\htmlhr
\chapterAndLabel{Generics and polymorphism}{polymorphism}

Section~\ref{generics} describes support for Java generics (also known as
``parametric polymorphism'').
Section~\ref{method-qualifier-polymorphism} describes polymorphism over
type qualifiers for methods. Section~\ref{class-qualifier-polymorphism} describes
polymorphism over type qualifiers for classes.


\sectionAndLabel{Generics (parametric polymorphism or type polymorphism)}{generics}

The Checker Framework fully supports
type-qualified Java generic types and methods (also known as ``parametric
polymorphism'').
When instantiating a generic type,
clients supply the qualifier along with the type argument, as in
\code{List<@NonNull String>}.
When using a type variable \code{T} within the implementation of a generic type,
typically no type qualifier is written (see Section~\ref{type-variable-use});
rather, the instantiation of the type parameter is restricted (see
Section~\ref{generics-instantiation}).


\subsectionAndLabel{Raw types}{generics-raw-types}

Before running any pluggable type-checker, we recommend that you eliminate
raw types from your code (e.g., your code should use \code{List<...>} as
opposed to \code{List}).
Your code should compile without warnings when using the standard Java
compiler and the \<-Xlint:unchecked -Xlint:rawtypes> command-line options.
Using generics helps prevent type errors just as using a pluggable
type-checker does, and makes the Checker Framework's warnings easier to
understand.

If your code uses raw types, then the Checker Framework will do its best to
infer the Java type arguments and the type qualifiers.  By default these
inferred types are ignored in subtyping checks. If you supply the
command-line option \<-AignoreRawTypeArguments=false> you will see errors
from raw types.


\subsectionAndLabel{Restricting instantiation of a generic class}{generics-instantiation}

When you define a generic class in Java, the \<extends> clause
of the generic type parameter (known as the ``upper bound'') requires that
the corresponding type argument must be a subtype of the bound.
For example, given the definition
\verb|class G<T extends Number> {...}|,
the upper bound is \<Number>
and a client can instantiate it as \code{G<Number>} or \code{G<Integer>}
but not \code{G<Date>}.

You can write a type qualifier on the \<extends> clause to make the upper
bound a qualified type.  For example, you can declare that a generic list class can hold only non-null values:
% Similarly, a generic map
% class might indicate it requires an immutable key type, but that it
% supports both nullable and non-null value types.

\begin{Verbatim}
  class MyList<T extends @NonNull Object> {...}

  MyList<@NonNull String> m1;       // OK
  MyList<@Nullable String> m2;      // error
\end{Verbatim}

That is, in the above example, all
arguments that replace \code{T} in \code{MyList<T>} must be subtypes of
\code{@NonNull Object}.


\subsubsectionAndLabel{Syntax for upper and lower bounds}{generics-bounds-syntax}

Conceptually, each generic type parameter has two bounds --- a lower bound
and an upper bound --- and at instantiation, the type argument must be
within the bounds.  Java only allows you to specify the upper bound; the
lower bound is implicitly the bottom type \<void>.  The Checker Framework
gives you more power:  you can specify both an upper and lower bound for
type parameters.
Write the upper bound on the \<extends> clause, and
write the lower bound on the type variable.

\begin{Verbatim}
  class MyList<@LowerBound T extends @UpperBound Object> { ... }
\end{Verbatim}

You may omit either the upper or the lower bound, and the Checker Framework
will use a default.

For a discussion of wildcards, see Section~\ref{annotations-on-wildcards}.

For a concrete example, consider the type system of the Regex Checker (see
Figure~\refwithpage{fig-regex-hierarchy}) in which
 \<@Regex(0)> :>
 \<@Regex(1)> :>
 \<@Regex(2)> :>
 \<@Regex(3)> :> \ldots.

\begin{Verbatim}
  class MyRegexes<@Regex(5) T extends @Regex(1) String> { ... }

  MyRegexes<@Regex(0) String> mu;   // error - @Regex(0) is not a subtype of @Regex(1)
  MyRegexes<@Regex(1) String> m1;   // OK
  MyRegexes<@Regex(3) String> m3;   // OK
  MyRegexes<@Regex(5) String> m5;   // OK
  MyRegexes<@Regex(6) String> m6;   // error - @Regex(6) is not a supertype of @Regex(5)
\end{Verbatim}

The above declaration states that the upper bound of the type variable
is \<@Regex(1) String> and the lower bound is \<@Regex(5) void>.  That is,
arguments that replace \code{T} in \code{MyList<T>} must be subtypes of
\code{@Regex(1) String} and supertypes of \code{@Regex(5) void}.
Since \<void> cannot be used to instantiate a generic class, \<MyList> may
be instantiated with \<@Regex(1) String> through \<@Regex(5) String>.


To specify an exact bound, place the same annotation on both bounds.  For example:

\begin{Verbatim}
  class MyListOfNonNulls<@NonNull T extends @NonNull Object> { ... }
  class MyListOfNullables<@Nullable T extends @Nullable Object> { ... }

  MyListOfNonNulls<@NonNull Number> v1;      // OK
  MyListOfNonNulls<@Nullable Number> v2;     // error
  MyListOfNullables<@NonNull Number> v4;     // error
  MyListOfNullables<@Nullable Number> v3;    // OK
\end{Verbatim}

It is an error if the lower bound is not a subtype of the upper bound.

