/// (without being so rigorous).
///
/// The core of the algorithm revolves about a "usefulness" check. In particular, we
-/// are trying to compute a predicate `U(P, p_{m + 1})` where `P` is a list of patterns
-/// of length `m` for a compound (product) type with `n` components (we refer to this as
-/// a matrix). `U(P, p_{m + 1})` represents whether, given an existing list of patterns
-/// `p_1 ..= p_m`, adding a new pattern will be "useful" (that is, cover previously-
+/// are trying to compute a predicate `U(P, p)` where `P` is a list of patterns (we refer to this as
+/// a matrix). `U(P, p)` represents whether, given an existing list of patterns
+/// `P_1 ..= P_m`, adding a new pattern `p` will be "useful" (that is, cover previously-
/// uncovered values of the type).
///
/// If we have this predicate, then we can easily compute both exhaustiveness of an
/// entire set of patterns and the individual usefulness of each one.
/// (a) the set of patterns is exhaustive iff `U(P, _)` is false (i.e., adding a wildcard
/// match doesn't increase the number of values we're matching)
-/// (b) a pattern `p_i` is not useful if `U(P[0..=(i-1), p_i)` is false (i.e., adding a
+/// (b) a pattern `P_i` is not useful if `U(P[0..=(i-1), P_i)` is false (i.e., adding a
/// pattern to those that have come before it doesn't increase the number of values
/// we're matching).
///
+/// During the course of the algorithm, the rows of the matrix won't just be individual patterns,
+/// but rather partially-deconstructed patterns in the form of a list of patterns. The paper
+/// calls those pattern-vectors, and we will call them pattern-stacks. The same holds for the
+/// new pattern `p`.
+///
/// For example, say we have the following:
/// ```
/// // x: (Option<bool>, Result<()>)
/// (None, Err(_)) => {}
/// }
/// ```
-/// Here, the matrix `P` is 3 x 2 (rows x columns).
+/// Here, the matrix `P` starts as:
/// [
-/// [Some(true), _],
-/// [None, Err(())],
-/// [None, Err(_)],
+/// [(Some(true), _)],
+/// [(None, Err(()))],
+/// [(None, Err(_))],
/// ]
/// We can tell it's not exhaustive, because `U(P, _)` is true (we're not covering
-/// `[Some(false), _]`, for instance). In addition, row 3 is not useful, because
+/// `[(Some(false), _)]`, for instance). In addition, row 3 is not useful, because
/// all the values it covers are already covered by row 2.
///
-/// To compute `U`, we must have two other concepts.
-/// 1. `S(c, P)` is a "specialized matrix", where `c` is a constructor (like `Some` or
-/// `None`). You can think of it as filtering `P` to just the rows whose *first* pattern
-/// can cover `c` (and expanding OR-patterns into distinct patterns), and then expanding
-/// the constructor into all of its components.
-/// The specialization of a row vector is computed by `specialize`.
+/// A list of patterns can be thought of as a stack, because we are mainly interested in the top of
+/// the stack at any given point, and we can pop or apply constructors to get new pattern-stacks.
+/// To match the paper, the top of the stack is at the beginning / on the left.
+///
+/// There are two important operations on pattern-stacks necessary to understand the algorithm:
+/// 1. We can pop a given constructor off the top of a stack. This operation is called
+/// `specialize`, and is denoted `S(c, p)` where `c` is a constructor (like `Some` or
+/// `None`) and `p` a pattern-stack.
+/// If the pattern on top of the stack can cover `c`, this removes the constructor and
+/// pushes its arguments onto the stack. It also expands OR-patterns into distinct patterns.
+/// Otherwise the pattern-stack is discarded.
+/// This essentially filters those pattern-stacks whose top covers the constructor `c` and
+/// discards the others.
+///
+/// For example, the first pattern above initially gives a stack `[(Some(true), _)]`. If we
+/// pop the tuple constructor, we are left with `[Some(true), _]`, and if we then pop the
+/// `Some` constructor we get `[true, _]`. If we had popped `None` instead, we would get
+/// nothing back.
///
-/// It is computed as follows. For each row `p_i` of P, we have four cases:
-/// 1.1. `p_(i,1) = c(r_1, .., r_a)`. Then `S(c, P)` has a corresponding row:
-/// r_1, .., r_a, p_(i,2), .., p_(i,n)
-/// 1.2. `p_(i,1) = c'(r_1, .., r_a')` where `c ≠ c'`. Then `S(c, P)` has no
-/// corresponding row.
-/// 1.3. `p_(i,1) = _`. Then `S(c, P)` has a corresponding row:
-/// _, .., _, p_(i,2), .., p_(i,n)
-/// 1.4. `p_(i,1) = r_1 | r_2`. Then `S(c, P)` has corresponding rows inlined from:
-/// S(c, (r_1, p_(i,2), .., p_(i,n)))
-/// S(c, (r_2, p_(i,2), .., p_(i,n)))
+/// This returns zero or more new pattern-stacks, as follows. We look at the pattern `p_1`
+/// on top of the stack, and we have four cases:
+/// 1.1. `p_1 = c(r_1, .., r_a)`, i.e. the top of the stack has constructor `c`. We
+/// push onto the stack the arguments of this constructor, and return the result:
+/// r_1, .., r_a, p_2, .., p_n
+/// 1.2. `p_1 = c'(r_1, .., r_a')` where `c ≠ c'`. We discard the current stack and
+/// return nothing.
