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Basis.debug; more restrictive type for Basis.form; weaken definition of polymorphic-ness for especialization
author Adam Chlipala <adamc@hcoop.net>
date Thu, 07 Jan 2010 14:02:58 -0500
parents 8d0f195710f1
children 81ddb010751e
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@1099 143 \item \texttt{include FILENAME} adds \texttt{FILENAME} to the list of files to be \texttt{\#include}d in C sources. This is most useful for interfacing with new FFI modules.
adamc@783 144 \item \texttt{jsFunc Module.ident=name} gives the JavaScript name of an FFI value.
adamc@1089 145 \item \texttt{library FILENAME} parses \texttt{FILENAME.urp} and merges its contents with the rest of the current file's contents. If \texttt{FILENAME.urp} doesn't exist, the compiler also tries \texttt{FILENAME/lib.urp}.
adamc@783 146 \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@1086 171 To stop the compilation process after type-checking, run
adamc@1086 172 \begin{verbatim}
adamc@1086 173 urweb -tc P
adamc@1086 174 \end{verbatim}
adamc@1086 175
adamc@896 176 Some other command-line parameters are accepted:
adamc@896 177 \begin{itemize}
adamc@896 178 \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@896 179
adamc@896 180 \item \texttt{-dbms [postgres|mysql|sqlite]}: Sets the database backend to use.
adamc@896 181 \begin{itemize}
adamc@896 182 \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@896 183
adamc@896 184 A command sequence like this can initialize a Postgres database, using a file \texttt{app.sql} generated by the compiler:
adamc@896 185 \begin{verbatim}
adamc@896 186 createdb app
adamc@896 187 psql -f app.sql app
adamc@896 188 \end{verbatim}
adamc@896 189
adamc@896 190 \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@896 191
adamc@896 192 A command sequence like this can initialize a MySQL database:
adamc@896 193 \begin{verbatim}
adamc@896 194 echo "CREATE DATABASE app" | mysql
adamc@896 195 mysql -D app <app.sql
adamc@896 196 \end{verbatim}
adamc@896 197
adamc@896 198 \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@896 199
adamc@896 200 A command like this can initialize an SQLite database:
adamc@896 201 \begin{verbatim}
adamc@896 202 sqlite3 path/to/database/file <app.sql
adamc@896 203 \end{verbatim}
adamc@896 204 \end{itemize}
adamc@896 205
adamc@896 206 \item \texttt{-output FILENAME}: Set where the application executable is written.
adamc@896 207
adamc@896 208 \item \texttt{-protocol [http|cgi|fastcgi]}: Set the protocol that the generated application speaks.
adamc@896 209 \begin{itemize}
adamc@896 210 \item \texttt{http}: This is the default. It is for building standalone web servers that can be accessed by web browsers directly.
adamc@896 211
adamc@896 212 \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@896 213
adamc@896 214 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@896 215 \begin{verbatim}
adamc@896 216 ScriptAlias /Hello /path/to/hello.exe
adamc@896 217 \end{verbatim}
adamc@896 218
adamc@896 219 \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@896 220
adamc@896 221 To configure a FastCGI program with Apache, one could combine the above \texttt{ScriptAlias} line with a line like this:
adamc@896 222 \begin{verbatim}
adamc@896 223 FastCgiServer /path/to/hello.exe -idle-timeout 99999
adamc@896 224 \end{verbatim}
adamc@896 225 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@896 226
adamc@896 227 Here is some lighttpd configuration for the same application.
adamc@896 228 \begin{verbatim}
adamc@896 229 fastcgi.server = (
adamc@896 230 "/Hello/" =>
adamc@896 231 (( "bin-path" => "/path/to/hello.exe",
adamc@896 232 "socket" => "/tmp/hello",
adamc@896 233 "check-local" => "disable",
adamc@896 234 "docroot" => "/",
adamc@896 235 "max-procs" => "1"
adamc@896 236 ))
adamc@896 237 )
adamc@896 238 \end{verbatim}
adamc@896 239 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@896 240
adamc@896 241 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@896 242 \end{itemize}
adamc@896 243
adamc@896 244 \item \texttt{-sql FILENAME}: Set where a database set-up SQL script is written.
adamc@1095 245
adamc@1095 246 \item \texttt{-static}: Link the runtime system statically. The default is to link against dynamic libraries.
adamc@896 247 \end{itemize}
adamc@896 248
adamc@556 249
adamc@529 250 \section{Ur Syntax}
adamc@529 251
adamc@784 252 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@524 253
adamc@524 254 \subsection{Lexical Conventions}
adamc@524 255
adamc@524 256 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@524 257
adamc@524 258 \begin{center}
adamc@524 259 \begin{tabular}{rl}
adamc@524 260 \textbf{\LaTeX} & \textbf{ASCII} \\
adamc@524 261 $\to$ & \cd{->} \\
adamc@652 262 $\longrightarrow$ & \cd{-->} \\
adamc@524 263 $\times$ & \cd{*} \\
adamc@524 264 $\lambda$ & \cd{fn} \\
adamc@524 265 $\Rightarrow$ & \cd{=>} \\
adamc@652 266 $\Longrightarrow$ & \cd{==>} \\
adamc@529 267 $\neq$ & \cd{<>} \\
adamc@529 268 $\leq$ & \cd{<=} \\
adamc@529 269 $\geq$ & \cd{>=} \\
adamc@524 270 \\
adamc@524 271 $x$ & Normal textual identifier, not beginning with an uppercase letter \\
adamc@525 272 $X$ & Normal textual identifier, beginning with an uppercase letter \\
adamc@524 273 \end{tabular}
adamc@524 274 \end{center}
adamc@524 275
adamc@525 276 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@524 277
adamc@873 278 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@526 279
adamc@527 280 This version of the manual doesn't include operator precedences; see \texttt{src/urweb.grm} for that.
adamc@527 281
adamc@552 282 \subsection{\label{core}Core Syntax}
adamc@524 283
adamc@524 284 \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@524 285 $$\begin{array}{rrcll}
adamc@524 286 \textrm{Kinds} & \kappa &::=& \mt{Type} & \textrm{proper types} \\
adamc@525 287 &&& \mt{Unit} & \textrm{the trivial constructor} \\
adamc@525 288 &&& \mt{Name} & \textrm{field names} \\
adamc@525 289 &&& \kappa \to \kappa & \textrm{type-level functions} \\
adamc@525 290 &&& \{\kappa\} & \textrm{type-level records} \\
adamc@525 291 &&& (\kappa\times^+) & \textrm{type-level tuples} \\
adamc@652 292 &&& X & \textrm{variable} \\
adamc@652 293 &&& X \longrightarrow k & \textrm{kind-polymorphic type-level function} \\
adamc@529 294 &&& \_\_ & \textrm{wildcard} \\
adamc@525 295 &&& (\kappa) & \textrm{explicit precedence} \\
adamc@524 296 \end{array}$$
adamc@524 297
adamc@524 298 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@524 299 $$\begin{array}{rrcll}
adamc@524 300 \textrm{Explicitness} & ? &::=& :: & \textrm{explicit} \\
adamc@558 301 &&& ::: & \textrm{implicit}
adamc@524 302 \end{array}$$
adamc@524 303
adamc@524 304 \emph{Constructors} are the main class of compile-time-only data. They include proper types and are classified by kinds.
adamc@524 305 $$\begin{array}{rrcll}
adamc@524 306 \textrm{Constructors} & c, \tau &::=& (c) :: \kappa & \textrm{kind annotation} \\
adamc@530 307 &&& \hat{x} & \textrm{constructor variable} \\
adamc@524 308 \\
adamc@525 309 &&& \tau \to \tau & \textrm{function type} \\
adamc@525 310 &&& x \; ? \; \kappa \to \tau & \textrm{polymorphic function type} \\
adamc@652 311 &&& X \longrightarrow \tau & \textrm{kind-polymorphic function type} \\
adamc@525 312 &&& \$ c & \textrm{record type} \\
adamc@524 313 \\
adamc@525 314 &&& c \; c & \textrm{type-level function application} \\
adamc@530 315 &&& \lambda x \; :: \; \kappa \Rightarrow c & \textrm{type-level function abstraction} \\
adamc@524 316 \\
adamc@652 317 &&& X \Longrightarrow c & \textrm{type-level kind-polymorphic function abstraction} \\
adamc@655 318 &&& c [\kappa] & \textrm{type-level kind-polymorphic function application} \\
adamc@652 319 \\
adamc@525 320 &&& () & \textrm{type-level unit} \\
adamc@525 321 &&& \#X & \textrm{field name} \\
adamc@524 322 \\
adamc@525 323 &&& [(c = c)^*] & \textrm{known-length type-level record} \\
adamc@525 324 &&& c \rc c & \textrm{type-level record concatenation} \\
adamc@652 325 &&& \mt{map} & \textrm{type-level record map} \\
adamc@524 326 \\
adamc@558 327 &&& (c,^+) & \textrm{type-level tuple} \\
adamc@525 328 &&& c.n & \textrm{type-level tuple projection ($n \in \mathbb N^+$)} \\
adamc@524 329 \\
adamc@652 330 &&& [c \sim c] \Rightarrow \tau & \textrm{guarded type} \\
adamc@524 331 \\
adamc@529 332 &&& \_ :: \kappa & \textrm{wildcard} \\
adamc@525 333 &&& (c) & \textrm{explicit precedence} \\
adamc@530 334 \\
adamc@530 335 \textrm{Qualified uncapitalized variables} & \hat{x} &::=& x & \textrm{not from a module} \\
adamc@530 336 &&& M.x & \textrm{projection from a module} \\
adamc@525 337 \end{array}$$
adamc@525 338
adamc@655 339 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@655 340
adamc@525 341 Modules of the module system are described by \emph{signatures}.
adamc@525 342 $$\begin{array}{rrcll}
adamc@525 343 \textrm{Signatures} & S &::=& \mt{sig} \; s^* \; \mt{end} & \textrm{constant} \\
adamc@525 344 &&& X & \textrm{variable} \\
adamc@525 345 &&& \mt{functor}(X : S) : S & \textrm{functor} \\
adamc@529 346 &&& S \; \mt{where} \; \mt{con} \; x = c & \textrm{concretizing an abstract constructor} \\
adamc@525 347 &&& M.X & \textrm{projection from a module} \\
adamc@525 348 \\
adamc@525 349 \textrm{Signature items} & s &::=& \mt{con} \; x :: \kappa & \textrm{abstract constructor} \\
adamc@525 350 &&& \mt{con} \; x :: \kappa = c & \textrm{concrete constructor} \\
adamc@528 351 &&& \mt{datatype} \; x \; x^* = dc\mid^+ & \textrm{algebraic datatype definition} \\
adamc@529 352 &&& \mt{datatype} \; x = \mt{datatype} \; M.x & \textrm{algebraic datatype import} \\
adamc@525 353 &&& \mt{val} \; x : \tau & \textrm{value} \\
adamc@525 354 &&& \mt{structure} \; X : S & \textrm{sub-module} \\
adamc@525 355 &&& \mt{signature} \; X = S & \textrm{sub-signature} \\
adamc@525 356 &&& \mt{include} \; S & \textrm{signature inclusion} \\
adamc@525 357 &&& \mt{constraint} \; c \sim c & \textrm{record disjointness constraint} \\
adamc@654 358 &&& \mt{class} \; x :: \kappa & \textrm{abstract constructor class} \\
adamc@654 359 &&& \mt{class} \; x :: \kappa = c & \textrm{concrete constructor class} \\
adamc@525 360 \\
adamc@525 361 \textrm{Datatype constructors} & dc &::=& X & \textrm{nullary constructor} \\
adamc@525 362 &&& X \; \mt{of} \; \tau & \textrm{unary constructor} \\
adamc@524 363 \end{array}$$
adamc@524 364
adamc@526 365 \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@526 366 $$\begin{array}{rrcll}
adamc@526 367 \textrm{Patterns} & p &::=& \_ & \textrm{wildcard} \\
adamc@526 368 &&& x & \textrm{variable} \\
adamc@526 369 &&& \ell & \textrm{constant} \\
adamc@526 370 &&& \hat{X} & \textrm{nullary constructor} \\
adamc@526 371 &&& \hat{X} \; p & \textrm{unary constructor} \\
adamc@526 372 &&& \{(x = p,)^*\} & \textrm{rigid record pattern} \\
adamc@526 373 &&& \{(x = p,)^+, \ldots\} & \textrm{flexible record pattern} \\
adamc@852 374 &&& p : \tau & \textrm{type annotation} \\
adamc@527 375 &&& (p) & \textrm{explicit precedence} \\
adamc@526 376 \\
adamc@529 377 \textrm{Qualified capitalized variables} & \hat{X} &::=& X & \textrm{not from a module} \\
adamc@526 378 &&& M.X & \textrm{projection from a module} \\
adamc@526 379 \end{array}$$
adamc@526 380
adamc@527 381 \emph{Expressions} are the main run-time entities, corresponding to both ``expressions'' and ``statements'' in mainstream imperative languages.
adamc@527 382 $$\begin{array}{rrcll}
adamc@527 383 \textrm{Expressions} & e &::=& e : \tau & \textrm{type annotation} \\
adamc@529 384 &&& \hat{x} & \textrm{variable} \\
adamc@529 385 &&& \hat{X} & \textrm{datatype constructor} \\
adamc@527 386 &&& \ell & \textrm{constant} \\
adamc@527 387 \\
adamc@527 388 &&& e \; e & \textrm{function application} \\
adamc@527 389 &&& \lambda x : \tau \Rightarrow e & \textrm{function abstraction} \\
adamc@527 390 &&& e [c] & \textrm{polymorphic function application} \\
adamc@852 391 &&& \lambda [x \; ? \; \kappa] \Rightarrow e & \textrm{polymorphic function abstraction} \\
adamc@655 392 &&& e [\kappa] & \textrm{kind-polymorphic function application} \\
adamc@652 393 &&& X \Longrightarrow e & \textrm{kind-polymorphic function abstraction} \\
adamc@527 394 \\
adamc@527 395 &&& \{(c = e,)^*\} & \textrm{known-length record} \\
adamc@527 396 &&& e.c & \textrm{record field projection} \\
adamc@527 397 &&& e \rc e & \textrm{record concatenation} \\
adamc@527 398 &&& e \rcut c & \textrm{removal of a single record field} \\
adamc@527 399 &&& e \rcutM c & \textrm{removal of multiple record fields} \\
adamc@527 400 \\
adamc@527 401 &&& \mt{let} \; ed^* \; \mt{in} \; e \; \mt{end} & \textrm{local definitions} \\
adamc@527 402 \\
adamc@527 403 &&& \mt{case} \; e \; \mt{of} \; (p \Rightarrow e|)^+ & \textrm{pattern matching} \\
adamc@527 404 \\
adamc@654 405 &&& \lambda [c \sim c] \Rightarrow e & \textrm{guarded expression abstraction} \\
adamc@654 406 &&& e \; ! & \textrm{guarded expression application} \\
adamc@527 407 \\
adamc@527 408 &&& \_ & \textrm{wildcard} \\
adamc@527 409 &&& (e) & \textrm{explicit precedence} \\
adamc@527 410 \\
adamc@527 411 \textrm{Local declarations} & ed &::=& \cd{val} \; x : \tau = e & \textrm{non-recursive value} \\
adamc@527 412 &&& \cd{val} \; \cd{rec} \; (x : \tau = e \; \cd{and})^+ & \textrm{mutually-recursive values} \\
adamc@527 413 \end{array}$$
adamc@527 414
adamc@655 415 As with constructors, we include both abstraction and application for kind polymorphism, but applications are only inferred internally.
