app1.tex

\section{Appendix: Derived Forms}
\label{derived-forms-app}
Several derived\index{66.1} grammatical forms are provided in the Core; they are presented
in Figures~\ref{der-exp}, \ref{der-pat} and \ref{der-dec}. Each derived form is
given with its equivalent form. Thus, each row of the tables should be
considered as a rewriting rule
\[ \mbox{Derived form \ $\Longrightarrow$\  Equivalent form} \]
and these rules may be applied repeatedly to a phrase until it is transformed
into a phrase of the bare language.
See Appendix~\ref{core-gram-app} for the full Core grammar, including all the
derived forms.

In the derived forms for tuples, in terms of records, we use $\overline{n}$ to
mean the ML numeral which stands for the natural number $n$.

Note that a new phrase class ~FvalBind~ of function-value bindings is introduced,
accompanied by a new declaration form ~\FUN\ \insertion{\theexplicittyvars}{\tyvarseq\ }\fvalbind~. The mixed forms
~\VAL\ \insertion{\theexplicittyvars}{\tyvarseq\ }\REC\ \fvalbind~, ~\VAL\ \insertion{\theexplicittyvars}{\tyvarseq\ }\fvalbind~ and ~\FUN\ \insertion{\theexplicittyvars}{\tyvarseq\ }\valbind~ are not
allowed -- though the first form arises during translation into the bare
language.

The following notes refer to Figure~\ref{der-dec}:
\begin{itemize}
%\item      In the equivalent form for a function-value binding, the
%           variables ~$\var_1$, $\cdots$, $\var_n$~ must be chosen not to
%           occur in the derived form.  The condition $m,n\geq 1$ applies.
\item      There is a version of the derived form for function-value binding
	   which allows the function identifier to be infixed;
	   see Figure~\ref{dec-gram} in Appendix~\ref{core-gram-app}.
\item      In the two forms involving ~\WITHTYPE~, the identifiers bound
           by ~\datbind~ and by ~\typbind~ must be distinct. Then the
           transformed binding ~\datbind$\/'$~ in the equivalent form is
           obtained from ~\datbind~ by expanding out all the definitions
           made by ~\typbind.  More precisely, if ~\typbind~ is
           \[ \tyvarseq_1\ \tycon_1\ \mbox{\ml{=}} \ty_1\ \ \AND
              \ \cdots\ \AND
            \ \ \tyvarseq_n\ \tycon_n\ \mbox{\ml{=}} \ty_n\ \]
           then $\hbox{\datbind}'$ is the result of simultaneous replacement
           (in ~\datbind) of every type expression ~$\tyseq_i\ \tycon_i$~
           ($1\leq i\leq n$)
           by the corresponding defining expression
           \[  \ty_i\{\tyseq_i/\tyvarseq_i\}\]
%\item      The abbreviation of ~\VAL\ {\tt it =} \exp~ to ~\exp~ is only
%           permitted at top-level, i.e. as a ~\program~.
\end{itemize}

Figure~\ref{functor-der-forms-fig} shows derived forms for functors.
They allow functors to take, say, a single type or value as a parameter,
in cases where it would seem clumsy to ``wrap up'' the argument as a
structure expression.
\deletion{\thenostrsharing}{
These forms are currently more experimental than the bare syntax of modules, 
but we recommend implementers to include them so that they can be
tested in practice.
In the derived forms for functor bindings and functor signature expressions,
$\strid$ is a new structure identifier and
the form of $\sigexp'$ depends
on the form of $\sigexp$ as follows. 
If $\sigexp$ is simply a signature identifier
$\sigid$, then $\sigexp'$ is also $\sigid$; otherwise $\sigexp$ must take
the form  ~$\SIG\ \spec_1\ \END$~,
and then $\sigexp'$ is
$\mbox{\SIG\ \LOCAL\ \OPEN\ \strid\ \IN\ $\spec_1$\ \END\ \END}$.
%(where $\strid$ is new).
}

\insertion{\thenostrsharing}{
Finally, Figure~\ref{spec-der-forms-fig} shows the derived forms for specifications and signature expressions.
The last derived form for specifications allows sharing between structure
identifiers as a shorthand for type sharing specifications. 
%Standard ML no 
%longer has a semantic notion of structure sharing; 
%however, for compatability with Standard ML '90, a weaker form
%of structure sharing specification is provided. 
The phrase
\[
\boxml{$\spec$ sharing $\longstrid_1$ = $\cdots$ = $\longstrid_k$}
\]
is a derived form whose equivalent form is
\pagebreak
%\medskip

