Added section about explicitly resolved (fka canonical) syntax

Author: eernstg

specification/dartLangSpec.tex

@@ -2675,7 +2675,7 @@ \subsection{Type of a Function}
   different type parameters, F-bounds, and the types of formal parameters.
   However, we do not wish to distinguish between two function types
   if they have the same structure and only differ in the choice of names.
-  This treatment of names is also known as alpha-equivalence.%
+  This treatment of names is also known as alpha equivalence.%
 }
 
 \LMHash{}%
@@ -22015,24 +22015,18 @@ \subsubsection{Subtype Rules}
   However, we must eliminate the difficulties associated with
   different syntax denoting the same type,
   and different types denoted by the same syntax.
-  We do this by assuming that every type in the program has been expressed
-  in a manner where those situations never occur,
-  because each type is denoted by the same globally unique syntax everywhere.%
+  We do this by defining in detail what it means to be the same type
+  (\ref{sameStaticType}).%
 }
 
 \LMHash{}%
 In section~\ref{subtypes} and its subsections,
 all designations of types are considered to be the same
-if{}f they have the same canonical syntax
-(\ref{theCanonicalSyntaxOfTypes}).
+type if{}f they satisfy the criteria in
+Section~\ref{sameStaticType}.
 
-\commentary{%
-  Note that the canonical syntax also implies
-  transitive expansion of all type aliases
-  (\ref{typedef}).
-  In other words, subtyping rules do not need to consider type aliases,
-  because all type aliases have been expanded.%
-}
+\LMHash{}%
+The subtype rules assume that all type aliases have been transitively expanded.
 
 \LMHash{}%
 Every \synt{typeName} used in a type mentioned
@@ -22418,19 +22412,13 @@ \subsection{Function Types}
   coincide in this case.%
 }
 
-\LMHash{}%
-Two function types are considered equal if consistent renaming of type
-parameters can make them identical.
-
 \commentary{%
-  A common way to say this is that we do not distinguish
-  function types which are alpha-equivalent.
-  For the subtyping rule below this means we can assume that
-  a suitable renaming has already taken place.
-  In cases where this is not possible
-  because the number of type parameters in the two types differ
-  or the bounds are different,
-  no subtype relationship exists.%
+  Two function types are the same type
+  if, roughly, consistent renaming of type
+  parameters can make them identical.
+  This is also known as alpha equivalence.
+  The detailed rules are described in
+  Section~\ref{sameStaticType}.%
 }
 