%BEGIN LATEX
\begin{smaller}
%END LATEX
\begin{Verbatim}
  class MyClass<@Nullable T extends @NonNull Object>  // error: @Nullable is not a subtype of @NonNull
\end{Verbatim}
%BEGIN LATEX
\end{smaller}
%END LATEX


\subsubsectionAndLabel{Defaults}{generics-defaults}

A generic type parameter or wildcard is written as \code{class
  MyClass<\emph{@LowerBound} T extends \emph{@UpperBound} JavaUpperBound>} or as
\code{MyClass<\emph{@UpperBound} ? super \emph{@LowerBound} JavaLowerBound>}, where
``\<\emph{@LowerBound}>'' and ``\<\emph{@UpperBound}>'' are type qualifiers.

For lower bounds:
If no type annotation is written in front of \<?>,
then the lower bound defaults to \<@\emph{BottomType} void>.

For upper bounds:
\begin{itemize}
\item
If the \<extends> clause is omitted,
then the upper bound defaults to \<@\emph{TopType} Object>.
\item
If the \<extends> clause is written but contains no type qualifier,
then the normal defaulting rules apply to the type in the \<extends>
clause (see Section~\ref{climb-to-top}).
\end{itemize}

The upper-bound rules mean that even though in Java the following two
declarations are equivalent:

\begin{Verbatim}
  class MyClass<T>
  class MyClass<T extends Object>
\end{Verbatim}

\noindent
they specify different type qualifiers on the upper bound,
if the type system's default annotation is not its top annotation.

The Nullness type system is an example.

\begin{Verbatim}
  class MyClass<T>                 ==  class MyClass<T extends @Nullable Object>
  class MyClass<T extends Object>  ==  class MyClass<T extends @NonNull Object>
\end{Verbatim}

The rationale for this choice is:
\begin{itemize}
\item
  The ``\code{<T>}'' in \code{MyClass<T>} means ``fully unconstrained'',
  and the rules maintain that, without the need for a programmer to
  change existing code.
\item
  The ``\code{Object}'' in \code{MyClass<T extends Object>} is treated
  exactly like every other occurrence of \code{Object} in the program ---
  it would be confusing for different occurrences of \code{Object} to mean
  different annotated types.
\end{itemize}

Here are some style guidelines:
\begin{itemize}
\item
  Use ``\code{<T>}'' when there are no constraints on the type qualifiers.
  This is short and is what already appears in source code.
\item
  Whenever you write an \<extends> clause, write an explicit type
  annotation on it.  For example, for the Nullness Checker, write
  \code{class MyClass<T>} rather than \code{class MyClass<T extends
    @Nullable Object>}, and write \code{class MyClass<T extends @NonNull
    Object>} rather than \code{class MyClass<T extends Object>}.
\end{itemize}

For further discussion, see Section~\ref{faq-implicit-bounds}.


\subsectionAndLabel{Type annotations on a use of a generic type variable}{type-variable-use}

A type annotation on a use of a generic type variable overrides/ignores any type
qualifier (in the same type hierarchy) on the corresponding actual type
argument.  For example, suppose that \code{T} is a formal type parameter.
Then using \code{@Nullable T} within the scope of \code{T} applies the type
qualifier \code{@Nullable} to the (unqualified) Java type of \code{T}\@.
This feature is sometimes useful, but more often the implementation of a
generic type just uses the type variable \code{T}, whose instantiation is
restricted (see Section~\ref{generics-instantiation}).

Here is an example of applying a type annotation to a generic type
variable:

\begin{Verbatim}
  class MyClass2<T> {
    ...
    @Nullable T myField = null;
    ...
  }
\end{Verbatim}

\noindent
The type annotation does not restrict how \code{MyClass2} may be
instantiated.  In other words, both
\code{MyClass2<@NonNull String>} and \code{MyClass2<@Nullable String>} are
legal, and in both cases \code{@Nullable T} means \code{@Nullable String}.
In \code{MyClass2<@Interned String>},
\code{@Nullable T} means \code{@Nullable @Interned String}.

% Note that a type annotation on a generic type variable does not act like
% other type qualifiers.  In both cases the type annotation acts as a type
% constructor, but as noted above they act slightly differently.


% %% This isn't quite right because a type qualifier is itself a type
% %% constructor.
% More formally, a type annotation on a generic type variable acts as a type
% constructor rather than a type qualifier.  Another example of a type
% constructor is \code{[]}.  Just as \code{T[]} is not the same type as
% \code{T}, \code{@Nullable T} is not (necessarily) the same type as
% \code{T}.


Defaulting never affects a use of a type variable, even if the type
variable use has no explicit annotation.  Defaulting helps to choose a
single type qualifier for a concrete Java class or interface.  By contrast,
a type variable use represents a set of possible types.


\subsectionAndLabel{Annotations on wildcards}{annotations-on-wildcards}

At an instantiation of a generic type, a Java wildcard indicates that some
constraints are known on the type argument, but the type argument is not known
exactly.
For example, you can indicate that the type parameter for variable \<ls> is
some unknown subtype of \<CharSequence>:

\begin{Verbatim}
  List<? extends CharSequence> ls;
  ls = new ArrayList<String>();      // OK
  ls = new ArrayList<Integer>();     // error: Integer is not a subtype of CharSequence
\end{Verbatim}

For more details about wildcards, see the
\href{https://docs.oracle.com/javase/tutorial/java/generics/wildcards.html}{Java
  tutorial on wildcards} or
\href{https://docs.oracle.com/javase/specs/jls/se17/html/jls-4.html#jls-4.5.1}{JLS
  \S 4.5.1}.

