Basic Declarations and Definitions
==================================


\syntax\begin{lstlisting}
  Dcl               ::=  `val' ValDcl
                      |  `var' VarDcl
                      |  `def' FunDcl
                      |  `type' {nl} TypeDcl
  PatVarDef         ::=  `val' PatDef
                      |  `var' VarDef
  Def               ::=  PatVarDef
                      |  `def' FunDef
                      |  `type' {nl} TypeDef
                      |  TmplDef
\end{lstlisting}

A {\em declaration} introduces names and assigns them types. It can
form part of a class definition (\sref{sec:templates}) or of a
refinement in a compound type (\sref{sec:refinements}).

A {\em definition} introduces names that denote terms or types. It can
form part of an object or class definition or it can be local to a
block.  Both declarations and definitions produce {\em bindings} that
associate type names with type definitions or bounds, and that
associate term names with types.

The scope of a name introduced by a declaration or definition is the
whole statement sequence containing the binding.  However, there is a
restriction on forward references in blocks: In a statement sequence
$s_1 \ldots s_n$ making up a block, if a simple name in $s_i$ refers
to an entity defined by $s_j$ where $j \geq i$, then for all $s_k$
between and including $s_i$ and $s_j$,
\begin{itemize}
\item $s_k$ cannot be a variable definition.
\item If $s_k$ is a value definition, it must be lazy.
\end{itemize}

\comment{
Every basic definition may introduce several defined names, separated
by commas. These are expanded according to the following scheme:
\bda{lcl}
\VAL;x, y: T = e && \VAL; x: T = e \\
                 && \VAL; y: T = x \\[0.5em]

\LET;x, y: T = e && \LET; x: T = e \\
                 && \VAL; y: T = x \\[0.5em]

\DEF;x, y (ps): T = e &\tab\mbox{expands to}\tab& \DEF; x(ps): T = e \\
                      && \DEF; y(ps): T = x(ps)\\[0.5em]

\VAR;x, y: T := e && \VAR;x: T := e\\
                  && \VAR;y: T := x\\[0.5em]

\TYPE;t,u = T && \TYPE; t = T\\
              && \TYPE; u = t\\[0.5em]
\eda

All definitions have a ``repeated form'' where the initial
definition keyword is followed by several constituent definitions
which are separated by commas.  A repeated definition is
always interpreted as a sequence formed from the
constituent definitions. E.g.\ the function definition
~\lstinline@def f(x) = x, g(y) = y@~ expands to
~\lstinline@def f(x) = x; def g(y) = y@~ and
the type definition
~\lstinline@type T, U <: B@~ expands to
~\lstinline@type T; type U <: B@.
}
\comment{
If an element in such a sequence introduces only the defined name,
possibly with some type or value parameters, but leaves out any
additional parts in the definition, then those parts are implicitly
copied from the next subsequent sequence element which consists of
more than just a defined name and parameters. Examples:
\begin{itemize}
\item[]
The variable declaration ~\lstinline@var x, y: Int@~ 
expands to ~\lstinline@var x: Int; var y: Int@.
\item[]
The value definition ~\lstinline@val x, y: Int = 1@~ 
expands to ~\lstinline@val x: Int = 1; val y: Int = 1@.
\item[]
The class definition ~\lstinline@case class X(), Y(n: Int) extends Z@~ expands to
~\lstinline@case class X extends Z; case class Y(n: Int) extends Z@.
\item
The object definition ~\lstinline@case object Red, Green, Blue extends Color@~
expands to  
\begin{lstlisting}
case object Red extends Color  
case object Green extends Color 
case object Blue extends Color .
\end{lstlisting}
\end{itemize}
}
\section{Value Declarations and Definitions}
\label{sec:valdef}

\syntax\begin{lstlisting}
  Dcl          ::=  `val' ValDcl
  ValDcl       ::=  ids `:' Type
  PatVarDef    ::=  `val' PatDef 
  PatDef       ::=  Pattern2 {`,' Pattern2} [`:' Type] `=' Expr
  ids          ::=  id {`,' id}
\end{lstlisting}

A value declaration ~\lstinline@val $x$: $T$@~ introduces $x$ as a name of a value of
type $T$.  

A value definition ~\lstinline@val $x$: $T$ = $e$@~ defines $x$ as a
name of the value that results from the evaluation of $e$. 
If the value definition is not recursive, the type
$T$ may be omitted, in which case the packed type (\sref{sec:expr-typing}) of expression $e$ is
assumed.  If a type $T$ is given, then $e$ is expected to conform to
it.

Evaluation of the value definition implies evaluation of its
right-hand side $e$, unless it has the modifier \lstinline@lazy@.  The
effect of the value definition is to bind $x$ to the value of $e$
converted to type $T$. A \lstinline@lazy@ value definition evaluates
its right hand side $e$ the first time the value is accessed.

A {\em constant value definition} is of the form
\begin{lstlisting}
final val x = e
\end{lstlisting}
where \lstinline@e@ is a constant expression
(\sref{sec:constant-expression}). 
The \lstinline@final@ modifier must be
present and no type annotation may be given. References to the
constant value \lstinline@x@ are themselves treated as constant expressions; in the
generated code they are replaced by the definition's right-hand side \lstinline@e@.

