scala 2.11.x/spec/03-types.md

scala/spec/03-types.md

title: Types

layout: default

chapter: 3

Types

  Type              ::=  FunctionArgTypes ‘=>’ Type
                      |  InfixType [ExistentialClause]
  FunctionArgTypes  ::=  InfixType
                      |  ‘(’ [ ParamType {‘,’ ParamType } ] ‘)’
  ExistentialClause ::=  ‘forSome’ ‘{’ ExistentialDcl
                             {semi ExistentialDcl} ‘}’
  ExistentialDcl    ::=  ‘type’ TypeDcl
                      |  ‘val’ ValDcl
  InfixType         ::=  CompoundType {id [nl] CompoundType}
  CompoundType      ::=  AnnotType {‘with’ AnnotType} [Refinement]
                      |  Refinement
  AnnotType         ::=  SimpleType {Annotation}
  SimpleType        ::=  SimpleType TypeArgs
                      |  SimpleType ‘#’ id
                      |  StableId
                      |  Path ‘.’ ‘type’
                      |  ‘(’ Types ‘)’
  TypeArgs          ::=  ‘[’ Types ‘]’
  Types             ::=  Type {‘,’ Type}

We distinguish between first-order types and type constructors, which

take type parameters and yield types. A subset of first-order types

called value types represents sets of (first-class) values.

Value types are either concrete or abstract.

Every concrete value type can be represented as a class type, i.e. a

type designator that refers to a

class or a trait ^[1], or as a

compound type representing an

intersection of types, possibly with a refinement

that further constrains the types of its members.

Abstract value types are introduced by type parameters

and abstract type bindings.

Parentheses in types can be used for grouping.

Non-value types capture properties of identifiers that

are not values. For example, a

type constructor does not directly specify a type of

values. However, when a type constructor is applied to the correct type

arguments, it yields a first-order type, which may be a value type.

Non-value types are expressed indirectly in Scala. E.g., a method type is

described by writing down a method signature, which in itself is not a real

type, although it gives rise to a corresponding method type.

Type constructors are another example, as one can write

type Swap[m[_, _], a,b] = m[b, a], but there is no syntax to write

the corresponding anonymous type function directly.

Paths

Path            ::=  StableId
                  |  [id ‘.’] this
StableId        ::=  id
                  |  Path ‘.’ id
                  |  [id ‘.’] ‘super’ [ClassQualifier] ‘.’ id
ClassQualifier  ::= ‘[’ id ‘]’

Paths are not types themselves, but they can be a part of named types

and in that function form a central role in Scala's type system.

A path is one of the following.

• The empty path ε (which cannot be written explicitly in user programs).
• \(C.\)this, where \(C\) references a class.
  
  The path this is taken as a shorthand for \(C.\)this where
  
  \(C\) is the name of the class directly enclosing the reference.
• \(p.x\) where \(p\) is a path and \(x\) is a stable member of \(p\).
  
  Stable members are packages or members introduced by object definitions or
  
  by value definitions of non-volatile types.
• \(C.\)super\(.x\) or \(C.\)super\([M].x\)
  
  where \(C\) references a class and \(x\) references a
  
  stable member of the super class or designated parent class \(M\) of \(C\).
  
  The prefix super is taken as a shorthand for \(C.\)super where
  
  \(C\) is the name of the class directly enclosing the reference.

A stable identifier is a path which ends in an identifier.

Value Types

Every value in Scala has a type which is of one of the following

forms.

Singleton Types

SimpleType  ::=  Path ‘.’ type

A singleton type is of the form \(p.\)type, where \(p\) is a

path pointing to a value expected to conform

to scala.AnyRef. The type denotes the set of values

consisting of null and the value denoted by \(p\).

A stable type is either a singleton type or a type which is

declared to be a subtype of trait scala.Singleton.

Type Projection

SimpleType  ::=  SimpleType ‘#’ id

A type projection \(T\)#\(x\) references the type member named

\(x\) of type \(T\).

Type Designators

SimpleType  ::=  StableId

A type designator refers to a named value type. It can be simple or

qualified. All such type designators are shorthands for type projections.

Specifically, the unqualified type name \(t\) where \(t\) is bound in some

class, object, or package \(C\) is taken as a shorthand for

\(C.\)this.type#\(t\). If \(t\) is

not bound in a class, object, or package, then \(t\) is taken as a

shorthand for ε.type#\(t\).

A qualified type designator has the form p.t where p is

a path and t is a type name. Such a type designator is

equivalent to the type projection p.type#t.