You can write a type annotation on the bound of a wildcard:

\begin{Verbatim}
  List<? extends @NonNull CharSequence> ls;
  ls = new ArrayList<@NonNull String>();    // OK
  ls = new ArrayList<@Nullable String>();   // error: @Nullable is not a subtype of @NonNull
\end{Verbatim}

Conceptually, every wildcard has two bounds --- an upper bound and a lower
bound.  Java only permits you to write one bound.
You can specify the upper bound with \code{<?\ extends SomeType>}, in which
case the lower bound is implicitly the bottom type \<void>.
You can specify the lower bound (with \code{<?\ super OtherType>}), in
which case the upper bound is implicitly the top type \<Object>.
The Checker Framework is more flexible:  it lets you similarly write
annotations on both the upper and lower bound.

To annotate the \emph{implicit} bound, write the type annotation
before the \<?>.  For example:

\begin{Verbatim}
  List<@LowerBound ? extends @UpperBound CharSequence> lo;
  List<@UpperBound ? super @NonNull Number> ls;
\end{Verbatim}

For an unbounded wildcard (\code{<?>}, with neither
bound specified), the annotation in front of a wildcard applies
to both bounds.  The following three declarations are equivalent (except
that you cannot write the bottom type \<void>; note that
\sunjavadoc{java.base/java/lang/Void.html}{Void} does not denote the bottom type):

\begin{Verbatim}
  List<@NonNull ?> lnn;
  List<@NonNull ? extends @NonNull Object> lnn;
  List<@NonNull ? super @NonNull void> lnn;
\end{Verbatim}

\noindent
Note that the annotation in front of a type parameter always applies to its
lower bound, because type parameters can only be written with \<extends>
and never \<super>.


% Defaults are as for type variables (see Section~\ref{generics-defaults}),
% with one exception.

The defaulting rules for
wildcards also differ from those of type parameters (see
Section~\ref{inherited-wildcard-annotations}).


%% Mike isn't sure that this section pulls its weight, especially since it
%% doesn't justify why it is desirable to be able to constrain both the
%% upper and the lower bound of a type.  If readers believe that, they will
%% be OK with the syntax.
% \subsubsectionAndLabel{Type parameter declaration annotation rationale}{type-parameter-rationale}
%
% It is desirable to be able to constrain both the upper and the lower bound
% of a type, as in
%
% \begin{Verbatim}
%   class MyClass<T extends @C MyUpperBound super @D void> { ... }
% \end{Verbatim}
%
% However, doing so is not possible due to two limitations of Java's syntax.
% First, it is illegal to specify both the upper and the lower bound of a
% type parameter or wildcard.
% Second, it is impossible to specify a type annotation for a lower
% bound without also specifying a type (use of \<void> is illegal).
%
% Thus, when you wish to specify both bounds, you write one of them
% explicitly, and you write the other one in front of the type variable name
% or \<?>.  When you wish to specify two identical bounds, you write a
% single annotation in front of the type variable name or \<?>.


\subsectionAndLabel{Examples of qualifiers on a type parameter}{type-parameter-qualifier-examples}

Recall that \<@Nullable \emph{X}> is a supertype of \<@NonNull \emph{X}>,
for any \emph{X}\@.
Most of the following types mean different things:

\begin{Verbatim}
  class MyList1<@Nullable T> { ... }
  class MyList1a<@Nullable T extends @Nullable Object> { ... } // same as MyList1
  class MyList2<@NonNull T extends @NonNull Object> { ... }
  class MyList2a<T extends @NonNull Object> { ... } // same as MyList2
  class MyList3<T extends @Nullable Object> { ... }
\end{Verbatim}

\<MyList1> and \<MyList1a> must be instantiated with a nullable type.
The implementation of \<MyList1> must be able to consume (store) a null
value and produce (retrieve) a null value.

\<MyList2> and \<MyList2a> must be instantiated with non-null type.
The implementation of \<MyList2> has to account for only non-null values --- it
does not have to account for consuming or producing null.

\<MyList3> may be instantiated either way:
with a nullable type or a non-null type.  The implementation of \<MyList3> must consider
that it may be instantiated either way --- flexible enough to support either
instantiation, yet rigorous enough to impose the correct constraints of the
specific instantiation.  It must also itself comply with the constraints of
the potential instantiations.

One way to express the difference among \<MyList1>, \<MyList2>, and
\<MyList3> is by comparing what expressions are legal in the implementation
of the list --- that is, what expressions may appear in the ellipsis in the
declarations above, such as inside a method's body.  Suppose each class
has, in the ellipsis, these declarations:

\begin{Verbatim}
  T t;
  @Nullable T nble;      // Section "Type annotations on a use of a generic type variable", below,
  @NonNull T nn;         // further explains the meaning of "@Nullable T" and "@NonNull T".
  void add(T arg) {}
  T get(int i) {}
\end{Verbatim}

\noindent
Then the following expressions would be legal, inside a given
implementation --- that is, also within the ellipses.
(Compilable source code appears as file
\<checker-framework/checker/tests/nullness/generics/GenericsExample.java>.)