\halign{\qquad\ignore{#}&#\hfil\cr
\boxml{$\spec$ sharing $\longstrid_1$ = $\cdots$} & \boxml{$\spec$}\cr
\boxml{\ \ \qquad\qquad\qquad\qquad\qquad = $\longstrid_k$} & \boxml{\ sharing type $\longtycon_{1}$ =  $\longtycon_{1}'$}\cr
  & \boxml{\ $\cdots$}\cr
  & \boxml{\ sharing type $\longtycon_{m}$ =  $\longtycon_{m}'$}\cr}
\medskip

\noindent
determined as follows. 
First, note that  $\spec$  specifies a set of 
     (possibly long) type constructors and structure identifiers, either 
     directly or via signature identifiers and $\INCLUDE$ specifications.  
     Then the equivalent form contains all type-sharing constraints 
     of the form 
\[\boxml{sharing type $\longstrid_i.\longtycon$ = $\longstrid_j.longtycon$}\]
     $(1\leq i<j\leq k)$,  such that both sides of the equation are long type 
     constructors specified by  $\spec$. 

     The meaning of the derived form does not depend on the order of the 
     type-sharing constraints in the equivalent form.

}


\begin{figure}
\adhocreplacementl{\theidstatus}{0cm}{\begin{tabular}{|l|l|l}
\multicolumn{1}{c}{Derived Form} & \multicolumn{1}{c}{Equivalent Form} &
\multicolumn{1}{c}{}\\
\multicolumn{3}{c}{}\\
\multicolumn{2}{l}{{\bf Expressions} \exp}\\
%\multicolumn{2}{l}{EXPRESSIONS \exp}\\
\cline{1-2}
\ml{()}         & \ml{\lttbrace\ \rttbrace} \\
\cline{1-2}
\ml{(}$\exp_1$ \ml{,} $\cdots$ \ml{,} $\exp_\n$\ml{)}
            & \ml{\lttbrace 1=}$\exp_1$\ml{,}\ $\cdots$\ml{,}\
                             $\overline{n}$\ml{=}$\exp_\n$\ml{\rttbrace}
                                                           & $(\n\geq 2)$\\
\cline{1-2}
\ml{\#}\ \lab      & \FN\ \ml{\lttbrace}\lab\ml{=}\var\ml{,...\rttbrace\  => }\var
                                                           & (\var\ new)\\
%\cline{1-2}
%\RAISE\ \longexn    & \RAISE\ \longexn\ \WITH\ \ml{()} \\
\cline{1-2}
\CASE\ \exp\ \OF\ \match
                & \ml{(}\FN\ \match\ml{)(}\exp\ml{)} \\
\cline{1-2}
\IF\ $\exp_1$\ \THEN\ $\exp_2$\ \ELSE\ $\exp_3$
                & \CASE\ $\exp_1$\ \OF\ \TRUE\ \ml{=>}\ \exp$_2$\\
                & \ \ \qquad\qquad\ml{|}\ \FALSE\ \ml{=>}\ \exp$_3$ \\
\cline{1-2}
\exp$_1$\ \ORELSE\ \exp$_2$
                & \IF\ \exp$_1$\ \THEN\ \TRUE\ \ELSE\ \exp$_2$ \\
\cline{1-2}
\exp$_1$\ \ANDALSO\ \exp$_2$
                & \IF\ \exp$_1$\ \THEN\ \exp$_2$\ \ELSE\ \FALSE \\
\cline{1-2}
\ml{(}$\exp_1$ \ml{;} $\cdots$ \ml{;} $\exp_\n$ \ml{;} \exp\ml{)}\
                & \CASE\ \exp$_1$\ \OF\ \ml{(\_) =>}
                                                           & $(\n\geq 1)$ \\
                & \qquad$\cdots$ \\
                & \CASE\ \exp$_n$\ \OF\ \ml{(\_) =>}\ \exp \\
\cline{1-2}
\LET\ \dec\ \IN
                & \LET\ \dec\ \IN                          & $(\n\geq 2)$ \\
\qquad$\exp_1$ \ml{;} $\cdots$ \ml{;} $\exp_\n$ \END
                & \ \ \ml{(}$\exp_1$ \ml{;} $\cdots$ \ml{;} $\exp_\n$\ml{)}\
                                                                         \END\\
\cline{1-2}
\WHILE\ \exp$_1$\ \DO\ \exp$_2$
                & \LET\ \VAL\ \REC\ \var\ \ml{=}\ \FN\ \ml{() =>}
