Elaborating where clauses on traits, structs about chalk

[nikomatsakis]

We now support a basic form of elaboration, but I am not happy with it. In particular, we have some hard-coded rules that take in the environment and "elaborate" it at various points, much like the compiler does. My ideal would be to find a solution that encodes these relationships "into the logic" as program-clauses. For example, you might imagine:

pub trait Bar: Foo { }

would imply a rule like

(T: Bar) :- (T: Foo).

However, that doesn't quite work. In particular, I also want a program like this to be illegal:

struct X;
impl Bar for X { }
// impl Foo for X { } <-- this impl is missing!

Now the question is, why would this be illegal? If the WF requirements of the impl are that X: Foo, then we can prove that by showing that X: Bar (because the impl exists). That's sort of circular reasoning, clearly. So you could imagine that the impl has an implied where-clause, i.e., it's sort of equivalent to:

impl Bar for X
    where X: Foo
{ }

The only way to prove that would be to have an actual impl of Foo for X so that things terminate. This seems fine, except that I would like to extend elaboration to all where-clauses on a trait, and we currently support circular where-clauses on traits:

trait Foo<T: ?Sized> where T: Bar<Self> { }
trait Bar<T: ?Sized> where T: Foo<Self> { }
impl Foo<i32> for i32 { }
impl Bar<i32> for i32 { }
fn main() { }

Under the approach I proposed, the impls would generate rules like this:

(i32: Foo<i32>) :- (i32: Bar<i32>).
(i32: Bar<i32>) :- (i32: Foo<i32>).

This would clearly never terminate.

I'm not sure yet how properly to think about this.

[withoutboats]

Note the first code block should be pub trait Bar: Foo { }

[withoutboats]

Can we assume all other impls are well formed when checking clauses generated by each impl?

[nikomatsakis]

@withoutboats

Can we assume all other impls are well formed when checking clauses generated by each impl?

I'm not sure what you mean by 'clauses generated by each impl'...

[withoutboats]

@nikomatsakis

trait Foo<T: ?Sized> where T: Foo<Self> { }

struct Bar;
struct Baz;

// impl A
impl Foo<Baz> for Bar { }

// impl B
impl Foo<Bar> for Baz { }

To prove the wellformedness of impl A, we need to prove Baz: Foo<Bar>; currently all we have available is a rule that Baz: Foo<Bar> :- Bar: Foo<Baz>. What if instead clauses carried a context and we had a rule available outside of impl B that Baz: Foo<Bar>, because we assume that impls are well formed when not checking those specific impls?

[withoutboats]

Okay learning a bit more about the internals I think what I mean is:

right now we are talking about translating impl A into (Bar: Foo<Baz>) :- (Baz: Foo<Bar>); I think instead we should translate it into:

• Bar: Foo<Baz>.
• WF(Bar: Foo<Baz>) :- Baz: Foo<Bar>.

That is, we add a fact the impl implies without checking the impl, but also check the impl through a distinct WF check. That avoids cycles.

[nikomatsakis]

Right, so I remember @aturon and I talking about something like this at one point. I thought there was a catch, but perhaps it does work out if we do it just right.

I think the key idea is to distinguish (as you suggest) T: Foo from WF(T: Foo), and then to say that an impl establishes T: Foo but not WF(T: Foo). We then say that:

• an impl can assume that its input types are well-formed, but not its entire trait-reference;
  • to be well-formed, the impl must be able to prove that the trait-reference is well-formed, given its where-clauses and the implied bounds from its input-types.
• every where-clause the user writes like T: Foo is effectively expanded into T: Foo and WF(T: Foo)
• every trait defines rules for what it means for a trait-reference to be well-formed (like you suggested, but we need a few more rules I think)

So let's play out this example a bit more. First, the trait definitions define various rules around well-formedness:

trait Bar { fn bar(&self); }
// WF(?Self: Bar) :-
//     WF(?Self). // input type must be well-formed

trait Foo: Bar { fn foo(&self); }
// WF(?Self: Foo) :-
//     ?Self: Bar,
//     WF(?Self: Bar),
//     WF(?Self).
// ?Self: Bar :-
//     WF(?Self: Foo).
// WF(?Self: Bar) :-
//     WF(?Self: Foo).

You see that I added inverse rules that say "if you know that WF(T: Foo), then you know that T: Bar". I think these rules are necessary, as we'll see later on.

Next, when you have an impl of Foo you get a rule defining T: Foo:

First, a where-clause like T: Foo actually translates into two requirements: WF(T: Foo) and T: Foo.

We create a set of WF(T: Foo) rules based on the trait declaration for Foo:

trait Bar { fn bar(&self) { } }
// WF(?Self: Bar).

trait Foo: Bar { fn foo(&self) { } }
// WF(?Self: Foo) :-
//     ?Self: Bar,
//     WF(?Self: Bar).

When you make an impl of Foo like so, you only get rules for establishing facts T: Foo:

impl Foo for u32 { .. } // impl A
// u32: Foo :-
//     WF(u32). // impl requires its input types are well-formed, along with any explicit where-clauses.

We also have rules that check whether the impl is well-formed. This amounts to showing the WF requirements are met:

// ImplAWellFormed :-
//     WF(u32) => // we get to assume that Self type is well-formed...
//         WF(u32: Foo). // ...we must prove that the trait-ref is well-formed.

Finally, let's imagine some generic code that has a T: Foo bound:

fn foo<T: Foo>(t: T) { ... }

this T: Foo winds up being "lowered" to T: Foo and WF(T: Foo). This is needed because we want the body of foo() to be able to assume that T: Bar. The same applies in other contexts.