Value definitions can alternatively have a pattern
(\sref{sec:patterns}) as left-hand side.  If $p$ is some pattern other
than a simple name or a name followed by a colon and a type, then the
value definition ~\lstinline@val $p$ = $e$@~ is expanded as follows:

1. If the pattern $p$ has bound variables $x_1 \commadots x_n$, where $n > 1$:
\begin{lstlisting}
val $\Dollar x$ = $e$ match {case $p$ => {$x_1 \commadots x_n$}}
val $x_1$ = $\Dollar x$._1
$\ldots$
val $x_n$ = $\Dollar x$._n  .
\end{lstlisting}
Here, $\Dollar x$ is a fresh name.  

2. If $p$ has a unique bound variable $x$:
\begin{lstlisting}
val $x$ = $e$ match { case $p$ => $x$ }
\end{lstlisting}

3. If $p$ has no bound variables:
\begin{lstlisting}
$e$ match { case $p$ => ()}
\end{lstlisting}

\example
The following are examples of value definitions
\begin{lstlisting}
val pi = 3.1415 
val pi: Double = 3.1415   // equivalent to first definition
val Some(x) = f()         // a pattern definition
val x :: xs = mylist      // an infix pattern definition
\end{lstlisting}

The last two definitions have the following expansions.
\begin{lstlisting}
val x = f() match { case Some(x) => x }

val x$\Dollar$ = mylist match { case x :: xs => {x, xs} }
val x = x$\Dollar$._1 
val xs = x$\Dollar$._2 
\end{lstlisting}

The name of any declared or defined value may not end in \lstinline@_=@.

A value declaration ~\lstinline@val $x_1 \commadots x_n$: $T$@~
is a
shorthand for the sequence of value declarations
~\lstinline@val $x_1$: $T$; ...; val $x_n$: $T$@.
A value definition ~\lstinline@val $p_1 \commadots p_n$ = $e$@~
is a
shorthand for the sequence of value definitions
~\lstinline@val $p_1$ = $e$; ...; val $p_n$ = $e$@.
A value definition ~\lstinline@val $p_1 \commadots p_n: T$ = $e$@~
is a
shorthand for the sequence of value definitions
~\lstinline@val $p_1: T$ = $e$; ...; val $p_n: T$ = $e$@.

\section{Variable Declarations and Definitions}
\label{sec:vardef}

\syntax\begin{lstlisting}
  Dcl            ::=  `var' VarDcl
  PatVarDef      ::=  `var' VarDef
  VarDcl         ::=  ids `:' Type
  VarDef         ::=  PatDef
                   |  ids `:' Type `=' `_'
\end{lstlisting}

A variable declaration ~\lstinline@var $x$: $T$@~ is equivalent to declarations
of a {\em getter function} $x$ and a {\em setter function}
\lstinline@$x$_=@, defined as follows:

\begin{lstlisting}
  def $x$: $T$ 
  def $x$_= ($y$: $T$): Unit
\end{lstlisting}

An implementation of a class containing variable declarations
may define these variables using variable definitions, or it may
define setter and getter functions directly.

A variable definition ~\lstinline@var $x$: $T$ = $e$@~ introduces a
mutable variable with type $T$ and initial value as given by the
expression $e$. The type $T$ can be omitted, in which case the type of
$e$ is assumed. If $T$ is given, then $e$ is expected to conform to it
(\sref{sec:expr-typing}).

Variable definitions can alternatively have a pattern
(\sref{sec:patterns}) as left-hand side.  A variable definition
 ~\lstinline@var $p$ = $e$@~ where $p$ is a pattern other
than a simple name or a name followed by a colon and a type is expanded in the same way 
(\sref{sec:valdef})
as a value definition  ~\lstinline@val $p$ = $e$@, except that
the free names in $p$ are introduced as mutable variables, not values.

The name of any declared or defined variable may not end in \lstinline@_=@.

A variable definition ~\lstinline@var $x$: $T$ = _@~ can appear only
as a member of a template. It introduces a mutable field with type
\ $T$ and a default initial value.  The default value depends on the
type $T$ as follows:
\begin{quote}\begin{tabular}{ll}
\code{0} & if $T$ is \code{Int} or one of its subrange types, \\
\code{0L} & if $T$ is \code{Long},\\
\lstinline@0.0f@ & if $T$ is \code{Float},\\
\lstinline@0.0d@ & if $T$ is \code{Double},\\
\code{false} & if $T$ is \code{Boolean},\\
\lstinline@{}@ & if $T$ is \code{Unit}, \\
\code{null} & for all other types $T$.
\end{tabular}\end{quote}
When they occur as members of a template, both forms of variable
definition also introduce a getter function $x$ which returns the
value currently assigned to the variable, as well as a setter function
\lstinline@$x$_=@ which changes the value currently assigned to the variable.
The functions have the same signatures as for a variable declaration.
The template then has these getter and setter functions as
members, whereas the original variable cannot be accessed directly as
a template member.

