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Fixing a bug in DInitializer elaboration
author Adam Chlipala <adamc@hcoop.net>
date Sun, 13 Dec 2009 14:34:18 -0500
parents 93315ac00394
children ae885ad70d83
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@554 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{row types}.
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@555 94 Some OSX users have reported needing to use this particular GCCARGS value.
adamc@555 95
adamc@555 96 The Emacs mode can be set to autoload by adding the following to your \texttt{.emacs} file.
adamc@555 97
adamc@555 98 \begin{verbatim}
adamc@555 99 (add-to-list 'load-path "/usr/local/share/emacs/site-lisp/urweb-mode")
adamc@555 100 (load "urweb-mode-startup")
adamc@555 101 \end{verbatim}
adamc@555 102
adamc@555 103 Change the path in the first line if you chose a different Emacs installation path during configuration.
adamc@555 104
adamc@555 105
adamc@556 106 \section{Command-Line Compiler}
adamc@556 107
adamc@556 108 \subsection{Project Files}
adamc@556 109
adamc@556 110 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@556 111
adamc@556 112 \begin{verbatim}
adamc@556 113 database dbname=test
adamc@556 114 sql crud1.sql
adamc@556 115
adamc@556 116 crud
adamc@556 117 crud1
adamc@556 118 \end{verbatim}
adamc@556 119
adamc@556 120 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@556 121
adamc@556 122 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@556 123
adamc@556 124 \begin{verbatim}
adamc@556 125 createdb test
adamc@556 126 psql -f crud1.sql test
adamc@556 127 \end{verbatim}
adamc@556 128
adamc@556 129 A blank line always separates the named directives from a list of modules to include in the project; if there are no named directives, a blank line must begin the file.
adamc@556 130
adamc@556 131 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@556 132
adamc@783 133 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@783 134 \begin{itemize}
adamc@783 135 \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@783 136 \item \texttt{clientOnly Module.ident} registers an FFI function or transaction that may only be run in client browsers.
adamc@783 137 \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@783 138 \item \texttt{database DBSTRING} sets the string to pass to libpq to open a database connection.
adamc@783 139 \item \texttt{debug} saves some intermediate C files, which is mostly useful to help in debugging the compiler itself.
adamc@783 140 \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@783 141 \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@783 142 \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@873 143 \item \texttt{header 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@783 144 \item \texttt{jsFunc Module.ident=name} gives the JavaScript name of an FFI value.
adamc@783 145 \item \texttt{library FILENAME} parses \texttt{FILENAME.urp} and merges its contents with the rest of the current file's contents.
adamc@783 146 \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@852 147 \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@783 148 \item \texttt{prefix PREFIX} sets the prefix included before every URI within the generated application. The default is \texttt{/}.
adamc@783 149 \item \texttt{profile} generates an executable that may be used with gprof.
adamc@783 150 \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@783 151 \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@783 152 \item \texttt{serverOnly Module.ident} registers an FFI function or transaction that may only be run on the server.
adamc@783 153 \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@783 154 \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@783 155 \end{itemize}
adamc@701 156
adamc@701 157
adamc@557 158 \subsection{Building an Application}
adamc@557 159
adamc@557 160 To compile project \texttt{P.urp}, simply run
adamc@557 161 \begin{verbatim}
adamc@557 162 urweb P
adamc@557 163 \end{verbatim}
adamc@558 164 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@557 165
adamc@557 166 To time how long the different compiler phases run, without generating an executable, run
adamc@557 167 \begin{verbatim}
adamc@557 168 urweb -timing P
adamc@557 169 \end{verbatim}
adamc@557 170
adamc@896 171 Some other command-line parameters are accepted:
adamc@896 172 \begin{itemize}
adamc@896 173 \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@896 174
adamc@896 175 \item \texttt{-dbms [postgres|mysql|sqlite]}: Sets the database backend to use.
adamc@896 176 \begin{itemize}
adamc@896 177 \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@896 178
adamc@896 179 A command sequence like this can initialize a Postgres database, using a file \texttt{app.sql} generated by the compiler:
adamc@896 180 \begin{verbatim}
adamc@896 181 createdb app
adamc@896 182 psql -f app.sql app
adamc@896 183 \end{verbatim}
adamc@896 184
adamc@896 185 \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@896 186
adamc@896 187 A command sequence like this can initialize a MySQL database:
adamc@896 188 \begin{verbatim}
adamc@896 189 echo "CREATE DATABASE app" | mysql
adamc@896 190 mysql -D app <app.sql
adamc@896 191 \end{verbatim}
adamc@896 192
adamc@896 193 \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@896 194
adamc@896 195 A command like this can initialize an SQLite database:
adamc@896 196 \begin{verbatim}
adamc@896 197 sqlite3 path/to/database/file <app.sql
adamc@896 198 \end{verbatim}
adamc@896 199 \end{itemize}
adamc@896 200
adamc@896 201 \item \texttt{-output FILENAME}: Set where the application executable is written.
adamc@896 202
adamc@896 203 \item \texttt{-protocol [http|cgi|fastcgi]}: Set the protocol that the generated application speaks.
adamc@896 204 \begin{itemize}
adamc@896 205 \item \texttt{http}: This is the default. It is for building standalone web servers that can be accessed by web browsers directly.
adamc@896 206
adamc@896 207 \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@896 208
adamc@896 209 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@896 210 \begin{verbatim}
adamc@896 211 ScriptAlias /Hello /path/to/hello.exe
adamc@896 212 \end{verbatim}
adamc@896 213
adamc@896 214 \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@896 215
adamc@896 216 To configure a FastCGI program with Apache, one could combine the above \texttt{ScriptAlias} line with a line like this:
adamc@896 217 \begin{verbatim}
adamc@896 218 FastCgiServer /path/to/hello.exe -idle-timeout 99999
adamc@896 219 \end{verbatim}
adamc@896 220 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@896 221
adamc@896 222 Here is some lighttpd configuration for the same application.
adamc@896 223 \begin{verbatim}
adamc@896 224 fastcgi.server = (
adamc@896 225 "/Hello/" =>
adamc@896 226 (( "bin-path" => "/path/to/hello.exe",
adamc@896 227 "socket" => "/tmp/hello",
adamc@896 228 "check-local" => "disable",
adamc@896 229 "docroot" => "/",
adamc@896 230 "max-procs" => "1"
adamc@896 231 ))
adamc@896 232 )
adamc@896 233 \end{verbatim}
adamc@896 234 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@896 235
adamc@896 236 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@896 237 \end{itemize}
adamc@896 238
adamc@896 239 \item \texttt{-sql FILENAME}: Set where a database set-up SQL script is written.
adamc@896 240 \end{itemize}
adamc@896 241
adamc@556 242
adamc@529 243 \section{Ur Syntax}
adamc@529 244
adamc@784 245 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@524 246
adamc@524 247 \subsection{Lexical Conventions}
adamc@524 248
adamc@524 249 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@524 250
adamc@524 251 \begin{center}
adamc@524 252 \begin{tabular}{rl}
adamc@524 253 \textbf{\LaTeX} & \textbf{ASCII} \\
adamc@524 254 $\to$ & \cd{->} \\
adamc@652 255 $\longrightarrow$ & \cd{-->} \\
adamc@524 256 $\times$ & \cd{*} \\
adamc@524 257 $\lambda$ & \cd{fn} \\
adamc@524 258 $\Rightarrow$ & \cd{=>} \\
adamc@652 259 $\Longrightarrow$ & \cd{==>} \\
adamc@529 260 $\neq$ & \cd{<>} \\
adamc@529 261 $\leq$ & \cd{<=} \\
adamc@529 262 $\geq$ & \cd{>=} \\
adamc@524 263 \\
adamc@524 264 $x$ & Normal textual identifier, not beginning with an uppercase letter \\
adamc@525 265 $X$ & Normal textual identifier, beginning with an uppercase letter \\
adamc@524 266 \end{tabular}
adamc@524 267 \end{center}
adamc@524 268
adamc@525 269 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@524 270
adamc@873 271 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@526 272
adamc@527 273 This version of the manual doesn't include operator precedences; see \texttt{src/urweb.grm} for that.
adamc@527 274
adamc@552 275 \subsection{\label{core}Core Syntax}
adamc@524 276
adamc@524 277 \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@524 278 $$\begin{array}{rrcll}
adamc@524 279 \textrm{Kinds} & \kappa &::=& \mt{Type} & \textrm{proper types} \\
adamc@525 280 &&& \mt{Unit} & \textrm{the trivial constructor} \\
adamc@525 281 &&& \mt{Name} & \textrm{field names} \\
adamc@525 282 &&& \kappa \to \kappa & \textrm{type-level functions} \\
adamc@525 283 &&& \{\kappa\} & \textrm{type-level records} \\
adamc@525 284 &&& (\kappa\times^+) & \textrm{type-level tuples} \\
adamc@652 285 &&& X & \textrm{variable} \\
adamc@652 286 &&& X \longrightarrow k & \textrm{kind-polymorphic type-level function} \\
adamc@529 287 &&& \_\_ & \textrm{wildcard} \\
adamc@525 288 &&& (\kappa) & \textrm{explicit precedence} \\
adamc@524 289 \end{array}$$
adamc@524 290
adamc@524 291 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@524 292 $$\begin{array}{rrcll}
adamc@524 293 \textrm{Explicitness} & ? &::=& :: & \textrm{explicit} \\
adamc@558 294 &&& ::: & \textrm{implicit}
adamc@524 295 \end{array}$$
adamc@524 296
adamc@524 297 \emph{Constructors} are the main class of compile-time-only data. They include proper types and are classified by kinds.
adamc@524 298 $$\begin{array}{rrcll}
adamc@524 299 \textrm{Constructors} & c, \tau &::=& (c) :: \kappa & \textrm{kind annotation} \\
adamc@530 300 &&& \hat{x} & \textrm{constructor variable} \\
adamc@524 301 \\
adamc@525 302 &&& \tau \to \tau & \textrm{function type} \\
adamc@525 303 &&& x \; ? \; \kappa \to \tau & \textrm{polymorphic function type} \\
adamc@652 304 &&& X \longrightarrow \tau & \textrm{kind-polymorphic function type} \\
adamc@525 305 &&& \$ c & \textrm{record type} \\
adamc@524 306 \\
adamc@525 307 &&& c \; c & \textrm{type-level function application} \\
adamc@530 308 &&& \lambda x \; :: \; \kappa \Rightarrow c & \textrm{type-level function abstraction} \\
adamc@524 309 \\
adamc@652 310 &&& X \Longrightarrow c & \textrm{type-level kind-polymorphic function abstraction} \\
adamc@655 311 &&& c [\kappa] & \textrm{type-level kind-polymorphic function application} \\
adamc@652 312 \\
adamc@525 313 &&& () & \textrm{type-level unit} \\
adamc@525 314 &&& \#X & \textrm{field name} \\
adamc@524 315 \\
adamc@525 316 &&& [(c = c)^*] & \textrm{known-length type-level record} \\
adamc@525 317 &&& c \rc c & \textrm{type-level record concatenation} \\
adamc@652 318 &&& \mt{map} & \textrm{type-level record map} \\
adamc@524 319 \\
adamc@558 320 &&& (c,^+) & \textrm{type-level tuple} \\
adamc@525 321 &&& c.n & \textrm{type-level tuple projection ($n \in \mathbb N^+$)} \\
adamc@524 322 \\
adamc@652 323 &&& [c \sim c] \Rightarrow \tau & \textrm{guarded type} \\
adamc@524 324 \\
adamc@529 325 &&& \_ :: \kappa & \textrm{wildcard} \\
adamc@525 326 &&& (c) & \textrm{explicit precedence} \\
adamc@530 327 \\
adamc@530 328 \textrm{Qualified uncapitalized variables} & \hat{x} &::=& x & \textrm{not from a module} \\
adamc@530 329 &&& M.x & \textrm{projection from a module} \\
adamc@525 330 \end{array}$$
adamc@525 331
adamc@655 332 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@655 333
adamc@525 334 Modules of the module system are described by \emph{signatures}.
adamc@525 335 $$\begin{array}{rrcll}
adamc@525 336 \textrm{Signatures} & S &::=& \mt{sig} \; s^* \; \mt{end} & \textrm{constant} \\
adamc@525 337 &&& X & \textrm{variable} \\
adamc@525 338 &&& \mt{functor}(X : S) : S & \textrm{functor} \\
adamc@529 339 &&& S \; \mt{where} \; \mt{con} \; x = c & \textrm{concretizing an abstract constructor} \\
adamc@525 340 &&& M.X & \textrm{projection from a module} \\
adamc@525 341 \\
adamc@525 342 \textrm{Signature items} & s &::=& \mt{con} \; x :: \kappa & \textrm{abstract constructor} \\
adamc@525 343 &&& \mt{con} \; x :: \kappa = c & \textrm{concrete constructor} \\
adamc@528 344 &&& \mt{datatype} \; x \; x^* = dc\mid^+ & \textrm{algebraic datatype definition} \\
adamc@529 345 &&& \mt{datatype} \; x = \mt{datatype} \; M.x & \textrm{algebraic datatype import} \\
adamc@525 346 &&& \mt{val} \; x : \tau & \textrm{value} \\
adamc@525 347 &&& \mt{structure} \; X : S & \textrm{sub-module} \\
adamc@525 348 &&& \mt{signature} \; X = S & \textrm{sub-signature} \\
adamc@525 349 &&& \mt{include} \; S & \textrm{signature inclusion} \\
adamc@525 350 &&& \mt{constraint} \; c \sim c & \textrm{record disjointness constraint} \\
adamc@654 351 &&& \mt{class} \; x :: \kappa & \textrm{abstract constructor class} \\
adamc@654 352 &&& \mt{class} \; x :: \kappa = c & \textrm{concrete constructor class} \\
adamc@525 353 \\
adamc@525 354 \textrm{Datatype constructors} & dc &::=& X & \textrm{nullary constructor} \\
adamc@525 355 &&& X \; \mt{of} \; \tau & \textrm{unary constructor} \\
adamc@524 356 \end{array}$$
adamc@524 357
adamc@526 358 \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@526 359 $$\begin{array}{rrcll}
adamc@526 360 \textrm{Patterns} & p &::=& \_ & \textrm{wildcard} \\
adamc@526 361 &&& x & \textrm{variable} \\
adamc@526 362 &&& \ell & \textrm{constant} \\
adamc@526 363 &&& \hat{X} & \textrm{nullary constructor} \\
adamc@526 364 &&& \hat{X} \; p & \textrm{unary constructor} \\
adamc@526 365 &&& \{(x = p,)^*\} & \textrm{rigid record pattern} \\
adamc@526 366 &&& \{(x = p,)^+, \ldots\} & \textrm{flexible record pattern} \\
adamc@852 367 &&& p : \tau & \textrm{type annotation} \\
adamc@527 368 &&& (p) & \textrm{explicit precedence} \\
adamc@526 369 \\
adamc@529 370 \textrm{Qualified capitalized variables} & \hat{X} &::=& X & \textrm{not from a module} \\
adamc@526 371 &&& M.X & \textrm{projection from a module} \\
adamc@526 372 \end{array}$$
adamc@526 373
adamc@527 374 \emph{Expressions} are the main run-time entities, corresponding to both ``expressions'' and ``statements'' in mainstream imperative languages.
adamc@527 375 $$\begin{array}{rrcll}
adamc@527 376 \textrm{Expressions} & e &::=& e : \tau & \textrm{type annotation} \\
adamc@529 377 &&& \hat{x} & \textrm{variable} \\
adamc@529 378 &&& \hat{X} & \textrm{datatype constructor} \\
adamc@527 379 &&& \ell & \textrm{constant} \\
adamc@527 380 \\
adamc@527 381 &&& e \; e & \textrm{function application} \\
adamc@527 382 &&& \lambda x : \tau \Rightarrow e & \textrm{function abstraction} \\
adamc@527 383 &&& e [c] & \textrm{polymorphic function application} \\
adamc@852 384 &&& \lambda [x \; ? \; \kappa] \Rightarrow e & \textrm{polymorphic function abstraction} \\
adamc@655 385 &&& e [\kappa] & \textrm{kind-polymorphic function application} \\
adamc@652 386 &&& X \Longrightarrow e & \textrm{kind-polymorphic function abstraction} \\
adamc@527 387 \\
adamc@527 388 &&& \{(c = e,)^*\} & \textrm{known-length record} \\
adamc@527 389 &&& e.c & \textrm{record field projection} \\
adamc@527 390 &&& e \rc e & \textrm{record concatenation} \\
adamc@527 391 &&& e \rcut c & \textrm{removal of a single record field} \\
adamc@527 392 &&& e \rcutM c & \textrm{removal of multiple record fields} \\
adamc@527 393 \\
adamc@527 394 &&& \mt{let} \; ed^* \; \mt{in} \; e \; \mt{end} & \textrm{local definitions} \\
adamc@527 395 \\
adamc@527 396 &&& \mt{case} \; e \; \mt{of} \; (p \Rightarrow e|)^+ & \textrm{pattern matching} \\
adamc@527 397 \\
adamc@654 398 &&& \lambda [c \sim c] \Rightarrow e & \textrm{guarded expression abstraction} \\
adamc@654 399 &&& e \; ! & \textrm{guarded expression application} \\
adamc@527 400 \\
adamc@527 401 &&& \_ & \textrm{wildcard} \\
adamc@527 402 &&& (e) & \textrm{explicit precedence} \\
adamc@527 403 \\
adamc@527 404 \textrm{Local declarations} & ed &::=& \cd{val} \; x : \tau = e & \textrm{non-recursive value} \\
adamc@527 405 &&& \cd{val} \; \cd{rec} \; (x : \tau = e \; \cd{and})^+ & \textrm{mutually-recursive values} \\
adamc@527 406 \end{array}$$
adamc@527 407
adamc@655 408 As with constructors, we include both abstraction and application for kind polymorphism, but applications are only inferred internally.
adamc@655 409
adamc@528 410 \emph{Declarations} primarily bring new symbols into context.