Perhaps this all works out nicely, then. Presumably we also have to ensure that we have enough guarantees to ensure that all types that appear (or are inferred) in the fn body are well-formed; from this it should follow that the resulting trait-references only refer to well-formed types.

[withoutboats]

Here's something which today is an unsupported cyclic reference:

trait Relation {
    type Related;
}

trait ToOne: Relation<Related = Option<Self::One>> {
    type One;
}

trait ToMany: Relation<Related = Vec<Self::Many>> {
    type Many;
}

I believe we could support it with the same cycle breaking technique of distinguishing between a T: Trait and WF(T: Trait). Do you think so?

[nikomatsakis]

@withoutboats

Here's something which today is an unsupported cyclic reference...I believe we could support it with the same cycle breaking technique of distinguishing between a T: Trait and WF(T: Trait). Do you think so?

This isn't really unsupported today. What actually happens is that we error out trying to resolve Self::One (and, in particular, to figure out what trait One comes from). If you write out the traits explicitly, it works:

trait Relation {
    type Related;
}

trait ToOne: Relation<Related = Option<<Self as ToOne>::One>> {
    type One;
}

trait ToMany: Relation<Related = Vec<<Self as ToMany>::Many>> {
    type Many;
}

fn main() { }

Chalk kind of sidesteps this problem by not supporting Self::One notation. =)

In any case, I think should work out fine in the setup. I guess the question is whether this setup might help rustc by separating out e.g. the fact that Self: ToOne from the construction of the facts around WF(Self: ToOne). Perhaps so.

[withoutboats]

This strategy should resolve recursive where clauses, I think (see this thread).

Something like this:

trait Foo { }
struct Bar;

impl Foo for Bar where Bar: Foo { }

Currently does not compile. But we should lower it to:

Foo: Bar.
WF(Foo: Bar) :- (Foo: Bar).

[nikomatsakis]

@scalexm this issue can be closed now, right?

[scalexm]

Well I don't know, currently we still have rules of the form WF(T: Trait) :- WF(T) (i.e. we still require that the input types are well formed), so maybe tweak this + add implied bounds -- I think I have a local branch implementing all this.

[nikomatsakis]

@scalexm I see. OK, want to open a PR with that branch?

[nikomatsakis]

OK, after some discussion with @arielb1 and @scalexm on gitter, it became clear that there is still another problem. This has to do with cycles that arise when trying to deal with associated types and bounds. Consider this setup:

struct Unit { }
trait Foo { type Assoc: Foo; }
impl Foo for Unit {  type Assoc = Unit; }

The actual Chalk syntax for that is as follows:

struct Unit { }
trait Foo where <Self as Foo>::Assoc: Foo { type Assoc; }
impl Foo for Unit {  type Assoc = Unit; }

which gives rise to lowered clauses in this gist, including notably this one:

WellFormed(?0 as Foo) :-
    WellFormed(?0),
    WellFormed(<?0 as Foo>::Assoc as Foo),
    <?0 as Foo>::Assoc: Foo.

So now imagine that we try to prove that the impl of Unit is valid. We have to prove WellFormed(Unit as Foo), which means proving WellFormed(<Unit as Foo>::Assoc as Foo). That leads us to this clause:

WellFormed(<?0 as Foo>::Assoc as Foo) :- WellFormed(?0 as Foo).

But, applying that here, means that we then have to prove WellFormed(Unit as Foo), which is precisely the thing we started out trying to prove. This is a problem.

It's not yet clear to me what step of this chain is wrong. =)

[scalexm]

Here is my last take at that. So we need to cleanly separate impls and WF requirements. The following impl block:

impl<...> Foo<...> for SomeType<...> where W1, ..., Wk {
    type Assoc = SomeOtherType<...>;
}

lowers to the following set of rules:

(SomeType<...>: Foo<...>) :- W1, ..., Wk
SomeType<...> as Foo>::Assoc ==> SomeOtherType<...> :- SomeType<...>: Foo<...>

So these are the same rules as before the WF PR.

Now say that the Foo trait has the following declaration:

trait Foo<...> where C1, ..., Cn {
     type Assoc;
}

Then the WF requirements for the previous impl block we have seen can be informally translated as follows:

FooImplWF :-
     assume that every bound implied by W1, ..., Wk is true,
    prove that every bound implied by C1, ..., Cn is true,
    (prove that SomeOtherType<...> is WF)

So the main problem is to be able to compute the set of bounds implied by another set of bounds. Actually, there are two sub-problems.

1. We must be able to compute the set of ALL bounds implied by C1, ..., Cn, since we want to prove them all.
2. When we want to prove such a bound, either we can prove it by using the existing impls, either we can prove it because it is implied by W1, ..., Wk. So actually what we need is knowing whether a given bound belongs to the set of bounds implied by W1, ..., Wk.

The first problem (computing all the implied bounds) can be formally described as a least fixed point computation problem. Hence, we introduce the following predicate, which has a natural co-inductive meaning:

WFco(Self: Trait<...>) :- (Self: Trait<...>), WFco(C1), ..., WFco(Cn)

Of course we introduce such a predicate for each trait. Note that WFco(Self: Trait1) is a different predicate from WFco(Self: Trait2) which means that only cycles of the form WFco(Self: Trait1) :- ... :- WFco(Self: Trait1) are accepted due to co-inductive semantics.
Why does it have a natural co-inductive meaning? Because if we reach a cycle, this means that we have reached a fixed point when computing the set of implied bounds, which means that we have enumerated all the implied bounds.