\example The following example shows how {\em properties} can be
simulated in Scala. It defines a class \code{TimeOfDayVar} of time
values with updatable integer fields representing hours, minutes, and
seconds. Its implementation contains tests that allow only legal
values to be assigned to these fields. The user code, on the other
hand, accesses these fields just like normal variables.

\begin{lstlisting}
class TimeOfDayVar {
  private var h: Int = 0 
  private var m: Int = 0 
  private var s: Int = 0 

  def hours              =  h 
  def hours_= (h: Int)   =  if (0 <= h && h < 24) this.h = h 
                            else throw new DateError() 

  def minutes            =  m 
  def minutes_= (m: Int) =  if (0 <= m && m < 60) this.m = m
                            else throw new DateError() 

  def seconds            =  s 
  def seconds_= (s: Int) =  if (0 <= s && s < 60) this.s = s
                            else throw new DateError() 
}
val d = new TimeOfDayVar 
d.hours = 8; d.minutes = 30; d.seconds = 0 
d.hours = 25                  // throws a DateError exception
\end{lstlisting}

A variable declaration ~\lstinline@var $x_1 \commadots x_n$: $T$@~
is a
shorthand for the sequence of variable declarations
~\lstinline@var $x_1$: $T$; ...; var $x_n$: $T$@.
A variable definition ~\lstinline@var $x_1 \commadots x_n$ = $e$@~
is a
shorthand for the sequence of variable definitions
~\lstinline@var $x_1$ = $e$; ...; var $x_n$ = $e$@.
A variable definition ~\lstinline@var $x_1 \commadots x_n: T$ = $e$@~
is a
shorthand for the sequence of variable definitions
~\lstinline@var $x_1: T$ = $e$; ...; var $x_n: T$ = $e$@.

Type Declarations and Type Aliases
----------------------------------

\label{sec:typedcl}
\label{sec:typealias}

\todo{Higher-kinded tdecls should have a separate section}

\syntax\begin{lstlisting}
  Dcl             ::=  `type' {nl} TypeDcl
  TypeDcl         ::=  id [TypeParamClause] [`>:' Type] [`<:' Type]
  Def             ::=  type {nl} TypeDef
  TypeDef         ::=  id [TypeParamClause] `=' Type
\end{lstlisting}

%@M
A {\em type declaration} ~\lstinline@type $t$[$\tps\,$] >: $L$ <: $U$@~ declares
$t$ to be an abstract type with lower bound type $L$ and upper bound
type $U$. If the type parameter clause \lstinline@[$\tps\,$]@ is omitted, $t$ abstracts over a first-order type, otherwise $t$ stands for a type constructor that accepts type arguments as described by the type parameter clause.

%@M
If a type declaration appears as a member declaration of a
type, implementations of the type may implement $t$ with any type $T$
for which $L \conforms T \conforms U$. It is a compile-time error if
$L$ does not conform to $U$.  Either or both bounds may be omitted.
If the lower bound $L$ is absent, the bottom type
\lstinline@scala.Nothing@ is assumed.  If the upper bound $U$ is absent,
the top type \lstinline@scala.Any@ is assumed.

%@M
A type constructor declaration imposes additional restrictions on the
concrete types for which $t$ may stand. Besides the bounds $L$ and
$U$, the type parameter clause may impose higher-order bounds and
variances, as governed by the conformance of type constructors
(\sref{sec:conformance}).

%@M
The scope of a type parameter extends over the bounds ~\lstinline@>: $L$ <: $U$@ and the type parameter clause $\tps$ itself. A
higher-order type parameter clause (of an abstract type constructor
$tc$) has the same kind of scope, restricted to the declaration of the
type parameter $tc$.

To illustrate nested scoping, these declarations are all equivalent: ~\lstinline@type t[m[x] <: Bound[x], Bound[x]]@, ~\lstinline@type t[m[x] <: Bound[x], Bound[y]]@ and ~\lstinline@type t[m[x] <: Bound[x], Bound[_]]@, as the scope of, e.g., the type parameter of $m$ is limited to the declaration of $m$. In all of them, $t$ is an abstract type member that abstracts over two type constructors: $m$ stands for a type constructor that takes one type parameter and that must be a subtype of $Bound$, $t$'s second type constructor parameter. ~\lstinline@t[MutableList, Iterable]@ is a valid use of $t$.

A {\em type alias} ~\lstinline@type $t$ = $T$@~ defines $t$ to be an alias
name for the type $T$.  The left hand side of a type alias may
have a type parameter clause, e.g. ~\lstinline@type $t$[$\tps\,$] = $T$@.  The scope
of a type parameter extends over the right hand side $T$ and the
type parameter clause $\tps$ itself.  

The scope rules for definitions (\sref{sec:defs}) and type parameters
(\sref{sec:funsigs}) make it possible that a type name appears in its
own bound or in its right-hand side.  However, it is a static error if
a type alias refers recursively to the defined type constructor itself.  
That is, the type $T$ in a type alias ~\lstinline@type $t$[$\tps\,$] = $T$@~ may not 
refer directly or indirectly to the name $t$.  It is also an error if
an abstract type is directly or indirectly its own upper or lower bound.

