Implied Bounds

Implied bounds remove the need to repeat where clauses written on a type declaration or a trait declaration. For example, say we have the following type declaration:

struct HashSet<K: Hash> {
    ...
}

then everywhere we use HashSet<K> as an "input" type, that is appearing in the receiver type of an impl or in the arguments of a function, we don't want to have to repeat the where K: Hash bound, as in:

// I don't want to have to repeat `where K: Hash` here.
impl<K> HashSet<K> {
    ...
}

// Same here.
fn loud_insert<K>(set: &mut HashSet<K>, item: K) {
    println!("inserting!");
    set.insert(item);
}

Note that in the loud_insert example, HashSet<K> is not the type of the set argument of loud_insert, it only appears in the argument type &mut HashSet<K>: we care about every type appearing in the function's header (the header is the signature without the return type), not only types of the function's arguments.

The rationale for applying implied bounds to input types is that, for example, in order to call the loud_insert function above, the programmer must have produced the type HashSet<K> already, hence the compiler already verified that HashSet<K> was well-formed, i.e. that K effectively implemented Hash, as in the following example:

fn main() {
    // I am producing a value of type `HashSet<i32>`.
    // If `i32` was not `Hash`, the compiler would report an error here.
    let set: HashSet<i32> = HashSet::new();
    loud_insert(&mut set, 5);
}

Hence, we don't want to repeat where clauses for input types because that would sort of duplicate the work of the programmer, having to verify that their types are well-formed both when calling the function and when using them in the arguments of their function. The same reasoning applies when using an impl.

Similarly, given the following trait declaration:

trait Copy where Self: Clone { // desugared version of `Copy: Clone`
    ...
}

then everywhere we bound over SomeType: Copy, we would like to be able to use the fact that SomeType: Clone without having to write it explicitly, as in:

fn loud_clone<T: Clone>(x: T) {
    println!("cloning!");
    x.clone();
}

fn fun_with_copy<T: Copy>(x: T) {
    println!("will clone a `Copy` type soon...");

    // I'm using `loud_clone<T: Clone>` with `T: Copy`, I know this
    // implies `T: Clone` so I don't want to have to write it explicitly.
    loud_clone(x);
}

The rationale for implied bounds for traits is that if a type implements Copy, that is, if there exists an impl Copy for that type, there ought to exist an impl Clone for that type, otherwise the compiler would have reported an error in the first place. So again, if we were forced to repeat the additional where SomeType: Clone everywhere whereas we already know that SomeType: Copy hold, we would kind of duplicate the verification work.

Implied bounds are not yet completely enforced in rustc, at the moment it only works for outlive requirements, super trait bounds, and bounds on associated types. The full RFC can be found here. We'll give here a brief view of how implied bounds work and why we chose to implement it that way. The complete set of lowering rules can be found in the corresponding chapter.

Implied bounds and lowering rules

Now we need to express implied bounds in terms of logical rules. We will start with exposing a naive way to do it. Suppose that we have the following traits:

trait Foo {
    ...
}

trait Bar where Self: Foo { } {
    ...
}

So we would like to say that if a type implements Bar, then necessarily it must also implement Foo. We might think that a clause like this would work:

forall<Type> {
    Implemented(Type: Foo) :- Implemented(Type: Bar).
}

Now suppose that we just write this impl:

struct X;

impl Bar for X { }

Clearly this should not be allowed: indeed, we wrote a Bar impl for X, but the Bar trait requires that we also implement Foo for X, which we never did. In terms of what the compiler does, this would look like this:

struct X;

impl Bar for X {
    // We are in a `Bar` impl for the type `X`.
    // There is a `where Self: Foo` bound on the `Bar` trait declaration.
    // Hence I need to prove that `X` also implements `Foo` for that impl
    // to be legal.
}

So the compiler would try to prove Implemented(X: Foo). Of course it will not find any impl Foo for X since we did not write any. However, it will see our implied bound clause:

forall<Type> {
    Implemented(Type: Foo) :- Implemented(Type: Bar).
}

so that it may be able to prove Implemented(X: Foo) if Implemented(X: Bar) holds. And it turns out that Implemented(X: Bar) does hold since we wrote a Bar impl for X! Hence the compiler will accept the Bar impl while it should not.