adamc@655 416
adamc@528 417 \emph{Declarations} primarily bring new symbols into context.
adamc@528 418 $$\begin{array}{rrcll}
adamc@528 419 \textrm{Declarations} & d &::=& \mt{con} \; x :: \kappa = c & \textrm{constructor synonym} \\
adamc@528 420 &&& \mt{datatype} \; x \; x^* = dc\mid^+ & \textrm{algebraic datatype definition} \\
adamc@529 421 &&& \mt{datatype} \; x = \mt{datatype} \; M.x & \textrm{algebraic datatype import} \\
adamc@528 422 &&& \mt{val} \; x : \tau = e & \textrm{value} \\
adamc@528 423 &&& \mt{val} \; \cd{rec} \; (x : \tau = e \; \mt{and})^+ & \textrm{mutually-recursive values} \\
adamc@528 424 &&& \mt{structure} \; X : S = M & \textrm{module definition} \\
adamc@528 425 &&& \mt{signature} \; X = S & \textrm{signature definition} \\
adamc@528 426 &&& \mt{open} \; M & \textrm{module inclusion} \\
adamc@528 427 &&& \mt{constraint} \; c \sim c & \textrm{record disjointness constraint} \\
adamc@528 428 &&& \mt{open} \; \mt{constraints} \; M & \textrm{inclusion of just the constraints from a module} \\
adamc@528 429 &&& \mt{table} \; x : c & \textrm{SQL table} \\
adamc@784 430 &&& \mt{view} \; x : c & \textrm{SQL view} \\
adamc@528 431 &&& \mt{sequence} \; x & \textrm{SQL sequence} \\
adamc@535 432 &&& \mt{cookie} \; x : \tau & \textrm{HTTP cookie} \\
adamc@784 433 &&& \mt{style} \; x : \tau & \textrm{CSS class} \\
adamc@654 434 &&& \mt{class} \; x :: \kappa = c & \textrm{concrete constructor class} \\
adamc@1085 435 &&& \mt{task} \; e = e & \textrm{recurring task} \\
adamc@528 436 \\
adamc@529 437 \textrm{Modules} & M &::=& \mt{struct} \; d^* \; \mt{end} & \textrm{constant} \\
adamc@529 438 &&& X & \textrm{variable} \\
adamc@529 439 &&& M.X & \textrm{projection} \\
adamc@529 440 &&& M(M) & \textrm{functor application} \\
adamc@529 441 &&& \mt{functor}(X : S) : S = M & \textrm{functor abstraction} \\
adamc@528 442 \end{array}$$
adamc@528 443
adamc@528 444 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@527 445
adamc@784 446 We omit some extra possibilities in $\mt{table}$ syntax, deferring them to Section \ref{tables}.
adamc@784 447
adamc@529 448 \subsection{Shorthands}
adamc@529 449
adamc@529 450 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@529 451
adamc@529 452 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@529 453
adamc@529 454 A record type may be written $\{(c = c,)^*\}$, which elaborates to $\$[(c = c,)^*]$.
adamc@529 455
adamc@533 456 The notation $[c_1, \ldots, c_n]$ is shorthand for $[c_1 = (), \ldots, c_n = ()]$.
adamc@533 457
adamc@529 458 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@529 459
adamc@852 460 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@529 461
adamc@529 462 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@529 463
adamc@529 464 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@529 465
adamc@654 466 A signature item or declaration $\mt{class} \; x = \lambda y \Rightarrow c$ may be abbreviated $\mt{class} \; x \; y = c$.
adamc@529 467
adamc@1102 468 Handling of implicit and explicit constructor arguments may be tweaked with some prefixes to variable references. An expression $@x$ is a version of $x$ where all implicit constructor arguments have been made explicit. An expression $@@x$ achieves the same effect, additionally halting automatic resolution of type class instances and automatic proving of disjointness constraints. The default is that implicit arguments are inserted automatically after any reference to a non-local variable, or after any application of a non-local variable to one or more arguments. For such an expression, implicit wildcard arguments are added for the longest prefix of the expression's type consisting only of implicit polymorphism, type class instances, and disjointness obligations. The same syntax works for variables projected out of modules and for capitalized variables (datatype constructors).
adamc@529 469
adamc@852 470 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@529 471
adamc@852 472 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@852 473
adamc@852 474 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@529 475
adamc@529 476 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@529 477
adamc@852 478 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@852 479
adamc@853 480 A declaration $\mt{table} \; x : \{(c = c,)^*\}$ is elaborated into $\mt{table} \; x : [(c = c,)^*]$.
adamc@529 481
adamc@529 482 The syntax $\mt{where} \; \mt{type}$ is an alternate form of $\mt{where} \; \mt{con}$.
adamc@529 483
adamc@529 484 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@529 485
adamc@529 486 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@529 487
adamc@784 488 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@529 489
adamc@530 490
adamc@530 491 \section{Static Semantics}
adamc@530 492
adamc@530 493 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@530 494
adamc@530 495 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@530 496 \begin{itemize}
adamc@655 497 \item $\Gamma \vdash \kappa$ expresses kind well-formedness.
adamc@530 498 \item $\Gamma \vdash c :: \kappa$ assigns a kind to a constructor in a context.
adamc@530 499 \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@531 500 \item $\Gamma \vdash c \hookrightarrow C$ proves that record constructor $c$ decomposes into set $C$ of field names and record constructors.
adamc@530 501 \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@530 502 \item $\Gamma \vdash e : \tau$ is a standard typing judgment.
adamc@534 503 \item $\Gamma \vdash p \leadsto \Gamma; \tau$ combines typing of patterns with calculation of which new variables they bind.
adamc@537 504 \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@537 505 \item $\Gamma \vdash S \equiv S$ is the signature equivalence judgment.
adamc@536 506 \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@530 507 \item $\Gamma \vdash M : S$ is the module signature checking judgment.
adamc@537 508 \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@539 509 \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@530 510 \end{itemize}
adamc@530 511
adamc@655 512
adamc@655 513 \subsection{Kind Well-Formedness}
adamc@655 514
adamc@655 515 $$\infer{\Gamma \vdash \mt{Type}}{}
adamc@655 516 \quad \infer{\Gamma \vdash \mt{Unit}}{}
adamc@655 517 \quad \infer{\Gamma \vdash \mt{Name}}{}
adamc@655 518 \quad \infer{\Gamma \vdash \kappa_1 \to \kappa_2}{
adamc@655 519 \Gamma \vdash \kappa_1
adamc@655 520 & \Gamma \vdash \kappa_2
adamc@655 521 }
adamc@655 522 \quad \infer{\Gamma \vdash \{\kappa\}}{
adamc@655 523 \Gamma \vdash \kappa
adamc@655 524 }
adamc@655 525 \quad \infer{\Gamma \vdash (\kappa_1 \times \ldots \times \kappa_n)}{
adamc@655 526 \forall i: \Gamma \vdash \kappa_i
adamc@655 527 }$$
adamc@655 528
adamc@655 529 $$\infer{\Gamma \vdash X}{
adamc@655 530 X \in \Gamma
adamc@655 531 }
adamc@655 532 \quad \infer{\Gamma \vdash X \longrightarrow \kappa}{
adamc@655 533 \Gamma, X \vdash \kappa
adamc@655 534 }$$
adamc@655 535
adamc@530 536 \subsection{Kinding}
adamc@530 537
adamc@655 538 We write $[X \mapsto \kappa_1]\kappa_2$ for capture-avoiding substitution of $\kappa_1$ for $X$ in $\kappa_2$.
adamc@655 539
adamc@530 540 $$\infer{\Gamma \vdash (c) :: \kappa :: \kappa}{
adamc@530 541 \Gamma \vdash c :: \kappa
adamc@530 542 }
adamc@530 543 \quad \infer{\Gamma \vdash x :: \kappa}{
adamc@530 544 x :: \kappa \in \Gamma
adamc@530 545 }
adamc@530 546 \quad \infer{\Gamma \vdash x :: \kappa}{
adamc@530 547 x :: \kappa = c \in \Gamma
adamc@530 548 }$$
adamc@530 549
adamc@530 550 $$\infer{\Gamma \vdash M.x :: \kappa}{
adamc@537 551 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 552 & \mt{proj}(M, \overline{s}, \mt{con} \; x) = \kappa
adamc@530 553 }
adamc@530 554 \quad \infer{\Gamma \vdash M.x :: \kappa}{
adamc@537 555 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 556 & \mt{proj}(M, \overline{s}, \mt{con} \; x) = (\kappa, c)
adamc@530 557 }$$
adamc@530 558
adamc@530 559 $$\infer{\Gamma \vdash \tau_1 \to \tau_2 :: \mt{Type}}{
adamc@530 560 \Gamma \vdash \tau_1 :: \mt{Type}
adamc@530 561 & \Gamma \vdash \tau_2 :: \mt{Type}
adamc@530 562 }
adamc@530 563 \quad \infer{\Gamma \vdash x \; ? \: \kappa \to \tau :: \mt{Type}}{
adamc@530 564 \Gamma, x :: \kappa \vdash \tau :: \mt{Type}
adamc@530 565 }
adamc@655 566 \quad \infer{\Gamma \vdash X \longrightarrow \tau :: \mt{Type}}{
adamc@655 567 \Gamma, X \vdash \tau :: \mt{Type}
adamc@655 568 }
adamc@530 569 \quad \infer{\Gamma \vdash \$c :: \mt{Type}}{
adamc@530 570 \Gamma \vdash c :: \{\mt{Type}\}
adamc@530 571 }$$
adamc@530 572
adamc@530 573 $$\infer{\Gamma \vdash c_1 \; c_2 :: \kappa_2}{
adamc@530 574 \Gamma \vdash c_1 :: \kappa_1 \to \kappa_2
adamc@530 575 & \Gamma \vdash c_2 :: \kappa_1
adamc@530 576 }
adamc@530 577 \quad \infer{\Gamma \vdash \lambda x \; :: \; \kappa_1 \Rightarrow c :: \kappa_1 \to \kappa_2}{
adamc@530 578 \Gamma, x :: \kappa_1 \vdash c :: \kappa_2
adamc@530 579 }$$
adamc@530 580
adamc@655 581 $$\infer{\Gamma \vdash c[\kappa'] :: [X \mapsto \kappa']\kappa}{
adamc@655 582 \Gamma \vdash c :: X \to \kappa
adamc@655 583 & \Gamma \vdash \kappa'
adamc@655 584 }
adamc@655 585 \quad \infer{\Gamma \vdash X \Longrightarrow c :: X \to \kappa}{
adamc@655 586 \Gamma, X \vdash c :: \kappa
adamc@655 587 }$$
adamc@655 588
adamc@530 589 $$\infer{\Gamma \vdash () :: \mt{Unit}}{}
adamc@530 590 \quad \infer{\Gamma \vdash \#X :: \mt{Name}}{}$$
adamc@530 591
adamc@530 592 $$\infer{\Gamma \vdash [\overline{c_i = c'_i}] :: \{\kappa\}}{
adamc@530 593 \forall i: \Gamma \vdash c_i : \mt{Name}
adamc@530 594 & \Gamma \vdash c'_i :: \kappa
adamc@530 595 & \forall i \neq j: \Gamma \vdash c_i \sim c_j
adamc@530 596 }
adamc@530 597 \quad \infer{\Gamma \vdash c_1 \rc c_2 :: \{\kappa\}}{
adamc@530 598 \Gamma \vdash c_1 :: \{\kappa\}
adamc@530 599 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@530 600 & \Gamma \vdash c_1 \sim c_2
adamc@530 601 }$$
adamc@530 602
adamc@655 603 $$\infer{\Gamma \vdash \mt{map} :: (\kappa_1 \to \kappa_2) \to \{\kappa_1\} \to \{\kappa_2\}}{}$$
adamc@530 604
adamc@573 605 $$\infer{\Gamma \vdash (\overline c) :: (\kappa_1 \times \ldots \times \kappa_n)}{
adamc@573 606 \forall i: \Gamma \vdash c_i :: \kappa_i
adamc@530 607 }
adamc@573 608 \quad \infer{\Gamma \vdash c.i :: \kappa_i}{
adamc@573 609 \Gamma \vdash c :: (\kappa_1 \times \ldots \times \kappa_n)
adamc@530 610 }$$
adamc@530 611
adamc@655 612 $$\infer{\Gamma \vdash \lambda [c_1 \sim c_2] \Rightarrow \tau :: \mt{Type}}{
adamc@655 613 \Gamma \vdash c_1 :: \{\kappa\}
adamc@530 614 & \Gamma \vdash c_2 :: \{\kappa'\}
adamc@655 615 & \Gamma, c_1 \sim c_2 \vdash \tau :: \mt{Type}
adamc@530 616 }$$
adamc@530 617
adamc@531 618 \subsection{Record Disjointness}
adamc@531 619
adamc@531 620 $$\infer{\Gamma \vdash c_1 \sim c_2}{
adamc@558 621 \Gamma \vdash c_1 \hookrightarrow C_1
adamc@558 622 & \Gamma \vdash c_2 \hookrightarrow C_2
adamc@558 623 & \forall c'_1 \in C_1, c'_2 \in C_2: \Gamma \vdash c'_1 \sim c'_2
adamc@531 624 }
adamc@531 625 \quad \infer{\Gamma \vdash X \sim X'}{
