Lowering rules

This section gives the complete lowering rules for Rust traits into program clauses. It is a kind of reference. These rules reference the domain goals defined in an earlier section.


The nonterminal Pi is used to mean some generic parameter, either a named lifetime like 'a or a type parameter like A.

The nonterminal Ai is used to mean some generic argument, which might be a lifetime like 'a or a type like Vec<A>.

When defining the lowering rules, we will give goals and clauses in the notation given in this section. We sometimes insert "macros" like LowerWhereClause! into these definitions; these macros reference other sections within this chapter.

Rule names and cross-references

Each of these lowering rules is given a name, documented with a comment like so:

// Rule Foo-Bar-Baz

The reference implementation of these rules is to be found in chalk/chalk-solve/src/clauses.rs. They are also ported in rustc in the rustc_traits crate.

Lowering where clauses

When used in a goal position, where clauses can be mapped directly to the Holds variant of domain goals, as follows:

  • A0: Foo<A1..An> maps to Implemented(A0: Foo<A1..An>)
  • T: 'r maps to Outlives(T, 'r)
  • 'a: 'b maps to Outlives('a, 'b)
  • A0: Foo<A1..An, Item = T> is a bit special and expands to two distinct goals, namely Implemented(A0: Foo<A1..An>) and AliasEq(<A0 as Foo<A1..An>>::Item = T)

In the rules below, we will use WC to indicate where clauses that appear in Rust syntax; we will then use the same WC to indicate where those where clauses appear as goals in the program clauses that we are producing. In that case, the mapping above is used to convert from the Rust syntax into goals.

Transforming the lowered where clauses

In addition, in the rules below, we sometimes do some transformations on the lowered where clauses, as defined here:

  • FromEnv(WC) – this indicates that:
    • Implemented(TraitRef) becomes FromEnv(TraitRef)
    • other where-clauses are left intact
  • WellFormed(WC) – this indicates that:
    • Implemented(TraitRef) becomes WellFormed(TraitRef)
    • other where-clauses are left intact

TODO: I suspect that we want to alter the outlives relations too, but Chalk isn't modeling those right now.

Lowering traits

Given a trait definition

trait Trait<P1..Pn> // P0 == Self
where WC
    // trait items

we will produce a number of declarations. This section is focused on the program clauses for the trait header (i.e., the stuff outside the {}); the section on trait items covers the stuff inside the {}.

Trait header

From the trait itself we mostly make "meta" rules that setup the relationships between different kinds of domain goals. The first such rule from the trait header creates the mapping between the FromEnv and Implemented predicates:

// Rule Implemented-From-Env
forall<Self, P1..Pn> {
  Implemented(Self: Trait<P1..Pn>) :- FromEnv(Self: Trait<P1..Pn>)

Implied bounds

The next few clauses have to do with implied bounds (see also RFC 2089 and the implied bounds chapter for a more in depth cover). For each trait, we produce two clauses:

// Rule Implied-Bound-From-Trait
// For each where clause WC:
forall<Self, P1..Pn> {
  FromEnv(WC) :- FromEnv(Self: Trait<P1..Pn>)

This clause says that if we are assuming that the trait holds, then we can also assume that its where-clauses hold. It's perhaps useful to see an example:

trait Eq: PartialEq { ... }

In this case, the PartialEq supertrait is equivalent to a where Self: PartialEq where clause, in our simplified model. The program clause above therefore states that if we can prove FromEnv(T: Eq) – e.g., if we are in some function with T: Eq in its where clauses – then we also know that FromEnv(T: PartialEq). Thus the set of things that follow from the environment are not only the direct where clauses but also things that follow from them.

The next rule is related; it defines what it means for a trait reference to be well-formed:

// Rule WellFormed-TraitRef
forall<Self, P1..Pn> {
  WellFormed(Self: Trait<P1..Pn>) :- Implemented(Self: Trait<P1..Pn>) && WellFormed(WC)

This WellFormed rule states that T: Trait is well-formed if (a) T: Trait is implemented and (b) all the where-clauses declared on Trait are well-formed (and hence they are implemented). Remember that the WellFormed predicate is coinductive; in this case, it is serving as a kind of "carrier" that allows us to enumerate all the where clauses that are transitively implied by T: Trait.

An example:

trait Foo: A + Bar { }
trait Bar: B + Foo { }
trait A { }
trait B { }

Here, the transitive set of implications for T: Foo are T: A, T: Bar, and T: B. And indeed if we were to try to prove WellFormed(T: Foo), we would have to prove each one of those:

  • WellFormed(T: Foo)
    • Implemented(T: Foo)
    • WellFormed(T: A)
      • Implemented(T: A)
    • WellFormed(T: Bar)
      • Implemented(T: Bar)
      • WellFormed(T: B)
        • Implemented(T: Bar)
      • WellFormed(T: Foo) -- cycle, true coinductively

This WellFormed predicate is only used when proving that impls are well-formed – basically, for each impl of some trait ref TraitRef, we must show that WellFormed(TraitRef). This in turn justifies the implied bounds rules that allow us to extend the set of FromEnv items.

Lowering type definitions

We also want to have some rules which define when a type is well-formed. For example, given this type:

struct Set<K> where K: Hash { ... }

then Set<i32> is well-formed because i32 implements Hash, but Set<NotHash> would not be well-formed. Basically, a type is well-formed if its parameters verify the where clauses written on the type definition.

