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| <h1 id="goals-and-clauses"><a class="header" href="#goals-and-clauses">Goals and clauses</a></h1> |
| <p>In logic programming terms, a <strong>goal</strong> is something that you must |
| prove and a <strong>clause</strong> is something that you know is true. As |
| described in the <a href="./lowering-to-logic.html">lowering to logic</a> |
| chapter, Rust’s trait solver is based on an extension of hereditary |
| harrop (HH) clauses, which extend traditional Prolog Horn clauses with |
| a few new superpowers.</p> |
| <h2 id="goals-and-clauses-meta-structure"><a class="header" href="#goals-and-clauses-meta-structure">Goals and clauses meta structure</a></h2> |
| <p>In Rust’s solver, <strong>goals</strong> and <strong>clauses</strong> have the following forms |
| (note that the two definitions reference one another):</p> |
| <pre><code class="language-text">Goal = DomainGoal // defined in the section below |
| | Goal && Goal |
| | Goal || Goal |
| | exists<K> { Goal } // existential quantification |
| | forall<K> { Goal } // universal quantification |
| | if (Clause) { Goal } // implication |
| | true // something that's trivially true |
| | ambiguous // something that's never provable |
| |
| Clause = DomainGoal |
| | Clause :- Goal // if can prove Goal, then Clause is true |
| | Clause && Clause |
| | forall<K> { Clause } |
| |
| K = <type> // a "kind" |
| | <lifetime> |
| </code></pre> |
| <p>The proof procedure for these sorts of goals is actually quite |
| straightforward. Essentially, it’s a form of depth-first search. The |
| paper |
| <a href="https://rust-lang.github.io/chalk/book/bibliography.html#pphhf">“A Proof Procedure for the Logic of Hereditary Harrop Formulas”</a> |
| gives the details.</p> |
| <p>In terms of code, these types are defined in |
| <a href="https://github.com/rust-lang/rust/blob/HEAD/compiler/rustc_middle/src/traits/mod.rs"><code>rustc_middle/src/traits/mod.rs</code></a> in rustc, and in |
| <a href="https://github.com/rust-lang/chalk/blob/master/chalk-ir/src/lib.rs"><code>chalk-ir/src/lib.rs</code></a> in chalk.</p> |
| <p><a id="domain-goals"></a></p> |
| <h2 id="domain-goals"><a class="header" href="#domain-goals">Domain goals</a></h2> |
| <p><em>Domain goals</em> are the atoms of the trait logic. As can be seen in the |
| definitions given above, general goals basically consist in a combination of |
| domain goals.</p> |
| <p>Moreover, flattening a bit the definition of clauses given previously, one can |
| see that clauses are always of the form:</p> |
| <pre><code class="language-text">forall<K1, ..., Kn> { DomainGoal :- Goal } |
| </code></pre> |
| <p>hence domain goals are in fact clauses’ LHS. That is, at the most granular level, |
| domain goals are what the trait solver will end up trying to prove.</p> |
| <p><a id="trait-ref"></a></p> |
| <p>To define the set of domain goals in our system, we need to first |
| introduce a few simple formulations. A <strong>trait reference</strong> consists of |
| the name of a trait along with a suitable set of inputs P0..Pn:</p> |
| <pre><code class="language-text">TraitRef = P0: TraitName<P1..Pn> |
| </code></pre> |
| <p>So, for example, <code>u32: Display</code> is a trait reference, as is <code>Vec<T>: IntoIterator</code>. Note that Rust surface syntax also permits some extra |
| things, like associated type bindings (<code>Vec<T>: IntoIterator<Item = T></code>), that are not part of a trait reference.</p> |
| <p><a id="projection"></a></p> |
| <p>A <strong>projection</strong> consists of an associated item reference along with |
| its inputs P0..Pm:</p> |
| <pre><code class="language-text">Projection = <P0 as TraitName<P1..Pn>>::AssocItem<Pn+1..Pm> |
| </code></pre> |
| <p>Given these, we can define a <code>DomainGoal</code> as follows:</p> |
| <pre><code class="language-text">DomainGoal = Holds(WhereClause) |
| | FromEnv(TraitRef) |
| | FromEnv(Type) |
| | WellFormed(TraitRef) |
| | WellFormed(Type) |
| | Normalize(Projection -> Type) |
| |
| WhereClause = Implemented(TraitRef) |
| | ProjectionEq(Projection = Type) |
| | Outlives(Type: Region) |
| | Outlives(Region: Region) |
| </code></pre> |
| <p><code>WhereClause</code> refers to a <code>where</code> clause that a Rust user would actually be able |
| to write in a Rust program. This abstraction exists only as a convenience as we |
