SIP - Sealed Types

_sips/sips/2021-05-18-sealed-types.md

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+---
+layout: sip
+title: Sealed Types
+vote-status: pending
+permalink: /sips/:title.html
+redirect_from: /sips/pending/sealed-types.html
+---
+
+**By: Dale Wijnand and Fengyun Liu**
+
+## History
+
+| Date          | Version   |
+|---------------|-----------|
+| May 18th 2021 | Submitted |
+
+## Introduction
+
+Exhaustivity checking is one of the safety belts for using pattern matching in functional
+programming.  However, if one wants to partition values of an existing type to reason about them in
+separate cases of a match or as separate types this requires wrapping the values in new classes,
+which incurs a boxing cost.
+
+As an alternative, one may define custom extractors and use them as the case patterns. However, there is no
+way in Scala to declare that these extractors are complementary, i.e. the match is exhaustive when
+the complete set of extractors are used together.
+
+Similarly, one can define `TypeTest`s for matching abstract types, but there is no way to determine
+if a match on an abstract type exhausts all its subtypes.
+
+This SIP solves these three problems by introducing *sealed types*.
+
+## Motivating Examples
+
+We've identified several real world use cases that calls for enhancing exhaustivity checking, and
+used them to stress the design proposed here.  You can find them [here][problems], but we'll present
+an example below.
+
+Using the opaque types and custom extractors we can work safely with positive integers and negative
+integers:
+
+```scala
+opaque type Pos <: Int = Int
+opaque type Neg <: Int = Int
+
+object Pos { def unapply(x: Int): Option[Pos] = if (x > 0) Some(x) else None }
+object Neg { def unapply(x: Int): Option[Neg] = if (x < 0) Some(x) else None }
+
+(n: Int) match
+  case 0      =>
+  case Pos(x) =>
+  case Neg(x) =>
+```
+
+With the above, when we get a `Pos` value, it's guaranteed to be a positive number. The same goes
+for `Neg`. Sadly the match is reported as not exhaustive because the two extractors and the value
+`0` aren't known to be complementary.
+
+## Design
+
+We identify two root causes:
+
+1. You can't define a type, that isn't a class, as `sealed`
+2. You can't define a mapping from values to types
+
+The *sealed type*, proposed by this SIP, allow partitioning of value of a given type into a sealed
+type hierarchy.  [Here][solutions] you can find how sealed types address the issues faced by all the
+motivating examples, but we'll present here how it fixes the number example above.
+
+In order to partition int into positive, zero, and negative numbers we'll define a new `sealed` type
+`Num` and how to distinguish its subtypes with match syntax:
+
+```scala
+sealed type Num = Int {
+  case 0          => val Zero
+  case n if n > 0 => type Pos
+  case _          => type Neg
+}
+```
+
+This sealed type definition desugars into the following type and value definitions:
+
+```scala
+type Num = Int
+val Zero: Num = 0
+opaque type Pos <: Num = Num
+opaque type Neg <: Num = Num
+```
+
+The match desugars into an ordinal method, that reuses the logic to associate an ordinal for each
+case:
+
+```scala
+extension (n: Num):
+  def ordinal: Int = (n: Int) match {
+    case 0          => 0
+    case n if n > 0 => 1
+    case _          => 2
+  }
+```
+
+Finally a series of `TypeTest`s are defined, allowing for values of both the underlying type `Int`
+and the sealed type `Num` to be tested against the subtypes and singleton subtypes `Pos`,
+`Zero.type`, and `Neg`, using the `ordinal` method:
+
+```scala
+given TypeTest[Int, Zero.type] = (n: Int) => if ((n: Num).ordinal == 0) Some(n) else None
+given TypeTest[Int, Pos]       = (n: Int) => if ((n: Num).ordinal == 1) Some(n) else None
+given TypeTest[Int, Neg]       = (n: Int) => if ((n: Num).ordinal == 2) Some(n) else None
+given TypeTest[Int, Num]       = (n: Int) => Some(n)
+given [T <: Num](using t: TypeTest[Int, T]): TypeTest[Num, T] = (n: Num) => t.unapply(n)
+```
+
