Subtyping and Variance

Subtyping is a relationship between types that allows statically typed languages to be a bit more flexible and permissive.

Subtyping in Rust is a bit different from subtyping in other languages. This makes it harder to give simple examples, which is a problem since subtyping, and especially variance, are already hard to understand properly. As in, even compiler writers mess it up all the time.

To keep things simple, this section will consider a small extension to the Rust language that adds a new and simpler subtyping relationship. After establishing concepts and issues under this simpler system, we will then relate it back to how subtyping actually occurs in Rust.

So here's our simple extension, Objective Rust, featuring three new types:

# #![allow(unused_variables)]
#fn main() {
trait Animal {
    fn snuggle(&self);
    fn eat(&mut self);

trait Cat: Animal {
    fn meow(&self);

trait Dog: Animal {
    fn bark(&self);

But unlike normal traits, we can use them as concrete and sized types, just like structs.

Now, say we have a very simple function that takes an Animal, like this:

fn love(pet: Animal) {

By default, static types must match exactly for a program to compile. As such, this code won't compile:

let mr_snuggles: Cat = ...;
love(mr_snuggles);         // ERROR: expected Animal, found Cat

Mr. Snuggles is a Cat, and Cats aren't exactly Animals, so we can't love him! 😿

This is annoying because Cats are Animals. They support every operation an Animal supports, so intuitively love shouldn't care if we pass it a Cat. We should be able to just forget the non-animal parts of our Cat, as they aren't necessary to love it.

This is exactly the problem that subtyping is intended to fix. Because Cats are just Animals and more, we say Cat is a subtype of Animal (because Cats are a subset of all the Animals). Equivalently, we say that Animal is a supertype of Cat. With subtypes, we can tweak our overly strict static type system with a simple rule: anywhere a value of type T is expected, we will also accept values that are subtypes of T.

Or more concretely: anywhere an Animal is expected, a Cat or Dog will also work.

As we will see throughout the rest of this section, subtyping is a lot more complicated and subtle than this, but this simple rule is a very good 99% intuition. And unless you write unsafe code, the compiler will automatically handle all the corner cases for you.

But this is the Rustonomicon. We're writing unsafe code, so we need to understand how this stuff really works, and how we can mess it up.

The core problem is that this rule, naively applied, will lead to meowing Dogs. That is, we can convince someone that a Dog is actually a Cat. This completely destroys the fabric of our static type system, making it worse than useless (and leading to Undefined Behaviour).

Here's a simple example of this happening when we apply subtyping in a completely naive "find and replace" way.

fn evil_feeder(pet: &mut Animal) {
    let spike: Dog = ...;

    // `pet` is an Animal, and Dog is a subtype of Animal,
    // so this should be fine, right..?
    *pet = spike;

fn main() {
    let mut mr_snuggles: Cat = ...;
    evil_feeder(&mut mr_snuggles);  // Replaces mr_snuggles with a Dog
    mr_snuggles.meow();             // OH NO, MEOWING DOG!

Clearly, we need a more robust system than "find and replace". That system is variance, which is a set of rules governing how subtyping should compose. Most importantly, variance defines situations where subtyping should be disabled.

But before we get into variance, let's take a quick peek at where subtyping actually occurs in Rust: lifetimes!

NOTE: The typed-ness of lifetimes is a fairly arbitrary construct that some disagree with. However it simplifies our analysis to treat lifetimes and types uniformly.

Lifetimes are just regions of code, and regions can be partially ordered with the contains (outlives) relationship. Subtyping on lifetimes is in terms of that relationship: if 'big: 'small ("big contains small" or "big outlives small"), then 'big is a subtype of 'small. This is a large source of confusion, because it seems backwards to many: the bigger region is a subtype of the smaller region. But it makes sense if you consider our Animal example: Cat is an Animal and more, just as 'big is 'small and more.

Put another way, if someone wants a reference that lives for 'small, usually what they actually mean is that they want a reference that lives for at least 'small. They don't actually care if the lifetimes match exactly. So it should be ok for us to forget that something lives for 'big and only remember that it lives for 'small.

The meowing dog problem for lifetimes will result in us being able to store a short-lived reference in a place that expects a longer-lived one, creating a dangling reference and letting us use-after-free.

It will be useful to note that 'static, the forever lifetime, is a subtype of every lifetime because by definition it outlives everything. We will be using this relationship in later examples to keep them as simple as possible.

With all that said, we still have no idea how to actually use subtyping of lifetimes, because nothing ever has type 'a. Lifetimes only occur as part of some larger type like &'a u32 or IterMut<'a, u32>. To apply lifetime subtyping, we need to know how to compose subtyping. Once again, we need variance.


Variance is where things get a bit complicated.

Variance is a property that type constructors have with respect to their arguments. A type constructor in Rust is any generic type with unbound arguments. For instance Vec is a type constructor that takes a type T and returns Vec<T>. & and &mut are type constructors that take two inputs: a lifetime, and a type to point to.

NOTE: For convenience we will often refer to F<T> as a type constructor just so that we can easily talk about T. Hopefully this is clear in context.