adamc@528 411 $$\begin{array}{rrcll}
adamc@528 412 \textrm{Declarations} & d &::=& \mt{con} \; x :: \kappa = c & \textrm{constructor synonym} \\
adamc@528 413 &&& \mt{datatype} \; x \; x^* = dc\mid^+ & \textrm{algebraic datatype definition} \\
adamc@529 414 &&& \mt{datatype} \; x = \mt{datatype} \; M.x & \textrm{algebraic datatype import} \\
adamc@528 415 &&& \mt{val} \; x : \tau = e & \textrm{value} \\
adamc@528 416 &&& \mt{val} \; \cd{rec} \; (x : \tau = e \; \mt{and})^+ & \textrm{mutually-recursive values} \\
adamc@528 417 &&& \mt{structure} \; X : S = M & \textrm{module definition} \\
adamc@528 418 &&& \mt{signature} \; X = S & \textrm{signature definition} \\
adamc@528 419 &&& \mt{open} \; M & \textrm{module inclusion} \\
adamc@528 420 &&& \mt{constraint} \; c \sim c & \textrm{record disjointness constraint} \\
adamc@528 421 &&& \mt{open} \; \mt{constraints} \; M & \textrm{inclusion of just the constraints from a module} \\
adamc@528 422 &&& \mt{table} \; x : c & \textrm{SQL table} \\
adamc@784 423 &&& \mt{view} \; x : c & \textrm{SQL view} \\
adamc@528 424 &&& \mt{sequence} \; x & \textrm{SQL sequence} \\
adamc@535 425 &&& \mt{cookie} \; x : \tau & \textrm{HTTP cookie} \\
adamc@784 426 &&& \mt{style} \; x : \tau & \textrm{CSS class} \\
adamc@654 427 &&& \mt{class} \; x :: \kappa = c & \textrm{concrete constructor class} \\
adamc@528 428 \\
adamc@529 429 \textrm{Modules} & M &::=& \mt{struct} \; d^* \; \mt{end} & \textrm{constant} \\
adamc@529 430 &&& X & \textrm{variable} \\
adamc@529 431 &&& M.X & \textrm{projection} \\
adamc@529 432 &&& M(M) & \textrm{functor application} \\
adamc@529 433 &&& \mt{functor}(X : S) : S = M & \textrm{functor abstraction} \\
adamc@528 434 \end{array}$$
adamc@528 435
adamc@528 436 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@527 437
adamc@784 438 We omit some extra possibilities in $\mt{table}$ syntax, deferring them to Section \ref{tables}.
adamc@784 439
adamc@529 440 \subsection{Shorthands}
adamc@529 441
adamc@529 442 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@529 443
adamc@529 444 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@529 445
adamc@529 446 A record type may be written $\{(c = c,)^*\}$, which elaborates to $\$[(c = c,)^*]$.
adamc@529 447
adamc@533 448 The notation $[c_1, \ldots, c_n]$ is shorthand for $[c_1 = (), \ldots, c_n = ()]$.
adamc@533 449
adamc@529 450 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@529 451
adamc@852 452 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@529 453
adamc@529 454 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@529 455
adamc@529 456 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@529 457
adamc@654 458 A signature item or declaration $\mt{class} \; x = \lambda y \Rightarrow c$ may be abbreviated $\mt{class} \; x \; y = c$.
adamc@529 459
adamc@654 460 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 any prefix of a variable's type consisting only of implicit polymorphism, type class instances, and disjointness obligations is resolved automatically, with the variable treated as having the type that starts after the last implicit element, with suitable unification variables substituted. The same syntax works for variables projected out of modules and for capitalized variables (datatype constructors).
adamc@529 461
adamc@852 462 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@529 463
adamc@852 464 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@852 465
adamc@852 466 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@529 467
adamc@529 468 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@529 469
adamc@852 470 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@852 471
adamc@853 472 A declaration $\mt{table} \; x : \{(c = c,)^*\}$ is elaborated into $\mt{table} \; x : [(c = c,)^*]$.
adamc@529 473
adamc@529 474 The syntax $\mt{where} \; \mt{type}$ is an alternate form of $\mt{where} \; \mt{con}$.
adamc@529 475
adamc@529 476 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@529 477
adamc@529 478 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@529 479
adamc@784 480 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@529 481
adamc@530 482
adamc@530 483 \section{Static Semantics}
adamc@530 484
adamc@530 485 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@530 486
adamc@530 487 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@530 488 \begin{itemize}
adamc@655 489 \item $\Gamma \vdash \kappa$ expresses kind well-formedness.
adamc@530 490 \item $\Gamma \vdash c :: \kappa$ assigns a kind to a constructor in a context.
adamc@530 491 \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@531 492 \item $\Gamma \vdash c \hookrightarrow C$ proves that record constructor $c$ decomposes into set $C$ of field names and record constructors.
adamc@530 493 \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@530 494 \item $\Gamma \vdash e : \tau$ is a standard typing judgment.
adamc@534 495 \item $\Gamma \vdash p \leadsto \Gamma; \tau$ combines typing of patterns with calculation of which new variables they bind.
adamc@537 496 \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@537 497 \item $\Gamma \vdash S \equiv S$ is the signature equivalence judgment.
adamc@536 498 \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@530 499 \item $\Gamma \vdash M : S$ is the module signature checking judgment.
adamc@537 500 \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@539 501 \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@530 502 \end{itemize}
adamc@530 503
adamc@655 504
adamc@655 505 \subsection{Kind Well-Formedness}
adamc@655 506
adamc@655 507 $$\infer{\Gamma \vdash \mt{Type}}{}
adamc@655 508 \quad \infer{\Gamma \vdash \mt{Unit}}{}
adamc@655 509 \quad \infer{\Gamma \vdash \mt{Name}}{}
adamc@655 510 \quad \infer{\Gamma \vdash \kappa_1 \to \kappa_2}{
adamc@655 511 \Gamma \vdash \kappa_1
adamc@655 512 & \Gamma \vdash \kappa_2
adamc@655 513 }
adamc@655 514 \quad \infer{\Gamma \vdash \{\kappa\}}{
adamc@655 515 \Gamma \vdash \kappa
adamc@655 516 }
adamc@655 517 \quad \infer{\Gamma \vdash (\kappa_1 \times \ldots \times \kappa_n)}{
adamc@655 518 \forall i: \Gamma \vdash \kappa_i
adamc@655 519 }$$
adamc@655 520
adamc@655 521 $$\infer{\Gamma \vdash X}{
adamc@655 522 X \in \Gamma
adamc@655 523 }
adamc@655 524 \quad \infer{\Gamma \vdash X \longrightarrow \kappa}{
adamc@655 525 \Gamma, X \vdash \kappa
adamc@655 526 }$$
adamc@655 527
adamc@530 528 \subsection{Kinding}
adamc@530 529
adamc@655 530 We write $[X \mapsto \kappa_1]\kappa_2$ for capture-avoiding substitution of $\kappa_1$ for $X$ in $\kappa_2$.
adamc@655 531
adamc@530 532 $$\infer{\Gamma \vdash (c) :: \kappa :: \kappa}{
adamc@530 533 \Gamma \vdash c :: \kappa
adamc@530 534 }
adamc@530 535 \quad \infer{\Gamma \vdash x :: \kappa}{
adamc@530 536 x :: \kappa \in \Gamma
adamc@530 537 }
adamc@530 538 \quad \infer{\Gamma \vdash x :: \kappa}{
adamc@530 539 x :: \kappa = c \in \Gamma
adamc@530 540 }$$
adamc@530 541
adamc@530 542 $$\infer{\Gamma \vdash M.x :: \kappa}{
adamc@537 543 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 544 & \mt{proj}(M, \overline{s}, \mt{con} \; x) = \kappa
adamc@530 545 }
adamc@530 546 \quad \infer{\Gamma \vdash M.x :: \kappa}{
adamc@537 547 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 548 & \mt{proj}(M, \overline{s}, \mt{con} \; x) = (\kappa, c)
adamc@530 549 }$$
adamc@530 550
adamc@530 551 $$\infer{\Gamma \vdash \tau_1 \to \tau_2 :: \mt{Type}}{
adamc@530 552 \Gamma \vdash \tau_1 :: \mt{Type}
adamc@530 553 & \Gamma \vdash \tau_2 :: \mt{Type}
adamc@530 554 }
adamc@530 555 \quad \infer{\Gamma \vdash x \; ? \: \kappa \to \tau :: \mt{Type}}{
adamc@530 556 \Gamma, x :: \kappa \vdash \tau :: \mt{Type}
adamc@530 557 }
adamc@655 558 \quad \infer{\Gamma \vdash X \longrightarrow \tau :: \mt{Type}}{
adamc@655 559 \Gamma, X \vdash \tau :: \mt{Type}
adamc@655 560 }
adamc@530 561 \quad \infer{\Gamma \vdash \$c :: \mt{Type}}{
adamc@530 562 \Gamma \vdash c :: \{\mt{Type}\}
adamc@530 563 }$$
adamc@530 564
adamc@530 565 $$\infer{\Gamma \vdash c_1 \; c_2 :: \kappa_2}{
adamc@530 566 \Gamma \vdash c_1 :: \kappa_1 \to \kappa_2
adamc@530 567 & \Gamma \vdash c_2 :: \kappa_1
adamc@530 568 }
adamc@530 569 \quad \infer{\Gamma \vdash \lambda x \; :: \; \kappa_1 \Rightarrow c :: \kappa_1 \to \kappa_2}{
adamc@530 570 \Gamma, x :: \kappa_1 \vdash c :: \kappa_2
adamc@530 571 }$$
adamc@530 572
adamc@655 573 $$\infer{\Gamma \vdash c[\kappa'] :: [X \mapsto \kappa']\kappa}{
adamc@655 574 \Gamma \vdash c :: X \to \kappa
adamc@655 575 & \Gamma \vdash \kappa'
adamc@655 576 }
adamc@655 577 \quad \infer{\Gamma \vdash X \Longrightarrow c :: X \to \kappa}{
adamc@655 578 \Gamma, X \vdash c :: \kappa
adamc@655 579 }$$
adamc@655 580
adamc@530 581 $$\infer{\Gamma \vdash () :: \mt{Unit}}{}
adamc@530 582 \quad \infer{\Gamma \vdash \#X :: \mt{Name}}{}$$
adamc@530 583
adamc@530 584 $$\infer{\Gamma \vdash [\overline{c_i = c'_i}] :: \{\kappa\}}{
adamc@530 585 \forall i: \Gamma \vdash c_i : \mt{Name}
adamc@530 586 & \Gamma \vdash c'_i :: \kappa
adamc@530 587 & \forall i \neq j: \Gamma \vdash c_i \sim c_j
adamc@530 588 }
adamc@530 589 \quad \infer{\Gamma \vdash c_1 \rc c_2 :: \{\kappa\}}{
adamc@530 590 \Gamma \vdash c_1 :: \{\kappa\}
adamc@530 591 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@530 592 & \Gamma \vdash c_1 \sim c_2
adamc@530 593 }$$
adamc@530 594
adamc@655 595 $$\infer{\Gamma \vdash \mt{map} :: (\kappa_1 \to \kappa_2) \to \{\kappa_1\} \to \{\kappa_2\}}{}$$
adamc@530 596
adamc@573 597 $$\infer{\Gamma \vdash (\overline c) :: (\kappa_1 \times \ldots \times \kappa_n)}{
adamc@573 598 \forall i: \Gamma \vdash c_i :: \kappa_i
adamc@530 599 }
adamc@573 600 \quad \infer{\Gamma \vdash c.i :: \kappa_i}{
adamc@573 601 \Gamma \vdash c :: (\kappa_1 \times \ldots \times \kappa_n)
adamc@530 602 }$$
adamc@530 603
adamc@655 604 $$\infer{\Gamma \vdash \lambda [c_1 \sim c_2] \Rightarrow \tau :: \mt{Type}}{
adamc@655 605 \Gamma \vdash c_1 :: \{\kappa\}
adamc@530 606 & \Gamma \vdash c_2 :: \{\kappa'\}
adamc@655 607 & \Gamma, c_1 \sim c_2 \vdash \tau :: \mt{Type}
adamc@530 608 }$$
adamc@530 609
adamc@531 610 \subsection{Record Disjointness}
adamc@531 611
adamc@531 612 $$\infer{\Gamma \vdash c_1 \sim c_2}{
adamc@558 613 \Gamma \vdash c_1 \hookrightarrow C_1
adamc@558 614 & \Gamma \vdash c_2 \hookrightarrow C_2
adamc@558 615 & \forall c'_1 \in C_1, c'_2 \in C_2: \Gamma \vdash c'_1 \sim c'_2
adamc@531 616 }
adamc@531 617 \quad \infer{\Gamma \vdash X \sim X'}{
adamc@531 618 X \neq X'
adamc@531 619 }$$
adamc@531 620
adamc@531 621 $$\infer{\Gamma \vdash c_1 \sim c_2}{
adamc@531 622 c'_1 \sim c'_2 \in \Gamma
adamc@558 623 & \Gamma \vdash c'_1 \hookrightarrow C_1
adamc@558 624 & \Gamma \vdash c'_2 \hookrightarrow C_2
adamc@558 625 & c_1 \in C_1
adamc@558 626 & c_2 \in C_2
adamc@531 627 }$$
adamc@531 628
adamc@531 629 $$\infer{\Gamma \vdash c \hookrightarrow \{c\}}{}
adamc@531 630 \quad \infer{\Gamma \vdash [\overline{c = c'}] \hookrightarrow \{\overline{c}\}}{}
adamc@531 631 \quad \infer{\Gamma \vdash c_1 \rc c_2 \hookrightarrow C_1 \cup C_2}{
adamc@531 632 \Gamma \vdash c_1 \hookrightarrow C_1
adamc@531 633 & \Gamma \vdash c_2 \hookrightarrow C_2
adamc@531 634 }
adamc@531 635 \quad \infer{\Gamma \vdash c \hookrightarrow C}{
adamc@531 636 \Gamma \vdash c \equiv c'
adamc@531 637 & \Gamma \vdash c' \hookrightarrow C
adamc@531 638 }
adamc@531 639 \quad \infer{\Gamma \vdash \mt{map} \; f \; c \hookrightarrow C}{
adamc@531 640 \Gamma \vdash c \hookrightarrow C
adamc@531 641 }$$
adamc@531 642
adamc@541 643 \subsection{\label{definitional}Definitional Equality}
adamc@532 644
adamc@655 645 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@532 646
adamc@532 647 $$\infer{\Gamma \vdash c \equiv c}{}
adamc@532 648 \quad \infer{\Gamma \vdash c_1 \equiv c_2}{
adamc@532 649 \Gamma \vdash c_2 \equiv c_1
adamc@532 650 }
adamc@532 651 \quad \infer{\Gamma \vdash c_1 \equiv c_3}{
adamc@532 652 \Gamma \vdash c_1 \equiv c_2
adamc@532 653 & \Gamma \vdash c_2 \equiv c_3
adamc@532 654 }
adamc@532 655 \quad \infer{\Gamma \vdash \mathcal C[c_1] \equiv \mathcal C[c_2]}{
adamc@532 656 \Gamma \vdash c_1 \equiv c_2
adamc@532 657 }$$
adamc@532 658
adamc@532 659 $$\infer{\Gamma \vdash x \equiv c}{
adamc@532 660 x :: \kappa = c \in \Gamma
adamc@532 661 }
adamc@532 662 \quad \infer{\Gamma \vdash M.x \equiv c}{
adamc@537 663 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 664 & \mt{proj}(M, \overline{s}, \mt{con} \; x) = (\kappa, c)
adamc@532 665 }
adamc@532 666 \quad \infer{\Gamma \vdash (\overline c).i \equiv c_i}{}$$