                                                           & (\var\ new)\\
                & \ \ \IF\ \exp$_1$\ \THEN\
                    \ml{(}\exp$_2$\ml{;}\var\ml{())}\ \ELSE\ \ml{()} \\
                & \ \ \IN\ \var\ml{()}\ \END\\
\cline{1-2}
\ml{[}$\exp_1$ \ml{,} $\cdots$ \ml{,} $\exp_\n$\ml{]}
                & \exp$_1$\ \ml{::}\ $\cdots$\ \ml{::}\ \exp$_n$\
                            \ml{::}\ \NIL                 & $(n\geq 0)$ \\
\cline{1-2}
\multicolumn{3}{c}{}\\
\end{tabular}}{\begin{tabular}{|l|l|l}
\multicolumn{1}{c}{Derived Form} & \multicolumn{1}{c}{Equivalent Form} &
\multicolumn{1}{c}{}\\
\multicolumn{3}{c}{}\\
\multicolumn{2}{l}{{\bf Expressions} \exp}\\
%\multicolumn{2}{l}{EXPRESSIONS \exp}\\
\cline{1-2}
\ml{()}         & \ml{\lttbrace\ \rttbrace} \\
\cline{1-2}
\ml{(}$\exp_1$ \ml{,} $\cdots$ \ml{,} $\exp_\n$\ml{)}
            & \ml{\lttbrace 1=}$\exp_1$\ml{,}\ $\cdots$\ml{,}\
                             $\overline{n}$\ml{=}$\exp_\n$\ml{\rttbrace}
                                                           & $(\n\geq 2)$\\
\cline{1-2}
\ml{\#}\ \lab      & \FN\ \ml{\lttbrace}\lab\ml{=}\vid\ml{,...\rttbrace\  => }\vid
                                                           & (\vid\ new)\\
%\cline{1-2}
%\RAISE\ \longexn    & \RAISE\ \longexn\ \WITH\ \ml{()} \\
\cline{1-2}
\CASE\ \exp\ \OF\ \match
                & \ml{(}\FN\ \match\ml{)(}\exp\ml{)} \\
\cline{1-2}
\IF\ $\exp_1$\ \THEN\ $\exp_2$\ \ELSE\ $\exp_3$
                & \CASE\ $\exp_1$\ \OF\ \TRUE\ \ml{=>}\ \exp$_2$\\
                & \ \ \qquad\qquad\ml{|}\ \FALSE\ \ml{=>}\ \exp$_3$ \\
\cline{1-2}
\exp$_1$\ \ORELSE\ \exp$_2$
                & \IF\ \exp$_1$\ \THEN\ \TRUE\ \ELSE\ \exp$_2$ \\
\cline{1-2}
\exp$_1$\ \ANDALSO\ \exp$_2$
                & \IF\ \exp$_1$\ \THEN\ \exp$_2$\ \ELSE\ \FALSE \\
\cline{1-2}
\ml{(}$\exp_1$ \ml{;} $\cdots$ \ml{;} $\exp_\n$ \ml{;} \exp\ml{)}\
                & \CASE\ \exp$_1$\ \OF\ \ml{(\_) =>}
                                                           & $(\n\geq 1)$ \\
                & \qquad$\cdots$ \\
                & \CASE\ \exp$_n$\ \OF\ \ml{(\_) =>}\ \exp \\
\cline{1-2}
\LET\ \dec\ \IN
                & \LET\ \dec\ \IN                          & $(\n\geq 2)$ \\
\qquad$\exp_1$ \ml{;} $\cdots$ \ml{;} $\exp_\n$ \END
                & \ \ \ml{(}$\exp_1$ \ml{;} $\cdots$ \ml{;} $\exp_\n$\ml{)}\
                                                                         \END\\
\cline{1-2}
\WHILE\ \exp$_1$\ \DO\ \exp$_2$
                & \LET\ \VAL\ \REC\ \vid\ \ml{=}\ \FN\ \ml{() =>}
                                                           & (\vid\ new)\\
                & \ \ \IF\ \exp$_1$\ \THEN\
                    \ml{(}\exp$_2$\ml{;}\vid\ml{())}\ \ELSE\ \ml{()} \\
                & \ \ \IN\ \vid\ml{()}\ \END\\
\cline{1-2}
\ml{[}$\exp_1$ \ml{,} $\cdots$ \ml{,} $\exp_\n$\ml{]}
                & \exp$_1$\ \ml{::}\ $\cdots$\ \ml{::}\ \exp$_n$\
                            \ml{::}\ \NIL                 & $(n\geq 0)$ \\
\cline{1-2}
\multicolumn{3}{c}{}\\
\end{tabular}}
\caption{Derived forms of Expressions\index{67.1}}
\label{der-exp}
\end{figure}
\begin{figure}

