Ur/Web: doc/manual.tex annotate

annotate doc/manual.tex @ 1179:c58453683bbb

Dead code elimination in Reduce code

author	Adam Chlipala <adamc@hcoop.net>
date	Thu, 04 Mar 2010 16:59:13 -0500
parents	7a2a7a8f9cab
children	9d3ccb8b39ac

rev	line source
adamc@524	1 \documentclass{article}
adamc@554	2 \usepackage{fullpage,amsmath,amssymb,proof,url}
adamc@524	3
adamc@524	4 \newcommand{\cd}[1]{\texttt{#1}}
adamc@524	5 \newcommand{\mt}[1]{\mathsf{#1}}
adamc@524	6
adamc@524	7 \newcommand{\rc}{+ \hspace{-.075in} + \;}
adamc@527	8 \newcommand{\rcut}{\; \texttt{--} \;}
adamc@527	9 \newcommand{\rcutM}{\; \texttt{---} \;}
adamc@524	10
adamc@524	11 \begin{document}
adamc@524	12
adamc@524	13 \title{The Ur/Web Manual}
adamc@524	14 \author{Adam Chlipala}
adamc@524	15
adamc@524	16 \maketitle
adamc@524	17
adamc@540	18 \tableofcontents
adamc@540	19
adamc@554	20
adamc@554	21 \section{Introduction}
adamc@554	22
adamc@1160	23 \emph{Ur} is a programming language designed to introduce richer type system features into functional programming in the tradition of ML and Haskell. Ur is functional, pure, statically-typed, and strict. Ur supports a powerful kind of \emph{metaprogramming} based on \emph{type-level computation with type-level records}.
adamc@554	24
adamc@554	25 \emph{Ur/Web} is Ur plus a special standard library and associated rules for parsing and optimization. Ur/Web supports construction of dynamic web applications backed by SQL databases. The signature of the standard library is such that well-typed Ur/Web programs ``don't go wrong'' in a very broad sense. Not only do they not crash during particular page generations, but they also may not:
adamc@554	26
adamc@554	27 \begin{itemize}
adamc@554	28 \item Suffer from any kinds of code-injection attacks
adamc@554	29 \item Return invalid HTML
adamc@554	30 \item Contain dead intra-application links
adamc@554	31 \item Have mismatches between HTML forms and the fields expected by their handlers
adamc@652	32 \item Include client-side code that makes incorrect assumptions about the ``AJAX''-style services that the remote web server provides
adamc@554	33 \item Attempt invalid SQL queries
adamc@652	34 \item Use improper marshaling or unmarshaling in communication with SQL databases or between browsers and web servers
adamc@554	35 \end{itemize}
adamc@554	36
adamc@554	37 This type safety is just the foundation of the Ur/Web methodology. It is also possible to use metaprogramming to build significant application pieces by analysis of type structure. For instance, the demo includes an ML-style functor for building an admin interface for an arbitrary SQL table. The type system guarantees that the admin interface sub-application that comes out will always be free of the above-listed bugs, no matter which well-typed table description is given as input.
adamc@554	38
adamc@652	39 The Ur/Web compiler also produces very efficient object code that does not use garbage collection. These compiled programs will often be even more efficient than what most programmers would bother to write in C. The compiler also generates JavaScript versions of client-side code, with no need to write those parts of applications in a different language.
adamc@554	40
adamc@554	41 \medskip
adamc@554	42
adamc@554	43 The official web site for Ur is:
adamc@554	44 \begin{center}
adamc@554	45 \url{http://www.impredicative.com/ur/}
adamc@554	46 \end{center}
adamc@554	47
adamc@555	48
adamc@555	49 \section{Installation}
adamc@555	50
adamc@555	51 If you are lucky, then the following standard command sequence will suffice for installation, in a directory to which you have unpacked the latest distribution tarball.
adamc@555	52
adamc@555	53 \begin{verbatim}
adamc@555	54 ./configure
adamc@555	55 make
adamc@555	56 sudo make install
adamc@555	57 \end{verbatim}
adamc@555	58
adamc@896	59 Some other packages must be installed for the above to work. At a minimum, you need a standard UNIX shell, with standard UNIX tools like sed and GCC in your execution path; MLton, the whole-program optimizing compiler for Standard ML; and the mhash C library. As of this writing, in the ``testing'' version of Debian Linux, this command will install the more uncommon of these dependencies:
adamc@896	60 \begin{verbatim}
adamc@896	61 apt-get install mlton libmhash-dev
adamc@896	62 \end{verbatim}
adamc@555	63
adamc@896	64 To build programs that access SQL databases, you also need one of these client libraries for supported backends.
adamc@555	65 \begin{verbatim}
adamc@896	66 apt-get install libpq-dev libmysqlclient15-dev libsqlite3-dev
adamc@555	67 \end{verbatim}
adamc@555	68
adamc@555	69 It is also possible to access the modules of the Ur/Web compiler interactively, within Standard ML of New Jersey. To install the prerequisites in Debian testing:
adamc@555	70 \begin{verbatim}
adamc@555	71 apt-get install smlnj libsmlnj-smlnj ml-yacc ml-lpt
adamc@555	72 \end{verbatim}
adamc@555	73
adamc@555	74 To begin an interactive session with the Ur compiler modules, run \texttt{make smlnj}, and then, from within an \texttt{sml} session, run \texttt{CM.make "src/urweb.cm";}. The \texttt{Compiler} module is the main entry point.
adamc@555	75
adamc@896	76 To run an SQL-backed application with a backend besides SQLite, you will probably want to install one of these servers.
adamc@555	77
adamc@555	78 \begin{verbatim}
adamc@896	79 apt-get install postgresql-8.3 mysql-server-5.0
adamc@555	80 \end{verbatim}
adamc@555	81
adamc@555	82 To use the Emacs mode, you must have a modern Emacs installed. We assume that you already know how to do this, if you're in the business of looking for an Emacs mode. The demo generation facility of the compiler will also call out to Emacs to syntax-highlight code, and that process depends on the \texttt{htmlize} module, which can be installed in Debian testing via:
adamc@555	83
adamc@555	84 \begin{verbatim}
adamc@555	85 apt-get install emacs-goodies-el
adamc@555	86 \end{verbatim}
adamc@555	87
adamc@555	88 Even with the right packages installed, configuration and building might fail to work. After you run \texttt{./configure}, you will see the values of some named environment variables printed. You may need to adjust these values to get proper installation for your system. To change a value, store your preferred alternative in the corresponding UNIX environment variable, before running \texttt{./configure}. For instance, here is how to change the list of extra arguments that the Ur/Web compiler will pass to GCC on every invocation.
adamc@555	89
adamc@555	90 \begin{verbatim}
adamc@555	91 GCCARGS=-fnested-functions ./configure
adamc@555	92 \end{verbatim}
adamc@555	93
adamc@1137	94 Some Mac OS X users have reported needing to use this particular GCCARGS value.
adamc@1137	95
adamc@1161	96 Since the author is still getting a handle on the GNU Autotools that provide the build system, you may need to do some further work to get started, especially in environments with significant differences from Linux (where most testing is done). One OS X user reported needing to run \texttt{./configure} with \texttt{CFLAGS=-I/opt/local/include}, since this directory wound up holding a header file associated with a \texttt{libmhash} package installed via DarwinPorts. Further, to get libpq to link, another user reported setting \texttt{GCCARGS="-I/opt/local/include -L/opt/local/lib/postgresql84"}, after creating a symbolic link with \texttt{ln -s /opt/local/include/postgresql84 /opt/local/include/postgresql}.
adamc@555	97
adamc@555	98 The Emacs mode can be set to autoload by adding the following to your \texttt{.emacs} file.
adamc@555	99
adamc@555	100 \begin{verbatim}
adamc@555	101 (add-to-list 'load-path "/usr/local/share/emacs/site-lisp/urweb-mode")
adamc@555	102 (load "urweb-mode-startup")
adamc@555	103 \end{verbatim}
adamc@555	104
adamc@555	105 Change the path in the first line if you chose a different Emacs installation path during configuration.
adamc@555	106
adamc@555	107
adamc@556	108 \section{Command-Line Compiler}
adamc@556	109
adamc@556	110 \subsection{Project Files}
adamc@556	111
adamc@556	112 The basic inputs to the \texttt{urweb} compiler are project files, which have the extension \texttt{.urp}. Here is a sample \texttt{.urp} file.
adamc@556	113
adamc@556	114 \begin{verbatim}
adamc@556	115 database dbname=test
adamc@556	116 sql crud1.sql
adamc@556	117
adamc@556	118 crud
adamc@556	119 crud1
adamc@556	120 \end{verbatim}
adamc@556	121
adamc@556	122 The \texttt{database} line gives the database information string to pass to libpq. In this case, the string only says to connect to a local database named \texttt{test}.
adamc@556	123
adamc@556	124 The \texttt{sql} line asks for an SQL source file to be generated, giving the commands to run to create the tables and sequences that this application expects to find. After building this \texttt{.urp} file, the following commands could be used to initialize the database, assuming that the current UNIX user exists as a Postgres user with database creation privileges:
adamc@556	125
adamc@556	126 \begin{verbatim}
adamc@556	127 createdb test
adamc@556	128 psql -f crud1.sql test
adamc@556	129 \end{verbatim}
adamc@556	130
adamc@1151	131 A blank line separates the named directives from a list of modules to include in the project.
adamc@556	132
adamc@556	133 For each entry \texttt{M} in the module list, the file \texttt{M.urs} is included in the project if it exists, and the file \texttt{M.ur} must exist and is always included.
adamc@556	134
adamc@783	135 Here is the complete list of directive forms. ``FFI'' stands for ``foreign function interface,'' Ur's facility for interaction between Ur programs and C and JavaScript libraries.
adamc@783	136 \begin{itemize}
adamc@783	137 \item \texttt{[allow\|deny] [url\|mime] PATTERN} registers a rule governing which URLs or MIME types are allowed in this application. The first such rule to match a URL or MIME type determines the verdict. If \texttt{PATTERN} ends in \texttt{*}, it is interpreted as a prefix rule. Otherwise, a string must match it exactly.
adamc@1171	138 \item \texttt{benignEffectful Module.ident} registers an FFI function or transaction as having side effects. The optimizer avoids removing, moving, or duplicating calls to such functions. Every effectful FFI function must be registered, or the optimizer may make invalid transformations. This version of the \texttt{effectful} directive registers that this function has only session-local side effects.
adamc@783	139 \item \texttt{clientOnly Module.ident} registers an FFI function or transaction that may only be run in client browsers.
adamc@783	140 \item \texttt{clientToServer Module.ident} adds FFI type \texttt{Module.ident} to the list of types that are OK to marshal from clients to servers. Values like XML trees and SQL queries are hard to marshal without introducing expensive validity checks, so it's easier to ensure that the server never trusts clients to send such values. The file \texttt{include/urweb.h} shows examples of the C support functions that are required of any type that may be marshalled. These include \texttt{attrify}, \texttt{urlify}, and \texttt{unurlify} functions.
adamc@783	141 \item \texttt{database DBSTRING} sets the string to pass to libpq to open a database connection.
adamc@783	142 \item \texttt{debug} saves some intermediate C files, which is mostly useful to help in debugging the compiler itself.
adamc@783	143 \item \texttt{effectful Module.ident} registers an FFI function or transaction as having side effects. The optimizer avoids removing, moving, or duplicating calls to such functions. Every effectful FFI function must be registered, or the optimizer may make invalid transformations.
adamc@783	144 \item \texttt{exe FILENAME} sets the filename to which to write the output executable. The default for file \texttt{P.urp} is \texttt{P.exe}.
adamc@783	145 \item \texttt{ffi FILENAME} reads the file \texttt{FILENAME.urs} to determine the interface to a new FFI module. The name of the module is calculated from \texttt{FILENAME} in the same way as for normal source files. See the files \texttt{include/urweb.h} and \texttt{src/c/urweb.c} for examples of C headers and implementations for FFI modules. In general, every type or value \texttt{Module.ident} becomes \texttt{uw\_Module\_ident} in C.
adamc@1099	146 \item \texttt{include FILENAME} adds \texttt{FILENAME} to the list of files to be \texttt{\#include}d in C sources. This is most useful for interfacing with new FFI modules.
adamc@783	147 \item \texttt{jsFunc Module.ident=name} gives the JavaScript name of an FFI value.
adamc@1089	148 \item \texttt{library FILENAME} parses \texttt{FILENAME.urp} and merges its contents with the rest of the current file's contents. If \texttt{FILENAME.urp} doesn't exist, the compiler also tries \texttt{FILENAME/lib.urp}.
adamc@783	149 \item \texttt{link FILENAME} adds \texttt{FILENAME} to the list of files to be passed to the GCC linker at the end of compilation. This is most useful for importing extra libraries needed by new FFI modules.
adamc@852	150 \item \texttt{path NAME=VALUE} creates a mapping from \texttt{NAME} to \texttt{VALUE}. This mapping may be used at the beginnings of filesystem paths given to various other configuration directives. A path like \texttt{\$NAME/rest} is expanded to \texttt{VALUE/rest}. There is an initial mapping from the empty name (for paths like \texttt{\$/list}) to the directory where the Ur/Web standard library is installed. If you accept the default \texttt{configure} options, this directory is \texttt{/usr/local/lib/urweb/ur}.
adamc@783	151 \item \texttt{prefix PREFIX} sets the prefix included before every URI within the generated application. The default is \texttt{/}.
adamc@783	152 \item \texttt{profile} generates an executable that may be used with gprof.
adamc@783	153 \item \texttt{rewrite KIND FROM TO} gives a rule for rewriting canonical module paths. For instance, the canonical path of a page may be \texttt{Mod1.Mod2.mypage}, while you would rather the page were accessed via a URL containing only \texttt{page}. The directive \texttt{rewrite url Mod1/Mod2/mypage page} would accomplish that. The possible values of \texttt{KIND} determine which kinds of objects are affected. The kind \texttt{all} matches any object, and \texttt{url} matches page URLs. The kinds \texttt{table}, \texttt{sequence}, and \texttt{view} match those sorts of SQL entities, and \texttt{relation} matches any of those three. \texttt{cookie} matches HTTP cookies, and \texttt{style} matches CSS class names. If \texttt{FROM} ends in \texttt{/*}, it is interpreted as a prefix matching rule, and rewriting occurs by replacing only the appropriate prefix of a path with \texttt{TO}. While the actual external names of relations and styles have parts separated by underscores instead of slashes, all rewrite rules must be written in terms of slashes.
adamc@783	154 \item \texttt{script URL} adds \texttt{URL} to the list of extra JavaScript files to be included at the beginning of any page that uses JavaScript. This is most useful for importing JavaScript versions of functions found in new FFI modules.
adamc@783	155 \item \texttt{serverOnly Module.ident} registers an FFI function or transaction that may only be run on the server.
adamc@1164	156 \item \texttt{sigfile PATH} sets a path where your application should look for a key to use in cryptographic signing. This is used to prevent cross-site request forgery attacks for any form handler that both reads a cookie and creates side effects. If the referenced file doesn't exist, an application will create it and read its saved data on future invocations. You can also initialize the file manually with any contents at least 16 bytes long; the first 16 bytes will be treated as the key.
adamc@783	157 \item \texttt{sql FILENAME} sets where to write an SQL file with the commands to create the expected database schema. The default is not to create such a file.
adamc@783	158 \item \texttt{timeout N} sets to \texttt{N} seconds the amount of time that the generated server will wait after the last contact from a client before determining that that client has exited the application. Clients that remain active will take the timeout setting into account in determining how often to ping the server, so it only makes sense to set a high timeout to cope with browser and network delays and failures. Higher timeouts can lead to more unnecessary client information taking up memory on the server. The timeout goes unused by any page that doesn't involve the \texttt{recv} function, since the server only needs to store per-client information for clients that receive asynchronous messages.
adamc@783	159 \end{itemize}
adamc@701	160
adamc@701	161
adamc@557	162 \subsection{Building an Application}
adamc@557	163
adamc@557	164 To compile project \texttt{P.urp}, simply run
adamc@557	165 \begin{verbatim}
adamc@557	166 urweb P
adamc@557	167 \end{verbatim}
adamc@558	168 The output executable is a standalone web server. Run it with the command-line argument \texttt{-h} to see which options it takes. If the project file lists a database, the web server will attempt to connect to that database on startup.
adamc@557	169
adamc@557	170 To time how long the different compiler phases run, without generating an executable, run
adamc@557	171 \begin{verbatim}
adamc@557	172 urweb -timing P
adamc@557	173 \end{verbatim}
adamc@557	174
adamc@1086	175 To stop the compilation process after type-checking, run
adamc@1086	176 \begin{verbatim}
adamc@1086	177 urweb -tc P
adamc@1086	178 \end{verbatim}
adamc@1086	179
adamc@1170	180 To output information relevant to CSS stylesheets (and not finish regular compilation), run
adamc@1170	181 \begin{verbatim}
adamc@1170	182 urweb -css P
adamc@1170	183 \end{verbatim}
adamc@1170	184 The first output line is a list of categories of CSS properties that would be worth setting on the document body. The remaining lines are space-separated pairs of CSS class names and categories of properties that would be worth setting for that class. The category codes are divided into two varieties. Codes that reveal properties of a tag or its (recursive) children are \cd{B} for block-level elements, \cd{C} for table captions, \cd{D} for table cells, \cd{L} for lists, and \cd{T} for tables. Codes that reveal properties of the precise tag that uses a class are \cd{b} for block-level elements, \cd{t} for tables, \cd{d} for table cells, \cd{-} for table rows, \cd{H} for the possibility to set a height, \cd{N} for non-replaced inline-level elements, \cd{R} for replaced inline elements, and \cd{W} for the possibility to set a width.
adamc@1170	185
adamc@896	186 Some other command-line parameters are accepted:
adamc@896	187 \begin{itemize}
adamc@896	188 \item \texttt{-db <DBSTRING>}: Set database connection information, using the format expected by Postgres's \texttt{PQconnectdb()}, which is \texttt{name1=value1 ... nameN=valueN}. The same format is also parsed and used to discover connection parameters for MySQL and SQLite. The only significant settings for MySQL are \texttt{host}, \texttt{hostaddr}, \texttt{port}, \texttt{dbname}, \texttt{user}, and \texttt{password}. The only significant setting for SQLite is \texttt{dbname}, which is interpreted as the filesystem path to the database. Additionally, when using SQLite, a database string may be just a file path.
adamc@896	189
adamc@896	190 \item \texttt{-dbms [postgres\|mysql\|sqlite]}: Sets the database backend to use.
adamc@896	191 \begin{itemize}
adamc@896	192 \item \texttt{postgres}: This is PostgreSQL, the default. Among the supported engines, Postgres best matches the design philosophy behind Ur, with a focus on consistent views of data, even in the face of much concurrency. Different database engines have different quirks of SQL syntax. Ur/Web tends to use Postgres idioms where there are choices to be made, though the compiler translates SQL as needed to support other backends.
adamc@896	193
adamc@896	194 A command sequence like this can initialize a Postgres database, using a file \texttt{app.sql} generated by the compiler:
adamc@896	195 \begin{verbatim}
adamc@896	196 createdb app
adamc@896	197 psql -f app.sql app
adamc@896	198 \end{verbatim}
adamc@896	199
adamc@896	200 \item \texttt{mysql}: This is MySQL, another popular relational database engine that uses persistent server processes. Ur/Web needs transactions to function properly. Many installations of MySQL use non-transactional storage engines by default. Ur/Web generates table definitions that try to use MySQL's InnoDB engine, which supports transactions. You can edit the first line of a generated \texttt{.sql} file to change this behavior, but it really is true that Ur/Web applications will exhibit bizarre behavior if you choose an engine that ignores transaction commands.
adamc@896	201
adamc@896	202 A command sequence like this can initialize a MySQL database:
adamc@896	203 \begin{verbatim}
adamc@896	204 echo "CREATE DATABASE app" \| mysql
adamc@896	205 mysql -D app <app.sql
adamc@896	206 \end{verbatim}
adamc@896	207
adamc@896	208 \item \texttt{sqlite}: This is SQLite, a simple filesystem-based transactional database engine. With this backend, Ur/Web applications can run without any additional server processes. The other engines are generally preferred for large-workload performance and full admin feature sets, while SQLite is popular for its low resource footprint and ease of set-up.
adamc@896	209
adamc@896	210 A command like this can initialize an SQLite database:
adamc@896	211 \begin{verbatim}
adamc@896	212 sqlite3 path/to/database/file <app.sql
adamc@896	213 \end{verbatim}
adamc@896	214 \end{itemize}
adamc@896	215
adamc@896	216 \item \texttt{-output FILENAME}: Set where the application executable is written.
adamc@896	217
adamc@1127	218 \item \texttt{-path NAME VALUE}: Set the value of path variable \texttt{\$NAME} to \texttt{VALUE}, for use in \texttt{.urp} files.
adamc@1127	219
adamc@896	220 \item \texttt{-protocol [http\|cgi\|fastcgi]}: Set the protocol that the generated application speaks.
adamc@896	221 \begin{itemize}
adamc@896	222 \item \texttt{http}: This is the default. It is for building standalone web servers that can be accessed by web browsers directly.
adamc@896	223
adamc@896	224 \item \texttt{cgi}: This is the classic protocol that web servers use to generate dynamic content by spawning new processes. While Ur/Web programs may in general use message-passing with the \texttt{send} and \texttt{recv} functions, that functionality is not yet supported in CGI, since CGI needs a fresh process for each request, and message-passing needs to use persistent sockets to deliver messages.
adamc@896	225
adamc@896	226 Since Ur/Web treats paths in an unusual way, a configuration line like this one can be used to configure an application that was built with URL prefix \texttt{/Hello}:
adamc@896	227 \begin{verbatim}
adamc@896	228 ScriptAlias /Hello /path/to/hello.exe
adamc@896	229 \end{verbatim}
adamc@896	230
adamc@1163	231 A different method can be used for, e.g., a shared host, where you can only configure Apache via \texttt{.htaccess} files. Drop the generated executable into your web space and mark it as CGI somehow. For instance, if the script ends in \texttt{.exe}, you might put this in \texttt{.htaccess} in the directory containing the script:
adamc@1163	232 \begin{verbatim}
adamc@1163	233 Options +ExecCGI
adamc@1163	234 AddHandler cgi-script .exe
adamc@1163	235 \end{verbatim}
adamc@1163	236
adamc@1163	237 Additionally, make sure that Ur/Web knows the proper URI prefix for your script. For instance, if the script is accessed via \texttt{http://somewhere/dir/script.exe}, then include this line in your \texttt{.urp} file:
adamc@1163	238 \begin{verbatim}
adamc@1163	239 prefix /dir/script.exe/
adamc@1163	240 \end{verbatim}
adamc@1163	241
adamc@1163	242 To access the \texttt{foo} function in the \texttt{Bar} module, you would then hit \texttt{http://somewhere/dir/script.exe/Bar/foo}.
adamc@1163	243
adamc@1164	244 If your application contains form handlers that read cookies before causing side effects, then you will need to use the \texttt{sigfile} \texttt{.urp} directive, too.
adamc@1164	245
adamc@896	246 \item \texttt{fastcgi}: This is a newer protocol inspired by CGI, wherein web servers can start and reuse persistent external processes to generate dynamic content. Ur/Web doesn't implement the whole protocol, but Ur/Web's support has been tested to work with the \texttt{mod\_fastcgi}s of Apache and lighttpd.
adamc@896	247
adamc@896	248 To configure a FastCGI program with Apache, one could combine the above \texttt{ScriptAlias} line with a line like this:
adamc@896	249 \begin{verbatim}
adamc@896	250 FastCgiServer /path/to/hello.exe -idle-timeout 99999
adamc@896	251 \end{verbatim}
adamc@896	252 The idle timeout is only important for applications that use message-passing. Client connections may go long periods without receiving messages, and Apache tries to be helpful and garbage collect them in such cases. To prevent that behavior, we specify how long a connection must be idle to be collected.
adamc@896	253
adamc@896	254 Here is some lighttpd configuration for the same application.
adamc@896	255 \begin{verbatim}
adamc@896	256 fastcgi.server = (
adamc@896	257 "/Hello/" =>
adamc@896	258 (( "bin-path" => "/path/to/hello.exe",
adamc@896	259 "socket" => "/tmp/hello",
adamc@896	260 "check-local" => "disable",
adamc@896	261 "docroot" => "/",
adamc@896	262 "max-procs" => "1"
adamc@896	263 ))
adamc@896	264 )
adamc@896	265 \end{verbatim}
adamc@896	266 The least obvious requirement is setting \texttt{max-procs} to 1, so that lighttpd doesn't try to multiplex requests across multiple external processes. This is required for message-passing applications, where a single database of client connections is maintained within a multi-threaded server process. Multiple processes may, however, be used safely with applications that don't use message-passing.
adamc@896	267
adamc@896	268 A FastCGI process reads the environment variable \texttt{URWEB\_NUM\_THREADS} to determine how many threads to spawn for handling client requests. The default is 1.
adamc@896	269 \end{itemize}
adamc@896	270
adamc@1127	271 \item \texttt{-root Name PATH}: Trigger an alternate module convention for all source files found in directory \texttt{PATH} or any of its subdirectories. Any file \texttt{PATH/foo.ur} defines a module \texttt{Name.Foo} instead of the usual \texttt{Foo}. Any file \texttt{PATH/subdir/foo.ur} defines a module \texttt{Name.Subdir.Foo}, and so on for arbitrary nesting of subdirectories.
adamc@1127	272
adamc@1164	273 \item \texttt{-sigfile PATH}: Same as the \texttt{sigfile} directive in \texttt{.urp} files
adamc@1164	274
adamc@896	275 \item \texttt{-sql FILENAME}: Set where a database set-up SQL script is written.
adamc@1095	276
adamc@1095	277 \item \texttt{-static}: Link the runtime system statically. The default is to link against dynamic libraries.
adamc@896	278 \end{itemize}
adamc@896	279
adamc@556	280
adamc@529	281 \section{Ur Syntax}
adamc@529	282
adamc@784	283 In this section, we describe the syntax of Ur, deferring to a later section discussion of most of the syntax specific to SQL and XML. The sole exceptions are the declaration forms for relations, cookies, and styles.
adamc@524	284
adamc@524	285 \subsection{Lexical Conventions}
adamc@524	286
adamc@524	287 We give the Ur language definition in \LaTeX $\;$ math mode, since that is prettier than monospaced ASCII. The corresponding ASCII syntax can be read off directly. Here is the key for mapping math symbols to ASCII character sequences.
