\section{Type expressions}
\pdfsection{Type expressions}
%HEVEA\cutname{types.html}
\ikwd{as\@\texttt{as}}

\begin{syntax}
typexpr:
        "'" ident
      | "_"
      | '(' typexpr ')'
      | [['?']label-name':'] typexpr '->' typexpr
      | typexpr {{ '*' typexpr }}
      | typeconstr
      | typexpr typeconstr
      | '(' typexpr { ',' typexpr } ')' typeconstr
      | typexpr 'as' "'" ident
      | polymorphic-variant-type
      | '<' ['..'] '>'
      | '<' method-type { ';' method-type } [';' || ';' '..'] '>'
      | '#' class-path
      | typexpr '#' class-path
      | '(' typexpr { ',' typexpr } ')' '#' class-path
;
poly-typexpr:
        typexpr
      | {{ "'" ident }} '.' typexpr
;
method-type:
    method-name ':' poly-typexpr
\end{syntax}
See also the following language extensions:
\hyperref[s-first-class-modules]{first-class modules},
\hyperref[s:attributes]{attributes} and
\hyperref[s:extension-nodes]{extension nodes}.

The table below shows the relative precedences and associativity of
operators and non-closed type constructions. The constructions with
higher precedences come first.
\ikwd{as\@\texttt{as}}
\begin{tableau}{|l|l|}{Operator}{Associativity}
\entree{Type constructor application}{--}
\entree{"#"}{--}
\entree{"*"}{--}
\entree{"->"}{right}
\entree{"as"}{--}
\end{tableau}

Type expressions denote types in definitions of data types as well as
in type constraints over patterns and expressions.

\subsubsection*{Type variables}

The type expression @"'" ident@ stands for the type variable named
@ident@. The type expression @"_"@ stands for either an anonymous type
variable or anonymous type parameters. In data type definitions, type
variables are names for the data type parameters. In type constraints,
they represent unspecified types that can be instantiated by any type
to satisfy the type constraint.  In general the scope of a named type
variable is the whole top-level phrase where it appears, and it can
only be generalized when leaving this scope.  Anonymous variables have
no such restriction. In the following cases, the scope of named type
variables is restricted to the type expression where they appear:
1) for universal (explicitly polymorphic) type variables;
2) for type variables that only appear in public method specifications
(as those variables will be made universal, as described in
section~\ref{sec-methspec});
3) for variables used as aliases, when the type they are aliased to
would be invalid in the scope of the enclosing definition ({\it i.e.}
when it contains free universal type variables, or locally
defined types.)

\subsubsection*{Parenthesized types}

The type expression @"(" typexpr ")"@ denotes the same type as
@typexpr@.

\subsubsection*{Function types}

The type expression @typexpr_1 '->' typexpr_2@ denotes the type of
functions mapping arguments of type @typexpr_1@ to results of type
@typexpr_2@.

@label-name ':' typexpr_1 '->' typexpr_2@ denotes the same function type, but
the argument is labeled @label@.

@'?' label-name ':' typexpr_1 '->' typexpr_2@ denotes the type of functions
mapping an optional labeled argument of type @typexpr_1@ to results of
type @typexpr_2@. That is, the physical type of the function will be
@typexpr_1 "option" '->' typexpr_2@.

\subsubsection*{Tuple types}

The type expression @typexpr_1 '*' \ldots '*' typexpr_n@
denotes the type of tuples whose elements belong to types @typexpr_1,
\ldots typexpr_n@ respectively.

\subsubsection*{Constructed types}

Type constructors with no parameter, as in @typeconstr@, are type
expressions.

The type expression @typexpr typeconstr@, where @typeconstr@ is a type
constructor with one parameter, denotes the application of the unary type
constructor @typeconstr@ to the type @typexpr@.

The type expression @(typexpr_1,\ldots,typexpr_n) typeconstr@, where
@typeconstr@ is a type constructor with $n$ parameters, denotes the
application of the $n$-ary type constructor @typeconstr@ to the types
@typexpr_1@ through @typexpr_n@.

In the type expression @ "_"  typeconstr @, the anonymous type expression
@ "_" @ stands in for anonymous type parameters and is equivalent to
@ ("_", \ldots,"_") @ with as many repetitions of "_" as the arity of
@typeconstr@.

