Setoid {
fn equals(&self, other: &Self) -> bool;
}
impl Setoid for Vec<i32> {
fn equals(&self, other: &Self) -> bool {
return self.len() == other.len();
}
}
assert_eq!(vec![1, 2].equals(&vec![1, 2]), true); // passes
```

An object that has a `combine`

function that combines it with another object of the same type.

It must obey following rules to be `Semigroup`

`a.combine(b).combine(c)`

is equivalent to`a.combine(b.combine(c))`

(associativity)

```
use itertools::concat;
trait Semigroup {
fn combine(&self, b: &Self) -> Self;
}
impl Semigroup for Vec<i32> {
fn combine(&self, b: &Self) -> Vec<i32> {
return concat(vec![self.clone(), b.clone()]);
}
}
assert_eq!(
vec![1, 2].combine(&vec![3, 4]),
vec![1, 2, 3, 4],
); // passes
assert_eq!(
a.combine(&b).combine(&c),
a.combine(&b.combine(&c)),
); // passes
```

An object that has a `foldr/l`

function that can transform that object into some other type. We are using `rats`

to give
an example but the crate only implements `fold_right`

.

`fold_right`

is equivalent to Fantasy Land Foldable's `reduce`

, which goes like:

`fantasy-land/reduce :: Foldable f => f a ~> ((b, a) -> b, b) -> b`

```
use rats::foldable::Foldable;
use rats::kind::IntoKind;
use rats::kinds::VecKind;
let k = vec![1, 2, 3].into_kind();
let result = VecKind::fold_right(k, 0, | (i, acc) | i + acc);
assert_eq!(result, 6);
```

A lens is a type that pairs a getter and a non-mutating setter for some other data structure.

```
trait Lens<S, A> {
fn over(s: &S, f: &Fn(Option<&A>) -> A) -> S {
let result: A = f(Self::get(s));
return Self::set(result, &s);
}
fn get(s: &S) -> Option<&A>;
fn set(a: A, s: &S) -> S;
}
#[derive(Debug, PartialEq, Clone)]
struct Person {
name: String,
}
#[derive(Debug)]
struct PersonNameLens;
impl Lens<Person, String> for PersonNameLens {
fn get(s: &Person) -> Option<&String> {
return Some(&s.name);
}
fn set(a: String, s: &Person) -> Person {
return Person {
name: a,
}
}
}
```

Having the pair of get and set for a given data structure enables a few key features.

```
let e1 = Person {
name: "Jason".to_string(),
};
let name = PersonNameLens::get(&e1);
let e2 = PersonNameLens::set("John".to_string(), &e1);
let expected = Person {
name: "John".to_string()
};
let e3 = PersonNameLens::over(&e1, &|x: Option<&String>| {
match x {
Some(y) => y.to_uppercase(),
None => panic!("T_T") // lol...
}
});
assert_eq!(*name.unwrap(), e1.name); // passes
assert_eq!(e2, expected); // passes
assert_eq!(e3, Person { name: "JASON".to_string() }); // passes
```

Lenses are also composable. This allows easy immutable updates to deeply nested data.

```
struct FirstLens;
impl<A> Lens<Vec<A>, A> for FirstLens {
fn get(s: &Vec<A>) -> Option<&A> {
return s.first();
}
fn set(a: A, s: &Vec<A>) -> Vec<A> {
unimplemented!();
}
}
let people = vec![Person { name: "Jason" }, Person { name: "John" }];
Lens::over(composeL!(FirstLens, NameLens), &|x: Option<&String>| {
match x {
Some(y) => y.to_uppercase(),
None => panic!("T_T")
}
}, people); // vec![Person { name: "JASON" }, Person { name: "John" }];
```

**Further Reading**

Every function in Rust will indicate the types of their arguments and return values.

```
// add :: i32 -> i32 -> i32
fn add(x: i32) -> impl Fn(i32)-> i32 {
return move |y| x + y;
}
// increment :: i32 -> i32
fn increment(x: i32) -> i32 {
return x + 1;
}
```

If a function accepts another function as an argument it is wrapped in parentheses.

```
// call :: (a -> b) -> a -> b
fn call<A, B>(f: &Fn(A) -> B) -> impl Fn(A) -> B + '_ {
return move |x| f(x);
}
```

The letters `a`

, `b`

, `c`

, `d`

are used to signify that the argument can be of any type.
The following version of map takes `a`

function that transforms `a`

value of some type `a`

into another type `b`

,
an array of values of type `a`

, and returns an array of values of type `b`

.

```
// map :: (a -> b) -> [a] -> [b]
fn map<A, B>(f: &Fn(A) -> B) -> impl Fn(A) -> B + '_ {
return move |x| f(x);
}
```

**Further Reading**

- [Mostly Adequate Guide](https://drboolean.gitbooks.io/mostly-adequate-guide-old/content/ch7.html# tales-from-the-cryptic)
- [What is Hindley-Milner?](https://stackoverflow.com/questions/399312/what-is-hindley-milner/399392# 399392)

A composite type made from putting other types together. Two common classes of algebraic types are [sum](# sum-type) and [product](# product-type).

A Sum type is the combination of two types together into another one. It is called sum because the number of possible values in the result type is the sum of the input types.

Rust has `enum`

that literally represent `sum`

in ADT.

```
enum WeakLogicValues {
True(bool),
False(bool),
HalfTrue(bool),
}
// WeakLogicValues = bool + otherbool + anotherbool
```

A product type combines types together in a way you're probably more familiar with:

```
struct Point {
x: i32,
y: i32,
}
// Point = i32 x i32
```

It's called a product because the total possible values of the data structure is the product of the different values. Many languages have a tuple type which is the simplest formulation of a product type.

See also Set Theory

**Further Reading**

Option is a [sum type](# sum-type) with two cases often called Some and None.

Option is useful for composing functions that might not return a value.

```
let mut cart = HashMap::new();
let mut item = HashMap::new();
item.insert(
"price".to_string(),
12
);
cart.insert(
"item".to_string(),
item,
);
fn get_item(cart: &HashMap<String, HashMap<String, i32>>) -> Option<&HashMap<String, i32>> {
return cart.get("item");
}
fn get_price(item: &HashMap<String, i32>) -> Option<&i32> {
return item.get("price");
}
```

Use [and_then](https://doc.rust-lang.org/std/option/enum.Option.html# method.and_then) or [map](https://doc.rust-lang.org/std/option/enum.Option.html# method.map) to sequence functions that return Options

```
fn get_nested_price(cart: &HashMap<String, HashMap<String, i32>>) -> Option<&i32> {
return get_item(cart).and_then(get_price);
}
let price = get_nested_price(&cart);
match price {
Some(v) => assert_eq!(v, &12),
None => panic!("T_T"),
}
```

`Option`

is also known as `Maybe`

. `Some`

is sometimes called `Just`

. `None`

is sometimes called `Nothing`

.

Please be mindful that the project does not fully cover every single definition with code examples since it does not worth investing too much time in a glossary project. I have put extra effort into adding complementary external references that you can check out for further study.

Did you like this article?

Jason Shin

Node, Python, Rust, Machine Learning, Functional Programming, React, Vue, Kubernetes and Scala.

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