| sometimes want to only deal with domain goals that are effectively writable in |
| Rust.</p> |
| <p>Let’s break down each one of these, one-by-one.</p> |
| <h4 id="implementedtraitref"><a class="header" href="#implementedtraitref">Implemented(TraitRef)</a></h4> |
| <p>e.g. <code>Implemented(i32: Copy)</code></p> |
| <p>True if the given trait is implemented for the given input types and lifetimes.</p> |
| <h4 id="projectioneqprojection--type"><a class="header" href="#projectioneqprojection--type">ProjectionEq(Projection = Type)</a></h4> |
| <p>e.g. <code>ProjectionEq<T as Iterator>::Item = u8</code></p> |
| <p>The given associated type <code>Projection</code> is equal to <code>Type</code>; this can be proved |
| with either normalization or using placeholder associated types. See |
| <a href="https://rust-lang.github.io/chalk/book/clauses/type_equality.html">the section on associated types in Chalk Book</a>.</p> |
| <h4 id="normalizeprojection---type"><a class="header" href="#normalizeprojection---type">Normalize(Projection -> Type)</a></h4> |
| <p>e.g. <code>ProjectionEq<T as Iterator>::Item -> u8</code></p> |
| <p>The given associated type <code>Projection</code> can be <a href="https://rust-lang.github.io/chalk/book/clauses/type_equality.html#normalize">normalized</a> to <code>Type</code>.</p> |
| <p>As discussed in <a href="https://rust-lang.github.io/chalk/book/clauses/type_equality.html">the section on associated |
| types in Chalk Book</a>, <code>Normalize</code> implies <code>ProjectionEq</code>, |
| but not vice versa. In general, proving <code>Normalize(<T as Trait>::Item -> U)</code> |
| also requires proving <code>Implemented(T: Trait)</code>.</p> |
| <h4 id="fromenvtraitref"><a class="header" href="#fromenvtraitref">FromEnv(TraitRef)</a></h4> |
| <p>e.g. <code>FromEnv(Self: Add<i32>)</code></p> |
| <p>True if the inner <code>TraitRef</code> is <em>assumed</em> to be true, |
| that is, if it can be derived from the in-scope where clauses.</p> |
| <p>For example, given the following function:</p> |
| <pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)] |
| </span><span class="boring">fn main() { |
| </span>fn loud_clone<T: Clone>(stuff: &T) -> T { |
| println!("cloning!"); |
| stuff.clone() |
| } |
| <span class="boring">}</span></code></pre> |
| <p>Inside the body of our function, we would have <code>FromEnv(T: Clone)</code>. In-scope |
| where clauses nest, so a function body inside an impl body inherits the |
| impl body’s where clauses, too.</p> |
| <p>This and the next rule are used to implement <a href="https://rust-lang.github.io/chalk/book/clauses/implied_bounds.html#implied-bounds">implied bounds</a>. As we’ll see |
| in the section on lowering, <code>FromEnv(TraitRef)</code> implies <code>Implemented(TraitRef)</code>, |
| but not vice versa. This distinction is crucial to implied bounds.</p> |
| <h4 id="fromenvtype"><a class="header" href="#fromenvtype">FromEnv(Type)</a></h4> |
| <p>e.g. <code>FromEnv(HashSet<K>)</code></p> |
| <p>True if the inner <code>Type</code> is <em>assumed</em> to be well-formed, that is, if it is an |
| input type of a function or an impl.</p> |
| <p>For example, given the following code:</p> |
| <pre><code class="language-rust ignore">struct HashSet<K> where K: Hash { ... } |
| |
| fn loud_insert<K>(set: &mut HashSet<K>, item: K) { |
| println!("inserting!"); |
| set.insert(item); |
| }</code></pre> |
| <p><code>HashSet<K></code> is an input type of the <code>loud_insert</code> function. Hence, we assume it |
| to be well-formed, so we would have <code>FromEnv(HashSet<K>)</code> inside the body of our |
| function. As we’ll see in the section on lowering, <code>FromEnv(HashSet<K>)</code> implies |
| <code>Implemented(K: Hash)</code> because the |
| <code>HashSet</code> declaration was written with a <code>K: Hash</code> where clause. Hence, we don’t |
| need to repeat that bound on the <code>loud_insert</code> function: we rather automatically |
| assume that it is true.</p> |
| <h4 id="wellformeditem"><a class="header" href="#wellformeditem">WellFormed(Item)</a></h4> |
| <p>These goals imply that the given item is <em>well-formed</em>.</p> |
| <p>We can talk about different types of items being well-formed:</p> |
| <ul> |
| <li> |
| <p><em>Types</em>, like <code>WellFormed(Vec<i32>)</code>, which is true in Rust, or |