Example

Some type designators and their expansions are listed below. We assume

a local type parameter \(t\), a value maintable

with a type member Node and the standard class scala.Int,

Designator	Expansion
t	ε.type#t
Int	scala.type#Int
scala.Int	scala.type#Int
data.maintable.Node	data.maintable.type#Node

Parameterized Types

SimpleType      ::=  SimpleType TypeArgs
TypeArgs        ::=  ‘[’ Types ‘]’

A parameterized type \(T[ T_1 , \ldots , T_n ]\) consists of a type

designator \(T\) and type parameters \(T_1 , \ldots , T_n\) where

\(n \geq 1\). \(T\) must refer to a type constructor which takes \(n\) type

parameters \(a_1 , \ldots , a_n\).

Say the type parameters have lower bounds \(L_1 , \ldots , L_n\) and

upper bounds \(U_1, \ldots, U_n\). The parameterized type is

well-formed if each actual type parameter

conforms to its bounds, i.e. \(\sigma L_i <: T_i <: \sigma U_i\) where \(\sigma\) is the

substitution \([ a_1 := T_1 , \ldots , a_n := T_n ]\).

Example Parameterized Types

Given the partial type definitions:

class TreeMap[A <: Comparable[A], B] { … }
class List[A] { … }
class I extends Comparable[I] { … }

class F[M[_], X] { … }
class S[K <: String] { … }
class G[M[ Z <: I ], I] { … }

the following parameterized types are well formed:

TreeMap[I, String]
List[I]
List[List[Boolean]]

F[List, Int]
G[S, String]

Example

Given the above type definitions,

the following types are ill-formed:

TreeMap[I]            // illegal: wrong number of parameters
TreeMap[List[I], Int] // illegal: type parameter not within bound

F[Int, Boolean]       // illegal: Int is not a type constructor
F[TreeMap, Int]       // illegal: TreeMap takes two parameters,
                      //   F expects a constructor taking one
G[S, Int]             // illegal: S constrains its parameter to
                      //   conform to String,
                      // G expects type constructor with a parameter
                      //   that conforms to Int

Tuple Types

SimpleType    ::=   ‘(’ Types ‘)’

A tuple type \((T_1 , \ldots , T_n)\) is an alias for the

class scala.Tuple$n$[$T_1$, … , $T_n$], where \(n \geq 2\).

Tuple classes are case classes whose fields can be accessed using

selectors _1 , … , _n. Their functionality is

abstracted in a corresponding Product trait. The n-ary tuple

class and product trait are defined at least as follows in the

standard Scala library (they might also add other methods and

implement other traits).

case class Tuple$n$[+$T_1$, … , +$T_n$](_1: $T_1$, … , _n: $T_n$)
extends Product_n[$T_1$, … , $T_n$]

trait Product_n[+$T_1$, … , +$T_n$] {
  override def productArity = $n$
  def _1: $T_1$
  …
  def _n: $T_n$
}

Annotated Types

AnnotType  ::=  SimpleType {Annotation}

An annotated type \(T\) \(a_1, \ldots, a_n\)

attaches annotations

\(a_1 , \ldots , a_n\) to the type \(T\).

Example

The following type adds the @suspendable annotation to the type String:

String @suspendable

Compound Types

CompoundType    ::=  AnnotType {‘with’ AnnotType} [Refinement]
                  |  Refinement
Refinement      ::=  [nl] ‘{’ RefineStat {semi RefineStat} ‘}’
RefineStat      ::=  Dcl
                  |  ‘type’ TypeDef
                  |

A compound type \(T_1\) with … with \(T_n \\{ R \\}\)

represents objects with members as given in the component types

\(T_1 , \ldots , T_n\) and the refinement \(\\{ R \\}\). A refinement

\(\\{ R \\}\) contains declarations and type definitions.

If a declaration or definition overrides a declaration or definition in

one of the component types \(T_1 , \ldots , T_n\), the usual rules for

overriding apply; otherwise the declaration

or definition is said to be “structural” ^[2].

Within a method declaration in a structural refinement, the type of

any value parameter may only refer to type parameters or abstract

types that are contained inside the refinement. That is, it must refer

either to a type parameter of the method itself, or to a type

definition within the refinement. This restriction does not apply to

the method's result type.

If no refinement is given, the empty refinement is implicitly added,

i.e. \(T_1\) with … with \(T_n\) is a shorthand for \(T_1\) with … with \(T_n \\{\\}\).

A compound type may also consist of just a refinement

\(\\{ R \\}\) with no preceding component types. Such a type is

equivalent to AnyRef \(\\{ R \\}\).

Example

The following example shows how to declare and use a method which has

a parameter type that contains a refinement with structural declarations.

case class Bird (val name: String) extends Object {
        def fly(height: Int) = …
…
}
case class Plane (val callsign: String) extends Object {
        def fly(height: Int) = …
…
}
def takeoff(
            runway: Int,
      r: { val callsign: String; def fly(height: Int) }) = {
  tower.print(r.callsign + " requests take-off on runway " + runway)
  tower.read(r.callsign + " is clear for take-off")
  r.fly(1000)
}
val bird = new Bird("Polly the parrot"){ val callsign = name }
val a380 = new Plane("TZ-987")
takeoff(42, bird)
takeoff(89, a380)

Although Bird and Plane do not share any parent class other than

Object, the parameter r of method takeoff is defined using a

refinement with structural declarations to accept any object that declares

a value callsign and a fly method.