The second problem is just walking up a graph: we start from what we want to prove, and we check if it is implied by something. If it is the case, then we try to know wether this new something is something we know. And so on. So we add an inductive predicate WF(Self: Trait<...>) associated to the following rules:

Self: Trait<...> :- WF(Self: Trait<...>)
WF(C1) :- WF(Self: Trait<...>)
...
WF(Cn) :- WF(Self: Trait<...>)

Again, we have such a predicate and rules for each trait.

Also, note that within the logic program that we just obtained, we never have cycles which mixes inductive and co-inductive predicates.

Now we can reformulate the WF requirements of the previous Foo impl:

FooImplWF :-
    if (WF(W1), ..., WF(Wk)) {
        WFco(SomeType<...>: Trait<...>)
    }

Some examples

trait Foo where Self: Bar {}
// WFco(Self: Foo) :- (Self: Foo), WFco(Self: Bar)
// (Self: Foo) :- WF(Self: Foo)
// WF(Self: Bar) :- WF(Self: Foo)

trait Bar where Self: Foo {}
// WFco(Self: Bar) :- (Self: Bar), WFco(Self: Foo)
// (Self: Bar) :- WF(Self: Bar)
// WF(Self: Foo) :- WF(Self: Bar)

impl Foo for i32 {}
// i32: Foo.
// i32FooImplWF :- WFco(i32: Foo)

impl Bar for i32 {}
// i32: Bar.
// i32BarImplWF :- WFco(i32: Bar)

The two WF requirements hold.

trait Clone {}
// WFco(Self: Clone) :- Self: Clone
// (Self: Clone) :- WF(Self: Clone)

trait Foo where Self: Bar, Self: Clone {}
// WFco(Self: Foo) :- (Self: Foo), WFco(Self: Bar), WFco(Self: Clone)
// (Self: Foo) :- WF(Self: Foo)
// WF(Self: Bar) :- WF(Self: Foo)
// WF(Self: Clone) :- WF(Self: Foo)

trait Bar where Self: Foo {}
// WFco(Self: Bar) :- (Self: Bar), WFco(Self: Foo)
// (Self: Bar) :- WF(Self: Bar)
// WF(Self: Foo) :- WF(Self: Bar)

impl<T> Foo for T {}
// T: Foo.
// TFooImplWF :- WFco(T: Foo)

impl<T> Bar for T where T: Foo {}
// T: Bar :- T: Foo.
// TBarImplWF :- if (WF(T: Foo)) { WFco(T: Bar) }

The Bar impl meets its WF requirements, but not the Foo impl. This can be fixed by adding a where T: Clone bound on the Foo impl.

trait Foo where <Self as Foo>::Item: Foo {
    type Item;
}
// WFco(Self: Foo) :- (Self: Foo), WFco(<Self as Foo>::Item: Foo)
// (Self: Foo) :- WF(Self: Foo)
// WF(<Self as Foo>::Item: Foo) :- WF(Self: Foo)

impl Foo for i32 {
    type Item = i32;
}
// i32: Foo.
// <i32 as Foo>::Item ==> i32 :- (i32: Foo)
// i32FooImplWF :- WFco(i32: Foo)

Since in this setting, we have <i32 as Foo>::Item ==> i32 then WFco(<i32 as Foo>::Item: Foo) unifies with WFco(i32: Foo) and we reach a co-inductive cycle when checking the WF requirements of the Foo impl.

How to use it

Inside each function:

fn f<T: Foo>(...) { ... }

just replace the T: Foo bound by a WF(T: Foo) bound.

Main problem of this approach

So we saw that this approach correctly handles cyclic impls. However, the problem is that the set of all implied bounds can be infinite, as in:

trait Foo where Box<Self>: Foo { }
// WFco(Self: Foo) :- (Self: Foo), WFco(Box<Self>: Foo)

In that case, the WFco(Self: Foo) predicate will run indefinitely since it will enumerate all the implied bounds, never reaching a co-inductive cycle. Of course, the only way to actually use this trait would be to have this impl:

impl<T> Foo for T {}

or at least an impl of the form:

impl<...> Foo for Family<...> {}

where Family<...> is an infinite family of types such that T belongs to Family<...> implies that Box<T> also belongs to Family<...>. Hence, the where Box<Self>: Foo is not needed actually. But this still seems like an actual limitation.

I think we are close to having something usable, because I wrote a very straightforward implementation in chalk, and I also sketched a formal proof that this setting is sound (where I gave a formal definition of everything used in this setting, in particular a formal definition of “implied bounds” in terms of least fixed point and a formal definition of “soundness”). I hope we could tweak this setting in order to leverage the previous limitation.

cc @nikomatsakis @arielb1

[arielb1]

Inside each function:

fn f<T: Foo>(...) { ... }

just replace the T: Foo bound by a WF(T: Foo) bound.

So the caller (actually, the expander of the type-scheme) has to prove that WFco(T: Foo)? Which they can prove by either enumerating all implied bounds, or by using an i32FooImplWF-style theorem?

[scalexm]

No, the caller just has to prove that T: Foo holds, because indeed i32FooImplWF has already been proven.

This is actually related to how I formally defined soundness: let A be a set of trait bounds and T a given type. If |- A and Implied(A) |- T: Trait, then |- T: Trait.

In other words, say you have a trait trait Foo: Copy and that String: Foo holds (i.e. an impl Foo for String has been written). And you call f::<String>(). Then f will assume that WF(T: Foo) holds where T = String (although f does not know what T is), hence deduce that T: Copy i.e. String: Copy. If you have the soundness property, then you know that |- String: Copy must hold, i.e. an impl Copy for String has been written. This means that you cannot trick the type-checker to derive trait bounds for a given type for which an actual impl does not exist. That seems like a good definition of soundness.