adamc@524	288
adamc@524	289 \begin{center}
adamc@524	290 \begin{tabular}{rl}
adamc@524	291 \textbf{\LaTeX} & \textbf{ASCII} \\
adamc@524	292 $\to$ & \cd{->} \\
adamc@652	293 $\longrightarrow$ & \cd{-->} \\
adamc@524	294 $\times$ & \cd{*} \\
adamc@524	295 $\lambda$ & \cd{fn} \\
adamc@524	296 $\Rightarrow$ & \cd{=>} \\
adamc@652	297 $\Longrightarrow$ & \cd{==>} \\
adamc@529	298 $\neq$ & \cd{<>} \\
adamc@529	299 $\leq$ & \cd{<=} \\
adamc@529	300 $\geq$ & \cd{>=} \\
adamc@524	301 \\
adamc@524	302 $x$ & Normal textual identifier, not beginning with an uppercase letter \\
adamc@525	303 $X$ & Normal textual identifier, beginning with an uppercase letter \\
adamc@524	304 \end{tabular}
adamc@524	305 \end{center}
adamc@524	306
adamc@525	307 We often write syntax like $e^$ to indicate zero or more copies of $e$, $e^+$ to indicate one or more copies, and $e,^$ and $e,^+$ to indicate multiple copies separated by commas. Another separator may be used in place of a comma. The $e$ term may be surrounded by parentheses to indicate grouping; those parentheses should not be included in the actual ASCII.
adamc@524	308
adamc@873	309 We write $\ell$ for literals of the primitive types, for the most part following C conventions. There are $\mt{int}$, $\mt{float}$, $\mt{char}$, and $\mt{string}$ literals. Character literals follow the SML convention instead of the C convention, written like \texttt{\#"a"} instead of \texttt{'a'}.
adamc@526	310
adamc@527	311 This version of the manual doesn't include operator precedences; see \texttt{src/urweb.grm} for that.
adamc@527	312
adamc@552	313 \subsection{\label{core}Core Syntax}
adamc@524	314
adamc@524	315 \emph{Kinds} classify types and other compile-time-only entities. Each kind in the grammar is listed with a description of the sort of data it classifies.
adamc@524	316 $$\begin{array}{rrcll}
adamc@524	317 \textrm{Kinds} & \kappa &::=& \mt{Type} & \textrm{proper types} \\
adamc@525	318 &&& \mt{Unit} & \textrm{the trivial constructor} \\
adamc@525	319 &&& \mt{Name} & \textrm{field names} \\
adamc@525	320 &&& \kappa \to \kappa & \textrm{type-level functions} \\
adamc@525	321 &&& \{\kappa\} & \textrm{type-level records} \\
adamc@525	322 &&& (\kappa\times^+) & \textrm{type-level tuples} \\
adamc@652	323 &&& X & \textrm{variable} \\
adamc@652	324 &&& X \longrightarrow k & \textrm{kind-polymorphic type-level function} \\
adamc@529	325 &&& \_\_ & \textrm{wildcard} \\
adamc@525	326 &&& (\kappa) & \textrm{explicit precedence} \\
adamc@524	327 \end{array}$$
adamc@524	328
adamc@524	329 Ur supports several different notions of functions that take types as arguments. These arguments can be either implicit, causing them to be inferred at use sites; or explicit, forcing them to be specified manually at use sites. There is a common explicitness annotation convention applied at the definitions of and in the types of such functions.
adamc@524	330 $$\begin{array}{rrcll}
adamc@524	331 \textrm{Explicitness} & ? &::=& :: & \textrm{explicit} \\
adamc@558	332 &&& ::: & \textrm{implicit}
adamc@524	333 \end{array}$$
adamc@524	334
adamc@524	335 \emph{Constructors} are the main class of compile-time-only data. They include proper types and are classified by kinds.
adamc@524	336 $$\begin{array}{rrcll}
adamc@524	337 \textrm{Constructors} & c, \tau &::=& (c) :: \kappa & \textrm{kind annotation} \\
adamc@530	338 &&& \hat{x} & \textrm{constructor variable} \\
adamc@524	339 \\
adamc@525	340 &&& \tau \to \tau & \textrm{function type} \\
adamc@525	341 &&& x \; ? \; \kappa \to \tau & \textrm{polymorphic function type} \\
adamc@652	342 &&& X \longrightarrow \tau & \textrm{kind-polymorphic function type} \\
adamc@525	343 &&& \$ c & \textrm{record type} \\
adamc@524	344 \\
adamc@525	345 &&& c \; c & \textrm{type-level function application} \\
adamc@530	346 &&& \lambda x \; :: \; \kappa \Rightarrow c & \textrm{type-level function abstraction} \\
adamc@524	347 \\
adamc@652	348 &&& X \Longrightarrow c & \textrm{type-level kind-polymorphic function abstraction} \\
adamc@655	349 &&& c [\kappa] & \textrm{type-level kind-polymorphic function application} \\
adamc@652	350 \\
adamc@525	351 &&& () & \textrm{type-level unit} \\
adamc@525	352 &&& \#X & \textrm{field name} \\
adamc@524	353 \\
adamc@525	354 &&& [(c = c)^*] & \textrm{known-length type-level record} \\
adamc@525	355 &&& c \rc c & \textrm{type-level record concatenation} \\
adamc@652	356 &&& \mt{map} & \textrm{type-level record map} \\
adamc@524	357 \\
adamc@558	358 &&& (c,^+) & \textrm{type-level tuple} \\
adamc@525	359 &&& c.n & \textrm{type-level tuple projection ($n \in \mathbb N^+$)} \\
adamc@524	360 \\
adamc@652	361 &&& [c \sim c] \Rightarrow \tau & \textrm{guarded type} \\
adamc@524	362 \\
adamc@529	363 &&& \_ :: \kappa & \textrm{wildcard} \\
adamc@525	364 &&& (c) & \textrm{explicit precedence} \\
adamc@530	365 \\
adamc@530	366 \textrm{Qualified uncapitalized variables} & \hat{x} &::=& x & \textrm{not from a module} \\
adamc@530	367 &&& M.x & \textrm{projection from a module} \\
adamc@525	368 \end{array}$$
adamc@525	369
adamc@655	370 We include both abstraction and application for kind polymorphism, but applications are only inferred internally; they may not be written explicitly in source programs.
adamc@655	371
adamc@525	372 Modules of the module system are described by \emph{signatures}.
adamc@525	373 $$\begin{array}{rrcll}
adamc@525	374 \textrm{Signatures} & S &::=& \mt{sig} \; s^* \; \mt{end} & \textrm{constant} \\
adamc@525	375 &&& X & \textrm{variable} \\
adamc@525	376 &&& \mt{functor}(X : S) : S & \textrm{functor} \\
adamc@529	377 &&& S \; \mt{where} \; \mt{con} \; x = c & \textrm{concretizing an abstract constructor} \\
adamc@525	378 &&& M.X & \textrm{projection from a module} \\
adamc@525	379 \\
adamc@525	380 \textrm{Signature items} & s &::=& \mt{con} \; x :: \kappa & \textrm{abstract constructor} \\
adamc@525	381 &&& \mt{con} \; x :: \kappa = c & \textrm{concrete constructor} \\
adamc@528	382 &&& \mt{datatype} \; x \; x^* = dc\mid^+ & \textrm{algebraic datatype definition} \\
adamc@529	383 &&& \mt{datatype} \; x = \mt{datatype} \; M.x & \textrm{algebraic datatype import} \\
adamc@525	384 &&& \mt{val} \; x : \tau & \textrm{value} \\
adamc@525	385 &&& \mt{structure} \; X : S & \textrm{sub-module} \\
adamc@525	386 &&& \mt{signature} \; X = S & \textrm{sub-signature} \\
adamc@525	387 &&& \mt{include} \; S & \textrm{signature inclusion} \\
adamc@525	388 &&& \mt{constraint} \; c \sim c & \textrm{record disjointness constraint} \\
adamc@654	389 &&& \mt{class} \; x :: \kappa & \textrm{abstract constructor class} \\
adamc@654	390 &&& \mt{class} \; x :: \kappa = c & \textrm{concrete constructor class} \\
adamc@525	391 \\
adamc@525	392 \textrm{Datatype constructors} & dc &::=& X & \textrm{nullary constructor} \\
adamc@525	393 &&& X \; \mt{of} \; \tau & \textrm{unary constructor} \\
adamc@524	394 \end{array}$$
adamc@524	395
adamc@526	396 \emph{Patterns} are used to describe structural conditions on expressions, such that expressions may be tested against patterns, generating assignments to pattern variables if successful.
adamc@526	397 $$\begin{array}{rrcll}
adamc@526	398 \textrm{Patterns} & p &::=& \_ & \textrm{wildcard} \\
adamc@526	399 &&& x & \textrm{variable} \\
adamc@526	400 &&& \ell & \textrm{constant} \\
adamc@526	401 &&& \hat{X} & \textrm{nullary constructor} \\
adamc@526	402 &&& \hat{X} \; p & \textrm{unary constructor} \\
adamc@526	403 &&& \{(x = p,)^*\} & \textrm{rigid record pattern} \\
adamc@526	404 &&& \{(x = p,)^+, \ldots\} & \textrm{flexible record pattern} \\
adamc@852	405 &&& p : \tau & \textrm{type annotation} \\
adamc@527	406 &&& (p) & \textrm{explicit precedence} \\
adamc@526	407 \\
adamc@529	408 \textrm{Qualified capitalized variables} & \hat{X} &::=& X & \textrm{not from a module} \\
adamc@526	409 &&& M.X & \textrm{projection from a module} \\
adamc@526	410 \end{array}$$
adamc@526	411
adamc@527	412 \emph{Expressions} are the main run-time entities, corresponding to both ``expressions'' and ``statements'' in mainstream imperative languages.
adamc@527	413 $$\begin{array}{rrcll}
adamc@527	414 \textrm{Expressions} & e &::=& e : \tau & \textrm{type annotation} \\
adamc@529	415 &&& \hat{x} & \textrm{variable} \\
adamc@529	416 &&& \hat{X} & \textrm{datatype constructor} \\
adamc@527	417 &&& \ell & \textrm{constant} \\
adamc@527	418 \\
adamc@527	419 &&& e \; e & \textrm{function application} \\
adamc@527	420 &&& \lambda x : \tau \Rightarrow e & \textrm{function abstraction} \\
adamc@527	421 &&& e [c] & \textrm{polymorphic function application} \\
adamc@852	422 &&& \lambda [x \; ? \; \kappa] \Rightarrow e & \textrm{polymorphic function abstraction} \\
adamc@655	423 &&& e [\kappa] & \textrm{kind-polymorphic function application} \\
adamc@652	424 &&& X \Longrightarrow e & \textrm{kind-polymorphic function abstraction} \\
adamc@527	425 \\
adamc@527	426 &&& \{(c = e,)^*\} & \textrm{known-length record} \\
adamc@527	427 &&& e.c & \textrm{record field projection} \\
adamc@527	428 &&& e \rc e & \textrm{record concatenation} \\
adamc@527	429 &&& e \rcut c & \textrm{removal of a single record field} \\
adamc@527	430 &&& e \rcutM c & \textrm{removal of multiple record fields} \\
adamc@527	431 \\
adamc@527	432 &&& \mt{let} \; ed^* \; \mt{in} \; e \; \mt{end} & \textrm{local definitions} \\
adamc@527	433 \\
adamc@527	434 &&& \mt{case} \; e \; \mt{of} \; (p \Rightarrow e\|)^+ & \textrm{pattern matching} \\
adamc@527	435 \\
adamc@654	436 &&& \lambda [c \sim c] \Rightarrow e & \textrm{guarded expression abstraction} \\
adamc@654	437 &&& e \; ! & \textrm{guarded expression application} \\
adamc@527	438 \\
adamc@527	439 &&& \_ & \textrm{wildcard} \\
adamc@527	440 &&& (e) & \textrm{explicit precedence} \\
adamc@527	441 \\
adamc@527	442 \textrm{Local declarations} & ed &::=& \cd{val} \; x : \tau = e & \textrm{non-recursive value} \\
adamc@527	443 &&& \cd{val} \; \cd{rec} \; (x : \tau = e \; \cd{and})^+ & \textrm{mutually-recursive values} \\
adamc@527	444 \end{array}$$
adamc@527	445
adamc@655	446 As with constructors, we include both abstraction and application for kind polymorphism, but applications are only inferred internally.
adamc@655	447
adamc@528	448 \emph{Declarations} primarily bring new symbols into context.
adamc@528	449 $$\begin{array}{rrcll}
adamc@528	450 \textrm{Declarations} & d &::=& \mt{con} \; x :: \kappa = c & \textrm{constructor synonym} \\
adamc@528	451 &&& \mt{datatype} \; x \; x^* = dc\mid^+ & \textrm{algebraic datatype definition} \\
adamc@529	452 &&& \mt{datatype} \; x = \mt{datatype} \; M.x & \textrm{algebraic datatype import} \\
adamc@528	453 &&& \mt{val} \; x : \tau = e & \textrm{value} \\
adamc@528	454 &&& \mt{val} \; \cd{rec} \; (x : \tau = e \; \mt{and})^+ & \textrm{mutually-recursive values} \\
adamc@528	455 &&& \mt{structure} \; X : S = M & \textrm{module definition} \\
adamc@528	456 &&& \mt{signature} \; X = S & \textrm{signature definition} \\
adamc@528	457 &&& \mt{open} \; M & \textrm{module inclusion} \\
adamc@528	458 &&& \mt{constraint} \; c \sim c & \textrm{record disjointness constraint} \\
adamc@528	459 &&& \mt{open} \; \mt{constraints} \; M & \textrm{inclusion of just the constraints from a module} \\
adamc@528	460 &&& \mt{table} \; x : c & \textrm{SQL table} \\
adamc@784	461 &&& \mt{view} \; x : c & \textrm{SQL view} \\
adamc@528	462 &&& \mt{sequence} \; x & \textrm{SQL sequence} \\
adamc@535	463 &&& \mt{cookie} \; x : \tau & \textrm{HTTP cookie} \\
adamc@784	464 &&& \mt{style} \; x : \tau & \textrm{CSS class} \\
adamc@654	465 &&& \mt{class} \; x :: \kappa = c & \textrm{concrete constructor class} \\
adamc@1085	466 &&& \mt{task} \; e = e & \textrm{recurring task} \\
adamc@528	467 \\
adamc@529	468 \textrm{Modules} & M &::=& \mt{struct} \; d^* \; \mt{end} & \textrm{constant} \\
adamc@529	469 &&& X & \textrm{variable} \\
adamc@529	470 &&& M.X & \textrm{projection} \\
adamc@529	471 &&& M(M) & \textrm{functor application} \\
adamc@529	472 &&& \mt{functor}(X : S) : S = M & \textrm{functor abstraction} \\
adamc@528	473 \end{array}$$
adamc@528	474
adamc@528	475 There are two kinds of Ur files. A file named $M\texttt{.ur}$ is an \emph{implementation file}, and it should contain a sequence of declarations $d^$. A file named $M\texttt{.urs}$ is an \emph{interface file}; it must always have a matching $M\texttt{.ur}$ and should contain a sequence of signature items $s^$. When both files are present, the overall effect is the same as a monolithic declaration $\mt{structure} \; M : \mt{sig} \; s^* \; \mt{end} = \mt{struct} \; d^* \; \mt{end}$. When no interface file is included, the overall effect is similar, with a signature for module $M$ being inferred rather than just checked against an interface.
adamc@527	476
adamc@784	477 We omit some extra possibilities in $\mt{table}$ syntax, deferring them to Section \ref{tables}.
adamc@784	478
adamc@529	479 \subsection{Shorthands}
adamc@529	480
adamc@529	481 There are a variety of derived syntactic forms that elaborate into the core syntax from the last subsection. We will present the additional forms roughly following the order in which we presented the constructs that they elaborate into.
adamc@529	482
adamc@529	483 In many contexts where record fields are expected, like in a projection $e.c$, a constant field may be written as simply $X$, rather than $\#X$.
adamc@529	484
adamc@529	485 A record type may be written $\{(c = c,)^\}$, which elaborates to $\$[(c = c,)^]$.
adamc@529	486
adamc@533	487 The notation $[c_1, \ldots, c_n]$ is shorthand for $[c_1 = (), \ldots, c_n = ()]$.
adamc@533	488
adamc@529	489 A tuple type $(\tau_1, \ldots, \tau_n)$ expands to a record type $\{1 = \tau_1, \ldots, n = \tau_n\}$, with natural numbers as field names. A tuple pattern $(p_1, \ldots, p_n)$ expands to a rigid record pattern $\{1 = p_1, \ldots, n = p_n\}$. Positive natural numbers may be used in most places where field names would be allowed.
adamc@529	490
adamc@852	491 In general, several adjacent $\lambda$ forms may be combined into one, and kind and type annotations may be omitted, in which case they are implicitly included as wildcards. More formally, for constructor-level abstractions, we can define a new non-terminal $b ::= x \mid (x :: \kappa) \mid X$ and allow composite abstractions of the form $\lambda b^+ \Rightarrow c$, elaborating into the obvious sequence of one core $\lambda$ per element of $b^+$.
adamc@529	492
adamc@529	493 For any signature item or declaration that defines some entity to be equal to $A$ with classification annotation $B$ (e.g., $\mt{val} \; x : B = A$), $B$ and the preceding colon (or similar punctuation) may be omitted, in which case it is filled in as a wildcard.
adamc@529	494
adamc@529	495 A signature item or declaration $\mt{type} \; x$ or $\mt{type} \; x = \tau$ is elaborated into $\mt{con} \; x :: \mt{Type}$ or $\mt{con} \; x :: \mt{Type} = \tau$, respectively.
adamc@529	496
adamc@654	497 A signature item or declaration $\mt{class} \; x = \lambda y \Rightarrow c$ may be abbreviated $\mt{class} \; x \; y = c$.
adamc@529	498
adamc@1102	499 Handling of implicit and explicit constructor arguments may be tweaked with some prefixes to variable references. An expression $@x$ is a version of $x$ where all implicit constructor arguments have been made explicit. An expression $@@x$ achieves the same effect, additionally halting automatic resolution of type class instances and automatic proving of disjointness constraints. The default is that implicit arguments are inserted automatically after any reference to a non-local variable, or after any application of a non-local variable to one or more arguments. For such an expression, implicit wildcard arguments are added for the longest prefix of the expression's type consisting only of implicit polymorphism, type class instances, and disjointness obligations. The same syntax works for variables projected out of modules and for capitalized variables (datatype constructors).
adamc@529	500
adamc@852	501 At the expression level, an analogue is available of the composite $\lambda$ form for constructors. We define the language of binders as $b ::= p \mid [x] \mid [x \; ? \; \kappa] \mid X \mid [c \sim c]$. A lone variable $[x]$ stands for an implicit constructor variable of unspecified kind. The standard value-level function binder is recovered as the type-annotated pattern form $x : \tau$. It is a compile-time error to include a pattern $p$ that does not match every value of the appropriate type.
adamc@529	502
adamc@852	503 A local $\mt{val}$ declaration may bind a pattern instead of just a plain variable. As for function arguments, only irrefutable patterns are legal.
adamc@852	504
adamc@852	505 The keyword $\mt{fun}$ is a shorthand for $\mt{val} \; \mt{rec}$ that allows arguments to be specified before the equal sign in the definition of each mutually-recursive function, as in SML. Each curried argument must follow the grammar of the $b$ non-terminal introduced two paragraphs ago. A $\mt{fun}$ declaration is elaborated into a version that adds additional $\lambda$s to the fronts of the righthand sides, as appropriate.
adamc@529	506
adamc@529	507 A signature item $\mt{functor} \; X_1 \; (X_2 : S_1) : S_2$ is elaborated into $\mt{structure} \; X_1 : \mt{functor}(X_2 : S_1) : S_2$. A declaration $\mt{functor} \; X_1 \; (X_2 : S_1) : S_2 = M$ is elaborated into $\mt{structure} \; X_1 : \mt{functor}(X_2 : S_1) : S_2 = \mt{functor}(X_2 : S_1) : S_2 = M$.
adamc@529	508
adamc@852	509 An $\mt{open} \; \mt{constraints}$ declaration is implicitly inserted for the argument of every functor at the beginning of the functor body. For every declaration of the form $\mt{structure} \; X : S = \mt{struct} \ldots \mt{end}$, an $\mt{open} \; \mt{constraints} \; X$ declaration is implicitly inserted immediately afterward.
adamc@852	510
adamc@853	511 A declaration $\mt{table} \; x : \{(c = c,)^\}$ is elaborated into $\mt{table} \; x : [(c = c,)^]$.
adamc@529	512
adamc@529	513 The syntax $\mt{where} \; \mt{type}$ is an alternate form of $\mt{where} \; \mt{con}$.
adamc@529	514
adamc@529	515 The syntax $\mt{if} \; e \; \mt{then} \; e_1 \; \mt{else} \; e_2$ expands to $\mt{case} \; e \; \mt{of} \; \mt{Basis}.\mt{True} \Rightarrow e_1 \mid \mt{Basis}.\mt{False} \Rightarrow e_2$.
adamc@529	516
adamc@529	517 There are infix operator syntaxes for a number of functions defined in the $\mt{Basis}$ module. There is $=$ for $\mt{eq}$, $\neq$ for $\mt{neq}$, $-$ for $\mt{neg}$ (as a prefix operator) and $\mt{minus}$, $+$ for $\mt{plus}$, $\times$ for $\mt{times}$, $/$ for $\mt{div}$, $\%$ for $\mt{mod}$, $<$ for $\mt{lt}$, $\leq$ for $\mt{le}$, $>$ for $\mt{gt}$, and $\geq$ for $\mt{ge}$.
adamc@529	518
adamc@784	519 A signature item $\mt{table} \; x : c$ is shorthand for $\mt{val} \; x : \mt{Basis}.\mt{sql\_table} \; c \; []$. $\mt{view} \; x : c$ is shorthand for $\mt{val} \; x : \mt{Basis}.\mt{sql\_view} \; c$, $\mt{sequence} \; x$ is short for $\mt{val} \; x : \mt{Basis}.\mt{sql\_sequence}$. $\mt{cookie} \; x : \tau$ is shorthand for $\mt{val} \; x : \mt{Basis}.\mt{http\_cookie} \; \tau$, and $\mt{style} \; x$ is shorthand for $\mt{val} \; x : \mt{Basis}.\mt{css\_class}$.
adamc@529	520
adamc@530	521
adamc@530	522 \section{Static Semantics}
adamc@530	523
adamc@530	524 In this section, we give a declarative presentation of Ur's typing rules and related judgments. Inference is the subject of the next section; here, we assume that an oracle has filled in all wildcards with concrete values.
adamc@530	525
adamc@530	526 Since there is significant mutual recursion among the judgments, we introduce them all before beginning to give rules. We use the same variety of contexts throughout this section, implicitly introducing new sorts of context entries as needed.
adamc@530	527 \begin{itemize}
adamc@655	528 \item $\Gamma \vdash \kappa$ expresses kind well-formedness.
adamc@530	529 \item $\Gamma \vdash c :: \kappa$ assigns a kind to a constructor in a context.
adamc@530	530 \item $\Gamma \vdash c \sim c$ proves the disjointness of two record constructors; that is, that they share no field names. We overload the judgment to apply to pairs of field names as well.
adamc@531	531 \item $\Gamma \vdash c \hookrightarrow C$ proves that record constructor $c$ decomposes into set $C$ of field names and record constructors.
adamc@530	532 \item $\Gamma \vdash c \equiv c$ proves the computational equivalence of two constructors. This is often called a \emph{definitional equality} in the world of type theory.
adamc@530	533 \item $\Gamma \vdash e : \tau$ is a standard typing judgment.
adamc@534	534 \item $\Gamma \vdash p \leadsto \Gamma; \tau$ combines typing of patterns with calculation of which new variables they bind.
adamc@537	535 \item $\Gamma \vdash d \leadsto \Gamma$ expresses how a declaration modifies a context. We overload this judgment to apply to sequences of declarations, as well as to signature items and sequences of signature items.
adamc@537	536 \item $\Gamma \vdash S \equiv S$ is the signature equivalence judgment.
adamc@536	537 \item $\Gamma \vdash S \leq S$ is the signature compatibility judgment. We write $\Gamma \vdash S$ as shorthand for $\Gamma \vdash S \leq S$.
adamc@530	538 \item $\Gamma \vdash M : S$ is the module signature checking judgment.
adamc@537	539 \item $\mt{proj}(M, \overline{s}, V)$ is a partial function for projecting a signature item from $\overline{s}$, given the module $M$ that we project from. $V$ may be $\mt{con} \; x$, $\mt{datatype} \; x$, $\mt{val} \; x$, $\mt{signature} \; X$, or $\mt{structure} \; X$. The parameter $M$ is needed because the projected signature item may refer to other items from $\overline{s}$.
adamc@539	540 \item $\mt{selfify}(M, \overline{s})$ adds information to signature items $\overline{s}$ to reflect the fact that we are concerned with the particular module $M$. This function is overloaded to work over individual signature items as well.
adamc@530	541 \end{itemize}
adamc@530	542
adamc@655	543
adamc@655	544 \subsection{Kind Well-Formedness}
adamc@655	545
adamc@655	546 $$\infer{\Gamma \vdash \mt{Type}}{}
adamc@655	547 \quad \infer{\Gamma \vdash \mt{Unit}}{}
adamc@655	548 \quad \infer{\Gamma \vdash \mt{Name}}{}
adamc@655	549 \quad \infer{\Gamma \vdash \kappa_1 \to \kappa_2}{
adamc@655	550 \Gamma \vdash \kappa_1
adamc@655	551 & \Gamma \vdash \kappa_2
adamc@655	552 }
adamc@655	553 \quad \infer{\Gamma \vdash \{\kappa\}}{
adamc@655	554 \Gamma \vdash \kappa
adamc@655	555 }
adamc@655	556 \quad \infer{\Gamma \vdash (\kappa_1 \times \ldots \times \kappa_n)}{
adamc@655	557 \forall i: \Gamma \vdash \kappa_i
adamc@655	558 }$$
adamc@655	559
adamc@655	560 $$\infer{\Gamma \vdash X}{
adamc@655	561 X \in \Gamma
adamc@655	562 }
adamc@655	563 \quad \infer{\Gamma \vdash X \longrightarrow \kappa}{
adamc@655	564 \Gamma, X \vdash \kappa
adamc@655	565 }$$
adamc@655	566
adamc@530	567 \subsection{Kinding}
adamc@530	568
adamc@655	569 We write $[X \mapsto \kappa_1]\kappa_2$ for capture-avoiding substitution of $\kappa_1$ for $X$ in $\kappa_2$.