\example The following are legal type declarations and definitions:
\begin{lstlisting}
type IntList = List[Integer]
type T <: Comparable[T]
type Two[A] = Tuple2[A, A]
type MyCollection[+X] <: Iterable[X]
\end{lstlisting}

The following are illegal:
\begin{lstlisting}
type Abs = Comparable[Abs]        // recursive type alias

type S <: T                       // S, T are bounded by themselves.
type T <: S

type T >: Comparable[T.That]      // Cannot select from T.
                                  // T is a type, not a value
type MyCollection <: Iterable     // Type constructor members must explicitly state their type parameters.
\end{lstlisting}

If a type alias ~\lstinline@type $t$[$\tps\,$] = $S$@~ refers to a class type
$S$, the name $t$ can also be used as a constructor for
objects of type $S$.

\example The \code{Predef} object contains a definition which establishes \code{Pair} 
as an alias of the parameterized class \code{Tuple2}:
\begin{lstlisting}
type Pair[+A, +B] = Tuple2[A, B] 
object Pair {
  def apply[A, B](x: A, y: B) = Tuple2(x, y)
  def unapply[A, B](x: Tuple2[A, B]): Option[Tuple2[A, B]] = Some(x)
}
\end{lstlisting}
As a consequence, for any two types $S$ and $T$, the type
~\lstinline@Pair[$S$, $T\,$]@~ is equivalent to the type ~\lstinline@Tuple2[$S$, $T\,$]@.
\code{Pair} can also be used as a constructor instead of \code{Tuple2}, as in:
\begin{lstlisting}
val x: Pair[Int, String] = new Pair(1, "abc")
\end{lstlisting}

\section{Type Parameters}\label{sec:type-params}

\syntax\begin{lstlisting}
  TypeParamClause  ::= `[' VariantTypeParam {`,' VariantTypeParam} `]'
  VariantTypeParam ::= {Annotation} [`+' | `-'] TypeParam
  TypeParam        ::= (id | `_') [TypeParamClause] [`>:' Type] [`<:' Type] [`:' Type]
\end{lstlisting}

Type parameters appear in type definitions, class definitions, and
function definitions.  In this section we consider only type parameter
definitions with lower bounds ~\lstinline@>: $L$@~ and upper bounds
~\lstinline@<: $U$@~ whereas a discussion of context bounds
~\lstinline@: $U$@~ and view bounds ~\lstinline@<% $U$@~ 
is deferred to Section~\ref{sec:context-bounds}.

The most general form of a first-order type parameter is
~\lstinline!$@a_1\ldots@a_n$ $\pm$ $t$ >: $L$ <: $U$!.  
Here, $L$, and $U$ are lower and upper bounds that
constrain possible type arguments for the parameter.  It is a
compile-time error if $L$ does not conform to $U$. $\pm$ is a {\em
variance}, i.e.\ an optional prefix of either \lstinline@+@, or
\lstinline@-@. One or more annotations may precede the type parameter.

\comment{
The upper bound $U$ in a type parameter clauses may not be a final
class. The lower bound may not denote a value type.\todo{Why}
}

\comment{@M TODO this is a pretty awkward description of scoping and distinctness of binders}
The names of all type parameters must be pairwise different in their enclosing type parameter clause.  The scope of a type parameter includes in each case the whole type parameter clause. Therefore it is possible that a type parameter appears as part of its own bounds or the bounds of other type parameters in the same clause.  However, a type parameter may not be bounded directly or indirectly by itself.\

A type constructor parameter adds a nested type parameter clause to the type parameter. The most general form of a type constructor parameter is ~\lstinline!$@a_1\ldots@a_n$ $\pm$ $t[\tps\,]$ >: $L$ <: $U$!.  

The above scoping restrictions are generalized to the case of nested type parameter clauses, which declare higher-order type parameters. Higher-order type parameters (the type parameters of a type parameter $t$) are only visible in their immediately surrounding parameter clause (possibly including clauses at a deeper nesting level) and in the bounds of $t$. Therefore, their names must only be pairwise different from the names of other visible parameters. Since the names of higher-order type parameters are thus often irrelevant, they may be denoted with a `\lstinline@_@', which is nowhere visible.

\example Here are some well-formed type parameter clauses:
\begin{lstlisting}
[S, T]
[@specialized T, U]
[Ex <: Throwable]
[A <: Comparable[B], B <: A]
[A, B >: A, C >: A <: B]
[M[X], N[X]]
[M[_], N[_]] // equivalent to previous clause
[M[X <: Bound[X]], Bound[_]]
[M[+X] <: Iterable[X]]
\end{lstlisting}
The following type parameter clauses are illegal:
\begin{lstlisting}
[A >: A]                  // illegal, `A' has itself as bound
[A <: B, B <: C, C <: A]  // illegal, `A' has itself as bound
[A, B, C >: A <: B]       // illegal lower bound `A' of `C' does
                          // not conform to upper bound `B'.
\end{lstlisting}

\section{Variance Annotations}\label{sec:variances}

Variance annotations indicate how instances of parameterized types
vary with respect to subtyping (\sref{sec:conformance}).  A
`\lstinline@+@' variance indicates a covariant dependency, a
`\lstinline@-@' variance indicates a contravariant dependency, and a
missing variance indication indicates an invariant dependency.