\begin{tabular}{|l|l|l}
\multicolumn{1}{c}{Derived Form} & \multicolumn{1}{c}{Equivalent Form} &
\multicolumn{1}{c}{}\\
\multicolumn{3}{c}{}\\
\multicolumn{2}{l}{{\bf Patterns} \pat}\\
%\multicolumn{2}{l}{PATTERNS \pat}\\
\cline{1-2}
\ml{()}         & \ml{\lttbrace\ \rttbrace} \\
\cline{1-2}
\ml{(}$\pat_1$ \ml{,} $\cdots$ \ml{,} $\pat_\n$\ml{)}
            & \ml{\lttbrace 1=}$\pat_1$\ml{,}\ $\cdots$ \ml{,}\
                             $\overline{n}$\ml{=}$\pat_\n$\ml{\rttbrace}
                                                           & $(\n\geq 2)$ \\
\cline{1-2}
\ml{[}$\pat_1$ \ml{,} $\cdots$ \ml{,} $\pat_\n$\ml{]}
                & \pat$_1$\ \ml{::}\ $\cdots$\ \ml{::}\ \pat$_n$\
                            \ml{::}\ \NIL                 & $(n\geq 0)$ \\
\cline{1-2}
\multicolumn{3}{c}{}\\
\multicolumn{2}{l}{{\bf Pattern Rows} \labpats}\\
%\multicolumn{2}{l}{PATTERN ROWS \labpats}\\
\cline{1-2}
\adhocreplacementl{\theidstatus}{1cm}{
\id$\langle$\ml{:}\ty$\rangle
    \ \langle\AS\ \pat\rangle
    \ \langle$\ml{,} \labpats$\rangle$
                }{\vid$\langle$\ml{:}\ty$\rangle
    \ \langle\AS\ \pat\rangle
    \ \langle$\ml{,} \labpats$\rangle$}& \adhocreplacementl{\theidstatus}{-9cm}{\id\ml{ = }\id$\langle$\ml{:}\ty$\rangle
                                 \ \langle\AS\ \pat\rangle
                                 \ \langle$\ml{,} \labpats$\rangle$}{\vid\ml{ = }\vid$\langle$\ml{:}\ty$\rangle
                                 \ \langle\AS\ \pat\rangle
                                 \ \langle$\ml{,} \labpats$\rangle$} \\
\cline{1-2}
\multicolumn{3}{c}{}\\
\multicolumn{2}{l}{{\bf Type Expressions} \ty}\\
%\multicolumn{2}{l}{TYPE EXPRESSIONS \ty}\\
\cline{1-2}
$\ty_1$ \ml{*} $\cdots$ \ml{*} $\ty_\n$
            & \ml{\lttbrace 1:}$\ty_1$\ml{,}\ $\cdots$ \ml{,}\
                             $\overline{n}$\ml{:}$\ty_\n$\ml{\rttbrace}
                                                           & $(\n\geq 2)$ \\
\cline{1-2}
\multicolumn{3}{c}{}\\
\end{tabular}
\caption{Derived forms of Patterns and Type Expressions\index{67.2}}
\label{der-pat}
\end{figure}
\begin{figure}
\adhocreplacementl{\theidstatus}{0cm}{
\begin{tabular}{|l|l|}
\multicolumn{1}{c}{Derived Form} & \multicolumn{1}{c}{Equivalent Form}\\
\multicolumn{2}{c}{}\\
\multicolumn{2}{l}{{\bf Function-value Bindings} \fvalbind}\\
%\multicolumn{2}{l}{FUNCTION-VALUE BINDINGS \fvalbind}\\
\hline
               & $\langle\OP\rangle$\var\ \ml{=} \FN\ \var$_1$\ml{=>} $\cdots$
                              \FN\ \var$_n$\ml{=>} \\
               & \CASE\
                 \ml{(}\var$_1$\ml{,} $\cdots$ \ml{,} \var$_n$\ml{)} \OF \\
\ \ $\langle\OP\rangle\var\ \atpat_{11}\cdots\atpat_{1n}
                                              \langle$\ml{:}\ty$\rangle$
                                              \ml{=} \exp$_1$
               & \ \ \ml{(}\atpat$_{11}$\ml{,}$\cdots$\ml{,}\atpat$_{1n}$
                             \ml{)=>}\exp$_1\langle$\ml{:}\ty$\rangle$\\
\ml{|}$\langle\OP\rangle\var\ \atpat_{21}\cdots\atpat_{2n}
                                              \langle$\ml{:}\ty$\rangle$
                                              \ml{=} \exp$_2$
               & \ml{|(}\atpat$_{21}$\ml{,}$\cdots$\ml{,}\atpat$_{2n}$
                             \ml{)=>}\exp$_2\langle$\ml{:}\ty$\rangle$\\
\ml{|}\qquad$\cdots\qquad\cdots$
               & \ml{|}\qquad$\cdots\qquad\cdots$\\
\ml{|}$\langle\OP\rangle\var\ \atpat_{m1}\cdots\atpat_{mn}
                                              \langle$\ml{:}\ty$\rangle$