[arielb1]

If |- A and Implied(A) |- T: Trait, then |- T: Trait.

This feels like it ignores the parametric context. While this should be enough for translation, rustc also wants the version with arguments:

"implication-elimination" theorem (here |- is recursive impl search):

Γ, WF(Γ) |- A
Γ, WF(Γ), Implied(A) |- T: Trait
------------
Γ, WF(Γ) |- T: Trait

[scalexm]

@arielb1 Yes I agree with you, and actually I did think about that when I sketched my proof. The thing is that we cannot ask for your definition in an intuitionistic model. Indeed, say we have the following traits and impls:

trait Foo: Copy {}
impl<T> Foo for T where T: Foo

Then if T is a given type, let Γ = { T: Foo } we have:

Γ |- T: Foo
Γ, Implied(T: Foo) |- T: Copy

But we do not have: Γ |- T: Copy. The point is that, if we had a function:

fn f<T: Foo>()

Then we would not be able to call f in a non-parametric context.

And, in general, if you are inside a parametric function, you can go up the call stack, and at one point you will be in a non-parametric context. So I think what we need is:

|- Γ1
Γ1, Implied(Γ1) |- Γ2
...
Γ(N-1), Implied(Γ(N-1)) |- ΓN
- - - - - - - - - -
|- ΓN

Which is a property that is true under my setting (if my soundness theorem is true under this setting), AS SOON AS Implied(Γ(K)) is finite for all K.

[arielb1]

@scalexm

That's why I added the WF(Γ) as a parameter to the parametric definition. We have WF(Γ) |- T: Copy, so order is restored.

Also, because impl-trees must be finite (and this should be the only case where we assume that impl trees are finite), we can recurse on them using the ImplWf rules to get this:

for every impl in the universe, ImplWf
Γ, WF(Γ) |- T: Trait
--------
Γ, WF(Γ) |- WF(T: Trait)

So after wf-checking, you can use this meta-theorem to avoid checking well-formedness in a "simple" semantics, and impl trees are restricted to be a finite tree of these forms (note: you can't get a use of WF here except at the root).

--------
T: Trait |- T: Trait

T2: Trait2 predicate of T: Trait
--------
WF(T: Trait) |- T2: Trait2

T2: Trait2 predicate of T: Trait
--------
WF(T: Trait) |- WF(T2: Trait2)

impl Trait for T where C1...Cn
--------
C1...Cn |- T: Trait

[scalexm]

Ah sorry, I responsed to the un-edited message. Is WF(Γ) the same thing as Implied(Γ)? If so, I agree.

[nikomatsakis]

OK, so, @scalexm and I were talking about this. I am going to write up the idea as I understand it -- we ultimately adopted somewhat different terminology that I found helpful -- and then walk through an illustrative example.

First, the terminology:

• Implemented(T: Trait) -- the type T is known to implement the trait Trait
  • this could be from an impl or by hypothesis
• WellFormed(T: Trait) -- the type T implements Trait and all of the where clauses on Trait are satisfied
  • @scalexm called this WFco originally
  • this is a coinductive "carrier predicate" that will ultimately expand to the full set of things that must hold for T: Trait to be implemented
• FromEnv(T: Trait) -- the environment tells us that T: Trait holds
  • @scalexm called this WF originally
  • this is an inductive predicate that tells us that T: Trait holds by hypothesis, either directly or indirectly via implied bounds

Then for a trait definition:

trait Trait: C1 + ..  + Cn { .. }

we lower to the following clauses:

WellFormed(T: Trait) :- Implemented(T: Trait) + WellFormed(C1) + .. + WellFormed(Cn).
Implemented(T: Trait) :- FromEnv(T: Trait).
FromEnv(T: C1) :- FromEnv(T: Trait).
...
FromEnv(T: Cn) :- FromEnv(T: Trait).

Then an impl like:

impl<..> Trait for .. where W1..Wk

lowers to an Implemented clause:

Implemented(T: Trait) :- Implemented(W1), ..., Implemented(Wk).

and further must prove the following goal to be well-formed:

forall<...> { // impl's type parameters
    if (FromEnv(W1) .. FromEnv(Wk)) {
        WellFormed(T: Trait)
    }
}

---

Let's walk through this example that @scalexm came up with and is derived from rust-lang/rust#43784:

trait A where Self: B, Self: Copy {}
trait B where Self: A { }
trait Copy { }

impl<T> A for T where T: B {}
impl<T> B for T {}

I believe that using the rules above, we get the following clauses:

// trait A where Self: B, Self: Copy {}
WellFormed(T: A) :- Implemented(T: A), WellFormed(T: B), WellFormed(T: Copy).
Implemented(T: A) :- FromEnv(T: A).
FromEnv(T: B) :- FromEnv(T: A).
FromEnv(T: Copy) :- FromEnv(T: A).

// trait B where Self: A { }
WellFormed(T: B) :- Implemented(T: B), WellFormed(T: A).
Implemented(T: B) :- FromEnv(T: B).
FromEnv(T: A) :- FromEnv(T: B).

// trait Copy { }
WellFormed(T: Copy) :- Implemented(T: Copy).
Implemented(T: Copy) :- FromEnv(T: Copy).

// impl<T> A for T where T: B {}
Implemented(T: A) :- Implemented(T: B).

// impl<T> B for T {}
Implemented(T: B).