adamc@532 667
adamc@532 668 $$\infer{\Gamma \vdash (\lambda x :: \kappa \Rightarrow c) \; c' \equiv [x \mapsto c'] c}{}
adamc@655 669 \quad \infer{\Gamma \vdash (X \Longrightarrow c) [\kappa] \equiv [X \mapsto \kappa] c}{}$$
adamc@655 670
adamc@655 671 $$\infer{\Gamma \vdash c_1 \rc c_2 \equiv c_2 \rc c_1}{}
adamc@532 672 \quad \infer{\Gamma \vdash c_1 \rc (c_2 \rc c_3) \equiv (c_1 \rc c_2) \rc c_3}{}$$
adamc@532 673
adamc@532 674 $$\infer{\Gamma \vdash [] \rc c \equiv c}{}
adamc@532 675 \quad \infer{\Gamma \vdash [\overline{c_1 = c'_1}] \rc [\overline{c_2 = c'_2}] \equiv [\overline{c_1 = c'_1}, \overline{c_2 = c'_2}]}{}$$
adamc@532 676
adamc@655 677 $$\infer{\Gamma \vdash \mt{map} \; f \; [] \equiv []}{}
adamc@655 678 \quad \infer{\Gamma \vdash \mt{map} \; f \; ([c_1 = c_2] \rc c) \equiv [c_1 = f \; c_2] \rc \mt{map} \; f \; c}{}$$
adamc@532 679
adamc@532 680 $$\infer{\Gamma \vdash \mt{map} \; (\lambda x \Rightarrow x) \; c \equiv c}{}
adamc@655 681 \quad \infer{\Gamma \vdash \mt{map} \; f \; (\mt{map} \; f' \; c)
adamc@655 682 \equiv \mt{map} \; (\lambda x \Rightarrow f \; (f' \; x)) \; c}{}$$
adamc@532 683
adamc@532 684 $$\infer{\Gamma \vdash \mt{map} \; f \; (c_1 \rc c_2) \equiv \mt{map} \; f \; c_1 \rc \mt{map} \; f \; c_2}{}$$
adamc@531 685
adamc@534 686 \subsection{Expression Typing}
adamc@533 687
adamc@873 688 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@533 689
adamc@533 690 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@533 691
adamc@533 692 $$\infer{\Gamma \vdash e : \tau : \tau}{
adamc@533 693 \Gamma \vdash e : \tau
adamc@533 694 }
adamc@533 695 \quad \infer{\Gamma \vdash e : \tau}{
adamc@533 696 \Gamma \vdash e : \tau'
adamc@533 697 & \Gamma \vdash \tau' \equiv \tau
adamc@533 698 }
adamc@533 699 \quad \infer{\Gamma \vdash \ell : T(\ell)}{}$$
adamc@533 700
adamc@533 701 $$\infer{\Gamma \vdash x : \mathcal I(\tau)}{
adamc@533 702 x : \tau \in \Gamma
adamc@533 703 }
adamc@533 704 \quad \infer{\Gamma \vdash M.x : \mathcal I(\tau)}{
adamc@537 705 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 706 & \mt{proj}(M, \overline{s}, \mt{val} \; x) = \tau
adamc@533 707 }
adamc@533 708 \quad \infer{\Gamma \vdash X : \mathcal I(\tau)}{
adamc@533 709 X : \tau \in \Gamma
adamc@533 710 }
adamc@533 711 \quad \infer{\Gamma \vdash M.X : \mathcal I(\tau)}{
adamc@537 712 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 713 & \mt{proj}(M, \overline{s}, \mt{val} \; X) = \tau
adamc@533 714 }$$
adamc@533 715
adamc@533 716 $$\infer{\Gamma \vdash e_1 \; e_2 : \tau_2}{
adamc@533 717 \Gamma \vdash e_1 : \tau_1 \to \tau_2
adamc@533 718 & \Gamma \vdash e_2 : \tau_1
adamc@533 719 }
adamc@533 720 \quad \infer{\Gamma \vdash \lambda x : \tau_1 \Rightarrow e : \tau_1 \to \tau_2}{
adamc@533 721 \Gamma, x : \tau_1 \vdash e : \tau_2
adamc@533 722 }$$
adamc@533 723
adamc@533 724 $$\infer{\Gamma \vdash e [c] : [x \mapsto c]\tau}{
adamc@533 725 \Gamma \vdash e : x :: \kappa \to \tau
adamc@533 726 & \Gamma \vdash c :: \kappa
adamc@533 727 }
adamc@852 728 \quad \infer{\Gamma \vdash \lambda [x \; ? \; \kappa] \Rightarrow e : x \; ? \; \kappa \to \tau}{
adamc@533 729 \Gamma, x :: \kappa \vdash e : \tau
adamc@533 730 }$$
adamc@533 731
adamc@655 732 $$\infer{\Gamma \vdash e [\kappa] : [X \mapsto \kappa]\tau}{
adamc@655 733 \Gamma \vdash e : X \longrightarrow \tau
adamc@655 734 & \Gamma \vdash \kappa
adamc@655 735 }
adamc@655 736 \quad \infer{\Gamma \vdash X \Longrightarrow e : X \longrightarrow \tau}{
adamc@655 737 \Gamma, X \vdash e : \tau
adamc@655 738 }$$
adamc@655 739
adamc@533 740 $$\infer{\Gamma \vdash \{\overline{c = e}\} : \{\overline{c : \tau}\}}{
adamc@533 741 \forall i: \Gamma \vdash c_i :: \mt{Name}
adamc@533 742 & \Gamma \vdash e_i : \tau_i
adamc@533 743 & \forall i \neq j: \Gamma \vdash c_i \sim c_j
adamc@533 744 }
adamc@533 745 \quad \infer{\Gamma \vdash e.c : \tau}{
adamc@533 746 \Gamma \vdash e : \$([c = \tau] \rc c')
adamc@533 747 }
adamc@533 748 \quad \infer{\Gamma \vdash e_1 \rc e_2 : \$(c_1 \rc c_2)}{
adamc@533 749 \Gamma \vdash e_1 : \$c_1
adamc@533 750 & \Gamma \vdash e_2 : \$c_2
adamc@573 751 & \Gamma \vdash c_1 \sim c_2
adamc@533 752 }$$
adamc@533 753
adamc@533 754 $$\infer{\Gamma \vdash e \rcut c : \$c'}{
adamc@533 755 \Gamma \vdash e : \$([c = \tau] \rc c')
adamc@533 756 }
adamc@533 757 \quad \infer{\Gamma \vdash e \rcutM c : \$c'}{
adamc@533 758 \Gamma \vdash e : \$(c \rc c')
adamc@533 759 }$$
adamc@533 760
adamc@533 761 $$\infer{\Gamma \vdash \mt{let} \; \overline{ed} \; \mt{in} \; e \; \mt{end} : \tau}{
adamc@533 762 \Gamma \vdash \overline{ed} \leadsto \Gamma'
adamc@533 763 & \Gamma' \vdash e : \tau
adamc@533 764 }
adamc@533 765 \quad \infer{\Gamma \vdash \mt{case} \; e \; \mt{of} \; \overline{p \Rightarrow e} : \tau}{
adamc@533 766 \forall i: \Gamma \vdash p_i \leadsto \Gamma_i, \tau'
adamc@533 767 & \Gamma_i \vdash e_i : \tau
adamc@533 768 }$$
adamc@533 769
adamc@573 770 $$\infer{\Gamma \vdash \lambda [c_1 \sim c_2] \Rightarrow e : \lambda [c_1 \sim c_2] \Rightarrow \tau}{
adamc@533 771 \Gamma \vdash c_1 :: \{\kappa\}
adamc@655 772 & \Gamma \vdash c_2 :: \{\kappa'\}
adamc@533 773 & \Gamma, c_1 \sim c_2 \vdash e : \tau
adamc@662 774 }
adamc@662 775 \quad \infer{\Gamma \vdash e \; ! : \tau}{
adamc@662 776 \Gamma \vdash e : [c_1 \sim c_2] \Rightarrow \tau
adamc@662 777 & \Gamma \vdash c_1 \sim c_2
adamc@533 778 }$$
adamc@533 779
adamc@534 780 \subsection{Pattern Typing}
adamc@534 781
adamc@534 782 $$\infer{\Gamma \vdash \_ \leadsto \Gamma; \tau}{}
adamc@534 783 \quad \infer{\Gamma \vdash x \leadsto \Gamma, x : \tau; \tau}{}
adamc@534 784 \quad \infer{\Gamma \vdash \ell \leadsto \Gamma; T(\ell)}{}$$
adamc@534 785
adamc@534 786 $$\infer{\Gamma \vdash X \leadsto \Gamma; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@534 787 X : \overline{x ::: \mt{Type}} \to \tau \in \Gamma
adamc@534 788 & \textrm{$\tau$ not a function type}
adamc@534 789 }
adamc@534 790 \quad \infer{\Gamma \vdash X \; p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@534 791 X : \overline{x ::: \mt{Type}} \to \tau'' \to \tau \in \Gamma
adamc@534 792 & \Gamma \vdash p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau''
adamc@534 793 }$$
adamc@534 794
adamc@534 795 $$\infer{\Gamma \vdash M.X \leadsto \Gamma; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@537 796 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 797 & \mt{proj}(M, \overline{s}, \mt{val} \; X) = \overline{x ::: \mt{Type}} \to \tau
adamc@534 798 & \textrm{$\tau$ not a function type}
adamc@534 799 }$$
adamc@534 800
adamc@534 801 $$\infer{\Gamma \vdash M.X \; p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@537 802 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 803 & \mt{proj}(M, \overline{s}, \mt{val} \; X) = \overline{x ::: \mt{Type}} \to \tau'' \to \tau
adamc@534 804 & \Gamma \vdash p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau''
adamc@534 805 }$$
adamc@534 806
adamc@534 807 $$\infer{\Gamma \vdash \{\overline{x = p}\} \leadsto \Gamma_n; \{\overline{x = \tau}\}}{
adamc@534 808 \Gamma_0 = \Gamma
adamc@534 809 & \forall i: \Gamma_i \vdash p_i \leadsto \Gamma_{i+1}; \tau_i
adamc@534 810 }
adamc@534 811 \quad \infer{\Gamma \vdash \{\overline{x = p}, \ldots\} \leadsto \Gamma_n; \$([\overline{x = \tau}] \rc c)}{
adamc@534 812 \Gamma_0 = \Gamma
adamc@534 813 & \forall i: \Gamma_i \vdash p_i \leadsto \Gamma_{i+1}; \tau_i
adamc@534 814 }$$
adamc@534 815
adamc@852 816 $$\infer{\Gamma \vdash p : \tau \leadsto \Gamma'; \tau}{
adamc@852 817 \Gamma \vdash p \leadsto \Gamma'; \tau'
adamc@852 818 & \Gamma \vdash \tau' \equiv \tau
adamc@852 819 }$$
adamc@852 820
adamc@535 821 \subsection{Declaration Typing}
adamc@535 822
adamc@535 823 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@535 824
adamc@655 825 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@535 826
adamc@558 827 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@537 828 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@535 829
adamc@535 830 $$\infer{\Gamma \vdash \cdot \leadsto \Gamma}{}
adamc@535 831 \quad \infer{\Gamma \vdash d, \overline{d} \leadsto \Gamma''}{
adamc@535 832 \Gamma \vdash d \leadsto \Gamma'
adamc@535 833 & \Gamma' \vdash \overline{d} \leadsto \Gamma''
adamc@535 834 }$$
adamc@535 835
adamc@535 836 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa = c \leadsto \Gamma, x :: \kappa = c}{
adamc@535 837 \Gamma \vdash c :: \kappa
adamc@535 838 }
adamc@535 839 \quad \infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leadsto \Gamma'}{
adamc@535 840 \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} \vdash \overline{dc} \leadsto \Gamma'
adamc@535 841 }$$
adamc@535 842
adamc@535 843 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leadsto \Gamma'}{
adamc@537 844 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 845 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@535 846 & \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} = M.z \vdash \overline{dc} \leadsto \Gamma'
adamc@535 847 }$$
adamc@535 848
adamc@535 849 $$\infer{\Gamma \vdash \mt{val} \; x : \tau = e \leadsto \Gamma, x : \tau}{
adamc@535 850 \Gamma \vdash e : \tau
adamc@535 851 }$$
adamc@535 852
adamc@535 853 $$\infer{\Gamma \vdash \mt{val} \; \mt{rec} \; \overline{x : \tau = e} \leadsto \Gamma, \overline{x : \tau}}{
adamc@535 854 \forall i: \Gamma, \overline{x : \tau} \vdash e_i : \tau_i
adamc@535 855 & \textrm{$e_i$ starts with an expression $\lambda$, optionally preceded by constructor and disjointness $\lambda$s}
adamc@535 856 }$$
adamc@535 857
adamc@535 858 $$\infer{\Gamma \vdash \mt{structure} \; X : S = M \leadsto \Gamma, X : S}{
adamc@535 859 \Gamma \vdash M : S
adamc@558 860 & \textrm{ $M$ not a constant or application}
adamc@535 861 }
adamc@558 862 \quad \infer{\Gamma \vdash \mt{structure} \; X : S = M \leadsto \Gamma, X : \mt{selfify}(X, \overline{s})}{
adamc@558 863 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@539 864 }$$
adamc@539 865
adamc@539 866 $$\infer{\Gamma \vdash \mt{signature} \; X = S \leadsto \Gamma, X = S}{
adamc@535 867 \Gamma \vdash S
adamc@535 868 }$$
adamc@535 869
adamc@537 870 $$\infer{\Gamma \vdash \mt{open} \; M \leadsto \Gamma, \mathcal O(M, \overline{s})}{
adamc@537 871 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@535 872 }$$
adamc@535 873
adamc@535 874 $$\infer{\Gamma \vdash \mt{constraint} \; c_1 \sim c_2 \leadsto \Gamma}{
adamc@535 875 \Gamma \vdash c_1 :: \{\kappa\}
adamc@535 876 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@535 877 & \Gamma \vdash c_1 \sim c_2
adamc@535 878 }
adamc@537 879 \quad \infer{\Gamma \vdash \mt{open} \; \mt{constraints} \; M \leadsto \Gamma, \mathcal O_c(M, \overline{s})}{
adamc@537 880 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@535 881 }$$
adamc@535 882
adamc@784 883 $$\infer{\Gamma \vdash \mt{table} \; x : c \leadsto \Gamma, x : \mt{Basis}.\mt{sql\_table} \; c \; []}{
adamc@535 884 \Gamma \vdash c :: \{\mt{Type}\}
adamc@535 885 }
adamc@784 886 \quad \infer{\Gamma \vdash \mt{view} \; x : c \leadsto \Gamma, x : \mt{Basis}.\mt{sql\_view} \; c}{
adamc@784 887 \Gamma \vdash c :: \{\mt{Type}\}
adamc@784 888 }$$
adamc@784 889
adamc@784 890 $$\infer{\Gamma \vdash \mt{sequence} \; x \leadsto \Gamma, x : \mt{Basis}.\mt{sql\_sequence}}{}$$
adamc@535 891
adamc@535 892 $$\infer{\Gamma \vdash \mt{cookie} \; x : \tau \leadsto \Gamma, x : \mt{Basis}.\mt{http\_cookie} \; \tau}{
adamc@535 893 \Gamma \vdash \tau :: \mt{Type}
adamc@784 894 }
adamc@784 895 \quad \infer{\Gamma \vdash \mt{style} \; x \leadsto \Gamma, x : \mt{Basis}.\mt{css\_class}}{}$$
adamc@535 896
adamc@784 897 $$\infer{\Gamma \vdash \mt{class} \; x :: \kappa = c \leadsto \Gamma, x :: \kappa = c}{
adamc@784 898 \Gamma \vdash c :: \kappa
adamc@535 899 }$$
adamc@535 900
adamc@535 901 $$\infer{\overline{y}; x; \Gamma \vdash \cdot \leadsto \Gamma}{}
adamc@535 902 \quad \infer{\overline{y}; x; \Gamma \vdash X \mid \overline{dc} \leadsto \Gamma', X : \overline{y ::: \mt{Type}} \to x \; \overline{y}}{
adamc@535 903 \overline{y}; x; \Gamma \vdash \overline{dc} \leadsto \Gamma'
adamc@535 904 }
adamc@535 905 \quad \infer{\overline{y}; x; \Gamma \vdash X \; \mt{of} \; \tau \mid \overline{dc} \leadsto \Gamma', X : \overline{y ::: \mt{Type}} \to \tau \to x \; \overline{y}}{
adamc@535 906 \overline{y}; x; \Gamma \vdash \overline{dc} \leadsto \Gamma'
adamc@535 907 }$$
adamc@535 908
adamc@537 909 \subsection{Signature Item Typing}
adamc@537 910
adamc@537 911 We appeal to a signature item analogue of the $\mathcal O$ function from the last subsection.