Implied bounds coming from the environment

So the naive approach does not work. What we need to do is to somehow decouple implied bounds from impls. Suppose we know that a type SomeType<...> implements Bar and we want to deduce that SomeType<...> must also implement Foo.

There are two possibilities: first, we have enough information about SomeType<...> to see that there exists a Bar impl in the program which covers SomeType<...>, for example a plain impl<...> Bar for SomeType<...>. Then if the compiler has done its job correctly, there must exist a Foo impl which covers SomeType<...>, e.g. another plain impl<...> Foo for SomeType<...>. In that case then, we can just use this impl and we do not need implied bounds at all.

Second possibility: we do not know enough about SomeType<...> in order to find a Bar impl which covers it, for example if SomeType<...> is just a type parameter in a function:

fn foo<T: Bar>() {
    // We'd like to deduce `Implemented(T: Foo)`.
}

That is, the information that T implements Bar here comes from the environment. The environment is the set of things that we assume to be true when we type check some Rust declaration. In that case, what we assume is that T: Bar. Then at that point, we might authorize ourselves to have some kind of "local" implied bound reasoning which would say Implemented(T: Foo) :- Implemented(T: Bar). This reasoning would only be done within our foo function in order to avoid the earlier problem where we had a global clause.

We can apply this local reasoning everywhere we can have an environment -- i.e. when we can write where clauses -- that is, inside impls, trait declarations, and type declarations.

Computing implied bounds with FromEnv

The previous subsection showed that it was only useful to compute implied bounds for facts coming from the environment. We talked about "local" rules, but there are multiple possible strategies to indeed implement the locality of implied bounds.

In rustc, the current strategy is to elaborate bounds: that is, each time we have a fact in the environment, we recursively derive all the other things that are implied by this fact until we reach a fixed point. For example, if we have the following declarations:

trait A { }
trait B where Self: A { }
trait C where Self: B { }

fn foo<T: C>() {
    ...
}

then inside the foo function, we start with an environment containing only Implemented(T: C). Then because of implied bounds for the C trait, we elaborate Implemented(T: B) and add it to our environment. Because of implied bounds for the B trait, we elaborate Implemented(T: A)and add it to our environment as well. We cannot elaborate anything else, so we conclude that our final environment consists of Implemented(T: A + B + C).

In the new-style trait system, we like to encode as many things as possible with logical rules. So rather than "elaborating", we have a set of global program clauses defined like so:

forall<T> { Implemented(T: A) :- FromEnv(T: A). }

forall<T> { Implemented(T: B) :- FromEnv(T: B). }
forall<T> { FromEnv(T: A) :- FromEnv(T: B). }

forall<T> { Implemented(T: C) :- FromEnv(T: C). }
forall<T> { FromEnv(T: B) :- FromEnv(T: C). }

So these clauses are defined globally (that is, they are available from everywhere in the program) but they cannot be used because the hypothesis is always of the form FromEnv(...) which is a bit special. Indeed, as indicated by the name, FromEnv(...) facts can only come from the environment. How it works is that in the foo function, instead of having an environment containing Implemented(T: C), we replace this environment with FromEnv(T: C). From here and thanks to the above clauses, we see that we are able to reach any of Implemented(T: A), Implemented(T: B) or Implemented(T: C), which is what we wanted.

Implied bounds and well-formedness checking

Implied bounds are tightly related with well-formedness checking. Well-formedness checking is the process of checking that the impls the programmer wrote are legal, what we referred to earlier as "the compiler doing its job correctly".