And we have the following proof obligations:

// impl<T> A for T where T: B {}
forall<T> {
    if (FromEnv(T: B)) {
        WellFormed(T: A).
    }
}

This one we can prove -- WellFormed(T: A) requires ultimately proving three things:
- Implemented(T: A) holds from the impl clause (and then hypothesis)
- Implemented(T: B) holds by hypothesis (FromEnv(T: B)) or from the impl clause.
- Implemented(T: Copy) holds transitively by hypothesis:
- FromEnv(T: B) => FromEnv(T: A) => FromEnv(T: Copy)

However the impl for B we cannot prove:

// impl<T> B for T {}
forall<T> {
    WellFormed(T: B).
}

It too must prove the same set of three things:
- Implemented(T: A) holds from the two impl clauses.
- Implemented(T: B) holds from the impl clause.
- However, Implemented(T: Copy) does not have a corresponding impl clause. And there is no FromEnv in scope so we can't use the implied bounds.

---

Of course the user could have written

impl<T> B for T where T: A { }

In that case, both impls would be well-formed, but neither is usable, because of infinite recursion.

[nikomatsakis]

@scalexm

In that case, the WFco(Self: Foo) predicate will run indefinitely since it will enumerate all the implied bounds, never reaching a co-inductive cycle. Of course, the only way to actually use this trait would be to have this impl:

Regarding this limitation, there is some work on recognizing such cycles and creating a richer notion of inductive hypothesis (as you probably recall). But I'm having trouble coming up with what could be a valid impl exactly or scenario where this limitation is problematic -- do you have any ideas?

[scalexm]

cc @nikomatsakis, @withoutboats

Plan for GATs

OK so we still need to figure out how to translate bounds on GATs, especially when it comes to WF requirements and implied bounds. Here's what I've come up with.

So let's write a fully general GAT example:

trait Foo {
    // `'a` and `T` may actually designate multiple parameters
    type Item<'a, T>: Bounds where WC;
}

impl Foo for Sometype {
    type Item<'a, T> = Value<'a, T>;
    // We don't repeat `WC`, and we cannot add more specific where clauses,
    // exactly like associated methods.

    // This may look like a limitation however, e.g.
    // you cannot implement `type Item<T> = HashSet<T>` unless there was a
    // `T: Hash` bound initially in the trait declaration, but it seems like a
    // necessary one.
}

and let us step directly onto the general rules, we'll see a simpler example later.

Projecting

We have two rules for projecting:

// Fallback rule
forall<Self, 'a, T> {
    ProjectionEq(<Self as Foo>::Item<'a, T> = (Foo::Item<'a, T>)<Self>).
}

forall<Self, 'a, T, U> {
    ProjectionEq(<Self as Foo>::Item<'a, T> = U) :-
        Normalize(<Self as Foo>::Item<'a, T> -> U).
}

Normalizing

The rule for normalizing is given by:

forall<'a, T> {
    Normalize(<SomeType as Foo>::Item<'a, T> -> Value<'a, T>) :- Implemented(Self: Foo), WC
}

WF requirements

The above impl typechecks only if the following goal can be proven to be true:

FooImplWF :-
    /* ... normal WF requirements if we add bounds on the impl or on the traits etc ... */
    forall<'a, T> {
        if (FromEnv(WC)) {
            WellFormed(Value<'a, T>), WellFormed(Value<'a, T>: Bounds)
        }
    }

Implied bounds

First of all, the where clause where T: Foo<Item = U> in the AST should now lower to two domain goals: Implemented(T: Item) and ProjectionEq(<T as Foo>::Item = U). We would also remove the WellFormed(ProjectionEq) domain goal.

Then, a new implied bounds rule would deal with (possibly higher-ranked) bounds on the associated type:

// New reverse rule
forall<Self, 'a, T, U> {
    FromEnv(<T as Foo>::Item<'a, T>: Bounds) :- FromEnv(Self: Foo)
}

Well-formedness of projection types

Finally, we add a rule about well-formedness of the skolemized projection type.

forall<Self, 'a, T> {
    WellFormed((Foo::Item<'a,T>)<Self>) :- WC, Implemented(Self: Foo).
    FromEnv(WC) :- FromEnv((Foo::Item<'a,T>)<Self>).
    FromEnv(Self: Foo) :- FromEnv((Foo::Item<'a,T>)<Self>).
}

Concrete example

Let's dig into a more concrete example and see how this all mixes up with the other ways of expressing bounds on associated types (i.e. in the trait header like where Self::Item: blabla.

trait Foo {
    type Item<T: Clone>: Clone where Self: Sized;
//            ^^^^^^^^^ this could be rewritten as `where T: Clone`
}

impl Foo for () {
    type Item<T> = Vec<T>;
}

Let's see the WF requirements for this impl:

FooImplWF :-
    forall<T> {
        if (FromEnv(T: Clone), FromEnv(Vec<T>: Sized)) {
//                                     ^^^^^^^^^^^^^ trivial bound
            WellFormed(Vec<T>: Clone), WellFormed(Vec<T>)
        }
    }

which are indeed satisfiable.

Example of usage of implied bounds:

fn foo<X: Foo>(arg: <X as Foo>::Item<T>) {
//                     ^^^^^^^^ no need for `T: Clone` because we assume
//                              `WellFormed(<X as Foo>::Item<T>)`
    let cloned = arg.clone(); // we can rely on the fact that
                              // `<X as Foo>::Item<T>: Clone` thanks to
                              // the reverse rule
}

The last question is how is this consistent with bounds expressed in the trait header. As long as there are no additional where clauses on the associated type, these two writings are perfectly equivalent when it comes to the rules we have seen before:

trait Foo {
     type Item<T>: Debug;
}

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

Now, in the first setting where we have where clauses on the associated type:

trait Foo {
    type Item<T>: Clone where T: Clone;
}

there is no other way to express the bound without adding support for where clauses in HRTBs. I don't really think this is a problem actually. Something you can do however is:

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

but this is probably not what you want since when writing an impl, you won't be able to use the fact that T: Clone in order to prove that <Self as Foo>::Item<T>: Clone (you will need to be able to prove it for all T without any other conditions).

