The project is a library for functional programming in Rust.

• fp-core.rs
  • installation
• functional programming jargon in rust

fp-core.rs

A library for functional programming in Rust.

It contains purely functional data structures to supplement the functional programming needs alongside with the Rust Standard Library.

Installation

Add below line to your Cargo.toml

fp-core = "0.1.9"

If you have Cargo Edit you may simply

$ cargo add fp-core

functional programming jargon in rust

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.

The content of this section was drawn from Functional Programming Jargon in Javascript and we sincerely appreciate them for providing the initial baseline.

Table of Contents

• Arity
• Higher-Order Functions (HOF)
• Closure
• Partial Application
• Currying
• Auto Currying
• Referential Transparency
• Lambda
• Lambda Calculus
• Purity
• Side effects
• Idempotent
• Function Composition
• Continuation
• Point-Free Style
• Predicate
• Contracts
• Category
• Value
• Constant
• Variance
• Higher Kinded Type
• Functor
• Pointed Functor
• Lifting
• Equational Reasoning
• Monoid
• Monad
• Comonad
• Applicative
• Morphism
  • Endomorphism
  • Isomorphism
  • Homomorphism
  • Catamorphism
  • Hylomorphism
  • Anamorphism
• Setoid
• Ord
• Semigroup
• Foldable
• Lens
• Type Signature
• Algebraic data type
  • Sum Type
  • Product Type
• Option
• Functional Programming References
• Function Programming development in Rust Language
• Inspiration

Arity

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

Higher-Order Functions (HOF)

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

let filter = | predicate: fn(&i32) -> bool, xs: Vec<i32> | {
    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]);

Closure

A closure is a scope which retains variables available to a function when it's created. This is important for 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 x 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

• Lambda Vs Closure
• How do JavaScript Closures Work?

Partial Application

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

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

• Partial Application in Haskell

Currying

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 {
    move |y| x + y
}

let add5 = add(5);
add5(10); // 15

Further reading

• Currying in Rust

Auto Currying

Transforming a function that takes multiple arguments into one that if 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

Referential Transparency

An expression that can be replaced with its value without changing the behavior 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.

Lambda

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.

Lambda Calculus

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

This is in contrast to a Turing machine, an equivalent model.

Lambda calculus has three key components: variables, abstraction, and application. A variable is just some symbol, say x. An abstraction is sort of a function: it binds variables into "formulae". Applications are function calls. This is meaningless without examples.

The identity function (|x| x in rust) looks like \ x. x in most literature (\ is a Lambda where Latex or Unicode make it available). It is an abstraction. If 1 were a value we could use, (\ x. x) 1 would be an application (and evaluating it gives you 1).

But there's more...

Computation in Pure Lambda Calculus

Let's invent booleans. \ x y. x can be true and \ x y. y can be false. If so, \ b1 b2. b1(b2,(\x y. y)) is and. Let's evaluate it to show how:

I'll leave or as an exercise. Furthermore, if can now be implemented: \c t e. c(t, e) where c is the condition, t the consequent (then) and e the else clause.

SICP leaves numbers as an exercise. They define 0 as \f . \x. x and adding one as \n. \f. \x. f(n(f)(x)). That isn't even ASCII art, so let's add: 0 + 1:

(\n. \f. \x. f(n(f)(x)))(\f. \x. x) = \f. \x. f((\x'. x')(x)) = \f. \x. f(x)

Basically, the number of fs in the expression is the number. I'll leave figuring out larger numbers as a exercise. With patience, you can show that \f. \x. f(f(x)) is two. This will help with addition: \n m. \f. \x. n(m(f)(x)) should add two numbers. Let's make 4:

(\n m. \f. \x. n(f)(m(f)(x)))(\f. x. f(f(x)), \f. \x. f(f(x)))
  = \f. \x. (\f'. \x'. f'(f'(x')))(f)((\f'. \x'. f'(f'(x')))(f)(x))
  = \f. \x. (\x'. f(f(x')))(f(f(x')))
  = \f. \x. f(f(f(f(x))))

Multiplication is harder and there's better exposition on Wikipedia. Another good reference is on stackoverflow.

