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<h1 id="canonicalization"><a class="header" href="#canonicalization">Canonicalization</a></h1>
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<blockquote>
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<p><strong>NOTE</strong>: FIXME: The content of this chapter has some overlap with
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<a href="../solve/canonicalization.html">Next-gen trait solving Canonicalization chapter</a>.
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It is suggested to reorganize these contents in the future.</p>
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</blockquote>
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<p>Canonicalization is the process of <strong>isolating</strong> an inference value
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from its context. It is a key part of implementing
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<a href="./canonical-queries.html">canonical queries</a>, and you may wish to read the parent chapter
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to get more context.</p>
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<p>Canonicalization is really based on a very simple concept: every
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<a href="../type-inference.html#vars">inference variable</a> is always in one of
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two states: either it is <strong>unbound</strong>, in which case we don't know yet
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what type it is, or it is <strong>bound</strong>, in which case we do. So to
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isolate some data-structure T that contains types/regions from its
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environment, we just walk down and find the unbound variables that
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appear in T; those variables get replaced with "canonical variables",
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starting from zero and numbered in a fixed order (left to right, for
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the most part, but really it doesn't matter as long as it is
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consistent).</p>
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<p>So, for example, if we have the type <code>X = (?T, ?U)</code>, where <code>?T</code> and
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<code>?U</code> are distinct, unbound inference variables, then the canonical
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form of <code>X</code> would be <code>(?0, ?1)</code>, where <code>?0</code> and <code>?1</code> represent these
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<strong>canonical placeholders</strong>. Note that the type <code>Y = (?U, ?T)</code> also
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canonicalizes to <code>(?0, ?1)</code>. But the type <code>Z = (?T, ?T)</code> would
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canonicalize to <code>(?0, ?0)</code> (as would <code>(?U, ?U)</code>). In other words, the
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exact identity of the inference variables is not important – unless
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they are repeated.</p>
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<p>We use this to improve caching as well as to detect cycles and other
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things during trait resolution. Roughly speaking, the idea is that if
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two trait queries have the same canonical form, then they will get
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the same answer. That answer will be expressed in terms of the
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canonical variables (<code>?0</code>, <code>?1</code>), which we can then map back to the
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original variables (<code>?T</code>, <code>?U</code>).</p>
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<h2 id="canonicalizing-the-query"><a class="header" href="#canonicalizing-the-query">Canonicalizing the query</a></h2>
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<p>To see how it works, imagine that we are asking to solve the following
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trait query: <code>?A: Foo<'static, ?B></code>, where <code>?A</code> and <code>?B</code> are unbound.
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This query contains two unbound variables, but it also contains the
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lifetime <code>'static</code>. The trait system generally ignores all lifetimes
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and treats them equally, so when canonicalizing, we will <em>also</em>
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replace any <a href="../appendix/background.html#free-vs-bound">free lifetime</a> with a
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canonical variable (Note that <code>'static</code> is actually a <em>free</em> lifetime
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variable here. We are not considering it in the typing context of the whole
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program but only in the context of this trait reference. Mathematically, we
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are not quantifying over the whole program, but only this obligation).
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Therefore, we get the following result:</p>
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<pre><code class="language-text">?0: Foo<'?1, ?2>
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</code></pre>
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<p>Sometimes we write this differently, like so:</p>
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<pre><code class="language-text">for<T,L,T> { ?0: Foo<'?1, ?2> }
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</code></pre>
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<p>This <code>for<></code> gives some information about each of the canonical
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variables within. In this case, each <code>T</code> indicates a type variable,
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so <code>?0</code> and <code>?2</code> are types; the <code>L</code> indicates a lifetime variable, so
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<code>?1</code> is a lifetime. The <code>canonicalize</code> method <em>also</em> gives back a
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<code>CanonicalVarValues</code> array OV with the "original values" for each
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canonicalized variable:</p>
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<pre><code class="language-text">[?A, 'static, ?B]
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</code></pre>
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<p>We'll need this vector OV later, when we process the query response.</p>
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<h2 id="executing-the-query"><a class="header" href="#executing-the-query">Executing the query</a></h2>
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<p>Once we've constructed the canonical query, we can try to solve it.
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To do so, we will wind up creating a fresh inference context and
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<strong>instantiating</strong> the canonical query in that context. The idea is that
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we create a substitution S from the canonical form containing a fresh
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inference variable (of suitable kind) for each canonical variable.