[nikomatsakis]

After some live discussion with @scalexm, it seems like we ought not to need the X: Sized and T: Clone bounds here, because knowing that the argument type is WF ought to let us infer them:

fn foo<X: Foo + Sized, T: Clone>(arg: <X as Foo>::Item<T>) {
//                     ^^^^^^^^ `T: Clone` needed for projecting out
    let cloned = arg.clone(); // we can rely on the fact that
                              // `<X as Foo>::Item<T>: Clone` thanks to
                              // the reverse rule
}

but we're not 100% sure how to set that up =)

UPDATE: Idea we think works is to:

1. Add FromEnv((Foo::Item)<X, T>) into the environment used to check the body of foo
   • note use of skolemized form here
2. Add FromEnv(WC) :- FromEnv(Foo::Item)<X, T>)
   • note use of skolemized form on RHS

The idea is that the caller must have been able to unify with skolemized form, so we can use it, which sidesteps infinite recursion that would otherwise occur.

[tmandry]

Looking at the current implementation in Chalk, I believe the Implemented(Self as Foo) condition ought to be moved from the WellFormed rule to the fallback rule, as follows:

// Fallback rule
forall<Self, 'a, T> {
    ProjectionEq(<Self as Foo>::Item<'a, T> = (Foo::Item<'a, T>)<Self>) :-
        Implemented(Self as Foo).
}

forall<Self, 'a, T> {
    WellFormed((Foo::Item<'a,T>)<Self>) :- WC.
}

Since <T as Foo>::Item will unify with (Foo::Item)<T> only if T: Foo, our WF clause is correct.

Is there any reason to do it one way or another?

[scalexm]

Actually this has been longly discussed :)
If you put it on the fallback clause instead of the WF clause, this allows for a slight inconsistency within implied bounds. That is, let’s say Implemented(Self: Foo) is on the fallback rule, and consider this function:

trait Foo {
    type Item<T>: Clone where T: Clone;
}

fn foo<X, T>(arg: <X as Foo>::Item<T>) {
    // let cloned = arg.clone();
}

The foo function assumes all its inputs are WF. Especially, it assumes that <X as Foo>::Item<T> is WF (note that this is the projection type and not the skolemized type, which does not really actually exist in rust code). As this point, this function is ok.

But if you uncomment the line, then you have to add a X: Foo bound on foo. Indeed, in order to use the fact that <X as Foo>::Item<T>: Clone, you would like to use the fact that X: Foo applies, which is in turn implied by WF((Foo::Item<T>)<Self>). Seems fine, but in fact the type in the WF predicate is the skolemized type this time, and in your environment you only have WF(<X as Foo>::Item<T>) (assumption that your inputs are WF). If you want this predicate to unify with the former, you need to use the fallback rule, which means you need to prove X: Foo. This is inconsistent and surprising.

This is related with niko’s comment above, when at first we used to have WC on the fallback rule and not on the WF rule, and we were dealing with another inconsistency. We were wondering whether we could juste replace every non-skolemized WF predicates in the environment by their skolemized form; it seemt to work but felt a bit awkward. Hence, we moved to this other design.

[tmandry]

Okay, interesting, thanks for the explanation.

As I understand these rules, they're basically saying that <X as Foo>::Item<T> and (Foo::Item<'a, T>)<Self> are equivalent representations of the same thing, and so should always be considered equal.

This at least intuitively makes sense, and gives the solver an option to prove facts about our type via normalization or unification on the skolemized type. (I'm not yet 100% understanding of where such equivalences can and cannot be applied, though.)

[scalexm]

@nikomatsakis there is a bug in the implied bounds rule for GATs as described above, I think it should be:

FromEnv(<T as Foo>::Item<‘a, T>: Bounds) :- FromEnv(Self: Foo), FromEnv(WC)

also @LukasKalbertodt raised interesting ergonomic questions and I don’t feel completely at ease with my answers, we should discuss this some more.
———-
ÉDIT: I am now at ease with the answers I provided to them :)

[arielb1]

Some comments from discord that someone might want to document permanently:

