Clarify and fix the explanation of the algorithm
There was a bit of confusion between individual patterns and lists of patterns, and index mismatches linked to that. This introduces a vocabulary of "pattern-stacks" to provide a clearer mental model of what is happening. This also adds examples.
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Nadrieril
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@ -11,20 +11,24 @@
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/// (without being so rigorous).
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///
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/// The core of the algorithm revolves about a "usefulness" check. In particular, we
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/// are trying to compute a predicate `U(P, p_{m + 1})` where `P` is a list of patterns
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/// of length `m` for a compound (product) type with `n` components (we refer to this as
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/// a matrix). `U(P, p_{m + 1})` represents whether, given an existing list of patterns
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/// `p_1 ..= p_m`, adding a new pattern will be "useful" (that is, cover previously-
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/// are trying to compute a predicate `U(P, p)` where `P` is a list of patterns (we refer to this as
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/// a matrix). `U(P, p)` represents whether, given an existing list of patterns
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/// `P_1 ..= P_m`, adding a new pattern `p` will be "useful" (that is, cover previously-
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/// uncovered values of the type).
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///
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/// If we have this predicate, then we can easily compute both exhaustiveness of an
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/// entire set of patterns and the individual usefulness of each one.
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/// (a) the set of patterns is exhaustive iff `U(P, _)` is false (i.e., adding a wildcard
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/// match doesn't increase the number of values we're matching)
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/// (b) a pattern `p_i` is not useful if `U(P[0..=(i-1), p_i)` is false (i.e., adding a
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/// (b) a pattern `P_i` is not useful if `U(P[0..=(i-1), P_i)` is false (i.e., adding a
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/// pattern to those that have come before it doesn't increase the number of values
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/// we're matching).
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///
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/// During the course of the algorithm, the rows of the matrix won't just be individual patterns,
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/// but rather partially-deconstructed patterns in the form of a list of patterns. The paper
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/// calls those pattern-vectors, and we will call them pattern-stacks. The same holds for the
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/// new pattern `p`.
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///
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/// For example, say we have the following:
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/// ```
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/// // x: (Option<bool>, Result<()>)
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@ -34,93 +38,155 @@
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/// (None, Err(_)) => {}
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/// }
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/// ```
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/// Here, the matrix `P` is 3 x 2 (rows x columns).
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/// Here, the matrix `P` starts as:
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/// [
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/// [Some(true), _],
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/// [None, Err(())],
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/// [None, Err(_)],
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/// [(Some(true), _)],
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/// [(None, Err(()))],
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/// [(None, Err(_))],
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/// ]
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/// We can tell it's not exhaustive, because `U(P, _)` is true (we're not covering
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/// `[Some(false), _]`, for instance). In addition, row 3 is not useful, because
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/// `[(Some(false), _)]`, for instance). In addition, row 3 is not useful, because
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/// all the values it covers are already covered by row 2.
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///
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/// To compute `U`, we must have two other concepts.
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/// 1. `S(c, P)` is a "specialized matrix", where `c` is a constructor (like `Some` or
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/// `None`). You can think of it as filtering `P` to just the rows whose *first* pattern
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/// can cover `c` (and expanding OR-patterns into distinct patterns), and then expanding
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/// the constructor into all of its components.
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/// The specialization of a row vector is computed by `specialize`.
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/// A list of patterns can be thought of as a stack, because we are mainly interested in the top of
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/// the stack at any given point, and we can pop or apply constructors to get new pattern-stacks.
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/// To match the paper, the top of the stack is at the beginning / on the left.
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///
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/// It is computed as follows. For each row `p_i` of P, we have four cases:
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/// 1.1. `p_(i,1) = c(r_1, .., r_a)`. Then `S(c, P)` has a corresponding row:
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/// r_1, .., r_a, p_(i,2), .., p_(i,n)
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/// 1.2. `p_(i,1) = c'(r_1, .., r_a')` where `c ≠ c'`. Then `S(c, P)` has no
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/// corresponding row.