                                              \ml{=} \exp$_m$
               & \ml{|(}\atpat$_{m1}$\ml{,}$\cdots$\ml{,}\atpat$_{mn}$
                             \ml{)=>}\exp$_m\langle$\ml{:}\ty$\rangle$\\
\qquad\qquad\qquad$\langle\AND\ \fvalbind\rangle$
               & \qquad\qquad\qquad$\langle\AND\ \fvalbind\rangle$\\
\hline
\multicolumn{2}{r}{($m,n\geq1$; $\var_1,\cdots,\var_n$ distinct and new)}\\
\multicolumn{2}{c}{}\\
\multicolumn{2}{l}{{\bf Declarations} \dec}\\
%\multicolumn{2}{l}{DECLARATIONS \dec}\\
\hline
\FUN\  \fvalbind
               & \VAL\ \REC\ \fvalbind  \\
\hline
\DATATYPE\ \datbind\ \WITHTYPE\ \typbind
               & \DATATYPE\ \datbind$\/'$\ \ml{;}\ \TYPE\ \typbind \\
\hline
\ABSTYPE\ \datbind\ \WITHTYPE\ \typbind
               & \ABSTYPE\ \datbind$\/'$ \\
\qquad\qquad\WITH\ \dec\ \END
               & \qquad\WITH\ \TYPE\ \typbind\ \ml{;}\ \dec\ \END\\
\hline
\multicolumn{2}{r}{(see note in text concerning \datbind$\/'$)}\\
\multicolumn{2}{c}{}\\
\end{tabular}}{\begin{tabular}{|l|l|}
\multicolumn{1}{c}{Derived Form} & \multicolumn{1}{c}{Equivalent Form}\\
\multicolumn{2}{c}{}\\
\multicolumn{2}{l}{{\bf Function-value Bindings} \fvalbind}\\
%\multicolumn{2}{l}{FUNCTION-VALUE BINDINGS \fvalbind}\\
\hline
               & $\langle\OP\rangle$\vid\ \ml{=} \FN\ \vid$_1$\ml{=>} $\cdots$
                              \FN\ \vid$_n$\ml{=>} \\
               & \CASE\
                 \ml{(}\vid$_1$\ml{,} $\cdots$ \ml{,} \vid$_n$\ml{)} \OF \\
\ \ $\langle\OP\rangle\vid\ \atpat_{11}\cdots\atpat_{1n}
                                              \langle$\ml{:}\ty$\rangle$
                                              \ml{=} \exp$_1$
               & \ \ \ml{(}\atpat$_{11}$\ml{,}$\cdots$\ml{,}\atpat$_{1n}$
                             \ml{)=>}\exp$_1\langle$\ml{:}\ty$\rangle$\\
\ml{|}$\langle\OP\rangle\vid\ \atpat_{21}\cdots\atpat_{2n}
                                              \langle$\ml{:}\ty$\rangle$
                                              \ml{=} \exp$_2$
               & \ml{|(}\atpat$_{21}$\ml{,}$\cdots$\ml{,}\atpat$_{2n}$
                             \ml{)=>}\exp$_2\langle$\ml{:}\ty$\rangle$\\
\ml{|}\qquad$\cdots\qquad\cdots$
               & \ml{|}\qquad$\cdots\qquad\cdots$\\
\ml{|}$\langle\OP\rangle\vid\ \atpat_{m1}\cdots\atpat_{mn}
                                              \langle$\ml{:}\ty$\rangle$
                                              \ml{=} \exp$_m$
               & \ml{|(}\atpat$_{m1}$\ml{,}$\cdots$\ml{,}\atpat$_{mn}$
                             \ml{)=>}\exp$_m\langle$\ml{:}\ty$\rangle$\\
\qquad\qquad\qquad$\langle\AND\ \fvalbind\rangle$
               & \qquad\qquad\qquad$\langle\AND\ \fvalbind\rangle$\\
\hline
\multicolumn{2}{r}{($m,n\geq1$; $\vid_1,\cdots,\vid_n$ distinct and new)}\\
\multicolumn{2}{c}{}\\
\multicolumn{2}{l}{{\bf Declarations} \dec}\\
%\multicolumn{2}{l}{DECLARATIONS \dec}\\
\hline
\FUN\ \tyvarseq\ \fvalbind
               & \VAL\ \tyvarseq\ \REC\ \fvalbind  \\
\hline
\DATATYPE\ \datbind\ \WITHTYPE\ \typbind
               & \DATATYPE\ \datbind$\/'$\ \ml{;}\ \TYPE\ \typbind \\
\hline
\ABSTYPE\ \datbind\ \WITHTYPE\ \typbind
               & \ABSTYPE\ \datbind$\/'$ \\
\qquad\qquad\WITH\ \dec\ \END
               & \qquad\WITH\ \TYPE\ \typbind\ \ml{;}\ \dec\ \END\\
\hline
\multicolumn{2}{r}{(see note in text concerning \datbind$\/'$)}\\
\multicolumn{2}{c}{}\\
\end{tabular}}
\caption{Derived forms of Function-value Bindings and Declarations\index{68.1}}
\label{der-dec}
\end{figure}