\begin{tabular}{|l|c|c|c|c|c|} \hline
                        & MyList1 & MyList2 & MyList3 \\ \hline
  t = null;             & OK      & error   & error   \\ \hline
  t = nble;             & OK      & error   & error   \\ \hline
  nble = null;          & OK      & OK      & OK      \\ \hline
  nn = null;            & error   & error   & error   \\ \hline
  t = this.get(0);      & OK      & OK      & OK      \\ \hline
  nble = this.get(0);   & OK      & OK      & OK      \\ \hline
  nn = this.get(0);     & error   & OK      & error   \\ \hline
  this.add(t);          & OK      & OK      & OK      \\ \hline
  this.add(nble);       & OK      & error   & error   \\ \hline
  this.add(nn);         & OK      & OK      & OK      \\ \hline
\end{tabular}


%% This text is not very helpful.
% The
% implementation of \code{MyList2} may only place non-null objects in the
% list and may assume that retrieved elements are non-null.  The
% implementation of \code{MyList3} is similar in that it may only place
% non-null objects in the list, because it might be instantiated as, say,
% \code{MyList3<@NonNull Date>}.  When retrieving elements from the list,
% the implementation of \code{MyList3} must account for the fact that
% elements of \code{MyList3} may be null, because it might be instantiated
% as, say, \code{MyList3<@Nullable Date>}.
The differences are more
significant when the qualifier hierarchy is more complicated than just
\<@Nullable> and \<@NonNull>.

\subsectionAndLabel{Covariant type parameters}{covariant-type-parameters}

Java types are \emph{invariant} in their type parameter.  This means that
\code{A<X>} is a subtype of \code{B<Y>} only if \<X> is identical to \<Y>.  For
example, \code{ArrayList<Number>} is a subtype of \code{List<Number>}, but
neither \code{ArrayList<Integer>} nor \code{List<Integer>} is a subtype of
\code{List<Number>}.  (If they were, there would be a loophole in the Java
type system.)  For the same reason, type parameter annotations are treated
invariantly.  For example, \code{List<@Nullable String>} is not a subtype
of \code{List<String>}.

When a type parameter is used in a read-only way --- that is, when clients
read values of that type from the class but never pass values of that type
to the class --- then it is safe for the
type to be \emph{covariant} in the type parameter.  Use the
\refqualclass{framework/qual}{Covariant} annotation to indicate this.
When a type parameter is covariant, two instantiations of the class with
different type arguments have the same subtyping relationship as the type
arguments do.

For example, consider \<Iterator>.  A client can read elements but not
write them, so \code{Iterator<@Nullable String>} can be a subtype of
\code{Iterator<String>} without introducing a hole in the type system.
Therefore, its type parameter is annotated with
\refqualclass{framework/qual}{Covariant}.
The first type parameter of \<Map.Entry> is also covariant.
Another example would be the type parameter of a hypothetical class
\<ImmutableList>.

The \<@Covariant> annotation is trusted but not checked.
If you incorrectly specify as covariant a type parameter that can be
written (say, the class supports a
\<set> operation or some other mutation on an object of that type), then
you have created an unsoundness in the type system.
For example, it would be incorrect to annotate the type parameter of
\<ListIterator> as covariant, because \<ListIterator> supports a \<set>
operation.


\subsectionAndLabel{Method type argument inference and type qualifiers}{infer-method-type-qualifiers}

Sometimes method type argument inference does not interact well with
type qualifiers. In such situations, you might need to provide
explicit method type arguments, for which the syntax is as follows:

\begin{alltt}
    Collections.<@MyTypeAnnotation Object>sort(l, c);
\end{alltt}

\noindent
This uses Java's existing syntax for specifying a method call's type arguments.


\subsectionAndLabel{The Bottom type}{bottom-type}

Many type systems have a \<*Bottom> type that is used only for the \<null>
value, dead code, and some erroneous situations.  A programmer should
rarely write the bottom type.

One use is on a lower bound, to indicate that any type qualifier is
permitted.  A lower-bounded wildcard indicates that a consumer method can
accept a collection containing any Java type above some Java type, and you
can add the bottom type qualifier as well:

\begin{Verbatim}
public static void addNumbers(List<? super @SignednessBottom Integer> list) { ... }
\end{Verbatim}


\sectionAndLabel{Qualifier polymorphism for methods\label{qualifier-polymorphism}}{method-qualifier-polymorphism}

Type qualifier polymorphism permits a single method to have multiple different qualified
type signatures.

Here is where a polymorphic qualifier (e.g., \<@PolyNull>) can be used:
\begin{itemize}
\item Polymorphic qualifiers are most often used in method signatures.
  See the examples below in Section~\ref{qualifier-polymorphism-examples}.
\item Polymorphic qualifiers can also be written in method bodies
  (implementations).
\item If you can use generics, you typically do not need to use a
  polymorphic qualifier.  Do not write a polymorphic qualifier on a type
  variable declaration.
\item Polymorphic qualifiers may not be used on a class declaration.
  To apply qualifier polymorphism to classes, use a class qualifier
  parameter; see Section~\ref{class-qualifier-polymorphism}.
\item A polymorphic qualifier may be used on a field declaration only in a
  class with a class qualifier parameter; see Section~\ref{class-qual-param-field}.
\item If a class has a class qualifier parameter, then a polymorphic
  qualifier written on a method in the class has a slightly different
  meaning, see Section~\ref{class-qual-param-method}.
\end{itemize}

Section~\ref{creating-declarative-hierarchy} explains how to define a
polymorphic qualifier.


\subsectionAndLabel{Using polymorphic qualifiers in a method signature}{qualifier-polymorphism-examples}

A method whose signature has a polymorphic qualifier (such as \<@PolyNull>) conceptually has multiple
versions, somewhat like the generics feature of Java or a template in C++.
In each version, each instance of the polymorphic qualifier has been
replaced by the same other qualifier from the hierarchy.