Expand

arielbyToday at 10:23 PM
In the Chalk WellFormed/FromEnv proposal
how do you get a FromEnv(ProjectionTy)?
do you have the "WFProjection" rule of old
(i.e. FromEnv(::C) := FromEnv(A: B))
ah that's ill-typed, because the LHS is a type and the RHS is a trait ref
but I suppose that the rule would be FromEnv(A: B) := all components ofA: Bare WF, Implemented(A: B)
or is it some sort of weird meta rule?
eddybToday at 10:33 PM
@nikomatsakis ^^
arielbyToday at 10:36 PM
or do functions need to prove only WFco on their callees, where the conversion from WFco to FromEnv occurs implicitly?
from how the rules look, if all the impls in the world are WF, then WFco(T: Tr) should impl FromEnv(T: Tr)
*should imply
nikomatsakisToday at 10:49 PM
I forget the details, @scalexm if they are around might remember; I think the rustc-guide contents are up to date though
arielbyToday at 10:54 PM
which rustc-guide
https://rust-lang-nursery.github.io/rustc-guide/traits/wf.html
Well-formedness checking - Guide to Rustc Development
A guide to developing rustc
"This chapter is mostly to be written."
scalexmToday at 10:56 PM
@arielby rules regarding projections are detailed here: #12 (comment)
GitHub
Elaborating where clauses on traits, structs · Issue #12 · rust-...
Related to #11, we don't really do anything clever with where clauses on traits/structs. We need to support "elaboration" -- ideally, a richer form than what the current compiler supp...
Note that what I once named WFco is now just WF (still co-recursive though)
arielbyToday at 10:58 PM
@scalexm don't you need WellFormed(Self) to prove WellFormed(::Item)?
scalexmToday at 10:58 PM
No
Self is a type parameter
And we assume all of them are WF
arielbyToday at 10:59 PM
who assumes that?
so WellFormed is only used when proving that associated types are WellFormed?
scalexmToday at 10:59 PM
Well basically there is no such thing as WF(T) where T is a type parameter
arielbyToday at 10:59 PM
*and other implied bounds
scalexmToday at 11:01 PM
WF(SomeType<P1,...Pn>) just means that the bounds declared onstruct SomeType<T1,...,Tn> hold for this instantiation of the type parameters
arielbyToday at 11:01 PM
so it also requires WF(P1), WF(P2), ... WF(Pn)?
scalexmToday at 11:02 PM
Actually not really
Because when type checking inside a function body we retrieve all types appearing and check that they are WF
arielbyToday at 11:02 PM
clearly when you have an associated type, it needs to be completely WF
having type Assoc=(SomeObviouslyNonWfType,); would be 100% bad
scalexmToday at 11:03 PM
So if SomeType<P1,...,Pn> is somewhere in your function, so are P1,...,Pn so we end up checking them as well
centrilToday at 11:03 PM
(how nice it would be to have a spec with abstract syntax, typing rules, and operational semantics in one place...)
scalexmToday at 11:03 PM
The WF-ness of your assoc type is checked in the impl that associates it
You can look at the various tests in chalk, in src/rules/wf/test.rs
It showcases exactly what typechecks and what does not, with implied bounds
arielbyToday at 11:04 PM
I mean, if you have impl Trait for () { type Assoc = Something; }, then Something must be hereditary WF
scalexmToday at 11:05 PM
Yes
arielbyToday at 11:05 PM
I mean, if it is Something, then T0 should also be WF
scalexmToday at 11:05 PM
But we avoided the expansion WF(Vec) :- WF(Something) in the rules because it leased to infinite branches in the proof tree
Rather, when typechecking, we aggregate all types appearing and we prove their WF-ness
So we will end up retrieving Something and proving its WF-ness
centrilToday at 11:06 PM
@scalexm just to double check, by WF we are referring to the same thing as when you write Δ ⊢ τ type e.g. τ type is WF(τ), yes?
arielbyToday at 11:06 PM
where?
I don't think you can do it when checking users of the impll
because you can have one fn prove (): Trait
and another fn project it as ::Assoc
or does proving (): Trait require exploring all the associated types that come from that?
scalexmToday at 11:08 PM
in chalk this is done in src/rules/wf/mod.rs or something
if (): Trait this means that at some point someone wrote an impl
and this impl is typechecked when typechecking the whole program
arielbyToday at 11:08 PM
so the impl needs to prove that the types in it are hereditary WF?
scalexmToday at 11:10 PM
@arielby yes, but this is not coded in the logical rules, but rather in the typechecking algorithm
arielbyToday at 11:10 PM
I mean it is a part of the rules
regardless of where it is coded
scalexmToday at 11:10 PM
yes sort of right
arielbyToday at 11:11 PM
so WF is only used in Chalk in checking associated types?
scalexmToday at 11:11 PM
No
it's also used for checking normal types appearing in e.g. function bodies
and in impls, and in trait defs etc
arielbyToday at 11:11 PM
but your WF predicate only applies for trait refs
the WF(trait predicate) function
scalexmToday at 11:12 PM
ah there lies the confusion maybe
we have two different WF predicates, with the same name
one for trait refs, the other for types
arielbyToday at 11:12 PM
maybe call the WF-for-trait-ref WFco?
so WFco is only used for proving projections?
and it does not require WF-for-types-inside-the-projection?
that feels a bit unsound with specialization
scalexmToday at 11:14 PM
mmh if by proving projections you mean proving WF(::Assoc) then I still consider ::Assoc as a "normal" type so it should use the WF predicate for types, although it is indeed specialized for associated types
arielbyToday at 11:14 PM
scalexm: I mean, where does chalk require that WFco(trait-ref)?
scalexmToday at 11:15 PM
when typechecking impls
arielbyToday at 11:15 PM
and only there?
scalexmToday at 11:15 PM
trait Foo where Self: Clone { }
impl Foo for () {
// inside here: prove that WFco((): Foo) hold, that is (): Clone
}
yes
only there
arielbyToday at 11:16 PM
and it requires WF(type) in a lot of places
scalexmToday at 11:16 PM
no you can never have WFco(traitref) :- WF(type)
because WFco comes from trait definitions which are fully parametric
arielbyToday at 11:18 PM
what do you mean by that?
scalexmToday at 11:19 PM
what I mean is that when you define
trait Foo where Self: Clone { }