adamc@531 626 X \neq X'
adamc@531 627 }$$
adamc@531 628
adamc@531 629 $$\infer{\Gamma \vdash c_1 \sim c_2}{
adamc@531 630 c'_1 \sim c'_2 \in \Gamma
adamc@558 631 & \Gamma \vdash c'_1 \hookrightarrow C_1
adamc@558 632 & \Gamma \vdash c'_2 \hookrightarrow C_2
adamc@558 633 & c_1 \in C_1
adamc@558 634 & c_2 \in C_2
adamc@531 635 }$$
adamc@531 636
adamc@531 637 $$\infer{\Gamma \vdash c \hookrightarrow \{c\}}{}
adamc@531 638 \quad \infer{\Gamma \vdash [\overline{c = c'}] \hookrightarrow \{\overline{c}\}}{}
adamc@531 639 \quad \infer{\Gamma \vdash c_1 \rc c_2 \hookrightarrow C_1 \cup C_2}{
adamc@531 640 \Gamma \vdash c_1 \hookrightarrow C_1
adamc@531 641 & \Gamma \vdash c_2 \hookrightarrow C_2
adamc@531 642 }
adamc@531 643 \quad \infer{\Gamma \vdash c \hookrightarrow C}{
adamc@531 644 \Gamma \vdash c \equiv c'
adamc@531 645 & \Gamma \vdash c' \hookrightarrow C
adamc@531 646 }
adamc@531 647 \quad \infer{\Gamma \vdash \mt{map} \; f \; c \hookrightarrow C}{
adamc@531 648 \Gamma \vdash c \hookrightarrow C
adamc@531 649 }$$
adamc@531 650
adamc@541 651 \subsection{\label{definitional}Definitional Equality}
adamc@532 652
adamc@655 653 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@532 654
adamc@532 655 $$\infer{\Gamma \vdash c \equiv c}{}
adamc@532 656 \quad \infer{\Gamma \vdash c_1 \equiv c_2}{
adamc@532 657 \Gamma \vdash c_2 \equiv c_1
adamc@532 658 }
adamc@532 659 \quad \infer{\Gamma \vdash c_1 \equiv c_3}{
adamc@532 660 \Gamma \vdash c_1 \equiv c_2
adamc@532 661 & \Gamma \vdash c_2 \equiv c_3
adamc@532 662 }
adamc@532 663 \quad \infer{\Gamma \vdash \mathcal C[c_1] \equiv \mathcal C[c_2]}{
adamc@532 664 \Gamma \vdash c_1 \equiv c_2
adamc@532 665 }$$
adamc@532 666
adamc@532 667 $$\infer{\Gamma \vdash x \equiv c}{
adamc@532 668 x :: \kappa = c \in \Gamma
adamc@532 669 }
adamc@532 670 \quad \infer{\Gamma \vdash M.x \equiv c}{
adamc@537 671 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 672 & \mt{proj}(M, \overline{s}, \mt{con} \; x) = (\kappa, c)
adamc@532 673 }
adamc@532 674 \quad \infer{\Gamma \vdash (\overline c).i \equiv c_i}{}$$
adamc@532 675
adamc@532 676 $$\infer{\Gamma \vdash (\lambda x :: \kappa \Rightarrow c) \; c' \equiv [x \mapsto c'] c}{}
adamc@655 677 \quad \infer{\Gamma \vdash (X \Longrightarrow c) [\kappa] \equiv [X \mapsto \kappa] c}{}$$
adamc@655 678
adamc@655 679 $$\infer{\Gamma \vdash c_1 \rc c_2 \equiv c_2 \rc c_1}{}
adamc@532 680 \quad \infer{\Gamma \vdash c_1 \rc (c_2 \rc c_3) \equiv (c_1 \rc c_2) \rc c_3}{}$$
adamc@532 681
adamc@532 682 $$\infer{\Gamma \vdash [] \rc c \equiv c}{}
adamc@532 683 \quad \infer{\Gamma \vdash [\overline{c_1 = c'_1}] \rc [\overline{c_2 = c'_2}] \equiv [\overline{c_1 = c'_1}, \overline{c_2 = c'_2}]}{}$$
adamc@532 684
adamc@655 685 $$\infer{\Gamma \vdash \mt{map} \; f \; [] \equiv []}{}
adamc@655 686 \quad \infer{\Gamma \vdash \mt{map} \; f \; ([c_1 = c_2] \rc c) \equiv [c_1 = f \; c_2] \rc \mt{map} \; f \; c}{}$$
adamc@532 687
adamc@532 688 $$\infer{\Gamma \vdash \mt{map} \; (\lambda x \Rightarrow x) \; c \equiv c}{}
adamc@655 689 \quad \infer{\Gamma \vdash \mt{map} \; f \; (\mt{map} \; f' \; c)
adamc@655 690 \equiv \mt{map} \; (\lambda x \Rightarrow f \; (f' \; x)) \; c}{}$$
adamc@532 691
adamc@532 692 $$\infer{\Gamma \vdash \mt{map} \; f \; (c_1 \rc c_2) \equiv \mt{map} \; f \; c_1 \rc \mt{map} \; f \; c_2}{}$$
adamc@531 693
adamc@534 694 \subsection{Expression Typing}
adamc@533 695
adamc@873 696 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@533 697
adamc@533 698 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@533 699
adamc@533 700 $$\infer{\Gamma \vdash e : \tau : \tau}{
adamc@533 701 \Gamma \vdash e : \tau
adamc@533 702 }
adamc@533 703 \quad \infer{\Gamma \vdash e : \tau}{
adamc@533 704 \Gamma \vdash e : \tau'
adamc@533 705 & \Gamma \vdash \tau' \equiv \tau
adamc@533 706 }
adamc@533 707 \quad \infer{\Gamma \vdash \ell : T(\ell)}{}$$
adamc@533 708
adamc@533 709 $$\infer{\Gamma \vdash x : \mathcal I(\tau)}{
adamc@533 710 x : \tau \in \Gamma
adamc@533 711 }
adamc@533 712 \quad \infer{\Gamma \vdash M.x : \mathcal I(\tau)}{
adamc@537 713 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 714 & \mt{proj}(M, \overline{s}, \mt{val} \; x) = \tau
adamc@533 715 }
adamc@533 716 \quad \infer{\Gamma \vdash X : \mathcal I(\tau)}{
adamc@533 717 X : \tau \in \Gamma
adamc@533 718 }
adamc@533 719 \quad \infer{\Gamma \vdash M.X : \mathcal I(\tau)}{
adamc@537 720 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 721 & \mt{proj}(M, \overline{s}, \mt{val} \; X) = \tau
adamc@533 722 }$$
adamc@533 723
adamc@533 724 $$\infer{\Gamma \vdash e_1 \; e_2 : \tau_2}{
adamc@533 725 \Gamma \vdash e_1 : \tau_1 \to \tau_2
adamc@533 726 & \Gamma \vdash e_2 : \tau_1
adamc@533 727 }
adamc@533 728 \quad \infer{\Gamma \vdash \lambda x : \tau_1 \Rightarrow e : \tau_1 \to \tau_2}{
adamc@533 729 \Gamma, x : \tau_1 \vdash e : \tau_2
adamc@533 730 }$$
adamc@533 731
adamc@533 732 $$\infer{\Gamma \vdash e [c] : [x \mapsto c]\tau}{
adamc@533 733 \Gamma \vdash e : x :: \kappa \to \tau
adamc@533 734 & \Gamma \vdash c :: \kappa
adamc@533 735 }
adamc@852 736 \quad \infer{\Gamma \vdash \lambda [x \; ? \; \kappa] \Rightarrow e : x \; ? \; \kappa \to \tau}{
adamc@533 737 \Gamma, x :: \kappa \vdash e : \tau
adamc@533 738 }$$
adamc@533 739
adamc@655 740 $$\infer{\Gamma \vdash e [\kappa] : [X \mapsto \kappa]\tau}{
adamc@655 741 \Gamma \vdash e : X \longrightarrow \tau
adamc@655 742 & \Gamma \vdash \kappa
adamc@655 743 }
adamc@655 744 \quad \infer{\Gamma \vdash X \Longrightarrow e : X \longrightarrow \tau}{
adamc@655 745 \Gamma, X \vdash e : \tau
adamc@655 746 }$$
adamc@655 747
adamc@533 748 $$\infer{\Gamma \vdash \{\overline{c = e}\} : \{\overline{c : \tau}\}}{
adamc@533 749 \forall i: \Gamma \vdash c_i :: \mt{Name}
adamc@533 750 & \Gamma \vdash e_i : \tau_i
adamc@533 751 & \forall i \neq j: \Gamma \vdash c_i \sim c_j
adamc@533 752 }
adamc@533 753 \quad \infer{\Gamma \vdash e.c : \tau}{
adamc@533 754 \Gamma \vdash e : \$([c = \tau] \rc c')
adamc@533 755 }
adamc@533 756 \quad \infer{\Gamma \vdash e_1 \rc e_2 : \$(c_1 \rc c_2)}{
adamc@533 757 \Gamma \vdash e_1 : \$c_1
adamc@533 758 & \Gamma \vdash e_2 : \$c_2
adamc@573 759 & \Gamma \vdash c_1 \sim c_2
adamc@533 760 }$$
adamc@533 761
adamc@533 762 $$\infer{\Gamma \vdash e \rcut c : \$c'}{
adamc@533 763 \Gamma \vdash e : \$([c = \tau] \rc c')
adamc@533 764 }
adamc@533 765 \quad \infer{\Gamma \vdash e \rcutM c : \$c'}{
adamc@533 766 \Gamma \vdash e : \$(c \rc c')
adamc@533 767 }$$
adamc@533 768
adamc@533 769 $$\infer{\Gamma \vdash \mt{let} \; \overline{ed} \; \mt{in} \; e \; \mt{end} : \tau}{
adamc@533 770 \Gamma \vdash \overline{ed} \leadsto \Gamma'
adamc@533 771 & \Gamma' \vdash e : \tau
adamc@533 772 }
adamc@533 773 \quad \infer{\Gamma \vdash \mt{case} \; e \; \mt{of} \; \overline{p \Rightarrow e} : \tau}{
adamc@533 774 \forall i: \Gamma \vdash p_i \leadsto \Gamma_i, \tau'
adamc@533 775 & \Gamma_i \vdash e_i : \tau
adamc@533 776 }$$
adamc@533 777
adamc@573 778 $$\infer{\Gamma \vdash \lambda [c_1 \sim c_2] \Rightarrow e : \lambda [c_1 \sim c_2] \Rightarrow \tau}{
adamc@533 779 \Gamma \vdash c_1 :: \{\kappa\}
adamc@655 780 & \Gamma \vdash c_2 :: \{\kappa'\}
adamc@533 781 & \Gamma, c_1 \sim c_2 \vdash e : \tau
adamc@662 782 }
adamc@662 783 \quad \infer{\Gamma \vdash e \; ! : \tau}{
adamc@662 784 \Gamma \vdash e : [c_1 \sim c_2] \Rightarrow \tau
adamc@662 785 & \Gamma \vdash c_1 \sim c_2
adamc@533 786 }$$
adamc@533 787
adamc@534 788 \subsection{Pattern Typing}
adamc@534 789
adamc@534 790 $$\infer{\Gamma \vdash \_ \leadsto \Gamma; \tau}{}
adamc@534 791 \quad \infer{\Gamma \vdash x \leadsto \Gamma, x : \tau; \tau}{}
adamc@534 792 \quad \infer{\Gamma \vdash \ell \leadsto \Gamma; T(\ell)}{}$$
adamc@534 793
adamc@534 794 $$\infer{\Gamma \vdash X \leadsto \Gamma; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@534 795 X : \overline{x ::: \mt{Type}} \to \tau \in \Gamma
adamc@534 796 & \textrm{$\tau$ not a function type}
adamc@534 797 }
adamc@534 798 \quad \infer{\Gamma \vdash X \; p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@534 799 X : \overline{x ::: \mt{Type}} \to \tau'' \to \tau \in \Gamma
adamc@534 800 & \Gamma \vdash p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau''
adamc@534 801 }$$
adamc@534 802
adamc@534 803 $$\infer{\Gamma \vdash M.X \leadsto \Gamma; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@537 804 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 805 & \mt{proj}(M, \overline{s}, \mt{val} \; X) = \overline{x ::: \mt{Type}} \to \tau
adamc@534 806 & \textrm{$\tau$ not a function type}
adamc@534 807 }$$
adamc@534 808
adamc@534 809 $$\infer{\Gamma \vdash M.X \; p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@537 810 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 811 & \mt{proj}(M, \overline{s}, \mt{val} \; X) = \overline{x ::: \mt{Type}} \to \tau'' \to \tau
adamc@534 812 & \Gamma \vdash p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau''
adamc@534 813 }$$
adamc@534 814
adamc@534 815 $$\infer{\Gamma \vdash \{\overline{x = p}\} \leadsto \Gamma_n; \{\overline{x = \tau}\}}{
adamc@534 816 \Gamma_0 = \Gamma
adamc@534 817 & \forall i: \Gamma_i \vdash p_i \leadsto \Gamma_{i+1}; \tau_i
adamc@534 818 }
adamc@534 819 \quad \infer{\Gamma \vdash \{\overline{x = p}, \ldots\} \leadsto \Gamma_n; \$([\overline{x = \tau}] \rc c)}{
adamc@534 820 \Gamma_0 = \Gamma
adamc@534 821 & \forall i: \Gamma_i \vdash p_i \leadsto \Gamma_{i+1}; \tau_i
adamc@534 822 }$$
adamc@534 823
adamc@852 824 $$\infer{\Gamma \vdash p : \tau \leadsto \Gamma'; \tau}{
adamc@852 825 \Gamma \vdash p \leadsto \Gamma'; \tau'
adamc@852 826 & \Gamma \vdash \tau' \equiv \tau
adamc@852 827 }$$
adamc@852 828
adamc@535 829 \subsection{Declaration Typing}
adamc@535 830
adamc@535 831 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@535 832
adamc@655 833 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@535 834
adamc@558 835 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@537 836 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@535 837
adamc@535 838 $$\infer{\Gamma \vdash \cdot \leadsto \Gamma}{}
adamc@535 839 \quad \infer{\Gamma \vdash d, \overline{d} \leadsto \Gamma''}{
adamc@535 840 \Gamma \vdash d \leadsto \Gamma'
adamc@535 841 & \Gamma' \vdash \overline{d} \leadsto \Gamma''
adamc@535 842 }$$
adamc@535 843
adamc@535 844 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa = c \leadsto \Gamma, x :: \kappa = c}{
adamc@535 845 \Gamma \vdash c :: \kappa
adamc@535 846 }
adamc@535 847 \quad \infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leadsto \Gamma'}{
adamc@535 848 \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} \vdash \overline{dc} \leadsto \Gamma'
adamc@535 849 }$$
adamc@535 850
adamc@535 851 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leadsto \Gamma'}{
adamc@537 852 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 853 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@535 854 & \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} = M.z \vdash \overline{dc} \leadsto \Gamma'
adamc@535 855 }$$
adamc@535 856