Infix Types

InfixType     ::=  CompoundType {id [nl] CompoundType}

An infix type \(T_1\) op \(T_2\) consists of an infix

operator op which gets applied to two type operands \(T_1\) and

\(T_2\). The type is equivalent to the type application

op\([T_1, T_2]\). The infix operator op may be an

arbitrary identifier.

All type infix operators have the same precedence; parentheses have to

be used for grouping. The associativity

of a type operator is determined as for term operators: type operators

ending in a colon ‘:’ are right-associative; all other

operators are left-associative.

In a sequence of consecutive type infix operations

\(t_0 \, \mathit{op} \, t_1 \, \mathit{op_2} \, \ldots \, \mathit{op_n} \, t_n\),

all operators \(\mathit{op}\_1 , \ldots , \mathit{op}\_n\) must have the same

associativity. If they are all left-associative, the sequence is

interpreted as

\((\ldots (t_0 \mathit{op_1} t_1) \mathit{op_2} \ldots) \mathit{op_n} t_n\),

otherwise it is interpreted as

\(t_0 \mathit{op_1} (t_1 \mathit{op_2} ( \ldots \mathit{op_n} t_n) \ldots)\).

Function Types

Type              ::=  FunctionArgs ‘=>’ Type
FunctionArgs      ::=  InfixType
                    |  ‘(’ [ ParamType {‘,’ ParamType } ] ‘)’

The type \((T_1 , \ldots , T_n) \Rightarrow U\) represents the set of function

values that take arguments of types \(T1 , \ldots , Tn\) and yield

results of type \(U\). In the case of exactly one argument type

\(T \Rightarrow U\) is a shorthand for \((T) \Rightarrow U\).

An argument type of the form \(\Rightarrow T\)

represents a call-by-name parameter of type \(T\).

Function types associate to the right, e.g.

\(S \Rightarrow T \Rightarrow U\) is the same as

\(S \Rightarrow (T \Rightarrow U)\).

Function types are shorthands for class types that define apply

functions. Specifically, the \(n\)-ary function type

\((T_1 , \ldots , T_n) \Rightarrow U\) is a shorthand for the class type

Function$_n$[T1 , … , $T_n$, U]. Such class

types are defined in the Scala library for \(n\) between 0 and 9 as follows.

package scala
trait Function_n[-T1 , … , -T$_n$, +R] {
  def apply(x1: T1 , … , x$_n$: T$_n$): R
  override def toString = "<function>"
}

Hence, function types are covariant in their

result type and contravariant in their argument types.

Existential Types

Type               ::= InfixType ExistentialClauses
ExistentialClauses ::= ‘forSome’ ‘{’ ExistentialDcl
                       {semi ExistentialDcl} ‘}’
ExistentialDcl     ::= ‘type’ TypeDcl
                    |  ‘val’ ValDcl

An existential type has the form $T$ forSome { $Q$ }

where \(Q\) is a sequence of

type declarations.

Let

\(t_1[\mathit{tps}\_1] >: L_1 <: U_1 , \ldots , t_n[\mathit{tps}\_n] >: L_n <: U_n\)

be the types declared in \(Q\) (any of the

type parameter sections [ $\mathit{tps}_i$ ] might be missing).

The scope of each type \(t_i\) includes the type \(T\) and the existential clause

\(Q\).

The type variables \(t_i\) are said to be bound in the type

$T$ forSome { $Q$ }.

Type variables which occur in a type \(T\) but which are not bound in \(T\) are said

to be free in \(T\).

A type instance of $T$ forSome { $Q$ }

is a type \(\sigma T\) where \(\sigma\) is a substitution over \(t_1 , \ldots , t_n\)

such that, for each \(i\), \(\sigma L_i <: \sigma t_i <: \sigma U_i\).

The set of values denoted by the existential type $T$ forSome {$\,Q\,$}

is the union of the set of values of all its type instances.

A skolemization of $T$ forSome { $Q$ } is

a type instance \(\sigma T\), where \(\sigma\) is the substitution

\([t_1'/t_1 , \ldots , t_n'/t_n]\) and each \(t_i'\) is a fresh abstract type

with lower bound \(\sigma L_i\) and upper bound \(\sigma U_i\).

