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_match.rs: fix module doc comment

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It was applied to a `use` item, not to the module
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Jonas Schievink 2020-06-14 14:53:36 +02:00
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src/librustc_mir_build/hair/pattern/_match.rs
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@ -1,274 +1,274 @@
/// Note: most of the tests relevant to this file can be found (at the time of writing) in
/// src/tests/ui/pattern/usefulness.
///
/// This file includes the logic for exhaustiveness and usefulness checking for
/// pattern-matching. Specifically, given a list of patterns for a type, we can
/// tell whether:
/// (a) the patterns cover every possible constructor for the type [exhaustiveness]
/// (b) each pattern is necessary [usefulness]
///
/// The algorithm implemented here is a modified version of the one described in:
/// http://moscova.inria.fr/~maranget/papers/warn/index.html
/// However, to save future implementors from reading the original paper, we
/// summarise the algorithm here to hopefully save time and be a little clearer
/// (without being so rigorous).
///
/// # Premise
///
/// The core of the algorithm revolves about a "usefulness" check. In particular, we
/// 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
/// pattern to those that have come before it doesn't increase the number of values
/// we're matching).
///
/// # Core concept
///
/// The idea that powers everything that is done in this file is the following: a value is made
/// from a constructor applied to some fields. Examples of constructors are `Some`, `None`, `(,)`
/// (the 2-tuple constructor), `Foo {..}` (the constructor for a struct `Foo`), and `2` (the
/// constructor for the number `2`). Fields are just a (possibly empty) list of values.
///
/// Some of the constructors listed above might feel weird: `None` and `2` don't take any
/// arguments. This is part of what makes constructors so general: we will consider plain values
/// like numbers and string literals to be constructors that take no arguments, also called "0-ary
/// constructors"; they are the simplest case of constructors. This allows us to see any value as
/// made up from a tree of constructors, each having a given number of children. For example:
/// `(None, Ok(0))` is made from 4 different constructors.
///
/// This idea can be extended to patterns: a pattern captures a set of possible values, and we can
/// describe this set using constructors. For example, `Err(_)` captures all values of the type
/// `Result<T, E>` that start with the `Err` constructor (for some choice of `T` and `E`). The
/// wildcard `_` captures all values of the given type starting with any of the constructors for
/// that type.
///
/// We use this to compute whether different patterns might capture a same value. Do the patterns
/// `Ok("foo")` and `Err(_)` capture a common value? The answer is no, because the first pattern
/// captures only values starting with the `Ok` constructor and the second only values starting
/// with the `Err` constructor. Do the patterns `Some(42)` and `Some(1..10)` intersect? They might,
/// since they both capture values starting with `Some`. To be certain, we need to dig under the
/// `Some` constructor and continue asking the question. This is the main idea behind the
/// exhaustiveness algorithm: by looking at patterns constructor-by-constructor, we can efficiently
/// figure out if some new pattern might capture a value that hadn't been captured by previous
/// patterns.
///
/// Constructors are represented by the `Constructor` enum, and its fields by the `Fields` enum.
/// Most of the complexity of this file resides in transforming between patterns and
/// (`Constructor`, `Fields`) pairs, handling all the special cases correctly.
///
/// Caveat: this constructors/fields distinction doesn't quite cover every Rust value. For example
/// a value of type `Rc<u64>` doesn't fit this idea very well, nor do various other things.
/// However, this idea covers most of the cases that are relevant to exhaustiveness checking.
///
///
/// # Algorithm
///
/// Recall that `U(P, p)` represents whether, given an existing list of patterns (aka matrix) `P`,
/// adding a new pattern `p` will cover previously-uncovered values of the type.
/// 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 fields. 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<()>)
/// match x {
/// (Some(true), _) => {}
/// (None, Err(())) => {}
/// (None, Err(_)) => {}
/// }
/// ```
/// Here, the matrix `P` starts as:
/// [
/// [(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
/// all the values it covers are already covered by row 2.
///
/// 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.
///
/// 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. 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. 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.
///
///
/// 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-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)` 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 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`.
///
/// 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.
///
/// Exhaustive integer matching
/// ---------------------------
/// An integer type can be thought of as a (huge) sum type: 1 | 2 | 3 | ...
/// So to support exhaustive integer matching, we can make use of the logic in the paper for
/// OR-patterns. However, we obviously can't just treat ranges x..=y as individual sums, because
/// they are likely gigantic. So we instead treat ranges as constructors of the integers. This means
/// that we have a constructor *of* constructors (the integers themselves). We then need to work
/// through all the inductive step rules above, deriving how the ranges would be treated as
/// OR-patterns, and making sure that they're treated in the same way even when they're ranges.
/// There are really only four special cases here:
/// - When we match on a constructor that's actually a range, we have to treat it as if we would
/// an OR-pattern.
/// + It turns out that we can simply extend the case for single-value patterns in
/// `specialize` to either be *equal* to a value constructor, or *contained within* a range
/// constructor.
/// + When the pattern itself is a range, you just want to tell whether any of the values in
/// the pattern range coincide with values in the constructor range, which is precisely
/// intersection.
/// Since when encountering a range pattern for a value constructor, we also use inclusion, it
/// means that whenever the constructor is a value/range and the pattern is also a value/range,
/// we can simply use intersection to test usefulness.
/// - When we're testing for usefulness of a pattern and the pattern's first component is a
/// wildcard.
/// + If all the constructors appear in the matrix, we have a slight complication. By default,
/// the behaviour (i.e., a disjunction over specialised matrices for each constructor) is
/// 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` 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
/// can't span wildcards using ranges).