adamc@537 912
adamc@537 913 $$\infer{\Gamma \vdash \cdot \leadsto \Gamma}{}
adamc@537 914 \quad \infer{\Gamma \vdash s, \overline{s} \leadsto \Gamma''}{
adamc@537 915 \Gamma \vdash s \leadsto \Gamma'
adamc@537 916 & \Gamma' \vdash \overline{s} \leadsto \Gamma''
adamc@537 917 }$$
adamc@537 918
adamc@537 919 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa \leadsto \Gamma, x :: \kappa}{}
adamc@537 920 \quad \infer{\Gamma \vdash \mt{con} \; x :: \kappa = c \leadsto \Gamma, x :: \kappa = c}{
adamc@537 921 \Gamma \vdash c :: \kappa
adamc@537 922 }
adamc@537 923 \quad \infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leadsto \Gamma'}{
adamc@537 924 \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} \vdash \overline{dc} \leadsto \Gamma'
adamc@537 925 }$$
adamc@537 926
adamc@537 927 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leadsto \Gamma'}{
adamc@537 928 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 929 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@537 930 & \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} = M.z \vdash \overline{dc} \leadsto \Gamma'
adamc@537 931 }$$
adamc@537 932
adamc@537 933 $$\infer{\Gamma \vdash \mt{val} \; x : \tau \leadsto \Gamma, x : \tau}{
adamc@537 934 \Gamma \vdash \tau :: \mt{Type}
adamc@537 935 }$$
adamc@537 936
adamc@537 937 $$\infer{\Gamma \vdash \mt{structure} \; X : S \leadsto \Gamma, X : S}{
adamc@537 938 \Gamma \vdash S
adamc@537 939 }
adamc@537 940 \quad \infer{\Gamma \vdash \mt{signature} \; X = S \leadsto \Gamma, X = S}{
adamc@537 941 \Gamma \vdash S
adamc@537 942 }$$
adamc@537 943
adamc@537 944 $$\infer{\Gamma \vdash \mt{include} \; S \leadsto \Gamma, \mathcal O(\overline{s})}{
adamc@537 945 \Gamma \vdash S
adamc@537 946 & \Gamma \vdash S \equiv \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 947 }$$
adamc@537 948
adamc@537 949 $$\infer{\Gamma \vdash \mt{constraint} \; c_1 \sim c_2 \leadsto \Gamma, c_1 \sim c_2}{
adamc@537 950 \Gamma \vdash c_1 :: \{\kappa\}
adamc@537 951 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@537 952 }$$
adamc@537 953
adamc@784 954 $$\infer{\Gamma \vdash \mt{class} \; x :: \kappa = c \leadsto \Gamma, x :: \kappa = c}{
adamc@784 955 \Gamma \vdash c :: \kappa
adamc@537 956 }
adamc@784 957 \quad \infer{\Gamma \vdash \mt{class} \; x :: \kappa \leadsto \Gamma, x :: \kappa}{}$$
adamc@537 958
adamc@536 959 \subsection{Signature Compatibility}
adamc@536 960
adamc@558 961 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@537 962
adamc@537 963 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@537 964
adamc@536 965 $$\infer{\Gamma \vdash S \equiv S}{}
adamc@536 966 \quad \infer{\Gamma \vdash S_1 \equiv S_2}{
adamc@536 967 \Gamma \vdash S_2 \equiv S_1
adamc@536 968 }
adamc@536 969 \quad \infer{\Gamma \vdash X \equiv S}{
adamc@536 970 X = S \in \Gamma
adamc@536 971 }
adamc@536 972 \quad \infer{\Gamma \vdash M.X \equiv S}{
adamc@537 973 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 974 & \mt{proj}(M, \overline{s}, \mt{signature} \; X) = S
adamc@536 975 }$$
adamc@536 976
adamc@536 977 $$\infer{\Gamma \vdash S \; \mt{where} \; \mt{con} \; x = c \equiv \mt{sig} \; \overline{s^1} \; \mt{con} \; x :: \kappa = c \; \overline{s_2} \; \mt{end}}{
adamc@536 978 \Gamma \vdash S \equiv \mt{sig} \; \overline{s^1} \; \mt{con} \; x :: \kappa \; \overline{s_2} \; \mt{end}
adamc@536 979 & \Gamma \vdash c :: \kappa
adamc@537 980 }
adamc@537 981 \quad \infer{\Gamma \vdash \mt{sig} \; \overline{s^1} \; \mt{include} \; S \; \overline{s^2} \; \mt{end} \equiv \mt{sig} \; \overline{s^1} \; \overline{s} \; \overline{s^2} \; \mt{end}}{
adamc@537 982 \Gamma \vdash S \equiv \mt{sig} \; \overline{s} \; \mt{end}
adamc@536 983 }$$
adamc@536 984
adamc@536 985 $$\infer{\Gamma \vdash S_1 \leq S_2}{
adamc@536 986 \Gamma \vdash S_1 \equiv S_2
adamc@536 987 }
adamc@536 988 \quad \infer{\Gamma \vdash \mt{sig} \; \overline{s} \; \mt{end} \leq \mt{sig} \; \mt{end}}{}
adamc@537 989 \quad \infer{\Gamma \vdash \mt{sig} \; \overline{s} \; \mt{end} \leq \mt{sig} \; s' \; \overline{s'} \; \mt{end}}{
adamc@537 990 \Gamma \vdash \overline{s} \leq s'
adamc@537 991 & \Gamma \vdash s' \leadsto \Gamma'
adamc@537 992 & \Gamma' \vdash \mt{sig} \; \overline{s} \; \mt{end} \leq \mt{sig} \; \overline{s'} \; \mt{end}
adamc@537 993 }$$
adamc@537 994
adamc@537 995 $$\infer{\Gamma \vdash s \; \overline{s} \leq s'}{
adamc@537 996 \Gamma \vdash s \leq s'
adamc@537 997 }
adamc@537 998 \quad \infer{\Gamma \vdash s \; \overline{s} \leq s'}{
adamc@537 999 \Gamma \vdash s \leadsto \Gamma'
adamc@537 1000 & \Gamma' \vdash \overline{s} \leq s'
adamc@536 1001 }$$
adamc@536 1002
adamc@536 1003 $$\infer{\Gamma \vdash \mt{functor} (X : S_1) : S_2 \leq \mt{functor} (X : S'_1) : S'_2}{
adamc@536 1004 \Gamma \vdash S'_1 \leq S_1
adamc@536 1005 & \Gamma, X : S'_1 \vdash S_2 \leq S'_2
adamc@536 1006 }$$
adamc@536 1007
adamc@537 1008 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa \leq \mt{con} \; x :: \kappa}{}
adamc@537 1009 \quad \infer{\Gamma \vdash \mt{con} \; x :: \kappa = c \leq \mt{con} \; x :: \kappa}{}
adamc@558 1010 \quad \infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leq \mt{con} \; x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type}}{}$$
adamc@537 1011
adamc@537 1012 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leq \mt{con} \; x :: \mt{Type}^{\mt{len}(y)} \to \mt{Type}}{
adamc@537 1013 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 1014 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@537 1015 }$$
adamc@537 1016
adamc@784 1017 $$\infer{\Gamma \vdash \mt{class} \; x :: \kappa \leq \mt{con} \; x :: \kappa}{}
adamc@784 1018 \quad \infer{\Gamma \vdash \mt{class} \; x :: \kappa = c \leq \mt{con} \; x :: \kappa}{}$$
adamc@537 1019
adamc@537 1020 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa = c_1 \leq \mt{con} \; x :: \mt{\kappa} = c_2}{
adamc@537 1021 \Gamma \vdash c_1 \equiv c_2
adamc@537 1022 }
adamc@784 1023 \quad \infer{\Gamma \vdash \mt{class} \; x :: \kappa = c_1 \leq \mt{con} \; x :: \kappa = c_2}{
adamc@537 1024 \Gamma \vdash c_1 \equiv c_2
adamc@537 1025 }$$
adamc@537 1026
adamc@537 1027 $$\infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leq \mt{datatype} \; x \; \overline{y} = \overline{dc'}}{
adamc@537 1028 \Gamma, \overline{y :: \mt{Type}} \vdash \overline{dc} \leq \overline{dc'}
adamc@537 1029 }$$
adamc@537 1030
adamc@537 1031 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leq \mt{datatype} \; x \; \overline{y} = \overline{dc'}}{
adamc@537 1032 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 1033 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@537 1034 & \Gamma, \overline{y :: \mt{Type}} \vdash \overline{dc} \leq \overline{dc'}
adamc@537 1035 }$$
adamc@537 1036
adamc@537 1037 $$\infer{\Gamma \vdash \cdot \leq \cdot}{}
adamc@537 1038 \quad \infer{\Gamma \vdash X; \overline{dc} \leq X; \overline{dc'}}{
adamc@537 1039 \Gamma \vdash \overline{dc} \leq \overline{dc'}
adamc@537 1040 }
adamc@537 1041 \quad \infer{\Gamma \vdash X \; \mt{of} \; \tau_1; \overline{dc} \leq X \; \mt{of} \; \tau_2; \overline{dc'}}{
adamc@537 1042 \Gamma \vdash \tau_1 \equiv \tau_2
adamc@537 1043 & \Gamma \vdash \overline{dc} \leq \overline{dc'}
adamc@537 1044 }$$
adamc@537 1045
adamc@537 1046 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leq \mt{datatype} \; x = \mt{datatype} \; M'.z'}{
adamc@537 1047 \Gamma \vdash M.z \equiv M'.z'
adamc@537 1048 }$$
adamc@537 1049
adamc@537 1050 $$\infer{\Gamma \vdash \mt{val} \; x : \tau_1 \leq \mt{val} \; x : \tau_2}{
adamc@537 1051 \Gamma \vdash \tau_1 \equiv \tau_2
adamc@537 1052 }
adamc@537 1053 \quad \infer{\Gamma \vdash \mt{structure} \; X : S_1 \leq \mt{structure} \; X : S_2}{
adamc@537 1054 \Gamma \vdash S_1 \leq S_2
adamc@537 1055 }
adamc@537 1056 \quad \infer{\Gamma \vdash \mt{signature} \; X = S_1 \leq \mt{signature} \; X = S_2}{
adamc@537 1057 \Gamma \vdash S_1 \leq S_2
adamc@537 1058 & \Gamma \vdash S_2 \leq S_1
adamc@537 1059 }$$
adamc@537 1060
adamc@537 1061 $$\infer{\Gamma \vdash \mt{constraint} \; c_1 \sim c_2 \leq \mt{constraint} \; c'_1 \sim c'_2}{
adamc@537 1062 \Gamma \vdash c_1 \equiv c'_1
adamc@537 1063 & \Gamma \vdash c_2 \equiv c'_2
adamc@537 1064 }$$
adamc@537 1065
adamc@655 1066 $$\infer{\Gamma \vdash \mt{class} \; x :: \kappa \leq \mt{class} \; x :: \kappa}{}
adamc@655 1067 \quad \infer{\Gamma \vdash \mt{class} \; x :: \kappa = c \leq \mt{class} \; x :: \kappa}{}
adamc@655 1068 \quad \infer{\Gamma \vdash \mt{class} \; x :: \kappa = c_1 \leq \mt{class} \; x :: \kappa = c_2}{
adamc@537 1069 \Gamma \vdash c_1 \equiv c_2
adamc@537 1070 }$$
adamc@537 1071
adamc@538 1072 \subsection{Module Typing}
adamc@538 1073
adamc@538 1074 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 1075
adamc@538 1076 $$\infer{\Gamma \vdash M : S}{
adamc@538 1077 \Gamma \vdash M : S'
adamc@538 1078 & \Gamma \vdash S' \leq S
adamc@538 1079 }
adamc@538 1080 \quad \infer{\Gamma \vdash \mt{struct} \; \overline{d} \; \mt{end} : \mt{sig} \; \mt{sigOf}(\overline{d}) \; \mt{end}}{
adamc@538 1081 \Gamma \vdash \overline{d} \leadsto \Gamma'
adamc@538 1082 }
adamc@538 1083 \quad \infer{\Gamma \vdash X : S}{
adamc@538 1084 X : S \in \Gamma
adamc@538 1085 }$$
adamc@538 1086
adamc@538 1087 $$\infer{\Gamma \vdash M.X : S}{
adamc@538 1088 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@538 1089 & \mt{proj}(M, \overline{s}, \mt{structure} \; X) = S
adamc@538 1090 }$$
adamc@538 1091
adamc@538 1092 $$\infer{\Gamma \vdash M_1(M_2) : [X \mapsto M_2]S_2}{
adamc@538 1093 \Gamma \vdash M_1 : \mt{functor}(X : S_1) : S_2
adamc@538 1094 & \Gamma \vdash M_2 : S_1
adamc@538 1095 }
adamc@538 1096 \quad \infer{\Gamma \vdash \mt{functor} (X : S_1) : S_2 = M : \mt{functor} (X : S_1) : S_2}{
adamc@538 1097 \Gamma \vdash S_1
adamc@538 1098 & \Gamma, X : S_1 \vdash S_2
adamc@538 1099 & \Gamma, X : S_1 \vdash M : S_2
adamc@538 1100 }$$
adamc@538 1101
adamc@538 1102 \begin{eqnarray*}
adamc@538 1103 \mt{sigOf}(\cdot) &=& \cdot \\
adamc@538 1104 \mt{sigOf}(s \; \overline{s'}) &=& \mt{sigOf}(s) \; \mt{sigOf}(\overline{s'}) \\
adamc@538 1105 \\
adamc@538 1106 \mt{sigOf}(\mt{con} \; x :: \kappa = c) &=& \mt{con} \; x :: \kappa = c \\
adamc@538 1107 \mt{sigOf}(\mt{datatype} \; x \; \overline{y} = \overline{dc}) &=& \mt{datatype} \; x \; \overline{y} = \overline{dc} \\
adamc@538 1108 \mt{sigOf}(\mt{datatype} \; x = \mt{datatype} \; M.z) &=& \mt{datatype} \; x = \mt{datatype} \; M.z \\
adamc@538 1109 \mt{sigOf}(\mt{val} \; x : \tau = e) &=& \mt{val} \; x : \tau \\
adamc@538 1110 \mt{sigOf}(\mt{val} \; \mt{rec} \; \overline{x : \tau = e}) &=& \overline{\mt{val} \; x : \tau} \\
adamc@538 1111 \mt{sigOf}(\mt{structure} \; X : S = M) &=& \mt{structure} \; X : S \\
adamc@538 1112 \mt{sigOf}(\mt{signature} \; X = S) &=& \mt{signature} \; X = S \\
adamc@538 1113 \mt{sigOf}(\mt{open} \; M) &=& \mt{include} \; S \textrm{ (where $\Gamma \vdash M : S$)} \\
adamc@538 1114 \mt{sigOf}(\mt{constraint} \; c_1 \sim c_2) &=& \mt{constraint} \; c_1 \sim c_2 \\
adamc@538 1115 \mt{sigOf}(\mt{open} \; \mt{constraints} \; M) &=& \cdot \\
adamc@538 1116 \mt{sigOf}(\mt{table} \; x : c) &=& \mt{table} \; x : c \\
adamc@784 1117 \mt{sigOf}(\mt{view} \; x : c) &=& \mt{view} \; x : c \\
adamc@538 1118 \mt{sigOf}(\mt{sequence} \; x) &=& \mt{sequence} \; x \\
adamc@538 1119 \mt{sigOf}(\mt{cookie} \; x : \tau) &=& \mt{cookie} \; x : \tau \\
adamc@784 1120 \mt{sigOf}(\mt{style} \; x) &=& \mt{style} \; x \\
adamc@655 1121 \mt{sigOf}(\mt{class} \; x :: \kappa = c) &=& \mt{class} \; x :: \kappa = c \\
adamc@538 1122 \end{eqnarray*}
adamc@539 1123 \begin{eqnarray*}
adamc@539 1124 \mt{selfify}(M, \cdot) &=& \cdot \\
adamc@558 1125 \mt{selfify}(M, s \; \overline{s'}) &=& \mt{selfify}(M, s) \; \mt{selfify}(M, \overline{s'}) \\
adamc@539 1126 \\
adamc@539 1127 \mt{selfify}(M, \mt{con} \; x :: \kappa) &=& \mt{con} \; x :: \kappa = M.x \\
adamc@539 1128 \mt{selfify}(M, \mt{con} \; x :: \kappa = c) &=& \mt{con} \; x :: \kappa = c \\
adamc@539 1129 \mt{selfify}(M, \mt{datatype} \; x \; \overline{y} = \overline{dc}) &=& \mt{datatype} \; x \; \overline{y} = \mt{datatype} \; M.x \\
adamc@539 1130 \mt{selfify}(M, \mt{datatype} \; x = \mt{datatype} \; M'.z) &=& \mt{datatype} \; x = \mt{datatype} \; M'.z \\
adamc@539 1131 \mt{selfify}(M, \mt{val} \; x : \tau) &=& \mt{val} \; x : \tau \\
adamc@539 1132 \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 1133 \mt{selfify}(M, \mt{signature} \; X = S) &=& \mt{signature} \; X = S \\
adamc@539 1134 \mt{selfify}(M, \mt{include} \; S) &=& \mt{include} \; S \\
adamc@539 1135 \mt{selfify}(M, \mt{constraint} \; c_1 \sim c_2) &=& \mt{constraint} \; c_1 \sim c_2 \\
adamc@655 1136 \mt{selfify}(M, \mt{class} \; x :: \kappa) &=& \mt{class} \; x :: \kappa = M.x \\
adamc@655 1137 \mt{selfify}(M, \mt{class} \; x :: \kappa = c) &=& \mt{class} \; x :: \kappa = c \\
adamc@539 1138 \end{eqnarray*}
adamc@539 1139
adamc@540 1140 \subsection{Module Projection}
adamc@540 1141
adamc@540 1142 \begin{eqnarray*}
adamc@540 1143 \mt{proj}(M, \mt{con} \; x :: \kappa \; \overline{s}, \mt{con} \; x) &=& \kappa \\
adamc@540 1144 \mt{proj}(M, \mt{con} \; x :: \kappa = c \; \overline{s}, \mt{con} \; x) &=& (\kappa, c) \\
adamc@540 1145 \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 1146 \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 1147 && \textrm{and $\mt{proj}(M', \overline{s'}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})$)} \\
adamc@655 1148 \mt{proj}(M, \mt{class} \; x :: \kappa \; \overline{s}, \mt{con} \; x) &=& \kappa \to \mt{Type} \\
adamc@655 1149 \mt{proj}(M, \mt{class} \; x :: \kappa = c \; \overline{s}, \mt{con} \; x) &=& (\kappa \to \mt{Type}, c) \\
adamc@540 1150 \\
adamc@540 1151 \mt{proj}(M, \mt{datatype} \; x \; \overline{y} = \overline{dc} \; \overline{s}, \mt{datatype} \; x) &=& (\overline{y}, \overline{dc}) \\
adamc@540 1152 \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 1153 \\
adamc@540 1154 \mt{proj}(M, \mt{val} \; x : \tau \; \overline{s}, \mt{val} \; x) &=& \tau \\
adamc@540 1155 \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 1156 \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 1157 \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 1158 && \textrm{and $\mt{proj}(M', \overline{s'}, \mt{datatype} \; z = (\overline{y}, \overline{dc})$ and $X \in \overline{dc}$)} \\
adamc@540 1159 \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 1160 && \textrm{and $\mt{proj}(M', \overline{s'}, \mt{datatype} \; z = (\overline{y}, \overline{dc})$ and $X \; \mt{of} \; \tau \in \overline{dc}$)} \\
adamc@540 1161 \\
adamc@540 1162 \mt{proj}(M, \mt{structure} \; X : S \; \overline{s}, \mt{structure} \; X) &=& S \\
adamc@540 1163 \\
adamc@540 1164 \mt{proj}(M, \mt{signature} \; X = S \; \overline{s}, \mt{signature} \; X) &=& S \\
adamc@540 1165 \\
adamc@540 1166 \mt{proj}(M, \mt{con} \; x :: \kappa \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@540 1167 \mt{proj}(M, \mt{con} \; x :: \kappa = c \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@540 1168 \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 1169 \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 1170 \mt{proj}(M, \mt{val} \; x : \tau \; \overline{s}, V) &=& \mt{proj}(M, \overline{s}, V) \\
adamc@540 1171 \mt{proj}(M, \mt{structure} \; X : S \; \overline{s}, V) &=& [X \mapsto M.X]\mt{proj}(M, \overline{s}, V) \\
adamc@540 1172 \mt{proj}(M, \mt{signature} \; X = S \; \overline{s}, V) &=& [X \mapsto M.X]\mt{proj}(M, \overline{s}, V) \\
adamc@540 1173 \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 1174 \mt{proj}(M, \mt{constraint} \; c_1 \sim c_2 \; \overline{s}, V) &=& \mt{proj}(M, \overline{s}, V) \\
adamc@655 1175 \mt{proj}(M, \mt{class} \; x :: \kappa \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@655 1176 \mt{proj}(M, \mt{class} \; x :: \kappa = c \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@540 1177 \end{eqnarray*}
adamc@540 1178
adamc@541 1179
adamc@541 1180 \section{Type Inference}
adamc@541 1181
adamc@541 1182 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 1183
adamc@541 1184 \subsection{Basic Unification}
adamc@541 1185
adamc@560 1186 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 1187
adamc@656 1188 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 1189
adamc@541 1190 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 1191
adamc@541 1192 \subsection{Unifying Record Types}
adamc@541 1193
adamc@570 1194 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 1195
adamc@656 1196 \subsection{\label{typeclasses}Constructor Classes}
adamc@541 1197
adamc@784 1198 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 1199
adamc@784 1200 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 1201
adamc@656 1202 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 1203
adamc@656 1204 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 1205
adamc@541 1206 \subsection{Reverse-Engineering Record Types}
adamc@541 1207
adamc@656 1208 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 1209
adamc@541 1210 \subsection{Implicit Arguments in Functor Applications}
adamc@541 1211
adamc@656 1212 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 1213
adamc@541 1214
adamc@542 1215 \section{The Ur Standard Library}
adamc@542 1216
adamc@542 1217 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 1218
adamc@542 1219 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 1220
adamc@542 1221 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 1222 $$\begin{array}{l}
adamc@542 1223 \mt{type} \; \mt{int} \\
adamc@542 1224 \mt{type} \; \mt{float} \\
adamc@873 1225 \mt{type} \; \mt{char} \\
adamc@542 1226 \mt{type} \; \mt{string} \\
adamc@542 1227 \mt{type} \; \mt{time} \\
adamc@785 1228 \mt{type} \; \mt{blob} \\
adamc@542 1229 \\
adamc@542 1230 \mt{type} \; \mt{unit} = \{\} \\
adamc@542 1231 \\
adamc@542 1232 \mt{datatype} \; \mt{bool} = \mt{False} \mid \mt{True} \\
adamc@542 1233 \\
adamc@785 1234 \mt{datatype} \; \mt{option} \; \mt{t} = \mt{None} \mid \mt{Some} \; \mt{of} \; \mt{t} \\
adamc@785 1235 \\
adamc@785 1236 \mt{datatype} \; \mt{list} \; \mt{t} = \mt{Nil} \mid \mt{Cons} \; \mt{of} \; \mt{t} \times \mt{list} \; \mt{t}
adamc@542 1237 \end{array}$$
adamc@542 1238
adamc@785 1239 The only unusual element of this list is the $\mt{blob}$ type, which stands for binary sequences.
adamc@785 1240
adamc@657 1241 Another important generic Ur element comes at the beginning of \texttt{top.urs}.