adamc@655	570
adamc@530	571 $$\infer{\Gamma \vdash (c) :: \kappa :: \kappa}{
adamc@530	572 \Gamma \vdash c :: \kappa
adamc@530	573 }
adamc@530	574 \quad \infer{\Gamma \vdash x :: \kappa}{
adamc@530	575 x :: \kappa \in \Gamma
adamc@530	576 }
adamc@530	577 \quad \infer{\Gamma \vdash x :: \kappa}{
adamc@530	578 x :: \kappa = c \in \Gamma
adamc@530	579 }$$
adamc@530	580
adamc@530	581 $$\infer{\Gamma \vdash M.x :: \kappa}{
adamc@537	582 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	583 & \mt{proj}(M, \overline{s}, \mt{con} \; x) = \kappa
adamc@530	584 }
adamc@530	585 \quad \infer{\Gamma \vdash M.x :: \kappa}{
adamc@537	586 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	587 & \mt{proj}(M, \overline{s}, \mt{con} \; x) = (\kappa, c)
adamc@530	588 }$$
adamc@530	589
adamc@530	590 $$\infer{\Gamma \vdash \tau_1 \to \tau_2 :: \mt{Type}}{
adamc@530	591 \Gamma \vdash \tau_1 :: \mt{Type}
adamc@530	592 & \Gamma \vdash \tau_2 :: \mt{Type}
adamc@530	593 }
adamc@530	594 \quad \infer{\Gamma \vdash x \; ? \: \kappa \to \tau :: \mt{Type}}{
adamc@530	595 \Gamma, x :: \kappa \vdash \tau :: \mt{Type}
adamc@530	596 }
adamc@655	597 \quad \infer{\Gamma \vdash X \longrightarrow \tau :: \mt{Type}}{
adamc@655	598 \Gamma, X \vdash \tau :: \mt{Type}
adamc@655	599 }
adamc@530	600 \quad \infer{\Gamma \vdash \$c :: \mt{Type}}{
adamc@530	601 \Gamma \vdash c :: \{\mt{Type}\}
adamc@530	602 }$$
adamc@530	603
adamc@530	604 $$\infer{\Gamma \vdash c_1 \; c_2 :: \kappa_2}{
adamc@530	605 \Gamma \vdash c_1 :: \kappa_1 \to \kappa_2
adamc@530	606 & \Gamma \vdash c_2 :: \kappa_1
adamc@530	607 }
adamc@530	608 \quad \infer{\Gamma \vdash \lambda x \; :: \; \kappa_1 \Rightarrow c :: \kappa_1 \to \kappa_2}{
adamc@530	609 \Gamma, x :: \kappa_1 \vdash c :: \kappa_2
adamc@530	610 }$$
adamc@530	611
adamc@655	612 $$\infer{\Gamma \vdash c[\kappa'] :: [X \mapsto \kappa']\kappa}{
adamc@655	613 \Gamma \vdash c :: X \to \kappa
adamc@655	614 & \Gamma \vdash \kappa'
adamc@655	615 }
adamc@655	616 \quad \infer{\Gamma \vdash X \Longrightarrow c :: X \to \kappa}{
adamc@655	617 \Gamma, X \vdash c :: \kappa
adamc@655	618 }$$
adamc@655	619
adamc@530	620 $$\infer{\Gamma \vdash () :: \mt{Unit}}{}
adamc@530	621 \quad \infer{\Gamma \vdash \#X :: \mt{Name}}{}$$
adamc@530	622
adamc@530	623 $$\infer{\Gamma \vdash [\overline{c_i = c'_i}] :: \{\kappa\}}{
adamc@530	624 \forall i: \Gamma \vdash c_i : \mt{Name}
adamc@530	625 & \Gamma \vdash c'_i :: \kappa
adamc@530	626 & \forall i \neq j: \Gamma \vdash c_i \sim c_j
adamc@530	627 }
adamc@530	628 \quad \infer{\Gamma \vdash c_1 \rc c_2 :: \{\kappa\}}{
adamc@530	629 \Gamma \vdash c_1 :: \{\kappa\}
adamc@530	630 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@530	631 & \Gamma \vdash c_1 \sim c_2
adamc@530	632 }$$
adamc@530	633
adamc@655	634 $$\infer{\Gamma \vdash \mt{map} :: (\kappa_1 \to \kappa_2) \to \{\kappa_1\} \to \{\kappa_2\}}{}$$
adamc@530	635
adamc@573	636 $$\infer{\Gamma \vdash (\overline c) :: (\kappa_1 \times \ldots \times \kappa_n)}{
adamc@573	637 \forall i: \Gamma \vdash c_i :: \kappa_i
adamc@530	638 }
adamc@573	639 \quad \infer{\Gamma \vdash c.i :: \kappa_i}{
adamc@573	640 \Gamma \vdash c :: (\kappa_1 \times \ldots \times \kappa_n)
adamc@530	641 }$$
adamc@530	642
adamc@655	643 $$\infer{\Gamma \vdash \lambda [c_1 \sim c_2] \Rightarrow \tau :: \mt{Type}}{
adamc@655	644 \Gamma \vdash c_1 :: \{\kappa\}
adamc@530	645 & \Gamma \vdash c_2 :: \{\kappa'\}
adamc@655	646 & \Gamma, c_1 \sim c_2 \vdash \tau :: \mt{Type}
adamc@530	647 }$$
adamc@530	648
adamc@531	649 \subsection{Record Disjointness}
adamc@531	650
adamc@531	651 $$\infer{\Gamma \vdash c_1 \sim c_2}{
adamc@558	652 \Gamma \vdash c_1 \hookrightarrow C_1
adamc@558	653 & \Gamma \vdash c_2 \hookrightarrow C_2
adamc@558	654 & \forall c'_1 \in C_1, c'_2 \in C_2: \Gamma \vdash c'_1 \sim c'_2
adamc@531	655 }
adamc@531	656 \quad \infer{\Gamma \vdash X \sim X'}{
adamc@531	657 X \neq X'
adamc@531	658 }$$
adamc@531	659
adamc@531	660 $$\infer{\Gamma \vdash c_1 \sim c_2}{
adamc@531	661 c'_1 \sim c'_2 \in \Gamma
adamc@558	662 & \Gamma \vdash c'_1 \hookrightarrow C_1
adamc@558	663 & \Gamma \vdash c'_2 \hookrightarrow C_2
adamc@558	664 & c_1 \in C_1
adamc@558	665 & c_2 \in C_2
adamc@531	666 }$$
adamc@531	667
adamc@531	668 $$\infer{\Gamma \vdash c \hookrightarrow \{c\}}{}
adamc@531	669 \quad \infer{\Gamma \vdash [\overline{c = c'}] \hookrightarrow \{\overline{c}\}}{}
adamc@531	670 \quad \infer{\Gamma \vdash c_1 \rc c_2 \hookrightarrow C_1 \cup C_2}{
adamc@531	671 \Gamma \vdash c_1 \hookrightarrow C_1
adamc@531	672 & \Gamma \vdash c_2 \hookrightarrow C_2
adamc@531	673 }
adamc@531	674 \quad \infer{\Gamma \vdash c \hookrightarrow C}{
adamc@531	675 \Gamma \vdash c \equiv c'
adamc@531	676 & \Gamma \vdash c' \hookrightarrow C
adamc@531	677 }
adamc@531	678 \quad \infer{\Gamma \vdash \mt{map} \; f \; c \hookrightarrow C}{
adamc@531	679 \Gamma \vdash c \hookrightarrow C
adamc@531	680 }$$
adamc@531	681
adamc@541	682 \subsection{\label{definitional}Definitional Equality}
adamc@532	683
adamc@655	684 We use $\mathcal C$ to stand for a one-hole context that, when filled, yields a constructor. The notation $\mathcal C[c]$ plugs $c$ into $\mathcal C$. We omit the standard definition of one-hole contexts. We write $[x \mapsto c_1]c_2$ for capture-avoiding substitution of $c_1$ for $x$ in $c_2$, with analogous notation for substituting a kind in a constructor.
adamc@532	685
adamc@532	686 $$\infer{\Gamma \vdash c \equiv c}{}
adamc@532	687 \quad \infer{\Gamma \vdash c_1 \equiv c_2}{
adamc@532	688 \Gamma \vdash c_2 \equiv c_1
adamc@532	689 }
adamc@532	690 \quad \infer{\Gamma \vdash c_1 \equiv c_3}{
adamc@532	691 \Gamma \vdash c_1 \equiv c_2
adamc@532	692 & \Gamma \vdash c_2 \equiv c_3
adamc@532	693 }
adamc@532	694 \quad \infer{\Gamma \vdash \mathcal C[c_1] \equiv \mathcal C[c_2]}{
adamc@532	695 \Gamma \vdash c_1 \equiv c_2
adamc@532	696 }$$
adamc@532	697
adamc@532	698 $$\infer{\Gamma \vdash x \equiv c}{
adamc@532	699 x :: \kappa = c \in \Gamma
adamc@532	700 }
adamc@532	701 \quad \infer{\Gamma \vdash M.x \equiv c}{
adamc@537	702 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	703 & \mt{proj}(M, \overline{s}, \mt{con} \; x) = (\kappa, c)
adamc@532	704 }
adamc@532	705 \quad \infer{\Gamma \vdash (\overline c).i \equiv c_i}{}$$
adamc@532	706
adamc@532	707 $$\infer{\Gamma \vdash (\lambda x :: \kappa \Rightarrow c) \; c' \equiv [x \mapsto c'] c}{}
adamc@655	708 \quad \infer{\Gamma \vdash (X \Longrightarrow c) [\kappa] \equiv [X \mapsto \kappa] c}{}$$
adamc@655	709
adamc@655	710 $$\infer{\Gamma \vdash c_1 \rc c_2 \equiv c_2 \rc c_1}{}
adamc@532	711 \quad \infer{\Gamma \vdash c_1 \rc (c_2 \rc c_3) \equiv (c_1 \rc c_2) \rc c_3}{}$$
adamc@532	712
adamc@532	713 $$\infer{\Gamma \vdash [] \rc c \equiv c}{}
adamc@532	714 \quad \infer{\Gamma \vdash [\overline{c_1 = c'_1}] \rc [\overline{c_2 = c'_2}] \equiv [\overline{c_1 = c'_1}, \overline{c_2 = c'_2}]}{}$$
adamc@532	715
adamc@655	716 $$\infer{\Gamma \vdash \mt{map} \; f \; [] \equiv []}{}
adamc@655	717 \quad \infer{\Gamma \vdash \mt{map} \; f \; ([c_1 = c_2] \rc c) \equiv [c_1 = f \; c_2] \rc \mt{map} \; f \; c}{}$$
adamc@532	718
adamc@532	719 $$\infer{\Gamma \vdash \mt{map} \; (\lambda x \Rightarrow x) \; c \equiv c}{}
adamc@655	720 \quad \infer{\Gamma \vdash \mt{map} \; f \; (\mt{map} \; f' \; c)
adamc@655	721 \equiv \mt{map} \; (\lambda x \Rightarrow f \; (f' \; x)) \; c}{}$$
adamc@532	722
adamc@532	723 $$\infer{\Gamma \vdash \mt{map} \; f \; (c_1 \rc c_2) \equiv \mt{map} \; f \; c_1 \rc \mt{map} \; f \; c_2}{}$$
adamc@531	724
adamc@534	725 \subsection{Expression Typing}
adamc@533	726
adamc@873	727 We assume the existence of a function $T$ assigning types to literal constants. It maps integer constants to $\mt{Basis}.\mt{int}$, float constants to $\mt{Basis}.\mt{float}$, character constants to $\mt{Basis}.\mt{char}$, and string constants to $\mt{Basis}.\mt{string}$.
adamc@533	728
adamc@533	729 We also refer to a function $\mathcal I$, such that $\mathcal I(\tau)$ ``uses an oracle'' to instantiate all constructor function arguments at the beginning of $\tau$ that are marked implicit; i.e., replace $x_1 ::: \kappa_1 \to \ldots \to x_n ::: \kappa_n \to \tau$ with $[x_1 \mapsto c_1]\ldots[x_n \mapsto c_n]\tau$, where the $c_i$s are inferred and $\tau$ does not start like $x ::: \kappa \to \tau'$.
adamc@533	730
adamc@533	731 $$\infer{\Gamma \vdash e : \tau : \tau}{
adamc@533	732 \Gamma \vdash e : \tau
adamc@533	733 }
adamc@533	734 \quad \infer{\Gamma \vdash e : \tau}{
adamc@533	735 \Gamma \vdash e : \tau'
adamc@533	736 & \Gamma \vdash \tau' \equiv \tau
adamc@533	737 }
adamc@533	738 \quad \infer{\Gamma \vdash \ell : T(\ell)}{}$$
adamc@533	739
adamc@533	740 $$\infer{\Gamma \vdash x : \mathcal I(\tau)}{
adamc@533	741 x : \tau \in \Gamma
adamc@533	742 }
adamc@533	743 \quad \infer{\Gamma \vdash M.x : \mathcal I(\tau)}{
adamc@537	744 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	745 & \mt{proj}(M, \overline{s}, \mt{val} \; x) = \tau
adamc@533	746 }
adamc@533	747 \quad \infer{\Gamma \vdash X : \mathcal I(\tau)}{
adamc@533	748 X : \tau \in \Gamma
adamc@533	749 }
adamc@533	750 \quad \infer{\Gamma \vdash M.X : \mathcal I(\tau)}{
adamc@537	751 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	752 & \mt{proj}(M, \overline{s}, \mt{val} \; X) = \tau
adamc@533	753 }$$
adamc@533	754
adamc@533	755 $$\infer{\Gamma \vdash e_1 \; e_2 : \tau_2}{
adamc@533	756 \Gamma \vdash e_1 : \tau_1 \to \tau_2
adamc@533	757 & \Gamma \vdash e_2 : \tau_1
adamc@533	758 }
adamc@533	759 \quad \infer{\Gamma \vdash \lambda x : \tau_1 \Rightarrow e : \tau_1 \to \tau_2}{
adamc@533	760 \Gamma, x : \tau_1 \vdash e : \tau_2
adamc@533	761 }$$
adamc@533	762
adamc@533	763 $$\infer{\Gamma \vdash e [c] : [x \mapsto c]\tau}{
adamc@533	764 \Gamma \vdash e : x :: \kappa \to \tau
adamc@533	765 & \Gamma \vdash c :: \kappa
adamc@533	766 }
adamc@852	767 \quad \infer{\Gamma \vdash \lambda [x \; ? \; \kappa] \Rightarrow e : x \; ? \; \kappa \to \tau}{
adamc@533	768 \Gamma, x :: \kappa \vdash e : \tau
adamc@533	769 }$$
adamc@533	770
adamc@655	771 $$\infer{\Gamma \vdash e [\kappa] : [X \mapsto \kappa]\tau}{
adamc@655	772 \Gamma \vdash e : X \longrightarrow \tau
adamc@655	773 & \Gamma \vdash \kappa
adamc@655	774 }
adamc@655	775 \quad \infer{\Gamma \vdash X \Longrightarrow e : X \longrightarrow \tau}{
adamc@655	776 \Gamma, X \vdash e : \tau
adamc@655	777 }$$
adamc@655	778
adamc@533	779 $$\infer{\Gamma \vdash \{\overline{c = e}\} : \{\overline{c : \tau}\}}{
adamc@533	780 \forall i: \Gamma \vdash c_i :: \mt{Name}
adamc@533	781 & \Gamma \vdash e_i : \tau_i
adamc@533	782 & \forall i \neq j: \Gamma \vdash c_i \sim c_j
adamc@533	783 }
adamc@533	784 \quad \infer{\Gamma \vdash e.c : \tau}{
adamc@533	785 \Gamma \vdash e : \$([c = \tau] \rc c')
adamc@533	786 }
adamc@533	787 \quad \infer{\Gamma \vdash e_1 \rc e_2 : \$(c_1 \rc c_2)}{
adamc@533	788 \Gamma \vdash e_1 : \$c_1
adamc@533	789 & \Gamma \vdash e_2 : \$c_2
adamc@573	790 & \Gamma \vdash c_1 \sim c_2
adamc@533	791 }$$
adamc@533	792
adamc@533	793 $$\infer{\Gamma \vdash e \rcut c : \$c'}{
adamc@533	794 \Gamma \vdash e : \$([c = \tau] \rc c')
adamc@533	795 }
adamc@533	796 \quad \infer{\Gamma \vdash e \rcutM c : \$c'}{
adamc@533	797 \Gamma \vdash e : \$(c \rc c')
adamc@533	798 }$$
adamc@533	799
adamc@533	800 $$\infer{\Gamma \vdash \mt{let} \; \overline{ed} \; \mt{in} \; e \; \mt{end} : \tau}{
adamc@533	801 \Gamma \vdash \overline{ed} \leadsto \Gamma'
adamc@533	802 & \Gamma' \vdash e : \tau
adamc@533	803 }
adamc@533	804 \quad \infer{\Gamma \vdash \mt{case} \; e \; \mt{of} \; \overline{p \Rightarrow e} : \tau}{
adamc@533	805 \forall i: \Gamma \vdash p_i \leadsto \Gamma_i, \tau'
adamc@533	806 & \Gamma_i \vdash e_i : \tau
adamc@533	807 }$$
adamc@533	808
adamc@573	809 $$\infer{\Gamma \vdash \lambda [c_1 \sim c_2] \Rightarrow e : \lambda [c_1 \sim c_2] \Rightarrow \tau}{
adamc@533	810 \Gamma \vdash c_1 :: \{\kappa\}
adamc@655	811 & \Gamma \vdash c_2 :: \{\kappa'\}
adamc@533	812 & \Gamma, c_1 \sim c_2 \vdash e : \tau
adamc@662	813 }
adamc@662	814 \quad \infer{\Gamma \vdash e \; ! : \tau}{
adamc@662	815 \Gamma \vdash e : [c_1 \sim c_2] \Rightarrow \tau
adamc@662	816 & \Gamma \vdash c_1 \sim c_2
adamc@533	817 }$$
adamc@533	818
adamc@534	819 \subsection{Pattern Typing}
adamc@534	820
adamc@534	821 $$\infer{\Gamma \vdash \_ \leadsto \Gamma; \tau}{}
adamc@534	822 \quad \infer{\Gamma \vdash x \leadsto \Gamma, x : \tau; \tau}{}
adamc@534	823 \quad \infer{\Gamma \vdash \ell \leadsto \Gamma; T(\ell)}{}$$
adamc@534	824
adamc@534	825 $$\infer{\Gamma \vdash X \leadsto \Gamma; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@534	826 X : \overline{x ::: \mt{Type}} \to \tau \in \Gamma
adamc@534	827 & \textrm{$\tau$ not a function type}
adamc@534	828 }
adamc@534	829 \quad \infer{\Gamma \vdash X \; p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@534	830 X : \overline{x ::: \mt{Type}} \to \tau'' \to \tau \in \Gamma
adamc@534	831 & \Gamma \vdash p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau''
adamc@534	832 }$$
adamc@534	833
adamc@534	834 $$\infer{\Gamma \vdash M.X \leadsto \Gamma; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@537	835 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	836 & \mt{proj}(M, \overline{s}, \mt{val} \; X) = \overline{x ::: \mt{Type}} \to \tau
adamc@534	837 & \textrm{$\tau$ not a function type}
adamc@534	838 }$$
adamc@534	839
adamc@534	840 $$\infer{\Gamma \vdash M.X \; p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@537	841 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	842 & \mt{proj}(M, \overline{s}, \mt{val} \; X) = \overline{x ::: \mt{Type}} \to \tau'' \to \tau
adamc@534	843 & \Gamma \vdash p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau''
adamc@534	844 }$$
adamc@534	845
adamc@534	846 $$\infer{\Gamma \vdash \{\overline{x = p}\} \leadsto \Gamma_n; \{\overline{x = \tau}\}}{
adamc@534	847 \Gamma_0 = \Gamma
adamc@534	848 & \forall i: \Gamma_i \vdash p_i \leadsto \Gamma_{i+1}; \tau_i
adamc@534	849 }
adamc@534	850 \quad \infer{\Gamma \vdash \{\overline{x = p}, \ldots\} \leadsto \Gamma_n; \$([\overline{x = \tau}] \rc c)}{
adamc@534	851 \Gamma_0 = \Gamma
adamc@534	852 & \forall i: \Gamma_i \vdash p_i \leadsto \Gamma_{i+1}; \tau_i
adamc@534	853 }$$
adamc@534	854
adamc@852	855 $$\infer{\Gamma \vdash p : \tau \leadsto \Gamma'; \tau}{
adamc@852	856 \Gamma \vdash p \leadsto \Gamma'; \tau'
adamc@852	857 & \Gamma \vdash \tau' \equiv \tau
adamc@852	858 }$$
adamc@852	859
adamc@535	860 \subsection{Declaration Typing}
adamc@535	861
adamc@535	862 We use an auxiliary judgment $\overline{y}; x; \Gamma \vdash \overline{dc} \leadsto \Gamma'$, expressing the enrichment of $\Gamma$ with the types of the datatype constructors $\overline{dc}$, when they are known to belong to datatype $x$ with type parameters $\overline{y}$.
adamc@535	863
adamc@655	864 This is the first judgment where we deal with constructor classes, for the $\mt{class}$ declaration form. We will omit their special handling in this formal specification. Section \ref{typeclasses} gives an informal description of how constructor classes influence type inference.
adamc@535	865
adamc@558	866 We presuppose the existence of a function $\mathcal O$, where $\mathcal O(M, \overline{s})$ implements the $\mt{open}$ declaration by producing a context with the appropriate entry for each available component of module $M$ with signature items $\overline{s}$. Where possible, $\mathcal O$ uses ``transparent'' entries (e.g., an abstract type $M.x$ is mapped to $x :: \mt{Type} = M.x$), so that the relationship with $M$ is maintained. A related function $\mathcal O_c$ builds a context containing the disjointness constraints found in $\overline s$.
adamc@537	867 We write $\kappa_1^n \to \kappa$ as a shorthand, where $\kappa_1^0 \to \kappa = \kappa$ and $\kappa_1^{n+1} \to \kappa_2 = \kappa_1 \to (\kappa_1^n \to \kappa_2)$. We write $\mt{len}(\overline{y})$ for the length of vector $\overline{y}$ of variables.
adamc@535	868
adamc@535	869 $$\infer{\Gamma \vdash \cdot \leadsto \Gamma}{}
adamc@535	870 \quad \infer{\Gamma \vdash d, \overline{d} \leadsto \Gamma''}{
adamc@535	871 \Gamma \vdash d \leadsto \Gamma'
adamc@535	872 & \Gamma' \vdash \overline{d} \leadsto \Gamma''
adamc@535	873 }$$
adamc@535	874
adamc@535	875 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa = c \leadsto \Gamma, x :: \kappa = c}{
adamc@535	876 \Gamma \vdash c :: \kappa
adamc@535	877 }
adamc@535	878 \quad \infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leadsto \Gamma'}{
adamc@535	879 \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} \vdash \overline{dc} \leadsto \Gamma'
adamc@535	880 }$$
adamc@535	881
adamc@535	882 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leadsto \Gamma'}{
adamc@537	883 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	884 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@535	885 & \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} = M.z \vdash \overline{dc} \leadsto \Gamma'
adamc@535	886 }$$
adamc@535	887
adamc@535	888 $$\infer{\Gamma \vdash \mt{val} \; x : \tau = e \leadsto \Gamma, x : \tau}{
adamc@535	889 \Gamma \vdash e : \tau
adamc@535	890 }$$
adamc@535	891
adamc@535	892 $$\infer{\Gamma \vdash \mt{val} \; \mt{rec} \; \overline{x : \tau = e} \leadsto \Gamma, \overline{x : \tau}}{
adamc@535	893 \forall i: \Gamma, \overline{x : \tau} \vdash e_i : \tau_i
adamc@535	894 & \textrm{$e_i$ starts with an expression $\lambda$, optionally preceded by constructor and disjointness $\lambda$s}
adamc@535	895 }$$
adamc@535	896
adamc@535	897 $$\infer{\Gamma \vdash \mt{structure} \; X : S = M \leadsto \Gamma, X : S}{
adamc@535	898 \Gamma \vdash M : S
adamc@558	899 & \textrm{ $M$ not a constant or application}
adamc@535	900 }
adamc@558	901 \quad \infer{\Gamma \vdash \mt{structure} \; X : S = M \leadsto \Gamma, X : \mt{selfify}(X, \overline{s})}{
adamc@558	902 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@539	903 }$$
adamc@539	904
adamc@539	905 $$\infer{\Gamma \vdash \mt{signature} \; X = S \leadsto \Gamma, X = S}{
adamc@535	906 \Gamma \vdash S
adamc@535	907 }$$
adamc@535	908
adamc@537	909 $$\infer{\Gamma \vdash \mt{open} \; M \leadsto \Gamma, \mathcal O(M, \overline{s})}{
adamc@537	910 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@535	911 }$$
adamc@535	912
adamc@535	913 $$\infer{\Gamma \vdash \mt{constraint} \; c_1 \sim c_2 \leadsto \Gamma}{
adamc@535	914 \Gamma \vdash c_1 :: \{\kappa\}
adamc@535	915 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@535	916 & \Gamma \vdash c_1 \sim c_2
adamc@535	917 }
adamc@537	918 \quad \infer{\Gamma \vdash \mt{open} \; \mt{constraints} \; M \leadsto \Gamma, \mathcal O_c(M, \overline{s})}{
adamc@537	919 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@535	920 }$$
adamc@535	921
adamc@784	922 $$\infer{\Gamma \vdash \mt{table} \; x : c \leadsto \Gamma, x : \mt{Basis}.\mt{sql\_table} \; c \; []}{
adamc@535	923 \Gamma \vdash c :: \{\mt{Type}\}
adamc@535	924 }
adamc@784	925 \quad \infer{\Gamma \vdash \mt{view} \; x : c \leadsto \Gamma, x : \mt{Basis}.\mt{sql\_view} \; c}{
adamc@784	926 \Gamma \vdash c :: \{\mt{Type}\}
adamc@784	927 }$$
adamc@784	928
adamc@784	929 $$\infer{\Gamma \vdash \mt{sequence} \; x \leadsto \Gamma, x : \mt{Basis}.\mt{sql\_sequence}}{}$$
adamc@535	930
adamc@535	931 $$\infer{\Gamma \vdash \mt{cookie} \; x : \tau \leadsto \Gamma, x : \mt{Basis}.\mt{http\_cookie} \; \tau}{
adamc@535	932 \Gamma \vdash \tau :: \mt{Type}
adamc@784	933 }
adamc@784	934 \quad \infer{\Gamma \vdash \mt{style} \; x \leadsto \Gamma, x : \mt{Basis}.\mt{css\_class}}{}$$
adamc@535	935
adamc@1085	936 $$\infer{\Gamma \vdash \mt{task} \; e_1 = e_2 \leadsto \Gamma}{
adamc@1085	937 \Gamma \vdash e_1 :: \mt{Basis}.\mt{task\_kind}
adamc@1085	938 & \Gamma \vdash e_2 :: \mt{Basis}.\mt{transaction} \; \{\}
adamc@1085	939 }$$
adamc@1085	940
adamc@784	941 $$\infer{\Gamma \vdash \mt{class} \; x :: \kappa = c \leadsto \Gamma, x :: \kappa = c}{
adamc@784	942 \Gamma \vdash c :: \kappa
adamc@535	943 }$$
adamc@535	944
adamc@535	945 $$\infer{\overline{y}; x; \Gamma \vdash \cdot \leadsto \Gamma}{}
adamc@535	946 \quad \infer{\overline{y}; x; \Gamma \vdash X \mid \overline{dc} \leadsto \Gamma', X : \overline{y ::: \mt{Type}} \to x \; \overline{y}}{
adamc@535	947 \overline{y}; x; \Gamma \vdash \overline{dc} \leadsto \Gamma'
adamc@535	948 }
adamc@535	949 \quad \infer{\overline{y}; x; \Gamma \vdash X \; \mt{of} \; \tau \mid \overline{dc} \leadsto \Gamma', X : \overline{y ::: \mt{Type}} \to \tau \to x \; \overline{y}}{
adamc@535	950 \overline{y}; x; \Gamma \vdash \overline{dc} \leadsto \Gamma'
adamc@535	951 }$$
adamc@535	952
adamc@537	953 \subsection{Signature Item Typing}
adamc@537	954
adamc@537	955 We appeal to a signature item analogue of the $\mathcal O$ function from the last subsection.
