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  • doc/bibliography/cfa.bib

    r8f5bf6d r55b2f5a  
    65776577}
    65786578
    6579 @unpublished{TIOBE,
    6580     contributer = {pabuhr@plg},
    6581     author      = {TIOBE Index},
    6582     title       = {},
    6583     year        = {March 2017},
    6584     note        = {\url{http://www.tiobe.com/tiobe_index}},
    6585 }
    6586 
    65876579@misc{Bumbulis90,
    65886580    keywords    = {parameter inference, ForceN},
    … …  
    65916583    title       = {Towards Making Signatures First-Class},
    65926584    howpublished= {personal communication},
    6593     month       = sep,
    6594     year        = 1990,
     6585    month       = sep, year = 1990,
    65956586    note        = {}
    65966587}

  • doc/generic_types/generic_types.tex

    r8f5bf6d r55b2f5a  
    2828\newcommand{\CCseventeen}{\rm C\kern-.1em\hbox{+\kern-.25em+}17\xspace} % C++17 symbolic name
    2929\newcommand{\CCtwenty}{\rm C\kern-.1em\hbox{+\kern-.25em+}20\xspace} % C++20 symbolic name
    30 \newcommand{\CS}{C\raisebox{-0.7ex}{\Large$^\sharp$}\xspace}
    31 \newcommand{\Textbf}[1]{{\color{red}\textbf{#1}}}
    3230
    3331\newcommand{\TODO}[1]{\textbf{TODO}: {\itshape #1}} % TODO included
    … …  
    126124\maketitle
    127125
    128 
    129 \section{Introduction and Background}
    130 
    131 The C programming language is a foundational technology for modern computing with millions of lines of code implementing everything from commercial operating-systems to hobby projects. This installation base and the programmers producing it represent a massive software-engineering investment spanning decades and likely to continue for decades more.
    132 TIOBE~\cite{TIOBE} ranks the top 5 most popular programming languages as: Java 16\%, \Textbf{C 7\%}, \Textbf{\CC 5\%}, \CS 4\%, Python 4\% = 36\%, where the next 50 languages are less than 3\% each with a long tail. The top 3 rankings over the past 30 years are:
    133 \lstDeleteShortInline@
    134 \begin{center}
    135 \setlength{\tabcolsep}{10pt}
    136 \begin{tabular}{@{}r|c|c|c|c|c|c|c@{}}
    137                 & 2017  & 2012  & 2007  & 2002  & 1997  & 1992  & 1987          \\
    138 \hline
    139 Java    & 1             & 1             & 1             & 3             & 13    & -             & -                     \\
    140 \hline
    141 \Textbf{C}      & \Textbf{2}& \Textbf{2}& \Textbf{2}& \Textbf{1}& \Textbf{1}& \Textbf{1}& \Textbf{1}    \\
    142 \hline
    143 \CC             & 3             & 3             & 3             & 3             & 2             & 2             & 4                     \\
    144 \end{tabular}
    145 \end{center}
    146 \lstMakeShortInline@
    147 Love it or hate it, C is extremely popular, highly used, and one of the few system's languages.
    148 In many cases, \CC is often used solely as a better C.
    149 Nonetheless, C, first standardized over thirty years ago, lacks many features that make programming in more modern languages safer and more productive.
    150 
    151 \CFA (pronounced ``C-for-all'', and written \CFA or Cforall) is an evolutionary extension of the C programming language that aims to add modern language features to C while maintaining both source compatibility with C and a familiar programming model for programmers. Four key design goals were set out in the original design of \CFA~\citep{Bilson03}:
    152 (1) The behaviour of standard C code must remain the same when translated by a \CFA compiler as when translated by a C compiler;
    153 (2) Standard C code must be as fast and as small when translated by a \CFA compiler as when translated by a C compiler;
    154 (3) \CFA code must be at least as portable as standard C code;
    155 (4) Extensions introduced by \CFA must be translated in the most efficient way possible.
     126\section{Introduction \& Background}
     127
     128\CFA\footnote{Pronounced ``C-for-all'', and written \CFA or Cforall.} is an evolutionary extension of the C programming language that aims to add modern language features to C while maintaining both source compatibility with C and a familiar programming model for programmers. Four key design goals were set out in the original design of \CFA~\citep{Bilson03}:
     129\begin{enumerate}
     130\item The behaviour of standard C code must remain the same when translated by a \CFA compiler as when translated by a C compiler.