adamc@657 1242
adamc@657 1243 $$\begin{array}{l}
adamc@657 1244 \mt{con} \; \mt{folder} :: \mt{K} \longrightarrow \{\mt{K}\} \to \mt{Type} \\
adamc@657 1245 \\
adamc@657 1246 \mt{val} \; \mt{fold} : \mt{K} \longrightarrow \mt{tf} :: (\{\mt{K}\} \to \mt{Type}) \\
adamc@657 1247 \hspace{.1in} \to (\mt{nm} :: \mt{Name} \to \mt{v} :: \mt{K} \to \mt{r} :: \{\mt{K}\} \to [[\mt{nm}] \sim \mt{r}] \Rightarrow \\
adamc@657 1248 \hspace{.2in} \mt{tf} \; \mt{r} \to \mt{tf} \; ([\mt{nm} = \mt{v}] \rc \mt{r})) \\
adamc@657 1249 \hspace{.1in} \to \mt{tf} \; [] \\
adamc@657 1250 \hspace{.1in} \to \mt{r} :: \{\mt{K}\} \to \mt{folder} \; \mt{r} \to \mt{tf} \; \mt{r}
adamc@657 1251 \end{array}$$
adamc@657 1252
adamc@657 1253 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 1254
adamc@664 1255 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 1256
adamc@542 1257
adamc@542 1258 \section{The Ur/Web Standard Library}
adamc@542 1259
adamc@658 1260 \subsection{Monads}
adamc@658 1261
adamc@658 1262 The Ur Basis defines the monad constructor class from Haskell.
adamc@658 1263
adamc@658 1264 $$\begin{array}{l}
adamc@658 1265 \mt{class} \; \mt{monad} :: \mt{Type} \to \mt{Type} \\
adamc@658 1266 \mt{val} \; \mt{return} : \mt{m} ::: (\mt{Type} \to \mt{Type}) \to \mt{t} ::: \mt{Type} \\
adamc@658 1267 \hspace{.1in} \to \mt{monad} \; \mt{m} \\
adamc@658 1268 \hspace{.1in} \to \mt{t} \to \mt{m} \; \mt{t} \\
adamc@658 1269 \mt{val} \; \mt{bind} : \mt{m} ::: (\mt{Type} \to \mt{Type}) \to \mt{t1} ::: \mt{Type} \to \mt{t2} ::: \mt{Type} \\
adamc@658 1270 \hspace{.1in} \to \mt{monad} \; \mt{m} \\
adamc@658 1271 \hspace{.1in} \to \mt{m} \; \mt{t1} \to (\mt{t1} \to \mt{m} \; \mt{t2}) \\
adamc@658 1272 \hspace{.1in} \to \mt{m} \; \mt{t2}
adamc@658 1273 \end{array}$$
adamc@658 1274
adamc@542 1275 \subsection{Transactions}
adamc@542 1276
adamc@542 1277 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 1278 $$\begin{array}{l}
adamc@542 1279 \mt{con} \; \mt{transaction} :: \mt{Type} \to \mt{Type} \\
adamc@658 1280 \mt{val} \; \mt{transaction\_monad} : \mt{monad} \; \mt{transaction}
adamc@542 1281 \end{array}$$
adamc@542 1282
adamc@542 1283 \subsection{HTTP}
adamc@542 1284
adamc@542 1285 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 1286 $$\begin{array}{l}
adamc@786 1287 \mt{val} \; \mt{requestHeader} : \mt{string} \to \mt{transaction} \; (\mt{option} \; \mt{string}) \\
adamc@786 1288 \\
adamc@786 1289 \mt{con} \; \mt{http\_cookie} :: \mt{Type} \to \mt{Type} \\
adamc@786 1290 \mt{val} \; \mt{getCookie} : \mt{t} ::: \mt{Type} \to \mt{http\_cookie} \; \mt{t} \to \mt{transaction} \; (\mt{option} \; \mt{t}) \\
adamc@1050 1291 \mt{val} \; \mt{setCookie} : \mt{t} ::: \mt{Type} \to \mt{http\_cookie} \; \mt{t} \to \{\mt{Value} : \mt{t}, \mt{Expires} : \mt{option} \; \mt{time}, \mt{Secure} : \mt{bool}\} \to \mt{transaction} \; \mt{unit} \\
adamc@1050 1292 \mt{val} \; \mt{clearCookie} : \mt{t} ::: \mt{Type} \to \mt{http\_cookie} \; \mt{t} \to \mt{transaction} \; \mt{unit}
adamc@786 1293 \end{array}$$
adamc@786 1294
adamc@786 1295 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 1296 $$\begin{array}{l}
adamc@786 1297 \mt{type} \; \mt{url} \\
adamc@786 1298 \mt{val} \; \mt{bless} : \mt{string} \to \mt{url} \\
adamc@786 1299 \mt{val} \; \mt{checkUrl} : \mt{string} \to \mt{option} \; \mt{url}
adamc@786 1300 \end{array}$$
adamc@786 1301 $\mt{bless}$ raises a runtime error if the string passed to it fails the URL policy.
adamc@786 1302
adamc@786 1303 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 1304 $$\begin{array}{l}
adamc@786 1305 \mt{type} \; \mt{file} \\
adamc@786 1306 \mt{val} \; \mt{fileName} : \mt{file} \to \mt{option} \; \mt{string} \\
adamc@786 1307 \mt{val} \; \mt{fileMimeType} : \mt{file} \to \mt{string} \\
adamc@786 1308 \mt{val} \; \mt{fileData} : \mt{file} \to \mt{blob}
adamc@786 1309 \end{array}$$
adamc@786 1310
adamc@786 1311 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 1312 $$\begin{array}{l}
adamc@786 1313 \mt{type} \; \mt{mimeType} \\
adamc@786 1314 \mt{val} \; \mt{blessMime} : \mt{string} \to \mt{mimeType} \\
adamc@786 1315 \mt{val} \; \mt{checkMime} : \mt{string} \to \mt{option} \; \mt{mimeType} \\
adamc@786 1316 \mt{val} \; \mt{returnBlob} : \mt{t} ::: \mt{Type} \to \mt{blob} \to \mt{mimeType} \to \mt{transaction} \; \mt{t}
adamc@542 1317 \end{array}$$
adamc@542 1318
adamc@543 1319 \subsection{SQL}
adamc@543 1320
adamc@543 1321 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 1322 $$\begin{array}{l}
adamc@785 1323 \mt{con} \; \mt{sql\_table} :: \{\mt{Type}\} \to \{\{\mt{Unit}\}\} \to \mt{Type}
adamc@785 1324 \end{array}$$
adamc@785 1325 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 1326
adamc@785 1327 We also have the simpler type family of SQL views, which have no keys.
adamc@785 1328 $$\begin{array}{l}
adamc@785 1329 \mt{con} \; \mt{sql\_view} :: \{\mt{Type}\} \to \mt{Type}
adamc@543 1330 \end{array}$$
adamc@543 1331
adamc@785 1332 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 1333 $$\begin{array}{l}
adamc@785 1334 \mt{class} \; \mt{fieldsOf} :: \mt{Type} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@785 1335 \mt{val} \; \mt{fieldsOf\_table} : \mt{fs} ::: \{\mt{Type}\} \to \mt{keys} ::: \{\{\mt{Unit}\}\} \to \mt{fieldsOf} \; (\mt{sql\_table} \; \mt{fs} \; \mt{keys}) \; \mt{fs} \\
adamc@785 1336 \mt{val} \; \mt{fieldsOf\_view} : \mt{fs} ::: \{\mt{Type}\} \to \mt{fieldsOf} \; (\mt{sql\_view} \; \mt{fs}) \; \mt{fs}
adamc@785 1337 \end{array}$$
adamc@785 1338
adamc@785 1339 \subsubsection{Table Constraints}
adamc@785 1340
adamc@785 1341 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 1342
adamc@785 1343 $$\begin{array}{l}
adamc@785 1344 \mt{con} \; \mt{primary\_key} :: \{\mt{Type}\} \to \{\{\mt{Unit}\}\} \to \mt{Type} \\
adamc@785 1345 \mt{val} \; \mt{no\_primary\_key} : \mt{fs} ::: \{\mt{Type}\} \to \mt{primary\_key} \; \mt{fs} \; [] \\
adamc@785 1346 \mt{val} \; \mt{primary\_key} : \mt{rest} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \to \mt{key1} :: \mt{Name} \to \mt{keys} :: \{\mt{Type}\} \\
adamc@785 1347 \hspace{.1in} \to [[\mt{key1}] \sim \mt{keys}] \Rightarrow [[\mt{key1} = \mt{t}] \rc \mt{keys} \sim \mt{rest}] \\
adamc@785 1348 \hspace{.1in} \Rightarrow \$([\mt{key1} = \mt{sql\_injectable\_prim} \; \mt{t}] \rc \mt{map} \; \mt{sql\_injectable\_prim} \; \mt{keys}) \\
adamc@785 1349 \hspace{.1in} \to \mt{primary\_key} \; ([\mt{key1} = \mt{t}] \rc \mt{keys} \rc \mt{rest}) \; [\mt{Pkey} = [\mt{key1}] \rc \mt{map} \; (\lambda \_ \Rightarrow ()) \; \mt{keys}]
adamc@785 1350 \end{array}$$
adamc@785 1351 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 1352
adamc@785 1353 A type family stands for sets of named constraints of the remaining varieties.
adamc@785 1354 $$\begin{array}{l}
adamc@785 1355 \mt{con} \; \mt{sql\_constraints} :: \{\mt{Type}\} \to \{\{\mt{Unit}\}\} \to \mt{Type}
adamc@785 1356 \end{array}$$
adamc@785 1357 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 1358
adamc@785 1359 There is a type family of individual, unnamed constraints.
adamc@785 1360 $$\begin{array}{l}
adamc@785 1361 \mt{con} \; \mt{sql\_constraint} :: \{\mt{Type}\} \to \{\mt{Unit}\} \to \mt{Type}
adamc@785 1362 \end{array}$$
adamc@785 1363 The first argument is the same as above, and the second argument gives the key columns for just this constraint.
adamc@785 1364
adamc@785 1365 We have operations for assembling constraints into constraint sets.
adamc@785 1366 $$\begin{array}{l}
adamc@785 1367 \mt{val} \; \mt{no\_constraint} : \mt{fs} ::: \{\mt{Type}\} \to \mt{sql\_constraints} \; \mt{fs} \; [] \\
adamc@785 1368 \mt{val} \; \mt{one\_constraint} : \mt{fs} ::: \{\mt{Type}\} \to \mt{unique} ::: \{\mt{Unit}\} \to \mt{name} :: \mt{Name} \\
adamc@785 1369 \hspace{.1in} \to \mt{sql\_constraint} \; \mt{fs} \; \mt{unique} \to \mt{sql\_constraints} \; \mt{fs} \; [\mt{name} = \mt{unique}] \\
adamc@785 1370 \mt{val} \; \mt{join\_constraints} : \mt{fs} ::: \{\mt{Type}\} \to \mt{uniques1} ::: \{\{\mt{Unit}\}\} \to \mt{uniques2} ::: \{\{\mt{Unit}\}\} \to [\mt{uniques1} \sim \mt{uniques2}] \\
adamc@785 1371 \hspace{.1in} \Rightarrow \mt{sql\_constraints} \; \mt{fs} \; \mt{uniques1} \to \mt{sql\_constraints} \; \mt{fs} \; \mt{uniques2} \to \mt{sql\_constraints} \; \mt{fs} \; (\mt{uniques1} \rc \mt{uniques2})
adamc@785 1372 \end{array}$$
adamc@785 1373
adamc@785 1374 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 1375 $$\begin{array}{l}
adamc@785 1376 \mt{val} \; \mt{unique} : \mt{rest} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \to \mt{unique1} :: \mt{Name} \to \mt{unique} :: \{\mt{Type}\} \\
adamc@785 1377 \hspace{.1in} \to [[\mt{unique1}] \sim \mt{unique}] \Rightarrow [[\mt{unique1} = \mt{t}] \rc \mt{unique} \sim \mt{rest}] \\
adamc@785 1378 \hspace{.1in} \Rightarrow \mt{sql\_constraint} \; ([\mt{unique1} = \mt{t}] \rc \mt{unique} \rc \mt{rest}) \; ([\mt{unique1}] \rc \mt{map} \; (\lambda \_ \Rightarrow ()) \; \mt{unique})
adamc@785 1379 \end{array}$$
adamc@785 1380
adamc@785 1381 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 1382 $$\begin{array}{l}
adamc@785 1383 \mt{class} \; \mt{linkable} :: \mt{Type} \to \mt{Type} \to \mt{Type} \\
adamc@785 1384 \mt{val} \; \mt{linkable\_same} : \mt{t} ::: \mt{Type} \to \mt{linkable} \; \mt{t} \; \mt{t} \\
adamc@785 1385 \mt{val} \; \mt{linkable\_from\_nullable} : \mt{t} ::: \mt{Type} \to \mt{linkable} \; (\mt{option} \; \mt{t}) \; \mt{t} \\
adamc@785 1386 \mt{val} \; \mt{linkable\_to\_nullable} : \mt{t} ::: \mt{Type} \to \mt{linkable} \; \mt{t} \; (\mt{option} \; \mt{t})
adamc@785 1387 \end{array}$$
adamc@785 1388
adamc@785 1389 The $\mt{matching}$ type family uses $\mt{linkable}$ to define when two keys match up type-wise.
adamc@785 1390 $$\begin{array}{l}
adamc@785 1391 \mt{con} \; \mt{matching} :: \{\mt{Type}\} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@785 1392 \mt{val} \; \mt{mat\_nil} : \mt{matching} \; [] \; [] \\
adamc@785 1393 \mt{val} \; \mt{mat\_cons} : \mt{t1} ::: \mt{Type} \to \mt{rest1} ::: \{\mt{Type}\} \to \mt{t2} ::: \mt{Type} \to \mt{rest2} ::: \{\mt{Type}\} \to \mt{nm1} :: \mt{Name} \to \mt{nm2} :: \mt{Name} \\
adamc@785 1394 \hspace{.1in} \to [[\mt{nm1}] \sim \mt{rest1}] \Rightarrow [[\mt{nm2}] \sim \mt{rest2}] \Rightarrow \mt{linkable} \; \mt{t1} \; \mt{t2} \to \mt{matching} \; \mt{rest1} \; \mt{rest2} \\
adamc@785 1395 \hspace{.1in} \to \mt{matching} \; ([\mt{nm1} = \mt{t1}] \rc \mt{rest1}) \; ([\mt{nm2} = \mt{t2}] \rc \mt{rest2})
adamc@785 1396 \end{array}$$
adamc@785 1397
adamc@785 1398 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 1399 $$\begin{array}{l}
adamc@785 1400 \mt{con} \; \mt{propagation\_mode} :: \{\mt{Type}\} \to \mt{Type} \\
adamc@785 1401 \mt{val} \; \mt{restrict} : \mt{fs} ::: \{\mt{Type}\} \to \mt{propagation\_mode} \; \mt{fs} \\
adamc@785 1402 \mt{val} \; \mt{cascade} : \mt{fs} ::: \{\mt{Type}\} \to \mt{propagation\_mode} \; \mt{fs} \\
adamc@785 1403 \mt{val} \; \mt{no\_action} : \mt{fs} ::: \{\mt{Type}\} \to \mt{propagation\_mode} \; \mt{fs} \\
adamc@785 1404 \mt{val} \; \mt{set\_null} : \mt{fs} ::: \{\mt{Type}\} \to \mt{propagation\_mode} \; (\mt{map} \; \mt{option} \; \mt{fs})
adamc@785 1405 \end{array}$$
adamc@785 1406
adamc@785 1407 Finally, we put these ingredient together to define the \texttt{FOREIGN KEY} constraint function.
adamc@785 1408 $$\begin{array}{l}
adamc@785 1409 \mt{val} \; \mt{foreign\_key} : \mt{mine1} ::: \mt{Name} \to \mt{t} ::: \mt{Type} \to \mt{mine} ::: \{\mt{Type}\} \to \mt{munused} ::: \{\mt{Type}\} \to \mt{foreign} ::: \{\mt{Type}\} \\
adamc@785 1410 \hspace{.1in} \to \mt{funused} ::: \{\mt{Type}\} \to \mt{nm} ::: \mt{Name} \to \mt{uniques} ::: \{\{\mt{Unit}\}\} \\
adamc@785 1411 \hspace{.1in} \to [[\mt{mine1}] \sim \mt{mine}] \Rightarrow [[\mt{mine1} = \mt{t}] \rc \mt{mine} \sim \mt{munused}] \Rightarrow [\mt{foreign} \sim \mt{funused}] \Rightarrow [[\mt{nm}] \sim \mt{uniques}] \\
adamc@785 1412 \hspace{.1in} \Rightarrow \mt{matching} \; ([\mt{mine1} = \mt{t}] \rc \mt{mine}) \; \mt{foreign} \\
adamc@785 1413 \hspace{.1in} \to \mt{sql\_table} \; (\mt{foreign} \rc \mt{funused}) \; ([\mt{nm} = \mt{map} \; (\lambda \_ \Rightarrow ()) \; \mt{foreign}] \rc \mt{uniques}) \\
adamc@785 1414 \hspace{.1in} \to \{\mt{OnDelete} : \mt{propagation\_mode} \; ([\mt{mine1} = \mt{t}] \rc \mt{mine}), \\
adamc@785 1415 \hspace{.2in} \mt{OnUpdate} : \mt{propagation\_mode} \; ([\mt{mine1} = \mt{t}] \rc \mt{mine})\} \\
adamc@785 1416 \hspace{.1in} \to \mt{sql\_constraint} \; ([\mt{mine1} = \mt{t}] \rc \mt{mine} \rc \mt{munused}) \; []
adamc@785 1417 \end{array}$$
adamc@785 1418
adamc@785 1419 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 1420 $$\begin{array}{l}
adamc@785 1421 \mt{val} \; \mt{check} : \mt{fs} ::: \{\mt{Type}\} \to \mt{sql\_exp} \; [] \; [] \; \mt{fs} \; \mt{bool} \to \mt{sql\_constraint} \; \mt{fs} \; []
adamc@785 1422 \end{array}$$
adamc@785 1423
adamc@785 1424 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 1425
adamc@784 1426
adamc@543 1427 \subsubsection{Queries}
adamc@543 1428
adamc@543 1429 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 1430 $$\begin{array}{l}
adamc@543 1431 \mt{con} \; \mt{sql\_query} :: \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@543 1432 \mt{val} \; \mt{sql\_query} : \mt{tables} ::: \{\{\mt{Type}\}\} \\
adamc@543 1433 \hspace{.1in} \to \mt{selectedFields} ::: \{\{\mt{Type}\}\} \\
adamc@543 1434 \hspace{.1in} \to \mt{selectedExps} ::: \{\mt{Type}\} \\
adamc@543 1435 \hspace{.1in} \to \{\mt{Rows} : \mt{sql\_query1} \; \mt{tables} \; \mt{selectedFields} \; \mt{selectedExps}, \\
adamc@543 1436 \hspace{.2in} \mt{OrderBy} : \mt{sql\_order\_by} \; \mt{tables} \; \mt{selectedExps}, \\
adamc@543 1437 \hspace{.2in} \mt{Limit} : \mt{sql\_limit}, \\
adamc@543 1438 \hspace{.2in} \mt{Offset} : \mt{sql\_offset}\} \\
adamc@543 1439 \hspace{.1in} \to \mt{sql\_query} \; \mt{selectedFields} \; \mt{selectedExps}
adamc@543 1440 \end{array}$$
adamc@543 1441
adamc@545 1442 Queries are used by folding over their results inside transactions.