| <code>WellFormed(Vec<str>)</code>, which is not (because <code>str</code> is not <code>Sized</code>.)</p> |
| </li> |
| <li> |
| <p><em>TraitRefs</em>, like <code>WellFormed(Vec<i32>: Clone)</code>.</p> |
| </li> |
| </ul> |
| <p>Well-formedness is important to <a href="https://rust-lang.github.io/chalk/book/clauses/implied_bounds.html#implied-bounds">implied bounds</a>. In particular, the reason |
| it is okay to assume <code>FromEnv(T: Clone)</code> in the <code>loud_clone</code> example is that we |
| <em>also</em> verify <code>WellFormed(T: Clone)</code> for each call site of <code>loud_clone</code>. |
| Similarly, it is okay to assume <code>FromEnv(HashSet<K>)</code> in the <code>loud_insert</code> |
| example because we will verify <code>WellFormed(HashSet<K>)</code> for each call site of |
| <code>loud_insert</code>.</p> |
| <h4 id="outlivestype-region-outlivesregion-region"><a class="header" href="#outlivestype-region-outlivesregion-region">Outlives(Type: Region), Outlives(Region: Region)</a></h4> |
| <p>e.g. <code>Outlives(&'a str: 'b)</code>, <code>Outlives('a: 'static)</code></p> |
| <p>True if the given type or region on the left outlives the right-hand region.</p> |
| <p><a id="coinductive"></a></p> |
| <h2 id="coinductive-goals"><a class="header" href="#coinductive-goals">Coinductive goals</a></h2> |
| <p>Most goals in our system are “inductive”. In an inductive goal, |
| circular reasoning is disallowed. Consider this example clause:</p> |
| <pre><code class="language-text"> Implemented(Foo: Bar) :- |
| Implemented(Foo: Bar). |
| </code></pre> |
| <p>Considered inductively, this clause is useless: if we are trying to |
| prove <code>Implemented(Foo: Bar)</code>, we would then recursively have to prove |
| <code>Implemented(Foo: Bar)</code>, and that cycle would continue ad infinitum |
| (the trait solver will terminate here, it would just consider that |
| <code>Implemented(Foo: Bar)</code> is not known to be true).</p> |
| <p>However, some goals are <em>co-inductive</em>. Simply put, this means that |
| cycles are OK. So, if <code>Bar</code> were a co-inductive trait, then the rule |
| above would be perfectly valid, and it would indicate that |
| <code>Implemented(Foo: Bar)</code> is true.</p> |
| <p><em>Auto traits</em> are one example in Rust where co-inductive goals are used. |
| Consider the <code>Send</code> trait, and imagine that we have this struct:</p> |
| <pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)] |
| </span><span class="boring">fn main() { |
| </span>struct Foo { |
| next: Option<Box<Foo>> |
| } |
| <span class="boring">}</span></code></pre> |
| <p>The default rules for auto traits say that <code>Foo</code> is <code>Send</code> if the |
| types of its fields are <code>Send</code>. Therefore, we would have a rule like</p> |
| <pre><code class="language-text">Implemented(Foo: Send) :- |
| Implemented(Option<Box<Foo>>: Send). |
| </code></pre> |
| <p>As you can probably imagine, proving that <code>Option<Box<Foo>>: Send</code> is |
| going to wind up circularly requiring us to prove that <code>Foo: Send</code> |
| again. So this would be an example where we wind up in a cycle – but |
| that’s ok, we <em>do</em> consider <code>Foo: Send</code> to hold, even though it |
| references itself.</p> |
| <p>In general, co-inductive traits are used in Rust trait solving when we |
| want to enumerate a fixed set of possibilities. In the case of auto |
| traits, we are enumerating the set of reachable types from a given |
| starting point (i.e., <code>Foo</code> can reach values of type |
| <code>Option<Box<Foo>></code>, which implies it can reach values of type |
| <code>Box<Foo></code>, and then of type <code>Foo</code>, and then the cycle is complete).</p> |
| <p>In addition to auto traits, <code>WellFormed</code> predicates are co-inductive. |
| These are used to achieve a similar “enumerate all the cases” pattern, |
| as described in the section on <a href="https://rust-lang.github.io/chalk/book/clauses/implied_bounds.html#implied-bounds">implied bounds</a>.</p> |
| <h2 id="incomplete-chapter"><a class="header" href="#incomplete-chapter">Incomplete chapter</a></h2> |
| <p>Some topics yet to be written:</p> |
| <ul> |
| <li>Elaborate on the proof procedure</li> |
| <li>SLG solving – introduce negative reasoning</li> |
| </ul> |
| |
| </main> |
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