The method body must type-check with all signatures.  A method call is
type-correct if it type-checks under any one of the signatures.  If a call
matches multiple signatures, then the compiler uses the most specific
matching signature for the purpose of type-checking.  This is the same as
Java's rule for resolving overloaded methods.


As an example of the use of \<@PolyNull>, method
\sunjavadoc{java.base/java/lang/Class.html\#cast(java.lang.Object)}{Class.cast}
returns null if and only if its argument is \<null>:

\begin{Verbatim}
  @PolyNull T cast(@PolyNull Object obj) { ... }
\end{Verbatim}

\noindent
This is like writing:

\begin{Verbatim}
   @NonNull T cast( @NonNull Object obj) { ... }
  @Nullable T cast(@Nullable Object obj) { ... }
\end{Verbatim}

\noindent
except that the latter is not legal Java, since it defines two
methods with the same Java signature.


As another example, consider

\begin{Verbatim}
  // Returns null if either argument is null.
  @PolyNull T max(@PolyNull T x, @PolyNull T y);
\end{Verbatim}

\noindent
which is like writing

\begin{Verbatim}
   @NonNull T max( @NonNull T x,  @NonNull T y);
  @Nullable T max(@Nullable T x, @Nullable T y);
\end{Verbatim}

\noindent
At a call site, the most specific applicable signature is selected.

Another way of thinking about which one of the two \code{max} variants is
selected is that the nullness annotations of (the declared types of) both
arguments are \emph{unified} to a type that is a supertype of both, also
known as the \emph{least upper bound} or lub.  If both
arguments are \code{@NonNull}, their unification (lub) is \<@NonNull>, and the
method return type is \<@NonNull>.  But if even one of the arguments is \<@Nullable>,
then the unification (lub) is \<@Nullable>, and so is the return type.



\subsectionAndLabel{Relationship to subtyping and generics}{qualifier-polymorphism-vs-subtyping}

Qualifier polymorphism has the same purpose and plays the same role as
Java's generics.  You use them for the similar reasons, such as:
\begin{itemize}
\item
  A method operates on collections with different types of
  elements.
\item
  Two different arguments have the same type, without constraining them to
  be one specific type.
\item
  A method returns a value of the same type as its argument.
\end{itemize}


If a method is written using Java generics, it usually does not need
qualifier polymorphism.  If you can use Java's generics, then that is often
better.  On the other hand, if you have legacy code that is not
written generically, and you cannot change it to use generics, then you can
use qualifier polymorphism to achieve a similar effect, with respect to
type qualifiers only.  The Java compiler still treats the base Java types
non-generically.

In some cases, you don't need qualifier polymorphism because subtyping
already provides the needed functionality.
\<String> is a supertype of \<@Interned String>, so a method \<toUpperCase>
that is declared to take a \<String> parameter can also be called on an
\<@Interned String> argument.

%% TODO: Polymorphic qualifiers do not yet take an optional argument.
% PolyAll in the section below should be changed to any poly qualifier.
%
% \subsectionAndLabel{Multiple instances of polymorphic qualifiers (the index argument)}{qualifier-polymorphism-multiple-instances}
%
% Each polymorphic qualifier such as \refqualclass{framework/qual}{PolyAll}
% takes an optional argument so that you can
% specify multiple, independent polymorphic type qualifiers.  For example,
% this signature is overly restrictive:
%
% \begin{Verbatim}
%   /**
%    * Returns true if the arrays are elementwise equal,
%    * testing for equality using == (not the equals method).
%    */
%   public static int eltwiseEqualUsingEq(@PolyAll Object[] a, @PolyAll Object elt) {
%     for (int i=0; i<a.length; i++) {
%       if (elt != a[i]) {
%         return false;
%       }
%     }
%     return true;
%   }
% \end{Verbatim}
%
% \noindent
% That signature requires the element type annotation to be identical for the
% two arguments.  For example, it forbids this invocation:
%
% \begin{Verbatim}
%   @Nullable Object[] x;
%    @NonNull Object   y;
%   ... indexOf(x, y) ...
% \end{Verbatim}
%
% \noindent
% A better signature lets the two arrays' element types vary independently:
%
% \begin{Verbatim}
%   public static int eltwiseEqualUsingEq(@PolyAll(1) Object[] a, @PolyAll(2) Object elt)
% \end{Verbatim}
%
% \noindent
% Note that in this case, the \<@Nullable> annotation on \<elt>'s type is no
% longer necessary, since it is subsumed by \<@PolyAll>.
%
% The \<@PolyAll> annotation at a location $l$ applies to every type
% qualifier hierarchy for which no explicit qualifier is written at location
% $l$.  For example, a declaration like
% \<@PolyAll @NonNull Object elt> is polymorphic over every type system
% \emph{except} the nullness type system, for which the type is fixed at
% \<@NonNull>.  That would be the proper declaration for \<elt> if the body
% had used \<elt.equals(a[i])> instead of \<elt == a[i]>.
%
%
% % Suppose that some type system has two qualifiers, such as
% % \<@Nullable> and \<@NonNull>.  When a polymorphic type qualifier such
% % as \<@PolyNull> is used in a method, then the method conceptually
% % has two different versions:  one in which every instance of
% % \<@PolyNull> has been replaced by \<@NonNull> and one in
% % which every instance of \<@PolyNull> has been replaced by
% % \<@Nullable>.
%
% If a method signature contains only indexless versions of a polymorphic
% qualifier such as \refqualclass{framework/qual}{PolyAll} or
% \refqualclass{checker/nullness/qual}{PolyNull}, then all of them refer to
% the same type as described in
% Section~\ref{qualifier-polymorphism-multiple-qualifiers}.  If any indexed
% version appears, then every occurrence of the polymorphic qualifier without
% an index is considered to use a fresh index.  For example, the following
% two declarations are equivalent (where \<@PA> means \<@PolyAll>, for brevity):
%
% \begin{smaller}
% \begin{Verbatim}
%   @PA(1) foo(@PA(1) Object a, @PA(2) Object b, @PA(2) Object c, @PA    Object d, @PA    Object e) {...}
%
%   @PA(1) foo(@PA(1) Object a, @PA(2) Object b, @PA(2) Object c, @PA(3) Object d, @PA(4) Object e) {...}
% \end{Verbatim}
% \end{smaller}
%
% As described in Section~\ref{qualifier-polymorphism-return-type}, the
% qualifier on a return type must be the same as that on some formal parameter.
% Therefore, the first of these declarations is legal because it is
% equivalent to the second, but the third is illegal because it is
% equivalent to the fourth.
%
% \begin{Verbatim}
%   @PolyAll    m1(@PolyAll    Object a, @PolyAll    Object b) { ... } // OK
%   @PolyAll(1) m2(@PolyAll(1) Object a, @PolyAll(1) Object b) { ... } // OK (same as m1)
%
%   @PolyAll    m3(@PolyAll    Object a, @PolyAll(1) Object b) { ... } // illegal
%   @PolyAll(2) m4(@PolyAll(3) Object a, @PolyAll(1) Object b) { ... } // illegal (same as m3)
% \end{Verbatim}