+/// 1.3. `p_1 = _`. We push onto the stack as many wildcards as the constructor `c` has
+/// arguments (its arity), and return the resulting stack:
+/// _, .., _, p_2, .., p_n
+/// 1.4. `p_1 = r_1 | r_2`. We expand the OR-pattern and then recurse on each resulting
+/// stack:
+/// S(c, (r_1, p_2, .., p_n))
+/// S(c, (r_2, p_2, .., p_n))
///
-/// 2. `D(P)` is a "default matrix". This is used when we know there are missing
-/// constructor cases, but there might be existing wildcard patterns, so to check the
-/// usefulness of the matrix, we have to check all its *other* components.
-/// The default matrix is computed inline in `is_useful`.
+/// 2. We can pop a wildcard off the top of the stack. This is called `D(p)`, where `p` is
+/// a pattern-stack.
+/// This is used when we know there are missing constructor cases, but there might be
+/// existing wildcard patterns, so to check the usefulness of the matrix, we have to check
+/// all its *other* components.
///
-/// It is computed as follows. For each row `p_i` of P, we have three cases:
-/// 1.1. `p_(i,1) = c(r_1, .., r_a)`. Then `D(P)` has no corresponding row.
-/// 1.2. `p_(i,1) = _`. Then `D(P)` has a corresponding row:
-/// p_(i,2), .., p_(i,n)
-/// 1.3. `p_(i,1) = r_1 | r_2`. Then `D(P)` has corresponding rows inlined from:
-/// D((r_1, p_(i,2), .., p_(i,n)))
-/// D((r_2, p_(i,2), .., p_(i,n)))
+/// It is computed as follows. We look at the pattern `p_1` on top of the stack,
+/// and we have three cases:
+/// 1.1. `p_1 = c(r_1, .., r_a)`. We discard the current stack and return nothing.
+/// 1.2. `p_1 = _`. We return the rest of the stack:
+/// p_2, .., p_n
+/// 1.3. `p_1 = r_1 | r_2`. We expand the OR-pattern and then recurse on each resulting
+/// stack.
+/// D((r_1, p_2, .., p_n))
+/// D((r_2, p_2, .., p_n))
+///
+/// Note that the OR-patterns are not always used directly in Rust, but are used to derive the
+/// exhaustive integer matching rules, so they're written here for posterity.
+///
+/// Both those operations extend straightforwardly to a list or pattern-stacks, i.e. a matrix, by
+/// working row-by-row. Popping a constructor ends up keeping only the matrix rows that start with
+/// the given constructor, and popping a wildcard keeps those rows that start with a wildcard.
///
-/// Note that the OR-patterns are not always used directly in Rust, but are used to derive
-/// the exhaustive integer matching rules, so they're written here for posterity.
///
/// The algorithm for computing `U`
/// -------------------------------
/// The algorithm is inductive (on the number of columns: i.e., components of tuple patterns).
/// That means we're going to check the components from left-to-right, so the algorithm
-/// operates principally on the first component of the matrix and new pattern `p_{m + 1}`.
+/// operates principally on the first component of the matrix and new pattern-stack `p`.
/// This algorithm is realised in the `is_useful` function.
///
/// Base case. (`n = 0`, i.e., an empty tuple pattern)
/// - If `P` already contains an empty pattern (i.e., if the number of patterns `m > 0`),
-/// then `U(P, p_{m + 1})` is false.
-/// - Otherwise, `P` must be empty, so `U(P, p_{m + 1})` is true.
+/// then `U(P, p)` is false.
+/// - Otherwise, `P` must be empty, so `U(P, p)` is true.
///
/// Inductive step. (`n > 0`, i.e., whether there's at least one column
/// [which may then be expanded into further columns later])
-/// We're going to match on the new pattern, `p_{m + 1}`.
-/// - If `p_{m + 1} == c(r_1, .., r_a)`, then we have a constructor pattern.
-/// Thus, the usefulness of `p_{m + 1}` can be reduced to whether it is useful when
-/// we ignore all the patterns in `P` that involve other constructors. This is where
-/// `S(c, P)` comes in:
-/// `U(P, p_{m + 1}) := U(S(c, P), S(c, p_{m + 1}))`
+/// We're going to match on the top of the new pattern-stack, `p_1`.
+/// - If `p_1 == c(r_1, .., r_a)`, i.e. we have a constructor pattern.
+/// Then, the usefulness of `p_1` can be reduced to whether it is useful when
+/// we ignore all the patterns in the first column of `P` that involve other constructors.