%@M
A variance annotation constrains the way the annotated type variable
may appear in the type or class which binds the type parameter.  In a
type definition ~\lstinline@type $T$[$\tps\,$] = $S$@, or a type 
declaration ~\lstinline@type $T$[$\tps\,$] >: $L$ <: $U$@~ type parameters labeled
`\lstinline@+@' must only appear in covariant position whereas
type parameters labeled `\lstinline@-@' must only appear in contravariant
position. Analogously, for a class definition
~\lstinline@class $C$[$\tps\,$]($\ps\,$) extends $T$ { $x$: $S$ => ...}@, 
type parameters labeled
`\lstinline@+@' must only appear in covariant position in the
self type $S$ and the template $T$, whereas type
parameters labeled `\lstinline@-@' must only appear in contravariant
position. 

The variance position of a type parameter in a type or template is
defined as follows.  Let the opposite of covariance be contravariance,
and the opposite of invariance be itself.  The top-level of the type
or template is always in covariant position. The variance position
changes at the following constructs.
\begin{itemize}
\item
The variance position of a method parameter is the opposite of the 
variance position of the enclosing parameter clause.
\item
The variance position of a type parameter is the opposite of the
variance position of the enclosing type parameter clause.
\item
The variance position of the lower bound of a type declaration or type parameter 
is the opposite of the variance position of the type declaration or parameter.  
\item
The type of a mutable variable is always in invariant position.
\item 
The prefix $S$ of a type selection \lstinline@$S$#$T$@ is always in invariant position.
\item
For a type argument $T$ of a type ~\lstinline@$S$[$\ldots T \ldots$ ]@: If the
corresponding type parameter is invariant, then $T$ is in
invariant position.  If the corresponding type parameter is
contravariant, the variance position of $T$ is the opposite of
the variance position of the enclosing type ~\lstinline@$S$[$\ldots T \ldots$ ]@.
\end{itemize}
\todo{handle type aliases}
References to the type parameters in object-private values, variables,
or methods (\sref{sec:modifiers}) of the class are not checked for their variance
position. In these members the type parameter may appear anywhere
without restricting its legal variance annotations.

\example The following variance annotation is legal. 
\begin{lstlisting}
abstract class P[+A, +B] {
  def fst: A; def snd: B
}
\end{lstlisting}
With this variance annotation, type instances
of $P$ subtype covariantly with respect to their arguments. 
For instance, 
\begin{lstlisting}
P[IOException, String] <: P[Throwable, AnyRef] .
\end{lstlisting}

If the members of $P$ are mutable variables, 
the same variance annotation becomes illegal. 
\begin{lstlisting}
abstract class Q[+A, +B](x: A, y: B) { 
  var fst: A = x           // **** error: illegal variance:
  var snd: B = y           // `A', `B' occur in invariant position.
}
\end{lstlisting}
If the mutable variables are object-private, the class definition
becomes legal again:
\begin{lstlisting}
abstract class R[+A, +B](x: A, y: B) { 
  private[this] var fst: A = x        // OK
  private[this] var snd: B = y        // OK
}
\end{lstlisting}

\example The following variance annotation is illegal, since $a$ appears
in contravariant position in the parameter of \code{append}:

\begin{lstlisting}
abstract class Sequence[+A] {
  def append(x: Sequence[A]): Sequence[A]  
                  // **** error: illegal variance: 
                  // `A' occurs in contravariant position.
}
\end{lstlisting} 
The problem can be avoided by generalizing the type of \code{append}
by means of a lower bound:

\begin{lstlisting}
abstract class Sequence[+A] {
  def append[B >: A](x: Sequence[B]): Sequence[B] 
}
\end{lstlisting}

\example Here is a case where a contravariant type parameter is useful.

\begin{lstlisting}
abstract class OutputChannel[-A] {
  def write(x: A): Unit
}
\end{lstlisting}
With that annotation, we have that
\lstinline@OutputChannel[AnyRef]@ conforms to \lstinline@OutputChannel[String]@.  
That is, a
channel on which one can write any object can substitute for a channel
on which one can write only strings.