We already saw examples of illegal and legal impls:

trait Foo { }
trait Bar where Self: Foo { }

struct X;
struct Y;

impl Bar for X {
    // This impl is not legal: the `Bar` trait requires that we also
    // implement `Foo`, and we didn't.
}

impl Foo for Y {
    // This impl is legal: there is nothing to check as there are no where
    // clauses on the `Foo` trait.
}

impl Bar for Y {
    // This impl is legal: we have a `Foo` impl for `Y`.
}

We must define what "legal" and "illegal" mean. For this, we introduce another predicate: WellFormed(Type: Trait). We say that the trait reference Type: Trait is well-formed if Type meets the bounds written on the Trait declaration. For each impl we write, assuming that the where clauses declared on the impl hold, the compiler tries to prove that the corresponding trait reference is well-formed. The impl is legal if the compiler manages to do so.

Coming to the definition of WellFormed(Type: Trait), it would be tempting to define it as:

trait Trait where WC1, WC2, ..., WCn {
    ...
}
forall<Type> {
    WellFormed(Type: Trait) :- WC1 && WC2 && .. && WCn.
}

and indeed this was basically what was done in rustc until it was noticed that this mixed badly with implied bounds. The key thing is that implied bounds allows someone to derive all bounds implied by a fact in the environment, and this transitively as we've seen with the A + B + C traits example. However, the WellFormed predicate as defined above only checks that the direct superbounds hold. That is, if we come back to our A + B + C example:

trait A { }
// No where clauses, always well-formed.
// forall<Type> { WellFormed(Type: A). }

trait B where Self: A { }
// We only check the direct superbound `Self: A`.
// forall<Type> { WellFormed(Type: B) :- Implemented(Type: A). }

trait C where Self: B { }
// We only check the direct superbound `Self: B`. We do not check
// the `Self: A` implied bound  coming from the `Self: B` superbound.
// forall<Type> { WellFormed(Type: C) :- Implemented(Type: B). }

There is an asymmetry between the recursive power of implied bounds and the shallow checking of WellFormed. It turns out that this asymmetry can be exploited. Indeed, suppose that we define the following traits:

trait Partial where Self: Copy { }
// WellFormed(Self: Partial) :- Implemented(Self: Copy).

trait Complete where Self: Partial { }
// WellFormed(Self: Complete) :- Implemented(Self: Partial).

impl<T> Partial for T where T: Complete { }

impl<T> Complete for T { }

For the Partial impl, what the compiler must prove is:

forall<T> {
    if (T: Complete) { // assume that the where clauses hold
        WellFormed(T: Partial) // show that the trait reference is well-formed
    }
}

Proving WellFormed(T: Partial) amounts to proving Implemented(T: Copy). However, we have Implemented(T: Complete) in our environment: thanks to implied bounds, we can deduce Implemented(T: Partial). Using implied bounds one level deeper, we can deduce Implemented(T: Copy). Finally, the Partial impl is legal.

For the Complete impl, what the compiler must prove is:

forall<T> {
    WellFormed(T: Complete) // show that the trait reference is well-formed
}

Proving WellFormed(T: Complete) amounts to proving Implemented(T: Partial). We see that the impl Partial for T applies if we can prove Implemented(T: Complete), and it turns out we can prove this fact since our impl<T> Complete for T is a blanket impl without any where clauses.

So both impls are legal and the compiler accepts the program. Moreover, thanks to the Complete blanket impl, all types implement Complete. So we could now use this impl like so:

fn eat<T>(x: T) { }

fn copy_everything<T: Complete>(x: T) {
    eat(x);
    eat(x);
}

fn main() {
    let not_copiable = vec![1, 2, 3, 4];
    copy_everything(not_copiable);
}

In this program, we use the fact that Vec<i32> implements Complete, as any other type. Hence we can call copy_everything with an argument of type Vec<i32>. Inside the copy_everything function, we have the Implemented(T: Complete) bound in our environment. Thanks to implied bounds, we can deduce Implemented(T: Partial). Using implied bounds again, we deduce Implemented(T: Copy) and we can indeed call the eat function which moves the argument twice since its argument is Copy. Problem: the T type was in fact Vec<i32> which is not copy at all, hence we will double-free the underlying vec storage so we have a memory unsoundness in safe Rust.

Of course, disregarding the asymmetry between WellFormed and implied bounds, this bug was possible only because we had some kind of self-referencing impls. But self-referencing impls are very useful in practice and are not the real culprits in this affair.