Purity

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.

Side effects

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!

Idempotent

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 x = x;
    x.sort();
    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 {
    x.abs()
};

let x: i32 = 10;
let result = abs(abs(x));
assert_eq!(result, x); // passes

Function Composition

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(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

Continuation

At any given point in a program, the part of the code that's yet to be executed is known as a continuation.

let print_as_string = |num: i32| println!("Given {}", num);

let add_one_and_continue = |num: i32, cc: fn(i32)| {
    let result = num + 1;
    cc(result)
};

add_one_and_continue(1, print_as_string); // Given 2

Continuations are often seen in asynchronous programming when the program needs to wait to receive data before it can continue. The response is often passed off to the rest of the program, which is the continuation, once it's been received.

Point-Free style

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.

Predicate

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

Contracts

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 {
    x > &10
};

let add_one = | x: &i32 | -> Result<i32, String> {
    if contract(x) {
        return Ok(x + 1);
    }
    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!")
}

Category

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:

1. 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.
2. 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).
3. 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. In particular, many see this as another foundation of mathematics (so, everything would be a category from this view of math). Various definitions in this guide are related to category theory since this approach applies elegantly to functional programming.

Examples of categories

When one specifies a category, the objects and morphisms are essential. Additionally, showing that the rules are met is nice though usually left to the reader as an exercise.

• Any type and pure functions on the type. (Note, we require purity since side-effects affect associativity (the third rule).)

These examples come up in mathematics:

• 1: the category with 1 object and its identity morphism.
• Monoidal categories: monoids are defined later but any monoid is a category with 1 object and many morphisms from the object to itself. (Yes, there is also a category of monoids -- this is not that -- this example is that any monoid is its own category.)

Further reading

• Category Theory for Programmers
• Awodey's introduction for those who like math.

Value

Anything that can be assigned to a variable.

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

Constant

A variable that cannot be reassigned once defined.

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

Constants are referentially transparent. That is, they can be replaced with the values that they represent without affecting the result.

Variance

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 or UnsafeCell in PhatomData

Further Reading

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

Higher Kinded Type (HKT)

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>.

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 {
    type URI;
    type Target;
}

// Lifted Option
impl HKT for Option {
    type URI = Self;
    type Target = Option;
}

object.map(x => x) ≍ object

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

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

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

impl Functor for Option {
    fn fmap(self, f: F) -> Self::Target
        where
            F: FnOnce(A) -> B
    {
        self.map(f)
    }
}

// This conflicts with the above, so isn't tested
// below and is commented out. TODO: add a careful
// implementation that makes sure Optional and this
// don't conflict.
impl HKT for T
where
    T: Sized + Iterator,
    U: Sized + Iterator,
{
    type URI = Self;
    type Target = U;
}

impl Functor for T
where
    T: Iterator,
{
    fn fmap(self, f: F) -> Self::Target
    where
        F: FnOnce(A) -> B,
        A: Sized,
        B: Sized,
    {
        self.map(f)
    }
}

#[test]
fn test_functor() {
    let z = Option::fmap(Some(1), |x| x + 1).fmap(|x| x + 1); // Return Option
    assert_eq!(z, Some(3)); // passes
}

#[derive(Debug, PartialEq, Eq)]
enum Maybe {
    Nothing,
    Just(T),
}


impl Maybe {
    fn of(x: T) -> Self {
        Maybe::Just(x)
    }
}

let pointed_functor = Maybe::of(1);

assert_eq!(pointed_functor, Maybe::Just(1));

1 + 1
// i32: 2

1 + 0
// i32: 1

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

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

[vec![1, 2], vec![]].concat();
// Vec: vec![1, 2]

fn identity(a: A) -> A {
    a
}

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

use crate::applicative_example::Applicative;

trait Empty {
    fn empty() -> A;
}

trait Monoid: Empty + Applicative
where
    F: FnOnce(A) -> B,
{
}

pub trait Chain: HKT {
    fn chain(self, f: F) -> <Self as HKT>::Target
        where F: FnOnce(A) -> <Self as HKT>::Target;
}