+Given the above, one can either change the usage from extractors to types:
+
+```scala
+(n: Int) match
+  case 0      =>
+  case x: Pos =>
+  case x: Neg =>
+```
+
+Or we can keep the usage the same by redefining the extractors (using a value class name-based
+extractors `PosExtractor` and `NegExtractor` to avoid allocating):
+
+```scala
+object Pos { def unapply(x: Pos): PosExtractor = new PosExtractor(x) }
+object Neg { def unapply(x: Neg): NegExtractor = new NegExtractor(x) }
+
+class PosExtractor(private val x: Pos) extends AnyVal { def isEmpty: false = false ; def get = x }
+class NegExtractor(private val x: Neg) extends AnyVal { def isEmpty: false = false ; def get = x }
+
+(n: Int) match
+  case 0      =>
+  case Pos(x) =>
+  case Neg(x) =>
+```
+
+## Syntax
+
+The existing syntax is enhanced as follows:
+
+```
+TypeDcl ::= `sealed` [`opaque`] `type` id [TypeParamClause]
+TypeDef ::= `sealed` [`opaque`] `type` id [TypeParamClause] [`>:` Type] [`<:` Type] `=` Type `{`
+    `case` Pattern [Guard] `=>` (`type` id [TypeParamClause] | `val` id [`:` Type])
+  `}`
+```
+
+Specifically:
+
+* the `sealed` modifier becomes available for type definitions and declarations
+* on the right-hand side of definitions is the underlying type of the sealed type
+* following the underlying type is a match that operates on a value of the
+  underlying type and defines the type or singleton type associated to that case.
+* the type is defined using the `type` keyword and singleton types are defined using `val`
+
+## Desugaring
+
+Using the example
+
+```
+sealed [opaque] type T[X..] [bounds] = U {
+  case p1 => type C[X..]
+  case p2 => val S
+}
+```
+
+That desugars into:
+* a type alias `type T[X..] [bounds] = U`, `opaque` if the sealed type is `opaque` (see Restrictions)
+* opaque type definitions, `opaque type C[X..] <: T[Y..] = T[Y..]`, for each non-singleton type case
+  - any type argument `Y` that isn't defined from `X..` will be:
+    + its lower bound, if the type parameter is covariant
+    + its upper bound, if the type parameter is contravariant
+    + a wildcard type, with the type parameter's bounds, if the type parameter is invariant
+* val definitions, `val S: T[Y..] = p`, for singleton type cases, using the same type argument rules
+* an ordinal method, `extension (t: T): def ordinal: Int = (t: U) match { case p1 => 0 p2 => 1 .. }`
+  - each of the sealed type's cases is associated with a unique ordinal
+  - ordinals starts from 0 and increase for each case, in the order of their definition
+  - the ordinal method adds a `case _ => -1` default case, if the sealed type's match is inexhaustive
+  - such an ordinal method may only be defined by the compiler, to preserve exhaustivity guarantees
+* a series a `TypeTest`s, defined in terms of the ordinal method
+  - a `TypeTest` between `U` and each case `A`, having ordinal `ordA`:
+    + `given TypeTest[U, C]      = (u: U) => if ((u: T).ordinal == $ordA) Some(u) else None`
+    + `given TypeTest[U, S.type] = (u: U) => if ((u: T).ordinal == $ordA) Some(u) else None`
+  - a type test between the underlying type and the sealed type:
+    + `given TypeTest[U, T] = (u: U) => if ((u: T).ordinal == -1) None else Some(u)`
+  - a generic type test between `T` and each case `A`, defined in terms of the above type tests:
+    + `given [A <: T](using t: TypeTest[U, A]): TypeTest[T, A] = (x: T) => t.unapply(x)`
+
+## Restrictions
+
+1. If the match on the value of the underlying type is not exhaustive, then the sealed type must be
+   declared `opaque`, in order to preserve the fact that the sealed type represents only a subset of
+   the values of the underlying type (e.g. positive integers)
+2. No other type may be declared to subtype the opaque type `T`
+3. For singleton types, the pattern `p` must be a stable value, e.g. a `val`, a `case object`, or a literal
+4. Each case must define a new type or singleton type
+
+## Alternative Design
+
+An alternative design is to introduce an annotation `@complete` to specify that
+a type can be partitioned into a list of subtypes.