\subsectionAndLabel{Using multiple polymorphic qualifiers in a method signature}{qualifier-polymorphism-multiple-qualifiers}

%% I can't think of a non-clumsy way to say this.
% Each method containing a polymorphic qualifier is (conceptually) expanded
% into multiple versions completely independently.

Usually, it does not make sense to write only a single instance of a polymorphic
qualifier in a method definition:  if you write one instance of (say)
\<@PolyNull>, then you should use at least two.
The main benefit of polymorphic qualifiers comes when one is used multiple times
in a method, since then each instance turns into the same type qualifier.
(Section~\ref{qualifier-polymorphism-single-qualifier} describes some
exceptions to this rule:  times when it makes sense to write a single
polymorphic qualifier in a signature.)

Most frequently, the polymorphic qualifier appears on at least one formal
parameter and also on the return type.



  It can also be useful to have
polymorphic qualifiers on (only) multiple formal parameters, especially if
the method side-effects one of its arguments.
For example, consider

\begin{Verbatim}
void moveBetweenStacks(Stack<@PolyNull Object> s1, Stack<@PolyNull Object> s2) {
  s1.push(s2.pop());
}
\end{Verbatim}

\noindent
In this particular example, it would be cleaner to rewrite your code to use
Java generics, if you can do so:

\begin{Verbatim}
<T> void moveBetweenStacks(Stack<T> s1, Stack<T> s2) {
  s1.push(s2.pop());
}
\end{Verbatim}


\subsectionAndLabel{Using a single polymorphic qualifier in a method signature}{qualifier-polymorphism-single-qualifier}

As explained in Section~\ref{qualifier-polymorphism-multiple-qualifiers},
you will usually use a polymorphic qualifier
multiple times in a signature.
This section describes situations when it makes sense to write just one
polymorphic qualifier in a method signature.
Some of these situations can be avoided by writing a generic method,
but in legacy code it may not be possible for you to change a method to be
generic.


\subsubsectionAndLabel{Using a single polymorphic qualifier on a return type}{qualifier-polymorphism-return-type}

It is unusual, but permitted, to write just one polymorphic qualifier, on a return type.

This is just like it is unusual, but permitted, to write just one
occurrence of a generic type parameter, on a return type.  An example of
such a method is
\sunjavadoc{java.base/java/util/Collections.html\#emptyList()}{Collections.emptyList()}.


\subsubsectionAndLabel{Using a single polymorphic qualifier on an element type}{qualifier-polymorphism-element-types}

It can make sense to use a polymorphic qualifier just once, on an array or
generic element type.

For example, consider a routine that returns the index, in an array, of a
given element:

\begin{Verbatim}
  public static int indexOf(@PolyNull Object[] a, @Nullable Object elt) { ... }
\end{Verbatim}

If \<@PolyNull> were replaced with either \<@Nullable> or \<@NonNull>, then
one of these safe client calls would be rejected:

\begin{Verbatim}
  @Nullable Object[] a1;
  @NonNull Object[] a2;

  indexOf(a1, someObject);
  indexOf(a2, someObject);
\end{Verbatim}

Of course, it would be better style to use a generic method, as in either
of these signatures:

\begin{Verbatim}
 public static <T extends @Nullable Object> int indexOf(T[] a, @Nullable Object elt) { ... }
 public static <T extends @Nullable Object> int indexOf(T[] a, T elt) { ... }
\end{Verbatim}


Another example is a method that writes bytes to a file.  It accepts an
array of signed or unsigned bytes, and it behaves identically for both:

\begin{Verbatim}
  void write(@PolySigned byte[] b) { ... }
\end{Verbatim}


These examples use arrays, but there are similar examples that
use collections.