adamc@545 1443 $$\begin{array}{l}
adamc@545 1444 \mt{val} \; \mt{query} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \lambda [\mt{tables} \sim \mt{exps}] \Rightarrow \mt{state} ::: \mt{Type} \to \mt{sql\_query} \; \mt{tables} \; \mt{exps} \\
adamc@658 1445 \hspace{.1in} \to (\$(\mt{exps} \rc \mt{map} \; (\lambda \mt{fields} :: \{\mt{Type}\} \Rightarrow \$\mt{fields}) \; \mt{tables}) \\
adamc@545 1446 \hspace{.2in} \to \mt{state} \to \mt{transaction} \; \mt{state}) \\
adamc@545 1447 \hspace{.1in} \to \mt{state} \to \mt{transaction} \; \mt{state}
adamc@545 1448 \end{array}$$
adamc@545 1449
adamc@543 1450 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 1451 $$\begin{array}{l}
adamc@543 1452 \mt{con} \; \mt{sql\_query1} :: \{\{\mt{Type}\}\} \to \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@543 1453 \\
adamc@543 1454 \mt{type} \; \mt{sql\_relop} \\
adamc@543 1455 \mt{val} \; \mt{sql\_union} : \mt{sql\_relop} \\
adamc@543 1456 \mt{val} \; \mt{sql\_intersect} : \mt{sql\_relop} \\
adamc@543 1457 \mt{val} \; \mt{sql\_except} : \mt{sql\_relop} \\
adamc@543 1458 \mt{val} \; \mt{sql\_relop} : \mt{tables1} ::: \{\{\mt{Type}\}\} \\
adamc@543 1459 \hspace{.1in} \to \mt{tables2} ::: \{\{\mt{Type}\}\} \\
adamc@543 1460 \hspace{.1in} \to \mt{selectedFields} ::: \{\{\mt{Type}\}\} \\
adamc@543 1461 \hspace{.1in} \to \mt{selectedExps} ::: \{\mt{Type}\} \\
adamc@543 1462 \hspace{.1in} \to \mt{sql\_relop} \\
adamc@543 1463 \hspace{.1in} \to \mt{sql\_query1} \; \mt{tables1} \; \mt{selectedFields} \; \mt{selectedExps} \\
adamc@543 1464 \hspace{.1in} \to \mt{sql\_query1} \; \mt{tables2} \; \mt{selectedFields} \; \mt{selectedExps} \\
adamc@543 1465 \hspace{.1in} \to \mt{sql\_query1} \; \mt{selectedFields} \; \mt{selectedFields} \; \mt{selectedExps}
adamc@543 1466 \end{array}$$
adamc@543 1467
adamc@543 1468 $$\begin{array}{l}
adamc@543 1469 \mt{val} \; \mt{sql\_query1} : \mt{tables} ::: \{\{\mt{Type}\}\} \\
adamc@543 1470 \hspace{.1in} \to \mt{grouped} ::: \{\{\mt{Type}\}\} \\
adamc@543 1471 \hspace{.1in} \to \mt{selectedFields} ::: \{\{\mt{Type}\}\} \\
adamc@543 1472 \hspace{.1in} \to \mt{selectedExps} ::: \{\mt{Type}\} \\
adamc@994 1473 \hspace{.1in} \to \{\mt{Distinct} : \mt{bool}, \\
adamc@994 1474 \hspace{.2in} \mt{From} : \mt{sql\_from\_items} \; \mt{tables}, \\
adamc@543 1475 \hspace{.2in} \mt{Where} : \mt{sql\_exp} \; \mt{tables} \; [] \; [] \; \mt{bool}, \\
adamc@543 1476 \hspace{.2in} \mt{GroupBy} : \mt{sql\_subset} \; \mt{tables} \; \mt{grouped}, \\
adamc@543 1477 \hspace{.2in} \mt{Having} : \mt{sql\_exp} \; \mt{grouped} \; \mt{tables} \; [] \; \mt{bool}, \\
adamc@543 1478 \hspace{.2in} \mt{SelectFields} : \mt{sql\_subset} \; \mt{grouped} \; \mt{selectedFields}, \\
adamc@658 1479 \hspace{.2in} \mt {SelectExps} : \$(\mt{map} \; (\mt{sql\_exp} \; \mt{grouped} \; \mt{tables} \; []) \; \mt{selectedExps}) \} \\
adamc@543 1480 \hspace{.1in} \to \mt{sql\_query1} \; \mt{tables} \; \mt{selectedFields} \; \mt{selectedExps}
adamc@543 1481 \end{array}$$
adamc@543 1482
adamc@543 1483 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 1484 $$\begin{array}{l}
adamc@543 1485 \mt{con} \; \mt{sql\_subset} :: \{\{\mt{Type}\}\} \to \{\{\mt{Type}\}\} \to \mt{Type} \\
adamc@543 1486 \mt{val} \; \mt{sql\_subset} : \mt{keep\_drop} :: \{(\{\mt{Type}\} \times \{\mt{Type}\})\} \\
adamc@543 1487 \hspace{.1in} \to \mt{sql\_subset} \\
adamc@658 1488 \hspace{.2in} (\mt{map} \; (\lambda \mt{fields} :: (\{\mt{Type}\} \times \{\mt{Type}\}) \Rightarrow \mt{fields}.1 \rc \mt{fields}.2)\; \mt{keep\_drop}) \\
adamc@658 1489 \hspace{.2in} (\mt{map} \; (\lambda \mt{fields} :: (\{\mt{Type}\} \times \{\mt{Type}\}) \Rightarrow \mt{fields}.1) \; \mt{keep\_drop}) \\
adamc@543 1490 \mt{val} \; \mt{sql\_subset\_all} : \mt{tables} :: \{\{\mt{Type}\}\} \to \mt{sql\_subset} \; \mt{tables} \; \mt{tables}
adamc@543 1491 \end{array}$$
adamc@543 1492
adamc@560 1493 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 1494 $$\begin{array}{l}
adamc@543 1495 \mt{con} \; \mt{sql\_exp} :: \{\{\mt{Type}\}\} \to \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \to \mt{Type}
adamc@543 1496 \end{array}$$
adamc@543 1497
adamc@543 1498 Any field in scope may be converted to an expression.
adamc@543 1499 $$\begin{array}{l}
adamc@543 1500 \mt{val} \; \mt{sql\_field} : \mt{otherTabs} ::: \{\{\mt{Type}\}\} \to \mt{otherFields} ::: \{\mt{Type}\} \\
adamc@543 1501 \hspace{.1in} \to \mt{fieldType} ::: \mt{Type} \to \mt{agg} ::: \{\{\mt{Type}\}\} \\
adamc@543 1502 \hspace{.1in} \to \mt{exps} ::: \{\mt{Type}\} \\
adamc@543 1503 \hspace{.1in} \to \mt{tab} :: \mt{Name} \to \mt{field} :: \mt{Name} \\
adamc@543 1504 \hspace{.1in} \to \mt{sql\_exp} \; ([\mt{tab} = [\mt{field} = \mt{fieldType}] \rc \mt{otherFields}] \rc \mt{otherTabs}) \; \mt{agg} \; \mt{exps} \; \mt{fieldType}
adamc@543 1505 \end{array}$$
adamc@543 1506
adamc@544 1507 There is an analogous function for referencing named expressions.
adamc@544 1508 $$\begin{array}{l}
adamc@544 1509 \mt{val} \; \mt{sql\_exp} : \mt{tabs} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{t} ::: \mt{Type} \to \mt{rest} ::: \{\mt{Type}\} \to \mt{nm} :: \mt{Name} \\
adamc@544 1510 \hspace{.1in} \to \mt{sql\_exp} \; \mt{tabs} \; \mt{agg} \; ([\mt{nm} = \mt{t}] \rc \mt{rest}) \; \mt{t}
adamc@544 1511 \end{array}$$
adamc@544 1512
adamc@544 1513 Ur values of appropriate types may be injected into SQL expressions.
adamc@544 1514 $$\begin{array}{l}
adamc@786 1515 \mt{class} \; \mt{sql\_injectable\_prim} \\
adamc@786 1516 \mt{val} \; \mt{sql\_bool} : \mt{sql\_injectable\_prim} \; \mt{bool} \\
adamc@786 1517 \mt{val} \; \mt{sql\_int} : \mt{sql\_injectable\_prim} \; \mt{int} \\
adamc@786 1518 \mt{val} \; \mt{sql\_float} : \mt{sql\_injectable\_prim} \; \mt{float} \\
adamc@786 1519 \mt{val} \; \mt{sql\_string} : \mt{sql\_injectable\_prim} \; \mt{string} \\
adamc@786 1520 \mt{val} \; \mt{sql\_time} : \mt{sql\_injectable\_prim} \; \mt{time} \\
adamc@786 1521 \mt{val} \; \mt{sql\_blob} : \mt{sql\_injectable\_prim} \; \mt{blob} \\
adamc@786 1522 \mt{val} \; \mt{sql\_channel} : \mt{t} ::: \mt{Type} \to \mt{sql\_injectable\_prim} \; (\mt{channel} \; \mt{t}) \\
adamc@786 1523 \mt{val} \; \mt{sql\_client} : \mt{sql\_injectable\_prim} \; \mt{client} \\
adamc@786 1524 \\
adamc@544 1525 \mt{class} \; \mt{sql\_injectable} \\
adamc@786 1526 \mt{val} \; \mt{sql\_prim} : \mt{t} ::: \mt{Type} \to \mt{sql\_injectable\_prim} \; \mt{t} \to \mt{sql\_injectable} \; \mt{t} \\
adamc@786 1527 \mt{val} \; \mt{sql\_option\_prim} : \mt{t} ::: \mt{Type} \to \mt{sql\_injectable\_prim} \; \mt{t} \to \mt{sql\_injectable} \; (\mt{option} \; \mt{t}) \\
adamc@786 1528 \\
adamc@544 1529 \mt{val} \; \mt{sql\_inject} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \to \mt{sql\_injectable} \; \mt{t} \\
adamc@544 1530 \hspace{.1in} \to \mt{t} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{t}
adamc@544 1531 \end{array}$$
adamc@544 1532
adamc@544 1533 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 1534 $$\begin{array}{l}
adamc@544 1535 \mt{val} \; \mt{sql\_is\_null} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544 1536 \hspace{.1in} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; (\mt{option} \; \mt{t}) \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{bool}
adamc@544 1537 \end{array}$$
adamc@544 1538
adamc@559 1539 We have generic nullary, unary, and binary operators.
adamc@544 1540 $$\begin{array}{l}
adamc@544 1541 \mt{con} \; \mt{sql\_nfunc} :: \mt{Type} \to \mt{Type} \\
adamc@544 1542 \mt{val} \; \mt{sql\_current\_timestamp} : \mt{sql\_nfunc} \; \mt{time} \\
adamc@544 1543 \mt{val} \; \mt{sql\_nfunc} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544 1544 \hspace{.1in} \to \mt{sql\_nfunc} \; \mt{t} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{t} \\\end{array}$$
adamc@544 1545
adamc@544 1546 $$\begin{array}{l}
adamc@544 1547 \mt{con} \; \mt{sql\_unary} :: \mt{Type} \to \mt{Type} \to \mt{Type} \\
adamc@544 1548 \mt{val} \; \mt{sql\_not} : \mt{sql\_unary} \; \mt{bool} \; \mt{bool} \\
adamc@544 1549 \mt{val} \; \mt{sql\_unary} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{arg} ::: \mt{Type} \to \mt{res} ::: \mt{Type} \\
adamc@544 1550 \hspace{.1in} \to \mt{sql\_unary} \; \mt{arg} \; \mt{res} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{arg} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{res} \\
adamc@544 1551 \end{array}$$
adamc@544 1552
adamc@544 1553 $$\begin{array}{l}
adamc@544 1554 \mt{con} \; \mt{sql\_binary} :: \mt{Type} \to \mt{Type} \to \mt{Type} \to \mt{Type} \\
adamc@544 1555 \mt{val} \; \mt{sql\_and} : \mt{sql\_binary} \; \mt{bool} \; \mt{bool} \; \mt{bool} \\
adamc@544 1556 \mt{val} \; \mt{sql\_or} : \mt{sql\_binary} \; \mt{bool} \; \mt{bool} \; \mt{bool} \\
adamc@544 1557 \mt{val} \; \mt{sql\_binary} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{arg_1} ::: \mt{Type} \to \mt{arg_2} ::: \mt{Type} \to \mt{res} ::: \mt{Type} \\
adamc@544 1558 \hspace{.1in} \to \mt{sql\_binary} \; \mt{arg_1} \; \mt{arg_2} \; \mt{res} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{arg_1} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{arg_2} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{res}
adamc@544 1559 \end{array}$$
adamc@544 1560
adamc@544 1561 $$\begin{array}{l}
adamc@559 1562 \mt{class} \; \mt{sql\_arith} \\
adamc@559 1563 \mt{val} \; \mt{sql\_int\_arith} : \mt{sql\_arith} \; \mt{int} \\
adamc@559 1564 \mt{val} \; \mt{sql\_float\_arith} : \mt{sql\_arith} \; \mt{float} \\
adamc@559 1565 \mt{val} \; \mt{sql\_neg} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_unary} \; \mt{t} \; \mt{t} \\
adamc@559 1566 \mt{val} \; \mt{sql\_plus} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559 1567 \mt{val} \; \mt{sql\_minus} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559 1568 \mt{val} \; \mt{sql\_times} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559 1569 \mt{val} \; \mt{sql\_div} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559 1570 \mt{val} \; \mt{sql\_mod} : \mt{sql\_binary} \; \mt{int} \; \mt{int} \; \mt{int}
adamc@559 1571 \end{array}$$
adamc@544 1572
adamc@656 1573 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 1574 $$\begin{array}{l}
adamc@544 1575 \mt{val} \; \mt{sql\_count} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{int}
adamc@544 1576 \end{array}$$
adamc@544 1577
adamc@544 1578 $$\begin{array}{l}
adamc@544 1579 \mt{con} \; \mt{sql\_aggregate} :: \mt{Type} \to \mt{Type} \\
adamc@544 1580 \mt{val} \; \mt{sql\_aggregate} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544 1581 \hspace{.1in} \to \mt{sql\_aggregate} \; \mt{t} \to \mt{sql\_exp} \; \mt{agg} \; \mt{agg} \; \mt{exps} \; \mt{t} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{t}
adamc@544 1582 \end{array}$$
adamc@544 1583
adamc@544 1584 $$\begin{array}{l}
adamc@544 1585 \mt{class} \; \mt{sql\_summable} \\
adamc@544 1586 \mt{val} \; \mt{sql\_summable\_int} : \mt{sql\_summable} \; \mt{int} \\
adamc@544 1587 \mt{val} \; \mt{sql\_summable\_float} : \mt{sql\_summable} \; \mt{float} \\
adamc@544 1588 \mt{val} \; \mt{sql\_avg} : \mt{t} ::: \mt{Type} \to \mt{sql\_summable} \; \mt{t} \to \mt{sql\_aggregate} \; \mt{t} \\
adamc@544 1589 \mt{val} \; \mt{sql\_sum} : \mt{t} ::: \mt{Type} \to \mt{sql\_summable} \mt{t} \to \mt{sql\_aggregate} \; \mt{t}
adamc@544 1590 \end{array}$$
adamc@544 1591
adamc@544 1592 $$\begin{array}{l}
adamc@544 1593 \mt{class} \; \mt{sql\_maxable} \\
adamc@544 1594 \mt{val} \; \mt{sql\_maxable\_int} : \mt{sql\_maxable} \; \mt{int} \\
adamc@544 1595 \mt{val} \; \mt{sql\_maxable\_float} : \mt{sql\_maxable} \; \mt{float} \\
adamc@544 1596 \mt{val} \; \mt{sql\_maxable\_string} : \mt{sql\_maxable} \; \mt{string} \\
adamc@544 1597 \mt{val} \; \mt{sql\_maxable\_time} : \mt{sql\_maxable} \; \mt{time} \\
adamc@544 1598 \mt{val} \; \mt{sql\_max} : \mt{t} ::: \mt{Type} \to \mt{sql\_maxable} \; \mt{t} \to \mt{sql\_aggregate} \; \mt{t} \\
adamc@544 1599 \mt{val} \; \mt{sql\_min} : \mt{t} ::: \mt{Type} \to \mt{sql\_maxable} \; \mt{t} \to \mt{sql\_aggregate} \; \mt{t}
adamc@544 1600 \end{array}$$
adamc@544 1601
adamc@786 1602 \texttt{FROM} clauses are specified using a type family.