Function Declarations and Definitions
-------------------------------------

\label{sec:funsigs}

\syntax\begin{lstlisting} 
Dcl                ::=  `def' FunDcl
FunDcl             ::=  FunSig `:' Type
Def                ::=  `def' FunDef
FunDef             ::=  FunSig [`:' Type] `=' Expr 
FunSig             ::=  id [FunTypeParamClause] ParamClauses
FunTypeParamClause ::=  `[' TypeParam {`,' TypeParam} `]' 
ParamClauses       ::=  {ParamClause} [[nl] `(' `implicit' Params `)']
ParamClause        ::=  [nl] `(' [Params] `)'} 
Params             ::=  Param {`,' Param}
Param              ::=  {Annotation} id [`:' ParamType] [`=' Expr]
ParamType          ::=  Type 
                     |  `=>' Type 
                     |  Type `*'
\end{lstlisting}

A function declaration has the form ~\lstinline@def $f\,\psig$: $T$@, where
$f$ is the function's name, $\psig$ is its parameter
signature and $T$ is its result type. A function definition
~\lstinline@def $f\,\psig$: $T$ = $e$@~ also includes a {\em function body} $e$,
i.e.\ an expression which defines the function's result.  A parameter
signature consists of an optional type parameter clause \lstinline@[$\tps\,$]@,
followed by zero or more value parameter clauses
~\lstinline@($\ps_1$)$\ldots$($\ps_n$)@.  Such a declaration or definition
introduces a value with a (possibly polymorphic) method type whose
parameter types and result type are as given.

The type of the function body is expected to conform (\sref{sec:expr-typing}) 
to the function's declared
result type, if one is given. If the function definition is not
recursive, the result type may be omitted, in which case it is
determined from the packed type of the function body.

A type parameter clause $\tps$ consists of one or more type
declarations (\sref{sec:typedcl}), which introduce type parameters,
possibly with bounds.  The scope of a type parameter includes
the whole signature, including any of the type parameter bounds as
well as the function body, if it is present.  

A value parameter clause $\ps$ consists of zero or more formal
parameter bindings such as \lstinline@$x$: $T$@ or \lstinline@$x: T = e$@, which bind value
parameters and associate them with their types. Each value parameter
declaration may optionally define a default argument. The default argument
expression $e$ is type-checked with an expected type $T'$ obtained
by replacing all occurences of the function's type parameters in $T$ by
the undefined type.

For every parameter $p_{i,j}$ with a default argument a method named
\lstinline@$f\Dollar$default$\Dollar$n@ is generated which computes the default argument
expression. Here, $n$ denotes the parameter's position in the method
declaration. These methods are parametrized by the type parameter clause
\lstinline@[$\tps\,$]@ and all value parameter clauses
~\lstinline@($\ps_1$)$\ldots$($\ps_{i-1}$)@ preceeding $p_{i,j}$.
The \lstinline@$f\Dollar$default$\Dollar$n@ methods are inaccessible for
user programs.

The scope of a formal value parameter name $x$ comprises all subsequent parameter
clauses, as well as the method return type and the function body, if
they are given.\footnote{However, at present singleton types of method
parameters may only appear in the method body; so {\em dependent method
types} are not supported.} Both type parameter names
and value parameter names must be pairwise distinct.

\example In the method
\begin{lstlisting}
def compare[T](a: T = 0)(b: T = a) = (a == b)
\end{lstlisting}
the default expression \code{0} is type-checked with an undefined expected
type. When applying \code{compare()}, the default value \code{0} is inserted
and \code{T} is instantiated to \code{Int}. The methods computing the default
arguments have the form:
\begin{lstlisting}
def compare$\Dollar$default$\Dollar$1[T]: Int = 0
def compare$\Dollar$default$\Dollar$2[T](a: T): T = a
\end{lstlisting}

\subsection{By-Name Parameters}\label{sec:by-name-params}

\syntax\begin{lstlisting} 
ParamType          ::=  `=>' Type
\end{lstlisting}

The type of a value parameter may be prefixed by \code{=>}, e.g.\
~\lstinline@$x$: => $T$@. The type of such a parameter is then the
parameterless method type ~\lstinline@=> $T$@. This indicates that the
corresponding argument is not evaluated at the point of function
application, but instead is evaluated at each use within the
function. That is, the argument is evaluated using {\em call-by-name}.

The by-name modifier is disallowed for parameters of classes that
carry a \code{val} or \code{var} prefix, including parameters of case
classes for which a \code{val} prefix is implicitly generated. The
by-name modifier is also disallowed for implicit parameters (\sref{sec:impl-params}).

\example The declaration
\begin{lstlisting}
def whileLoop (cond: => Boolean) (stat: => Unit): Unit
\end{lstlisting}
indicates that both parameters of \code{whileLoop} are evaluated using
call-by-name.

\subsection{Repeated Parameters}\label{sec:repeated-params}

\syntax\begin{lstlisting} 
ParamType          ::=  Type `*'
\end{lstlisting}

The last value parameter of a parameter section may be suffixed by
``\code{*}'', e.g.\ ~\lstinline@(..., $x$:$T$*)@.  The type of such a
{\em repeated} parameter inside the method is then the sequence type
\lstinline@scala.Seq[$T$]@.  Methods with repeated parameters
\lstinline@$T$*@ take a variable number of arguments of type $T$.
That is, if a method $m$ with type ~\lstinline@($p_1:T_1 \commadots p_n:T_n,
p_s:S$*)$U$@~ is applied to arguments $(e_1 \commadots e_k)$ where $k \geq
n$, then $m$ is taken in that application to have type $(p_1:T_1
\commadots p_n:T_n, p_s:S \commadots p_{s'}S)U$, with $k - n$ occurrences of type
$S$ where any parameter names beyond $p_s$ are fresh. The only exception to this rule is if the last argument is
marked to be a {\em sequence argument} via a \lstinline@_*@ type
annotation. If $m$ above is applied to arguments
~\lstinline@($e_1 \commadots e_n, e'$: _*)@, then the type of $m$ in
that application is taken to be 
~\lstinline@($p_1:T_1\commadots p_n:T_n,p_{s}:$scala.Seq[$S$])@.