A type constructor F's variance is how the subtyping of its inputs affects the subtyping of its outputs. There are three kinds of variance in Rust. Given two types Sub and Super, where Sub is a subtype of Super:

  • F is covariant if F<Sub> is a subtype of F<Super> (subtyping "passes through")
  • F is contravariant if F<Super> is a subtype of F<Sub> (subtyping is "inverted")
  • F is invariant otherwise (no subtyping relationship exists)

If F has multiple type parameters, we can talk about the individual variances by saying that, for example, F<T, U> is covariant over T and invariant over U.

It is very useful to keep in mind that covariance is, in practical terms, "the" variance. Almost all consideration of variance is in terms of whether something should be covariant or invariant. Actually witnessing contravariance is quite difficult in Rust, though it does in fact exist.

Here is a table of important variances which the rest of this section will be devoted to trying to explain:

'a T U
* &'a T covariant covariant
* &'a mut T covariant invariant
* Box<T> covariant
Vec<T> covariant
* UnsafeCell<T> invariant
Cell<T> invariant
* fn(T) -> U contravariant covariant
*const T covariant
*mut T invariant

The types with *'s are the ones we will be focusing on, as they are in some sense "fundamental". All the others can be understood by analogy to the others:

  • Vec and all other owning pointers and collections follow the same logic as Box
  • Cell and all other interior mutability types follow the same logic as UnsafeCell
  • *const follows the logic of &T
  • *mut follows the logic of &mut T (or UnsafeCell<T>)

NOTE: the only source of contravariance in the language is the arguments to a function, which is why it really doesn't come up much in practice. Invoking contravariance involves higher-order programming with function pointers that take references with specific lifetimes (as opposed to the usual "any lifetime", which gets into higher rank lifetimes, which work independently of subtyping).

Ok, that's enough type theory! Let's try to apply the concept of variance to Rust and look at some examples.

First off, let's revisit the meowing dog example:

fn evil_feeder(pet: &mut Animal) {
    let spike: Dog = ...;

    // `pet` is an Animal, and Dog is a subtype of Animal,
    // so this should be fine, right..?
    *pet = spike;

fn main() {
    let mut mr_snuggles: Cat = ...;
    evil_feeder(&mut mr_snuggles);  // Replaces mr_snuggles with a Dog
    mr_snuggles.meow();             // OH NO, MEOWING DOG!

If we look at our table of variances, we see that &mut T is invariant over T. As it turns out, this completely fixes the issue! With invariance, the fact that Cat is a subtype of Animal doesn't matter; &mut Cat still won't be a subtype of &mut Animal. The static type checker will then correctly stop us from passing a Cat into evil_feeder.

The soundness of subtyping is based on the idea that it's ok to forget unnecessary details. But with references, there's always someone that remembers those details: the value being referenced. That value expects those details to keep being true, and may behave incorrectly if its expectations are violated.

The problem with making &mut T covariant over T is that it gives us the power to modify the original value when we don't remember all of its constraints. And so, we can make someone have a Dog when they're certain they still have a Cat.

With that established, we can easily see why &T being covariant over T is sound: it doesn't let you modify the value, only look at it. Without any way to mutate, there's no way for us to mess with any details. We can also see why UnsafeCell and all the other interior mutability types must be invariant: they make &T work like &mut T!

Now what about the lifetime on references? Why is it ok for both kinds of references to be covariant over their lifetimes? Well, here's a two-pronged argument:

First and foremost, subtyping references based on their lifetimes is the entire point of subtyping in Rust. The only reason we have subtyping is so we can pass long-lived things where short-lived things are expected. So it better work!

Second, and more seriously, lifetimes are only a part of the reference itself. The type of the referent is shared knowledge, which is why adjusting that type in only one place (the reference) can lead to issues. But if you shrink down a reference's lifetime when you hand it to someone, that lifetime information isn't shared in anyway. There are now two independent references with independent lifetimes. There's no way to mess with original reference's lifetime using the other one.

Or rather, the only way to mess with someone's lifetime is to build a meowing dog. But as soon as you try to build a meowing dog, the lifetime should be wrapped up in an invariant type, preventing the lifetime from being shrunk. To understand this better, let's port the meowing dog problem over to real Rust.

In the meowing dog problem we take a subtype (Cat), convert it into a supertype (Animal), and then use that fact to overwrite the subtype with a value that satisfies the constraints of the supertype but not the subtype (Dog).

So with lifetimes, we want to take a long-lived thing, convert it into a short-lived thing, and then use that to write something that doesn't live long enough into the place expecting something long-lived.

Here it is:

fn evil_feeder<T>(input: &mut T, val: T) {
    *input = val;

fn main() {
    let mut mr_snuggles: &'static str = "meow! :3";  // mr. snuggles forever!!
        let spike = String::from("bark! >:V");
        let spike_str: &str = &spike;                // Only lives for the block
        evil_feeder(&mut mr_snuggles, spike_str);    // EVIL!
    println!("{}", mr_snuggles);                     // Use after free?