Simplification Rules

Existential types obey the following four equivalences:

1. Multiple for-clauses in an existential type can be merged. E.g.,
   
   $T$ forSome { $Q$ } forSome { $Q'$ }
   
   is equivalent to
   
   $T$ forSome { $Q$ ; $Q'$}.
2. Unused quantifications can be dropped. E.g.,
   
   $T$ forSome { $Q$ ; $Q'$}
   
   where none of the types defined in \(Q'\) are referred to by \(T\) or \(Q\),
   
   is equivalent to
   
   $T$ forSome {$ Q $}.
3. An empty quantification can be dropped. E.g.,
   
   $T$ forSome { } is equivalent to \(T\).
4. An existential type $T$ forSome { $Q$ } where \(Q\) contains
   
   a clause type $t[\mathit{tps}] >: L <: U$ is equivalent
   
   to the type $T'$ forSome { $Q$ } where \(T'\) results from \(T\) by replacing
   
   every covariant occurrence of \(t\) in \(T\) by \(U\) and by
   
   replacing every contravariant occurrence of \(t\) in \(T\) by \(L\).

Existential Quantification over Values

As a syntactic convenience, the bindings clause

in an existential type may also contain

value declarations val $x$: $T$.

An existential type $T$ forSome { $Q$; val $x$: $S\,$;$\,Q'$ }

is treated as a shorthand for the type

$T'$ forSome { $Q$; type $t$ <: $S$ with Singleton; $Q'$ }, where \(t\) is a

fresh type name and \(T'\) results from \(T\) by replacing every occurrence of

$x$.type with \(t\).

Placeholder Syntax for Existential Types

WildcardType   ::=  ‘_’ TypeBounds

Scala supports a placeholder syntax for existential types.

A wildcard type is of the form _$\;$>:$\,L\,$<:$\,U$. Both bound

clauses may be omitted. If a lower bound clause >:$\,L$ is missing,

>:$\,$scala.Nothing

is assumed. If an upper bound clause <:$\,U$ is missing,

<:$\,$scala.Any is assumed. A wildcard type is a shorthand for an

existentially quantified type variable, where the existential quantification is

implicit.

A wildcard type must appear as type argument of a parameterized type.

Let \(T = p.c[\mathit{targs},T,\mathit{targs}']\) be a parameterized type where

\(\mathit{targs}, \mathit{targs}'\) may be empty and

\(T\) is a wildcard type _$\;$>:$\,L\,$<:$\,U$. Then \(T\) is equivalent to the

existential

type

$p.c[\mathit{targs},t,\mathit{targs}']$ forSome { type $t$ >: $L$ <: $U$ }

where \(t\) is some fresh type variable.

Wildcard types may also appear as parts of infix types

, function types,

or tuple types.

Their expansion is then the expansion in the equivalent parameterized

type.

Example

Assume the class definitions

class Ref[T]
abstract class Outer { type T } .

Here are some examples of existential types:

Ref[T] forSome { type T <: java.lang.Number }
Ref[x.T] forSome { val x: Outer }
Ref[x_type # T] forSome { type x_type <: Outer with Singleton }

The last two types in this list are equivalent.

An alternative formulation of the first type above using wildcard syntax is:

Ref[_ <: java.lang.Number]

Example

The type List[List[_]] is equivalent to the existential type

List[List[t] forSome { type t }] .

Example

Assume a covariant type

class List[+T]

The type

List[T] forSome { type T <: java.lang.Number }

is equivalent (by simplification rule 4 above) to

List[java.lang.Number] forSome { type T <: java.lang.Number }

which is in turn equivalent (by simplification rules 2 and 3 above) to

List[java.lang.Number].

Non-Value Types

The types explained in the following do not denote sets of values, nor

do they appear explicitly in programs. They are introduced in this

report as the internal types of defined identifiers.

Method Types

A method type is denoted internally as \((\mathit{Ps})U\), where \((\mathit{Ps})\)

is a sequence of parameter names and types \((p_1:T_1 , \ldots , p_n:T_n)\)

for some \(n \geq 0\) and \(U\) is a (value or method) type. This type

represents named methods that take arguments named \(p_1 , \ldots , p_n\)

of types \(T_1 , \ldots , T_n\)

and that return a result of type \(U\).

Method types associate to the right: \((\mathit{Ps}\_1)(\mathit{Ps}\_2)U\) is

treated as \((\mathit{Ps}\_1)((\mathit{Ps}\_2)U)\).

A special case are types of methods without any parameters. They are

written here => T. Parameterless methods name expressions

that are re-evaluated each time the parameterless method name is

referenced.

Method types do not exist as types of values. If a method name is used

as a value, its type is implicitly converted to a

corresponding function type.

Example

The declarations

def a: Int
def b (x: Int): Boolean
def c (x: Int) (y: String, z: String): String

produce the typings

a: => Int
b: (Int) Boolean
c: (Int) (String, String) String

Polymorphic Method Types

A polymorphic method type is denoted internally as [$\mathit{tps}\,$]$T$ where

[$\mathit{tps}\,$] is a type parameter section

[$a_1$ >: $L_1$ <: $U_1 , \ldots , a_n$ >: $L_n$ <: $U_n$]

for some \(n \geq 0\) and \(T\) is a

(value or method) type. This type represents named methods that

take type arguments $S_1 , \ldots , S_n$ which

conform to the lower bounds

$L_1 , \ldots , L_n$ and the upper bounds

$U_1 , \ldots , U_n$ and that yield results of type \(T\).