It is not allowed to define any default arguments in a parameter section
with a repeated parameter.

\example The following method definition computes the sum of the squares of a variable number
of integer arguments.
\begin{lstlisting}
def sum(args: Int*) = {
  var result = 0
  for (arg <- args) result += arg * arg
  result
}
\end{lstlisting}
The following applications of this method yield \code{0}, \code{1},
\code{6}, in that order.
\begin{lstlisting}
sum()
sum(1)
sum(1, 2, 3)
\end{lstlisting}
Furthermore, assume the definition:
\begin{lstlisting}
val xs = List(1, 2, 3)
\end{lstlisting}
The following application of method \lstinline@sum@ is ill-formed:
\begin{lstlisting}
sum(xs)       // ***** error: expected: Int, found: List[Int]
\end{lstlisting}
By contrast, the following application is well formed and yields again
the result \code{6}:
\begin{lstlisting}
sum(xs: _*) 
\end{lstlisting}

\subsection{Procedures}\label{sec:procedures}

\syntax\begin{lstlisting} 
  FunDcl   ::=  FunSig
  FunDef   ::=  FunSig [nl] `{' Block `}'
\end{lstlisting}

Special syntax exists for procedures, i.e.\ functions that return the
\verb@Unit@ value \verb@{}@. 
A procedure declaration is a function declaration where the result type
is omitted. The result type is then implicitly completed to the
\verb@Unit@ type. E.g., ~\lstinline@def $f$($\ps$)@~ is equivalent to
~\lstinline@def $f$($\ps$): Unit@.

A procedure definition is a function definition where the result type
and the equals sign are omitted; its defining expression must be a block.
E.g., ~\lstinline@def $f$($\ps$) {$\stats$}@~ is equivalent to
~\lstinline@def $f$($\ps$): Unit = {$\stats$}@.

\example Here is a declaration and a definition of a procedure named \lstinline@write@: 
\begin{lstlisting}
trait Writer { 
  def write(str: String)
}
object Terminal extends Writer {
  def write(str: String) { System.out.println(str) }
}
\end{lstlisting}
The code above is implicitly completed to the following code:
\begin{lstlisting}
trait Writer { 
  def write(str: String): Unit
}
object Terminal extends Writer {
  def write(str: String): Unit = { System.out.println(str) }
}
\end{lstlisting}

\subsection{Method Return Type Inference}\label{sec:meth-type-inf}

\comment{
Functions that are members of a class $C$ may define parameters
without type annotations. The types of such parameters are inferred as
follows.  Say, a method $m$ in a class $C$ has a parameter $p$ which
does not have a type annotation. We first determine methods $m'$ in
$C$ that might be overridden (\sref{sec:overriding}) by $m$, assuming
that appropriate types are assigned to all parameters of $m$ whose
types are missing.  If there is exactly one such method, the type of
the parameter corresponding to $p$ in that method -- seen as a member
of $C$ -- is assigned to $p$. It is a compile-time error if there are
several such overridden methods $m'$, or if there is none.
}

A class member definition $m$ that overrides some other function $m'$
in a base class of $C$ may leave out the return type, even if it is
recursive. In this case, the return type $R'$ of the overridden
function $m'$, seen as a member of $C$, is taken as the return type of
$m$ for each recursive invocation of $m$. That way, a type $R$ for the
right-hand side of $m$ can be determined, which is then taken as the
return type of $m$. Note that $R$ may be different from $R'$, as long
as $R$ conforms to $R'$.

\comment{
\example Assume the following definitions:
\begin{lstlisting}
trait I[A] {
  def f(x: A)(y: A): A
}
class C extends I[Int] {
  def f(x)(y) = x + y
}
\end{lstlisting}
Here, the parameter and return types of \lstinline@f@ in \lstinline@C@ are
inferred from the corresponding types of \lstinline@f@ in \lstinline@I@. The 
signature of \lstinline@f@ in \lstinline@C@ is thus inferred to be
\begin{lstlisting}
  def f(x: Int)(y: Int): Int
\end{lstlisting}
}

\example Assume the following definitions:
\begin{lstlisting}
trait I {
  def factorial(x: Int): Int
}
class C extends I {
  def factorial(x: Int) = if (x == 0) 1 else x * factorial(x - 1)
}
\end{lstlisting}
Here, it is OK to leave out the result type of \lstinline@factorial@
in \lstinline@C@, even though the method is recursive. 