the rule that Chalk builds from that trait def is:
WFCo(Self: Foo) :- Implemented(Self: Clone)
it does not require WF-type(Self)
arielbyToday at 11:19 PM
because otherwise things would explode?
scalexmToday at 11:19 PM
yes
we started with this
but indeed infinite branches were ruining everything
so we just side-stepped the problem to the typechecking phase
arielbyToday at 11:20 PM
what sort of infinite branches?
scalexmToday at 11:20 PM
e.g. if you have WFtype(Vec) :- WFtype(T)
mmh
no bad example
arielbyToday at 11:21 PM
the one thing I remember is that ProjectionEq can't require any WF because otherwise things really explode
scalexmToday at 11:22 PM
there may be an example somewhere in the chalk issues and/or various dropbox papers
but yeah basically things were exploding as you said
arielbyToday at 11:22 PM
so why is putting it in type-checking different than putting it in WFco?
is WFco blocking anything?
scalexmToday at 11:23 PM
because type-checking occurs "outside" of the proof solver
we can just retrieve all concrete types appearing in the definition that we want to type-check, using e.g. some fixed point algorithm
and then prove WF(TheType) for each type we got
(and proving WF(TheType) then happens in the proof solver of course)
arielbyToday at 11:24 PM
where if it is done inside the proof checker, the proof checker will try weird strategies?
scalexmToday at 11:24 PM
exactly
arielbyToday at 11:25 PM
so it is not a deadlock
but rather a proof checker "weakness"
scalexmToday at 11:25 PM
yes
the approach WF(Vec) :- WF(T) should be right but this was indeed a technical limitation with the solver
arielbyToday at 11:25 PM
deadlock = like what happened when you had ProjectionEq depend on FromEnv, you can't FromEnv because you can't project, and you can't project because you can't FromEnv
@scalexm - so that's an argument against having Chalk be a spec
you have implementation limitations showing up as logic rules :_)
scalexmToday at 11:27 PM
@arielby well now I remember it was indeed a limitation when we used to have no distinction between WF and FromEnv and that we did not use the co-recursive setup
but I think after making this distinction, we discussed that it might now work to have the hereditary thing directly in the rules
(we = @nikomatsakis + me)
but since we implemented everything in the typechecking and that it was working well we did not think much more about it
it could be worth to re-try though
arielbyToday at 11:29 PM
also, is there a way of proving FromEnv from knowing that you ahve an impl?
or is that another "meta rule only" thing?
scalexmToday at 11:31 PM
meta rule only
FromEnv can only come from the environment and implied bounds
arielbyToday at 11:32 PM
thanks
so to summarize
WF(type) - destruction rule is by the type-checker, introduction rule is inductive & structural except for prover limitation, relied on when generating FromEnv
WFco(trait-ref) - destruction rule is when WF-checking impls, introduction rule is coinductive & structural,
relied on when generating FromEnv
FromEnv(trait-ref) - destruction rule gives you Implemented + FromEnv, introduction rule is either from a FromEnv or a meta rule
Implemented(trait-ref) - destruction rule is both during type-checking and for projecting, introduction rule is from FromEnv or from impls
having to differentiate between "Chalk destruction rules" and "logic destruction rules" is confusing
if the type-checker can use a rule, then it is not really a meta-rule
scalexmToday at 11:40 PM
@arielby sounds like a good summary
arielbyToday at 11:41 PM
WFco needs to be coinductive because of the "circular supertraits" case
and FromEnv needs to be inductive?
or something
scalexmToday at 11:42 PM
yes that's it
arielbyToday at 11:42 PM
because otherwise you have FromEnv(T: Trait) :- FromEnv(T: Trait) giving you FromEnv(T : Trait) oops massive unsoundness
and they need to be distinct from Implemented so that normalization won't be blocked on WF
and probably some other solver trouble
so also, you don't have a rule FromEnv(implied-bound) :- WF(ty)
only a meta-rule?
scalexmToday at 11:43 PM
yes, basically we introduced the distinction because of one of your former (simple) examples
trait Foo {
type Assoc: Foo
}
impl Foo for () { type Assoc = (); }
@arielby there is also a FromEnv(ty)
and we have a rule FromEnv(implied-bound) :- FromEnv(ty)
but there is a "meta rule" which says "the typechecking can assume FromEnv(Type) for each input Type of the function"
arielbyToday at 11:45 PM
that's a nice explanation
now the only thing needed is to write it down somewhere
and to rename WF(trait-ref) to WFco
or some other name
because it has like 0% to do with WF(type)
scalexmToday at 11:46 PM
it should indeed be written in the rustc-guide 😃
but never took the time to do it
hopefully I'll write everything down during October
arielbyToday at 11:46 PM
(actually, I'm not that sure about the 0% thing)
scalexmToday at 11:47 PM
but for now, the only thing written down is chalk's code
arielbyToday at 11:47 PM
@scalexm what will you do in October?
scalexmToday at 11:47 PM
I'll probably be working a bit on chalk/rustc during October -> November
as I have some time off
@Centril sorry missed your message
@Centril I'm not sure this is the same, that is WF is not really related to typing but rather to "do the bounds declared on the type hold"
arielbyToday at 11:50 PM
@scalexm, btw do you handle object types?
in Chalk
how are they called?
centrilToday at 11:50 PM
@scalexm ah; so it's a different sort of well-formedness
scalexmToday at 11:51 PM
@arielby not yet, especially because we don't have a clear plan, remember this compiler bug rust-lang/rust#44454
GitHub
unsoundness relating to WF requirements on trait object types · I...
So @scalexm pointed me at this curious example: trait Animal: 'static { } fn foo<Y, X>() where Y: Animal + ?Sized { // Y implements Animal<X> so Y is 'st...
arielbyToday at 11:51 PM
sure
I just thought you had a solution
"handle object types" = having a Chalk solution for the problem
scalexmToday at 11:51 PM
I must admit I didn't find the time to think too much about it yet :p
arielbyToday at 11:55 PM
k