adamc@535 857 $$\infer{\Gamma \vdash \mt{val} \; x : \tau = e \leadsto \Gamma, x : \tau}{
adamc@535 858 \Gamma \vdash e : \tau
adamc@535 859 }$$
adamc@535 860
adamc@535 861 $$\infer{\Gamma \vdash \mt{val} \; \mt{rec} \; \overline{x : \tau = e} \leadsto \Gamma, \overline{x : \tau}}{
adamc@535 862 \forall i: \Gamma, \overline{x : \tau} \vdash e_i : \tau_i
adamc@535 863 & \textrm{$e_i$ starts with an expression $\lambda$, optionally preceded by constructor and disjointness $\lambda$s}
adamc@535 864 }$$
adamc@535 865
adamc@535 866 $$\infer{\Gamma \vdash \mt{structure} \; X : S = M \leadsto \Gamma, X : S}{
adamc@535 867 \Gamma \vdash M : S
adamc@558 868 & \textrm{ $M$ not a constant or application}
adamc@535 869 }
adamc@558 870 \quad \infer{\Gamma \vdash \mt{structure} \; X : S = M \leadsto \Gamma, X : \mt{selfify}(X, \overline{s})}{
adamc@558 871 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@539 872 }$$
adamc@539 873
adamc@539 874 $$\infer{\Gamma \vdash \mt{signature} \; X = S \leadsto \Gamma, X = S}{
adamc@535 875 \Gamma \vdash S
adamc@535 876 }$$
adamc@535 877
adamc@537 878 $$\infer{\Gamma \vdash \mt{open} \; M \leadsto \Gamma, \mathcal O(M, \overline{s})}{
adamc@537 879 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@535 880 }$$
adamc@535 881
adamc@535 882 $$\infer{\Gamma \vdash \mt{constraint} \; c_1 \sim c_2 \leadsto \Gamma}{
adamc@535 883 \Gamma \vdash c_1 :: \{\kappa\}
adamc@535 884 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@535 885 & \Gamma \vdash c_1 \sim c_2
adamc@535 886 }
adamc@537 887 \quad \infer{\Gamma \vdash \mt{open} \; \mt{constraints} \; M \leadsto \Gamma, \mathcal O_c(M, \overline{s})}{
adamc@537 888 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@535 889 }$$
adamc@535 890
adamc@784 891 $$\infer{\Gamma \vdash \mt{table} \; x : c \leadsto \Gamma, x : \mt{Basis}.\mt{sql\_table} \; c \; []}{
adamc@535 892 \Gamma \vdash c :: \{\mt{Type}\}
adamc@535 893 }
adamc@784 894 \quad \infer{\Gamma \vdash \mt{view} \; x : c \leadsto \Gamma, x : \mt{Basis}.\mt{sql\_view} \; c}{
adamc@784 895 \Gamma \vdash c :: \{\mt{Type}\}
adamc@784 896 }$$
adamc@784 897
adamc@784 898 $$\infer{\Gamma \vdash \mt{sequence} \; x \leadsto \Gamma, x : \mt{Basis}.\mt{sql\_sequence}}{}$$
adamc@535 899
adamc@535 900 $$\infer{\Gamma \vdash \mt{cookie} \; x : \tau \leadsto \Gamma, x : \mt{Basis}.\mt{http\_cookie} \; \tau}{
adamc@535 901 \Gamma \vdash \tau :: \mt{Type}
adamc@784 902 }
adamc@784 903 \quad \infer{\Gamma \vdash \mt{style} \; x \leadsto \Gamma, x : \mt{Basis}.\mt{css\_class}}{}$$
adamc@535 904
adamc@1085 905 $$\infer{\Gamma \vdash \mt{task} \; e_1 = e_2 \leadsto \Gamma}{
adamc@1085 906 \Gamma \vdash e_1 :: \mt{Basis}.\mt{task\_kind}
adamc@1085 907 & \Gamma \vdash e_2 :: \mt{Basis}.\mt{transaction} \; \{\}
adamc@1085 908 }$$
adamc@1085 909
adamc@784 910 $$\infer{\Gamma \vdash \mt{class} \; x :: \kappa = c \leadsto \Gamma, x :: \kappa = c}{
adamc@784 911 \Gamma \vdash c :: \kappa
adamc@535 912 }$$
adamc@535 913
adamc@535 914 $$\infer{\overline{y}; x; \Gamma \vdash \cdot \leadsto \Gamma}{}
adamc@535 915 \quad \infer{\overline{y}; x; \Gamma \vdash X \mid \overline{dc} \leadsto \Gamma', X : \overline{y ::: \mt{Type}} \to x \; \overline{y}}{
adamc@535 916 \overline{y}; x; \Gamma \vdash \overline{dc} \leadsto \Gamma'
adamc@535 917 }
adamc@535 918 \quad \infer{\overline{y}; x; \Gamma \vdash X \; \mt{of} \; \tau \mid \overline{dc} \leadsto \Gamma', X : \overline{y ::: \mt{Type}} \to \tau \to x \; \overline{y}}{
adamc@535 919 \overline{y}; x; \Gamma \vdash \overline{dc} \leadsto \Gamma'
adamc@535 920 }$$
adamc@535 921
adamc@537 922 \subsection{Signature Item Typing}
adamc@537 923
adamc@537 924 We appeal to a signature item analogue of the $\mathcal O$ function from the last subsection.
adamc@537 925
adamc@537 926 $$\infer{\Gamma \vdash \cdot \leadsto \Gamma}{}
adamc@537 927 \quad \infer{\Gamma \vdash s, \overline{s} \leadsto \Gamma''}{
adamc@537 928 \Gamma \vdash s \leadsto \Gamma'
adamc@537 929 & \Gamma' \vdash \overline{s} \leadsto \Gamma''
adamc@537 930 }$$
adamc@537 931
adamc@537 932 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa \leadsto \Gamma, x :: \kappa}{}
adamc@537 933 \quad \infer{\Gamma \vdash \mt{con} \; x :: \kappa = c \leadsto \Gamma, x :: \kappa = c}{
adamc@537 934 \Gamma \vdash c :: \kappa
adamc@537 935 }
adamc@537 936 \quad \infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leadsto \Gamma'}{
adamc@537 937 \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} \vdash \overline{dc} \leadsto \Gamma'
adamc@537 938 }$$
adamc@537 939
adamc@537 940 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leadsto \Gamma'}{
adamc@537 941 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 942 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@537 943 & \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} = M.z \vdash \overline{dc} \leadsto \Gamma'
adamc@537 944 }$$
adamc@537 945
adamc@537 946 $$\infer{\Gamma \vdash \mt{val} \; x : \tau \leadsto \Gamma, x : \tau}{
adamc@537 947 \Gamma \vdash \tau :: \mt{Type}
adamc@537 948 }$$
adamc@537 949
adamc@537 950 $$\infer{\Gamma \vdash \mt{structure} \; X : S \leadsto \Gamma, X : S}{
adamc@537 951 \Gamma \vdash S
adamc@537 952 }
adamc@537 953 \quad \infer{\Gamma \vdash \mt{signature} \; X = S \leadsto \Gamma, X = S}{
adamc@537 954 \Gamma \vdash S
adamc@537 955 }$$
adamc@537 956
adamc@537 957 $$\infer{\Gamma \vdash \mt{include} \; S \leadsto \Gamma, \mathcal O(\overline{s})}{
adamc@537 958 \Gamma \vdash S
adamc@537 959 & \Gamma \vdash S \equiv \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 960 }$$
adamc@537 961
adamc@537 962 $$\infer{\Gamma \vdash \mt{constraint} \; c_1 \sim c_2 \leadsto \Gamma, c_1 \sim c_2}{
adamc@537 963 \Gamma \vdash c_1 :: \{\kappa\}
adamc@537 964 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@537 965 }$$
adamc@537 966
adamc@784 967 $$\infer{\Gamma \vdash \mt{class} \; x :: \kappa = c \leadsto \Gamma, x :: \kappa = c}{
adamc@784 968 \Gamma \vdash c :: \kappa
adamc@537 969 }
adamc@784 970 \quad \infer{\Gamma \vdash \mt{class} \; x :: \kappa \leadsto \Gamma, x :: \kappa}{}$$
adamc@537 971
adamc@536 972 \subsection{Signature Compatibility}
adamc@536 973
adamc@558 974 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@537 975
adamc@537 976 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@537 977
adamc@536 978 $$\infer{\Gamma \vdash S \equiv S}{}
adamc@536 979 \quad \infer{\Gamma \vdash S_1 \equiv S_2}{
adamc@536 980 \Gamma \vdash S_2 \equiv S_1
adamc@536 981 }
adamc@536 982 \quad \infer{\Gamma \vdash X \equiv S}{
adamc@536 983 X = S \in \Gamma
adamc@536 984 }
adamc@536 985 \quad \infer{\Gamma \vdash M.X \equiv S}{
adamc@537 986 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 987 & \mt{proj}(M, \overline{s}, \mt{signature} \; X) = S
adamc@536 988 }$$
adamc@536 989
adamc@536 990 $$\infer{\Gamma \vdash S \; \mt{where} \; \mt{con} \; x = c \equiv \mt{sig} \; \overline{s^1} \; \mt{con} \; x :: \kappa = c \; \overline{s_2} \; \mt{end}}{
adamc@536 991 \Gamma \vdash S \equiv \mt{sig} \; \overline{s^1} \; \mt{con} \; x :: \kappa \; \overline{s_2} \; \mt{end}
adamc@536 992 & \Gamma \vdash c :: \kappa
adamc@537 993 }
adamc@537 994 \quad \infer{\Gamma \vdash \mt{sig} \; \overline{s^1} \; \mt{include} \; S \; \overline{s^2} \; \mt{end} \equiv \mt{sig} \; \overline{s^1} \; \overline{s} \; \overline{s^2} \; \mt{end}}{
adamc@537 995 \Gamma \vdash S \equiv \mt{sig} \; \overline{s} \; \mt{end}
adamc@536 996 }$$
adamc@536 997
adamc@536 998 $$\infer{\Gamma \vdash S_1 \leq S_2}{
adamc@536 999 \Gamma \vdash S_1 \equiv S_2
adamc@536 1000 }
adamc@536 1001 \quad \infer{\Gamma \vdash \mt{sig} \; \overline{s} \; \mt{end} \leq \mt{sig} \; \mt{end}}{}
adamc@537 1002 \quad \infer{\Gamma \vdash \mt{sig} \; \overline{s} \; \mt{end} \leq \mt{sig} \; s' \; \overline{s'} \; \mt{end}}{
adamc@537 1003 \Gamma \vdash \overline{s} \leq s'
adamc@537 1004 & \Gamma \vdash s' \leadsto \Gamma'
adamc@537 1005 & \Gamma' \vdash \mt{sig} \; \overline{s} \; \mt{end} \leq \mt{sig} \; \overline{s'} \; \mt{end}
adamc@537 1006 }$$
adamc@537 1007
adamc@537 1008 $$\infer{\Gamma \vdash s \; \overline{s} \leq s'}{
adamc@537 1009 \Gamma \vdash s \leq s'
adamc@537 1010 }
adamc@537 1011 \quad \infer{\Gamma \vdash s \; \overline{s} \leq s'}{
adamc@537 1012 \Gamma \vdash s \leadsto \Gamma'
adamc@537 1013 & \Gamma' \vdash \overline{s} \leq s'
adamc@536 1014 }$$
adamc@536 1015
adamc@536 1016 $$\infer{\Gamma \vdash \mt{functor} (X : S_1) : S_2 \leq \mt{functor} (X : S'_1) : S'_2}{
adamc@536 1017 \Gamma \vdash S'_1 \leq S_1
adamc@536 1018 & \Gamma, X : S'_1 \vdash S_2 \leq S'_2
adamc@536 1019 }$$
adamc@536 1020
adamc@537 1021 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa \leq \mt{con} \; x :: \kappa}{}
adamc@537 1022 \quad \infer{\Gamma \vdash \mt{con} \; x :: \kappa = c \leq \mt{con} \; x :: \kappa}{}
adamc@558 1023 \quad \infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leq \mt{con} \; x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type}}{}$$
adamc@537 1024
adamc@537 1025 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leq \mt{con} \; x :: \mt{Type}^{\mt{len}(y)} \to \mt{Type}}{
adamc@537 1026 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 1027 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@537 1028 }$$
adamc@537 1029
adamc@784 1030 $$\infer{\Gamma \vdash \mt{class} \; x :: \kappa \leq \mt{con} \; x :: \kappa}{}
adamc@784 1031 \quad \infer{\Gamma \vdash \mt{class} \; x :: \kappa = c \leq \mt{con} \; x :: \kappa}{}$$
adamc@537 1032
adamc@537 1033 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa = c_1 \leq \mt{con} \; x :: \mt{\kappa} = c_2}{
adamc@537 1034 \Gamma \vdash c_1 \equiv c_2
adamc@537 1035 }
adamc@784 1036 \quad \infer{\Gamma \vdash \mt{class} \; x :: \kappa = c_1 \leq \mt{con} \; x :: \kappa = c_2}{
adamc@537 1037 \Gamma \vdash c_1 \equiv c_2
adamc@537 1038 }$$
adamc@537 1039
adamc@537 1040 $$\infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leq \mt{datatype} \; x \; \overline{y} = \overline{dc'}}{
adamc@537 1041 \Gamma, \overline{y :: \mt{Type}} \vdash \overline{dc} \leq \overline{dc'}
adamc@537 1042 }$$
adamc@537 1043
adamc@537 1044 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leq \mt{datatype} \; x \; \overline{y} = \overline{dc'}}{
adamc@537 1045 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537 1046 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@537 1047 & \Gamma, \overline{y :: \mt{Type}} \vdash \overline{dc} \leq \overline{dc'}
adamc@537 1048 }$$
adamc@537 1049
adamc@537 1050 $$\infer{\Gamma \vdash \cdot \leq \cdot}{}
adamc@537 1051 \quad \infer{\Gamma \vdash X; \overline{dc} \leq X; \overline{dc'}}{
adamc@537 1052 \Gamma \vdash \overline{dc} \leq \overline{dc'}
adamc@537 1053 }
adamc@537 1054 \quad \infer{\Gamma \vdash X \; \mt{of} \; \tau_1; \overline{dc} \leq X \; \mt{of} \; \tau_2; \overline{dc'}}{
adamc@537 1055 \Gamma \vdash \tau_1 \equiv \tau_2
adamc@537 1056 & \Gamma \vdash \overline{dc} \leq \overline{dc'}
adamc@537 1057 }$$
adamc@537 1058