\comment{
For any index $i$ let $\fsig_i$ be a function signature consisting of a function
name, an optional type parameter section, and zero or more parameter
sections.  Then a function declaration 
~\lstinline@def $\fsig_1 \commadots \fsig_n$: $T$@~ 
is a shorthand for the sequence of function
declarations ~\lstinline@def $\fsig_1$: $T$; ...; def $\fsig_n$: $T$@.  
A function definition ~\lstinline@def $\fsig_1 \commadots \fsig_n$ = $e$@~ is a
shorthand for the sequence of function definitions 
~\lstinline@def $\fsig_1$ = $e$; ...; def $\fsig_n$ = $e$@.  
A function definition
~\lstinline@def $\fsig_1 \commadots \fsig_n: T$ = $e$@~ is a shorthand for the
sequence of function definitions 
~\lstinline@def $\fsig_1: T$ = $e$; ...; def $\fsig_n: T$ = $e$@.
}

\comment{
\section{Overloaded Definitions}
\label{sec:overloaded-defs}
\todo{change}

An overloaded definition is a set of $n > 1$ value or function
definitions in the same statement sequence that define the same name,
binding it to types ~\lstinline@$T_1 \commadots T_n$@, respectively.
The individual definitions are called {\em alternatives}.  Overloaded
definitions may only appear in the statement sequence of a template.
Alternatives always need to specify the type of the defined entity
completely.  It is an error if the types of two alternatives $T_i$ and
$T_j$ have the same erasure (\sref{sec:erasure}).

\todo{Say something about bridge methods.}
%This must be a well-formed
%overloaded type
}

\section{Import Clauses}
\label{sec:import}

\syntax\begin{lstlisting}
  Import          ::= `import' ImportExpr {`,' ImportExpr}
  ImportExpr      ::= StableId `.' (id | `_' | ImportSelectors)
  ImportSelectors ::= `{' {ImportSelector `,'} 
                      (ImportSelector | `_') `}'
  ImportSelector  ::= id [`=>' id | `=>' `_']
\end{lstlisting}

An import clause has the form ~\lstinline@import $p$.$I$@~ where $p$ is a stable
identifier (\sref{sec:paths}) and $I$ is an import expression.
The import expression determines a set of names of importable members of $p$
which are made available without qualification.  A member $m$ of $p$ is
{\em importable} if it is not object-private (\sref{sec:modifiers}).
The most general form of an import expression is a list of {\em import
selectors}
\begin{lstlisting}
{ $x_1$ => $y_1 \commadots x_n$ => $y_n$, _ } .
\end{lstlisting}
for $n \geq 0$, where the final wildcard `\lstinline@_@' may be absent.  It
makes available each importable member \lstinline@$p$.$x_i$@ under the unqualified name
$y_i$. I.e.\ every import selector ~\lstinline@$x_i$ => $y_i$@~ renames
\lstinline@$p$.$x_i$@ to
$y_i$.  If a final wildcard is present, all importable members $z$ of
$p$ other than ~\lstinline@$x_1 \commadots x_n,y_1 \commadots y_n$@~ are also made available
under their own unqualified names.

Import selectors work in the same way for type and term members. For
instance, an import clause ~\lstinline@import $p$.{$x$ => $y\,$}@~ renames the term
name \lstinline@$p$.$x$@ to the term name $y$ and the type name \lstinline@$p$.$x$@
to the type name $y$. At least one of these two names must
reference an importable member of $p$.

If the target in an import selector is a wildcard, the import selector
hides access to the source member. For instance, the import selector
~\lstinline@$x$ => _@~ ``renames'' $x$ to the wildcard symbol (which is
unaccessible as a name in user programs), and thereby effectively
prevents unqualified access to $x$. This is useful if there is a
final wildcard in the same import selector list, which imports all
members not mentioned in previous import selectors.

The scope of a binding introduced by an import-clause starts
immediately after the import clause and extends to the end of the
enclosing block, template, package clause, or compilation unit,
whichever comes first.

Several shorthands exist. An import selector may be just a simple name
$x$. In this case, $x$ is imported without renaming, so the
import selector is equivalent to ~\lstinline@$x$ => $x$@. Furthermore, it is
possible to replace the whole import selector list by a single
identifier or wildcard. The import clause ~\lstinline@import $p$.$x$@~ is
equivalent to ~\lstinline@import $p$.{$x\,$}@~, i.e.\ it makes available without
qualification the member $x$ of $p$. The import clause
~\lstinline@import $p$._@~ is equivalent to
~\lstinline@import $p$.{_}@, 
i.e.\ it makes available without qualification all members of $p$
(this is analogous to ~\lstinline@import $p$.*@~ in Java).

An import clause with multiple import expressions
~\lstinline@import $p_1$.$I_1 \commadots p_n$.$I_n$@~ is interpreted as a
sequence of import clauses 
~\lstinline@import $p_1$.$I_1$; $\ldots$; import $p_n$.$I_n$@.

\example Consider the object definition:
\begin{lstlisting}
object M { 
  def z = 0, one = 1  
  def add(x: Int, y: Int): Int = x + y 
}
\end{lstlisting}
Then the block
\begin{lstlisting}
{ import M.{one, z => zero, _}; add(zero, one) }
\end{lstlisting}
is equivalent to the block 
\begin{lstlisting}
{ M.add(M.z, M.one) } .
\end{lstlisting}