adamc@537	956
adamc@537	957 $$\infer{\Gamma \vdash \cdot \leadsto \Gamma}{}
adamc@537	958 \quad \infer{\Gamma \vdash s, \overline{s} \leadsto \Gamma''}{
adamc@537	959 \Gamma \vdash s \leadsto \Gamma'
adamc@537	960 & \Gamma' \vdash \overline{s} \leadsto \Gamma''
adamc@537	961 }$$
adamc@537	962
adamc@537	963 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa \leadsto \Gamma, x :: \kappa}{}
adamc@537	964 \quad \infer{\Gamma \vdash \mt{con} \; x :: \kappa = c \leadsto \Gamma, x :: \kappa = c}{
adamc@537	965 \Gamma \vdash c :: \kappa
adamc@537	966 }
adamc@537	967 \quad \infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leadsto \Gamma'}{
adamc@537	968 \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} \vdash \overline{dc} \leadsto \Gamma'
adamc@537	969 }$$
adamc@537	970
adamc@537	971 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leadsto \Gamma'}{
adamc@537	972 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	973 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@537	974 & \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} = M.z \vdash \overline{dc} \leadsto \Gamma'
adamc@537	975 }$$
adamc@537	976
adamc@537	977 $$\infer{\Gamma \vdash \mt{val} \; x : \tau \leadsto \Gamma, x : \tau}{
adamc@537	978 \Gamma \vdash \tau :: \mt{Type}
adamc@537	979 }$$
adamc@537	980
adamc@537	981 $$\infer{\Gamma \vdash \mt{structure} \; X : S \leadsto \Gamma, X : S}{
adamc@537	982 \Gamma \vdash S
adamc@537	983 }
adamc@537	984 \quad \infer{\Gamma \vdash \mt{signature} \; X = S \leadsto \Gamma, X = S}{
adamc@537	985 \Gamma \vdash S
adamc@537	986 }$$
adamc@537	987
adamc@537	988 $$\infer{\Gamma \vdash \mt{include} \; S \leadsto \Gamma, \mathcal O(\overline{s})}{
adamc@537	989 \Gamma \vdash S
adamc@537	990 & \Gamma \vdash S \equiv \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	991 }$$
adamc@537	992
adamc@537	993 $$\infer{\Gamma \vdash \mt{constraint} \; c_1 \sim c_2 \leadsto \Gamma, c_1 \sim c_2}{
adamc@537	994 \Gamma \vdash c_1 :: \{\kappa\}
adamc@537	995 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@537	996 }$$
adamc@537	997
adamc@784	998 $$\infer{\Gamma \vdash \mt{class} \; x :: \kappa = c \leadsto \Gamma, x :: \kappa = c}{
adamc@784	999 \Gamma \vdash c :: \kappa
adamc@537	1000 }
adamc@784	1001 \quad \infer{\Gamma \vdash \mt{class} \; x :: \kappa \leadsto \Gamma, x :: \kappa}{}$$
adamc@537	1002
adamc@536	1003 \subsection{Signature Compatibility}
adamc@536	1004
adamc@558	1005 To simplify the judgments in this section, we assume that all signatures are alpha-varied as necessary to avoid including multiple bindings for the same identifier. This is in addition to the usual alpha-variation of locally-bound variables.
adamc@537	1006
adamc@537	1007 We rely on a judgment $\Gamma \vdash \overline{s} \leq s'$, which expresses the occurrence in signature items $\overline{s}$ of an item compatible with $s'$. We also use a judgment $\Gamma \vdash \overline{dc} \leq \overline{dc}$, which expresses compatibility of datatype definitions.
adamc@537	1008
adamc@536	1009 $$\infer{\Gamma \vdash S \equiv S}{}
adamc@536	1010 \quad \infer{\Gamma \vdash S_1 \equiv S_2}{
adamc@536	1011 \Gamma \vdash S_2 \equiv S_1
adamc@536	1012 }
adamc@536	1013 \quad \infer{\Gamma \vdash X \equiv S}{
adamc@536	1014 X = S \in \Gamma
adamc@536	1015 }
adamc@536	1016 \quad \infer{\Gamma \vdash M.X \equiv S}{
adamc@537	1017 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	1018 & \mt{proj}(M, \overline{s}, \mt{signature} \; X) = S
adamc@536	1019 }$$
adamc@536	1020
adamc@536	1021 $$\infer{\Gamma \vdash S \; \mt{where} \; \mt{con} \; x = c \equiv \mt{sig} \; \overline{s^1} \; \mt{con} \; x :: \kappa = c \; \overline{s_2} \; \mt{end}}{
adamc@536	1022 \Gamma \vdash S \equiv \mt{sig} \; \overline{s^1} \; \mt{con} \; x :: \kappa \; \overline{s_2} \; \mt{end}
adamc@536	1023 & \Gamma \vdash c :: \kappa
adamc@537	1024 }
adamc@537	1025 \quad \infer{\Gamma \vdash \mt{sig} \; \overline{s^1} \; \mt{include} \; S \; \overline{s^2} \; \mt{end} \equiv \mt{sig} \; \overline{s^1} \; \overline{s} \; \overline{s^2} \; \mt{end}}{
adamc@537	1026 \Gamma \vdash S \equiv \mt{sig} \; \overline{s} \; \mt{end}
adamc@536	1027 }$$
adamc@536	1028
adamc@536	1029 $$\infer{\Gamma \vdash S_1 \leq S_2}{
adamc@536	1030 \Gamma \vdash S_1 \equiv S_2
adamc@536	1031 }
adamc@536	1032 \quad \infer{\Gamma \vdash \mt{sig} \; \overline{s} \; \mt{end} \leq \mt{sig} \; \mt{end}}{}
adamc@537	1033 \quad \infer{\Gamma \vdash \mt{sig} \; \overline{s} \; \mt{end} \leq \mt{sig} \; s' \; \overline{s'} \; \mt{end}}{
adamc@537	1034 \Gamma \vdash \overline{s} \leq s'
adamc@537	1035 & \Gamma \vdash s' \leadsto \Gamma'
adamc@537	1036 & \Gamma' \vdash \mt{sig} \; \overline{s} \; \mt{end} \leq \mt{sig} \; \overline{s'} \; \mt{end}
adamc@537	1037 }$$
adamc@537	1038
adamc@537	1039 $$\infer{\Gamma \vdash s \; \overline{s} \leq s'}{
adamc@537	1040 \Gamma \vdash s \leq s'
adamc@537	1041 }
adamc@537	1042 \quad \infer{\Gamma \vdash s \; \overline{s} \leq s'}{
adamc@537	1043 \Gamma \vdash s \leadsto \Gamma'
adamc@537	1044 & \Gamma' \vdash \overline{s} \leq s'
adamc@536	1045 }$$
adamc@536	1046
adamc@536	1047 $$\infer{\Gamma \vdash \mt{functor} (X : S_1) : S_2 \leq \mt{functor} (X : S'_1) : S'_2}{
adamc@536	1048 \Gamma \vdash S'_1 \leq S_1
adamc@536	1049 & \Gamma, X : S'_1 \vdash S_2 \leq S'_2
adamc@536	1050 }$$
adamc@536	1051
adamc@537	1052 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa \leq \mt{con} \; x :: \kappa}{}
adamc@537	1053 \quad \infer{\Gamma \vdash \mt{con} \; x :: \kappa = c \leq \mt{con} \; x :: \kappa}{}
adamc@558	1054 \quad \infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leq \mt{con} \; x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type}}{}$$
adamc@537	1055
adamc@537	1056 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leq \mt{con} \; x :: \mt{Type}^{\mt{len}(y)} \to \mt{Type}}{
adamc@537	1057 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	1058 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@537	1059 }$$
adamc@537	1060
adamc@784	1061 $$\infer{\Gamma \vdash \mt{class} \; x :: \kappa \leq \mt{con} \; x :: \kappa}{}
adamc@784	1062 \quad \infer{\Gamma \vdash \mt{class} \; x :: \kappa = c \leq \mt{con} \; x :: \kappa}{}$$
adamc@537	1063
adamc@537	1064 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa = c_1 \leq \mt{con} \; x :: \mt{\kappa} = c_2}{
adamc@537	1065 \Gamma \vdash c_1 \equiv c_2
adamc@537	1066 }
adamc@784	1067 \quad \infer{\Gamma \vdash \mt{class} \; x :: \kappa = c_1 \leq \mt{con} \; x :: \kappa = c_2}{
adamc@537	1068 \Gamma \vdash c_1 \equiv c_2
adamc@537	1069 }$$
adamc@537	1070
adamc@537	1071 $$\infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leq \mt{datatype} \; x \; \overline{y} = \overline{dc'}}{
adamc@537	1072 \Gamma, \overline{y :: \mt{Type}} \vdash \overline{dc} \leq \overline{dc'}
adamc@537	1073 }$$
adamc@537	1074
adamc@537	1075 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leq \mt{datatype} \; x \; \overline{y} = \overline{dc'}}{
adamc@537	1076 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	1077 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@537	1078 & \Gamma, \overline{y :: \mt{Type}} \vdash \overline{dc} \leq \overline{dc'}
adamc@537	1079 }$$
adamc@537	1080
adamc@537	1081 $$\infer{\Gamma \vdash \cdot \leq \cdot}{}
adamc@537	1082 \quad \infer{\Gamma \vdash X; \overline{dc} \leq X; \overline{dc'}}{
adamc@537	1083 \Gamma \vdash \overline{dc} \leq \overline{dc'}
adamc@537	1084 }
adamc@537	1085 \quad \infer{\Gamma \vdash X \; \mt{of} \; \tau_1; \overline{dc} \leq X \; \mt{of} \; \tau_2; \overline{dc'}}{
adamc@537	1086 \Gamma \vdash \tau_1 \equiv \tau_2
adamc@537	1087 & \Gamma \vdash \overline{dc} \leq \overline{dc'}
adamc@537	1088 }$$
adamc@537	1089
adamc@537	1090 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leq \mt{datatype} \; x = \mt{datatype} \; M'.z'}{
adamc@537	1091 \Gamma \vdash M.z \equiv M'.z'
adamc@537	1092 }$$
adamc@537	1093
adamc@537	1094 $$\infer{\Gamma \vdash \mt{val} \; x : \tau_1 \leq \mt{val} \; x : \tau_2}{
adamc@537	1095 \Gamma \vdash \tau_1 \equiv \tau_2
adamc@537	1096 }
adamc@537	1097 \quad \infer{\Gamma \vdash \mt{structure} \; X : S_1 \leq \mt{structure} \; X : S_2}{
adamc@537	1098 \Gamma \vdash S_1 \leq S_2
adamc@537	1099 }
adamc@537	1100 \quad \infer{\Gamma \vdash \mt{signature} \; X = S_1 \leq \mt{signature} \; X = S_2}{
adamc@537	1101 \Gamma \vdash S_1 \leq S_2
adamc@537	1102 & \Gamma \vdash S_2 \leq S_1
adamc@537	1103 }$$
adamc@537	1104
adamc@537	1105 $$\infer{\Gamma \vdash \mt{constraint} \; c_1 \sim c_2 \leq \mt{constraint} \; c'_1 \sim c'_2}{
adamc@537	1106 \Gamma \vdash c_1 \equiv c'_1
adamc@537	1107 & \Gamma \vdash c_2 \equiv c'_2
adamc@537	1108 }$$
adamc@537	1109
adamc@655	1110 $$\infer{\Gamma \vdash \mt{class} \; x :: \kappa \leq \mt{class} \; x :: \kappa}{}
adamc@655	1111 \quad \infer{\Gamma \vdash \mt{class} \; x :: \kappa = c \leq \mt{class} \; x :: \kappa}{}
adamc@655	1112 \quad \infer{\Gamma \vdash \mt{class} \; x :: \kappa = c_1 \leq \mt{class} \; x :: \kappa = c_2}{
adamc@537	1113 \Gamma \vdash c_1 \equiv c_2
adamc@537	1114 }$$
adamc@537	1115
adamc@538	1116 \subsection{Module Typing}
adamc@538	1117
adamc@538	1118 We use a helper function $\mt{sigOf}$, which converts declarations and sequences of declarations into their principal signature items and sequences of signature items, respectively.
adamc@538	1119
adamc@538	1120 $$\infer{\Gamma \vdash M : S}{
adamc@538	1121 \Gamma \vdash M : S'
adamc@538	1122 & \Gamma \vdash S' \leq S
adamc@538	1123 }
adamc@538	1124 \quad \infer{\Gamma \vdash \mt{struct} \; \overline{d} \; \mt{end} : \mt{sig} \; \mt{sigOf}(\overline{d}) \; \mt{end}}{
adamc@538	1125 \Gamma \vdash \overline{d} \leadsto \Gamma'
adamc@538	1126 }
adamc@538	1127 \quad \infer{\Gamma \vdash X : S}{
adamc@538	1128 X : S \in \Gamma
adamc@538	1129 }$$
adamc@538	1130
adamc@538	1131 $$\infer{\Gamma \vdash M.X : S}{
adamc@538	1132 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@538	1133 & \mt{proj}(M, \overline{s}, \mt{structure} \; X) = S
adamc@538	1134 }$$
adamc@538	1135
adamc@538	1136 $$\infer{\Gamma \vdash M_1(M_2) : [X \mapsto M_2]S_2}{
adamc@538	1137 \Gamma \vdash M_1 : \mt{functor}(X : S_1) : S_2
adamc@538	1138 & \Gamma \vdash M_2 : S_1
adamc@538	1139 }
adamc@538	1140 \quad \infer{\Gamma \vdash \mt{functor} (X : S_1) : S_2 = M : \mt{functor} (X : S_1) : S_2}{
adamc@538	1141 \Gamma \vdash S_1
adamc@538	1142 & \Gamma, X : S_1 \vdash S_2
adamc@538	1143 & \Gamma, X : S_1 \vdash M : S_2
adamc@538	1144 }$$
adamc@538	1145
adamc@538	1146 \begin{eqnarray*}
adamc@538	1147 \mt{sigOf}(\cdot) &=& \cdot \\
adamc@538	1148 \mt{sigOf}(s \; \overline{s'}) &=& \mt{sigOf}(s) \; \mt{sigOf}(\overline{s'}) \\
adamc@538	1149 \\
adamc@538	1150 \mt{sigOf}(\mt{con} \; x :: \kappa = c) &=& \mt{con} \; x :: \kappa = c \\
adamc@538	1151 \mt{sigOf}(\mt{datatype} \; x \; \overline{y} = \overline{dc}) &=& \mt{datatype} \; x \; \overline{y} = \overline{dc} \\
adamc@538	1152 \mt{sigOf}(\mt{datatype} \; x = \mt{datatype} \; M.z) &=& \mt{datatype} \; x = \mt{datatype} \; M.z \\
adamc@538	1153 \mt{sigOf}(\mt{val} \; x : \tau = e) &=& \mt{val} \; x : \tau \\
adamc@538	1154 \mt{sigOf}(\mt{val} \; \mt{rec} \; \overline{x : \tau = e}) &=& \overline{\mt{val} \; x : \tau} \\
adamc@538	1155 \mt{sigOf}(\mt{structure} \; X : S = M) &=& \mt{structure} \; X : S \\
adamc@538	1156 \mt{sigOf}(\mt{signature} \; X = S) &=& \mt{signature} \; X = S \\
adamc@538	1157 \mt{sigOf}(\mt{open} \; M) &=& \mt{include} \; S \textrm{ (where $\Gamma \vdash M : S$)} \\
adamc@538	1158 \mt{sigOf}(\mt{constraint} \; c_1 \sim c_2) &=& \mt{constraint} \; c_1 \sim c_2 \\
adamc@538	1159 \mt{sigOf}(\mt{open} \; \mt{constraints} \; M) &=& \cdot \\
adamc@538	1160 \mt{sigOf}(\mt{table} \; x : c) &=& \mt{table} \; x : c \\
adamc@784	1161 \mt{sigOf}(\mt{view} \; x : c) &=& \mt{view} \; x : c \\
adamc@538	1162 \mt{sigOf}(\mt{sequence} \; x) &=& \mt{sequence} \; x \\
adamc@538	1163 \mt{sigOf}(\mt{cookie} \; x : \tau) &=& \mt{cookie} \; x : \tau \\
adamc@784	1164 \mt{sigOf}(\mt{style} \; x) &=& \mt{style} \; x \\
adamc@655	1165 \mt{sigOf}(\mt{class} \; x :: \kappa = c) &=& \mt{class} \; x :: \kappa = c \\
adamc@538	1166 \end{eqnarray*}
adamc@539	1167 \begin{eqnarray*}
adamc@539	1168 \mt{selfify}(M, \cdot) &=& \cdot \\
adamc@558	1169 \mt{selfify}(M, s \; \overline{s'}) &=& \mt{selfify}(M, s) \; \mt{selfify}(M, \overline{s'}) \\
adamc@539	1170 \\
adamc@539	1171 \mt{selfify}(M, \mt{con} \; x :: \kappa) &=& \mt{con} \; x :: \kappa = M.x \\
adamc@539	1172 \mt{selfify}(M, \mt{con} \; x :: \kappa = c) &=& \mt{con} \; x :: \kappa = c \\
adamc@539	1173 \mt{selfify}(M, \mt{datatype} \; x \; \overline{y} = \overline{dc}) &=& \mt{datatype} \; x \; \overline{y} = \mt{datatype} \; M.x \\
adamc@539	1174 \mt{selfify}(M, \mt{datatype} \; x = \mt{datatype} \; M'.z) &=& \mt{datatype} \; x = \mt{datatype} \; M'.z \\
adamc@539	1175 \mt{selfify}(M, \mt{val} \; x : \tau) &=& \mt{val} \; x : \tau \\
adamc@539	1176 \mt{selfify}(M, \mt{structure} \; X : S) &=& \mt{structure} \; X : \mt{selfify}(M.X, \overline{s}) \textrm{ (where $\Gamma \vdash S \equiv \mt{sig} \; \overline{s} \; \mt{end}$)} \\
adamc@539	1177 \mt{selfify}(M, \mt{signature} \; X = S) &=& \mt{signature} \; X = S \\
adamc@539	1178 \mt{selfify}(M, \mt{include} \; S) &=& \mt{include} \; S \\
adamc@539	1179 \mt{selfify}(M, \mt{constraint} \; c_1 \sim c_2) &=& \mt{constraint} \; c_1 \sim c_2 \\
adamc@655	1180 \mt{selfify}(M, \mt{class} \; x :: \kappa) &=& \mt{class} \; x :: \kappa = M.x \\
adamc@655	1181 \mt{selfify}(M, \mt{class} \; x :: \kappa = c) &=& \mt{class} \; x :: \kappa = c \\
adamc@539	1182 \end{eqnarray*}
adamc@539	1183
adamc@540	1184 \subsection{Module Projection}
adamc@540	1185
adamc@540	1186 \begin{eqnarray*}
adamc@540	1187 \mt{proj}(M, \mt{con} \; x :: \kappa \; \overline{s}, \mt{con} \; x) &=& \kappa \\
adamc@540	1188 \mt{proj}(M, \mt{con} \; x :: \kappa = c \; \overline{s}, \mt{con} \; x) &=& (\kappa, c) \\
adamc@540	1189 \mt{proj}(M, \mt{datatype} \; x \; \overline{y} = \overline{dc} \; \overline{s}, \mt{con} \; x) &=& \mt{Type}^{\mt{len}(\overline{y})} \to \mt{Type} \\
adamc@540	1190 \mt{proj}(M, \mt{datatype} \; x = \mt{datatype} \; M'.z \; \overline{s}, \mt{con} \; x) &=& (\mt{Type}^{\mt{len}(\overline{y})} \to \mt{Type}, M'.z) \textrm{ (where $\Gamma \vdash M' : \mt{sig} \; \overline{s'} \; \mt{end}$} \\
adamc@540	1191 && \textrm{and $\mt{proj}(M', \overline{s'}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})$)} \\
adamc@655	1192 \mt{proj}(M, \mt{class} \; x :: \kappa \; \overline{s}, \mt{con} \; x) &=& \kappa \to \mt{Type} \\
adamc@655	1193 \mt{proj}(M, \mt{class} \; x :: \kappa = c \; \overline{s}, \mt{con} \; x) &=& (\kappa \to \mt{Type}, c) \\
adamc@540	1194 \\
adamc@540	1195 \mt{proj}(M, \mt{datatype} \; x \; \overline{y} = \overline{dc} \; \overline{s}, \mt{datatype} \; x) &=& (\overline{y}, \overline{dc}) \\
adamc@540	1196 \mt{proj}(M, \mt{datatype} \; x = \mt{datatype} \; M'.z \; \overline{s}, \mt{con} \; x) &=& \mt{proj}(M', \overline{s'}, \mt{datatype} \; z) \textrm{ (where $\Gamma \vdash M' : \mt{sig} \; \overline{s'} \; \mt{end}$)} \\
adamc@540	1197 \\
adamc@540	1198 \mt{proj}(M, \mt{val} \; x : \tau \; \overline{s}, \mt{val} \; x) &=& \tau \\
adamc@540	1199 \mt{proj}(M, \mt{datatype} \; x \; \overline{y} = \overline{dc} \; \overline{s}, \mt{val} \; X) &=& \overline{y ::: \mt{Type}} \to M.x \; \overline y \textrm{ (where $X \in \overline{dc}$)} \\
adamc@540	1200 \mt{proj}(M, \mt{datatype} \; x \; \overline{y} = \overline{dc} \; \overline{s}, \mt{val} \; X) &=& \overline{y ::: \mt{Type}} \to \tau \to M.x \; \overline y \textrm{ (where $X \; \mt{of} \; \tau \in \overline{dc}$)} \\
adamc@540	1201 \mt{proj}(M, \mt{datatype} \; x = \mt{datatype} \; M'.z, \mt{val} \; X) &=& \overline{y ::: \mt{Type}} \to M.x \; \overline y \textrm{ (where $\Gamma \vdash M' : \mt{sig} \; \overline{s'} \; \mt{end}$} \\
adamc@540	1202 && \textrm{and $\mt{proj}(M', \overline{s'}, \mt{datatype} \; z = (\overline{y}, \overline{dc})$ and $X \in \overline{dc}$)} \\
adamc@540	1203 \mt{proj}(M, \mt{datatype} \; x = \mt{datatype} \; M'.z, \mt{val} \; X) &=& \overline{y ::: \mt{Type}} \to \tau \to M.x \; \overline y \textrm{ (where $\Gamma \vdash M' : \mt{sig} \; \overline{s'} \; \mt{end}$} \\
adamc@558	1204 && \textrm{and $\mt{proj}(M', \overline{s'}, \mt{datatype} \; z = (\overline{y}, \overline{dc})$ and $X \; \mt{of} \; \tau \in \overline{dc}$)} \\
adamc@540	1205 \\
adamc@540	1206 \mt{proj}(M, \mt{structure} \; X : S \; \overline{s}, \mt{structure} \; X) &=& S \\
adamc@540	1207 \\
adamc@540	1208 \mt{proj}(M, \mt{signature} \; X = S \; \overline{s}, \mt{signature} \; X) &=& S \\
adamc@540	1209 \\
adamc@540	1210 \mt{proj}(M, \mt{con} \; x :: \kappa \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@540	1211 \mt{proj}(M, \mt{con} \; x :: \kappa = c \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@540	1212 \mt{proj}(M, \mt{datatype} \; x \; \overline{y} = \overline{dc} \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@540	1213 \mt{proj}(M, \mt{datatype} \; x = \mt{datatype} \; M'.z \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@540	1214 \mt{proj}(M, \mt{val} \; x : \tau \; \overline{s}, V) &=& \mt{proj}(M, \overline{s}, V) \\
adamc@540	1215 \mt{proj}(M, \mt{structure} \; X : S \; \overline{s}, V) &=& [X \mapsto M.X]\mt{proj}(M, \overline{s}, V) \\
adamc@540	1216 \mt{proj}(M, \mt{signature} \; X = S \; \overline{s}, V) &=& [X \mapsto M.X]\mt{proj}(M, \overline{s}, V) \\
adamc@540	1217 \mt{proj}(M, \mt{include} \; S \; \overline{s}, V) &=& \mt{proj}(M, \overline{s'} \; \overline{s}, V) \textrm{ (where $\Gamma \vdash S \equiv \mt{sig} \; \overline{s'} \; \mt{end}$)} \\
adamc@540	1218 \mt{proj}(M, \mt{constraint} \; c_1 \sim c_2 \; \overline{s}, V) &=& \mt{proj}(M, \overline{s}, V) \\
adamc@655	1219 \mt{proj}(M, \mt{class} \; x :: \kappa \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@655	1220 \mt{proj}(M, \mt{class} \; x :: \kappa = c \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@540	1221 \end{eqnarray*}
adamc@540	1222
adamc@541	1223
adamc@541	1224 \section{Type Inference}
adamc@541	1225
adamc@541	1226 The Ur/Web compiler uses \emph{heuristic type inference}, with no claims of completeness with respect to the declarative specification of the last section. The rules in use seem to work well in practice. This section summarizes those rules, to help Ur programmers predict what will work and what won't.
adamc@541	1227
adamc@541	1228 \subsection{Basic Unification}
adamc@541	1229
adamc@560	1230 Type-checkers for languages based on the Hindley-Milner type discipline, like ML and Haskell, take advantage of \emph{principal typing} properties, making complete type inference relatively straightforward. Inference algorithms are traditionally implemented using type unification variables, at various points asserting equalities between types, in the process discovering the values of type variables. The Ur/Web compiler uses the same basic strategy, but the complexity of the type system rules out easy completeness.
adamc@541	1231
adamc@656	1232 Type-checking can require evaluating recursive functional programs, thanks to the type-level $\mt{map}$ operator. When a unification variable appears in such a type, the next step of computation can be undetermined. The value of that variable might be determined later, but this would be ``too late'' for the unification problems generated at the first occurrence. This is the essential source of incompleteness.
adamc@541	1233
adamc@541	1234 Nonetheless, the unification engine tends to do reasonably well. Unlike in ML, polymorphism is never inferred in definitions; it must be indicated explicitly by writing out constructor-level parameters. By writing these and other annotations, the programmer can generally get the type inference engine to do most of the type reconstruction work.