adamc@537 1059 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leq \mt{datatype} \; x = \mt{datatype} \; M'.z'}{
adamc@537 1060 \Gamma \vdash M.z \equiv M'.z'
adamc@537 1061 }$$
adamc@537 1062
adamc@537 1063 $$\infer{\Gamma \vdash \mt{val} \; x : \tau_1 \leq \mt{val} \; x : \tau_2}{
adamc@537 1064 \Gamma \vdash \tau_1 \equiv \tau_2
adamc@537 1065 }
adamc@537 1066 \quad \infer{\Gamma \vdash \mt{structure} \; X : S_1 \leq \mt{structure} \; X : S_2}{
adamc@537 1067 \Gamma \vdash S_1 \leq S_2
adamc@537 1068 }
adamc@537 1069 \quad \infer{\Gamma \vdash \mt{signature} \; X = S_1 \leq \mt{signature} \; X = S_2}{
adamc@537 1070 \Gamma \vdash S_1 \leq S_2
adamc@537 1071 & \Gamma \vdash S_2 \leq S_1
adamc@537 1072 }$$
adamc@537 1073
adamc@537 1074 $$\infer{\Gamma \vdash \mt{constraint} \; c_1 \sim c_2 \leq \mt{constraint} \; c'_1 \sim c'_2}{
adamc@537 1075 \Gamma \vdash c_1 \equiv c'_1
adamc@537 1076 & \Gamma \vdash c_2 \equiv c'_2
adamc@537 1077 }$$
adamc@537 1078
adamc@655 1079 $$\infer{\Gamma \vdash \mt{class} \; x :: \kappa \leq \mt{class} \; x :: \kappa}{}
adamc@655 1080 \quad \infer{\Gamma \vdash \mt{class} \; x :: \kappa = c \leq \mt{class} \; x :: \kappa}{}
adamc@655 1081 \quad \infer{\Gamma \vdash \mt{class} \; x :: \kappa = c_1 \leq \mt{class} \; x :: \kappa = c_2}{
adamc@537 1082 \Gamma \vdash c_1 \equiv c_2
adamc@537 1083 }$$
adamc@537 1084
adamc@538 1085 \subsection{Module Typing}
adamc@538 1086
adamc@538 1087 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 1088
adamc@538 1089 $$\infer{\Gamma \vdash M : S}{
adamc@538 1090 \Gamma \vdash M : S'
adamc@538 1091 & \Gamma \vdash S' \leq S
adamc@538 1092 }
adamc@538 1093 \quad \infer{\Gamma \vdash \mt{struct} \; \overline{d} \; \mt{end} : \mt{sig} \; \mt{sigOf}(\overline{d}) \; \mt{end}}{
adamc@538 1094 \Gamma \vdash \overline{d} \leadsto \Gamma'
adamc@538 1095 }
adamc@538 1096 \quad \infer{\Gamma \vdash X : S}{
adamc@538 1097 X : S \in \Gamma
adamc@538 1098 }$$
adamc@538 1099
adamc@538 1100 $$\infer{\Gamma \vdash M.X : S}{
adamc@538 1101 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@538 1102 & \mt{proj}(M, \overline{s}, \mt{structure} \; X) = S
adamc@538 1103 }$$
adamc@538 1104
adamc@538 1105 $$\infer{\Gamma \vdash M_1(M_2) : [X \mapsto M_2]S_2}{
adamc@538 1106 \Gamma \vdash M_1 : \mt{functor}(X : S_1) : S_2
adamc@538 1107 & \Gamma \vdash M_2 : S_1
adamc@538 1108 }
adamc@538 1109 \quad \infer{\Gamma \vdash \mt{functor} (X : S_1) : S_2 = M : \mt{functor} (X : S_1) : S_2}{
adamc@538 1110 \Gamma \vdash S_1
adamc@538 1111 & \Gamma, X : S_1 \vdash S_2
adamc@538 1112 & \Gamma, X : S_1 \vdash M : S_2
adamc@538 1113 }$$
adamc@538 1114
adamc@538 1115 \begin{eqnarray*}
adamc@538 1116 \mt{sigOf}(\cdot) &=& \cdot \\
adamc@538 1117 \mt{sigOf}(s \; \overline{s'}) &=& \mt{sigOf}(s) \; \mt{sigOf}(\overline{s'}) \\
adamc@538 1118 \\
adamc@538 1119 \mt{sigOf}(\mt{con} \; x :: \kappa = c) &=& \mt{con} \; x :: \kappa = c \\
adamc@538 1120 \mt{sigOf}(\mt{datatype} \; x \; \overline{y} = \overline{dc}) &=& \mt{datatype} \; x \; \overline{y} = \overline{dc} \\
adamc@538 1121 \mt{sigOf}(\mt{datatype} \; x = \mt{datatype} \; M.z) &=& \mt{datatype} \; x = \mt{datatype} \; M.z \\
adamc@538 1122 \mt{sigOf}(\mt{val} \; x : \tau = e) &=& \mt{val} \; x : \tau \\
adamc@538 1123 \mt{sigOf}(\mt{val} \; \mt{rec} \; \overline{x : \tau = e}) &=& \overline{\mt{val} \; x : \tau} \\
adamc@538 1124 \mt{sigOf}(\mt{structure} \; X : S = M) &=& \mt{structure} \; X : S \\
adamc@538 1125 \mt{sigOf}(\mt{signature} \; X = S) &=& \mt{signature} \; X = S \\
adamc@538 1126 \mt{sigOf}(\mt{open} \; M) &=& \mt{include} \; S \textrm{ (where $\Gamma \vdash M : S$)} \\
adamc@538 1127 \mt{sigOf}(\mt{constraint} \; c_1 \sim c_2) &=& \mt{constraint} \; c_1 \sim c_2 \\
adamc@538 1128 \mt{sigOf}(\mt{open} \; \mt{constraints} \; M) &=& \cdot \\
adamc@538 1129 \mt{sigOf}(\mt{table} \; x : c) &=& \mt{table} \; x : c \\
adamc@784 1130 \mt{sigOf}(\mt{view} \; x : c) &=& \mt{view} \; x : c \\
adamc@538 1131 \mt{sigOf}(\mt{sequence} \; x) &=& \mt{sequence} \; x \\
adamc@538 1132 \mt{sigOf}(\mt{cookie} \; x : \tau) &=& \mt{cookie} \; x : \tau \\
adamc@784 1133 \mt{sigOf}(\mt{style} \; x) &=& \mt{style} \; x \\
adamc@655 1134 \mt{sigOf}(\mt{class} \; x :: \kappa = c) &=& \mt{class} \; x :: \kappa = c \\
adamc@538 1135 \end{eqnarray*}
adamc@539 1136 \begin{eqnarray*}
adamc@539 1137 \mt{selfify}(M, \cdot) &=& \cdot \\
adamc@558 1138 \mt{selfify}(M, s \; \overline{s'}) &=& \mt{selfify}(M, s) \; \mt{selfify}(M, \overline{s'}) \\
adamc@539 1139 \\
adamc@539 1140 \mt{selfify}(M, \mt{con} \; x :: \kappa) &=& \mt{con} \; x :: \kappa = M.x \\
adamc@539 1141 \mt{selfify}(M, \mt{con} \; x :: \kappa = c) &=& \mt{con} \; x :: \kappa = c \\
adamc@539 1142 \mt{selfify}(M, \mt{datatype} \; x \; \overline{y} = \overline{dc}) &=& \mt{datatype} \; x \; \overline{y} = \mt{datatype} \; M.x \\
adamc@539 1143 \mt{selfify}(M, \mt{datatype} \; x = \mt{datatype} \; M'.z) &=& \mt{datatype} \; x = \mt{datatype} \; M'.z \\
adamc@539 1144 \mt{selfify}(M, \mt{val} \; x : \tau) &=& \mt{val} \; x : \tau \\
adamc@539 1145 \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 1146 \mt{selfify}(M, \mt{signature} \; X = S) &=& \mt{signature} \; X = S \\
adamc@539 1147 \mt{selfify}(M, \mt{include} \; S) &=& \mt{include} \; S \\
adamc@539 1148 \mt{selfify}(M, \mt{constraint} \; c_1 \sim c_2) &=& \mt{constraint} \; c_1 \sim c_2 \\
adamc@655 1149 \mt{selfify}(M, \mt{class} \; x :: \kappa) &=& \mt{class} \; x :: \kappa = M.x \\
adamc@655 1150 \mt{selfify}(M, \mt{class} \; x :: \kappa = c) &=& \mt{class} \; x :: \kappa = c \\
adamc@539 1151 \end{eqnarray*}
adamc@539 1152
adamc@540 1153 \subsection{Module Projection}
adamc@540 1154
adamc@540 1155 \begin{eqnarray*}
adamc@540 1156 \mt{proj}(M, \mt{con} \; x :: \kappa \; \overline{s}, \mt{con} \; x) &=& \kappa \\
adamc@540 1157 \mt{proj}(M, \mt{con} \; x :: \kappa = c \; \overline{s}, \mt{con} \; x) &=& (\kappa, c) \\
adamc@540 1158 \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 1159 \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 1160 && \textrm{and $\mt{proj}(M', \overline{s'}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})$)} \\
adamc@655 1161 \mt{proj}(M, \mt{class} \; x :: \kappa \; \overline{s}, \mt{con} \; x) &=& \kappa \to \mt{Type} \\
adamc@655 1162 \mt{proj}(M, \mt{class} \; x :: \kappa = c \; \overline{s}, \mt{con} \; x) &=& (\kappa \to \mt{Type}, c) \\
adamc@540 1163 \\
adamc@540 1164 \mt{proj}(M, \mt{datatype} \; x \; \overline{y} = \overline{dc} \; \overline{s}, \mt{datatype} \; x) &=& (\overline{y}, \overline{dc}) \\
adamc@540 1165 \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 1166 \\
adamc@540 1167 \mt{proj}(M, \mt{val} \; x : \tau \; \overline{s}, \mt{val} \; x) &=& \tau \\
adamc@540 1168 \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 1169 \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 1170 \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 1171 && \textrm{and $\mt{proj}(M', \overline{s'}, \mt{datatype} \; z = (\overline{y}, \overline{dc})$ and $X \in \overline{dc}$)} \\
adamc@540 1172 \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 1173 && \textrm{and $\mt{proj}(M', \overline{s'}, \mt{datatype} \; z = (\overline{y}, \overline{dc})$ and $X \; \mt{of} \; \tau \in \overline{dc}$)} \\
adamc@540 1174 \\
adamc@540 1175 \mt{proj}(M, \mt{structure} \; X : S \; \overline{s}, \mt{structure} \; X) &=& S \\
adamc@540 1176 \\
adamc@540 1177 \mt{proj}(M, \mt{signature} \; X = S \; \overline{s}, \mt{signature} \; X) &=& S \\
adamc@540 1178 \\
adamc@540 1179 \mt{proj}(M, \mt{con} \; x :: \kappa \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@540 1180 \mt{proj}(M, \mt{con} \; x :: \kappa = c \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@540 1181 \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 1182 \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 1183 \mt{proj}(M, \mt{val} \; x : \tau \; \overline{s}, V) &=& \mt{proj}(M, \overline{s}, V) \\
adamc@540 1184 \mt{proj}(M, \mt{structure} \; X : S \; \overline{s}, V) &=& [X \mapsto M.X]\mt{proj}(M, \overline{s}, V) \\
adamc@540 1185 \mt{proj}(M, \mt{signature} \; X = S \; \overline{s}, V) &=& [X \mapsto M.X]\mt{proj}(M, \overline{s}, V) \\
adamc@540 1186 \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 1187 \mt{proj}(M, \mt{constraint} \; c_1 \sim c_2 \; \overline{s}, V) &=& \mt{proj}(M, \overline{s}, V) \\
adamc@655 1188 \mt{proj}(M, \mt{class} \; x :: \kappa \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@655 1189 \mt{proj}(M, \mt{class} \; x :: \kappa = c \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@540 1190 \end{eqnarray*}
adamc@540 1191
adamc@541 1192
adamc@541 1193 \section{Type Inference}
adamc@541 1194
adamc@541 1195 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 1196
adamc@541 1197 \subsection{Basic Unification}
adamc@541 1198
adamc@560 1199 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 1200
adamc@656 1201 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 1202
adamc@541 1203 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 1204
adamc@541 1205 \subsection{Unifying Record Types}
adamc@541 1206
adamc@570 1207 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 1208
adamc@656 1209 \subsection{\label{typeclasses}Constructor Classes}
adamc@541 1210
adamc@784 1211 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 1212
adamc@784 1213 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 1214
adamc@656 1215 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 1216
adamc@656 1217 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 1218
adamc@541 1219 \subsection{Reverse-Engineering Record Types}
adamc@541 1220
adamc@656 1221 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 1222
adamc@541 1223 \subsection{Implicit Arguments in Functor Applications}
adamc@541 1224
adamc@656 1225 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 1226
adamc@541 1227
adamc@542 1228 \section{The Ur Standard Library}
adamc@542 1229
adamc@542 1230 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 1231
adamc@542 1232 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 1233
adamc@542 1234 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 1235 $$\begin{array}{l}
adamc@542 1236 \mt{type} \; \mt{int} \\
adamc@542 1237 \mt{type} \; \mt{float} \\
adamc@873 1238 \mt{type} \; \mt{char} \\
adamc@542 1239 \mt{type} \; \mt{string} \\
adamc@542 1240 \mt{type} \; \mt{time} \\
adamc@785 1241 \mt{type} \; \mt{blob} \\
adamc@542 1242 \\
adamc@542 1243 \mt{type} \; \mt{unit} = \{\} \\
adamc@542 1244 \\
adamc@542 1245 \mt{datatype} \; \mt{bool} = \mt{False} \mid \mt{True} \\
adamc@542 1246 \\
adamc@785 1247 \mt{datatype} \; \mt{option} \; \mt{t} = \mt{None} \mid \mt{Some} \; \mt{of} \; \mt{t} \\
adamc@785 1248 \\
adamc@785 1249 \mt{datatype} \; \mt{list} \; \mt{t} = \mt{Nil} \mid \mt{Cons} \; \mt{of} \; \mt{t} \times \mt{list} \; \mt{t}
adamc@542 1250 \end{array}$$
adamc@542 1251
adamc@785 1252 The only unusual element of this list is the $\mt{blob}$ type, which stands for binary sequences.