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/// 1.3. `p_(i,1) = _`. Then `S(c, P)` has a corresponding row:
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/// _, .., _, p_(i,2), .., p_(i,n)
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/// 1.4. `p_(i,1) = r_1 | r_2`. Then `S(c, P)` has corresponding rows inlined from:
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/// S(c, (r_1, p_(i,2), .., p_(i,n)))
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/// S(c, (r_2, p_(i,2), .., p_(i,n)))
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/// There are two important operations on pattern-stacks necessary to understand the algorithm:
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/// 1. We can pop a given constructor off the top of a stack. This operation is called
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/// `specialize`, and is denoted `S(c, p)` where `c` is a constructor (like `Some` or
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/// `None`) and `p` a pattern-stack.
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/// If the pattern on top of the stack can cover `c`, this removes the constructor and
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/// pushes its arguments onto the stack. It also expands OR-patterns into distinct patterns.
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/// Otherwise the pattern-stack is discarded.
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/// This essentially filters those pattern-stacks whose top covers the constructor `c` and
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/// discards the others.
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///
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/// 2. `D(P)` is a "default matrix". This is used when we know there are missing
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/// constructor cases, but there might be existing wildcard patterns, so to check the
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/// usefulness of the matrix, we have to check all its *other* components.
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/// The default matrix is computed inline in `is_useful`.
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/// For example, the first pattern above initially gives a stack `[(Some(true), _)]`. If we
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/// pop the tuple constructor, we are left with `[Some(true), _]`, and if we then pop the
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/// `Some` constructor we get `[true, _]`. If we had popped `None` instead, we would get
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/// nothing back.
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///
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/// It is computed as follows. For each row `p_i` of P, we have three cases:
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/// 1.1. `p_(i,1) = c(r_1, .., r_a)`. Then `D(P)` has no corresponding row.
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/// 1.2. `p_(i,1) = _`. Then `D(P)` has a corresponding row:
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/// p_(i,2), .., p_(i,n)
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/// 1.3. `p_(i,1) = r_1 | r_2`. Then `D(P)` has corresponding rows inlined from:
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/// D((r_1, p_(i,2), .., p_(i,n)))
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/// D((r_2, p_(i,2), .., p_(i,n)))
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/// This returns zero or more new pattern-stacks, as follows. We look at the pattern `p_1`
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/// on top of the stack, and we have four cases:
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/// 1.1. `p_1 = c(r_1, .., r_a)`, i.e. the top of the stack has constructor `c`. We
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/// push onto the stack the arguments of this constructor, and return the result:
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/// r_1, .., r_a, p_2, .., p_n
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/// 1.2. `p_1 = c'(r_1, .., r_a')` where `c ≠ c'`. We discard the current stack and
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/// return nothing.
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/// 1.3. `p_1 = _`. We push onto the stack as many wildcards as the constructor `c` has
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/// arguments (its arity), and return the resulting stack:
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/// _, .., _, p_2, .., p_n
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/// 1.4. `p_1 = r_1 | r_2`. We expand the OR-pattern and then recurse on each resulting
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/// stack:
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/// S(c, (r_1, p_2, .., p_n))
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/// S(c, (r_2, p_2, .., p_n))
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///
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/// 2. We can pop a wildcard off the top of the stack. This is called `D(p)`, where `p` is
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/// a pattern-stack.
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/// This is used when we know there are missing constructor cases, but there might be
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/// existing wildcard patterns, so to check the usefulness of the matrix, we have to check
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/// all its *other* components.
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///
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/// It is computed as follows. We look at the pattern `p_1` on top of the stack,
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/// and we have three cases:
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/// 1.1. `p_1 = c(r_1, .., r_a)`. We discard the current stack and return nothing.