%               Derived forms of functors:

\begin{figure}\adhocreplacementl{\thenostrsharing}{0mm}{
\begin{tabular}{|l|l|}
\multicolumn{1}{c}{Derived Form} & \multicolumn{1}{c}{Equivalent Form} \\
\multicolumn{2}{c}{}\\
\multicolumn{2}{l}{{\bf Structure  Expressions} \strexp}\\
%\multicolumn{2}{l}{STRUCTURE EXPRESSIONS \strexp}\\
\cline{1-2}
\funappdec & \mbox{\funid\ \ml{(} \STRUCT\ \strdec\ \END\ \ml{)}}\\
\cline{1-2}
\multicolumn{2}{c}{}\\
%\multicolumn{2}{l}{FUNCTOR BINDINGS \funbind}\\
\multicolumn{2}{l}{{\bf Functor Bindings} \funbind}\\
\cline{1-2}        
\mbox{\funid\ \ml{(}\ \spec\ \ml{)}\ $\langle$\ml{:}\ \sigexp$\rangle$\ \ml{=}}&
\mbox{\funid\ \ml{(}\ \strid\ \ml{:} \SIG\ \spec\ \END\ \ml{)} 
              $\langle$\ml{:}\ $\sigexp'\rangle$\ \ml{=}}\\
\mbox{\ \ \strexp\ $\langle$\AND\ \funbind$\rangle$} &
  \mbox{\ \ \LET\ \OPEN\ \strid\ \IN\ \strexp\ \END\ $\langle$\AND\ \funbind$\rangle$} \\
\cline{1-2}
\multicolumn{2}{r}{($\strid$ new; see note in text concerning $\sigexp'$)}\\
\multicolumn{2}{c}{}\\
\multicolumn{2}{l}{{\bf Functor Signature Expressions} \funsigexp}\\
%\multicolumn{2}{l}{FUNCTOR SIGNATURES \funsigexp}\\
\cline{1-2}
\longfunsigexp & \mbox{\ml{(} \strid\ \ml{:}\ \SIG\ \spec\ \END\ \ml{)}
                \ml{:}\ \sigexp$'$} \\
\cline{1-2}
\multicolumn{2}{r}{($\strid$ new; see note in text concerning $\sigexp'$)}\\
\multicolumn{2}{c}{}\\
\multicolumn{2}{l}{{\bf Top-level Declarations} \topdec}\\
\cline{1-2}
\exp           & \VAL\ \ml{it =} \exp  \\
\cline{1-2}
\multicolumn{2}{c}{}\\
\end{tabular}}{
\begin{tabular}{|l|l|}
\multicolumn{1}{c}{Derived Form} & \multicolumn{1}{c}{Equivalent Form} \\
\multicolumn{2}{c}{}\\
\multicolumn{2}{l}{{\bf Structure  Bindings} \strbind}\\
\cline{1-2}
\derivedstrbinder & \equivalentstrbinder\\
\cline{1-2}
\derivedabststrbinder & \equivalentabststrbinder\\
\cline{1-2}
\multicolumn{2}{c}{}\\
\multicolumn{2}{l}{{\bf Structure  Expressions} \strexp}\\
\cline{1-2}