     131\item Standard C code must be as fast and as small when translated by a \CFA compiler as when translated by a C compiler.
     132\item \CFA code must be at least as portable as standard C code.
     133\item Extensions introduced by \CFA must be translated in the most efficient way possible.
     134\end{enumerate}
    156135These goals ensure existing C code-bases can be converted to \CFA incrementally and with minimal effort, and C programmers can productively generate \CFA code without training beyond the features they wish to employ. In its current implementation, \CFA is compiled by translating it to the GCC-dialect of C~\citep{GCCExtensions}, allowing it to leverage the portability and code optimizations provided by GCC, meeting goals (1)-(3). Ultimately, a compiler is necessary for advanced features and optimal performance.
    157136
    158 \CFA has been previously extended with polymorphic functions and name overloading (including operator overloading) by \citet{Bilson03}, and deterministically-executed constructors and destructors by \citet{Schluntz17}. This paper builds on those contributions, identifying shortcomings in existing approaches to generic and variadic data types in C-like languages and presenting a design for generic and variadic types avoiding those shortcomings. Specifically, the solution is both reusable and type-checked, as well as conforming to the design goals of \CFA with ergonomic use of existing C abstractions. The new constructs are empirically compared with both standard C and \CC; the results show the new design is comparable in performance.
    159 
     137\CFA has been previously extended with polymorphic functions and name overloading (including operator overloading) by \citet{Bilson03}, and deterministically-executed constructors and destructors by \citet{Schluntz17}. This paper builds on those contributions, identifying shortcomings in existing approaches to generic and variadic data types in C-like languages and presenting a design of generic and variadic types as as extension of the \CFA language that avoids those shortcomings. Particularly, the solution we present is both reusable and type-checked, as well as conforming to the design goals of \CFA and ergonomically using existing C abstractions. We have empirically compared our new design to both standard C and \CC; the results show that this design is \TODO{awesome, I hope}.
    160138
    161139\subsection{Polymorphic Functions}
    162140\label{sec:poly-fns}
    163141
    164 \CFA's polymorphism was originally formalized by \citet{Ditchfield92}, and first implemented by \citet{Bilson03}. The signature feature of \CFA is parametric-polymorphic functions where functions are generalized using a @forall@ clause (giving the language its name):
     142\CFA's polymorphism was originally formalized by \citet{Ditchfield92}, and first implemented by \citet{Bilson03}. The signature feature of \CFA is parametric-polymorphic functions; such functions are written using a @forall@ clause (which gives the language its name):
    165143\begin{lstlisting}
    166144`forall( otype T )` T identity( T val ) { return val; }
    167145int forty_two = identity( 42 );                         $\C{// T is bound to int, forty\_two == 42}$
    168146\end{lstlisting}
    169 The @identity@ function above can be applied to any complete object-type (or ``@otype@''). The type variable @T@ is transformed into a set of additional implicit parameters encoding sufficient information about @T@ to create and return a variable of that type. The \CFA implementation passes the size and alignment of the type represented by an @otype@ parameter, as well as an assignment operator, constructor, copy constructor and destructor. If this extra information is not needed, \eg for a pointer, the type parameter can be declared as @dtype T@, where @dtype@ is short for ``data type''.
    170 
    171 Here, the runtime cost of polymorphism is spread over each polymorphic call, due to passing more arguments to polymorphic functions; preliminary experiments have shown this overhead is similar to \CC virtual function calls. An advantage of this design is that, unlike \CC template functions, \CFA @forall@ functions are compatible with C \emph{separate} compilation.
    172 
    173 Since bare polymorphic-types provide only a narrow set of available operations, \CFA provides a \emph{type assertion} mechanism to provide further type information, where type assertions may be variable or function declarations that depend on a polymorphic type-variable. For example, the function @twice@ can be defined using the \CFA syntax for operator overloading:
    174 \begin{lstlisting}
    175 forall( otype T `| { T ?+?(T, T); }` ) T twice( T x ) { return x + x; } $\C{// ? denotes operands}$
     147The @identity@ function above can be applied to any complete object-type (or ``@otype@''). The type variable @T@ is transformed into a set of additional implicit parameters to @identity@ that encode sufficient information about @T@ to create and return a variable of that type. The \CFA implementation passes the size and alignment of the type represented by an @otype@ parameter, as well as an assignment operator, constructor, copy constructor and destructor. If this extra information is not needed, \eg for a pointer, the type parameter can be declared as @dtype T@, where @dtype@ is short for ``data type''.
