Add back the canonicalization chapter.

src/SUMMARY.md

@@ -123,6 +123,7 @@
         - [Lowering to logic](./traits/lowering-to-logic.md)
         - [Goals and clauses](./traits/goals-and-clauses.md)
         - [Canonical queries](./traits/canonical-queries.md)
+        - [Canonicalization](./traits/canonicalization.md)
     - [Next-gen trait solving](./solve/trait-solving.md)
         - [Invariants of the type system](./solve/invariants.md)
         - [The solver](./solve/the-solver.md)

src/traits/canonicalization.md

@@ -0,0 +1,256 @@
+# Canonicalization
+
+Canonicalization is the process of **isolating** an inference value
+from its context. It is a key part of implementing
+[canonical queries][cq], and you may wish to read the parent chapter
+to get more context.
+
+Canonicalization is really based on a very simple concept: every
+[inference variable](../type-inference.html#vars) is always in one of
+two states: either it is **unbound**, in which case we don't know yet
+what type it is, or it is **bound**, in which case we do. So to
+isolate some data-structure T that contains types/regions from its
+environment, we just walk down and find the unbound variables that
+appear in T; those variables get replaced with "canonical variables",
+starting from zero and numbered in a fixed order (left to right, for
+the most part, but really it doesn't matter as long as it is
+consistent).
+
+[cq]: ./canonical-queries.html
+
+So, for example, if we have the type `X = (?T, ?U)`, where `?T` and
+`?U` are distinct, unbound inference variables, then the canonical
+form of `X` would be `(?0, ?1)`, where `?0` and `?1` represent these
+**canonical placeholders**. Note that the type `Y = (?U, ?T)` also
+canonicalizes to `(?0, ?1)`. But the type `Z = (?T, ?T)` would
+canonicalize to `(?0, ?0)` (as would `(?U, ?U)`). In other words, the
+exact identity of the inference variables is not important – unless
+they are repeated.
+
+We use this to improve caching as well as to detect cycles and other
+things during trait resolution. Roughly speaking, the idea is that if
+two trait queries have the same canonical form, then they will get
+the same answer. That answer will be expressed in terms of the
+canonical variables (`?0`, `?1`), which we can then map back to the
+original variables (`?T`, `?U`).
+
+## Canonicalizing the query
+
+To see how it works, imagine that we are asking to solve the following
+trait query: `?A: Foo<'static, ?B>`, where `?A` and `?B` are unbound.
+This query contains two unbound variables, but it also contains the
+lifetime `'static`. The trait system generally ignores all lifetimes
+and treats them equally, so when canonicalizing, we will *also*
+replace any [free lifetime](../appendix/background.html#free-vs-bound) with a
+canonical variable (Note that `'static` is actually a _free_ lifetime 
+variable here. We are not considering it in the typing context of the whole 
+program but only in the context of this trait reference. Mathematically, we
+are not quantifying over the whole program, but only this obligation).
+Therefore, we get the following result:
+
+```text
+?0: Foo<'?1, ?2>
+```
+
+Sometimes we write this differently, like so:
+
+```text
+for<T,L,T> { ?0: Foo<'?1, ?2> }
+```
+
+This `for<>` gives some information about each of the canonical
+variables within.  In this case, each `T` indicates a type variable,
+so `?0` and `?2` are types; the `L` indicates a lifetime variable, so
+`?1` is a lifetime. The `canonicalize` method *also* gives back a
+`CanonicalVarValues` array OV with the "original values" for each
+canonicalized variable:
+
+```text
+[?A, 'static, ?B]
+```
+
+We'll need this vector OV later, when we process the query response.
+
+## Executing the query
+
+Once we've constructed the canonical query, we can try to solve it.
+To do so, we will wind up creating a fresh inference context and
+**instantiating** the canonical query in that context. The idea is that
+we create a substitution S from the canonical form containing a fresh
+inference variable (of suitable kind) for each canonical variable.
+So, for our example query:
+
+```text
+for<T,L,T> { ?0: Foo<'?1, ?2> }
+```
+
+the substitution S might be:
+
+```text
+S = [?A, '?B, ?C]
+```
+
+We can then replace the bound canonical variables (`?0`, etc) with
+these inference variables, yielding the following fully instantiated
+query:
+
+```text
+?A: Foo<'?B, ?C>
+```
+
+Remember that substitution S though! We're going to need it later.
+
+OK, now that we have a fresh inference context and an instantiated