\subsubsection*{Aliased and recursive types}

\ikwd{as\@\texttt{as}}

The type expression @typexpr 'as' "'" ident@ denotes the same type as
@typexpr@, and also binds the type variable @ident@ to type @typexpr@ both
in @typexpr@ and in other types.  In general the scope of an alias is
the same as for a named type variable, and covers the whole enclosing
definition. If the type variable
@ident@ actually occurs in @typexpr@, a recursive type is created. Recursive
types for which  there exists a recursive path that does not contain
an object or polymorphic variant type constructor are rejected, except
when the "-rectypes" mode is selected.

If @"'" ident@ denotes an explicit polymorphic variable, and @typexpr@
denotes either an object or polymorphic variant type, the row variable
of @typexpr@ is captured by @"'" ident@, and quantified upon.

\subsubsection*{Polymorphic variant types}
\ikwd{of\@\texttt{of}}

\begin{syntax}
polymorphic-variant-type:
        '[' tag-spec-first { '|' tag-spec } ']'
      | '[>' [ tag-spec ] { '|' tag-spec } ']'
      | '[<' ['|'] tag-spec-full { '|' tag-spec-full }
             [ '>' {{ '`'tag-name }} ] ']'
;
%\end{syntax} \begin{syntax}
tag-spec-first:
        '`'tag-name [ 'of' typexpr ]
      | [ typexpr ] '|' tag-spec
;
tag-spec:
        '`'tag-name [ 'of' typexpr ]
      | typexpr
;
tag-spec-full:
        '`'tag-name [ 'of' ['&'] typexpr { '&' typexpr } ]
      | typexpr
\end{syntax}

Polymorphic variant types describe the values a polymorphic variant
may take.

The first case is an exact variant type: all possible tags are
known, with their associated types, and they can all be present.
Its structure is fully known.

The second case is an open variant type, describing a polymorphic
variant value: it gives the list of all tags the value could take,
with their associated types. This type is still compatible with a
variant type containing more tags. A special case is the unknown
type, which does not define any tag, and is compatible with any
variant type.

The third case is a closed variant type. It gives information about
all the possible tags and their associated types, and which tags are
known to potentially appear in values. The exact variant type (first
case) is
just an abbreviation for a closed variant type where all possible tags
are also potentially present.

In all three cases, tags may be either specified directly in the
@'`'tag-name ["of" typexpr]@ form, or indirectly through a type
expression, which must expand to an
exact variant type, whose tag specifications are inserted in its
place.

Full specifications of variant tags are only used for non-exact closed
types. They can be understood as a conjunctive type for the argument:
it is intended to have all the types enumerated in the
specification.

Such conjunctive constraints may be unsatisfiable. In such a case the
corresponding tag may not be used in a value of this type. This
does not mean that the whole type is not valid: one can still use
other available tags.
Conjunctive constraints are mainly intended as output from the type
checker. When they are used in source programs, unsolvable constraints
may cause early failures.

\subsubsection*{Object types}

An object type
@'<' [method-type { ';' method-type }] '>'@
is a record of method types.

Each method may have an explicit polymorphic type: @{{ "'" ident }}
'.' typexpr@. Explicit polymorphic variables have a local scope, and
an explicit polymorphic type can only be unified to an
equivalent one, where only the order and names of polymorphic
variables may change.

The type @'<' {method-type ';'} '..'  '>'@ is the
type of an object whose method names and types are described by
@method-type_1, \ldots, method-type_n@, and possibly some other
methods represented by the ellipsis.  This ellipsis actually is
a special kind of type variable (called {\em row variable} in the
literature) that stands for any number of extra method types.

\subsubsection*{\#-types}
\label{s:sharp-types}

The type @'#' class-path@ is a special kind of abbreviation. This
abbreviation unifies with the type of any object belonging to a subclass
of class @class-path@.
%
It is handled in a special way as it usually hides a type variable (an
ellipsis, representing the methods that may be added in a subclass).
In particular, it vanishes when the ellipsis gets instantiated.
%
Each type expression @'#' class-path@ defines a new type variable, so
type @'#' class-path '->' '#' class-path@ is usually not the same as
type @('#' class-path 'as' "'" ident) '->' "'" ident@.
%

Use of \#-types to abbreviate polymorphic variant types is deprecated.
If @@t@@ is an exact variant type then @"#"@t@@ translates to @"[<" @t@"]"@,
and @"#"@t@"[>" "`"tag_1 \dots"`"tag_k"]"@ translates to
@"[<" @t@ ">" "`"tag_1 \dots"`"tag_k"]"@

\subsubsection*{Variant and record types}

There are no type expressions describing (defined) variant types nor
record types, since those are always named, i.e. defined before use
and referred to by name.  Type definitions are described in
section~\ref{s:type-defs}.