adamc@785 1253
adamc@657 1254 Another important generic Ur element comes at the beginning of \texttt{top.urs}.
adamc@657 1255
adamc@657 1256 $$\begin{array}{l}
adamc@657 1257 \mt{con} \; \mt{folder} :: \mt{K} \longrightarrow \{\mt{K}\} \to \mt{Type} \\
adamc@657 1258 \\
adamc@657 1259 \mt{val} \; \mt{fold} : \mt{K} \longrightarrow \mt{tf} :: (\{\mt{K}\} \to \mt{Type}) \\
adamc@657 1260 \hspace{.1in} \to (\mt{nm} :: \mt{Name} \to \mt{v} :: \mt{K} \to \mt{r} :: \{\mt{K}\} \to [[\mt{nm}] \sim \mt{r}] \Rightarrow \\
adamc@657 1261 \hspace{.2in} \mt{tf} \; \mt{r} \to \mt{tf} \; ([\mt{nm} = \mt{v}] \rc \mt{r})) \\
adamc@657 1262 \hspace{.1in} \to \mt{tf} \; [] \\
adamc@657 1263 \hspace{.1in} \to \mt{r} :: \{\mt{K}\} \to \mt{folder} \; \mt{r} \to \mt{tf} \; \mt{r}
adamc@657 1264 \end{array}$$
adamc@657 1265
adamc@657 1266 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 1267
adamc@664 1268 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 1269
adamc@542 1270
adamc@542 1271 \section{The Ur/Web Standard Library}
adamc@542 1272
adamc@658 1273 \subsection{Monads}
adamc@658 1274
adamc@658 1275 The Ur Basis defines the monad constructor class from Haskell.
adamc@658 1276
adamc@658 1277 $$\begin{array}{l}
adamc@658 1278 \mt{class} \; \mt{monad} :: \mt{Type} \to \mt{Type} \\
adamc@658 1279 \mt{val} \; \mt{return} : \mt{m} ::: (\mt{Type} \to \mt{Type}) \to \mt{t} ::: \mt{Type} \\
adamc@658 1280 \hspace{.1in} \to \mt{monad} \; \mt{m} \\
adamc@658 1281 \hspace{.1in} \to \mt{t} \to \mt{m} \; \mt{t} \\
adamc@658 1282 \mt{val} \; \mt{bind} : \mt{m} ::: (\mt{Type} \to \mt{Type}) \to \mt{t1} ::: \mt{Type} \to \mt{t2} ::: \mt{Type} \\
adamc@658 1283 \hspace{.1in} \to \mt{monad} \; \mt{m} \\
adamc@658 1284 \hspace{.1in} \to \mt{m} \; \mt{t1} \to (\mt{t1} \to \mt{m} \; \mt{t2}) \\
adamc@658 1285 \hspace{.1in} \to \mt{m} \; \mt{t2}
adamc@658 1286 \end{array}$$
adamc@658 1287
adamc@542 1288 \subsection{Transactions}
adamc@542 1289
adamc@542 1290 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 1291 $$\begin{array}{l}
adamc@542 1292 \mt{con} \; \mt{transaction} :: \mt{Type} \to \mt{Type} \\
adamc@658 1293 \mt{val} \; \mt{transaction\_monad} : \mt{monad} \; \mt{transaction}
adamc@542 1294 \end{array}$$
adamc@542 1295
adamc@542 1296 \subsection{HTTP}
adamc@542 1297
adamc@542 1298 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 1299 $$\begin{array}{l}
adamc@786 1300 \mt{val} \; \mt{requestHeader} : \mt{string} \to \mt{transaction} \; (\mt{option} \; \mt{string}) \\
adamc@786 1301 \\
adamc@786 1302 \mt{con} \; \mt{http\_cookie} :: \mt{Type} \to \mt{Type} \\
adamc@786 1303 \mt{val} \; \mt{getCookie} : \mt{t} ::: \mt{Type} \to \mt{http\_cookie} \; \mt{t} \to \mt{transaction} \; (\mt{option} \; \mt{t}) \\
adamc@1050 1304 \mt{val} \; \mt{setCookie} : \mt{t} ::: \mt{Type} \to \mt{http\_cookie} \; \mt{t} \to \{\mt{Value} : \mt{t}, \mt{Expires} : \mt{option} \; \mt{time}, \mt{Secure} : \mt{bool}\} \to \mt{transaction} \; \mt{unit} \\
adamc@1050 1305 \mt{val} \; \mt{clearCookie} : \mt{t} ::: \mt{Type} \to \mt{http\_cookie} \; \mt{t} \to \mt{transaction} \; \mt{unit}
adamc@786 1306 \end{array}$$
adamc@786 1307
adamc@786 1308 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 1309 $$\begin{array}{l}
adamc@786 1310 \mt{type} \; \mt{url} \\
adamc@786 1311 \mt{val} \; \mt{bless} : \mt{string} \to \mt{url} \\
adamc@786 1312 \mt{val} \; \mt{checkUrl} : \mt{string} \to \mt{option} \; \mt{url}
adamc@786 1313 \end{array}$$
adamc@786 1314 $\mt{bless}$ raises a runtime error if the string passed to it fails the URL policy.
adamc@786 1315
adamc@1085 1316 It is possible to grab the current page's URL or to build a URL for an arbitrary transaction that would also be an acceptable value of a \texttt{link} attribute of the \texttt{a} tag.
adamc@1085 1317 $$\begin{array}{l}
adamc@1085 1318 \mt{val} \; \mt{currentUrl} : \mt{transaction} \; \mt{url} \\
adamc@1085 1319 \mt{val} \; \mt{url} : \mt{transaction} \; \mt{page} \to \mt{url}
adamc@1085 1320 \end{array}$$
adamc@1085 1321
adamc@1085 1322 Page generation may be interrupted at any time with a request to redirect to a particular URL instead.
adamc@1085 1323 $$\begin{array}{l}
adamc@1085 1324 \mt{val} \; \mt{redirect} : \mt{t} ::: \mt{Type} \to \mt{url} \to \mt{transaction} \; \mt{t}
adamc@1085 1325 \end{array}$$
adamc@1085 1326
adamc@786 1327 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 1328 $$\begin{array}{l}
adamc@786 1329 \mt{type} \; \mt{file} \\
adamc@786 1330 \mt{val} \; \mt{fileName} : \mt{file} \to \mt{option} \; \mt{string} \\
adamc@786 1331 \mt{val} \; \mt{fileMimeType} : \mt{file} \to \mt{string} \\
adamc@786 1332 \mt{val} \; \mt{fileData} : \mt{file} \to \mt{blob}
adamc@786 1333 \end{array}$$
adamc@786 1334
adamc@786 1335 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 1336 $$\begin{array}{l}
adamc@786 1337 \mt{type} \; \mt{mimeType} \\
adamc@786 1338 \mt{val} \; \mt{blessMime} : \mt{string} \to \mt{mimeType} \\
adamc@786 1339 \mt{val} \; \mt{checkMime} : \mt{string} \to \mt{option} \; \mt{mimeType} \\
adamc@786 1340 \mt{val} \; \mt{returnBlob} : \mt{t} ::: \mt{Type} \to \mt{blob} \to \mt{mimeType} \to \mt{transaction} \; \mt{t}
adamc@542 1341 \end{array}$$
adamc@542 1342
adamc@543 1343 \subsection{SQL}
adamc@543 1344
adamc@543 1345 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 1346 $$\begin{array}{l}
adamc@785 1347 \mt{con} \; \mt{sql\_table} :: \{\mt{Type}\} \to \{\{\mt{Unit}\}\} \to \mt{Type}
adamc@785 1348 \end{array}$$
adamc@785 1349 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 1350
adamc@785 1351 We also have the simpler type family of SQL views, which have no keys.
adamc@785 1352 $$\begin{array}{l}
adamc@785 1353 \mt{con} \; \mt{sql\_view} :: \{\mt{Type}\} \to \mt{Type}
adamc@543 1354 \end{array}$$
adamc@543 1355
adamc@785 1356 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 1357 $$\begin{array}{l}
adamc@785 1358 \mt{class} \; \mt{fieldsOf} :: \mt{Type} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@785 1359 \mt{val} \; \mt{fieldsOf\_table} : \mt{fs} ::: \{\mt{Type}\} \to \mt{keys} ::: \{\{\mt{Unit}\}\} \to \mt{fieldsOf} \; (\mt{sql\_table} \; \mt{fs} \; \mt{keys}) \; \mt{fs} \\
adamc@785 1360 \mt{val} \; \mt{fieldsOf\_view} : \mt{fs} ::: \{\mt{Type}\} \to \mt{fieldsOf} \; (\mt{sql\_view} \; \mt{fs}) \; \mt{fs}
adamc@785 1361 \end{array}$$
adamc@785 1362
adamc@785 1363 \subsubsection{Table Constraints}
adamc@785 1364
adamc@785 1365 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 1366
adamc@785 1367 $$\begin{array}{l}
adamc@785 1368 \mt{con} \; \mt{primary\_key} :: \{\mt{Type}\} \to \{\{\mt{Unit}\}\} \to \mt{Type} \\
adamc@785 1369 \mt{val} \; \mt{no\_primary\_key} : \mt{fs} ::: \{\mt{Type}\} \to \mt{primary\_key} \; \mt{fs} \; [] \\
adamc@785 1370 \mt{val} \; \mt{primary\_key} : \mt{rest} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \to \mt{key1} :: \mt{Name} \to \mt{keys} :: \{\mt{Type}\} \\
adamc@785 1371 \hspace{.1in} \to [[\mt{key1}] \sim \mt{keys}] \Rightarrow [[\mt{key1} = \mt{t}] \rc \mt{keys} \sim \mt{rest}] \\
adamc@785 1372 \hspace{.1in} \Rightarrow \$([\mt{key1} = \mt{sql\_injectable\_prim} \; \mt{t}] \rc \mt{map} \; \mt{sql\_injectable\_prim} \; \mt{keys}) \\
adamc@785 1373 \hspace{.1in} \to \mt{primary\_key} \; ([\mt{key1} = \mt{t}] \rc \mt{keys} \rc \mt{rest}) \; [\mt{Pkey} = [\mt{key1}] \rc \mt{map} \; (\lambda \_ \Rightarrow ()) \; \mt{keys}]
adamc@785 1374 \end{array}$$
adamc@785 1375 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 1376
adamc@785 1377 A type family stands for sets of named constraints of the remaining varieties.
adamc@785 1378 $$\begin{array}{l}
adamc@785 1379 \mt{con} \; \mt{sql\_constraints} :: \{\mt{Type}\} \to \{\{\mt{Unit}\}\} \to \mt{Type}
adamc@785 1380 \end{array}$$
adamc@785 1381 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 1382
adamc@785 1383 There is a type family of individual, unnamed constraints.
adamc@785 1384 $$\begin{array}{l}
adamc@785 1385 \mt{con} \; \mt{sql\_constraint} :: \{\mt{Type}\} \to \{\mt{Unit}\} \to \mt{Type}
adamc@785 1386 \end{array}$$
adamc@785 1387 The first argument is the same as above, and the second argument gives the key columns for just this constraint.
adamc@785 1388
adamc@785 1389 We have operations for assembling constraints into constraint sets.
adamc@785 1390 $$\begin{array}{l}
adamc@785 1391 \mt{val} \; \mt{no\_constraint} : \mt{fs} ::: \{\mt{Type}\} \to \mt{sql\_constraints} \; \mt{fs} \; [] \\
adamc@785 1392 \mt{val} \; \mt{one\_constraint} : \mt{fs} ::: \{\mt{Type}\} \to \mt{unique} ::: \{\mt{Unit}\} \to \mt{name} :: \mt{Name} \\
adamc@785 1393 \hspace{.1in} \to \mt{sql\_constraint} \; \mt{fs} \; \mt{unique} \to \mt{sql\_constraints} \; \mt{fs} \; [\mt{name} = \mt{unique}] \\
adamc@785 1394 \mt{val} \; \mt{join\_constraints} : \mt{fs} ::: \{\mt{Type}\} \to \mt{uniques1} ::: \{\{\mt{Unit}\}\} \to \mt{uniques2} ::: \{\{\mt{Unit}\}\} \to [\mt{uniques1} \sim \mt{uniques2}] \\
adamc@785 1395 \hspace{.1in} \Rightarrow \mt{sql\_constraints} \; \mt{fs} \; \mt{uniques1} \to \mt{sql\_constraints} \; \mt{fs} \; \mt{uniques2} \to \mt{sql\_constraints} \; \mt{fs} \; (\mt{uniques1} \rc \mt{uniques2})
adamc@785 1396 \end{array}$$
adamc@785 1397
adamc@785 1398 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 1399 $$\begin{array}{l}
adamc@785 1400 \mt{val} \; \mt{unique} : \mt{rest} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \to \mt{unique1} :: \mt{Name} \to \mt{unique} :: \{\mt{Type}\} \\
adamc@785 1401 \hspace{.1in} \to [[\mt{unique1}] \sim \mt{unique}] \Rightarrow [[\mt{unique1} = \mt{t}] \rc \mt{unique} \sim \mt{rest}] \\
adamc@785 1402 \hspace{.1in} \Rightarrow \mt{sql\_constraint} \; ([\mt{unique1} = \mt{t}] \rc \mt{unique} \rc \mt{rest}) \; ([\mt{unique1}] \rc \mt{map} \; (\lambda \_ \Rightarrow ()) \; \mt{unique})
adamc@785 1403 \end{array}$$
adamc@785 1404
adamc@785 1405 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 1406 $$\begin{array}{l}
adamc@785 1407 \mt{class} \; \mt{linkable} :: \mt{Type} \to \mt{Type} \to \mt{Type} \\
adamc@785 1408 \mt{val} \; \mt{linkable\_same} : \mt{t} ::: \mt{Type} \to \mt{linkable} \; \mt{t} \; \mt{t} \\
adamc@785 1409 \mt{val} \; \mt{linkable\_from\_nullable} : \mt{t} ::: \mt{Type} \to \mt{linkable} \; (\mt{option} \; \mt{t}) \; \mt{t} \\
adamc@785 1410 \mt{val} \; \mt{linkable\_to\_nullable} : \mt{t} ::: \mt{Type} \to \mt{linkable} \; \mt{t} \; (\mt{option} \; \mt{t})
adamc@785 1411 \end{array}$$
adamc@785 1412
adamc@785 1413 The $\mt{matching}$ type family uses $\mt{linkable}$ to define when two keys match up type-wise.