     148
     149Here, the runtime cost of polymorphism is spread over each polymorphic call, due to passing more arguments to polymorphic functions; preliminary experiments have shown this overhead to be similar to \CC virtual function calls. An advantage of this design is that, unlike \CC template functions, \CFA @forall@ functions are compatible with C separate compilation.
     150
     151Since bare polymorphic-types provide only a narrow set of available operations, \CFA provides a \emph{type assertion} mechanism to provide further type information, where type assertions may be variable or function declarations that depend on a polymorphic type variable. For instance, @twice@ can be defined using the \CFA syntax for operator overloading:
     152\begin{lstlisting}
     153forall( otype T | { T `?`+`?`(T, T); } )        $\C{// ? denotes operands}$
     154  T twice( T x ) { return x + x; }                      $\C{// (2)}$
    176155int val = twice( twice( 3.7 ) );
    177156\end{lstlisting}
    178 which works for any type @T@ with a matching addition operator. The polymorphism is achieved by creating a wrapper function for calling @+@ with @T@ bound to @double@, then passing this function to the first call of @twice@. There is now the option of using the same @twice@ and converting the result to @int@ on assignment, or creating another @twice@ with type parameter @T@ bound to @int@ because \CFA uses the return type in its type analysis. The first approach has a late conversion from @int@ to @double@ on the final assignment, while the second has an eager conversion to @int@. \CFA minimizes the number of conversions and their potential to lose information, so it selects the first approach, which corresponds with C-programmer intuition.
     157which works for any type @T@ with an addition operator defined. The translator accomplishes this polymorphism by creating a wrapper function for calling @+@ with @T@ bound to @double@, then providing this function to the first call of @twice@. It then has the option of using the same @twice@ again and converting the result to @int@ on assignment, or creating another @twice@ with type parameter @T@ bound to @int@ because \CFA uses the return type in its type analysis. The first approach has a late conversion from integer to floating-point on the final assignment, while the second has an eager conversion to integer. \CFA minimizes the number of conversions and their potential to lose information, so it selects the first approach.
    179158
    180159Monomorphic specializations of polymorphic functions can satisfy polymorphic type-assertions.
    … …  
    186165% \end{lstlisting}
    187166\begin{lstlisting}
    188 forall( otype T `| { int ?<?( T, T ); }` ) void qsort( const T * arr, size_t size );
    189 forall( otype T `| { int ?<?( T, T ); }` ) T * bsearch( T key, const T * arr, size_t size );
     167forall( otype T `| { int ?<?( T, T ); }` )      $\C{// type assertion}$
     168  void qsort( const T * arr, size_t size );
     169forall( otype T `| { int ?<?( T, T ); }` )      $\C{// type assertion}$
     170  T * bsearch( T key, const T * arr, size_t size );
    190171double vals[10] = { /* 10 floating-point values */ };
    191172qsort( vals, 10 );                                                      $\C{// sort array}$
    192173double * val = bsearch( 5.0, vals, 10 );        $\C{// binary search sorted array for key}$
    193174\end{lstlisting}
    194 @qsort@ and @bsearch@ work for any type @T@ with a matching @<@ operator, and the built-in monomorphic specialization of @<@ for type @double@ is passed as an implicit parameter to the calls of @qsort@ and @bsearch@.
     175@qsort@ and @bsearch@ can only be called with arguments for which there exists a function named @<@ taking two arguments of the same type and returning an @int@ value.
     176Here, the built-in monomorphic specialization of @<@ for type @double@ is passed as an additional implicit parameter to the calls of @qsort@ and @bsearch@.
    195177
    196178Crucial to the design of a new programming language are the libraries to access thousands of external features.
    … …  
    205187double * val = (double *)bsearch( &key, vals, size, sizeof(vals[0]), comp );
    206188\end{lstlisting}
    207 which can be augmented simply with a generalized, type-safe, \CFA-overloaded wrapper:
     189but providing a type-safe \CFA overloaded wrapper.
    208190\begin{lstlisting}
    209191forall( otype T | { int ?<?( T, T ); } ) T * bsearch( T key, const T * arr, size_t size ) {
    … …  
    219201\end{lstlisting}
    220202The nested routine @comp@ provides the hidden interface from typed \CFA to untyped (@void *@) C, plus the cast of the result.