+query, we can go ahead and try to solve it. The trait solver itself is
+explained in more detail in [another section](./slg.html), but
+suffice to say that it will compute a [certainty value][cqqr] (`Proven` or
+`Ambiguous`) and have side-effects on the inference variables we've
+created. For example, if there were only one impl of `Foo`, like so:
+
+[cqqr]: ./canonical-queries.html#query-response
+
+```rust,ignore
+impl<'a, X> Foo<'a, X> for Vec<X>
+where X: 'a
+{ ... }
+```
+
+then we might wind up with a certainty value of `Proven`, as well as
+creating fresh inference variables `'?D` and `?E` (to represent the
+parameters on the impl) and unifying as follows:
+
+- `'?B = '?D`
+- `?A = Vec<?E>`
+- `?C = ?E`
+
+We would also accumulate the region constraint `?E: '?D`, due to the
+where clause.
+
+In order to create our final query result, we have to "lift" these
+values out of the query's inference context and into something that
+can be reapplied in our original inference context. We do that by
+**re-applying canonicalization**, but to the **query result**.
+
+## Canonicalizing the query result
+
+As discussed in [the parent section][cqqr], most trait queries wind up
+with a result that brings together a "certainty value" `certainty`, a
+result substitution `var_values`, and some region constraints. To
+create this, we wind up re-using the substitution S that we created
+when first instantiating our query. To refresh your memory, we had a query
+
+```text
+for<T,L,T> { ?0: Foo<'?1, ?2> }
+```
+
+for which we made a substutition S:
+
+```text
+S = [?A, '?B, ?C]
+```
+
+We then did some work which unified some of those variables with other things.
+If we "refresh" S with the latest results, we get:
+
+```text
+S = [Vec<?E>, '?D, ?E]
+```
+
+These are precisely the new values for the three input variables from
+our original query. Note though that they include some new variables
+(like `?E`). We can make those go away by canonicalizing again! We don't
+just canonicalize S, though, we canonicalize the whole query response QR:
+
+```text
+QR = {
+    certainty: Proven,             // or whatever
+    var_values: [Vec<?E>, '?D, ?E] // this is S
+    region_constraints: [?E: '?D], // from the impl
+    value: (),                     // for our purposes, just (), but
+                                   // in some cases this might have
+                                   // a type or other info
+}
+```
+
+The result would be as follows:
+
+```text
+Canonical(QR) = for<T, L> {
+    certainty: Proven,
+    var_values: [Vec<?0>, '?1, ?0]
+    region_constraints: [?0: '?1],
+    value: (),
+}
+```
+
+(One subtle point: when we canonicalize the query **result**, we do not
+use any special treatment for free lifetimes. Note that both
+references to `'?D`, for example, were converted into the same
+canonical variable (`?1`). This is in contrast to the original query,
+where we canonicalized every free lifetime into a fresh canonical
+variable.)
+
+Now, this result must be reapplied in each context where needed.
+
+## Processing the canonicalized query result
+
+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:
+
+```text
+?A: Foo<'static, ?B>
+```
+
+We canonicalized that into this:
+
+```text
+for<T,L,T> { ?0: Foo<'?1, ?2> }
+```
+
+and now we got back a canonical response:
+
+```text
+for<T, L> {
+    certainty: Proven,
+    var_values: [Vec<?0>, '?1, ?0]
+    region_constraints: [?0: '?1],
+    value: (),
+}
+```
+
+We now want to apply that response to our context. Conceptually, how
+we do that is to (a) instantiate each of the canonical variables in
+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
+
+```text
+{
+      certainty: Proven,
+      var_values: [Vec<?C>, '?D, ?C]
+                       ^^   ^^^ fresh inference variables
+      region_constraints: [?C: '?D],
+      value: (),
+}
+```
+
+Step (b) would then unify:
+
+```text
+?A with Vec<?C>
+'static with '?D
+?B with ?C
+```
+
+And finally the region constraint of `?C: 'static` would be recorded
+for later verification.
+
+(What we *actually* do is a mildly optimized variant of that: Rather
+than eagerly instantiating all of the canonical values in the result
+with variables, we instead walk the vector of values, looking for
+cases where the value is just a canonical variable. In our example,
+`values[2]` is `?C`, so that means we can deduce that `?C := ?B` and
+`'?D := 'static`. This gives us a partial set of values. Anything for
+which we do not find a value, we create an inference variable.)
+