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/// 1.2. `p_1 = _`. We return the rest of the stack:
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/// p_2, .., p_n
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/// 1.3. `p_1 = r_1 | r_2`. We expand the OR-pattern and then recurse on each resulting
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/// stack.
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/// D((r_1, p_2, .., p_n))
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/// D((r_2, p_2, .., p_n))
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///
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/// Note that the OR-patterns are not always used directly in Rust, but are used to derive the
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/// exhaustive integer matching rules, so they're written here for posterity.
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///
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/// Both those operations extend straightforwardly to a list or pattern-stacks, i.e. a matrix, by
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/// working row-by-row. Popping a constructor ends up keeping only the matrix rows that start with
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/// the given constructor, and popping a wildcard keeps those rows that start with a wildcard.
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///
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/// Note that the OR-patterns are not always used directly in Rust, but are used to derive
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/// the exhaustive integer matching rules, so they're written here for posterity.
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///
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/// The algorithm for computing `U`
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/// -------------------------------
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/// The algorithm is inductive (on the number of columns: i.e., components of tuple patterns).
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/// That means we're going to check the components from left-to-right, so the algorithm
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/// operates principally on the first component of the matrix and new pattern `p_{m + 1}`.
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/// operates principally on the first component of the matrix and new pattern-stack `p`.
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/// This algorithm is realised in the `is_useful` function.
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///
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/// Base case. (`n = 0`, i.e., an empty tuple pattern)
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/// - If `P` already contains an empty pattern (i.e., if the number of patterns `m > 0`),
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/// then `U(P, p_{m + 1})` is false.
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/// - Otherwise, `P` must be empty, so `U(P, p_{m + 1})` is true.
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/// then `U(P, p)` is false.
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/// - Otherwise, `P` must be empty, so `U(P, p)` is true.
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///
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/// Inductive step. (`n > 0`, i.e., whether there's at least one column
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/// [which may then be expanded into further columns later])
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/// We're going to match on the new pattern, `p_{m + 1}`.
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/// - If `p_{m + 1} == c(r_1, .., r_a)`, then we have a constructor pattern.
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/// Thus, the usefulness of `p_{m + 1}` can be reduced to whether it is useful when
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/// we ignore all the patterns in `P` that involve other constructors. This is where
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/// `S(c, P)` comes in:
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/// `U(P, p_{m + 1}) := U(S(c, P), S(c, p_{m + 1}))`
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/// We're going to match on the top of the new pattern-stack, `p_1`.
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/// - If `p_1 == c(r_1, .., r_a)`, i.e. we have a constructor pattern.
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/// Then, the usefulness of `p_1` can be reduced to whether it is useful when
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/// we ignore all the patterns in the first column of `P` that involve other constructors.
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/// This is where `S(c, P)` comes in:
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/// `U(P, p) := U(S(c, P), S(c, p))`
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/// This special case is handled in `is_useful_specialized`.
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/// - If `p_{m + 1} == _`, then we have two more cases:
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/// + All the constructors of the first component of the type exist within
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/// all the rows (after having expanded OR-patterns). In this case:
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/// `U(P, p_{m + 1}) := ∨(k ϵ constructors) U(S(k, P), S(k, p_{m + 1}))`
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/// I.e., the pattern `p_{m + 1}` is only useful when all the constructors are
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/// present *if* its later components are useful for the respective constructors
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/// covered by `p_{m + 1}` (usually a single constructor, but all in the case of `_`).
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/// + Some constructors are not present in the existing rows (after having expanded
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/// OR-patterns). However, there might be wildcard patterns (`_`) present. Thus, we
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/// are only really concerned with the other patterns leading with wildcards. This is
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/// where `D` comes in:
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/// `U(P, p_{m + 1}) := U(D(P), p_({m + 1},2), .., p_({m + 1},n))`
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/// - If `p_{m + 1} == r_1 | r_2`, then the usefulness depends on each separately:
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/// `U(P, p_{m + 1}) := U(P, (r_1, p_({m + 1},2), .., p_({m + 1},n)))
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/// || U(P, (r_2, p_({m + 1},2), .., p_({m + 1},n)))`
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///
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/// For example, if `P` is:
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/// [
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/// [Some(true), _],
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/// [None, 0],
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/// ]
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/// and `p` is [Some(false), 0], then we don't care about row 2 since we know `p` only
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/// matches values that row 2 doesn't. For row 1 however, we need to dig into the
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/// arguments of `Some` to know whether some new value is covered. So we compute
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/// `U([[true, _]], [false, 0])`.