+
+For example, given the following definition:
+
+```scala
+  opaque type Nat <: Int = Int
+  opaque type Neg <: Int = Int
+
+  @complete[(Nat, Neg)] // Num decomposes to `Nat` and `Neg`
+  type Num = Int
+
+  given TypeTest[Int, Neg] = (n: Int) => if (x < 0) Some(n) else None
+  given TypeTest[Int, Nat] = (n: Int) => if (x >= 0) Some(n) else None
+```
+
+The user now can write code as follows:
+
+``` Scala
+  def foo(n: Num) =
+    n match
+    case x: Neg =>
+    case x: Nat =>
+```
+
+Knowing that the type `Num` can be decomposed to `Neg` and `Nat`, the compiler
+can verify that the pattern match above is exhaustive.
+
+This approach, however, is relatively low-level and the compiler does not
+provide any guarantee that the annotation is actually correct.
+
+You can find more examples [here][complete-gist]
+
+## Related Work
+
+Haskell has a `COMPLETE` pragma which allows patterns and type constructors to be
+defined to be a complete set, relying on the programmer getting it right.
+
+```haskell
+data Choice a = Choice Bool a
+
+pattern LeftChoice :: a -> Choice a
+pattern LeftChoice a = Choice False a
+
+pattern RightChoice :: a -> Choice a
+pattern RightChoice a = Choice True a
+
+{-# COMPLETE LeftChoice, RightChoice #-}
+
+foo :: Choice Int -> Int
+foo (LeftChoice n) = n * 2
+foo (RightChoice n) = n - 2
+```
+
+## References
+
+1. [Opaque types][1]
+2. [Forum discussion about Opt[T]][2]
+3. [Github discussion about enhancing exhaustivity check][3]
+4. [_Lightweight static capabilities_][4], Oleg Kiselyov, Chung-chieh Shan, 2007
+5. [TypeTest documentation][5]
+
+[1]: https://docs.scala-lang.org/sips/opaque-types.html
+[2]: https://contributors.scala-lang.org/t/trouble-with-2-13-4-exhaustivity-checking-being-too-strict/4817
+[3]: https://github.com/lampepfl/dotty/issues/10961
+[4]: http://okmij.org/ftp/Computation/lightweight-guarantees/lightweight-static-capabilities.pdf
+[5]: http://dotty.epfl.ch/docs/reference/other-new-features/type-test.html
+[6]: https://github.com/lampepfl/dotty/pull/11186
+[problems]:  https://gist.github.com/dwijnand/d33436cf197daa15216b3cd35d03ba1c#file-sealedtypeproblems-scala
+[solutions]: https://gist.github.com/dwijnand/d33436cf197daa15216b3cd35d03ba1c#file-sealedtypesolutions-scala
+[complete-gist]: https://gist.github.com/dwijnand/d33436cf197daa15216b3cd35d03ba1c#file-z-complete-scala
+
+* https://github.com/lampepfl/dotty/issues/10961 False “match may not be exhaustive warning”
+* https://github.com/lampepfl/dotty/pull/11186 Implement @covers annotation for partial irrefutable specification
+* https://downloads.haskell.org/~ghc/9.0.1/docs/html/users_guide/exts/pragmas.html#complete-pragma
+* https://downloads.haskell.org/~ghc/9.0.1/docs/html/users_guide/exts/pattern_synonyms.html
+* https://dotty.epfl.ch/docs/reference/other-new-features/opaques.html