\subsubsectionAndLabel{Don't use a single polymorphic qualifier on a formal parameter type}{qualifier-polymorphism-formal-parameter}

There is no point to writing just one polymorphic qualifier in a method
signature, as the main type qualifier on a formal parameter type.
Consider this signature:

\begin{Verbatim}
  void m(@PolyNull Object obj)
\end{Verbatim}

\noindent
which expands to

\begin{Verbatim}
  void m(@NonNull Object obj)
  void m(@Nullable Object obj)
\end{Verbatim}

This is no different (in terms of which calls to the method will
type-check) than writing just

\begin{Verbatim}
  void m(@Nullable Object obj)
\end{Verbatim}

However, it does make sense to write a single polymorphic qualifier
\emph{within} a formal parameter type, as in
\begin{Verbatim}
  void m2a(List<@PolyNull MySubClass> strings)
\end{Verbatim}
which is similar to
\begin{Verbatim}
  void m2b(List<? extends @Nullable MySubClass> strings)
\end{Verbatim}
Method \<m2b()> can be called on (for example) \code{List<@Nullable
  MySubClass>} and \code{List<@NonNull MySubClass>}, which are not legal
arguments to method \<m2a()>.


\sectionAndLabel{Class qualifier parameters}{class-qualifier-polymorphism}

Class qualifier parameters permit you to supply a type qualifier (only,
without a Java basetype) to any class (generic or not).

When a generic class represents a collection, a user can write a type
qualifier on the type argument, as in \code{List<@Tainted Character>}
versus \code{List<@Untainted Character>}.  When a non-generic class
represents a collection with a hard-coded type (as \<StringBuffer>
hard-codes \<Character>), you can use a class qualifier parameter to
distinguish \<StringBuffer>s that contain different types of characters.
% (For a discussion of tainting, see \chapterpageref{tainting-checker}.)

To add a qualifier parameter to a class, annotate its declaration with \refqualclass{framework/qual}{HasQualifierParameter}
and write the class of the top qualifier as its element.

\begin{Verbatim}
@HasQualifierParameter(Tainted.class)
class StringBuffer { ... }
\end{Verbatim}

A qualifier on a use of \<StringBuffer> is treated as
appearing both on the \<StringBuffer> and on its conceptual type argument.  That
is:\\
\begin{tt}
  @Tainted StringBuffer $\approx$ @Tainted Collection<@Tainted Character>\\
  @Untainted StringBuffer $\approx$ @Untainted Collection<@Untainted Character>
\end{tt}

If two types have different qualifier arguments, they have no subtyping
relationship.  (This is ``invariant subtyping'', also used by Java for
generic classes.)  In particular, \<@Untainted StringBuffer> is not a
subtype of \<@Tainted StringBuffer>; an attempt to cast between them, in
either direction, will yield an \<invariant.cast.unsafe> error.

\<@HasQualifierParameter> is inherited.  If type $T$ has a
\<@HasQualifierParameter> annotation, then its subtypes are automatically
treated as having the same annotation.

%% TODO: Inheritance of classes with a qualifier parameter below.

Within a class with a qualifier parameter,
the default qualifier for uses of that class is the polymorphic qualifier.


\subsectionAndLabel{Resolving polymorphism when the receiver type has a polymorphic qualifier}{class-qual-param-method}

Qualifier polymorphism changes the rules for instantiating polymorphic
qualifiers (Section~\ref{method-qualifier-polymorphism}).
If the receiver type has a qualifier parameter and is annotated with a polymorphic qualifier,
then at a call site all polymorphic annotations are instantiated to
the same qualifier as the type of the receiver expression of the method call.
Otherwise, use the rules of Section~\ref{method-qualifier-polymorphism}

For example, consider

\begin{Verbatim}
@HasQualifierParameter(Tainted.class)
class Buffer {
  void append(@PolyTainted Buffer this, @PolyTainted String s) { ... }
}
\end{Verbatim}

\noindent
Because \<@PolyTainted> applies to a type (\<Buffer>) with a qualifier parameter, all
uses of \<@PolyTainted> are instantiated to the qualifiers on the type of
the receiver expression at call sites to \<append>. For example,

\begin{Verbatim}
@Untainted Buffer untaintedBuffer = ...;
@Tainted String taintedString = ...;
untaintedBuffer.append(taintedString); // error: (argument)
\end{Verbatim}
The above \<append> call is illegal because the \<@PolyTainted> is instantiated to \<@Untainted> and
the type of the argument is \<@Tainted> which is a supertype of \<@Untainted>.  If the type of
\<untaintedBuffer> were \<@Tainted> then the call would be legal.

\subsectionAndLabel{Using class qualifier parameters in the type of a field}{class-qual-param-field}

To express that the type of a field
should have the same qualifier as the class qualifier parameter,
annotate the field type with the polymorphic qualifier for the type system.

\begin{Verbatim}
@HasQualifierParameter(Tainted.class)
class Buffer {
  @PolyTainted String field;
}
\end{Verbatim}

At a field access where the declared type of the field has a polymorphic
qualifier, that polymorphic qualifier is instantiated to the qualifier on the
type of the receiver of the field access (or in the case of type variables, the
qualifier on the upper bound).  That is, the qualifier on \<myBuffer.field>
is that same as that on \<myBuffer>.