Example

The declarations

def empty[A]: List[A]
def union[A <: Comparable[A]] (x: Set[A], xs: Set[A]): Set[A]

produce the typings

empty : [A >: Nothing <: Any] List[A]
union : [A >: Nothing <: Comparable[A]] (x: Set[A], xs: Set[A]) Set[A]

Type Constructors

A type constructor is represented internally much like a polymorphic method type.

[$\pm$ $a_1$ >: $L_1$ <: $U_1 , \ldots , \pm a_n$ >: $L_n$ <: $U_n$] $T$

represents a type that is expected by a

type constructor parameter or an

abstract type constructor binding with

the corresponding type parameter clause.

Example

Consider this fragment of the Iterable[+X] class:

trait Iterable[+X] {
  def flatMap[newType[+X] <: Iterable[X], S](f: X => newType[S]): newType[S]
}

Conceptually, the type constructor Iterable is a name for the

anonymous type [+X] Iterable[X], which may be passed to the

newType type constructor parameter in flatMap.

String): Unit = $\ldots$
```
define a single function `println` which has an overloaded
type.
```
println: => Unit $\overload$
(String) Unit $\overload$
(Float) Unit $\overload$
(Float, Int) Unit $\overload$
[A] (A) (A => String) Unit
```

###### Example
```
def f(x: T): T = $\ldots$
val f = 0
```
define a function `f} which has type `(x: T)T $\overload$ Int`.
-->

Base Types and Member Definitions

Types of class members depend on the way the members are referenced.

Central here are three notions, namely:

1. the notion of the set of base types of a type \(T\),
2. the notion of a type \(T\) in some class \(C\) seen from some
   
   prefix type \(S\),
3. the notion of the set of member bindings of some type \(T\).

These notions are defined mutually recursively as follows.

1. The set of base types of a type is a set of class types,
   
   given as follows.

• The base types of a class type \(C\) with parents \(T_1 , \ldots , T_n\) are
  
  \(C\) itself, as well as the base types of the compound type
  
  $T_1$ with … with $T_n$ { $R$ }.
• The base types of an aliased type are the base types of its alias.
• The base types of an abstract type are the base types of its upper bound.
• The base types of a parameterized type
  
  $C$[$T_1 , \ldots , T_n$] are the base types
  
  of type \(C\), where every occurrence of a type parameter \(a_i\)
  
  of \(C\) has been replaced by the corresponding parameter type \(T_i\).
• The base types of a singleton type $p$.type are the base types of
  
  the type of \(p\).
• The base types of a compound type
  
  $T_1$ with $\ldots$ with $T_n$ { $R$ }
  
  are the reduced union of the base
  
  classes of all \(T_i\)'s. This means:
  
  Let the multi-set \(\mathscr{S}\) be the multi-set-union of the
  
  base types of all \(T_i\)'s.
  
  If \(\mathscr{S}\) contains several type instances of the same class, say
  
  $S^i$#$C$[$T^i_1 , \ldots , T^i_n$] \((i \in I)\), then
  
  all those instances
  
  are replaced by one of them which conforms to all
  
  others. It is an error if no such instance exists. It follows that the
  
  reduced union, if it exists,
  
  produces a set of class types, where different types are instances of
  
  different classes.
• The base types of a type selection $S$#$T$ are
  
  determined as follows. If \(T\) is an alias or abstract type, the
  
  previous clauses apply. Otherwise, \(T\) must be a (possibly
  
  parameterized) class type, which is defined in some class \(B\). Then
  
  the base types of $S$#$T$ are the base types of \(T\)
  
  in \(B\) seen from the prefix type \(S\).
• The base types of an existential type $T$ forSome { $Q$ } are
  
  all types $S$ forSome { $Q$ } where \(S\) is a base type of \(T\).

1. The notion of a type \(T\) in class \(C\) seen from some prefix type \(S\)
   
   makes sense only if the prefix type \(S\)
   
   has a type instance of class \(C\) as a base type, say
   
   $S'$#$C$[$T_1 , \ldots , T_n$]. Then we define as follows.

   • If $S$ = $\epsilon$.type, then \(T\) in \(C\) seen from \(S\) is
     
     \(T\) itself.
   • Otherwise, if \(S\) is an existential type $S'$ forSome { $Q$ }, and
     
     \(T\) in \(C\) seen from \(S'\) is \(T'\),
     
     then \(T\) in \(C\) seen from \(S\) is $T'$ forSome {$\,Q\,$}.
   • Otherwise, if \(T\) is the \(i\)'th type parameter of some class \(D\), then
     • If \(S\) has a base type $D$[$U_1 , \ldots , U_n$], for some type
       
       parameters [$U_1 , \ldots , U_n$], then \(T\) in \(C\) seen from \(S\)
       
       is \(U_i\).
     • Otherwise, if \(C\) is defined in a class \(C'\), then
       
