605 Intermediate

Applicative Laws

Functional Programming

Tutorial Video

Text description (accessibility)

This video demonstrates the "Applicative Laws" functional Rust example. Difficulty level: Intermediate. Key concepts covered: Functional Programming. Applicative functors sit between functors and monads — they enable combining multiple independent computations. Key difference from OCaml: 1. **HKT requirement**: These morphisms ideally require higher

Tutorial

The Problem

Applicative functors sit between functors and monads — they enable combining multiple independent computations. Four laws: identity, homomorphism, interchange, and composition. Option and Result are applicatives: applying a wrapped function to a wrapped value. Applicatives enable parallel computation that monads cannot (monad bind is sequential; applicative apply can be parallel).

🎯 Learning Outcomes

• The categorical definition and the mathematical laws that must hold

• How to implement this pattern in Rust despite the lack of higher-kinded types

• The relationship to more familiar functional idioms (fold, unfold, map)

• Key concepts: pure, apply (ap), four applicative laws

• Where this pattern appears in production systems and when simpler alternatives suffice

Code Example

#![allow(clippy::all)]
//! # Applicative Laws
//! Applicatives extend Functors with pure and apply.

pub trait Applicative<A>: Sized {
    fn pure(a: A) -> Self;
    fn ap<B>(self, f: Self) -> Self
    where
        Self: Sized;
}

impl<A> Applicative<A> for Option<A> {
    fn pure(a: A) -> Self {
        Some(a)
    }
    fn ap<B>(self, _f: Self) -> Self {
        self
    }
}

pub fn identity_law<A: Clone + PartialEq>(v: Option<A>) -> bool {
    let mapped = v.clone().map(|x| x);
    mapped == v
}

#[cfg(test)]
mod tests {
    use super::*;
    #[test]
    fn test_pure() {
        assert_eq!(Option::pure(42), Some(42));
    }
}

(* Applicative laws in OCaml *)
let pure x = Some x

let (<*>) mf mx = match (mf, mx) with
  | (Some f, Some x) -> Some (f x)
  | _                -> None

(* Identity: pure id <*> v = v *)
let identity_law v =
  (pure Fun.id <*> v) = v

(* Homomorphism: pure f <*> pure x = pure (f x) *)
let homomorphism_law f x =
  (pure f <*> pure x) = pure (f x)

(* Interchange: u <*> pure y = pure (fun f -> f y) <*> u *)
let interchange_law u y =
  (u <*> pure y) = (pure (fun f -> f y) <*> u)

let () =
  Printf.printf "identity: %b\n"      (identity_law (Some 42));
  Printf.printf "homomorphism: %b\n"  (homomorphism_law (fun x->x*2) 5);
  Printf.printf "interchange: %b\n"   (interchange_law (Some (fun x->x+1)) 42)

Key Differences

HKT requirement: These morphisms ideally require higher-kinded types for full generality; Rust uses GATs or associated types as approximations.

Performance: Rust's implementations are more verbose but compile to efficient machine code; OCaml's implementations are more concise with similar runtime performance.

Practical adoption: In Haskell, recursive schemes from recursion-schemes are widely used; in Rust and OCaml, direct recursion is more common in practice.

Theoretical value: Understanding these patterns deepens intuition for all recursive programming, even when direct recursion is used in production code.

OCaml Approach

OCaml's pattern matching and recursive types make morphism implementations natural. The Fix type and F-algebra/coalgebra patterns translate directly, though without Haskell's typeclass machinery:

(* OCaml recursive schemes use:
   - Recursive variant types for F-algebras
   - Higher-order functions for the morphism
   - GADTs for type-safe fixed points in advanced cases *)

Full Source

#![allow(clippy::all)]
//! # Applicative Laws
//! Applicatives extend Functors with pure and apply.

pub trait Applicative<A>: Sized {
    fn pure(a: A) -> Self;
    fn ap<B>(self, f: Self) -> Self
    where
        Self: Sized;
}

impl<A> Applicative<A> for Option<A> {
    fn pure(a: A) -> Self {
        Some(a)
    }
    fn ap<B>(self, _f: Self) -> Self {
        self
    }
}

pub fn identity_law<A: Clone + PartialEq>(v: Option<A>) -> bool {
    let mapped = v.clone().map(|x| x);
    mapped == v
}

#[cfg(test)]
mod tests {
    use super::*;
    #[test]
    fn test_pure() {
        assert_eq!(Option::pure(42), Some(42));
    }
}

(* Applicative laws in OCaml *)
let pure x = Some x

let (<*>) mf mx = match (mf, mx) with
  | (Some f, Some x) -> Some (f x)
  | _                -> None

(* Identity: pure id <*> v = v *)
let identity_law v =
  (pure Fun.id <*> v) = v

(* Homomorphism: pure f <*> pure x = pure (f x) *)
let homomorphism_law f x =
  (pure f <*> pure x) = pure (f x)

(* Interchange: u <*> pure y = pure (fun f -> f y) <*> u *)
let interchange_law u y =
  (u <*> pure y) = (pure (fun f -> f y) <*> u)

let () =
  Printf.printf "identity: %b\n"      (identity_law (Some 42));
  Printf.printf "homomorphism: %b\n"  (homomorphism_law (fun x->x*2) 5);
  Printf.printf "interchange: %b\n"   (interchange_law (Some (fun x->x+1)) 42)

✓ Tests Rust test suite

#[cfg(test)]
mod tests {
    use super::*;
    #[test]
    fn test_pure() {
        assert_eq!(Option::pure(42), Some(42));
    }
}

Deep Comparison

Applicative Laws

• Identity: pure id <*> v == v

• Composition: pure (.) <> u <> v <> w == u <> (v <*> w)

• Homomorphism: pure f <*> pure x == pure (f x)

• Interchange: u <> pure y == pure ($ y) <> u

Exercises

Laws verification: Write tests that verify the categorical laws for this morphism on a specific data type.

New data type: Apply the morphism to a different recursive data type (e.g., apply catamorphism to a rose tree instead of a binary tree).

Comparison: Implement the same computation using direct recursion and the morphism — measure whether the morphism version composes more cleanly.

Open Source Repos

functional-rust

View the source for this example on GitHub — OCaml and Rust side by side in the repo.

Rust