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

pub trait Monad: Chain + Applicative
    where F: FnOnce(A) -> B {}

impl Monad for Option
    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
}

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

trait Extract {
    fn extract(self) -> A;
}

trait Comonad: Extend + Extract {}

impl Extend for Option {
    fn extend(self, f: W) -> Self::Target
    where
        W: FnOnce(Self) -> B,
    {
        self.map(|x| f(Some(x)))
    }
}

impl Extract for Option {
    fn extract(self) -> A {
        self.unwrap() // is there a better way to achieve this?
    }
}

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

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

trait HKT3 {
    type Target2;
}

impl HKT3 for Option {
    type Target2 = Option;
}

// Apply
trait Apply : Functor + HKT3
    where F: FnOnce(A) -> B,
{
    fn ap(self, f: <Self as HKT3>::Target2) -> <Self as HKT>::Target;
}

impl Apply for Option
    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: HKT {
    fn of(self) -> <Self as HKT>::Target;
}

impl Pure for Option {
    fn of(self) -> Self::Target {
        self
    }
}

// Applicative
trait Applicative : Apply + Pure
    where F: FnOnce(A) -> B,
{
} // Simply derives Apply and Pure

impl Applicative for Option
    where F: FnOnce(A) -> B,
{
}

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

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

// decrement :: i32 -> i32
let decrement = |x: i32| x - 1;

#[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

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

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);

let count_down = unfold((8_u32, 1_u32), |state| {
    let (ref mut x1, ref mut x2) = *state;

    if *x1 == 0 {
        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],
);

trait Setoid {
    fn equals(&self, other: &Self) -> bool;
}

impl Setoid for Vec<i32> {
    fn equals(&self, other: &Self) -> bool {
        self.len() == other.len()
    }
}

assert_eq!(vec![1, 2].equals(&vec![1, 2]), true); // passes

use std::ops::Add;

pub trait Semigroup: Add {
}

assert_eq!(
    vec![1, 2].add(&vec![3, 4]),
    vec![1, 2, 3, 4],
); // passes

assert_eq!(
    a.add(&b).add(&c),
    a.add(&b.add(&c)),
); // passes

use fp_core::foldable::*;

let k = vec![1, 2, 3];
let result = k.reduce(0, |i, acc| i + acc);
assert_eq!(result, 6);

use crate::hkt::HKT;
use crate::monoid::Monoid;

pub trait Foldable: HKT {
    fn reduce(b: B, ba: F) -> <Self as HKT>::Target
    where
        F: FnOnce(B, A) -> (B, B);

    fn fold_map(m: M, fa: F) -> M
    where
        M: Monoid,
        F: FnOnce(<Self as HKT>::URI) -> M;

    fn reduce_right(b: B, f: F) -> <Self as HKT>::Target
    where
        F: FnOnce(A, B) -> (B, B);
}

trait Lens {
    fn over(s: &S, f: &Fn(Option<&A>) -> A) -> S {
        let result: A = f(Self::get(s));
        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 LensString> for PersonNameLens {
    fn get(s: &Person) -> Option<&String> {
       Some(&s.name)
    }

    fn set(a: String, s: &Person) -> Person {
        Person {
            name: a,
        }
    }
}

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

struct FirstLens;

impl Lens<Vec, A> for FirstLens {
  fn get(s: &Vec) -> Option<&A> {
     s.first()
  }

  fn set(a: A, s: &Vec) -> Vec {
      unimplemented!() // Nothing to set in FirstLens
  }
}

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" }];

// add :: i32 -> i32 -> i32
fn add(x: i32) -> impl Fn(i32)-> i32 {
    move |y| x + y
}

// increment :: i32 -> i32
fn increment(x: i32) -> i32 {
    x + 1
}

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

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

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

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

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>> {
    cart.get("item")
}

fn get_price(item: &HashMap<String, i32>) -> Option<&i32> {
    item.get("price")
}

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"),
}