adamc@785 1414 $$\begin{array}{l}
adamc@785 1415 \mt{con} \; \mt{matching} :: \{\mt{Type}\} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@785 1416 \mt{val} \; \mt{mat\_nil} : \mt{matching} \; [] \; [] \\
adamc@785 1417 \mt{val} \; \mt{mat\_cons} : \mt{t1} ::: \mt{Type} \to \mt{rest1} ::: \{\mt{Type}\} \to \mt{t2} ::: \mt{Type} \to \mt{rest2} ::: \{\mt{Type}\} \to \mt{nm1} :: \mt{Name} \to \mt{nm2} :: \mt{Name} \\
adamc@785 1418 \hspace{.1in} \to [[\mt{nm1}] \sim \mt{rest1}] \Rightarrow [[\mt{nm2}] \sim \mt{rest2}] \Rightarrow \mt{linkable} \; \mt{t1} \; \mt{t2} \to \mt{matching} \; \mt{rest1} \; \mt{rest2} \\
adamc@785 1419 \hspace{.1in} \to \mt{matching} \; ([\mt{nm1} = \mt{t1}] \rc \mt{rest1}) \; ([\mt{nm2} = \mt{t2}] \rc \mt{rest2})
adamc@785 1420 \end{array}$$
adamc@785 1421
adamc@785 1422 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 1423 $$\begin{array}{l}
adamc@785 1424 \mt{con} \; \mt{propagation\_mode} :: \{\mt{Type}\} \to \mt{Type} \\
adamc@785 1425 \mt{val} \; \mt{restrict} : \mt{fs} ::: \{\mt{Type}\} \to \mt{propagation\_mode} \; \mt{fs} \\
adamc@785 1426 \mt{val} \; \mt{cascade} : \mt{fs} ::: \{\mt{Type}\} \to \mt{propagation\_mode} \; \mt{fs} \\
adamc@785 1427 \mt{val} \; \mt{no\_action} : \mt{fs} ::: \{\mt{Type}\} \to \mt{propagation\_mode} \; \mt{fs} \\
adamc@785 1428 \mt{val} \; \mt{set\_null} : \mt{fs} ::: \{\mt{Type}\} \to \mt{propagation\_mode} \; (\mt{map} \; \mt{option} \; \mt{fs})
adamc@785 1429 \end{array}$$
adamc@785 1430
adamc@785 1431 Finally, we put these ingredient together to define the \texttt{FOREIGN KEY} constraint function.
adamc@785 1432 $$\begin{array}{l}
adamc@785 1433 \mt{val} \; \mt{foreign\_key} : \mt{mine1} ::: \mt{Name} \to \mt{t} ::: \mt{Type} \to \mt{mine} ::: \{\mt{Type}\} \to \mt{munused} ::: \{\mt{Type}\} \to \mt{foreign} ::: \{\mt{Type}\} \\
adamc@785 1434 \hspace{.1in} \to \mt{funused} ::: \{\mt{Type}\} \to \mt{nm} ::: \mt{Name} \to \mt{uniques} ::: \{\{\mt{Unit}\}\} \\
adamc@785 1435 \hspace{.1in} \to [[\mt{mine1}] \sim \mt{mine}] \Rightarrow [[\mt{mine1} = \mt{t}] \rc \mt{mine} \sim \mt{munused}] \Rightarrow [\mt{foreign} \sim \mt{funused}] \Rightarrow [[\mt{nm}] \sim \mt{uniques}] \\
adamc@785 1436 \hspace{.1in} \Rightarrow \mt{matching} \; ([\mt{mine1} = \mt{t}] \rc \mt{mine}) \; \mt{foreign} \\
adamc@785 1437 \hspace{.1in} \to \mt{sql\_table} \; (\mt{foreign} \rc \mt{funused}) \; ([\mt{nm} = \mt{map} \; (\lambda \_ \Rightarrow ()) \; \mt{foreign}] \rc \mt{uniques}) \\
adamc@785 1438 \hspace{.1in} \to \{\mt{OnDelete} : \mt{propagation\_mode} \; ([\mt{mine1} = \mt{t}] \rc \mt{mine}), \\
adamc@785 1439 \hspace{.2in} \mt{OnUpdate} : \mt{propagation\_mode} \; ([\mt{mine1} = \mt{t}] \rc \mt{mine})\} \\
adamc@785 1440 \hspace{.1in} \to \mt{sql\_constraint} \; ([\mt{mine1} = \mt{t}] \rc \mt{mine} \rc \mt{munused}) \; []
adamc@785 1441 \end{array}$$
adamc@785 1442
adamc@785 1443 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 1444 $$\begin{array}{l}
adamc@785 1445 \mt{val} \; \mt{check} : \mt{fs} ::: \{\mt{Type}\} \to \mt{sql\_exp} \; [] \; [] \; \mt{fs} \; \mt{bool} \to \mt{sql\_constraint} \; \mt{fs} \; []
adamc@785 1446 \end{array}$$
adamc@785 1447
adamc@785 1448 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 1449
adamc@784 1450
adamc@543 1451 \subsubsection{Queries}
adamc@543 1452
adamc@543 1453 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 1454 $$\begin{array}{l}
adamc@543 1455 \mt{con} \; \mt{sql\_query} :: \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@543 1456 \mt{val} \; \mt{sql\_query} : \mt{tables} ::: \{\{\mt{Type}\}\} \\
adamc@543 1457 \hspace{.1in} \to \mt{selectedFields} ::: \{\{\mt{Type}\}\} \\
adamc@543 1458 \hspace{.1in} \to \mt{selectedExps} ::: \{\mt{Type}\} \\
adamc@543 1459 \hspace{.1in} \to \{\mt{Rows} : \mt{sql\_query1} \; \mt{tables} \; \mt{selectedFields} \; \mt{selectedExps}, \\
adamc@543 1460 \hspace{.2in} \mt{OrderBy} : \mt{sql\_order\_by} \; \mt{tables} \; \mt{selectedExps}, \\
adamc@543 1461 \hspace{.2in} \mt{Limit} : \mt{sql\_limit}, \\
adamc@543 1462 \hspace{.2in} \mt{Offset} : \mt{sql\_offset}\} \\
adamc@543 1463 \hspace{.1in} \to \mt{sql\_query} \; \mt{selectedFields} \; \mt{selectedExps}
adamc@543 1464 \end{array}$$
adamc@543 1465
adamc@545 1466 Queries are used by folding over their results inside transactions.
adamc@545 1467 $$\begin{array}{l}
adamc@545 1468 \mt{val} \; \mt{query} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \lambda [\mt{tables} \sim \mt{exps}] \Rightarrow \mt{state} ::: \mt{Type} \to \mt{sql\_query} \; \mt{tables} \; \mt{exps} \\
adamc@658 1469 \hspace{.1in} \to (\$(\mt{exps} \rc \mt{map} \; (\lambda \mt{fields} :: \{\mt{Type}\} \Rightarrow \$\mt{fields}) \; \mt{tables}) \\
adamc@545 1470 \hspace{.2in} \to \mt{state} \to \mt{transaction} \; \mt{state}) \\
adamc@545 1471 \hspace{.1in} \to \mt{state} \to \mt{transaction} \; \mt{state}
adamc@545 1472 \end{array}$$
adamc@545 1473
adamc@543 1474 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 1475 $$\begin{array}{l}
adamc@543 1476 \mt{con} \; \mt{sql\_query1} :: \{\{\mt{Type}\}\} \to \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@543 1477 \\
adamc@543 1478 \mt{type} \; \mt{sql\_relop} \\
adamc@543 1479 \mt{val} \; \mt{sql\_union} : \mt{sql\_relop} \\
adamc@543 1480 \mt{val} \; \mt{sql\_intersect} : \mt{sql\_relop} \\
adamc@543 1481 \mt{val} \; \mt{sql\_except} : \mt{sql\_relop} \\
adamc@543 1482 \mt{val} \; \mt{sql\_relop} : \mt{tables1} ::: \{\{\mt{Type}\}\} \\
adamc@543 1483 \hspace{.1in} \to \mt{tables2} ::: \{\{\mt{Type}\}\} \\
adamc@543 1484 \hspace{.1in} \to \mt{selectedFields} ::: \{\{\mt{Type}\}\} \\
adamc@543 1485 \hspace{.1in} \to \mt{selectedExps} ::: \{\mt{Type}\} \\
adamc@543 1486 \hspace{.1in} \to \mt{sql\_relop} \\
adamc@543 1487 \hspace{.1in} \to \mt{sql\_query1} \; \mt{tables1} \; \mt{selectedFields} \; \mt{selectedExps} \\
adamc@543 1488 \hspace{.1in} \to \mt{sql\_query1} \; \mt{tables2} \; \mt{selectedFields} \; \mt{selectedExps} \\
adamc@543 1489 \hspace{.1in} \to \mt{sql\_query1} \; \mt{selectedFields} \; \mt{selectedFields} \; \mt{selectedExps}
adamc@543 1490 \end{array}$$
adamc@543 1491
adamc@543 1492 $$\begin{array}{l}
adamc@543 1493 \mt{val} \; \mt{sql\_query1} : \mt{tables} ::: \{\{\mt{Type}\}\} \\
adamc@543 1494 \hspace{.1in} \to \mt{grouped} ::: \{\{\mt{Type}\}\} \\
adamc@543 1495 \hspace{.1in} \to \mt{selectedFields} ::: \{\{\mt{Type}\}\} \\
adamc@543 1496 \hspace{.1in} \to \mt{selectedExps} ::: \{\mt{Type}\} \\
adamc@1085 1497 \hspace{.1in} \to \mt{empties} :: \{\mt{Unit}\} \\
adamc@1085 1498 \hspace{.1in} \to [\mt{empties} \sim \mt{selectedFields}] \\
adamc@1085 1499 \hspace{.1in} \Rightarrow \{\mt{Distinct} : \mt{bool}, \\
adamc@994 1500 \hspace{.2in} \mt{From} : \mt{sql\_from\_items} \; \mt{tables}, \\
adamc@543 1501 \hspace{.2in} \mt{Where} : \mt{sql\_exp} \; \mt{tables} \; [] \; [] \; \mt{bool}, \\
adamc@543 1502 \hspace{.2in} \mt{GroupBy} : \mt{sql\_subset} \; \mt{tables} \; \mt{grouped}, \\
adamc@543 1503 \hspace{.2in} \mt{Having} : \mt{sql\_exp} \; \mt{grouped} \; \mt{tables} \; [] \; \mt{bool}, \\
adamc@1085 1504 \hspace{.2in} \mt{SelectFields} : \mt{sql\_subset} \; \mt{grouped} \; (\mt{map} \; (\lambda \_ \Rightarrow []) \; \mt{empties} \rc \mt{selectedFields}), \\
adamc@658 1505 \hspace{.2in} \mt {SelectExps} : \$(\mt{map} \; (\mt{sql\_exp} \; \mt{grouped} \; \mt{tables} \; []) \; \mt{selectedExps}) \} \\
adamc@543 1506 \hspace{.1in} \to \mt{sql\_query1} \; \mt{tables} \; \mt{selectedFields} \; \mt{selectedExps}
adamc@543 1507 \end{array}$$
adamc@543 1508
adamc@543 1509 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 1510 $$\begin{array}{l}
adamc@543 1511 \mt{con} \; \mt{sql\_subset} :: \{\{\mt{Type}\}\} \to \{\{\mt{Type}\}\} \to \mt{Type} \\
adamc@543 1512 \mt{val} \; \mt{sql\_subset} : \mt{keep\_drop} :: \{(\{\mt{Type}\} \times \{\mt{Type}\})\} \\
adamc@543 1513 \hspace{.1in} \to \mt{sql\_subset} \\
adamc@658 1514 \hspace{.2in} (\mt{map} \; (\lambda \mt{fields} :: (\{\mt{Type}\} \times \{\mt{Type}\}) \Rightarrow \mt{fields}.1 \rc \mt{fields}.2)\; \mt{keep\_drop}) \\
adamc@658 1515 \hspace{.2in} (\mt{map} \; (\lambda \mt{fields} :: (\{\mt{Type}\} \times \{\mt{Type}\}) \Rightarrow \mt{fields}.1) \; \mt{keep\_drop}) \\
adamc@543 1516 \mt{val} \; \mt{sql\_subset\_all} : \mt{tables} :: \{\{\mt{Type}\}\} \to \mt{sql\_subset} \; \mt{tables} \; \mt{tables}
adamc@543 1517 \end{array}$$
adamc@543 1518
adamc@560 1519 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 1520 $$\begin{array}{l}
adamc@543 1521 \mt{con} \; \mt{sql\_exp} :: \{\{\mt{Type}\}\} \to \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \to \mt{Type}
adamc@543 1522 \end{array}$$
adamc@543 1523
adamc@543 1524 Any field in scope may be converted to an expression.
adamc@543 1525 $$\begin{array}{l}
adamc@543 1526 \mt{val} \; \mt{sql\_field} : \mt{otherTabs} ::: \{\{\mt{Type}\}\} \to \mt{otherFields} ::: \{\mt{Type}\} \\
adamc@543 1527 \hspace{.1in} \to \mt{fieldType} ::: \mt{Type} \to \mt{agg} ::: \{\{\mt{Type}\}\} \\
adamc@543 1528 \hspace{.1in} \to \mt{exps} ::: \{\mt{Type}\} \\
adamc@543 1529 \hspace{.1in} \to \mt{tab} :: \mt{Name} \to \mt{field} :: \mt{Name} \\
adamc@543 1530 \hspace{.1in} \to \mt{sql\_exp} \; ([\mt{tab} = [\mt{field} = \mt{fieldType}] \rc \mt{otherFields}] \rc \mt{otherTabs}) \; \mt{agg} \; \mt{exps} \; \mt{fieldType}
adamc@543 1531 \end{array}$$
adamc@543 1532
adamc@544 1533 There is an analogous function for referencing named expressions.