 \LMHash{}%
@@ -23303,6 +23291,208 @@ \subsubsection{Least Upper Bounds}
 }
 
 
+\subsection{Same Static Type}
+\LMLabel{sameStaticType}
+
+\LMHash{}%
+This section specifies how to soundly determine whether or not two types
+that are encountered during static analysis are the same type.
+
+\LMHash{}%
+Concrete syntax denoting types gives rise to several difficulties
+when used to determine static semantic properties,
+like subtyping relationships
+(\ref{subtypes})
+or upper and lower bounds
+(\ref{standardUpperBoundsAndStandardLowerBounds}).
+
+\commentary{%
+Note that the notion of being the same type at run time is different from
+the notion of being the same type during static analysis.
+For example, two distinct type variables \code{X} and \code{Y}
+might at run time be bound to the same type, e.g.,
+\code{List<int>}.
+However, during static analysis we must maintain that
+\code{X} and \code{Y} are different types
+because there is no guarantee that they are always bound to
+the same type at run time.%
+}
+
+\commentary{%
+The phrases `same type' and `identical syntax' deserves some extra scrutiny.
+If a library $L$ imports the libraries $L_1$ and $L_2$
+(where $L_1$ and $L_2$ are not the same library),
+and both $L_1$ and $L_2$ declare a class \code{C},
+then the syntax \code{C} may occur as a type during static analysis of $L$
+in situations where it refers to two distinct types.
+
+For instance, $L_1$ could declare a variable \code{v}
+of type \code{C}-in-$L_1$,
+and $L_2$ could declare a function
+\code{\VOID\,foo(C\,\,c)\,\,\{\}}
+which uses the type \code{C}-in-$L_2$,
+and $L$ could contain the expression \code{foo(v)}.
+
+Note that even though it would be a compile-time error to use \code{C} in $L$
+(because it is ambiguous),
+it is not an error to have an expression like \code{foo(v)},
+and the static analysis of this expression \emph{must}
+be able to determine that the two types whose name is \code{C} are
+not the same type. The invocation may or may not be an error,
+depending on the subtyping relations.%
+}
+
+\rationale{%
+This shows that concrete syntax behaves in such a manner that it is
+unsafe to consider two types to be the same type,
+based on the fact that they are denoted by the same syntax,
+even during the static analysis of a single expression.
+
+Similarly, it is incorrect to consider two terms derived from \synt{type}
+as different types based on the fact that they are syntactically different.
+For example, they could be the same type imported with different import
+prefixes.
+
+We could say that two identifiers
+(possibly qualified, e.g., \code{myPrefix.C} and \code{C})
+denote the same type
+if they have been resolved to the same declaration.
+
+However, we introduce \emph{the explicitly resolved syntax for a type},
+which is basically an explicit representation of this idea,
+in order to make the underlying issues more explicit.
+The explicitly resolved syntax has the property that
+each type has a unique syntactic form
+(except for alpha equivalence, which is defined below).
+We may then consider this unique syntactic form as a static semantic value,
+rather than just a syntactic form which is dependent on
+its location in the program.%
+}
+
+\LMHash{}%
+The
+\IndexCustom{explicitly resolved syntax}{type!explicitly resolved syntax of}
+of the types in a given library $L_1$
+and all libraries \List{L}{2}{n} reachable from $L_1$ via
+one or more import links
+is determined as follows.
+First, choose a set of distinct, globally fresh identifiers
+\List{\metavar{prefix}}{1}{n}.
+Then transform each library $L_i$, $i \in 1 .. n$ as follows:
+
+\begin{enumerate}
+\item
+  For each type variable declared in $L_i$, consistently rename
+  it to a globally fresh name.
+\item
+  If $D_T$ is a declaration of a class, mixin, or type alias in $L_i$
+  whose name $n$ is private,
+  and an occurrence of $n$ that resolves to $D$
+  exists in a type alias declaration $D_A$ whose name is non-private,
+  then perform a consistent renaming of
+  all occurrences of $n$ in $L_i$ that resolve to $D_T$
+  to a fresh, non-private identifier.
+  \commentary{%
+    We make $D_T$ public, because it is being leaked anyway.%
+  }
+\item
+  Add a set of import directives to $L_i$ that imports
+  each of the libraries \List{L}{1}{n} with
+  the corresponding prefix $\metavar{prefix}_j$, $j \in 1 .. n$.
+
+  \commentary{%
+    This means that every library in the set
+    $\{\,\List{L}{1}{n}\,\}$
+    imports every other library in that set,
+    even itself and system libraries like \code{dart:core}.%
+  }
+\item
+  Let \id{} be an occurrence of
+  a non-private type identifier derived from \synt{typeName}
+  that resolves to a library declaration in the library $L_j$
+  in the original program;
+  \id{} is then transformed to \code{$\metavar{prefix}_j$.\id}.
+  Let \code{$p$.\id} be derived from \synt{typeName} such that $p$ is
+  an import prefix in the original program
+  and \id{} is a non-private identifier,
+  and consider the case where \code{$p$.\id} resolves to
+  a library declaration in the library $L_j$ in the original program,
+  for some $j$;
+  \code{$p$.\id} is then transformed to \code{$\metavar{prefix}_j$.\id}.
+\item
+  Replace every type in $L_i$ that denotes a type alias
+  along with its actual type arguments, if any,
+  by its transitive alias expansion
+  (\ref{typedef}).
+  \commentary{%
+    Note that the bodies of type alias declarations
+    already use the new prefixes,
+    so the results of the alias expansion will also use
+    the new prefixes consistently.%
+  }
+\end{enumerate}
+
+\commentary{%
+In short, this transformation adds a unique prefix to every type name
+which is resolved to a top-level declaration
+(in the same library or in an imported library).
+  
+This transformation does not change any occurrence of \VOID,
+and does not need to;
+\VOID{} is a reserved word, not a type identifier.
+Also, \code{$\metavar{prefix}_j$.\VOID} would be a syntax error.
+
+Note that the transformation changes terms derived from \synt{type},
+but it does not change expressions, or any other program element
+(except that a \synt{type} can occur in an expression, e.g., \code{<int>[]}).
+In particular, it does not change type literals
+(that is, expressions denoting types).%
+}
+
+\LMHash{}%
+Every \synt{type} in the resulting set of libraries
+is now expressed in a globally unique syntactic form,
+which is the form that we call the
+\IndexCustom{explicitly resolved syntax of}{type!explicitly resolved syntax of}
+said types.
+
+\LMHash{}%
+When we say that two types $T_1$ and $T_2$ have the
+\IndexCustom{same explicitly resolved syntax}{type!same explicitly resolved syntax},
+it refers to the situation where the current library
+and all libraries which are reachable via one or more imports
+have been transformed as described above,
+and the resulting explicitly resolved syntaxes are textually identical.
+
+\LMHash{}%
+We need to introduce one more concept:
+An \Index{alpha conversion} of a type is a consistent renaming
+of the type variables declared in that type.
+
+\commentary{%
+In Dart, only function types can be subject to an alpha conversion:
+A function type is the only kind of type that declares type variables.
+For example,
+\code{List<X>\,\,\FUNCTION<X>({X\,\,arg})}
+can be turned into 
+\code{List<Y>\,\,\FUNCTION<Y>({Y\,\,arg})}
+by alpha conversion.%
+}
+
+\LMHash{}%
+Two function types are \Index{alpha equivalent} if{}f
+there exists an alpha conversion that makes them textually identical.
+
+\LMHash{}%
+Finally, we define that two type denotations $T_1$ and $T_2$ are the
+\IndexCustom{same static type}{type!same static}
+if there exist alpha conversions
+that can be applied to the function types
+that occur as subterms of $T_1$ and $T_2$, if any,
+such that the resulting terms $T'_1$ and $T'_2$ have
+the same explicitly resolved syntax.
+
+
 \section{Reference}
 \LMLabel{reference}