adamc@541	1235
adamc@541	1236 \subsection{Unifying Record Types}
adamc@541	1237
adamc@570	1238 The type inference engine tries to take advantage of the algebraic rules governing type-level records, as shown in Section \ref{definitional}. When two constructors of record kind are unified, they are reduced to normal forms, with like terms crossed off from each normal form until, hopefully, nothing remains. This cannot be complete, with the inclusion of unification variables. The type-checker can help you understand what goes wrong when the process fails, as it outputs the unmatched remainders of the two normal forms.
adamc@541	1239
adamc@656	1240 \subsection{\label{typeclasses}Constructor Classes}
adamc@541	1241
adamc@784	1242 Ur includes a constructor class facility inspired by Haskell's. The current version is experimental, with very general Prolog-like facilities that can lead to compile-time non-termination.
adamc@541	1243
adamc@784	1244 Constructor classes are integrated with the module system. A constructor class of kind $\kappa$ is just a constructor of kind $\kappa$. By marking such a constructor $c$ as a constructor class, the programmer instructs the type inference engine to, in each scope, record all values of types $c \; c_1 \; \ldots \; c_n$ as \emph{instances}. Any function argument whose type is of such a form is treated as implicit, to be determined by examining the current instance database.
adamc@541	1245
adamc@656	1246 The ``dictionary encoding'' often used in Haskell implementations is made explicit in Ur. Constructor class instances are just properly-typed values, and they can also be considered as ``proofs'' of membership in the class. In some cases, it is useful to pass these proofs around explicitly. An underscore written where a proof is expected will also be inferred, if possible, from the current instance database.
adamc@541	1247
adamc@656	1248 Just as for constructors, constructors classes may be exported from modules, and they may be exported as concrete or abstract. Concrete constructor classes have their ``real'' definitions exposed, so that client code may add new instances freely. Abstract constructor classes are useful as ``predicates'' that can be used to enforce invariants, as we will see in some definitions of SQL syntax in the Ur/Web standard library.
adamc@541	1249
adamc@541	1250 \subsection{Reverse-Engineering Record Types}
adamc@541	1251
adamc@656	1252 It's useful to write Ur functions and functors that take record constructors as inputs, but these constructors can grow quite long, even though their values are often implied by other arguments. The compiler uses a simple heuristic to infer the values of unification variables that are mapped over, yielding known results. If the result is empty, we're done; if it's not empty, we replace a single unification variable with a new constructor formed from three new unification variables, as in $[\alpha = \beta] \rc \gamma$. This process can often be repeated to determine a unification variable fully.
adamc@541	1253
adamc@541	1254 \subsection{Implicit Arguments in Functor Applications}
adamc@541	1255
adamc@656	1256 Constructor, constraint, and constructor class witness members of structures may be omitted, when those structures are used in contexts where their assigned signatures imply how to fill in those missing members. This feature combines well with reverse-engineering to allow for uses of complicated meta-programming functors with little more code than would be necessary to invoke an untyped, ad-hoc code generator.
adamc@541	1257
adamc@541	1258
adamc@542	1259 \section{The Ur Standard Library}
adamc@542	1260
adamc@542	1261 The built-in parts of the Ur/Web standard library are described by the signature in \texttt{lib/basis.urs} in the distribution. A module $\mt{Basis}$ ascribing to that signature is available in the initial environment, and every program is implicitly prefixed by $\mt{open} \; \mt{Basis}$.
adamc@542	1262
adamc@542	1263 Additionally, other common functions that are definable within Ur are included in \texttt{lib/top.urs} and \texttt{lib/top.ur}. This $\mt{Top}$ module is also opened implicitly.
adamc@542	1264
adamc@542	1265 The idea behind Ur is to serve as the ideal host for embedded domain-specific languages. For now, however, the ``generic'' functionality is intermixed with Ur/Web-specific functionality, including in these two library modules. We hope that these generic library components have types that speak for themselves. The next section introduces the Ur/Web-specific elements. Here, we only give the type declarations from the beginning of $\mt{Basis}$.
adamc@542	1266 $$\begin{array}{l}
adamc@542	1267 \mt{type} \; \mt{int} \\
adamc@542	1268 \mt{type} \; \mt{float} \\
adamc@873	1269 \mt{type} \; \mt{char} \\
adamc@542	1270 \mt{type} \; \mt{string} \\
adamc@542	1271 \mt{type} \; \mt{time} \\
adamc@785	1272 \mt{type} \; \mt{blob} \\
adamc@542	1273 \\
adamc@542	1274 \mt{type} \; \mt{unit} = \{\} \\
adamc@542	1275 \\
adamc@542	1276 \mt{datatype} \; \mt{bool} = \mt{False} \mid \mt{True} \\
adamc@542	1277 \\
adamc@785	1278 \mt{datatype} \; \mt{option} \; \mt{t} = \mt{None} \mid \mt{Some} \; \mt{of} \; \mt{t} \\
adamc@785	1279 \\
adamc@785	1280 \mt{datatype} \; \mt{list} \; \mt{t} = \mt{Nil} \mid \mt{Cons} \; \mt{of} \; \mt{t} \times \mt{list} \; \mt{t}
adamc@542	1281 \end{array}$$
adamc@542	1282
adamc@1123	1283 The only unusual element of this list is the $\mt{blob}$ type, which stands for binary sequences. Simple blobs can be created from strings via $\mt{Basis.textBlob}$. Blobs will also be generated from HTTP file uploads.
adamc@785	1284
adamc@657	1285 Another important generic Ur element comes at the beginning of \texttt{top.urs}.
adamc@657	1286
adamc@657	1287 $$\begin{array}{l}
adamc@657	1288 \mt{con} \; \mt{folder} :: \mt{K} \longrightarrow \{\mt{K}\} \to \mt{Type} \\
adamc@657	1289 \\
adamc@657	1290 \mt{val} \; \mt{fold} : \mt{K} \longrightarrow \mt{tf} :: (\{\mt{K}\} \to \mt{Type}) \\
adamc@657	1291 \hspace{.1in} \to (\mt{nm} :: \mt{Name} \to \mt{v} :: \mt{K} \to \mt{r} :: \{\mt{K}\} \to [[\mt{nm}] \sim \mt{r}] \Rightarrow \\
adamc@657	1292 \hspace{.2in} \mt{tf} \; \mt{r} \to \mt{tf} \; ([\mt{nm} = \mt{v}] \rc \mt{r})) \\
adamc@657	1293 \hspace{.1in} \to \mt{tf} \; [] \\
adamc@657	1294 \hspace{.1in} \to \mt{r} :: \{\mt{K}\} \to \mt{folder} \; \mt{r} \to \mt{tf} \; \mt{r}
adamc@657	1295 \end{array}$$
adamc@657	1296
adamc@657	1297 For a type-level record $\mt{r}$, a $\mt{folder} \; \mt{r}$ encodes a permutation of $\mt{r}$'s elements. The $\mt{fold}$ function can be called on a $\mt{folder}$ to iterate over the elements of $\mt{r}$ in that order. $\mt{fold}$ is parameterized on a type-level function to be used to calculate the type of each intermediate result of folding. After processing a subset $\mt{r'}$ of $\mt{r}$'s entries, the type of the accumulator should be $\mt{tf} \; \mt{r'}$. The next two expression arguments to $\mt{fold}$ are the usual step function and initial accumulator, familiar from fold functions over lists. The final two arguments are the record to fold over and a $\mt{folder}$ for it.
adamc@657	1298
adamc@664	1299 The Ur compiler treats $\mt{folder}$ like a constructor class, using built-in rules to infer $\mt{folder}$s for records with known structure. The order in which field names are mentioned in source code is used as a hint about the permutation that the programmer would like.
adamc@657	1300
adamc@542	1301
adamc@542	1302 \section{The Ur/Web Standard Library}
adamc@542	1303
adamc@658	1304 \subsection{Monads}
adamc@658	1305
adamc@658	1306 The Ur Basis defines the monad constructor class from Haskell.
adamc@658	1307
adamc@658	1308 $$\begin{array}{l}
adamc@658	1309 \mt{class} \; \mt{monad} :: \mt{Type} \to \mt{Type} \\
adamc@658	1310 \mt{val} \; \mt{return} : \mt{m} ::: (\mt{Type} \to \mt{Type}) \to \mt{t} ::: \mt{Type} \\
adamc@658	1311 \hspace{.1in} \to \mt{monad} \; \mt{m} \\
adamc@658	1312 \hspace{.1in} \to \mt{t} \to \mt{m} \; \mt{t} \\
adamc@658	1313 \mt{val} \; \mt{bind} : \mt{m} ::: (\mt{Type} \to \mt{Type}) \to \mt{t1} ::: \mt{Type} \to \mt{t2} ::: \mt{Type} \\
adamc@658	1314 \hspace{.1in} \to \mt{monad} \; \mt{m} \\
adamc@658	1315 \hspace{.1in} \to \mt{m} \; \mt{t1} \to (\mt{t1} \to \mt{m} \; \mt{t2}) \\
adamc@658	1316 \hspace{.1in} \to \mt{m} \; \mt{t2}
adamc@658	1317 \end{array}$$
adamc@658	1318
adamc@542	1319 \subsection{Transactions}
adamc@542	1320
adamc@542	1321 Ur is a pure language; we use Haskell's trick to support controlled side effects. The standard library defines a monad $\mt{transaction}$, meant to stand for actions that may be undone cleanly. By design, no other kinds of actions are supported.
adamc@542	1322 $$\begin{array}{l}
adamc@542	1323 \mt{con} \; \mt{transaction} :: \mt{Type} \to \mt{Type} \\
adamc@658	1324 \mt{val} \; \mt{transaction\_monad} : \mt{monad} \; \mt{transaction}
adamc@542	1325 \end{array}$$
adamc@542	1326
adamc@1123	1327 For debugging purposes, a transactional function is provided for outputting a string on the server process' \texttt{stderr}.
adamc@1123	1328 $$\begin{array}{l}
adamc@1123	1329 \mt{val} \; \mt{debug} : \mt{string} \to \mt{transaction} \; \mt{unit}
adamc@1123	1330 \end{array}$$
adamc@1123	1331
adamc@542	1332 \subsection{HTTP}
adamc@542	1333
adamc@542	1334 There are transactions for reading an HTTP header by name and for getting and setting strongly-typed cookies. Cookies may only be created by the $\mt{cookie}$ declaration form, ensuring that they be named consistently based on module structure.
adamc@542	1335 $$\begin{array}{l}
adamc@786	1336 \mt{val} \; \mt{requestHeader} : \mt{string} \to \mt{transaction} \; (\mt{option} \; \mt{string}) \\
adamc@786	1337 \\
adamc@786	1338 \mt{con} \; \mt{http\_cookie} :: \mt{Type} \to \mt{Type} \\
adamc@786	1339 \mt{val} \; \mt{getCookie} : \mt{t} ::: \mt{Type} \to \mt{http\_cookie} \; \mt{t} \to \mt{transaction} \; (\mt{option} \; \mt{t}) \\
adamc@1050	1340 \mt{val} \; \mt{setCookie} : \mt{t} ::: \mt{Type} \to \mt{http\_cookie} \; \mt{t} \to \{\mt{Value} : \mt{t}, \mt{Expires} : \mt{option} \; \mt{time}, \mt{Secure} : \mt{bool}\} \to \mt{transaction} \; \mt{unit} \\
adamc@1050	1341 \mt{val} \; \mt{clearCookie} : \mt{t} ::: \mt{Type} \to \mt{http\_cookie} \; \mt{t} \to \mt{transaction} \; \mt{unit}
adamc@786	1342 \end{array}$$
adamc@786	1343
adamc@786	1344 There are also an abstract $\mt{url}$ type and functions for converting to it, based on the policy defined by \texttt{[allow\|deny] url} directives in the project file.
adamc@786	1345 $$\begin{array}{l}
adamc@786	1346 \mt{type} \; \mt{url} \\
adamc@786	1347 \mt{val} \; \mt{bless} : \mt{string} \to \mt{url} \\
adamc@786	1348 \mt{val} \; \mt{checkUrl} : \mt{string} \to \mt{option} \; \mt{url}
adamc@786	1349 \end{array}$$
adamc@786	1350 $\mt{bless}$ raises a runtime error if the string passed to it fails the URL policy.
adamc@786	1351
adamc@1085	1352 It is possible to grab the current page's URL or to build a URL for an arbitrary transaction that would also be an acceptable value of a \texttt{link} attribute of the \texttt{a} tag.
adamc@1085	1353 $$\begin{array}{l}
adamc@1085	1354 \mt{val} \; \mt{currentUrl} : \mt{transaction} \; \mt{url} \\
adamc@1085	1355 \mt{val} \; \mt{url} : \mt{transaction} \; \mt{page} \to \mt{url}
adamc@1085	1356 \end{array}$$
adamc@1085	1357
adamc@1085	1358 Page generation may be interrupted at any time with a request to redirect to a particular URL instead.
adamc@1085	1359 $$\begin{array}{l}
adamc@1085	1360 \mt{val} \; \mt{redirect} : \mt{t} ::: \mt{Type} \to \mt{url} \to \mt{transaction} \; \mt{t}
adamc@1085	1361 \end{array}$$
adamc@1085	1362
adamc@786	1363 It's possible for pages to return files of arbitrary MIME types. A file can be input from the user using this data type, along with the $\mt{upload}$ form tag.
adamc@786	1364 $$\begin{array}{l}
adamc@786	1365 \mt{type} \; \mt{file} \\
adamc@786	1366 \mt{val} \; \mt{fileName} : \mt{file} \to \mt{option} \; \mt{string} \\
adamc@786	1367 \mt{val} \; \mt{fileMimeType} : \mt{file} \to \mt{string} \\
adamc@786	1368 \mt{val} \; \mt{fileData} : \mt{file} \to \mt{blob}
adamc@786	1369 \end{array}$$
adamc@786	1370
adamc@786	1371 A blob can be extracted from a file and returned as the page result. There are bless and check functions for MIME types analogous to those for URLs.
adamc@786	1372 $$\begin{array}{l}
adamc@786	1373 \mt{type} \; \mt{mimeType} \\
adamc@786	1374 \mt{val} \; \mt{blessMime} : \mt{string} \to \mt{mimeType} \\
adamc@786	1375 \mt{val} \; \mt{checkMime} : \mt{string} \to \mt{option} \; \mt{mimeType} \\
adamc@786	1376 \mt{val} \; \mt{returnBlob} : \mt{t} ::: \mt{Type} \to \mt{blob} \to \mt{mimeType} \to \mt{transaction} \; \mt{t}
adamc@542	1377 \end{array}$$
adamc@542	1378
adamc@543	1379 \subsection{SQL}
adamc@543	1380
adamc@543	1381 The fundamental unit of interest in the embedding of SQL is tables, described by a type family and creatable only via the $\mt{table}$ declaration form.
adamc@543	1382 $$\begin{array}{l}
adamc@785	1383 \mt{con} \; \mt{sql\_table} :: \{\mt{Type}\} \to \{\{\mt{Unit}\}\} \to \mt{Type}
adamc@785	1384 \end{array}$$
adamc@785	1385 The first argument to this constructor gives the names and types of a table's columns, and the second argument gives the set of valid keys. Keys are the only subsets of the columns that may be referenced as foreign keys. Each key has a name.
adamc@785	1386
adamc@785	1387 We also have the simpler type family of SQL views, which have no keys.
adamc@785	1388 $$\begin{array}{l}
adamc@785	1389 \mt{con} \; \mt{sql\_view} :: \{\mt{Type}\} \to \mt{Type}
adamc@543	1390 \end{array}$$
adamc@543	1391
adamc@785	1392 A multi-parameter type class is used to allow tables and views to be used interchangeably, with a way of extracting the set of columns from each.
adamc@785	1393 $$\begin{array}{l}
adamc@785	1394 \mt{class} \; \mt{fieldsOf} :: \mt{Type} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@785	1395 \mt{val} \; \mt{fieldsOf\_table} : \mt{fs} ::: \{\mt{Type}\} \to \mt{keys} ::: \{\{\mt{Unit}\}\} \to \mt{fieldsOf} \; (\mt{sql\_table} \; \mt{fs} \; \mt{keys}) \; \mt{fs} \\
adamc@785	1396 \mt{val} \; \mt{fieldsOf\_view} : \mt{fs} ::: \{\mt{Type}\} \to \mt{fieldsOf} \; (\mt{sql\_view} \; \mt{fs}) \; \mt{fs}
adamc@785	1397 \end{array}$$
adamc@785	1398
adamc@785	1399 \subsubsection{Table Constraints}
adamc@785	1400
adamc@785	1401 Tables may be declared with constraints, such that database modifications that violate the constraints are blocked. A table may have at most one \texttt{PRIMARY KEY} constraint, which gives the subset of columns that will most often be used to look up individual rows in the table.
adamc@785	1402
adamc@785	1403 $$\begin{array}{l}
adamc@785	1404 \mt{con} \; \mt{primary\_key} :: \{\mt{Type}\} \to \{\{\mt{Unit}\}\} \to \mt{Type} \\
adamc@785	1405 \mt{val} \; \mt{no\_primary\_key} : \mt{fs} ::: \{\mt{Type}\} \to \mt{primary\_key} \; \mt{fs} \; [] \\
adamc@785	1406 \mt{val} \; \mt{primary\_key} : \mt{rest} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \to \mt{key1} :: \mt{Name} \to \mt{keys} :: \{\mt{Type}\} \\
adamc@785	1407 \hspace{.1in} \to [[\mt{key1}] \sim \mt{keys}] \Rightarrow [[\mt{key1} = \mt{t}] \rc \mt{keys} \sim \mt{rest}] \\
adamc@785	1408 \hspace{.1in} \Rightarrow \$([\mt{key1} = \mt{sql\_injectable\_prim} \; \mt{t}] \rc \mt{map} \; \mt{sql\_injectable\_prim} \; \mt{keys}) \\
adamc@785	1409 \hspace{.1in} \to \mt{primary\_key} \; ([\mt{key1} = \mt{t}] \rc \mt{keys} \rc \mt{rest}) \; [\mt{Pkey} = [\mt{key1}] \rc \mt{map} \; (\lambda \_ \Rightarrow ()) \; \mt{keys}]
adamc@785	1410 \end{array}$$
adamc@785	1411 The type class $\mt{sql\_injectable\_prim}$ characterizes which types are allowed in SQL and are not $\mt{option}$ types. In SQL, a \texttt{PRIMARY KEY} constraint enforces after-the-fact that a column may not contain \texttt{NULL}s, but Ur/Web forces that information to be included in table types from the beginning. Thus, the only effect of this kind of constraint in Ur/Web is to enforce uniqueness of the given key within the table.
adamc@785	1412
adamc@785	1413 A type family stands for sets of named constraints of the remaining varieties.
adamc@785	1414 $$\begin{array}{l}
adamc@785	1415 \mt{con} \; \mt{sql\_constraints} :: \{\mt{Type}\} \to \{\{\mt{Unit}\}\} \to \mt{Type}
adamc@785	1416 \end{array}$$
adamc@785	1417 The first argument gives the column types of the table being constrained, and the second argument maps constraint names to the keys that they define. Constraints that don't define keys are mapped to ``empty keys.''
adamc@785	1418
adamc@785	1419 There is a type family of individual, unnamed constraints.
adamc@785	1420 $$\begin{array}{l}
adamc@785	1421 \mt{con} \; \mt{sql\_constraint} :: \{\mt{Type}\} \to \{\mt{Unit}\} \to \mt{Type}
adamc@785	1422 \end{array}$$
adamc@785	1423 The first argument is the same as above, and the second argument gives the key columns for just this constraint.
adamc@785	1424
adamc@785	1425 We have operations for assembling constraints into constraint sets.
adamc@785	1426 $$\begin{array}{l}
adamc@785	1427 \mt{val} \; \mt{no\_constraint} : \mt{fs} ::: \{\mt{Type}\} \to \mt{sql\_constraints} \; \mt{fs} \; [] \\
adamc@785	1428 \mt{val} \; \mt{one\_constraint} : \mt{fs} ::: \{\mt{Type}\} \to \mt{unique} ::: \{\mt{Unit}\} \to \mt{name} :: \mt{Name} \\
adamc@785	1429 \hspace{.1in} \to \mt{sql\_constraint} \; \mt{fs} \; \mt{unique} \to \mt{sql\_constraints} \; \mt{fs} \; [\mt{name} = \mt{unique}] \\
adamc@785	1430 \mt{val} \; \mt{join\_constraints} : \mt{fs} ::: \{\mt{Type}\} \to \mt{uniques1} ::: \{\{\mt{Unit}\}\} \to \mt{uniques2} ::: \{\{\mt{Unit}\}\} \to [\mt{uniques1} \sim \mt{uniques2}] \\
adamc@785	1431 \hspace{.1in} \Rightarrow \mt{sql\_constraints} \; \mt{fs} \; \mt{uniques1} \to \mt{sql\_constraints} \; \mt{fs} \; \mt{uniques2} \to \mt{sql\_constraints} \; \mt{fs} \; (\mt{uniques1} \rc \mt{uniques2})
adamc@785	1432 \end{array}$$
adamc@785	1433
adamc@785	1434 A \texttt{UNIQUE} constraint forces a set of columns to be a key, which means that no combination of column values may occur more than once in the table. The $\mt{unique1}$ and $\mt{unique}$ arguments are separated out only to ensure that empty \texttt{UNIQUE} constraints are rejected.
adamc@785	1435 $$\begin{array}{l}
adamc@785	1436 \mt{val} \; \mt{unique} : \mt{rest} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \to \mt{unique1} :: \mt{Name} \to \mt{unique} :: \{\mt{Type}\} \\
adamc@785	1437 \hspace{.1in} \to [[\mt{unique1}] \sim \mt{unique}] \Rightarrow [[\mt{unique1} = \mt{t}] \rc \mt{unique} \sim \mt{rest}] \\
adamc@785	1438 \hspace{.1in} \Rightarrow \mt{sql\_constraint} \; ([\mt{unique1} = \mt{t}] \rc \mt{unique} \rc \mt{rest}) \; ([\mt{unique1}] \rc \mt{map} \; (\lambda \_ \Rightarrow ()) \; \mt{unique})
adamc@785	1439 \end{array}$$
adamc@785	1440
adamc@785	1441 A \texttt{FOREIGN KEY} constraint connects a set of local columns to a local or remote key, enforcing that the local columns always reference an existent row of the foreign key's table. A local column of type $\mt{t}$ may be linked to a foreign column of type $\mt{option} \; \mt{t}$, and vice versa. We formalize that notion with a type class.
adamc@785	1442 $$\begin{array}{l}
adamc@785	1443 \mt{class} \; \mt{linkable} :: \mt{Type} \to \mt{Type} \to \mt{Type} \\
adamc@785	1444 \mt{val} \; \mt{linkable\_same} : \mt{t} ::: \mt{Type} \to \mt{linkable} \; \mt{t} \; \mt{t} \\
adamc@785	1445 \mt{val} \; \mt{linkable\_from\_nullable} : \mt{t} ::: \mt{Type} \to \mt{linkable} \; (\mt{option} \; \mt{t}) \; \mt{t} \\
adamc@785	1446 \mt{val} \; \mt{linkable\_to\_nullable} : \mt{t} ::: \mt{Type} \to \mt{linkable} \; \mt{t} \; (\mt{option} \; \mt{t})
adamc@785	1447 \end{array}$$
adamc@785	1448
adamc@785	1449 The $\mt{matching}$ type family uses $\mt{linkable}$ to define when two keys match up type-wise.
adamc@785	1450 $$\begin{array}{l}
adamc@785	1451 \mt{con} \; \mt{matching} :: \{\mt{Type}\} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@785	1452 \mt{val} \; \mt{mat\_nil} : \mt{matching} \; [] \; [] \\
adamc@785	1453 \mt{val} \; \mt{mat\_cons} : \mt{t1} ::: \mt{Type} \to \mt{rest1} ::: \{\mt{Type}\} \to \mt{t2} ::: \mt{Type} \to \mt{rest2} ::: \{\mt{Type}\} \to \mt{nm1} :: \mt{Name} \to \mt{nm2} :: \mt{Name} \\
adamc@785	1454 \hspace{.1in} \to [[\mt{nm1}] \sim \mt{rest1}] \Rightarrow [[\mt{nm2}] \sim \mt{rest2}] \Rightarrow \mt{linkable} \; \mt{t1} \; \mt{t2} \to \mt{matching} \; \mt{rest1} \; \mt{rest2} \\
adamc@785	1455 \hspace{.1in} \to \mt{matching} \; ([\mt{nm1} = \mt{t1}] \rc \mt{rest1}) \; ([\mt{nm2} = \mt{t2}] \rc \mt{rest2})
adamc@785	1456 \end{array}$$
adamc@785	1457
adamc@785	1458 SQL provides a number of different propagation modes for \texttt{FOREIGN KEY} constraints, governing what happens when a row containing a still-referenced foreign key value is deleted or modified to have a different key value. The argument of a propagation mode's type gives the local key type.
adamc@785	1459 $$\begin{array}{l}
adamc@785	1460 \mt{con} \; \mt{propagation\_mode} :: \{\mt{Type}\} \to \mt{Type} \\
adamc@785	1461 \mt{val} \; \mt{restrict} : \mt{fs} ::: \{\mt{Type}\} \to \mt{propagation\_mode} \; \mt{fs} \\
adamc@785	1462 \mt{val} \; \mt{cascade} : \mt{fs} ::: \{\mt{Type}\} \to \mt{propagation\_mode} \; \mt{fs} \\
adamc@785	1463 \mt{val} \; \mt{no\_action} : \mt{fs} ::: \{\mt{Type}\} \to \mt{propagation\_mode} \; \mt{fs} \\
adamc@785	1464 \mt{val} \; \mt{set\_null} : \mt{fs} ::: \{\mt{Type}\} \to \mt{propagation\_mode} \; (\mt{map} \; \mt{option} \; \mt{fs})
adamc@785	1465 \end{array}$$
adamc@785	1466
adamc@785	1467 Finally, we put these ingredient together to define the \texttt{FOREIGN KEY} constraint function.