\funappdec & \mbox{\funid\ \ml{(} \STRUCT\ \strdec\ \END\ \ml{)}}\\
\cline{1-2}
\multicolumn{2}{c}{}\\
%\multicolumn{2}{l}{FUNCTOR BINDINGS \funbind}\\
\multicolumn{2}{l}{{\bf Functor Bindings} \funbind}\\
\cline{1-2}        
\mbox{\funid\ \ml{(}\strid\ml{:}\sigexp\ml{)}\ml{:} $\sigexp'$ \ml{=}}&
\mbox{\funid\ \ml{(}\strid\ \ml{:} \sigexp\ml{)} \ \ml{=}}\\
\mbox{\ \ \strexp\ $\langle$\AND\ \funbind$\rangle$} &
  \mbox{\ \ \strexp\ml{:}$\sigexp'$\  $\langle$\AND\ \funbind$\rangle$} \\
\cline{1-2}        
\mbox{\funid\ \ml{(}\strid\ml{:}\sigexp\ml{)}\ABSTRACT $\sigexp'$ \ml{=}}&
\mbox{\funid\ \ml{(}\strid\ \ml{:} \sigexp\ml{)} \ \ml{=}}\\
\mbox{\ \ \strexp\ $\langle$\AND\ \funbind$\rangle$} &
  \mbox{\ \ \strexp\ABSTRACT$\sigexp'$\  $\langle$\AND\ \funbind$\rangle$} \\
\cline{1-2}        
\mbox{\funid\ \ml{(}\ \spec\ \ml{)}\ $\langle$\ml{:}\ \sigexp$\rangle$\ \ml{=}}&
\mbox{\funid\ \ml{(}\ $\strid_\nu$\ \ml{:} \SIG\ \spec\ \END\ \ml{)} 
              \ \ml{=}}\\
\mbox{\ \ \strexp\ $\langle$\AND\ \funbind$\rangle$} &
  \mbox{\ \ \LET\ \OPEN\ $\strid_\nu$ \IN\ \strexp$\langle$\ml{:}\ $\sigexp\rangle$}\\
& \mbox{\ \ \END\ $\langle$\AND\ \funbind$\rangle$} \\
\cline{1-2}        
\mbox{\funid\ \ml{(}\ \spec\ \ml{)}\ $\langle$\ABSTRACT\ \sigexp$\rangle$\ \ml{=}}&
\mbox{\funid\ \ml{(}\ $\strid_\nu$\ \ml{:} \SIG\ \spec\ \END\ \ml{)} 
              \ \ml{=}}\\
\mbox{\ \ \strexp\ $\langle$\AND\ \funbind$\rangle$} &
  \mbox{\ \ \LET\ \OPEN\ $\strid_\nu$ \IN\ \strexp$\langle$\ABSTRACT $\sigexp\rangle$}\\
& \mbox{\ \ \END\ $\langle$\AND\ \funbind$\rangle$} \\
\cline{1-2}
\multicolumn{2}{r}{($\strid_\nu$ new)}\\
\multicolumn{2}{c}{}\\
\multicolumn{2}{l}{{\bf Programs} \program}\\
\cline{1-2}
$\exp\boxml{;}\langle\program\rangle$           & $\VAL\ \boxml{it =}\; \exp\boxml{;}\langle\program\rangle$\\
\cline{1-2}
\multicolumn{2}{c}{}\\
\end{tabular}}
\caption{Derived forms of Functors, Structure Bindings and Programs\index{68.2}}
\label{functor-der-forms-fig}
\end{figure}