+/// This is where `S(c, P)` comes in:
+/// `U(P, p) := U(S(c, P), S(c, p))`
/// This special case is handled in `is_useful_specialized`.
-/// - If `p_{m + 1} == _`, then we have two more cases:
-/// + All the constructors of the first component of the type exist within
-/// all the rows (after having expanded OR-patterns). In this case:
-/// `U(P, p_{m + 1}) := ∨(k ϵ constructors) U(S(k, P), S(k, p_{m + 1}))`
-/// I.e., the pattern `p_{m + 1}` is only useful when all the constructors are
-/// present *if* its later components are useful for the respective constructors
-/// covered by `p_{m + 1}` (usually a single constructor, but all in the case of `_`).
-/// + Some constructors are not present in the existing rows (after having expanded
-/// OR-patterns). However, there might be wildcard patterns (`_`) present. Thus, we
-/// are only really concerned with the other patterns leading with wildcards. This is
-/// where `D` comes in:
-/// `U(P, p_{m + 1}) := U(D(P), p_({m + 1},2), .., p_({m + 1},n))`
-/// - If `p_{m + 1} == r_1 | r_2`, then the usefulness depends on each separately:
-/// `U(P, p_{m + 1}) := U(P, (r_1, p_({m + 1},2), .., p_({m + 1},n)))
-/// || U(P, (r_2, p_({m + 1},2), .., p_({m + 1},n)))`
+///
+/// For example, if `P` is:
+/// [
+/// [Some(true), _],
+/// [None, 0],
+/// ]
+/// and `p` is [Some(false), 0], then we don't care about row 2 since we know `p` only
+/// matches values that row 2 doesn't. For row 1 however, we need to dig into the
+/// arguments of `Some` to know whether some new value is covered. So we compute
+/// `U([[true, _]], [false, 0])`.
+///
+/// - If `p_1 == _`, then we look at the list of constructors that appear in the first
+/// component of the rows of `P`:
+/// + If there are some constructors that aren't present, then we might think that the
+/// wildcard `_` is useful, since it covers those constructors that weren't covered
+/// before.
+/// That's almost correct, but only works if there were no wildcards in those first
+/// components. So we need to check that `p` is useful with respect to the rows that
+/// start with a wildcard, if there are any. This is where `D` comes in:
+/// `U(P, p) := U(D(P), D(p))`
+///
+/// For example, if `P` is:
+/// [
+/// [_, true, _],
+/// [None, false, 1],
+/// ]
+/// and `p` is [_, false, _], the `Some` constructor doesn't appear in `P`. So if we
+/// only had row 2, we'd know that `p` is useful. However row 1 starts with a
+/// wildcard, so we need to check whether `U([[true, _]], [false, 1])`.
+///
+/// + Otherwise, all possible constructors (for the relevant type) are present. In this
+/// case we must check whether the wildcard pattern covers any unmatched value. For
+/// that, we can think of the `_` pattern as a big OR-pattern that covers all
+/// possible constructors. For `Option`, that would mean `_ = None | Some(_)` for
+/// example. The wildcard pattern is useful in this case if it is useful when
+/// specialized to one of the possible constructors. So we compute:
+/// `U(P, p) := ∃(k ϵ constructors) U(S(k, P), S(k, p))`
+///
+/// For example, if `P` is:
+/// [
+/// [Some(true), _],
+/// [None, false],
+/// ]
+/// and `p` is [_, false], both `None` and `Some` constructors appear in the first
+/// components of `P`. We will therefore try popping both constructors in turn: we
+/// compute U([[true, _]], [_, false]) for the `Some` constructor, and U([[false]],
+/// [false]) for the `None` constructor. The first case returns true, so we know that
+/// `p` is useful for `P`. Indeed, it matches `[Some(false), _]` that wasn't matched
+/// before.
+///
+/// - If `p_1 == r_1 | r_2`, then the usefulness depends on each `r_i` separately:
+/// `U(P, p) := U(P, (r_1, p_2, .., p_n))
+/// || U(P, (r_2, p_2, .., p_n))`
///
/// Modifications to the algorithm
/// ------------------------------
/// The algorithm in the paper doesn't cover some of the special cases that arise in Rust, for
/// example uninhabited types and variable-length slice patterns. These are drawn attention to
-/// throughout the code below. I'll make a quick note here about how exhaustive integer matching
-/// is accounted for, though.
+/// throughout the code below. I'll make a quick note here about how exhaustive integer matching is
+/// accounted for, though.
///
/// Exhaustive integer matching
/// ---------------------------
/// invalid, because we want a disjunction over every *integer* in each range, not just a
/// disjunction over every range. This is a bit more tricky to deal with: essentially we need
/// to form equivalence classes of subranges of the constructor range for which the behaviour
-/// of the matrix `P` and new pattern `p_{m + 1}` are the same. This is described in more
+/// of the matrix `P` and new pattern `p` are the same. This is described in more
/// detail in `split_grouped_constructors`.
/// + If some constructors are missing from the matrix, it turns out we don't need to do
/// anything special (because we know none of the integers are actually wildcards: i.e., we
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