adamc@785	1468 $$\begin{array}{l}
adamc@785	1469 \mt{val} \; \mt{foreign\_key} : \mt{mine1} ::: \mt{Name} \to \mt{t} ::: \mt{Type} \to \mt{mine} ::: \{\mt{Type}\} \to \mt{munused} ::: \{\mt{Type}\} \to \mt{foreign} ::: \{\mt{Type}\} \\
adamc@785	1470 \hspace{.1in} \to \mt{funused} ::: \{\mt{Type}\} \to \mt{nm} ::: \mt{Name} \to \mt{uniques} ::: \{\{\mt{Unit}\}\} \\
adamc@785	1471 \hspace{.1in} \to [[\mt{mine1}] \sim \mt{mine}] \Rightarrow [[\mt{mine1} = \mt{t}] \rc \mt{mine} \sim \mt{munused}] \Rightarrow [\mt{foreign} \sim \mt{funused}] \Rightarrow [[\mt{nm}] \sim \mt{uniques}] \\
adamc@785	1472 \hspace{.1in} \Rightarrow \mt{matching} \; ([\mt{mine1} = \mt{t}] \rc \mt{mine}) \; \mt{foreign} \\
adamc@785	1473 \hspace{.1in} \to \mt{sql\_table} \; (\mt{foreign} \rc \mt{funused}) \; ([\mt{nm} = \mt{map} \; (\lambda \_ \Rightarrow ()) \; \mt{foreign}] \rc \mt{uniques}) \\
adamc@785	1474 \hspace{.1in} \to \{\mt{OnDelete} : \mt{propagation\_mode} \; ([\mt{mine1} = \mt{t}] \rc \mt{mine}), \\
adamc@785	1475 \hspace{.2in} \mt{OnUpdate} : \mt{propagation\_mode} \; ([\mt{mine1} = \mt{t}] \rc \mt{mine})\} \\
adamc@785	1476 \hspace{.1in} \to \mt{sql\_constraint} \; ([\mt{mine1} = \mt{t}] \rc \mt{mine} \rc \mt{munused}) \; []
adamc@785	1477 \end{array}$$
adamc@785	1478
adamc@785	1479 The last kind of constraint is a \texttt{CHECK} constraint, which attaches a boolean invariant over a row's contents. It is defined using the $\mt{sql\_exp}$ type family, which we discuss in more detail below.
adamc@785	1480 $$\begin{array}{l}
adamc@785	1481 \mt{val} \; \mt{check} : \mt{fs} ::: \{\mt{Type}\} \to \mt{sql\_exp} \; [] \; [] \; \mt{fs} \; \mt{bool} \to \mt{sql\_constraint} \; \mt{fs} \; []
adamc@785	1482 \end{array}$$
adamc@785	1483
adamc@785	1484 Section \ref{tables} shows the expanded syntax of the $\mt{table}$ declaration and signature item that includes constraints. There is no other way to use constraints with SQL in Ur/Web.
adamc@785	1485
adamc@784	1486
adamc@543	1487 \subsubsection{Queries}
adamc@543	1488
adamc@543	1489 A final query is constructed via the $\mt{sql\_query}$ function. Constructor arguments respectively specify the table fields we select (as records mapping tables to the subsets of their fields that we choose) and the (always named) extra expressions that we select.
adamc@543	1490 $$\begin{array}{l}
adamc@543	1491 \mt{con} \; \mt{sql\_query} :: \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@543	1492 \mt{val} \; \mt{sql\_query} : \mt{tables} ::: \{\{\mt{Type}\}\} \\
adamc@543	1493 \hspace{.1in} \to \mt{selectedFields} ::: \{\{\mt{Type}\}\} \\
adamc@543	1494 \hspace{.1in} \to \mt{selectedExps} ::: \{\mt{Type}\} \\
adamc@543	1495 \hspace{.1in} \to \{\mt{Rows} : \mt{sql\_query1} \; \mt{tables} \; \mt{selectedFields} \; \mt{selectedExps}, \\
adamc@543	1496 \hspace{.2in} \mt{OrderBy} : \mt{sql\_order\_by} \; \mt{tables} \; \mt{selectedExps}, \\
adamc@543	1497 \hspace{.2in} \mt{Limit} : \mt{sql\_limit}, \\
adamc@543	1498 \hspace{.2in} \mt{Offset} : \mt{sql\_offset}\} \\
adamc@543	1499 \hspace{.1in} \to \mt{sql\_query} \; \mt{selectedFields} \; \mt{selectedExps}
adamc@543	1500 \end{array}$$
adamc@543	1501
adamc@545	1502 Queries are used by folding over their results inside transactions.
adamc@545	1503 $$\begin{array}{l}
adamc@545	1504 \mt{val} \; \mt{query} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \lambda [\mt{tables} \sim \mt{exps}] \Rightarrow \mt{state} ::: \mt{Type} \to \mt{sql\_query} \; \mt{tables} \; \mt{exps} \\
adamc@658	1505 \hspace{.1in} \to (\$(\mt{exps} \rc \mt{map} \; (\lambda \mt{fields} :: \{\mt{Type}\} \Rightarrow \$\mt{fields}) \; \mt{tables}) \\
adamc@545	1506 \hspace{.2in} \to \mt{state} \to \mt{transaction} \; \mt{state}) \\
adamc@545	1507 \hspace{.1in} \to \mt{state} \to \mt{transaction} \; \mt{state}
adamc@545	1508 \end{array}$$
adamc@545	1509
adamc@543	1510 Most of the complexity of the query encoding is in the type $\mt{sql\_query1}$, which includes simple queries and derived queries based on relational operators. Constructor arguments respectively specify the tables we select from, the subset of fields that we keep from each table for the result rows, and the extra expressions that we select.
adamc@543	1511 $$\begin{array}{l}
adamc@543	1512 \mt{con} \; \mt{sql\_query1} :: \{\{\mt{Type}\}\} \to \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@543	1513 \\
adamc@543	1514 \mt{type} \; \mt{sql\_relop} \\
adamc@543	1515 \mt{val} \; \mt{sql\_union} : \mt{sql\_relop} \\
adamc@543	1516 \mt{val} \; \mt{sql\_intersect} : \mt{sql\_relop} \\
adamc@543	1517 \mt{val} \; \mt{sql\_except} : \mt{sql\_relop} \\
adamc@543	1518 \mt{val} \; \mt{sql\_relop} : \mt{tables1} ::: \{\{\mt{Type}\}\} \\
adamc@543	1519 \hspace{.1in} \to \mt{tables2} ::: \{\{\mt{Type}\}\} \\
adamc@543	1520 \hspace{.1in} \to \mt{selectedFields} ::: \{\{\mt{Type}\}\} \\
adamc@543	1521 \hspace{.1in} \to \mt{selectedExps} ::: \{\mt{Type}\} \\
adamc@543	1522 \hspace{.1in} \to \mt{sql\_relop} \\
adamc@543	1523 \hspace{.1in} \to \mt{sql\_query1} \; \mt{tables1} \; \mt{selectedFields} \; \mt{selectedExps} \\
adamc@543	1524 \hspace{.1in} \to \mt{sql\_query1} \; \mt{tables2} \; \mt{selectedFields} \; \mt{selectedExps} \\
adamc@543	1525 \hspace{.1in} \to \mt{sql\_query1} \; \mt{selectedFields} \; \mt{selectedFields} \; \mt{selectedExps}
adamc@543	1526 \end{array}$$
adamc@543	1527
adamc@543	1528 $$\begin{array}{l}
adamc@543	1529 \mt{val} \; \mt{sql\_query1} : \mt{tables} ::: \{\{\mt{Type}\}\} \\
adamc@543	1530 \hspace{.1in} \to \mt{grouped} ::: \{\{\mt{Type}\}\} \\
adamc@543	1531 \hspace{.1in} \to \mt{selectedFields} ::: \{\{\mt{Type}\}\} \\
adamc@543	1532 \hspace{.1in} \to \mt{selectedExps} ::: \{\mt{Type}\} \\
adamc@1085	1533 \hspace{.1in} \to \mt{empties} :: \{\mt{Unit}\} \\
adamc@1085	1534 \hspace{.1in} \to [\mt{empties} \sim \mt{selectedFields}] \\
adamc@1085	1535 \hspace{.1in} \Rightarrow \{\mt{Distinct} : \mt{bool}, \\
adamc@994	1536 \hspace{.2in} \mt{From} : \mt{sql\_from\_items} \; \mt{tables}, \\
adamc@543	1537 \hspace{.2in} \mt{Where} : \mt{sql\_exp} \; \mt{tables} \; [] \; [] \; \mt{bool}, \\
adamc@543	1538 \hspace{.2in} \mt{GroupBy} : \mt{sql\_subset} \; \mt{tables} \; \mt{grouped}, \\
adamc@543	1539 \hspace{.2in} \mt{Having} : \mt{sql\_exp} \; \mt{grouped} \; \mt{tables} \; [] \; \mt{bool}, \\
adamc@1085	1540 \hspace{.2in} \mt{SelectFields} : \mt{sql\_subset} \; \mt{grouped} \; (\mt{map} \; (\lambda \_ \Rightarrow []) \; \mt{empties} \rc \mt{selectedFields}), \\
adamc@658	1541 \hspace{.2in} \mt {SelectExps} : \$(\mt{map} \; (\mt{sql\_exp} \; \mt{grouped} \; \mt{tables} \; []) \; \mt{selectedExps}) \} \\
adamc@543	1542 \hspace{.1in} \to \mt{sql\_query1} \; \mt{tables} \; \mt{selectedFields} \; \mt{selectedExps}
adamc@543	1543 \end{array}$$
adamc@543	1544
adamc@543	1545 To encode projection of subsets of fields in $\mt{SELECT}$ clauses, and to encode $\mt{GROUP} \; \mt{BY}$ clauses, we rely on a type family $\mt{sql\_subset}$, capturing what it means for one record of table fields to be a subset of another. The main constructor $\mt{sql\_subset}$ ``proves subset facts'' by requiring a split of a record into kept and dropped parts. The extra constructor $\mt{sql\_subset\_all}$ is a convenience for keeping all fields of a record.
adamc@543	1546 $$\begin{array}{l}
adamc@543	1547 \mt{con} \; \mt{sql\_subset} :: \{\{\mt{Type}\}\} \to \{\{\mt{Type}\}\} \to \mt{Type} \\
adamc@543	1548 \mt{val} \; \mt{sql\_subset} : \mt{keep\_drop} :: \{(\{\mt{Type}\} \times \{\mt{Type}\})\} \\
adamc@543	1549 \hspace{.1in} \to \mt{sql\_subset} \\
adamc@658	1550 \hspace{.2in} (\mt{map} \; (\lambda \mt{fields} :: (\{\mt{Type}\} \times \{\mt{Type}\}) \Rightarrow \mt{fields}.1 \rc \mt{fields}.2)\; \mt{keep\_drop}) \\
adamc@658	1551 \hspace{.2in} (\mt{map} \; (\lambda \mt{fields} :: (\{\mt{Type}\} \times \{\mt{Type}\}) \Rightarrow \mt{fields}.1) \; \mt{keep\_drop}) \\
adamc@543	1552 \mt{val} \; \mt{sql\_subset\_all} : \mt{tables} :: \{\{\mt{Type}\}\} \to \mt{sql\_subset} \; \mt{tables} \; \mt{tables}
adamc@543	1553 \end{array}$$
adamc@543	1554
adamc@560	1555 SQL expressions are used in several places, including $\mt{SELECT}$, $\mt{WHERE}$, $\mt{HAVING}$, and $\mt{ORDER} \; \mt{BY}$ clauses. They reify a fragment of the standard SQL expression language, while making it possible to inject ``native'' Ur values in some places. The arguments to the $\mt{sql\_exp}$ type family respectively give the unrestricted-availability table fields, the table fields that may only be used in arguments to aggregate functions, the available selected expressions, and the type of the expression.
adamc@543	1556 $$\begin{array}{l}
adamc@543	1557 \mt{con} \; \mt{sql\_exp} :: \{\{\mt{Type}\}\} \to \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \to \mt{Type}
adamc@543	1558 \end{array}$$
adamc@543	1559
adamc@543	1560 Any field in scope may be converted to an expression.
adamc@543	1561 $$\begin{array}{l}
adamc@543	1562 \mt{val} \; \mt{sql\_field} : \mt{otherTabs} ::: \{\{\mt{Type}\}\} \to \mt{otherFields} ::: \{\mt{Type}\} \\
adamc@543	1563 \hspace{.1in} \to \mt{fieldType} ::: \mt{Type} \to \mt{agg} ::: \{\{\mt{Type}\}\} \\
adamc@543	1564 \hspace{.1in} \to \mt{exps} ::: \{\mt{Type}\} \\
adamc@543	1565 \hspace{.1in} \to \mt{tab} :: \mt{Name} \to \mt{field} :: \mt{Name} \\
adamc@543	1566 \hspace{.1in} \to \mt{sql\_exp} \; ([\mt{tab} = [\mt{field} = \mt{fieldType}] \rc \mt{otherFields}] \rc \mt{otherTabs}) \; \mt{agg} \; \mt{exps} \; \mt{fieldType}
adamc@543	1567 \end{array}$$
adamc@543	1568
adamc@544	1569 There is an analogous function for referencing named expressions.
adamc@544	1570 $$\begin{array}{l}
adamc@544	1571 \mt{val} \; \mt{sql\_exp} : \mt{tabs} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{t} ::: \mt{Type} \to \mt{rest} ::: \{\mt{Type}\} \to \mt{nm} :: \mt{Name} \\
adamc@544	1572 \hspace{.1in} \to \mt{sql\_exp} \; \mt{tabs} \; \mt{agg} \; ([\mt{nm} = \mt{t}] \rc \mt{rest}) \; \mt{t}
adamc@544	1573 \end{array}$$
adamc@544	1574
adamc@544	1575 Ur values of appropriate types may be injected into SQL expressions.
adamc@544	1576 $$\begin{array}{l}
adamc@786	1577 \mt{class} \; \mt{sql\_injectable\_prim} \\
adamc@786	1578 \mt{val} \; \mt{sql\_bool} : \mt{sql\_injectable\_prim} \; \mt{bool} \\
adamc@786	1579 \mt{val} \; \mt{sql\_int} : \mt{sql\_injectable\_prim} \; \mt{int} \\
adamc@786	1580 \mt{val} \; \mt{sql\_float} : \mt{sql\_injectable\_prim} \; \mt{float} \\
adamc@786	1581 \mt{val} \; \mt{sql\_string} : \mt{sql\_injectable\_prim} \; \mt{string} \\
adamc@786	1582 \mt{val} \; \mt{sql\_time} : \mt{sql\_injectable\_prim} \; \mt{time} \\
adamc@786	1583 \mt{val} \; \mt{sql\_blob} : \mt{sql\_injectable\_prim} \; \mt{blob} \\
adamc@786	1584 \mt{val} \; \mt{sql\_channel} : \mt{t} ::: \mt{Type} \to \mt{sql\_injectable\_prim} \; (\mt{channel} \; \mt{t}) \\
adamc@786	1585 \mt{val} \; \mt{sql\_client} : \mt{sql\_injectable\_prim} \; \mt{client} \\
adamc@786	1586 \\
adamc@544	1587 \mt{class} \; \mt{sql\_injectable} \\
adamc@786	1588 \mt{val} \; \mt{sql\_prim} : \mt{t} ::: \mt{Type} \to \mt{sql\_injectable\_prim} \; \mt{t} \to \mt{sql\_injectable} \; \mt{t} \\
adamc@786	1589 \mt{val} \; \mt{sql\_option\_prim} : \mt{t} ::: \mt{Type} \to \mt{sql\_injectable\_prim} \; \mt{t} \to \mt{sql\_injectable} \; (\mt{option} \; \mt{t}) \\
adamc@786	1590 \\
adamc@544	1591 \mt{val} \; \mt{sql\_inject} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \to \mt{sql\_injectable} \; \mt{t} \\
adamc@544	1592 \hspace{.1in} \to \mt{t} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{t}
adamc@544	1593 \end{array}$$
adamc@544	1594
adamc@1123	1595 Additionally, most function-free types may be injected safely, via the $\mt{serialized}$ type family.
adamc@1123	1596 $$\begin{array}{l}
adamc@1123	1597 \mt{con} \; \mt{serialized} :: \mt{Type} \to \mt{Type} \\
adamc@1123	1598 \mt{val} \; \mt{serialize} : \mt{t} ::: \mt{Type} \to \mt{t} \to \mt{serialized} \; \mt{t} \\
adamc@1123	1599 \mt{val} \; \mt{deserialize} : \mt{t} ::: \mt{Type} \to \mt{serialized} \; \mt{t} \to \mt{t} \\
adamc@1123	1600 \mt{val} \; \mt{sql\_serialized} : \mt{t} ::: \mt{Type} \to \mt{sql\_injectable\_prim} \; (\mt{serialized} \; \mt{t})
adamc@1123	1601 \end{array}$$
adamc@1123	1602
adamc@544	1603 We have the SQL nullness test, which is necessary because of the strange SQL semantics of equality in the presence of null values.
adamc@544	1604 $$\begin{array}{l}
adamc@544	1605 \mt{val} \; \mt{sql\_is\_null} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544	1606 \hspace{.1in} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; (\mt{option} \; \mt{t}) \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{bool}
adamc@544	1607 \end{array}$$
adamc@544	1608
adamc@559	1609 We have generic nullary, unary, and binary operators.
adamc@544	1610 $$\begin{array}{l}
adamc@544	1611 \mt{con} \; \mt{sql\_nfunc} :: \mt{Type} \to \mt{Type} \\
adamc@544	1612 \mt{val} \; \mt{sql\_current\_timestamp} : \mt{sql\_nfunc} \; \mt{time} \\
adamc@544	1613 \mt{val} \; \mt{sql\_nfunc} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544	1614 \hspace{.1in} \to \mt{sql\_nfunc} \; \mt{t} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{t} \\\end{array}$$
adamc@544	1615
adamc@544	1616 $$\begin{array}{l}
adamc@544	1617 \mt{con} \; \mt{sql\_unary} :: \mt{Type} \to \mt{Type} \to \mt{Type} \\
adamc@544	1618 \mt{val} \; \mt{sql\_not} : \mt{sql\_unary} \; \mt{bool} \; \mt{bool} \\
adamc@544	1619 \mt{val} \; \mt{sql\_unary} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{arg} ::: \mt{Type} \to \mt{res} ::: \mt{Type} \\
adamc@544	1620 \hspace{.1in} \to \mt{sql\_unary} \; \mt{arg} \; \mt{res} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{arg} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{res} \\
adamc@544	1621 \end{array}$$
adamc@544	1622
adamc@544	1623 $$\begin{array}{l}
adamc@544	1624 \mt{con} \; \mt{sql\_binary} :: \mt{Type} \to \mt{Type} \to \mt{Type} \to \mt{Type} \\
adamc@544	1625 \mt{val} \; \mt{sql\_and} : \mt{sql\_binary} \; \mt{bool} \; \mt{bool} \; \mt{bool} \\
adamc@544	1626 \mt{val} \; \mt{sql\_or} : \mt{sql\_binary} \; \mt{bool} \; \mt{bool} \; \mt{bool} \\
adamc@544	1627 \mt{val} \; \mt{sql\_binary} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{arg_1} ::: \mt{Type} \to \mt{arg_2} ::: \mt{Type} \to \mt{res} ::: \mt{Type} \\
adamc@544	1628 \hspace{.1in} \to \mt{sql\_binary} \; \mt{arg_1} \; \mt{arg_2} \; \mt{res} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{arg_1} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{arg_2} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{res}
adamc@544	1629 \end{array}$$
adamc@544	1630
adamc@544	1631 $$\begin{array}{l}
adamc@559	1632 \mt{class} \; \mt{sql\_arith} \\
adamc@559	1633 \mt{val} \; \mt{sql\_int\_arith} : \mt{sql\_arith} \; \mt{int} \\
adamc@559	1634 \mt{val} \; \mt{sql\_float\_arith} : \mt{sql\_arith} \; \mt{float} \\
adamc@559	1635 \mt{val} \; \mt{sql\_neg} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_unary} \; \mt{t} \; \mt{t} \\
adamc@559	1636 \mt{val} \; \mt{sql\_plus} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559	1637 \mt{val} \; \mt{sql\_minus} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559	1638 \mt{val} \; \mt{sql\_times} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559	1639 \mt{val} \; \mt{sql\_div} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559	1640 \mt{val} \; \mt{sql\_mod} : \mt{sql\_binary} \; \mt{int} \; \mt{int} \; \mt{int}
adamc@559	1641 \end{array}$$
adamc@544	1642
adamc@656	1643 Finally, we have aggregate functions. The $\mt{COUNT(\ast)}$ syntax is handled specially, since it takes no real argument. The other aggregate functions are placed into a general type family, using constructor classes to restrict usage to properly-typed arguments. The key aspect of the $\mt{sql\_aggregate}$ function's type is the shift of aggregate-function-only fields into unrestricted fields.
adamc@544	1644 $$\begin{array}{l}
adamc@544	1645 \mt{val} \; \mt{sql\_count} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{int}
adamc@544	1646 \end{array}$$
adamc@544	1647
adamc@544	1648 $$\begin{array}{l}
adamc@544	1649 \mt{con} \; \mt{sql\_aggregate} :: \mt{Type} \to \mt{Type} \\
adamc@544	1650 \mt{val} \; \mt{sql\_aggregate} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544	1651 \hspace{.1in} \to \mt{sql\_aggregate} \; \mt{t} \to \mt{sql\_exp} \; \mt{agg} \; \mt{agg} \; \mt{exps} \; \mt{t} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{t}
adamc@544	1652 \end{array}$$
adamc@544	1653
adamc@544	1654 $$\begin{array}{l}
adamc@544	1655 \mt{class} \; \mt{sql\_summable} \\
adamc@544	1656 \mt{val} \; \mt{sql\_summable\_int} : \mt{sql\_summable} \; \mt{int} \\
adamc@544	1657 \mt{val} \; \mt{sql\_summable\_float} : \mt{sql\_summable} \; \mt{float} \\
adamc@544	1658 \mt{val} \; \mt{sql\_avg} : \mt{t} ::: \mt{Type} \to \mt{sql\_summable} \; \mt{t} \to \mt{sql\_aggregate} \; \mt{t} \\
adamc@544	1659 \mt{val} \; \mt{sql\_sum} : \mt{t} ::: \mt{Type} \to \mt{sql\_summable} \mt{t} \to \mt{sql\_aggregate} \; \mt{t}
adamc@544	1660 \end{array}$$
adamc@544	1661
adamc@544	1662 $$\begin{array}{l}
adamc@544	1663 \mt{class} \; \mt{sql\_maxable} \\
adamc@544	1664 \mt{val} \; \mt{sql\_maxable\_int} : \mt{sql\_maxable} \; \mt{int} \\
adamc@544	1665 \mt{val} \; \mt{sql\_maxable\_float} : \mt{sql\_maxable} \; \mt{float} \\
adamc@544	1666 \mt{val} \; \mt{sql\_maxable\_string} : \mt{sql\_maxable} \; \mt{string} \\
adamc@544	1667 \mt{val} \; \mt{sql\_maxable\_time} : \mt{sql\_maxable} \; \mt{time} \\
adamc@544	1668 \mt{val} \; \mt{sql\_max} : \mt{t} ::: \mt{Type} \to \mt{sql\_maxable} \; \mt{t} \to \mt{sql\_aggregate} \; \mt{t} \\
adamc@544	1669 \mt{val} \; \mt{sql\_min} : \mt{t} ::: \mt{Type} \to \mt{sql\_maxable} \; \mt{t} \to \mt{sql\_aggregate} \; \mt{t}
adamc@544	1670 \end{array}$$
adamc@544	1671
adamc@786	1672 \texttt{FROM} clauses are specified using a type family.
adamc@786	1673 $$\begin{array}{l}
adamc@786	1674 \mt{con} \; \mt{sql\_from\_items} :: \{\{\mt{Type}\}\} \to \mt{Type} \\
adamc@786	1675 \mt{val} \; \mt{sql\_from\_table} : \mt{t} ::: \mt{Type} \to \mt{fs} ::: \{\mt{Type}\} \to \mt{fieldsOf} \; \mt{t} \; \mt{fs} \to \mt{name} :: \mt{Name} \to \mt{t} \to \mt{sql\_from\_items} \; [\mt{name} = \mt{fs}] \\
adamc@786	1676 \mt{val} \; \mt{sql\_from\_comma} : \mt{tabs1} ::: \{\{\mt{Type}\}\} \to \mt{tabs2} ::: \{\{\mt{Type}\}\} \to [\mt{tabs1} \sim \mt{tabs2}] \\
adamc@786	1677 \hspace{.1in} \Rightarrow \mt{sql\_from\_items} \; \mt{tabs1} \to \mt{sql\_from\_items} \; \mt{tabs2} \\
adamc@786	1678 \hspace{.1in} \to \mt{sql\_from\_items} \; (\mt{tabs1} \rc \mt{tabs2}) \\
adamc@786	1679 \mt{val} \; \mt{sql\_inner\_join} : \mt{tabs1} ::: \{\{\mt{Type}\}\} \to \mt{tabs2} ::: \{\{\mt{Type}\}\} \to [\mt{tabs1} \sim \mt{tabs2}] \\
adamc@786	1680 \hspace{.1in} \Rightarrow \mt{sql\_from\_items} \; \mt{tabs1} \to \mt{sql\_from\_items} \; \mt{tabs2} \\
adamc@786	1681 \hspace{.1in} \to \mt{sql\_exp} \; (\mt{tabs1} \rc \mt{tabs2}) \; [] \; [] \; \mt{bool} \\
adamc@786	1682 \hspace{.1in} \to \mt{sql\_from\_items} \; (\mt{tabs1} \rc \mt{tabs2})
adamc@786	1683 \end{array}$$
adamc@786	1684
adamc@786	1685 Besides these basic cases, outer joins are supported, which requires a type class for turning non-$\mt{option}$ columns into $\mt{option}$ columns.
adamc@786	1686 $$\begin{array}{l}
adamc@786	1687 \mt{class} \; \mt{nullify} :: \mt{Type} \to \mt{Type} \to \mt{Type} \\
adamc@786	1688 \mt{val} \; \mt{nullify\_option} : \mt{t} ::: \mt{Type} \to \mt{nullify} \; (\mt{option} \; \mt{t}) \; (\mt{option} \; \mt{t}) \\
adamc@786	1689 \mt{val} \; \mt{nullify\_prim} : \mt{t} ::: \mt{Type} \to \mt{sql\_injectable\_prim} \; \mt{t} \to \mt{nullify} \; \mt{t} \; (\mt{option} \; \mt{t})
adamc@786	1690 \end{array}$$
adamc@786	1691
adamc@786	1692 Left, right, and full outer joins can now be expressed using functions that accept records of $\mt{nullify}$ instances. Here, we give only the type for a left join as an example.