       \(T\) in \(C\) seen from \(S\) is the same as \(T\) in \(C'\) seen from \(S'\).
     • Otherwise, if \(C\) is not defined in another class, then
       
       \(T\) in \(C\) seen from \(S\) is \(T\) itself.
   • Otherwise, if \(T\) is the singleton type $D$.this.type for some class \(D\)
     
     then
     • If \(D\) is a subclass of \(C\) and \(S\) has a type instance of class \(D\)
       
       among its base types, then \(T\) in \(C\) seen from \(S\) is \(S\).
     • Otherwise, if \(C\) is defined in a class \(C'\), then
       
       \(T\) in \(C\) seen from \(S\) is the same as \(T\) in \(C'\) seen from \(S'\).
     • Otherwise, if \(C\) is not defined in another class, then
       
       \(T\) in \(C\) seen from \(S\) is \(T\) itself.
   • If \(T\) is some other type, then the described mapping is performed
     
     to all its type components.

   If \(T\) is a possibly parameterized class type, where \(T\)'s class
   
   is defined in some other class \(D\), and \(S\) is some prefix type,
   
   then we use "\(T\) seen from \(S\)" as a shorthand for
   
   "\(T\) in \(D\) seen from \(S\)".

2. The member bindings of a type \(T\) are

   1. all bindings \(d\) such that there exists a type instance of some class \(C\) among the base types of \(T\)
      
      and there exists a definition or declaration \(d'\) in \(C\)
      
      such that \(d\) results from \(d'\) by replacing every
      
      type \(T'\) in \(d'\) by \(T'\) in \(C\) seen from \(T\), and
   2. all bindings of the type's refinement, if it has one.

   The definition of a type projection S#T is the member
   
   binding \(d_T\) of the type T in S. In that case, we also say
   
   that S#T is defined by \(d_T\).

Relations between types

We define two relations between types.

Name	Symbolically	Interpretation
Equivalence	\(T \equiv U\)	\(T\) and \(U\) are interchangeable in all contexts.
Conformance	\(T <: U\)	Type \(T\) conforms to type \(U\).

Equivalence

Equivalence \((\equiv)\) between types is the smallest congruence <sup>[3]</sup> such that

the following holds:

• If \(t\) is defined by a type alias type $t$ = $T$, then \(t\) is
  
  equivalent to \(T\).
• If a path \(p\) has a singleton type $q$.type, then
  
  $p$.type $\equiv q$.type.
• If \(O\) is defined by an object definition, and \(p\) is a path
  
  consisting only of package or object selectors and ending in \(O\), then
  
  $O$.this.type $\equiv p$.type.
• Two compound types are equivalent if the sequences
  
  of their component are pairwise equivalent, and occur in the same order, and
  
  their refinements are equivalent. Two refinements are equivalent if they
  
  bind the same names and the modifiers, types and bounds of every
  
  declared entity are equivalent in both refinements.
• Two method types are equivalent if:
  • neither are implicit, or they both are <sup>[4]</sup>;
  • they have equivalent result types;
  • they have the same number of parameters; and
  • corresponding parameters have equivalent types.
    
    Note that the names of parameters do not matter for method type equivalence.
• Two polymorphic method types are equivalent if
  
  they have the same number of type parameters, and, after renaming one set of
  
  type parameters by another, the result types as well as lower and upper bounds
  
  of corresponding type parameters are equivalent.
• Two existential types
  
  are equivalent if they have the same number of
  
  quantifiers, and, after renaming one list of type quantifiers by
  
  another, the quantified types as well as lower and upper bounds of
  
  corresponding quantifiers are equivalent.
• Two type constructors are equivalent if they have the
  
  same number of type parameters, and, after renaming one list of type
  
  parameters by another, the result types as well as variances, lower and upper
  
  bounds of corresponding type parameters are equivalent.

Conformance

The conformance relation \((<:)\) is the smallest

transitive relation that satisfies the following conditions.

• Conformance includes equivalence. If \(T \equiv U\) then \(T <: U\).

• For every value type \(T\), scala.Nothing <: $T$ <: scala.Any.

• For every type constructor \(T\) (with any number of type parameters),
  
  scala.Nothing <: $T$ <: scala.Any.

• For every class type \(T\) such that $T$ <: scala.AnyRef one has scala.Null <: $T$.

• A type variable or abstract type \(t\) conforms to its upper bound and
  
  its lower bound conforms to \(t\).

• A class type or parameterized type conforms to any of its base-types.