\begin{figure}
\adhocinsertion{\thetypabbr}{-10mm}{
\begin{tabular}{|l|l|}
\multicolumn{1}{c}{Derived Form} & \multicolumn{1}{c}{Equivalent Form} \\
\multicolumn{2}{c}{}\\
\multicolumn{2}{l}{{\bf Specifications} \spec}\\ 
\cline{1-2}
$\TYPE\;\tyvarseq\;\tycon\,\boxml{=}\,\ty$ & \boxml{include}\\
 &\boxml{\ sig $\TYPE\;\tyvarseq\;\tycon$}\\
 &\boxml{\ end where type $\tyvarseq\;\tycon\,\boxml{=}\,\ty$} \\
\cline{1-2}
\boxml{type $\tyvarseq_1\;\tycon_1$ = $\ty_1$} & \boxml{type $\tyvarseq_1\;\tycon_1$ = $\ty_1$}\\
\boxml{ and $\cdots$} & \boxml{type $\cdots$}\\
\boxml{ $\cdots$} & \boxml{ $\cdots$}\\
\boxml{ and $\tyvarseq_n\;\tycon_n$ = $\ty_n$} & \boxml{type $\tyvarseq_n\;\tycon_n$ = $\ty_n$}\\
\cline{1-2}
$\inclspec$  & $\INCLUDE\,\sigid_1; \cdots\, ; \INCLUDE \,sigid_n$\\
\cline{1-2}
\boxml{$\spec$ sharing $\longstrid_1$ = $\cdots$} & \boxml{$\spec$}\\
\boxml{\ \ \qquad\qquad\qquad\qquad\qquad = $\longstrid_k$} & \boxml{\ sharing type $\longtycon_{1}$ = }\\
  & \boxml{\ \ \ \ \ \qquad\qquad\qquad\qquad $\longtycon_{1}'$}\\
  & \boxml{\ $\cdots$}\\
  & \boxml{\ sharing type $\longtycon_{m}$ = }\\
  & \boxml{\ \ \ \ \ \qquad\qquad\qquad\qquad  $\longtycon_{m}'$}\\
\cline{1-2}
\multicolumn{2}{r}{\vrule height14pt depth0pt width0pt(see note in text concerning $\longtycon_{1},\ldots,\longtycon_{m}'$)}\\
%\multicolumn{2}{r}{($n\geq 1$)}\\
\multicolumn{2}{c}{}\\
\multicolumn{2}{l}{{\bf Signature Expressions} \sigexp}\\ 
\cline{1-2}
\boxml{$\sigexp$} & \boxml{$\sigexp$}\\
\boxml{where type $\tyvarseq_1\; \longtycon_1$ = $\ty_1$} & \boxml{\ where type $\tyvarseq_1\; \longtycon_1$ = $\ty_1$}\\
\boxml{\ \ and\ type $\cdots$}  & \boxml{\ where type $\cdots$}\\
\boxml{\ \ $\cdots$}  & \boxml{\ $\cdots$}\\
\boxml{\ \ and\ type $\tyvarseq_n\;\longtycon_n$ = $\ty_n$}        &  \boxml{\ where type $\tyvarseq_n\;\longtycon_n$ = $\ty_n$}\\
\cline{1-2}
\end{tabular}}
\caption{Derived forms of Specifications and Signature Expressions}
\label{spec-der-forms-fig}
\end{figure}