And what do we get when we run this?

error[E0597]: `spike` does not live long enough
  --> src/
9  |         let spike_str: &str = &spike;
   |                                ^^^^^ borrowed value does not live long enough
10 |         evil_feeder(&mut mr_snuggles, spike_str);
11 |     }
   |     - borrowed value only lives until here
   = note: borrowed value must be valid for the static lifetime...

Good, it doesn't compile! Let's break down what's happening here in detail.

First let's look at the new evil_feeder function:

# #![allow(unused_variables)]
#fn main() {
fn evil_feeder<T>(input: &mut T, val: T) {
    *input = val;

All it does it take a mutable reference and a value and overwrite the referent with it. What's important about this function is that it creates a type equality constraint. It clearly says in its signature the referent and the value must be the exact same type.

Meanwhile, in the caller we pass in &mut &'static str and &'spike_str str.

Because &mut T is invariant over T, the compiler concludes it can't apply any subtyping to the first argument, and so T must be exactly &'static str.

The other argument is only an &'a str, which is covariant over 'a. So the compiler adopts a constraint: &'spike_str str must be a subtype of &'static str (inclusive), which in turn implies 'spike_str must be a subtype of 'static (inclusive). Which is to say, 'spike_str must contain 'static. But only one thing contains 'static -- 'static itself!

This is why we get an error when we try to assign &spike to spike_str. The compiler has worked backwards to conclude spike_str must live forever, and &spike simply can't live that long.

So even though references are covariant over their lifetimes, they "inherit" invariance whenever they're put into a context that could do something bad with that. In this case, we inherited invariance as soon as we put our reference inside an &mut T.

As it turns out, the argument for why it's ok for Box (and Vec, Hashmap, etc.) to be covariant is pretty similar to the argument for why it's ok for lifetimes to be covariant: as soon as you try to stuff them in something like a mutable reference, they inherit invariance and you're prevented from doing anything bad.

However Box makes it easier to focus on by-value aspect of references that we partially glossed over.

Unlike a lot of languages which allow values to be freely aliased at all times, Rust has a very strict rule: if you're allowed to mutate or move a value, you are guaranteed to be the only one with access to it.

Consider the following code:

let mr_snuggles: Box<Cat> = ..;
let spike: Box<Dog> = ..;

let mut pet: Box<Animal>;
pet = mr_snuggles;
pet = spike;

There is no problem at all with the fact that we have forgotten that mr_snuggles was a Cat, or that we overwrote him with a Dog, because as soon as we moved mr_snuggles to a variable that only knew he was an Animal, we destroyed the only thing in the universe that remembered he was a Cat!

In contrast to the argument about immutable references being soundly covariant because they don't let you change anything, owned values can be covariant because they make you change everything. There is no connection between old locations and new locations. Applying by-value subtyping is an irreversible act of knowledge destruction, and without any memory of how things used to be, no one can be tricked into acting on that old information!

Only one thing left to explain: function pointers.

To see why fn(T) -> U should be covariant over U, consider the following signature:

fn get_animal() -> Animal;

This function claims to produce an Animal. As such, it is perfectly valid to provide a function with the following signature instead:

fn get_animal() -> Cat;

After all, Cats are Animals, so always producing a Cat is a perfectly valid way to produce Animals. Or to relate it back to real Rust: if we need a function that is supposed to produce something that lives for 'short, it's perfectly fine for it to produce something that lives for 'long. We don't care, we can just forget that fact.

However, the same logic does not apply to arguments. Consider trying to satisfy:

fn handle_animal(Animal);


fn handle_animal(Cat);

The first function can accept Dogs, but the second function absolutely can't. Covariance doesn't work here. But if we flip it around, it actually does work! If we need a function that can handle Cats, a function that can handle any Animal will surely work fine. Or to relate it back to real Rust: if we need a function that can handle anything that lives for at least 'long, it's perfectly fine for it to be able to handle anything that lives for at least 'short.

And that's why function types, unlike anything else in the language, are contravariant over their arguments.

Now, this is all well and good for the types the standard library provides, but how is variance determined for type that you define? A struct, informally speaking, inherits the variance of its fields. If a struct MyType has a generic argument A that is used in a field a, then MyType's variance over A is exactly a's variance over A.

However if A is used in multiple fields:

  • If all uses of A are covariant, then MyType is covariant over A
  • If all uses of A are contravariant, then MyType is contravariant over A
  • Otherwise, MyType is invariant over A

# #![allow(unused_variables)]
#fn main() {
use std::cell::Cell;

struct MyType<'a, 'b, A: 'a, B: 'b, C, D, E, F, G, H, In, Out, Mixed> {
    a: &'a A,     // covariant over 'a and A
    b: &'b mut B, // covariant over 'b and invariant over B

    c: *const C,  // covariant over C
    d: *mut D,    // invariant over D

    e: E,         // covariant over E
    f: Vec<F>,    // covariant over F
    g: Cell<G>,   // invariant over G

    h1: H,        // would also be variant over H except...
    h2: Cell<H>,  // invariant over H, because invariance wins all conflicts

    i: fn(In) -> Out,       // contravariant over In, covariant over Out

    k1: fn(Mixed) -> usize, // would be contravariant over Mixed except..
    k2: Mixed,              // invariant over Mixed, because invariance wins all conflicts