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So, for our example query:</p>
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<pre><code class="language-text">for<T,L,T> { ?0: Foo<'?1, ?2> }
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</code></pre>
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<p>the substitution S might be:</p>
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<pre><code class="language-text">S = [?A, '?B, ?C]
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</code></pre>
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<p>We can then replace the bound canonical variables (<code>?0</code>, etc) with
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these inference variables, yielding the following fully instantiated
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query:</p>
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<pre><code class="language-text">?A: Foo<'?B, ?C>
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</code></pre>
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<p>Remember that substitution S though! We're going to need it later.</p>
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<p>OK, now that we have a fresh inference context and an instantiated
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query, we can go ahead and try to solve it. The trait solver itself is
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explained in more detail in <a href="../solve/the-solver.html">another section</a>, but
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suffice to say that it will compute a <a href="./canonical-queries.html#query-response">certainty value</a> (<code>Proven</code> or
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<code>Ambiguous</code>) and have side-effects on the inference variables we've
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created. For example, if there were only one impl of <code>Foo</code>, like so:</p>
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<pre><code class="language-rust ignore">impl<'a, X> Foo<'a, X> for Vec<X>
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where X: 'a
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{ ... }</code></pre>
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<p>then we might wind up with a certainty value of <code>Proven</code>, as well as
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creating fresh inference variables <code>'?D</code> and <code>?E</code> (to represent the
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parameters on the impl) and unifying as follows:</p>
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<ul>
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<li><code>'?B = '?D</code></li>
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<li><code>?A = Vec<?E></code></li>
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<li><code>?C = ?E</code></li>
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</ul>
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<p>We would also accumulate the region constraint <code>?E: '?D</code>, due to the
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where clause.</p>
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<p>In order to create our final query result, we have to "lift" these
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values out of the query's inference context and into something that
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can be reapplied in our original inference context. We do that by
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<strong>re-applying canonicalization</strong>, but to the <strong>query result</strong>.</p>
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<h2 id="canonicalizing-the-query-result"><a class="header" href="#canonicalizing-the-query-result">Canonicalizing the query result</a></h2>
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<p>As discussed in <a href="./canonical-queries.html#query-response">the parent section</a>, most trait queries wind up
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with a result that brings together a "certainty value" <code>certainty</code>, a
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result substitution <code>var_values</code>, and some region constraints. To
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create this, we wind up re-using the substitution S that we created
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when first instantiating our query. To refresh your memory, we had a query</p>
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<pre><code class="language-text">for<T,L,T> { ?0: Foo<'?1, ?2> }
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</code></pre>
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<p>for which we made a substutition S:</p>
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<pre><code class="language-text">S = [?A, '?B, ?C]
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</code></pre>
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<p>We then did some work which unified some of those variables with other things.
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If we "refresh" S with the latest results, we get:</p>
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<pre><code class="language-text">S = [Vec<?E>, '?D, ?E]
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</code></pre>
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<p>These are precisely the new values for the three input variables from
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our original query. Note though that they include some new variables
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(like <code>?E</code>). We can make those go away by canonicalizing again! We don't
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just canonicalize S, though, we canonicalize the whole query response QR:</p>
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<pre><code class="language-text">QR = {
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certainty: Proven, // or whatever
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var_values: [Vec<?E>, '?D, ?E] // this is S
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region_constraints: [?E: '?D], // from the impl
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value: (), // for our purposes, just (), but
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// in some cases this might have
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||
// a type or other info
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}
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</code></pre>
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<p>The result would be as follows:</p>
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<pre><code class="language-text">Canonical(QR) = for<T, L> {
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certainty: Proven,
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var_values: [Vec<?0>, '?1, ?0]
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region_constraints: [?0: '?1],
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value: (),
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||
}
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||
</code></pre>
|
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<p>(One subtle point: when we canonicalize the query <strong>result</strong>, we do not
|
||
use any special treatment for free lifetimes. Note that both
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references to <code>'?D</code>, for example, were converted into the same
|
||
canonical variable (<code>?1</code>). This is in contrast to the original query,
|
||
where we canonicalized every free lifetime into a fresh canonical
|
||
variable.)</p>
|
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<p>Now, this result must be reapplied in each context where needed.</p>
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<h2 id="processing-the-canonicalized-query-result"><a class="header" href="#processing-the-canonicalized-query-result">Processing the canonicalized query result</a></h2>
|
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<p>In the previous section we produced a canonical query result. We now have
|
||
to apply that result in our original context. If you recall, way back in the
|
||
beginning, we were trying to prove this query:</p>
|
||
<pre><code class="language-text">?A: Foo<'static, ?B>
|
||
</code></pre>
|
||
<p>We canonicalized that into this:</p>
|
||
<pre><code class="language-text">for<T,L,T> { ?0: Foo<'?1, ?2> }
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||
</code></pre>
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||
<p>and now we got back a canonical response:</p>
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||
<pre><code class="language-text">for<T, L> {
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certainty: Proven,
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||
var_values: [Vec<?0>, '?1, ?0]
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||
region_constraints: [?0: '?1],
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||
value: (),
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||
}
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||
</code></pre>
|
||
<p>We now want to apply that response to our context. Conceptually, how
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||
we do that is to (a) instantiate each of the canonical variables in
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||
the result with a fresh inference variable, (b) unify the values in
|
||
the result with the original values, and then (c) record the region
|
||
constraints for later. Doing step (a) would yield a result of</p>
|
||
<pre><code class="language-text">{
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||
certainty: Proven,
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||
var_values: [Vec<?C>, '?D, ?C]
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||
^^ ^^^ fresh inference variables
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||
region_constraints: [?C: '?D],
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||
value: (),
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||
}
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||
</code></pre>
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||
<p>Step (b) would then unify:</p>
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||
<pre><code class="language-text">?A with Vec<?C>
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'static with '?D
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?B with ?C
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||
</code></pre>
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<p>And finally the region constraint of <code>?C: 'static</code> would be recorded
|
||
for later verification.</p>
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<p>(What we <em>actually</em> do is a mildly optimized variant of that: Rather
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||
than eagerly instantiating all of the canonical values in the result
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||
with variables, we instead walk the vector of values, looking for
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||
cases where the value is just a canonical variable. In our example,
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||
<code>values[2]</code> is <code>?C</code>, so that means we can deduce that <code>?C := ?B</code> and
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||
<code>'?D := 'static</code>. This gives us a partial set of values. Anything for
|
||
which we do not find a value, we create an inference variable.)</p>
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