//! Note: most of the tests relevant to this file can be found (at the time of writing) in
//! src/tests/ui/pattern/usefulness.
//!
//! This file includes the logic for exhaustiveness and usefulness checking for
//! pattern-matching. Specifically, given a list of patterns for a type, we can
//! tell whether:
//! (a) the patterns cover every possible constructor for the type [exhaustiveness]
//! (b) each pattern is necessary [usefulness]
//!
//! The algorithm implemented here is a modified version of the one described in:
//! http://moscova.inria.fr/~maranget/papers/warn/index.html
//! However, to save future implementors from reading the original paper, we
//! summarise the algorithm here to hopefully save time and be a little clearer
//! (without being so rigorous).
//!
//! # Premise
//!
//! The core of the algorithm revolves about a "usefulness" check. In particular, we
//! 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
//! pattern to those that have come before it doesn't increase the number of values
//! we're matching).
//!
//! # Core concept
//!
//! The idea that powers everything that is done in this file is the following: a value is made
//! from a constructor applied to some fields. Examples of constructors are `Some`, `None`, `(,)`
//! (the 2-tuple constructor), `Foo {..}` (the constructor for a struct `Foo`), and `2` (the
//! constructor for the number `2`). Fields are just a (possibly empty) list of values.
//!
//! Some of the constructors listed above might feel weird: `None` and `2` don't take any
//! arguments. This is part of what makes constructors so general: we will consider plain values
//! like numbers and string literals to be constructors that take no arguments, also called "0-ary
//! constructors"; they are the simplest case of constructors. This allows us to see any value as
//! made up from a tree of constructors, each having a given number of children. For example:
//! `(None, Ok(0))` is made from 4 different constructors.
//!
//! This idea can be extended to patterns: a pattern captures a set of possible values, and we can
//! describe this set using constructors. For example, `Err(_)` captures all values of the type
//! `Result<T, E>` that start with the `Err` constructor (for some choice of `T` and `E`). The
//! wildcard `_` captures all values of the given type starting with any of the constructors for
//! that type.
//!
//! We use this to compute whether different patterns might capture a same value. Do the patterns
//! `Ok("foo")` and `Err(_)` capture a common value? The answer is no, because the first pattern
//! captures only values starting with the `Ok` constructor and the second only values starting
//! with the `Err` constructor. Do the patterns `Some(42)` and `Some(1..10)` intersect? They might,
//! since they both capture values starting with `Some`. To be certain, we need to dig under the
//! `Some` constructor and continue asking the question. This is the main idea behind the
//! exhaustiveness algorithm: by looking at patterns constructor-by-constructor, we can efficiently
//! figure out if some new pattern might capture a value that hadn't been captured by previous
//! patterns.
//!
//! Constructors are represented by the `Constructor` enum, and its fields by the `Fields` enum.
//! Most of the complexity of this file resides in transforming between patterns and
//! (`Constructor`, `Fields`) pairs, handling all the special cases correctly.
//!
//! Caveat: this constructors/fields distinction doesn't quite cover every Rust value. For example
//! a value of type `Rc<u64>` doesn't fit this idea very well, nor do various other things.
//! However, this idea covers most of the cases that are relevant to exhaustiveness checking.
//!
//!
//! # Algorithm
//!
//! Recall that `U(P, p)` represents whether, given an existing list of patterns (aka matrix) `P`,
//! adding a new pattern `p` will cover previously-uncovered values of the type.
//! 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 fields. 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<()>)
//! match x {
//! (Some(true), _) => {}
//! (None, Err(())) => {}
//! (None, Err(_)) => {}
//! }
//! ```
//! Here, the matrix `P` starts as:
//! [
//! [(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
//! all the values it covers are already covered by row 2.
//!
//! 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.
//!
//! 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. 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. 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.
//!
//!
//! 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-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)` 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 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`.
//!
//! 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.
//!
//! Exhaustive integer matching
//! ---------------------------
//! An integer type can be thought of as a (huge) sum type: 1 | 2 | 3 | ...
//! So to support exhaustive integer matching, we can make use of the logic in the paper for
//! OR-patterns. However, we obviously can't just treat ranges x..=y as individual sums, because
//! they are likely gigantic. So we instead treat ranges as constructors of the integers. This means
//! that we have a constructor *of* constructors (the integers themselves). We then need to work
//! through all the inductive step rules above, deriving how the ranges would be treated as
//! OR-patterns, and making sure that they're treated in the same way even when they're ranges.
//! There are really only four special cases here:
//! - When we match on a constructor that's actually a range, we have to treat it as if we would
//! an OR-pattern.
//! + It turns out that we can simply extend the case for single-value patterns in
//! `specialize` to either be *equal* to a value constructor, or *contained within* a range
//! constructor.
//! + When the pattern itself is a range, you just want to tell whether any of the values in
//! the pattern range coincide with values in the constructor range, which is precisely
//! intersection.
//! Since when encountering a range pattern for a value constructor, we also use inclusion, it
//! means that whenever the constructor is a value/range and the pattern is also a value/range,
//! we can simply use intersection to test usefulness.
//! - When we're testing for usefulness of a pattern and the pattern's first component is a
//! wildcard.
//! + If all the constructors appear in the matrix, we have a slight complication. By default,
//! the behaviour (i.e., a disjunction over specialised matrices for each constructor) is
//! 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` 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
//! can't span wildcards using ranges).
use self::Constructor::*;
use self::SliceKind::*;
use self::Usefulness::*;