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///
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/// - If `p_1 == _`, then we look at the list of constructors that appear in the first
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/// component of the rows of `P`:
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/// + If there are some constructors that aren't present, then we might think that the
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/// wildcard `_` is useful, since it covers those constructors that weren't covered
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/// before.
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/// That's almost correct, but only works if there were no wildcards in those first
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/// components. So we need to check that `p` is useful with respect to the rows that
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/// start with a wildcard, if there are any. This is where `D` comes in:
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/// `U(P, p) := U(D(P), D(p))`
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///
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/// For example, if `P` is:
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/// [
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/// [_, true, _],
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/// [None, false, 1],
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/// ]
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/// and `p` is [_, false, _], the `Some` constructor doesn't appear in `P`. So if we
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/// only had row 2, we'd know that `p` is useful. However row 1 starts with a
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/// wildcard, so we need to check whether `U([[true, _]], [false, 1])`.
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///
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/// + Otherwise, all possible constructors (for the relevant type) are present. In this
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/// case we must check whether the wildcard pattern covers any unmatched value. For
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/// that, we can think of the `_` pattern as a big OR-pattern that covers all
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/// possible constructors. For `Option`, that would mean `_ = None | Some(_)` for
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/// example. The wildcard pattern is useful in this case if it is useful when
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/// specialized to one of the possible constructors. So we compute:
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/// `U(P, p) := ∃(k ϵ constructors) U(S(k, P), S(k, p))`
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///
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/// For example, if `P` is:
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/// [
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/// [Some(true), _],
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/// [None, false],
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/// ]
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/// and `p` is [_, false], both `None` and `Some` constructors appear in the first
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/// components of `P`. We will therefore try popping both constructors in turn: we
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/// compute U([[true, _]], [_, false]) for the `Some` constructor, and U([[false]],
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/// [false]) for the `None` constructor. The first case returns true, so we know that
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/// `p` is useful for `P`. Indeed, it matches `[Some(false), _]` that wasn't matched
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/// before.
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///
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/// - If `p_1 == r_1 | r_2`, then the usefulness depends on each `r_i` separately:
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/// `U(P, p) := U(P, (r_1, p_2, .., p_n))
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/// || U(P, (r_2, p_2, .., p_n))`
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///
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/// Modifications to the algorithm
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/// ------------------------------
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/// The algorithm in the paper doesn't cover some of the special cases that arise in Rust, for
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/// example uninhabited types and variable-length slice patterns. These are drawn attention to
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/// throughout the code below. I'll make a quick note here about how exhaustive integer matching
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/// is accounted for, though.
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/// throughout the code below. I'll make a quick note here about how exhaustive integer matching is
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/// accounted for, though.
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///
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/// Exhaustive integer matching
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/// ---------------------------
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@ -150,7 +216,7 @@
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/// invalid, because we want a disjunction over every *integer* in each range, not just a
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/// disjunction over every range. This is a bit more tricky to deal with: essentially we need
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/// to form equivalence classes of subranges of the constructor range for which the behaviour
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/// of the matrix `P` and new pattern `p_{m + 1}` are the same. This is described in more
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/// of the matrix `P` and new pattern `p` are the same. This is described in more
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/// detail in `split_grouped_constructors`.
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/// + If some constructors are missing from the matrix, it turns out we don't need to do
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/// anything special (because we know none of the integers are actually wildcards: i.e., we
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