    221 As well, an alternate kind of return is made available: position versus pointer to found element.
     203As well, an alternate kind of return is made available, position versus pointer to found element.
    222204\CC's type-system cannot disambiguate between the two versions of @bsearch@ because it does not use the return type in overload resolution, nor can \CC separately compile a templated @bsearch@.
    223205
    … …  
    229211\end{lstlisting}
    230212Within the block, the nested version of @<@ performs @>@ and this local version overrides the built-in @<@ so it is passed to @qsort@.
    231 Hence, programmers can easily form a local environments, adding and modifying appropriate functions, to maximize reuse of other existing functions and types.
    232 
    233 Finally, \CFA allows variable overloading:
     213Hence, programmers can easily form new local environments to maximize reuse of existing functions and types.
     214
     215Finally, variables may be overloaded:
    234216\lstDeleteShortInline@
    235217\par\smallskip
    … …  
    251233Hence, the single name @MAX@ replaces all the C type-specific names: @SHRT_MAX@, @INT_MAX@, @DBL_MAX@.
    252234
    253 
    254235\subsection{Traits}
    255236
    256 \CFA provides \emph{traits} to name a group of type assertions, where the trait name allows specifying the same set of assertions in multiple locations, preventing repetition mistakes at each function declaration:
     237\CFA provides \emph{traits} to name a group of type assertions:
     238% \begin{lstlisting}
     239% trait has_magnitude(otype T) {
     240%     _Bool ?<?(T, T);                                          $\C{// comparison operator for T}$
     241%     T -?(T);                                                          $\C{// negation operator for T}$
     242%     void ?{}(T*, zero_t);                                     $\C{// constructor from 0 literal}$
     243% };
     244% forall(otype M | has_magnitude(M))
     245% M abs( M m ) {
     246%     M zero = { 0 };                                                   $\C{// uses zero\_t constructor from trait}$
     247%     return m < zero ? -m : m;
     248% }
     249% forall(otype M | has_magnitude(M))
     250% M max_magnitude( M a, M b ) {
     251%     return abs(a) < abs(b) ? b : a;
     252% }
     253% \end{lstlisting}
    257254\begin{lstlisting}
    258255trait summable( otype T ) {
    … …  
    265262forall( otype T | summable( T ) )
    266263  T sum( T a[$\,$], size_t size ) {
    267         `T` total = { `0` };                                    $\C{// instantiate T from 0 but calling its constructor}$
     264        T total = { 0 };                                                $\C{// instantiate T from 0}$
    268265        for ( unsigned int i = 0; i < size; i += 1 )
    269                 total `+=` a[i];                                        $\C{// select appropriate +}$
     266                total += a[i];                                          $\C{// select appropriate +}$
    270267        return total;
    271268}
    272269\end{lstlisting}
     270The trait name allows specifying the same set of assertions in multiple locations, preventing repetition mistakes at each function declaration.
    273271
    274272In fact, the set of operators is incomplete, \eg no assignment, but @otype@ is syntactic sugar for the following implicit trait:
    275273\begin{lstlisting}
    276 trait otype( dtype T | sized(T) ) {  // sized is a pseudo-trait for types with known size and alignment
     274trait otype( dtype T | sized(T) ) {
     275        // sized is a compiler-provided pseudo-trait for types with known size and alignment}
    277276        void ?{}( T * );                                                $\C{// default constructor}$
    278277        void ?{}( T *, T );                                             $\C{// copy constructor}$
    … …  
    281280};
    282281\end{lstlisting}
    283 Given the information provided for an @otype@, variables of polymorphic type can be treated as if they were a complete type: stack-allocatable, default or copy-initialized, assigned, and deleted.