\subsectionAndLabel{Local variable defaults for types with qualifier parameters}{local-vars-qual-param-defaults}

Local variables default to the top type (see
Section~\ref{climb-to-top}). Type refinement determines if a variable can be
treated as a suitable subtype, and annotations on local variables are rarely
needed as a result. However, since qualifier parameters add invariant subtyping,
type refinement is no longer valid. For example, suppose in the following code
that \<StringBuffer> is annotated with \<@HasQualifierParameter(Tainted.class)>.

\begin{Verbatim}
    void method(@Untainted StringBuffer buffer) {
        StringBuffer local = buffer;
        executeSql(local.toString());
    }

    void executeSql(@Untainted String code) {
        // ...
    }
\end{Verbatim}

Normally, the framework would determine that \<local> has type \<@Untainted
StringBuffer> and the call to \<executeSql> would be valid. However, since by
default \<local> has type \<@Tainted StringBuffer>, and
\<@Untainted StringBuffer> is not a subtype, no type refinement would be
performed, leading to an error. Fixing this would require manually annotating
\<local> as an \<@Untainted StringBuffer>, increasing the annotation burden on
programmers.

For this reason, local variables with types that have a qualifier parameter use
different defaulting rules. When a local variable has an initializer, the type
of that initializer is used as the default type of that variable if no other
annotations are written. For example, in the above code, the type of \<local>
would be \<@Untainted StringBuffer>. This eliminates the need for type
refinement.

\subsectionAndLabel{Qualifier parameters by default}{default-has-qualifier-parameter}

If many classes in a project should have \<@HasQualifierParameter>, it's
possible to enable it on all classes in a package by default. Writing
\<@HasQualifierParameter> on a package is equivalent to writing
\<@HasQualifierParameter> on each class in that package and all subpackages with
the same arguments.

For example, writing this annotation enables \<@HasQualifierParameter>
for all classes in \code{mypackage}.

\begin{Verbatim}
@HasQualifierParameter(Tainted.class)
package mypackage;
\end{Verbatim}

When using \<@HasQualifierParameter> on a package, it's possible to disable it
for a specific class using \refqualclass{framework/qual}{NoQualifierParameter}.
Writing this on a class indicates it has no class qualifier parameter and
\<@HasQualifierParameter> will not be enabled by default. Like
\<@HasQualifierParameter>, it takes one or more top annotations. It is illegal
to explicitly write both \<@HasQualifierParameter> and \<@NoQualifierParameter>
on the same class for the same hierarchy.

%% TODO: implement this.
%% https://github.com/typetools/checker-framework/issues/3400
%\subsectionAndLabel{Inheritance of classes with a qualifier parameter}{class-qual-param-inheritance}
%
%If a class extends a \<@HasQualifierParameter> class (or implements a
%\<@HasQualifierParameter> interface), then that class must also be marked
%\<@HasQualifierParameter>. A particular subclass can specify which qualifier
%parameter should be used for the super class.  (This is similar to \<class
%StringList implements List<String> \{...\}>.) For example,
%
%\begin{Verbatim}
%  @HasQualifierParameter(Tainted.class)
%  class Buffer {
%  void append(@PolyTainted Buffer this, @PolyTainted String s) { ... }
%  }
%  @HasQualifierParameter(Tainted.class)
%  @Tainted class TaintedBuffer extends @Tainted Buffer {
%  @Override
%  void append(@Tainted TaintedBuffer this, @Tainted String s) { ... } // legal override
%  }
%\end{Verbatim}
%
% \<@Untainted TaintedBuffer> is an invalid type.

\subsectionAndLabel{Types with qualifier parameters as type arguments}{class-qual-param-type-arg}

Types with qualifier parameters are only allowed as type arguments to type parameters whose upper bound
have a qualifier parameter. If they were allowed for as type arguments for any type parameter, then
unsound casts would be permitted. For example:

\begin{Verbatim}
    @HasQualifierParameter(Tainted.class)
    interface Buffer {
        void append(@PolyTainted String s);
    }

    public class ClassQPTypeVarTest {
        <T> @Tainted T cast(T param) {
            return param;
        }

        void bug(@Untainted Buffer b, @Tainted String s) {
            cast(b).append(s);  // error
        }
    }


\end{Verbatim}

% LocalWords:  nullable MyList nble nn Nullness DefaultQualifier MyClass quals
% LocalWords:  DefaultLocation subtype's ImmutableList ListIterator nullness
% LocalWords:  PolyNull java lub invariantly supertype's
% LocalWords:  MyList1 MyList2 MyList4 MyList3 MyClass2 toUpperCase elt
% LocalWords:  PolyAll arrays' Xlint rawtypes AignoreRawTypeArguments s1
% LocalWords:  call's Regex taintedHolder untaintedHolder taintedHolder2
% LocalWords:  wildcards holderExtends holderSuper getTaintedString s2 JLS
% LocalWords:  getUntaintedString Typesystem Param ClassTaintingParam h1
% LocalWords:  arg param MethodTaintingParam Util meth value1 h2 TopType
% LocalWords:  nestedHolder nestedHolder2 extendsHolder PolyTainted Poly
%%  LocalWords:  BottomType CharSequence SomeType OtherType MyList1a
%%  LocalWords:  MyList2a polymorphism'' bound'' LowerBound UpperBound
% LocalWords:  JavaUpperBound JavaLowerBound MyTypeAnnotation myBuffer
% LocalWords:  HasQualifierParameter myUntaintedBuffer myTaintedString
% LocalWords:  myTaintedBuffer myUntaintedString