• A singleton type $p$.type conforms to the type of the path \(p\).

• A singleton type $p$.type conforms to the type scala.Singleton.

• A type projection $T$#$t$ conforms to $U$#$t$ if \(T\) conforms to \(U\).

• A parameterized type $T$[$T_1$ , … , $T_n$] conforms to
  
  $T$[$U_1$ , … , $U_n$] if
  
  the following three conditions hold for \(i \in \{ 1 , \ldots , n \}\):

1. If the \(i\)'th type parameter of \(T\) is declared covariant, then
   
   \(T_i <: U_i\).
2. If the \(i\)'th type parameter of \(T\) is declared contravariant, then
   
   \(U_i <: T_i\).
3. If the \(i\)'th type parameter of \(T\) is declared neither covariant
   
   nor contravariant, then \(U_i \equiv T_i\).

• A compound type $T_1$ with $\ldots$ with $T_n$ {$R\,$} conforms to
  
  each of its component types \(T_i\).
• If \(T <: U_i\) for \(i \in \{ 1 , \ldots , n \}\) and for every
  
  binding \(d\) of a type or value \(x\) in \(R\) there exists a member
  
  binding of \(x\) in \(T\) which subsumes \(d\), then \(T\) conforms to the
  
  compound type $U_1$ with $\ldots$ with $U_n$ {$R\,$}.
• The existential type $T$ forSome {$\,Q\,$} conforms to
  
  \(U\) if its skolemization
  
  conforms to \(U\).
• The type \(T\) conforms to the existential type $U$ forSome {$\,Q\,$}
  
  if \(T\) conforms to one of the type instances
  
  of $U$ forSome {$\,Q\,$}.
• If
  
  \(T_i \equiv T_i'\) for \(i \in \{ 1 , \ldots , n\}\) and \(U\) conforms to \(U'\)
  
  then the method type \((p_1:T_1 , \ldots , p_n:T_n) U\) conforms to
  
  \((p_1':T_1' , \ldots , p_n':T_n') U'\).
• The polymorphic type
  
  \([a_1 >: L_1 <: U_1 , \ldots , a_n >: L_n <: U_n] T\) conforms to the
  
  polymorphic type
  
  \([a_1 >: L_1' <: U_1' , \ldots , a_n >: L_n' <: U_n'] T'\) if, assuming
  
  \(L_1' <: a_1 <: U_1' , \ldots , L_n' <: a_n <: U_n'\)
  
  one has \(T <: T'\) and \(L_i <: L_i'\) and \(U_i' <: U_i\)
  
  for \(i \in \{ 1 , \ldots , n \}\).
• Type constructors \(T\) and \(T'\) follow a similar discipline. We characterize
  
  \(T\) and \(T'\) by their type parameter clauses
  
  \([a_1 , \ldots , a_n]\) and
  
  \([a_1' , \ldots , a_n']\), where an \(a_i\) or \(a_i'\) may include a variance
  
  annotation, a higher-order type parameter clause, and bounds. Then, \(T\)
  
  conforms to \(T'\) if any list \([t_1 , \ldots , t_n]\) -- with declared
  
  variances, bounds and higher-order type parameter clauses -- of valid type
  
  arguments for \(T'\) is also a valid list of type arguments for \(T\) and
  
  \(T[t_1 , \ldots , t_n] <: T'[t_1 , \ldots , t_n]\). Note that this entails
  
  that:
  • The bounds on \(a_i\) must be weaker than the corresponding bounds declared
    
    for \(a'_i\).
  • The variance of \(a_i\) must match the variance of \(a'_i\), where covariance
    
    matches covariance, contravariance matches contravariance and any variance
    
    matches invariance.
  • Recursively, these restrictions apply to the corresponding higher-order
    
    type parameter clauses of \(a_i\) and \(a'_i\).

A declaration or definition in some compound type of class type \(C\)

subsumes another declaration of the same name in some compound type or class

type \(C'\), if one of the following holds.

• A value declaration or definition that defines a name \(x\) with type \(T\)
  
  subsumes a value or method declaration that defines \(x\) with type \(T'\), provided
  
  \(T <: T'\).
• A method declaration or definition that defines a name \(x\) with type \(T\)
  
  subsumes a method declaration that defines \(x\) with type \(T'\), provided
  
  \(T <: T'\).
• A type alias
  
  type $t$[$T_1$ , … , $T_n$] = $T$ subsumes a type alias
  
  type $t$[$T_1$ , … , $T_n$] = $T'$ if \(T \equiv T'\).
• A type declaration type $t$[$T_1$ , … , $T_n$] >: $L$ <: $U$ subsumes
  
  a type declaration type $t$[$T_1$ , … , $T_n$] >: $L'$ <: $U'$ if
  
  \(L' <: L\) and \(U <: U'\).
• A type or class definition that binds a type name \(t\) subsumes an abstract
  
  type declaration type t[$T_1$ , … , $T_n$] >: L <: U if
  
  \(L <: t <: U\).

The \((<:)\) relation forms pre-order between types,

i.e. it is transitive and reflexive. least upper bounds and

greatest lower bounds of a set of types

are understood to be relative to that order.

Note

The least upper bound or greatest lower bound

of a set of types does not always exist. For instance, consider

the class definitions

class A[+T] {}
class B extends A[B]
class C extends A[C]

Then the types A[Any], A[A[Any]], A[A[A[Any]]], ... form

a descending sequence of upper bounds for B and C. The