Co-inductiveness of WellFormed

So the solution is to fix this asymmetry between WellFormed and implied bounds. For that, we need for the WellFormed predicate to not only require that the direct superbounds hold, but also all the bounds transitively implied by the superbounds. What we can do is to have the following rules for the WellFormed predicate:

trait A { }
// WellFormed(Self: A) :- Implemented(Self: A).

trait B where Self: A { }
// WellFormed(Self: B) :- Implemented(Self: B) && WellFormed(Self: A).

trait C where Self: B { }
// WellFormed(Self: C) :- Implemented(Self: C) && WellFormed(Self: B).

Notice that we are now also requiring Implemented(Self: Trait) for WellFormed(Self: Trait) to be true: this is to simplify the process of traversing all the implied bounds transitively. This does not change anything when checking whether impls are legal, because since we assume that the where clauses hold inside the impl, we know that the corresponding trait reference does hold. Thanks to this setup, you can see that we indeed require to prove the set of all bounds transitively implied by the where clauses.

However there is still a catch. Suppose that we have the following trait definition:

trait Foo where <Self as Foo>::Item: Foo {
    type Item;
}

so this definition is a bit more involved than the ones we've seen already because it defines an associated item. However, the well-formedness rule would not be more complicated:

WellFormed(Self: Foo) :-
    Implemented(Self: Foo) &&
    WellFormed(<Self as Foo>::Item: Foo).

Now we would like to write the following impl:

impl Foo for i32 {
    type Item = i32;
}

The Foo trait definition and the impl Foo for i32 are perfectly valid Rust: we're kind of recursively using our Foo impl in order to show that the associated value indeed implements Foo, but that's ok. But if we translate this to our well-formedness setting, the compiler proof process inside the Foo impl is the following: it starts with proving that the well-formedness goal WellFormed(i32: Foo) is true. In order to do that, it must prove the following goals: Implemented(i32: Foo) and WellFormed(<i32 as Foo>::Item: Foo). Implemented(i32: Foo) holds because there is our impl and there are no where clauses on it so it's always true. However, because of the associated type value we used, WellFormed(<i32 as Foo>::Item: Foo) simplifies to just WellFormed(i32: Foo). So in order to prove its original goal WellFormed(i32: Foo), the compiler needs to prove WellFormed(i32: Foo): this clearly is a cycle and cycles are usually rejected by the trait solver, unless... if the WellFormed predicate was made to be co-inductive.

A co-inductive predicate, as discussed in the chapter on goals and clauses, are predicates for which the trait solver accepts cycles. In our setting, this would be a valid thing to do: indeed, the WellFormed predicate just serves as a way of enumerating all the implied bounds. Hence, it's like a fixed point algorithm: it tries to grow the set of implied bounds until there is nothing more to add. Here, a cycle in the chain of WellFormed predicates just means that there is no more bounds to add in that direction, so we can just accept this cycle and focus on other directions. It's easy to prove that under these co-inductive semantics, we are effectively visiting all the transitive implied bounds, and only these.

Implied bounds on types

We mainly talked about implied bounds for traits, but implied bounds on types are very similar. Suppose we have the following definition:

struct Type<...> where WC1, ..., WCn {
    ...
}

To prove that Type<...> is well-formed, we would need to prove a goal of the form WellFormed(Type<...>).. The WellFormed(Type<...>) predicate is defined by the rule:

forall<...> {
    WellFormed(Type<...>) :- WellFormed(WC1), ..., WellFormed(WCn).
}

Conversely, if we know a type is well-formed from our environment (for example because it appears as an argument of one of our functions), we can have implied bounds thanks to the below set of rules:

forall<...> {
    FromEnv(WC1) :- FromEnv(Type<...>).
    ...
    FromEnv(WCn) :- FromEnv(Type<...>).
}

Looking at the above rules, we see that we can never encounter a chain of deductions of the form WellFormed(Type<...>) :- ... :- WellFormed(Type<...>). So in contrast with traits, the WellFormed(Type<...>) predicate does not need to be co-inductive.