adamc@786 1603 $$\begin{array}{l}
adamc@786 1604 \mt{con} \; \mt{sql\_from\_items} :: \{\{\mt{Type}\}\} \to \mt{Type} \\
adamc@786 1605 \mt{val} \; \mt{sql\_from\_table} : \mt{t} ::: \mt{Type} \to \mt{fs} ::: \{\mt{Type}\} \to \mt{fieldsOf} \; \mt{t} \; \mt{fs} \to \mt{name} :: \mt{Name} \to \mt{t} \to \mt{sql\_from\_items} \; [\mt{name} = \mt{fs}] \\
adamc@786 1606 \mt{val} \; \mt{sql\_from\_comma} : \mt{tabs1} ::: \{\{\mt{Type}\}\} \to \mt{tabs2} ::: \{\{\mt{Type}\}\} \to [\mt{tabs1} \sim \mt{tabs2}] \\
adamc@786 1607 \hspace{.1in} \Rightarrow \mt{sql\_from\_items} \; \mt{tabs1} \to \mt{sql\_from\_items} \; \mt{tabs2} \\
adamc@786 1608 \hspace{.1in} \to \mt{sql\_from\_items} \; (\mt{tabs1} \rc \mt{tabs2}) \\
adamc@786 1609 \mt{val} \; \mt{sql\_inner\_join} : \mt{tabs1} ::: \{\{\mt{Type}\}\} \to \mt{tabs2} ::: \{\{\mt{Type}\}\} \to [\mt{tabs1} \sim \mt{tabs2}] \\
adamc@786 1610 \hspace{.1in} \Rightarrow \mt{sql\_from\_items} \; \mt{tabs1} \to \mt{sql\_from\_items} \; \mt{tabs2} \\
adamc@786 1611 \hspace{.1in} \to \mt{sql\_exp} \; (\mt{tabs1} \rc \mt{tabs2}) \; [] \; [] \; \mt{bool} \\
adamc@786 1612 \hspace{.1in} \to \mt{sql\_from\_items} \; (\mt{tabs1} \rc \mt{tabs2})
adamc@786 1613 \end{array}$$
adamc@786 1614
adamc@786 1615 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 1616 $$\begin{array}{l}
adamc@786 1617 \mt{class} \; \mt{nullify} :: \mt{Type} \to \mt{Type} \to \mt{Type} \\
adamc@786 1618 \mt{val} \; \mt{nullify\_option} : \mt{t} ::: \mt{Type} \to \mt{nullify} \; (\mt{option} \; \mt{t}) \; (\mt{option} \; \mt{t}) \\
adamc@786 1619 \mt{val} \; \mt{nullify\_prim} : \mt{t} ::: \mt{Type} \to \mt{sql\_injectable\_prim} \; \mt{t} \to \mt{nullify} \; \mt{t} \; (\mt{option} \; \mt{t})
adamc@786 1620 \end{array}$$
adamc@786 1621
adamc@786 1622 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 1623
adamc@786 1624 $$\begin{array}{l}
adamc@786 1625 \mt{val} \; \mt{sql\_left\_join} : \mt{tabs1} ::: \{\{\mt{Type}\}\} \to \mt{tabs2} ::: \{\{(\mt{Type} \times \mt{Type})\}\} \to [\mt{tabs1} \sim \mt{tabs2}] \\
adamc@786 1626 \hspace{.1in} \Rightarrow \$(\mt{map} \; (\lambda \mt{r} \Rightarrow \$(\mt{map} \; (\lambda \mt{p} :: (\mt{Type} \times \mt{Type}) \Rightarrow \mt{nullify} \; \mt{p}.1 \; \mt{p}.2) \; \mt{r})) \; \mt{tabs2}) \\
adamc@786 1627 \hspace{.1in} \to \mt{sql\_from\_items} \; \mt{tabs1} \to \mt{sql\_from\_items} \; (\mt{map} \; (\mt{map} \; (\lambda \mt{p} :: (\mt{Type} \times \mt{Type}) \Rightarrow \mt{p}.1)) \; \mt{tabs2}) \\
adamc@786 1628 \hspace{.1in} \to \mt{sql\_exp} \; (\mt{tabs1} \rc \mt{map} \; (\mt{map} \; (\lambda \mt{p} :: (\mt{Type} \times \mt{Type}) \Rightarrow \mt{p}.1)) \; \mt{tabs2}) \; [] \; [] \; \mt{bool} \\
adamc@786 1629 \hspace{.1in} \to \mt{sql\_from\_items} \; (\mt{tabs1} \rc \mt{map} \; (\mt{map} \; (\lambda \mt{p} :: (\mt{Type} \times \mt{Type}) \Rightarrow \mt{p}.2)) \; \mt{tabs2})
adamc@786 1630 \end{array}$$
adamc@786 1631
adamc@544 1632 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 1633 $$\begin{array}{l}
adamc@544 1634 \mt{type} \; \mt{sql\_direction} \\
adamc@544 1635 \mt{val} \; \mt{sql\_asc} : \mt{sql\_direction} \\
adamc@544 1636 \mt{val} \; \mt{sql\_desc} : \mt{sql\_direction} \\
adamc@544 1637 \\
adamc@544 1638 \mt{con} \; \mt{sql\_order\_by} :: \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@544 1639 \mt{val} \; \mt{sql\_order\_by\_Nil} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{exps} :: \{\mt{Type}\} \to \mt{sql\_order\_by} \; \mt{tables} \; \mt{exps} \\
adamc@544 1640 \mt{val} \; \mt{sql\_order\_by\_Cons} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544 1641 \hspace{.1in} \to \mt{sql\_exp} \; \mt{tables} \; [] \; \mt{exps} \; \mt{t} \to \mt{sql\_direction} \to \mt{sql\_order\_by} \; \mt{tables} \; \mt{exps} \to \mt{sql\_order\_by} \; \mt{tables} \; \mt{exps} \\
adamc@544 1642 \\
adamc@544 1643 \mt{type} \; \mt{sql\_limit} \\
adamc@544 1644 \mt{val} \; \mt{sql\_no\_limit} : \mt{sql\_limit} \\
adamc@544 1645 \mt{val} \; \mt{sql\_limit} : \mt{int} \to \mt{sql\_limit} \\
adamc@544 1646 \\
adamc@544 1647 \mt{type} \; \mt{sql\_offset} \\
adamc@544 1648 \mt{val} \; \mt{sql\_no\_offset} : \mt{sql\_offset} \\
adamc@544 1649 \mt{val} \; \mt{sql\_offset} : \mt{int} \to \mt{sql\_offset}
adamc@544 1650 \end{array}$$
adamc@544 1651
adamc@545 1652
adamc@545 1653 \subsubsection{DML}
adamc@545 1654
adamc@545 1655 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 1656
adamc@545 1657 $$\begin{array}{l}
adamc@545 1658 \mt{type} \; \mt{dml} \\
adamc@545 1659 \mt{val} \; \mt{dml} : \mt{dml} \to \mt{transaction} \; \mt{unit}
adamc@545 1660 \end{array}$$
adamc@545 1661
adamc@545 1662 Properly-typed records may be used to form $\mt{INSERT}$ commands.
adamc@545 1663 $$\begin{array}{l}
adamc@545 1664 \mt{val} \; \mt{insert} : \mt{fields} ::: \{\mt{Type}\} \to \mt{sql\_table} \; \mt{fields} \\
adamc@658 1665 \hspace{.1in} \to \$(\mt{map} \; (\mt{sql\_exp} \; [] \; [] \; []) \; \mt{fields}) \to \mt{dml}
adamc@545 1666 \end{array}$$
adamc@545 1667
adamc@545 1668 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 1669 $$\begin{array}{l}
adamc@545 1670 \mt{val} \; \mt{update} : \mt{unchanged} ::: \{\mt{Type}\} \to \mt{changed} :: \{\mt{Type}\} \to \lambda [\mt{changed} \sim \mt{unchanged}] \\
adamc@658 1671 \hspace{.1in} \Rightarrow \$(\mt{map} \; (\mt{sql\_exp} \; [\mt{T} = \mt{changed} \rc \mt{unchanged}] \; [] \; []) \; \mt{changed}) \\
adamc@545 1672 \hspace{.1in} \to \mt{sql\_table} \; (\mt{changed} \rc \mt{unchanged}) \to \mt{sql\_exp} \; [\mt{T} = \mt{changed} \rc \mt{unchanged}] \; [] \; [] \; \mt{bool} \to \mt{dml}
adamc@545 1673 \end{array}$$
adamc@545 1674
adamc@545 1675 A $\mt{DELETE}$ command is formed from a table and a $\mt{WHERE}$ clause.
adamc@545 1676 $$\begin{array}{l}
adamc@545 1677 \mt{val} \; \mt{delete} : \mt{fields} ::: \{\mt{Type}\} \to \mt{sql\_table} \; \mt{fields} \to \mt{sql\_exp} \; [\mt{T} = \mt{fields}] \; [] \; [] \; \mt{bool} \to \mt{dml}
adamc@545 1678 \end{array}$$
adamc@545 1679
adamc@546 1680 \subsubsection{Sequences}
adamc@546 1681
adamc@546 1682 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 1683
adamc@546 1684 $$\begin{array}{l}
adamc@546 1685 \mt{type} \; \mt{sql\_sequence} \\
adamc@546 1686 \mt{val} \; \mt{nextval} : \mt{sql\_sequence} \to \mt{transaction} \; \mt{int}
adamc@546 1687 \end{array}$$
adamc@546 1688
adamc@546 1689
adamc@547 1690 \subsection{XML}
adamc@547 1691
adamc@547 1692 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 1693
adamc@547 1694 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 1695 $$\begin{array}{l}
adamc@547 1696 \mt{con} \; \mt{xml} :: \{\mt{Unit}\} \to \{\mt{Type}\} \to \{\mt{Type}\} \to \mt{Type}
adamc@547 1697 \end{array}$$
adamc@547 1698
adamc@547 1699 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 1700 $$\begin{array}{l}
adamc@547 1701 \mt{con} \; \mt{tag} :: \{\mt{Type}\} \to \{\mt{Unit}\} \to \{\mt{Unit}\} \to \{\mt{Type}\} \to \{\mt{Type}\} \to \mt{Type}
adamc@547 1702 \end{array}$$
adamc@547 1703
adamc@547 1704 Literal text may be injected into XML as ``CDATA.''
adamc@547 1705 $$\begin{array}{l}
adamc@547 1706 \mt{val} \; \mt{cdata} : \mt{ctx} ::: \{\mt{Unit}\} \to \mt{use} ::: \{\mt{Type}\} \to \mt{string} \to \mt{xml} \; \mt{ctx} \; \mt{use} \; []
adamc@547 1707 \end{array}$$
adamc@547 1708
adamc@547 1709 There is a function for producing an XML tree with a particular tag at its root.
adamc@547 1710 $$\begin{array}{l}
adamc@547 1711 \mt{val} \; \mt{tag} : \mt{attrsGiven} ::: \{\mt{Type}\} \to \mt{attrsAbsent} ::: \{\mt{Type}\} \to \mt{ctxOuter} ::: \{\mt{Unit}\} \to \mt{ctxInner} ::: \{\mt{Unit}\} \\
adamc@547 1712 \hspace{.1in} \to \mt{useOuter} ::: \{\mt{Type}\} \to \mt{useInner} ::: \{\mt{Type}\} \to \mt{bindOuter} ::: \{\mt{Type}\} \to \mt{bindInner} ::: \{\mt{Type}\} \\
adamc@787 1713 \hspace{.1in} \to \lambda [\mt{attrsGiven} \sim \mt{attrsAbsent}] \; [\mt{useOuter} \sim \mt{useInner}] \; [\mt{bindOuter} \sim \mt{bindInner}] \\
adamc@787 1714 \hspace{.1in} \Rightarrow \mt{option} \; \mt{css\_class} \\
adamc@787 1715 \hspace{.1in} \to \$\mt{attrsGiven} \\
adamc@547 1716 \hspace{.1in} \to \mt{tag} \; (\mt{attrsGiven} \rc \mt{attrsAbsent}) \; \mt{ctxOuter} \; \mt{ctxInner} \; \mt{useOuter} \; \mt{bindOuter} \\
adamc@547 1717 \hspace{.1in} \to \mt{xml} \; \mt{ctxInner} \; \mt{useInner} \; \mt{bindInner} \to \mt{xml} \; \mt{ctxOuter} \; (\mt{useOuter} \rc \mt{useInner}) \; (\mt{bindOuter} \rc \mt{bindInner})
adamc@547 1718 \end{array}$$
adamc@787 1719 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 1720
adamc@547 1721 Two XML fragments may be concatenated.
adamc@547 1722 $$\begin{array}{l}
adamc@547 1723 \mt{val} \; \mt{join} : \mt{ctx} ::: \{\mt{Unit}\} \to \mt{use_1} ::: \{\mt{Type}\} \to \mt{bind_1} ::: \{\mt{Type}\} \to \mt{bind_2} ::: \{\mt{Type}\} \\
adamc@547 1724 \hspace{.1in} \to \lambda [\mt{use_1} \sim \mt{bind_1}] \; [\mt{bind_1} \sim \mt{bind_2}] \\
adamc@547 1725 \hspace{.1in} \Rightarrow \mt{xml} \; \mt{ctx} \; \mt{use_1} \; \mt{bind_1} \to \mt{xml} \; \mt{ctx} \; (\mt{use_1} \rc \mt{bind_1}) \; \mt{bind_2} \to \mt{xml} \; \mt{ctx} \; \mt{use_1} \; (\mt{bind_1} \rc \mt{bind_2})
adamc@547 1726 \end{array}$$
adamc@547 1727
adamc@547 1728 Finally, any XML fragment may be updated to ``claim'' to use more form fields than it does.
adamc@547 1729 $$\begin{array}{l}
adamc@547 1730 \mt{val} \; \mt{useMore} : \mt{ctx} ::: \{\mt{Unit}\} \to \mt{use_1} ::: \{\mt{Type}\} \to \mt{use_2} ::: \{\mt{Type}\} \to \mt{bind} ::: \{\mt{Type}\} \to \lambda [\mt{use_1} \sim \mt{use_2}] \\
adamc@547 1731 \hspace{.1in} \Rightarrow \mt{xml} \; \mt{ctx} \; \mt{use_1} \; \mt{bind} \to \mt{xml} \; \mt{ctx} \; (\mt{use_1} \rc \mt{use_2}) \; \mt{bind}
adamc@547 1732 \end{array}$$
adamc@547 1733
adamc@547 1734 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 1735
adamc@547 1736 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 1737 $$\begin{array}{l}
adamc@547 1738 \mt{val} \; \mt{error} : \mt{t} ::: \mt{Type} \to \mt{xml} \; [\mt{Body}] \; [] \; [] \to \mt{t}
adamc@547 1739 \end{array}$$
adamc@547 1740
adamc@549 1741
adamc@701 1742 \subsection{Client-Side Programming}
adamc@659 1743
adamc@701 1744 Ur/Web supports running code on web browsers, via automatic compilation to JavaScript.
adamc@701 1745
adamc@701 1746 \subsubsection{The Basics}
adamc@701 1747
adamc@701 1748 Clients can open alert dialog boxes, in the usual annoying JavaScript way.
adamc@701 1749 $$\begin{array}{l}
adamc@701 1750 \mt{val} \; \mt{alert} : \mt{string} \to \mt{transaction} \; \mt{unit}
adamc@701 1751 \end{array}$$
adamc@701 1752
adamc@701 1753 Any transaction may be run in a new thread with the $\mt{spawn}$ function.
adamc@701 1754 $$\begin{array}{l}
adamc@701 1755 \mt{val} \; \mt{spawn} : \mt{transaction} \; \mt{unit} \to \mt{transaction} \; \mt{unit}
adamc@701 1756 \end{array}$$
adamc@701 1757
adamc@701 1758 The current thread can be paused for at least a specified number of milliseconds.
adamc@701 1759 $$\begin{array}{l}
adamc@701 1760 \mt{val} \; \mt{sleep} : \mt{int} \to \mt{transaction} \; \mt{unit}
adamc@701 1761 \end{array}$$
adamc@701 1762
adamc@787 1763 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 1764 $$\begin{array}{l}
adamc@787 1765 \mt{val} \; \mt{onError} : (\mt{xbody} \to \mt{transaction} \; \mt{unit}) \to \mt{transaction} \; \mt{unit} \\
adamc@787 1766 \mt{val} \; \mt{onFail} : (\mt{string} \to \mt{transaction} \; \mt{unit}) \to \mt{transaction} \; \mt{unit} \\
adamc@787 1767 \mt{val} \; \mt{onConnectFail} : \mt{transaction} \; \mt{unit} \to \mt{transaction} \; \mt{unit} \\
adamc@787 1768 \mt{val} \; \mt{onDisconnect} : \mt{transaction} \; \mt{unit} \to \mt{transaction} \; \mt{unit} \\
adamc@787 1769 \mt{val} \; \mt{onServerError} : (\mt{string} \to \mt{transaction} \; \mt{unit}) \to \mt{transaction} \; \mt{unit}
adamc@787 1770 \end{array}$$
adamc@787 1771
adamc@701 1772 \subsubsection{Functional-Reactive Page Generation}
adamc@701 1773
adamc@701 1774 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 1775
adamc@659 1776 $$\begin{array}{l}
adamc@659 1777 \mt{con} \; \mt{source} :: \mt{Type} \to \mt{Type} \\
adamc@659 1778 \mt{val} \; \mt{source} : \mt{t} ::: \mt{Type} \to \mt{t} \to \mt{transaction} \; (\mt{source} \; \mt{t}) \\
adamc@659 1779 \mt{val} \; \mt{set} : \mt{t} ::: \mt{Type} \to \mt{source} \; \mt{t} \to \mt{t} \to \mt{transaction} \; \mt{unit} \\
adamc@659 1780 \mt{val} \; \mt{get} : \mt{t} ::: \mt{Type} \to \mt{source} \; \mt{t} \to \mt{transaction} \; \mt{t}
adamc@659 1781 \end{array}$$
adamc@659 1782
adamc@659 1783 Pure functions over sources are represented in a monad of \emph{signals}.
adamc@659 1784
adamc@659 1785 $$\begin{array}{l}
adamc@659 1786 \mt{con} \; \mt{signal} :: \mt{Type} \to \mt{Type} \\
adamc@659 1787 \mt{val} \; \mt{signal\_monad} : \mt{monad} \; \mt{signal} \\
adamc@659 1788 \mt{val} \; \mt{signal} : \mt{t} ::: \mt{Type} \to \mt{source} \; \mt{t} \to \mt{signal} \; \mt{t}
adamc@659 1789 \end{array}$$
adamc@659 1790
adamc@659 1791 A reactive portion of an HTML page is injected with a $\mt{dyn}$ tag, which has a signal-valued attribute $\mt{Signal}$.