Hence, for every type definition:

struct Type<P1..Pn> where WC { ... }

we produce the following rule:

// Rule WellFormed-Type
forall<P1..Pn> {
  WellFormed(Type<P1..Pn>) :- WellFormed(WC)

Note that we use struct to define a type, but this should be understood as a general type definition (it could be e.g. a generic enum).

Conversely, we define rules which say that if we assume that a type is well-formed, we can also assume that its where clauses hold. That is, we produce the following family of rules:

// Rule Implied-Bound-From-Type
// For each where clause `WC`
forall<P1..Pn> {
  FromEnv(WC) :- FromEnv(Type<P1..Pn>)

As for the implied bounds RFC, functions will assume that their arguments are well-formed. For example, suppose we have the following bit of code:

trait Hash: Eq { }
struct Set<K: Hash> { ... }

fn foo<K>(collection: Set<K>, x: K, y: K) {
    // `x` and `y` can be equalized even if we did not explicitly write
    // `where K: Eq`
    if x == y {

In the foo function, we assume that Set<K> is well-formed, i.e. we have FromEnv(Set<K>) in our environment. Because of the previous rule, we get FromEnv(K: Hash) without needing an explicit where clause. And because of the Hash trait definition, there also exists a rule which says:

forall<K> {
  FromEnv(K: Eq) :- FromEnv(K: Hash)

which means that we finally get FromEnv(K: Eq) and then can compare x and y without needing an explicit where clause.

Lowering trait items

Associated type declarations

Given a trait that declares a (possibly generic) associated type:

trait Trait<P1..Pn> // P0 == Self
where WC
    type AssocType<Pn+1..Pm>: Bounds where WC1;

We will produce a number of program clauses. The first two define the rules by which AliasEq for associated type projections can succeed; these two clauses are discussed in detail in the section on associated types, but reproduced here for reference:

// Rule AliasEq-Normalize
// AliasEq can succeed by normalizing:
forall<Self, P1..Pn, Pn+1..Pm, U> {
  AliasEq(<Self as Trait<P1..Pn>>::AssocType<Pn+1..Pm> = U) :-
      Normalize(<Self as Trait<P1..Pn>>::AssocType<Pn+1..Pm> -> U)
// Rule AliasEq-Placeholder
// AliasEq can succeed through the placeholder associated type,
// see "associated type" chapter for more:
forall<Self, P1..Pn, Pn+1..Pm> {
    <Self as Trait<P1..Pn>>::AssocType<Pn+1..Pm> =
    (Trait::AssocType)<Self, P1..Pn, Pn+1..Pm>

The next rule covers implied bounds for the projection. In particular, the Bounds declared on the associated type must have been proven to hold to show that the impl is well-formed, and hence we can rely on them elsewhere.

// Rule Implied-Bound-From-AssocTy
// For each `Bound` in `Bounds`:
forall<Self, P1..Pn, Pn+1..Pm> {
    FromEnv(<Self as Trait<P1..Pn>>::AssocType<Pn+1..Pm>>: Bound) :-
      FromEnv(Self: Trait<P1..Pn>) && WC1

Next, we define the requirements for an instantiation of our associated type to be well-formed...

// Rule WellFormed-AssocTy
forall<Self, P1..Pn, Pn+1..Pm> {
    WellFormed((Trait::AssocType)<Self, P1..Pn, Pn+1..Pm>) :-
      WellFormed(Self: Trait<P1..Pn>) && WellFormed(WC1)

...along with the reverse implications, when we can assume that it is well-formed.

// Rule Implied-WC-From-AssocTy
// For each where clause WC1:
forall<Self, P1..Pn, Pn+1..Pm> {
    FromEnv(WC1) :- FromEnv((Trait::AssocType)<Self, P1..Pn, Pn+1..Pm>)
// Rule Implied-Trait-From-AssocTy
forall<Self, P1..Pn, Pn+1..Pm> {
    FromEnv(Self: Trait<P1..Pn>) :-
      FromEnv((Trait::AssocType)<Self, P1..Pn, Pn+1..Pm>)

Lowering function and constant declarations

Chalk didn't model functions and constants, but I would eventually like to treat them exactly like normalization. See the section on function/constant values below for more details.

Lowering impls

Given an impl of a trait:

impl<P0..Pn> Trait<A1..An> for A0
where WC
    // zero or more impl items

Let TraitRef be the trait reference A0: Trait<A1..An>. Then we will create the following rules:

// Rule Implemented-From-Impl
forall<P0..Pn> {
  Implemented(TraitRef) :- WC

In addition, we will lower all of the impl items.

Lowering impl items

Associated type values

Given an impl that contains:

impl<P0..Pn> Trait<P1..Pn> for P0
where WC_impl
    type AssocType<Pn+1..Pm> = T;

and our where clause WC1 on the trait associated type from above, we produce the following rule:

// Rule Normalize-From-Impl
forall<P0..Pm> {
  forall<Pn+1..Pm> {
    Normalize(<P0 as Trait<P1..Pn>>::AssocType<Pn+1..Pm> -> T) :-
      Implemented(P0 as Trait) && WC1

Note that WC_impl and WC1 both encode where-clauses that the impl can rely on. (WC_impl is not used here, because it is implied by Implemented(P0 as Trait).)

Function and constant values

Chalk didn't model functions and constants, but I would eventually like to treat them exactly like normalization. This presumably involves adding a new kind of parameter (constant), and then having a NormalizeValue domain goal. This is to be written because the details are a bit up in the air.