    284 % As an example, the @sum@ function produces generated code something like the following (simplified for clarity and brevity)\TODO{fix example, maybe elide, it's likely too long with the more complicated function}:
    285 % \begin{lstlisting}
    286 % void abs( size_t _sizeof_M, size_t _alignof_M,
    287 %               void (*_ctor_M)(void*), void (*_copy_M)(void*, void*),
    288 %               void (*_assign_M)(void*, void*), void (*_dtor_M)(void*),
    289 %               _Bool (*_lt_M)(void*, void*), void (*_neg_M)(void*, void*),
    290 %       void (*_ctor_M_zero)(void*, int),
    291 %               void* m, void* _rtn ) {                         $\C{// polymorphic parameter and return passed as void*}$
    292 %                                                                                       $\C{// M zero = { 0 };}$
    293 %       void* zero = alloca(_sizeof_M);                 $\C{// stack allocate zero temporary}$
    294 %       _ctor_M_zero(zero, 0);                                  $\C{// initialize using zero\_t constructor}$
    295 %                                                                                       $\C{// return m < zero ? -m : m;}$
    296 %       void *_tmp = alloca(_sizeof_M);
    297 %       _copy_M( _rtn,                                                  $\C{// copy-initialize return value}$
    298 %               _lt_M( m, zero ) ?                                      $\C{// check condition}$
    299 %                (_neg_M(m, _tmp), _tmp) :                      $\C{// negate m}$
    300 %                m);
    301 %       _dtor_M(_tmp); _dtor_M(zero);                   $\C{// destroy temporaries}$
    302 % }
    303 % \end{lstlisting}
    304 
    305 Traits may be used for many of the same purposes as interfaces in Java or abstract base classes in \CC. Unlike Java interfaces or \CC base classes, \CFA types do not explicitly state any inheritance relationship to traits they satisfy, which is a form of structural inheritance, similar to the implementation of an interface in Go~\citep{Go}, as opposed to the nominal inheritance model of Java and \CC.
    306 
    307 Nominal inheritance can be simulated with traits using marker variables or functions:
     282Given the information provided for an @otype@, variables of polymorphic type can be treated as if they were a complete struct type -- they can be stack-allocated using the @alloca@ compiler builtin, default or copy-initialized, assigned, and deleted. As an example, the @sum@ function produces generated code something like the following (simplified for clarity and brevity)\TODO{fix example, maybe elide, it's likely too long with the more complicated function}:
     283\begin{lstlisting}
     284void abs( size_t _sizeof_M, size_t _alignof_M,
     285                void (*_ctor_M)(void*), void (*_copy_M)(void*, void*),
     286                void (*_assign_M)(void*, void*), void (*_dtor_M)(void*),
     287                _Bool (*_lt_M)(void*, void*), void (*_neg_M)(void*, void*),
     288        void (*_ctor_M_zero)(void*, int),
     289                void* m, void* _rtn ) {                         $\C{// polymorphic parameter and return passed as void*}$
     290                                                                                        $\C{// M zero = { 0 };}$
     291        void* zero = alloca(_sizeof_M);                 $\C{// stack allocate zero temporary}$
     292        _ctor_M_zero(zero, 0);                                  $\C{// initialize using zero\_t constructor}$
     293                                                                                        $\C{// return m < zero ? -m : m;}$
     294        void *_tmp = alloca(_sizeof_M);
     295        _copy_M( _rtn,                                                  $\C{// copy-initialize return value}$
     296                _lt_M( m, zero ) ?                                      $\C{// check condition}$
     297                 (_neg_M(m, _tmp), _tmp) :                      $\C{// negate m}$
     298                 m);
     299        _dtor_M(_tmp); _dtor_M(zero);                   $\C{// destroy temporaries}$
     300}
     301\end{lstlisting}
     302
     303Semantically, traits are simply a named lists of type assertions, but they may be used for many of the same purposes that interfaces in Java or abstract base classes in \CC are used for. Unlike Java interfaces or \CC base classes, \CFA types do not explicitly state any inheritance relationship to traits they satisfy; this can be considered a form of structural inheritance, similar to implementation of an interface in Go, as opposed to the nominal inheritance model of Java and \CC. Nominal inheritance can be simulated with traits using marker variables or functions:
    308304\begin{lstlisting}
    309305trait nominal(otype T) {
    310306    T is_nominal;
    311307};
     308
    312309int is_nominal;                                                         $\C{// int now satisfies the nominal trait}$
    313310\end{lstlisting}
    314311
    315 Traits, however, are significantly more powerful than nominal-inheritance interfaces; most notably, traits may be used to declare a relationship \emph{among} multiple types, a property that may be difficult or impossible to represent in nominal-inheritance type systems:
     312Traits, however, are significantly more powerful than nominal-inheritance interfaces; most notably, traits may be used to declare a relationship among multiple types, a property that may be difficult or impossible to represent in nominal-inheritance type systems:
    316313\begin{lstlisting}
    317314trait pointer_like(otype Ptr, otype El) {
    318315    lvalue El *?(Ptr);                                          $\C{// Ptr can be dereferenced into a modifiable value of type El}$
    319316}
     317
    320318struct list {
    321319    int value;
    322     list *next;                                                         $\C{// may omit "struct" on type names as in \CC}$
     320    list *next;                                                         $\C{// may omit "struct" on type names}$
    323321};
     322
    324323typedef list *list_iterator;
    325324
    … …  
    334333One of the known shortcomings of standard C is that it does not provide reusable type-safe abstractions for generic data structures and algorithms. Broadly speaking, there are three approaches to create data structures in C. One approach is to write bespoke data structures for each context in which they are needed. While this approach is flexible and supports integration with the C type-checker and tooling, it is also tedious and error-prone, especially for more complex data structures. A second approach is to use @void*@-based polymorphism. This approach is taken by the C standard library functions @qsort@ and @bsearch@, and does allow the use of common code for common functionality. However, basing all polymorphism on @void*@ eliminates the type-checker's ability to ensure that argument types are properly matched, often requires a number of extra function parameters, and also adds pointer indirection and dynamic allocation to algorithms and data structures that would not otherwise require them. A third approach to generic code is to use pre-processor macros to generate it -- this approach does allow the generated code to be both generic and type-checked, though any errors produced may be difficult to interpret. Furthermore, writing and invoking C code as preprocessor macros is unnatural and somewhat inflexible.
    335334
    336 Other C-like languages such as \CC and Java use \emph{generic types} to produce type-safe abstract data types. \CFA implements generic types with some care taken that the generic types design for \CFA integrates efficiently and naturally with the existing polymorphic functions in \CFA while retaining backwards compatibility with C; maintaining separate compilation is a particularly important constraint on the design. However, where the concrete parameters of the generic type are known, there is no extra overhead for the use of a generic type, as for \CC templates.
     335Other C-like languages such as \CC and Java use \emph{generic types} to produce type-safe abstract data types. The authors have chosen to implement generic types as well, with some care taken that the generic types design for \CFA integrates efficiently and naturally with the existing polymorphic functions in \CFA while retaining backwards compatibility with C; maintaining separate compilation is a particularly important constraint on the design. However, where the concrete parameters of the generic type are known, there is not extra overhead for the use of a generic type.
    337336
    338337A generic type can be declared by placing a @forall@ specifier on a @struct@ or @union@ declaration, and instantiated using a parenthesized list of types after the type name:
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    359358\end{lstlisting}
    360359
    361 \CFA classifies generic types as either \emph{concrete} or \emph{dynamic}. Concrete generic types have a fixed memory layout regardless of type parameters, while dynamic generic types vary in their in-memory layout depending on their type parameters. A type may have polymorphic parameters but still be concrete; in \CFA such types are called \emph{dtype-static}. Polymorphic pointers are an example of dtype-static types -- @forall(dtype T) T*@ is a polymorphic type, but for any @T@ chosen, @T*@ has exactly the same in-memory representation as a @void*@, and can therefore be represented by a @void*@ in code generation.
    362 
    363 \CFA generic types may also specify constraints on their argument type to be checked by the compiler. For example, consider the following declaration of a sorted set-type, which ensures that the set key supports equality and relational comparison:
     360\CFA classifies generic types as either \emph{concrete} or \emph{dynamic}. Dynamic generic types vary in their in-memory layout depending on their type parameters, while concrete generic types have a fixed memory layout regardless of type parameters. A type may have polymorphic parameters but still be concrete; in \CFA such types are called \emph{dtype-static}. Polymorphic pointers are an example of dtype-static types -- @forall(dtype T) T*@ is a polymorphic type, but for any @T@ chosen, @T*@ has exactly the same in-memory representation as a @void*@, and can therefore be represented by a @void*@ in code generation.
     361
     362\CFA generic types may also specify constraints on their argument type to be checked by the compiler. For example, consider the following declaration of a sorted set type, which ensures that the set key supports comparison and tests for equality:
    364363\begin{lstlisting}
    365364forall(otype Key | { _Bool ?==?(Key, Key); _Bool ?<?(Key, Key); })
    366   struct sorted_set;
     365struct sorted_set;
    367366\end{lstlisting}
    368367
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