adamc@659 1792
adamc@659 1793 $$\begin{array}{l}
adamc@701 1794 \mt{val} \; \mt{dyn} : \mt{use} ::: \{\mt{Type}\} \to \mt{bind} ::: \{\mt{Type}\} \to \mt{unit} \\
adamc@701 1795 \hspace{.1in} \to \mt{tag} \; [\mt{Signal} = \mt{signal} \; (\mt{xml} \; \mt{body} \; \mt{use} \; \mt{bind})] \; \mt{body} \; [] \; \mt{use} \; \mt{bind}
adamc@659 1796 \end{array}$$
adamc@659 1797
adamc@701 1798 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 1799
adamc@914 1800 \subsubsection{Remote Procedure Calls}
adamc@914 1801
adamc@914 1802 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 1803
adamc@914 1804 $$\begin{array}{l}
adamc@914 1805 \mt{val} \; \mt{rpc} : \mt{t} ::: \mt{Type} \to \mt{transaction} \; \mt{t} \to \mt{transaction} \; \mt{t}
adamc@914 1806 \end{array}$$
adamc@914 1807
adamc@701 1808 \subsubsection{Asynchronous Message-Passing}
adamc@701 1809
adamc@701 1810 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 1811
adamc@701 1812 $$\begin{array}{l}
adamc@701 1813 \mt{type} \; \mt{client} \\
adamc@701 1814 \mt{val} \; \mt{self} : \mt{transaction} \; \mt{client}
adamc@701 1815 \end{array}$$
adamc@701 1816
adamc@701 1817 \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 1818
adamc@701 1819 $$\begin{array}{l}
adamc@701 1820 \mt{con} \; \mt{channel} :: \mt{Type} \to \mt{Type} \\
adamc@701 1821 \mt{val} \; \mt{channel} : \mt{t} ::: \mt{Type} \to \mt{transaction} \; (\mt{channel} \; \mt{t}) \\
adamc@701 1822 \mt{val} \; \mt{send} : \mt{t} ::: \mt{Type} \to \mt{channel} \; \mt{t} \to \mt{t} \to \mt{transaction} \; \mt{unit} \\
adamc@701 1823 \mt{val} \; \mt{recv} : \mt{t} ::: \mt{Type} \to \mt{channel} \; \mt{t} \to \mt{transaction} \; \mt{t}
adamc@701 1824 \end{array}$$
adamc@701 1825
adamc@701 1826 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 1827
adamc@701 1828 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 1829
adamc@659 1830
adamc@549 1831 \section{Ur/Web Syntax Extensions}
adamc@549 1832
adamc@549 1833 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 1834
adamc@549 1835 \subsection{SQL}
adamc@549 1836
adamc@786 1837 \subsubsection{\label{tables}Table Declarations}
adamc@786 1838
adamc@788 1839 $\mt{table}$ declarations may include constraints, via these grammar rules.
adamc@788 1840 $$\begin{array}{rrcll}
adamc@788 1841 \textrm{Declarations} & d &::=& \mt{table} \; x : c \; [pk[,]] \; cts \\
adamc@788 1842 \textrm{Primary key constraints} & pk &::=& \mt{PRIMARY} \; \mt{KEY} \; K \\
adamc@788 1843 \textrm{Keys} & K &::=& f \mid (f, (f,)^+) \\
adamc@788 1844 \textrm{Constraint sets} & cts &::=& \mt{CONSTRAINT} f \; ct \mid cts, cts \mid \{\{e\}\} \\
adamc@788 1845 \textrm{Constraints} & ct &::=& \mt{UNIQUE} \; K \mid \mt{CHECK} \; E \\
adamc@788 1846 &&& \mid \mt{FOREIGN} \; \mt{KEY} \; K \; \mt{REFERENCES} \; F \; (K) \; [\mt{ON} \; \mt{DELETE} \; pr] \; [\mt{ON} \; \mt{UPDATE} \; pr] \\
adamc@788 1847 \textrm{Foreign tables} & F &::=& x \mid \{\{e\}\} \\
adamc@788 1848 \textrm{Propagation modes} & pr &::=& \mt{NO} \; \mt{ACTION} \mid \mt{RESTRICT} \mid \mt{CASCADE} \mid \mt{SET} \; \mt{NULL}
adamc@788 1849 \end{array}$$
adamc@788 1850
adamc@788 1851 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 1852
adamc@788 1853
adamc@549 1854 \subsubsection{Queries}
adamc@549 1855
adamc@550 1856 Queries $Q$ are added to the rules for expressions $e$.
adamc@550 1857
adamc@549 1858 $$\begin{array}{rrcll}
adamc@550 1859 \textrm{Queries} & Q &::=& (q \; [\mt{ORDER} \; \mt{BY} \; (E \; [o],)^+] \; [\mt{LIMIT} \; N] \; [\mt{OFFSET} \; N]) \\
adamc@994 1860 \textrm{Pre-queries} & q &::=& \mt{SELECT} \; [\mt{DISTINCT}] \; P \; \mt{FROM} \; T,^+ \; [\mt{WHERE} \; E] \; [\mt{GROUP} \; \mt{BY} \; p,^+] \; [\mt{HAVING} \; E] \\
adamc@549 1861 &&& \mid q \; R \; q \\
adamc@549 1862 \textrm{Relational operators} & R &::=& \mt{UNION} \mid \mt{INTERSECT} \mid \mt{EXCEPT}
adamc@549 1863 \end{array}$$
adamc@549 1864
adamc@549 1865 $$\begin{array}{rrcll}
adamc@549 1866 \textrm{Projections} & P &::=& \ast & \textrm{all columns} \\
adamc@549 1867 &&& p,^+ & \textrm{particular columns} \\
adamc@549 1868 \textrm{Pre-projections} & p &::=& t.f & \textrm{one column from a table} \\
adamc@558 1869 &&& t.\{\{c\}\} & \textrm{a record of columns from a table (of kind $\{\mt{Type}\}$)} \\
adamc@549 1870 \textrm{Table names} & t &::=& x & \textrm{constant table name (automatically capitalized)} \\
adamc@549 1871 &&& X & \textrm{constant table name} \\
adamc@549 1872 &&& \{\{c\}\} & \textrm{computed table name (of kind $\mt{Name}$)} \\
adamc@549 1873 \textrm{Column names} & f &::=& X & \textrm{constant column name} \\
adamc@549 1874 &&& \{c\} & \textrm{computed column name (of kind $\mt{Name}$)} \\
adamc@549 1875 \textrm{Tables} & T &::=& x & \textrm{table variable, named locally by its own capitalization} \\
adamc@549 1876 &&& x \; \mt{AS} \; t & \textrm{table variable, with local name} \\
adamc@549 1877 &&& \{\{e\}\} \; \mt{AS} \; t & \textrm{computed table expression, with local name} \\
adamc@549 1878 \textrm{SQL expressions} & E &::=& p & \textrm{column references} \\
adamc@549 1879 &&& X & \textrm{named expression references} \\
adamc@549 1880 &&& \{\{e\}\} & \textrm{injected native Ur expressions} \\
adamc@549 1881 &&& \{e\} & \textrm{computed expressions, probably using $\mt{sql\_exp}$ directly} \\
adamc@549 1882 &&& \mt{TRUE} \mid \mt{FALSE} & \textrm{boolean constants} \\
adamc@549 1883 &&& \ell & \textrm{primitive type literals} \\
adamc@549 1884 &&& \mt{NULL} & \textrm{null value (injection of $\mt{None}$)} \\
adamc@549 1885 &&& E \; \mt{IS} \; \mt{NULL} & \textrm{nullness test} \\
adamc@549 1886 &&& n & \textrm{nullary operators} \\
adamc@549 1887 &&& u \; E & \textrm{unary operators} \\
adamc@549 1888 &&& E \; b \; E & \textrm{binary operators} \\
adamc@549 1889 &&& \mt{COUNT}(\ast) & \textrm{count number of rows} \\
adamc@549 1890 &&& a(E) & \textrm{other aggregate function} \\
adamc@549 1891 &&& (E) & \textrm{explicit precedence} \\
adamc@549 1892 \textrm{Nullary operators} & n &::=& \mt{CURRENT\_TIMESTAMP} \\
adamc@549 1893 \textrm{Unary operators} & u &::=& \mt{NOT} \\
adamc@549 1894 \textrm{Binary operators} & b &::=& \mt{AND} \mid \mt{OR} \mid \neq \mid < \mid \leq \mid > \mid \geq \\
adamc@549 1895 \textrm{Aggregate functions} & a &::=& \mt{AVG} \mid \mt{SUM} \mid \mt{MIN} \mid \mt{MAX} \\
adamc@550 1896 \textrm{Directions} & o &::=& \mt{ASC} \mid \mt{DESC} \\
adamc@549 1897 \textrm{SQL integer} & N &::=& n \mid \{e\} \\
adamc@549 1898 \end{array}$$
adamc@549 1899
adamc@549 1900 Additionally, an SQL expression may be inserted into normal Ur code with the syntax $(\mt{SQL} \; E)$ or $(\mt{WHERE} \; E)$.
adamc@549 1901
adamc@550 1902 \subsubsection{DML}
adamc@550 1903
adamc@550 1904 DML commands $D$ are added to the rules for expressions $e$.
adamc@550 1905
adamc@550 1906 $$\begin{array}{rrcll}
adamc@550 1907 \textrm{Commands} & D &::=& (\mt{INSERT} \; \mt{INTO} \; T^E \; (f,^+) \; \mt{VALUES} \; (E,^+)) \\
adamc@550 1908 &&& (\mt{UPDATE} \; T^E \; \mt{SET} \; (f = E,)^+ \; \mt{WHERE} \; E) \\
adamc@550 1909 &&& (\mt{DELETE} \; \mt{FROM} \; T^E \; \mt{WHERE} \; E) \\
adamc@550 1910 \textrm{Table expressions} & T^E &::=& x \mid \{\{e\}\}
adamc@550 1911 \end{array}$$
adamc@550 1912
adamc@550 1913 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 1914
adamc@551 1915 \subsection{XML}
adamc@551 1916
adamc@551 1917 XML fragments $L$ are added to the rules for expressions $e$.
adamc@551 1918
adamc@551 1919 $$\begin{array}{rrcll}
adamc@551 1920 \textrm{XML fragments} & L &::=& \texttt{<xml/>} \mid \texttt{<xml>}l^*\texttt{</xml>} \\
adamc@551 1921 \textrm{XML pieces} & l &::=& \textrm{text} & \textrm{cdata} \\
adamc@551 1922 &&& \texttt{<}g\texttt{/>} & \textrm{tag with no children} \\
adamc@551 1923 &&& \texttt{<}g\texttt{>}l^*\texttt{</}x\texttt{>} & \textrm{tag with children} \\
adamc@559 1924 &&& \{e\} & \textrm{computed XML fragment} \\
adamc@559 1925 &&& \{[e]\} & \textrm{injection of an Ur expression, via the $\mt{Top}.\mt{txt}$ function} \\
adamc@551 1926 \textrm{Tag} & g &::=& h \; (x = v)^* \\
adamc@551 1927 \textrm{Tag head} & h &::=& x & \textrm{tag name} \\
adamc@551 1928 &&& h\{c\} & \textrm{constructor parameter} \\
adamc@551 1929 \textrm{Attribute value} & v &::=& \ell & \textrm{literal value} \\
adamc@551 1930 &&& \{e\} & \textrm{computed value} \\
adamc@551 1931 \end{array}$$
adamc@551 1932
adamc@552 1933
adamc@553 1934 \section{The Structure of Web Applications}
adamc@553 1935
adamc@553 1936 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 $\mt{unit} \to \mt{transaction} \; \mt{page}$, 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 $M$, it will be accessible at URI \texttt{/M/f}, and so on for more deeply-nested functions, as described in Section \ref{tag} below.
adamc@553 1937
adamc@553 1938 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 1939
adamc@553 1940 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 1941
adamc@558 1942 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 1943
adamc@660 1944 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 1945
adamc@789 1946 \medskip
adamc@789 1947
adamc@789 1948 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 1949
adamc@789 1950 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 1951
adamc@553 1952
adamc@897 1953 \section{The Foreign Function Interface}
adamc@897 1954
adamc@897 1955 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 1956
adamc@897 1957 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 1958
adamc@897 1959 \begin{itemize}
adamc@897 1960 \item \texttt{clientOnly Module.ident} registers a value as being allowed only in client-side code.
adamc@897 1961 \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 1962 \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 1963 \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@897 1964 \item \texttt{header FILE} requests inclusion of a C header file.
adamc@897 1965 \item \texttt{jsFunc Module.ident=name} gives a mapping from an Ur name for a value to a JavaScript name.
adamc@897 1966 \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 1967 \item \texttt{script URL} requests inclusion of a JavaScript source file within application HTML.
adamc@897 1968 \item \texttt{serverOnly Module.ident} registers a value as being allowed only in server-side code.
adamc@897 1969 \end{itemize}
adamc@897 1970
adamc@897 1971 \subsection{Writing C FFI Code}
adamc@897 1972
adamc@897 1973 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 1974
adamc@897 1975 \begin{itemize}
adamc@897 1976 \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 1977 \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 1978 \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 1979 \end{itemize}
adamc@897 1980
adamc@897 1981 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 1982 \begin{itemize}
adamc@897 1983 \item \begin{verbatim}
adamc@897 1984 void uw_error(uw_context, failure_kind, const char *fmt, ...);
adamc@897 1985 \end{verbatim}
adamc@897 1986 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 1987
adamc@897 1988 \item \begin{verbatim}
adamc@897 1989 void uw_push_cleanup(uw_context, void (*func)(void *), void *arg);
adamc@897 1990 void uw_pop_cleanup(uw_context);
adamc@897 1991 \end{verbatim}
adamc@897 1992 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 1993
adamc@897 1994 \item \begin{verbatim}
adamc@897 1995 void *uw_malloc(uw_context, size_t);
adamc@897 1996 \end{verbatim}
adamc@897 1997 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 1998
adamc@897 1999 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 2000
adamc@897 2001 \item \begin{verbatim}
adamc@897 2002 typedef void (*uw_callback)(void *);
adamc@897 2003 void uw_register_transactional(uw_context, void *data, uw_callback commit,
adamc@897 2004 uw_callback rollback, uw_callback free);
adamc@897 2005 \end{verbatim}
adamc@897 2006 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 2007
adamc@897 2008 To accommodate some stubbornly non-transactional real-world actions like sending an e-mail message, Ur/Web allows the \texttt{rollback} parameter to be \texttt{NULL}. 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@897 2009 \end{itemize}
adamc@897 2010
adamc@897 2011
adamc@897 2012 \subsection{Writing JavaScript FFI Code}
adamc@897 2013
adamc@897 2014 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 2015
adamc@897 2016 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 2017
adamc@897 2018 \begin{itemize}
adamc@897 2019 \item Integers, floats, strings, characters, and booleans are represented in the usual JavaScript way.
adamc@985 2020 \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 2021 \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 2022 \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 2023 \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 2024 \end{itemize}
adamc@897 2025
adamc@897 2026 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 2027
adamc@897 2028
adamc@552 2029 \section{Compiler Phases}
adamc@552 2030
adamc@552 2031 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 2032
adamc@552 2033 In this section, we step through the main phases of compilation, noting what consequences each phase has for effective programming.
adamc@552 2034
adamc@552 2035 \subsection{Parse}
adamc@552 2036
adamc@552 2037 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 2038
adamc@552 2039 \subsection{Elaborate}
adamc@552 2040
adamc@552 2041 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 2042
adamc@552 2043 \subsection{Unnest}
adamc@552 2044
adamc@552 2045 Named local function definitions are moved to the top level, to avoid the need to generate closures.
adamc@552 2046
adamc@552 2047 \subsection{Corify}
adamc@552 2048
adamc@552 2049 Module system features are compiled away, through inlining of functor definitions at application sites. Afterward, most abstraction boundaries are broken, facilitating optimization.
adamc@552 2050
adamc@552 2051 \subsection{Especialize}
adamc@552 2052
adamc@552 2053 Functions are specialized to particular argument patterns. This is an important trick for avoiding the need to maintain any closures at runtime.
adamc@552 2054
adamc@552 2055 \subsection{Untangle}
adamc@552 2056
adamc@552 2057 Remove unnecessary mutual recursion, splitting recursive groups into strongly-connected components.
adamc@552 2058
adamc@552 2059 \subsection{Shake}
adamc@552 2060
adamc@552 2061 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 2062
adamc@661 2063 \subsection{Rpcify}
adamc@661 2064
adamc@661 2065 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 2066
adamc@661 2067 \subsection{Untangle, Shake}
adamc@661 2068
adamc@661 2069 Repeat these simplifications.
adamc@661 2070
adamc@553 2071 \subsection{\label{tag}Tag}
adamc@552 2072
adamc@552 2073 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 2074
adamc@552 2075 \subsection{Reduce}
adamc@552 2076
adamc@552 2077 Apply definitional equality rules to simplify the program as much as possible. This effectively includes inlining of every non-recursive definition.
adamc@552 2078
adamc@552 2079 \subsection{Unpoly}
adamc@552 2080
adamc@552 2081 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 2082
adamc@552 2083 \subsection{Specialize}
adamc@552 2084
adamc@558 2085 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 2086
adamc@552 2087 \subsection{Shake}
adamc@552 2088
adamc@558 2089 Here the compiler repeats the earlier Shake phase.
adamc@552 2090
adamc@552 2091 \subsection{Monoize}
adamc@552 2092
adamc@552 2093 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 2094
adamc@552 2095 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 2096
adamc@552 2097 \subsection{MonoOpt}
adamc@552 2098
adamc@552 2099 Simple algebraic laws are applied to simplify the program, focusing especially on efficient imperative generation of HTML pages.
adamc@552 2100
adamc@552 2101 \subsection{MonoUntangle}
adamc@552 2102
adamc@552 2103 Unnecessary mutual recursion is broken up again.
adamc@552 2104
adamc@552 2105 \subsection{MonoReduce}
adamc@552 2106
adamc@552 2107 Equivalents of the definitional equality rules are applied to simplify programs, with inlining again playing a major role.
adamc@552 2108
adamc@552 2109 \subsection{MonoShake, MonoOpt}
adamc@552 2110
adamc@552 2111 Unneeded declarations are removed, and basic optimizations are repeated.
adamc@552 2112
adamc@552 2113 \subsection{Fuse}
adamc@552 2114
adamc@552 2115 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 2116
adamc@552 2117 \subsection{MonoUntangle, MonoShake}
adamc@552 2118
adamc@552 2119 Fuse often creates more opportunities to remove spurious mutual recursion.
adamc@552 2120
adamc@552 2121 \subsection{Pathcheck}
adamc@552 2122
adamc@552 2123 The compiler checks that no link or action name has been used more than once.
adamc@552 2124
adamc@552 2125 \subsection{Cjrize}
adamc@552 2126
adamc@552 2127 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 2128
adamc@552 2129 \subsection{C Compilation and Linking}
adamc@552 2130
adamc@552 2131 The output of the last phase is pretty-printed as C source code and passed to GCC.
adamc@552 2132
adamc@552 2133
adamc@524 2134 \end{document}