adamc@786	1693
adamc@786	1694 $$\begin{array}{l}
adamc@786	1695 \mt{val} \; \mt{sql\_left\_join} : \mt{tabs1} ::: \{\{\mt{Type}\}\} \to \mt{tabs2} ::: \{\{(\mt{Type} \times \mt{Type})\}\} \to [\mt{tabs1} \sim \mt{tabs2}] \\
adamc@786	1696 \hspace{.1in} \Rightarrow \$(\mt{map} \; (\lambda \mt{r} \Rightarrow \$(\mt{map} \; (\lambda \mt{p} :: (\mt{Type} \times \mt{Type}) \Rightarrow \mt{nullify} \; \mt{p}.1 \; \mt{p}.2) \; \mt{r})) \; \mt{tabs2}) \\
adamc@786	1697 \hspace{.1in} \to \mt{sql\_from\_items} \; \mt{tabs1} \to \mt{sql\_from\_items} \; (\mt{map} \; (\mt{map} \; (\lambda \mt{p} :: (\mt{Type} \times \mt{Type}) \Rightarrow \mt{p}.1)) \; \mt{tabs2}) \\
adamc@786	1698 \hspace{.1in} \to \mt{sql\_exp} \; (\mt{tabs1} \rc \mt{map} \; (\mt{map} \; (\lambda \mt{p} :: (\mt{Type} \times \mt{Type}) \Rightarrow \mt{p}.1)) \; \mt{tabs2}) \; [] \; [] \; \mt{bool} \\
adamc@786	1699 \hspace{.1in} \to \mt{sql\_from\_items} \; (\mt{tabs1} \rc \mt{map} \; (\mt{map} \; (\lambda \mt{p} :: (\mt{Type} \times \mt{Type}) \Rightarrow \mt{p}.2)) \; \mt{tabs2})
adamc@786	1700 \end{array}$$
adamc@786	1701
adamc@544	1702 We wrap up the definition of query syntax with the types used in representing $\mt{ORDER} \; \mt{BY}$, $\mt{LIMIT}$, and $\mt{OFFSET}$ clauses.
adamc@544	1703 $$\begin{array}{l}
adamc@544	1704 \mt{type} \; \mt{sql\_direction} \\
adamc@544	1705 \mt{val} \; \mt{sql\_asc} : \mt{sql\_direction} \\
adamc@544	1706 \mt{val} \; \mt{sql\_desc} : \mt{sql\_direction} \\
adamc@544	1707 \\
adamc@544	1708 \mt{con} \; \mt{sql\_order\_by} :: \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@544	1709 \mt{val} \; \mt{sql\_order\_by\_Nil} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{exps} :: \{\mt{Type}\} \to \mt{sql\_order\_by} \; \mt{tables} \; \mt{exps} \\
adamc@544	1710 \mt{val} \; \mt{sql\_order\_by\_Cons} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544	1711 \hspace{.1in} \to \mt{sql\_exp} \; \mt{tables} \; [] \; \mt{exps} \; \mt{t} \to \mt{sql\_direction} \to \mt{sql\_order\_by} \; \mt{tables} \; \mt{exps} \to \mt{sql\_order\_by} \; \mt{tables} \; \mt{exps} \\
adamc@544	1712 \\
adamc@544	1713 \mt{type} \; \mt{sql\_limit} \\
adamc@544	1714 \mt{val} \; \mt{sql\_no\_limit} : \mt{sql\_limit} \\
adamc@544	1715 \mt{val} \; \mt{sql\_limit} : \mt{int} \to \mt{sql\_limit} \\
adamc@544	1716 \\
adamc@544	1717 \mt{type} \; \mt{sql\_offset} \\
adamc@544	1718 \mt{val} \; \mt{sql\_no\_offset} : \mt{sql\_offset} \\
adamc@544	1719 \mt{val} \; \mt{sql\_offset} : \mt{int} \to \mt{sql\_offset}
adamc@544	1720 \end{array}$$
adamc@544	1721
adamc@545	1722
adamc@545	1723 \subsubsection{DML}
adamc@545	1724
adamc@545	1725 The Ur/Web library also includes an embedding of a fragment of SQL's DML, the Data Manipulation Language, for modifying database tables. Any piece of DML may be executed in a transaction.
adamc@545	1726
adamc@545	1727 $$\begin{array}{l}
adamc@545	1728 \mt{type} \; \mt{dml} \\
adamc@545	1729 \mt{val} \; \mt{dml} : \mt{dml} \to \mt{transaction} \; \mt{unit}
adamc@545	1730 \end{array}$$
adamc@545	1731
adamc@545	1732 Properly-typed records may be used to form $\mt{INSERT}$ commands.
adamc@545	1733 $$\begin{array}{l}
adamc@545	1734 \mt{val} \; \mt{insert} : \mt{fields} ::: \{\mt{Type}\} \to \mt{sql\_table} \; \mt{fields} \\
adamc@658	1735 \hspace{.1in} \to \$(\mt{map} \; (\mt{sql\_exp} \; [] \; [] \; []) \; \mt{fields}) \to \mt{dml}
adamc@545	1736 \end{array}$$
adamc@545	1737
adamc@545	1738 An $\mt{UPDATE}$ command is formed from a choice of which table fields to leave alone and which to change, along with an expression to use to compute the new value of each changed field and a $\mt{WHERE}$ clause.
adamc@545	1739 $$\begin{array}{l}
adamc@545	1740 \mt{val} \; \mt{update} : \mt{unchanged} ::: \{\mt{Type}\} \to \mt{changed} :: \{\mt{Type}\} \to \lambda [\mt{changed} \sim \mt{unchanged}] \\
adamc@658	1741 \hspace{.1in} \Rightarrow \$(\mt{map} \; (\mt{sql\_exp} \; [\mt{T} = \mt{changed} \rc \mt{unchanged}] \; [] \; []) \; \mt{changed}) \\
adamc@545	1742 \hspace{.1in} \to \mt{sql\_table} \; (\mt{changed} \rc \mt{unchanged}) \to \mt{sql\_exp} \; [\mt{T} = \mt{changed} \rc \mt{unchanged}] \; [] \; [] \; \mt{bool} \to \mt{dml}
adamc@545	1743 \end{array}$$
adamc@545	1744
adamc@545	1745 A $\mt{DELETE}$ command is formed from a table and a $\mt{WHERE}$ clause.
adamc@545	1746 $$\begin{array}{l}
adamc@545	1747 \mt{val} \; \mt{delete} : \mt{fields} ::: \{\mt{Type}\} \to \mt{sql\_table} \; \mt{fields} \to \mt{sql\_exp} \; [\mt{T} = \mt{fields}] \; [] \; [] \; \mt{bool} \to \mt{dml}
adamc@545	1748 \end{array}$$
adamc@545	1749
adamc@546	1750 \subsubsection{Sequences}
adamc@546	1751
adamc@546	1752 SQL sequences are counters with concurrency control, often used to assign unique IDs. Ur/Web supports them via a simple interface. The only way to create a sequence is with the $\mt{sequence}$ declaration form.
adamc@546	1753
adamc@546	1754 $$\begin{array}{l}
adamc@546	1755 \mt{type} \; \mt{sql\_sequence} \\
adamc@1085	1756 \mt{val} \; \mt{nextval} : \mt{sql\_sequence} \to \mt{transaction} \; \mt{int} \\
adamc@1085	1757 \mt{val} \; \mt{setval} : \mt{sql\_sequence} \to \mt{int} \to \mt{transaction} \; \mt{unit}
adamc@546	1758 \end{array}$$
adamc@546	1759
adamc@546	1760
adamc@547	1761 \subsection{XML}
adamc@547	1762
adamc@547	1763 Ur/Web's library contains an encoding of XML syntax and semantic constraints. We make no effort to follow the standards governing XML schemas. Rather, XML fragments are viewed more as values of ML datatypes, and we only track which tags are allowed inside which other tags.
adamc@547	1764
adamc@547	1765 The basic XML type family has arguments respectively indicating the \emph{context} of a fragment, the fields that the fragment expects to be bound on entry (and their types), and the fields that the fragment will bind (and their types). Contexts are a record-based ``poor man's subtyping'' encoding, with each possible set of valid tags corresponding to a different context record. The arguments dealing with field binding are only relevant to HTML forms.
adamc@547	1766 $$\begin{array}{l}
adamc@547	1767 \mt{con} \; \mt{xml} :: \{\mt{Unit}\} \to \{\mt{Type}\} \to \{\mt{Type}\} \to \mt{Type}
adamc@547	1768 \end{array}$$
adamc@547	1769
adamc@547	1770 We also have a type family of XML tags, indexed respectively by the record of optional attributes accepted by the tag, the context in which the tag may be placed, the context required of children of the tag, which form fields the tag uses, and which fields the tag defines.
adamc@547	1771 $$\begin{array}{l}
adamc@547	1772 \mt{con} \; \mt{tag} :: \{\mt{Type}\} \to \{\mt{Unit}\} \to \{\mt{Unit}\} \to \{\mt{Type}\} \to \{\mt{Type}\} \to \mt{Type}
adamc@547	1773 \end{array}$$
adamc@547	1774
adamc@547	1775 Literal text may be injected into XML as ``CDATA.''
adamc@547	1776 $$\begin{array}{l}
adamc@547	1777 \mt{val} \; \mt{cdata} : \mt{ctx} ::: \{\mt{Unit}\} \to \mt{use} ::: \{\mt{Type}\} \to \mt{string} \to \mt{xml} \; \mt{ctx} \; \mt{use} \; []
adamc@547	1778 \end{array}$$
adamc@547	1779
adamc@547	1780 There is a function for producing an XML tree with a particular tag at its root.
adamc@547	1781 $$\begin{array}{l}
adamc@547	1782 \mt{val} \; \mt{tag} : \mt{attrsGiven} ::: \{\mt{Type}\} \to \mt{attrsAbsent} ::: \{\mt{Type}\} \to \mt{ctxOuter} ::: \{\mt{Unit}\} \to \mt{ctxInner} ::: \{\mt{Unit}\} \\
adamc@547	1783 \hspace{.1in} \to \mt{useOuter} ::: \{\mt{Type}\} \to \mt{useInner} ::: \{\mt{Type}\} \to \mt{bindOuter} ::: \{\mt{Type}\} \to \mt{bindInner} ::: \{\mt{Type}\} \\
adamc@787	1784 \hspace{.1in} \to \lambda [\mt{attrsGiven} \sim \mt{attrsAbsent}] \; [\mt{useOuter} \sim \mt{useInner}] \; [\mt{bindOuter} \sim \mt{bindInner}] \\
adamc@787	1785 \hspace{.1in} \Rightarrow \mt{option} \; \mt{css\_class} \\
adamc@787	1786 \hspace{.1in} \to \$\mt{attrsGiven} \\
adamc@547	1787 \hspace{.1in} \to \mt{tag} \; (\mt{attrsGiven} \rc \mt{attrsAbsent}) \; \mt{ctxOuter} \; \mt{ctxInner} \; \mt{useOuter} \; \mt{bindOuter} \\
adamc@547	1788 \hspace{.1in} \to \mt{xml} \; \mt{ctxInner} \; \mt{useInner} \; \mt{bindInner} \to \mt{xml} \; \mt{ctxOuter} \; (\mt{useOuter} \rc \mt{useInner}) \; (\mt{bindOuter} \rc \mt{bindInner})
adamc@547	1789 \end{array}$$
adamc@787	1790 Note that any tag may be assigned a CSS class. This is the sole way of making use of the values produced by $\mt{style}$ declarations. Ur/Web itself doesn't deal with the syntax or semantics of style sheets; they can be linked via URLs with \texttt{link} tags. However, Ur/Web does make it easy to calculate upper bounds on usage of CSS classes through program analysis.
adamc@547	1791
adamc@547	1792 Two XML fragments may be concatenated.
adamc@547	1793 $$\begin{array}{l}
adamc@547	1794 \mt{val} \; \mt{join} : \mt{ctx} ::: \{\mt{Unit}\} \to \mt{use_1} ::: \{\mt{Type}\} \to \mt{bind_1} ::: \{\mt{Type}\} \to \mt{bind_2} ::: \{\mt{Type}\} \\
adamc@547	1795 \hspace{.1in} \to \lambda [\mt{use_1} \sim \mt{bind_1}] \; [\mt{bind_1} \sim \mt{bind_2}] \\
adamc@547	1796 \hspace{.1in} \Rightarrow \mt{xml} \; \mt{ctx} \; \mt{use_1} \; \mt{bind_1} \to \mt{xml} \; \mt{ctx} \; (\mt{use_1} \rc \mt{bind_1}) \; \mt{bind_2} \to \mt{xml} \; \mt{ctx} \; \mt{use_1} \; (\mt{bind_1} \rc \mt{bind_2})
adamc@547	1797 \end{array}$$
adamc@547	1798
adamc@547	1799 Finally, any XML fragment may be updated to ``claim'' to use more form fields than it does.
adamc@547	1800 $$\begin{array}{l}
adamc@547	1801 \mt{val} \; \mt{useMore} : \mt{ctx} ::: \{\mt{Unit}\} \to \mt{use_1} ::: \{\mt{Type}\} \to \mt{use_2} ::: \{\mt{Type}\} \to \mt{bind} ::: \{\mt{Type}\} \to \lambda [\mt{use_1} \sim \mt{use_2}] \\
adamc@547	1802 \hspace{.1in} \Rightarrow \mt{xml} \; \mt{ctx} \; \mt{use_1} \; \mt{bind} \to \mt{xml} \; \mt{ctx} \; (\mt{use_1} \rc \mt{use_2}) \; \mt{bind}
adamc@547	1803 \end{array}$$
adamc@547	1804
adamc@547	1805 We will not list here the different HTML tags and related functions from the standard library. They should be easy enough to understand from the code in \texttt{basis.urs}. The set of tags in the library is not yet claimed to be complete for HTML standards.
adamc@547	1806
adamc@547	1807 One last useful function is for aborting any page generation, returning some XML as an error message. This function takes the place of some uses of a general exception mechanism.
adamc@547	1808 $$\begin{array}{l}
adamc@547	1809 \mt{val} \; \mt{error} : \mt{t} ::: \mt{Type} \to \mt{xml} \; [\mt{Body}] \; [] \; [] \to \mt{t}
adamc@547	1810 \end{array}$$
adamc@547	1811
adamc@549	1812
adamc@701	1813 \subsection{Client-Side Programming}
adamc@659	1814
adamc@701	1815 Ur/Web supports running code on web browsers, via automatic compilation to JavaScript.
adamc@701	1816
adamc@701	1817 \subsubsection{The Basics}
adamc@701	1818
adamc@701	1819 Clients can open alert dialog boxes, in the usual annoying JavaScript way.
adamc@701	1820 $$\begin{array}{l}
adamc@701	1821 \mt{val} \; \mt{alert} : \mt{string} \to \mt{transaction} \; \mt{unit}
adamc@701	1822 \end{array}$$
adamc@701	1823
adamc@701	1824 Any transaction may be run in a new thread with the $\mt{spawn}$ function.
adamc@701	1825 $$\begin{array}{l}
adamc@701	1826 \mt{val} \; \mt{spawn} : \mt{transaction} \; \mt{unit} \to \mt{transaction} \; \mt{unit}
adamc@701	1827 \end{array}$$
adamc@701	1828
adamc@701	1829 The current thread can be paused for at least a specified number of milliseconds.
adamc@701	1830 $$\begin{array}{l}
adamc@701	1831 \mt{val} \; \mt{sleep} : \mt{int} \to \mt{transaction} \; \mt{unit}
adamc@701	1832 \end{array}$$
adamc@701	1833
adamc@787	1834 A few functions are available to registers callbacks for particular error events. Respectively, they are triggered on calls to $\mt{error}$, uncaught JavaScript exceptions, failure of remote procedure calls, the severance of the connection serving asynchronous messages, or the occurrence of some other error with that connection. If no handlers are registered for a kind of error, then occurrences of that error are ignored silently.
adamc@787	1835 $$\begin{array}{l}
adamc@787	1836 \mt{val} \; \mt{onError} : (\mt{xbody} \to \mt{transaction} \; \mt{unit}) \to \mt{transaction} \; \mt{unit} \\
adamc@787	1837 \mt{val} \; \mt{onFail} : (\mt{string} \to \mt{transaction} \; \mt{unit}) \to \mt{transaction} \; \mt{unit} \\
adamc@787	1838 \mt{val} \; \mt{onConnectFail} : \mt{transaction} \; \mt{unit} \to \mt{transaction} \; \mt{unit} \\
adamc@787	1839 \mt{val} \; \mt{onDisconnect} : \mt{transaction} \; \mt{unit} \to \mt{transaction} \; \mt{unit} \\
adamc@787	1840 \mt{val} \; \mt{onServerError} : (\mt{string} \to \mt{transaction} \; \mt{unit}) \to \mt{transaction} \; \mt{unit}
adamc@787	1841 \end{array}$$
adamc@787	1842
adamc@701	1843 \subsubsection{Functional-Reactive Page Generation}
adamc@701	1844
adamc@701	1845 Most approaches to ``AJAX''-style coding involve imperative manipulation of the DOM tree representing an HTML document's structure. Ur/Web follows the \emph{functional-reactive} approach instead. Programs may allocate mutable \emph{sources} of arbitrary types, and an HTML page is effectively a pure function over the latest values of the sources. The page is not mutated directly, but rather it changes automatically as the sources are mutated.
adamc@659	1846
adamc@659	1847 $$\begin{array}{l}
adamc@659	1848 \mt{con} \; \mt{source} :: \mt{Type} \to \mt{Type} \\
adamc@659	1849 \mt{val} \; \mt{source} : \mt{t} ::: \mt{Type} \to \mt{t} \to \mt{transaction} \; (\mt{source} \; \mt{t}) \\
adamc@659	1850 \mt{val} \; \mt{set} : \mt{t} ::: \mt{Type} \to \mt{source} \; \mt{t} \to \mt{t} \to \mt{transaction} \; \mt{unit} \\
adamc@659	1851 \mt{val} \; \mt{get} : \mt{t} ::: \mt{Type} \to \mt{source} \; \mt{t} \to \mt{transaction} \; \mt{t}
adamc@659	1852 \end{array}$$
adamc@659	1853
adamc@659	1854 Pure functions over sources are represented in a monad of \emph{signals}.
adamc@659	1855
adamc@659	1856 $$\begin{array}{l}
adamc@659	1857 \mt{con} \; \mt{signal} :: \mt{Type} \to \mt{Type} \\
adamc@659	1858 \mt{val} \; \mt{signal\_monad} : \mt{monad} \; \mt{signal} \\
adamc@659	1859 \mt{val} \; \mt{signal} : \mt{t} ::: \mt{Type} \to \mt{source} \; \mt{t} \to \mt{signal} \; \mt{t}
adamc@659	1860 \end{array}$$
adamc@659	1861
adamc@659	1862 A reactive portion of an HTML page is injected with a $\mt{dyn}$ tag, which has a signal-valued attribute $\mt{Signal}$.
adamc@659	1863
adamc@659	1864 $$\begin{array}{l}
adamc@701	1865 \mt{val} \; \mt{dyn} : \mt{use} ::: \{\mt{Type}\} \to \mt{bind} ::: \{\mt{Type}\} \to \mt{unit} \\
adamc@701	1866 \hspace{.1in} \to \mt{tag} \; [\mt{Signal} = \mt{signal} \; (\mt{xml} \; \mt{body} \; \mt{use} \; \mt{bind})] \; \mt{body} \; [] \; \mt{use} \; \mt{bind}
adamc@659	1867 \end{array}$$
adamc@659	1868
adamc@701	1869 Transactions can be run on the client by including them in attributes like the $\mt{Onclick}$ attribute of $\mt{button}$, and GUI widgets like $\mt{ctextbox}$ have $\mt{Source}$ attributes that can be used to connect them to sources, so that their values can be read by code running because of, e.g., an $\mt{Onclick}$ event.
adamc@701	1870
adamc@914	1871 \subsubsection{Remote Procedure Calls}
adamc@914	1872
adamc@914	1873 Any function call may be made a client-to-server ``remote procedure call'' if the function being called needs no features that are only available to client code. To make a function call an RPC, pass that function call as the argument to $\mt{Basis.rpc}$:
adamc@914	1874
adamc@914	1875 $$\begin{array}{l}
adamc@914	1876 \mt{val} \; \mt{rpc} : \mt{t} ::: \mt{Type} \to \mt{transaction} \; \mt{t} \to \mt{transaction} \; \mt{t}
adamc@914	1877 \end{array}$$
adamc@914	1878
adamc@701	1879 \subsubsection{Asynchronous Message-Passing}
adamc@701	1880
adamc@701	1881 To support asynchronous, ``server push'' delivery of messages to clients, any client that might need to receive an asynchronous message is assigned a unique ID. These IDs may be retrieved both on the client and on the server, during execution of code related to a client.
adamc@701	1882
adamc@701	1883 $$\begin{array}{l}
adamc@701	1884 \mt{type} \; \mt{client} \\
adamc@701	1885 \mt{val} \; \mt{self} : \mt{transaction} \; \mt{client}
adamc@701	1886 \end{array}$$
adamc@701	1887
adamc@701	1888 \emph{Channels} are the means of message-passing. Each channel is created in the context of a client and belongs to that client; no other client may receive the channel's messages. Each channel type includes the type of values that may be sent over the channel. Sending and receiving are asynchronous, in the sense that a client need not be ready to receive a message right away. Rather, sent messages may queue up, waiting to be processed.
adamc@701	1889
adamc@701	1890 $$\begin{array}{l}
adamc@701	1891 \mt{con} \; \mt{channel} :: \mt{Type} \to \mt{Type} \\
adamc@701	1892 \mt{val} \; \mt{channel} : \mt{t} ::: \mt{Type} \to \mt{transaction} \; (\mt{channel} \; \mt{t}) \\
adamc@701	1893 \mt{val} \; \mt{send} : \mt{t} ::: \mt{Type} \to \mt{channel} \; \mt{t} \to \mt{t} \to \mt{transaction} \; \mt{unit} \\
adamc@701	1894 \mt{val} \; \mt{recv} : \mt{t} ::: \mt{Type} \to \mt{channel} \; \mt{t} \to \mt{transaction} \; \mt{t}
adamc@701	1895 \end{array}$$
adamc@701	1896
adamc@701	1897 The $\mt{channel}$ and $\mt{send}$ operations may only be executed on the server, and $\mt{recv}$ may only be executed on a client. Neither clients nor channels may be passed as arguments from clients to server-side functions, so persistent channels can only be maintained by storing them in the database and looking them up using the current client ID or some application-specific value as a key.
adamc@701	1898
adamc@701	1899 Clients and channels live only as long as the web browser page views that they are associated with. When a user surfs away, his client and its channels will be garbage-collected, after that user is not heard from for the timeout period. Garbage collection deletes any database row that contains a client or channel directly. Any reference to one of these types inside an $\mt{option}$ is set to $\mt{None}$ instead. Both kinds of handling have the flavor of weak pointers, and that is a useful way to think about clients and channels in the database.
adamc@701	1900
adamc@659	1901
adamc@549	1902 \section{Ur/Web Syntax Extensions}
adamc@549	1903
adamc@549	1904 Ur/Web features some syntactic shorthands for building values using the functions from the last section. This section sketches the grammar of those extensions. We write spans of syntax inside brackets to indicate that they are optional.
adamc@549	1905
adamc@549	1906 \subsection{SQL}
adamc@549	1907
adamc@786	1908 \subsubsection{\label{tables}Table Declarations}
adamc@786	1909
adamc@788	1910 $\mt{table}$ declarations may include constraints, via these grammar rules.
adamc@788	1911 $$\begin{array}{rrcll}
adamc@788	1912 \textrm{Declarations} & d &::=& \mt{table} \; x : c \; [pk[,]] \; cts \\
adamc@788	1913 \textrm{Primary key constraints} & pk &::=& \mt{PRIMARY} \; \mt{KEY} \; K \\
adamc@788	1914 \textrm{Keys} & K &::=& f \mid (f, (f,)^+) \\
adamc@788	1915 \textrm{Constraint sets} & cts &::=& \mt{CONSTRAINT} f \; ct \mid cts, cts \mid \{\{e\}\} \\
adamc@788	1916 \textrm{Constraints} & ct &::=& \mt{UNIQUE} \; K \mid \mt{CHECK} \; E \\
adamc@788	1917 &&& \mid \mt{FOREIGN} \; \mt{KEY} \; K \; \mt{REFERENCES} \; F \; (K) \; [\mt{ON} \; \mt{DELETE} \; pr] \; [\mt{ON} \; \mt{UPDATE} \; pr] \\
adamc@788	1918 \textrm{Foreign tables} & F &::=& x \mid \{\{e\}\} \\
adamc@788	1919 \textrm{Propagation modes} & pr &::=& \mt{NO} \; \mt{ACTION} \mid \mt{RESTRICT} \mid \mt{CASCADE} \mid \mt{SET} \; \mt{NULL}
adamc@788	1920 \end{array}$$
adamc@788	1921
adamc@788	1922 A signature item $\mt{table} \; \mt{x} : \mt{c}$ is actually elaborated into two signature items: $\mt{con} \; \mt{x\_hidden\_constraints} :: \{\{\mt{Unit}\}\}$ and $\mt{val} \; \mt{x} : \mt{sql\_table} \; \mt{c} \; \mt{x\_hidden\_constraints}$. This is appropriate for common cases where client code doesn't care which keys a table has. It's also possible to include constraints after a $\mt{table}$ signature item, with the same syntax as for $\mt{table}$ declarations. This may look like dependent typing, but it's just a convenience. The constraints are type-checked to determine a constructor $u$ to include in $\mt{val} \; \mt{x} : \mt{sql\_table} \; \mt{c} \; (u \rc \mt{x\_hidden\_constraints})$, and then the expressions are thrown away. Nonetheless, it can be useful for documentation purposes to include table constraint details in signatures. Note that the automatic generation of $\mt{x\_hidden\_constraints}$ leads to a kind of free subtyping with respect to which constraints are defined.
adamc@788	1923
adamc@788	1924
adamc@549	1925 \subsubsection{Queries}
adamc@549	1926
adamc@550	1927 Queries $Q$ are added to the rules for expressions $e$.