least upper bound would be the infinite limit of that sequence, which

does not exist as a Scala type. Since cases like this are in general

impossible to detect, a Scala compiler is free to reject a term

which has a type specified as a least upper or greatest lower bound,

and that bound would be more complex than some compiler-set

limit <sup>[5]</sup>.

The least upper bound or greatest lower bound might also not be

unique. For instance A with B and B with A are both

greatest lower bounds of A and B. If there are several

least upper bounds or greatest lower bounds, the Scala compiler is

free to pick any one of them.

Weak Conformance

In some situations Scala uses a more general conformance relation. A

type \(S\) weakly conforms

to a type \(T\), written \(S <:_w
T\), if \(S <: T\) or both \(S\) and \(T\) are primitive number types

and \(S\) precedes \(T\) in the following ordering.

Byte  $<:_w$ Short
Short $<:_w$ Int
Char  $<:_w$ Int
Int   $<:_w$ Long
Long  $<:_w$ Float
Float $<:_w$ Double

A weak least upper bound is a least upper bound with respect to

weak conformance.

Volatile Types

Type volatility approximates the possibility that a type parameter or abstract

type instance

of a type does not have any non-null values. A value member of a volatile type

cannot appear in a path.

A type is volatile if it falls into one of four categories:

A compound type $T_1$ with … with $T_n$ {$R\,$}

is volatile if one of the following two conditions hold.

1. One of \(T_2 , \ldots , T_n\) is a type parameter or abstract type, or
2. \(T_1\) is an abstract type and and either the refinement \(R\)
   
   or a type \(T_j\) for \(j > 1\) contributes an abstract member
   
   to the compound type, or
3. one of \(T_1 , \ldots , T_n\) is a singleton type.

Here, a type \(S\) contributes an abstract member to a type \(T\) if

\(S\) contains an abstract member that is also a member of \(T\).

A refinement \(R\) contributes an abstract member to a type \(T\) if \(R\)

contains an abstract declaration which is also a member of \(T\).

A type designator is volatile if it is an alias of a volatile type, or

if it designates a type parameter or abstract type that has a volatile type as

its upper bound.

A singleton type $p$.type is volatile, if the underlying

type of path \(p\) is volatile.

An existential type $T$ forSome {$\,Q\,$} is volatile if

\(T\) is volatile.

Type Erasure

A type is called generic if it contains type arguments or type variables.

Type erasure is a mapping from (possibly generic) types to

non-generic types. We write \(|T|\) for the erasure of type \(T\).

The erasure mapping is defined as follows.

• The erasure of an alias type is the erasure of its right-hand side.
• The erasure of an abstract type is the erasure of its upper bound.
• The erasure of the parameterized type scala.Array$[T_1]$ is
  
  scala.Array$[|T_1|]$.
• The erasure of every other parameterized type \(T[T_1 , \ldots , T_n]\) is \(|T|\).
• The erasure of a singleton type $p$.type is the
  
  erasure of the type of \(p\).
• The erasure of a type projection $T$#$x$ is |$T$|#$x$.
• The erasure of a compound type
  
  $T_1$ with $\ldots$ with $T_n$ {$R\,$} is the erasure of the intersection
  
  dominator of \(T_1 , \ldots , T_n\).
• The erasure of an existential type $T$ forSome {$\,Q\,$} is \(|T|\).

The intersection dominator of a list of types \(T_1 , \ldots , T_n\) is computed

as follows.

Let \(T_{i_1} , \ldots , T_{i_m}\) be the subsequence of types \(T_i\)

which are not supertypes of some other type \(T_j\).

If this subsequence contains a type designator \(T_c\) that refers to a class

which is not a trait,

the intersection dominator is \(T_c\). Otherwise, the intersection

dominator is the first element of the subsequence, \(T_{i_1}\).

---

1. We assume that objects and packages also implicitly
   
   define a class (of the same name as the object or package, but
   
   inaccessible to user programs). ↩︎

2. A reference to a structurally defined member (method call or access
   
   to a value or variable) may generate binary code that is significantly
   
   slower than an equivalent code to a non-structural member. ↩︎

3. A congruence is an equivalence relation which is closed under formation of contexts. ↩︎

4. A method type is implicit if the parameter section that defines it starts with the implicit keyword. ↩︎

5. The current Scala compiler limits the nesting level
   
   of parameterization in such bounds to be at most two deeper than the
   
   maximum nesting level of the operand types ↩︎