adamc@550	1928
adamc@549	1929 $$\begin{array}{rrcll}
adamc@550	1930 \textrm{Queries} & Q &::=& (q \; [\mt{ORDER} \; \mt{BY} \; (E \; [o],)^+] \; [\mt{LIMIT} \; N] \; [\mt{OFFSET} \; N]) \\
adamc@1085	1931 \textrm{Pre-queries} & q &::=& \mt{SELECT} \; [\mt{DISTINCT}] \; P \; \mt{FROM} \; F,^+ \; [\mt{WHERE} \; E] \; [\mt{GROUP} \; \mt{BY} \; p,^+] \; [\mt{HAVING} \; E] \\
adamc@1085	1932 &&& \mid q \; R \; q \mid \{\{\{e\}\}\} \\
adamc@549	1933 \textrm{Relational operators} & R &::=& \mt{UNION} \mid \mt{INTERSECT} \mid \mt{EXCEPT}
adamc@549	1934 \end{array}$$
adamc@549	1935
adamc@549	1936 $$\begin{array}{rrcll}
adamc@549	1937 \textrm{Projections} & P &::=& \ast & \textrm{all columns} \\
adamc@549	1938 &&& p,^+ & \textrm{particular columns} \\
adamc@549	1939 \textrm{Pre-projections} & p &::=& t.f & \textrm{one column from a table} \\
adamc@558	1940 &&& t.\{\{c\}\} & \textrm{a record of columns from a table (of kind $\{\mt{Type}\}$)} \\
adamc@549	1941 \textrm{Table names} & t &::=& x & \textrm{constant table name (automatically capitalized)} \\
adamc@549	1942 &&& X & \textrm{constant table name} \\
adamc@549	1943 &&& \{\{c\}\} & \textrm{computed table name (of kind $\mt{Name}$)} \\
adamc@549	1944 \textrm{Column names} & f &::=& X & \textrm{constant column name} \\
adamc@549	1945 &&& \{c\} & \textrm{computed column name (of kind $\mt{Name}$)} \\
adamc@549	1946 \textrm{Tables} & T &::=& x & \textrm{table variable, named locally by its own capitalization} \\
adamc@549	1947 &&& x \; \mt{AS} \; t & \textrm{table variable, with local name} \\
adamc@549	1948 &&& \{\{e\}\} \; \mt{AS} \; t & \textrm{computed table expression, with local name} \\
adamc@1085	1949 \textrm{$\mt{FROM}$ items} & F &::=& T \mid \{\{e\}\} \mid F \; J \; \mt{JOIN} \; F \; \mt{ON} \; E \\
adamc@1085	1950 &&& \mid F \; \mt{CROSS} \; \mt{JOIN} \ F \\
adamc@1085	1951 \textrm{Joins} & J &::=& [\mt{INNER}] \\
adamc@1085	1952 &&& \mid [\mt{LEFT} \mid \mt{RIGHT} \mid \mt{FULL}] \; [\mt{OUTER}] \\
adamc@549	1953 \textrm{SQL expressions} & E &::=& p & \textrm{column references} \\
adamc@549	1954 &&& X & \textrm{named expression references} \\
adamc@549	1955 &&& \{\{e\}\} & \textrm{injected native Ur expressions} \\
adamc@549	1956 &&& \{e\} & \textrm{computed expressions, probably using $\mt{sql\_exp}$ directly} \\
adamc@549	1957 &&& \mt{TRUE} \mid \mt{FALSE} & \textrm{boolean constants} \\
adamc@549	1958 &&& \ell & \textrm{primitive type literals} \\
adamc@549	1959 &&& \mt{NULL} & \textrm{null value (injection of $\mt{None}$)} \\
adamc@549	1960 &&& E \; \mt{IS} \; \mt{NULL} & \textrm{nullness test} \\
adamc@549	1961 &&& n & \textrm{nullary operators} \\
adamc@549	1962 &&& u \; E & \textrm{unary operators} \\
adamc@549	1963 &&& E \; b \; E & \textrm{binary operators} \\
adamc@549	1964 &&& \mt{COUNT}(\ast) & \textrm{count number of rows} \\
adamc@549	1965 &&& a(E) & \textrm{other aggregate function} \\
adamc@549	1966 &&& (E) & \textrm{explicit precedence} \\
adamc@549	1967 \textrm{Nullary operators} & n &::=& \mt{CURRENT\_TIMESTAMP} \\
adamc@549	1968 \textrm{Unary operators} & u &::=& \mt{NOT} \\
adamc@549	1969 \textrm{Binary operators} & b &::=& \mt{AND} \mid \mt{OR} \mid \neq \mid < \mid \leq \mid > \mid \geq \\
adamc@549	1970 \textrm{Aggregate functions} & a &::=& \mt{AVG} \mid \mt{SUM} \mid \mt{MIN} \mid \mt{MAX} \\
adamc@550	1971 \textrm{Directions} & o &::=& \mt{ASC} \mid \mt{DESC} \\
adamc@549	1972 \textrm{SQL integer} & N &::=& n \mid \{e\} \\
adamc@549	1973 \end{array}$$
adamc@549	1974
adamc@1085	1975 Additionally, an SQL expression may be inserted into normal Ur code with the syntax $(\mt{SQL} \; E)$ or $(\mt{WHERE} \; E)$. Similar shorthands exist for other nonterminals, with the prefix $\mt{FROM}$ for $\mt{FROM}$ items and $\mt{SELECT1}$ for pre-queries.
adamc@549	1976
adamc@550	1977 \subsubsection{DML}
adamc@550	1978
adamc@550	1979 DML commands $D$ are added to the rules for expressions $e$.
adamc@550	1980
adamc@550	1981 $$\begin{array}{rrcll}
adamc@550	1982 \textrm{Commands} & D &::=& (\mt{INSERT} \; \mt{INTO} \; T^E \; (f,^+) \; \mt{VALUES} \; (E,^+)) \\
adamc@550	1983 &&& (\mt{UPDATE} \; T^E \; \mt{SET} \; (f = E,)^+ \; \mt{WHERE} \; E) \\
adamc@550	1984 &&& (\mt{DELETE} \; \mt{FROM} \; T^E \; \mt{WHERE} \; E) \\
adamc@550	1985 \textrm{Table expressions} & T^E &::=& x \mid \{\{e\}\}
adamc@550	1986 \end{array}$$
adamc@550	1987
adamc@550	1988 Inside $\mt{UPDATE}$ and $\mt{DELETE}$ commands, lone variables $X$ are interpreted as references to columns of the implicit table $\mt{T}$, rather than to named expressions.
adamc@549	1989
adamc@551	1990 \subsection{XML}
adamc@551	1991
adamc@551	1992 XML fragments $L$ are added to the rules for expressions $e$.
adamc@551	1993
adamc@551	1994 $$\begin{array}{rrcll}
adamc@551	1995 \textrm{XML fragments} & L &::=& \texttt{<xml/>} \mid \texttt{<xml>}l^*\texttt{</xml>} \\
adamc@551	1996 \textrm{XML pieces} & l &::=& \textrm{text} & \textrm{cdata} \\
adamc@551	1997 &&& \texttt{<}g\texttt{/>} & \textrm{tag with no children} \\
adamc@551	1998 &&& \texttt{<}g\texttt{>}l^*\texttt{</}x\texttt{>} & \textrm{tag with children} \\
adamc@559	1999 &&& \{e\} & \textrm{computed XML fragment} \\
adamc@559	2000 &&& \{[e]\} & \textrm{injection of an Ur expression, via the $\mt{Top}.\mt{txt}$ function} \\
adamc@551	2001 \textrm{Tag} & g &::=& h \; (x = v)^* \\
adamc@551	2002 \textrm{Tag head} & h &::=& x & \textrm{tag name} \\
adamc@551	2003 &&& h\{c\} & \textrm{constructor parameter} \\
adamc@551	2004 \textrm{Attribute value} & v &::=& \ell & \textrm{literal value} \\
adamc@551	2005 &&& \{e\} & \textrm{computed value} \\
adamc@551	2006 \end{array}$$
adamc@551	2007
adamc@552	2008
adamc@553	2009 \section{The Structure of Web Applications}
adamc@553	2010
adamc@1127	2011 A web application is built from a series of modules, with one module, the last one appearing in the \texttt{.urp} file, designated as the main module. The signature of the main module determines the URL entry points to the application. Such an entry point should have type $\mt{t1} \to \ldots \to \mt{tn} \to \mt{transaction} \; \mt{page}$, for any integer $n \geq 0$, where $\mt{page}$ is a type synonym for top-level HTML pages, defined in $\mt{Basis}$. If such a function is at the top level of main module $M$, with $n = 0$, it will be accessible at URI \texttt{/M/f}, and so on for more deeply-nested functions, as described in Section \ref{tag} below. Arguments to an entry-point function are deserialized from the part of the URI following \texttt{f}.
adamc@553	2012
adamc@553	2013 When the standalone web server receives a request for a known page, it calls the function for that page, ``running'' the resulting transaction to produce the page to return to the client. Pages link to other pages with the \texttt{link} attribute of the \texttt{a} HTML tag. A link has type $\mt{transaction} \; \mt{page}$, and the semantics of a link are that this transaction should be run to compute the result page, when the link is followed. Link targets are assigned URL names in the same way as top-level entry points.
adamc@553	2014
adamc@553	2015 HTML forms are handled in a similar way. The $\mt{action}$ attribute of a $\mt{submit}$ form tag takes a value of type $\$\mt{use} \to \mt{transaction} \; \mt{page}$, where $\mt{use}$ is a kind-$\{\mt{Type}\}$ record of the form fields used by this action handler. Action handlers are assigned URL patterns in the same way as above.
adamc@553	2016
adamc@558	2017 For both links and actions, direct arguments and local variables mentioned implicitly via closures are automatically included in serialized form in URLs, in the order in which they appear in the source code.
adamc@553	2018
adamc@660	2019 Ur/Web programs generally mix server- and client-side code in a fairly transparent way. The one important restriction is that mixed client-server code must encapsulate all server-side pieces within named functions. This is because execution of such pieces will be implemented by explicit calls to the remote web server, and it is useful to get the programmer's help in designing the interface to be used. For example, this makes it easier to allow a client running an old version of an application to continue interacting with a server that has been upgraded to a new version, if the programmer took care to keep the interfaces of all of the old remote calls the same. The functions implementing these services are assigned names in the same way as normal web entry points, by using module structure.
adamc@660	2020
adamc@789	2021 \medskip
adamc@789	2022
adamc@789	2023 The HTTP standard suggests that GET requests only be used in ways that generate no side effects. Side effecting operations should use POST requests instead. The Ur/Web compiler enforces this rule strictly, via a simple conservative program analysis. Any page that may have a side effect must be accessed through a form, all of which use POST requests. A page is judged to have a side effect if its code depends syntactically on any of the side-effecting, server-side FFI functions. Links, forms, and most client-side event handlers are not followed during this syntactic traversal, but \texttt{<body onload=\{...\}>} handlers \emph{are} examined, since they run right away and could just as well be considered parts of main page handlers.
adamc@789	2024
adamc@789	2025 Ur/Web includes a kind of automatic protection against cross site request forgery attacks. Whenever any page execution can have side effects and can also read at least one cookie value, all cookie values must be signed cryptographically, to ensure that the user has come to the current page by submitting a form on a real page generated by the proper server. Signing and signature checking are inserted automatically by the compiler. This prevents attacks like phishing schemes where users are directed to counterfeit pages with forms that submit to your application, where a user's cookies might be submitted without his knowledge, causing some undesired side effect.
adamc@789	2026
adamc@553	2027
adamc@897	2028 \section{The Foreign Function Interface}
adamc@897	2029
adamc@897	2030 It is possible to call your own C and JavaScript code from Ur/Web applications, via the foreign function interface (FFI). The starting point for a new binding is a \texttt{.urs} signature file that presents your external library as a single Ur/Web module (with no nested modules). Compilation conventions map the types and values that you use into C and/or JavaScript types and values.
adamc@897	2031
adamc@897	2032 It is most convenient to encapsulate an FFI binding with a new \texttt{.urp} file, which applications can include with the \texttt{library} directive in their own \texttt{.urp} files. A number of directives are likely to show up in the library's project file.
adamc@897	2033
adamc@897	2034 \begin{itemize}
adamc@897	2035 \item \texttt{clientOnly Module.ident} registers a value as being allowed only in client-side code.
adamc@897	2036 \item \texttt{clientToServer Module.ident} declares a type as OK to marshal between clients and servers. By default, abstract FFI types are not allowed to be marshalled, since your library might be maintaining invariants that the simple serialization code doesn't check.
adamc@897	2037 \item \texttt{effectful Module.ident} registers a function that can have side effects. It is important to remember to use this directive for each such function, or else the optimizer might change program semantics.
adamc@897	2038 \item \texttt{ffi FILE.urs} names the file giving your library's signature. You can include multiple such files in a single \texttt{.urp} file, and each file \texttt{mod.urp} defines an FFI module \texttt{Mod}.
adamc@1099	2039 \item \texttt{include FILE} requests inclusion of a C header file.
adamc@897	2040 \item \texttt{jsFunc Module.ident=name} gives a mapping from an Ur name for a value to a JavaScript name.
adamc@897	2041 \item \texttt{link FILE} requests that \texttt{FILE} be linked into applications. It should be a C object or library archive file, and you are responsible for generating it with your own build process.
adamc@897	2042 \item \texttt{script URL} requests inclusion of a JavaScript source file within application HTML.
adamc@897	2043 \item \texttt{serverOnly Module.ident} registers a value as being allowed only in server-side code.
adamc@897	2044 \end{itemize}
adamc@897	2045
adamc@897	2046 \subsection{Writing C FFI Code}
adamc@897	2047
adamc@897	2048 A server-side FFI type or value \texttt{Module.ident} must have a corresponding type or value definition \texttt{uw\_Module\_ident} in C code. With the current Ur/Web version, it's not generally possible to work with Ur records or complex datatypes in C code, but most other kinds of types are fair game.
adamc@897	2049
adamc@897	2050 \begin{itemize}
adamc@897	2051 \item Primitive types defined in \texttt{Basis} are themselves using the standard FFI interface, so you may refer to them like \texttt{uw\_Basis\_t}. See \texttt{include/types.h} for their definitions.
adamc@897	2052 \item Enumeration datatypes, which have only constructors that take no arguments, should be defined using C \texttt{enum}s. The type is named as for any other type identifier, and each constructor \texttt{c} gets an enumeration constant named \texttt{uw\_Module\_c}.
adamc@897	2053 \item A datatype \texttt{dt} (such as \texttt{Basis.option}) that has one non-value-carrying constructor \texttt{NC} and one value-carrying constructor \texttt{C} gets special treatment. Where \texttt{T} is the type of \texttt{C}'s argument, and where we represent \texttt{T} as \texttt{t} in C, we represent \texttt{NC} with \texttt{NULL}. The representation of \texttt{C} depends on whether we're sure that we don't need to use \texttt{NULL} to represent \texttt{t} values; this condition holds only for strings and complex datatypes. For such types, \texttt{C v} is represented with the C encoding of \texttt{v}, such that the translation of \texttt{dt} is \texttt{t}. For other types, \texttt{C v} is represented with a pointer to the C encoding of v, such that the translation of \texttt{dt} is \texttt{t*}.
adamc@897	2054 \end{itemize}
adamc@897	2055
adamc@897	2056 The C FFI version of a Ur function with type \texttt{T1 -> ... -> TN -> R} or \texttt{T1 -> ... -> TN -> transaction R} has a C prototype like \texttt{R uw\_Module\_ident(uw\_context, T1, ..., TN)}. Only functions with types of the second form may have side effects. \texttt{uw\_context} is the type of state that persists across handling a client request. Many functions that operate on contexts are prototyped in \texttt{include/urweb.h}. Most should only be used internally by the compiler. A few are useful in general FFI implementation:
adamc@897	2057 \begin{itemize}
adamc@897	2058 \item \begin{verbatim}
adamc@897	2059 void uw_error(uw_context, failure_kind, const char *fmt, ...);
adamc@897	2060 \end{verbatim}
adamc@897	2061 Abort the current request processing, giving a \texttt{printf}-style format string and arguments for generating an error message. The \texttt{failure\_kind} argument can be \texttt{FATAL}, to abort the whole execution; \texttt{BOUNDED\_RETRY}, to try processing the request again from the beginning, but failing if this happens too many times; or \texttt{UNLIMITED\_RETRY}, to repeat processing, with no cap on how many times this can recur.
adamc@897	2062
adamc@897	2063 \item \begin{verbatim}
adamc@897	2064 void uw_push_cleanup(uw_context, void (func)(void ), void *arg);
adamc@897	2065 void uw_pop_cleanup(uw_context);
adamc@897	2066 \end{verbatim}
adamc@897	2067 Manipulate a stack of actions that should be taken if any kind of error condition arises. Calling the ``pop'' function both removes an action from the stack and executes it.
adamc@897	2068
adamc@897	2069 \item \begin{verbatim}
adamc@897	2070 void *uw_malloc(uw_context, size_t);
adamc@897	2071 \end{verbatim}
adamc@897	2072 A version of \texttt{malloc()} that allocates memory inside a context's heap, which is managed with region allocation. Thus, there is no \texttt{uw\_free()}, but you need to be careful not to keep ad-hoc C pointers to this area of memory.
adamc@897	2073
adamc@897	2074 For performance and correctness reasons, it is usually preferable to use \texttt{uw\_malloc()} instead of \texttt{malloc()}. The former manipulates a local heap that can be kept allocated across page requests, while the latter uses global data structures that may face contention during concurrent execution.
adamc@897	2075
adamc@897	2076 \item \begin{verbatim}
adamc@897	2077 typedef void (uw_callback)(void );
adamc@897	2078 void uw_register_transactional(uw_context, void *data, uw_callback commit,
adamc@897	2079 uw_callback rollback, uw_callback free);
adamc@897	2080 \end{verbatim}
adamc@897	2081 All side effects in Ur/Web programs need to be compatible with transactions, such that any set of actions can be undone at any time. Thus, you should not perform actions with non-local side effects directly; instead, register handlers to be called when the current transaction is committed or rolled back. The arguments here give an arbitary piece of data to be passed to callbacks, a function to call on commit, a function to call on rollback, and a function to call afterward in either case to clean up any allocated resources. A rollback handler may be called after the associated commit handler has already been called, if some later part of the commit process fails.
adamc@897	2082
adamc@1085	2083 Any of the callbacks may be \texttt{NULL}. To accommodate some stubbornly non-transactional real-world actions like sending an e-mail message, Ur/Web treats \texttt{NULL} \texttt{rollback} callbacks specially. When a transaction commits, all \texttt{commit} actions that have non-\texttt{NULL} rollback actions are tried before any \texttt{commit} actions that have \texttt{NULL} rollback actions. Thus, if a single execution uses only one non-transactional action, and if that action never fails partway through its execution while still causing an observable side effect, then Ur/Web can maintain the transactional abstraction.
adamc@1085	2084
adamc@1085	2085 \item \begin{verbatim}
adamc@1085	2086 void uw_get_global(uw_context, char name);
adamc@1085	2087 void uw_set_global(uw_context, char name, void data, uw_callback free);
adamc@1085	2088 \end{verbatim}
adamc@1085	2089 Different FFI-based extensions may want to associate their own pieces of data with contexts. The global interface provides a way of doing that, where each extension must come up with its own unique key. The \texttt{free} argument to \texttt{uw\_set\_global()} explains how to deallocate the saved data.
adamc@1085	2090
adamc@897	2091 \end{itemize}
adamc@897	2092
adamc@897	2093 \subsection{Writing JavaScript FFI Code}
adamc@897	2094
adamc@897	2095 JavaScript is dynamically typed, so Ur/Web type definitions imply no JavaScript code. The JavaScript identifier for each FFI function is set with the \texttt{jsFunc} directive. Each identifier can be defined in any JavaScript file that you ask to include with the \texttt{script} directive.
adamc@897	2096
adamc@897	2097 In contrast to C FFI code, JavaScript FFI functions take no extra context argument. Their argument lists are as you would expect from their Ur types. Only functions whose ranges take the form \texttt{transaction T} should have side effects; the JavaScript ``return type'' of such a function is \texttt{T}. Here are the conventions for representing Ur values in JavaScript.
adamc@897	2098
adamc@897	2099 \begin{itemize}
adamc@897	2100 \item Integers, floats, strings, characters, and booleans are represented in the usual JavaScript way.
adamc@985	2101 \item Ur functions are represented in an unspecified way. This means that you should not rely on any details of function representation. Named FFI functions are represented as JavaScript functions with as many arguments as their Ur types specify. To call a non-FFI function \texttt{f} on argument \texttt{x}, run \texttt{execF(f, x)}.
adamc@897	2102 \item An Ur record is represented with a JavaScript record, where Ur field name \texttt{N} translates to JavaScript field name \texttt{\_N}. An exception to this rule is that the empty record is encoded as \texttt{null}.
adamc@897	2103 \item \texttt{option}-like types receive special handling similar to their handling in C. The ``\texttt{None}'' constructor is \texttt{null}, and a use of the ``\texttt{Some}'' constructor on a value \texttt{v} is either \texttt{v}, if the underlying type doesn't need to use \texttt{null}; or \texttt{\{v:v\}} otherwise.
adamc@985	2104 \item Any other datatypes represent a non-value-carrying constructor \texttt{C} as \texttt{"C"} and an application of a constructor \texttt{C} to value \texttt{v} as \texttt{\{n:"C", v:v\}}. This rule only applies to datatypes defined in FFI module signatures; the compiler is free to optimize the representations of other, non-\texttt{option}-like datatypes in arbitrary ways.
adamc@897	2105 \end{itemize}
adamc@897	2106
adamc@897	2107 It is possible to write JavaScript FFI code that interacts with the functional-reactive structure of a document, but this version of the manual doesn't cover the details.
adamc@897	2108
adamc@897	2109
adamc@552	2110 \section{Compiler Phases}
adamc@552	2111
adamc@552	2112 The Ur/Web compiler is unconventional in that it relies on a kind of \emph{heuristic compilation}. Not all valid programs will compile successfully. Informally, programs fail to compile when they are ``too higher order.'' Compiler phases do their best to eliminate different kinds of higher order-ness, but some programs just won't compile. This is a trade-off for producing very efficient executables. Compiled Ur/Web programs use native C representations and require no garbage collection.
adamc@552	2113
adamc@552	2114 In this section, we step through the main phases of compilation, noting what consequences each phase has for effective programming.
adamc@552	2115
adamc@552	2116 \subsection{Parse}
adamc@552	2117
adamc@552	2118 The compiler reads a \texttt{.urp} file, figures out which \texttt{.urs} and \texttt{.ur} files it references, and combines them all into what is conceptually a single sequence of declarations in the core language of Section \ref{core}.
adamc@552	2119
adamc@552	2120 \subsection{Elaborate}
adamc@552	2121
adamc@552	2122 This is where type inference takes place, translating programs into an explicit form with no more wildcards. This phase is the most likely source of compiler error messages.
adamc@552	2123
adamc@552	2124 \subsection{Unnest}
adamc@552	2125
adamc@552	2126 Named local function definitions are moved to the top level, to avoid the need to generate closures.
adamc@552	2127
adamc@552	2128 \subsection{Corify}
adamc@552	2129
adamc@552	2130 Module system features are compiled away, through inlining of functor definitions at application sites. Afterward, most abstraction boundaries are broken, facilitating optimization.
adamc@552	2131
adamc@552	2132 \subsection{Especialize}
adamc@552	2133
adamc@552	2134 Functions are specialized to particular argument patterns. This is an important trick for avoiding the need to maintain any closures at runtime.
adamc@552	2135
adamc@552	2136 \subsection{Untangle}
adamc@552	2137
adamc@552	2138 Remove unnecessary mutual recursion, splitting recursive groups into strongly-connected components.
adamc@552	2139
adamc@552	2140 \subsection{Shake}
adamc@552	2141
adamc@552	2142 Remove all definitions not needed to run the page handlers that are visible in the signature of the last module listed in the \texttt{.urp} file.
adamc@552	2143
adamc@661	2144 \subsection{Rpcify}
adamc@661	2145
adamc@661	2146 Pieces of code are determined to be client-side, server-side, neither, or both, by figuring out which standard library functions might be needed to execute them. Calls to server-side functions (e.g., $\mt{query}$) within mixed client-server code are identified and replaced with explicit remote calls. Some mixed functions may be converted to continuation-passing style to facilitate this transformation.
adamc@661	2147
adamc@661	2148 \subsection{Untangle, Shake}
adamc@661	2149
adamc@661	2150 Repeat these simplifications.
adamc@661	2151
adamc@553	2152 \subsection{\label{tag}Tag}
adamc@552	2153
adamc@552	2154 Assign a URL name to each link and form action. It is important that these links and actions are written as applications of named functions, because such names are used to generate URL patterns. A URL pattern has a name built from the full module path of the named function, followed by the function name, with all pieces separated by slashes. The path of a functor application is based on the name given to the result, rather than the path of the functor itself.
adamc@552	2155
adamc@552	2156 \subsection{Reduce}
adamc@552	2157
adamc@552	2158 Apply definitional equality rules to simplify the program as much as possible. This effectively includes inlining of every non-recursive definition.
adamc@552	2159
adamc@552	2160 \subsection{Unpoly}
adamc@552	2161
adamc@552	2162 This phase specializes polymorphic functions to the specific arguments passed to them in the program. If the program contains real polymorphic recursion, Unpoly will be insufficient to avoid later error messages about too much polymorphism.
adamc@552	2163
adamc@552	2164 \subsection{Specialize}
adamc@552	2165
adamc@558	2166 Replace uses of parameterized datatypes with versions specialized to specific parameters. As for Unpoly, this phase will not be effective enough in the presence of polymorphic recursion or other fancy uses of impredicative polymorphism.
adamc@552	2167
adamc@552	2168 \subsection{Shake}
adamc@552	2169
adamc@558	2170 Here the compiler repeats the earlier Shake phase.
adamc@552	2171
adamc@552	2172 \subsection{Monoize}
adamc@552	2173
adamc@552	2174 Programs are translated to a new intermediate language without polymorphism or non-$\mt{Type}$ constructors. Error messages may pop up here if earlier phases failed to remove such features.
adamc@552	2175
adamc@552	2176 This is the stage at which concrete names are generated for cookies, tables, and sequences. They are named following the same convention as for links and actions, based on module path information saved from earlier stages. Table and sequence names separate path elements with underscores instead of slashes, and they are prefixed by \texttt{uw\_}.
adamc@664	2177
adamc@552	2178 \subsection{MonoOpt}
adamc@552	2179
adamc@552	2180 Simple algebraic laws are applied to simplify the program, focusing especially on efficient imperative generation of HTML pages.
adamc@552	2181
adamc@552	2182 \subsection{MonoUntangle}
adamc@552	2183
adamc@552	2184 Unnecessary mutual recursion is broken up again.
adamc@552	2185
adamc@552	2186 \subsection{MonoReduce}
adamc@552	2187
adamc@552	2188 Equivalents of the definitional equality rules are applied to simplify programs, with inlining again playing a major role.
adamc@552	2189
adamc@552	2190 \subsection{MonoShake, MonoOpt}
adamc@552	2191
adamc@552	2192 Unneeded declarations are removed, and basic optimizations are repeated.
adamc@552	2193
adamc@552	2194 \subsection{Fuse}
adamc@552	2195
adamc@552	2196 The compiler tries to simplify calls to recursive functions whose results are immediately written as page output. The write action is pushed inside the function definitions to avoid allocation of intermediate results.
adamc@552	2197
adamc@552	2198 \subsection{MonoUntangle, MonoShake}
adamc@552	2199
adamc@552	2200 Fuse often creates more opportunities to remove spurious mutual recursion.
adamc@552	2201
adamc@552	2202 \subsection{Pathcheck}
adamc@552	2203
adamc@552	2204 The compiler checks that no link or action name has been used more than once.
adamc@552	2205
adamc@552	2206 \subsection{Cjrize}
adamc@552	2207
adamc@552	2208 The program is translated to what is more or less a subset of C. If any use of functions as data remains at this point, the compiler will complain.
adamc@552	2209
adamc@552	2210 \subsection{C Compilation and Linking}
adamc@552	2211
adamc@552	2212 The output of the last phase is pretty-printed as C source code and passed to GCC.
adamc@552	2213
adamc@552	2214
adamc@524	2215 \end{document}

Mercurial > urweb

annotate doc/manual.tex @ 1179:c58453683bbb