adamc@544 1534 $$\begin{array}{l}
adamc@544 1535 \mt{val} \; \mt{sql\_exp} : \mt{tabs} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{t} ::: \mt{Type} \to \mt{rest} ::: \{\mt{Type}\} \to \mt{nm} :: \mt{Name} \\
adamc@544 1536 \hspace{.1in} \to \mt{sql\_exp} \; \mt{tabs} \; \mt{agg} \; ([\mt{nm} = \mt{t}] \rc \mt{rest}) \; \mt{t}
adamc@544 1537 \end{array}$$
adamc@544 1538
adamc@544 1539 Ur values of appropriate types may be injected into SQL expressions.
adamc@544 1540 $$\begin{array}{l}
adamc@786 1541 \mt{class} \; \mt{sql\_injectable\_prim} \\
adamc@786 1542 \mt{val} \; \mt{sql\_bool} : \mt{sql\_injectable\_prim} \; \mt{bool} \\
adamc@786 1543 \mt{val} \; \mt{sql\_int} : \mt{sql\_injectable\_prim} \; \mt{int} \\
adamc@786 1544 \mt{val} \; \mt{sql\_float} : \mt{sql\_injectable\_prim} \; \mt{float} \\
adamc@786 1545 \mt{val} \; \mt{sql\_string} : \mt{sql\_injectable\_prim} \; \mt{string} \\
adamc@786 1546 \mt{val} \; \mt{sql\_time} : \mt{sql\_injectable\_prim} \; \mt{time} \\
adamc@786 1547 \mt{val} \; \mt{sql\_blob} : \mt{sql\_injectable\_prim} \; \mt{blob} \\
adamc@786 1548 \mt{val} \; \mt{sql\_channel} : \mt{t} ::: \mt{Type} \to \mt{sql\_injectable\_prim} \; (\mt{channel} \; \mt{t}) \\
adamc@786 1549 \mt{val} \; \mt{sql\_client} : \mt{sql\_injectable\_prim} \; \mt{client} \\
adamc@786 1550 \\
adamc@544 1551 \mt{class} \; \mt{sql\_injectable} \\
adamc@786 1552 \mt{val} \; \mt{sql\_prim} : \mt{t} ::: \mt{Type} \to \mt{sql\_injectable\_prim} \; \mt{t} \to \mt{sql\_injectable} \; \mt{t} \\
adamc@786 1553 \mt{val} \; \mt{sql\_option\_prim} : \mt{t} ::: \mt{Type} \to \mt{sql\_injectable\_prim} \; \mt{t} \to \mt{sql\_injectable} \; (\mt{option} \; \mt{t}) \\
adamc@786 1554 \\
adamc@544 1555 \mt{val} \; \mt{sql\_inject} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \to \mt{sql\_injectable} \; \mt{t} \\
adamc@544 1556 \hspace{.1in} \to \mt{t} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{t}
adamc@544 1557 \end{array}$$
adamc@544 1558
adamc@544 1559 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 1560 $$\begin{array}{l}
adamc@544 1561 \mt{val} \; \mt{sql\_is\_null} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544 1562 \hspace{.1in} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; (\mt{option} \; \mt{t}) \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{bool}
adamc@544 1563 \end{array}$$
adamc@544 1564
adamc@559 1565 We have generic nullary, unary, and binary operators.
adamc@544 1566 $$\begin{array}{l}
adamc@544 1567 \mt{con} \; \mt{sql\_nfunc} :: \mt{Type} \to \mt{Type} \\
adamc@544 1568 \mt{val} \; \mt{sql\_current\_timestamp} : \mt{sql\_nfunc} \; \mt{time} \\
adamc@544 1569 \mt{val} \; \mt{sql\_nfunc} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544 1570 \hspace{.1in} \to \mt{sql\_nfunc} \; \mt{t} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{t} \\\end{array}$$
adamc@544 1571
adamc@544 1572 $$\begin{array}{l}
adamc@544 1573 \mt{con} \; \mt{sql\_unary} :: \mt{Type} \to \mt{Type} \to \mt{Type} \\
adamc@544 1574 \mt{val} \; \mt{sql\_not} : \mt{sql\_unary} \; \mt{bool} \; \mt{bool} \\
adamc@544 1575 \mt{val} \; \mt{sql\_unary} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{arg} ::: \mt{Type} \to \mt{res} ::: \mt{Type} \\
adamc@544 1576 \hspace{.1in} \to \mt{sql\_unary} \; \mt{arg} \; \mt{res} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{arg} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{res} \\
adamc@544 1577 \end{array}$$
adamc@544 1578
adamc@544 1579 $$\begin{array}{l}
adamc@544 1580 \mt{con} \; \mt{sql\_binary} :: \mt{Type} \to \mt{Type} \to \mt{Type} \to \mt{Type} \\
adamc@544 1581 \mt{val} \; \mt{sql\_and} : \mt{sql\_binary} \; \mt{bool} \; \mt{bool} \; \mt{bool} \\
adamc@544 1582 \mt{val} \; \mt{sql\_or} : \mt{sql\_binary} \; \mt{bool} \; \mt{bool} \; \mt{bool} \\
adamc@544 1583 \mt{val} \; \mt{sql\_binary} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{arg_1} ::: \mt{Type} \to \mt{arg_2} ::: \mt{Type} \to \mt{res} ::: \mt{Type} \\
adamc@544 1584 \hspace{.1in} \to \mt{sql\_binary} \; \mt{arg_1} \; \mt{arg_2} \; \mt{res} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{arg_1} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{arg_2} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{res}
adamc@544 1585 \end{array}$$
adamc@544 1586
adamc@544 1587 $$\begin{array}{l}
adamc@559 1588 \mt{class} \; \mt{sql\_arith} \\
adamc@559 1589 \mt{val} \; \mt{sql\_int\_arith} : \mt{sql\_arith} \; \mt{int} \\
adamc@559 1590 \mt{val} \; \mt{sql\_float\_arith} : \mt{sql\_arith} \; \mt{float} \\
adamc@559 1591 \mt{val} \; \mt{sql\_neg} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_unary} \; \mt{t} \; \mt{t} \\
adamc@559 1592 \mt{val} \; \mt{sql\_plus} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559 1593 \mt{val} \; \mt{sql\_minus} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559 1594 \mt{val} \; \mt{sql\_times} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559 1595 \mt{val} \; \mt{sql\_div} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559 1596 \mt{val} \; \mt{sql\_mod} : \mt{sql\_binary} \; \mt{int} \; \mt{int} \; \mt{int}
adamc@559 1597 \end{array}$$
adamc@544 1598
adamc@656 1599 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 1600 $$\begin{array}{l}
adamc@544 1601 \mt{val} \; \mt{sql\_count} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{int}
adamc@544 1602 \end{array}$$
adamc@544 1603
adamc@544 1604 $$\begin{array}{l}
adamc@544 1605 \mt{con} \; \mt{sql\_aggregate} :: \mt{Type} \to \mt{Type} \\
adamc@544 1606 \mt{val} \; \mt{sql\_aggregate} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544 1607 \hspace{.1in} \to \mt{sql\_aggregate} \; \mt{t} \to \mt{sql\_exp} \; \mt{agg} \; \mt{agg} \; \mt{exps} \; \mt{t} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{t}
adamc@544 1608 \end{array}$$
adamc@544 1609
adamc@544 1610 $$\begin{array}{l}
adamc@544 1611 \mt{class} \; \mt{sql\_summable} \\
adamc@544 1612 \mt{val} \; \mt{sql\_summable\_int} : \mt{sql\_summable} \; \mt{int} \\
adamc@544 1613 \mt{val} \; \mt{sql\_summable\_float} : \mt{sql\_summable} \; \mt{float} \\
adamc@544 1614 \mt{val} \; \mt{sql\_avg} : \mt{t} ::: \mt{Type} \to \mt{sql\_summable} \; \mt{t} \to \mt{sql\_aggregate} \; \mt{t} \\
adamc@544 1615 \mt{val} \; \mt{sql\_sum} : \mt{t} ::: \mt{Type} \to \mt{sql\_summable} \mt{t} \to \mt{sql\_aggregate} \; \mt{t}
adamc@544 1616 \end{array}$$
adamc@544 1617
adamc@544 1618 $$\begin{array}{l}
adamc@544 1619 \mt{class} \; \mt{sql\_maxable} \\
adamc@544 1620 \mt{val} \; \mt{sql\_maxable\_int} : \mt{sql\_maxable} \; \mt{int} \\
adamc@544 1621 \mt{val} \; \mt{sql\_maxable\_float} : \mt{sql\_maxable} \; \mt{float} \\
adamc@544 1622 \mt{val} \; \mt{sql\_maxable\_string} : \mt{sql\_maxable} \; \mt{string} \\
adamc@544 1623 \mt{val} \; \mt{sql\_maxable\_time} : \mt{sql\_maxable} \; \mt{time} \\
adamc@544 1624 \mt{val} \; \mt{sql\_max} : \mt{t} ::: \mt{Type} \to \mt{sql\_maxable} \; \mt{t} \to \mt{sql\_aggregate} \; \mt{t} \\
adamc@544 1625 \mt{val} \; \mt{sql\_min} : \mt{t} ::: \mt{Type} \to \mt{sql\_maxable} \; \mt{t} \to \mt{sql\_aggregate} \; \mt{t}
adamc@544 1626 \end{array}$$
adamc@544 1627
adamc@786 1628 \texttt{FROM} clauses are specified using a type family.
adamc@786 1629 $$\begin{array}{l}
adamc@786 1630 \mt{con} \; \mt{sql\_from\_items} :: \{\{\mt{Type}\}\} \to \mt{Type} \\
adamc@786 1631 \mt{val} \; \mt{sql\_from\_table} : \mt{t} ::: \mt{Type} \to \mt{fs} ::: \{\mt{Type}\} \to \mt{fieldsOf} \; \mt{t} \; \mt{fs} \to \mt{name} :: \mt{Name} \to \mt{t} \to \mt{sql\_from\_items} \; [\mt{name} = \mt{fs}] \\
adamc@786 1632 \mt{val} \; \mt{sql\_from\_comma} : \mt{tabs1} ::: \{\{\mt{Type}\}\} \to \mt{tabs2} ::: \{\{\mt{Type}\}\} \to [\mt{tabs1} \sim \mt{tabs2}] \\
adamc@786 1633 \hspace{.1in} \Rightarrow \mt{sql\_from\_items} \; \mt{tabs1} \to \mt{sql\_from\_items} \; \mt{tabs2} \\
adamc@786 1634 \hspace{.1in} \to \mt{sql\_from\_items} \; (\mt{tabs1} \rc \mt{tabs2}) \\
adamc@786 1635 \mt{val} \; \mt{sql\_inner\_join} : \mt{tabs1} ::: \{\{\mt{Type}\}\} \to \mt{tabs2} ::: \{\{\mt{Type}\}\} \to [\mt{tabs1} \sim \mt{tabs2}] \\
adamc@786 1636 \hspace{.1in} \Rightarrow \mt{sql\_from\_items} \; \mt{tabs1} \to \mt{sql\_from\_items} \; \mt{tabs2} \\
adamc@786 1637 \hspace{.1in} \to \mt{sql\_exp} \; (\mt{tabs1} \rc \mt{tabs2}) \; [] \; [] \; \mt{bool} \\
adamc@786 1638 \hspace{.1in} \to \mt{sql\_from\_items} \; (\mt{tabs1} \rc \mt{tabs2})
adamc@786 1639 \end{array}$$
adamc@786 1640
adamc@786 1641 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 1642 $$\begin{array}{l}
adamc@786 1643 \mt{class} \; \mt{nullify} :: \mt{Type} \to \mt{Type} \to \mt{Type} \\
adamc@786 1644 \mt{val} \; \mt{nullify\_option} : \mt{t} ::: \mt{Type} \to \mt{nullify} \; (\mt{option} \; \mt{t}) \; (\mt{option} \; \mt{t}) \\
adamc@786 1645 \mt{val} \; \mt{nullify\_prim} : \mt{t} ::: \mt{Type} \to \mt{sql\_injectable\_prim} \; \mt{t} \to \mt{nullify} \; \mt{t} \; (\mt{option} \; \mt{t})
adamc@786 1646 \end{array}$$
adamc@786 1647
adamc@786 1648 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 1649
adamc@786 1650 $$\begin{array}{l}
adamc@786 1651 \mt{val} \; \mt{sql\_left\_join} : \mt{tabs1} ::: \{\{\mt{Type}\}\} \to \mt{tabs2} ::: \{\{(\mt{Type} \times \mt{Type})\}\} \to [\mt{tabs1} \sim \mt{tabs2}] \\
adamc@786 1652 \hspace{.1in} \Rightarrow \$(\mt{map} \; (\lambda \mt{r} \Rightarrow \$(\mt{map} \; (\lambda \mt{p} :: (\mt{Type} \times \mt{Type}) \Rightarrow \mt{nullify} \; \mt{p}.1 \; \mt{p}.2) \; \mt{r})) \; \mt{tabs2}) \\
adamc@786 1653 \hspace{.1in} \to \mt{sql\_from\_items} \; \mt{tabs1} \to \mt{sql\_from\_items} \; (\mt{map} \; (\mt{map} \; (\lambda \mt{p} :: (\mt{Type} \times \mt{Type}) \Rightarrow \mt{p}.1)) \; \mt{tabs2}) \\
adamc@786 1654 \hspace{.1in} \to \mt{sql\_exp} \; (\mt{tabs1} \rc \mt{map} \; (\mt{map} \; (\lambda \mt{p} :: (\mt{Type} \times \mt{Type}) \Rightarrow \mt{p}.1)) \; \mt{tabs2}) \; [] \; [] \; \mt{bool} \\
adamc@786 1655 \hspace{.1in} \to \mt{sql\_from\_items} \; (\mt{tabs1} \rc \mt{map} \; (\mt{map} \; (\lambda \mt{p} :: (\mt{Type} \times \mt{Type}) \Rightarrow \mt{p}.2)) \; \mt{tabs2})
adamc@786 1656 \end{array}$$
adamc@786 1657
adamc@544 1658 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 1659 $$\begin{array}{l}
adamc@544 1660 \mt{type} \; \mt{sql\_direction} \\
adamc@544 1661 \mt{val} \; \mt{sql\_asc} : \mt{sql\_direction} \\
adamc@544 1662 \mt{val} \; \mt{sql\_desc} : \mt{sql\_direction} \\
adamc@544 1663 \\
adamc@544 1664 \mt{con} \; \mt{sql\_order\_by} :: \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@544 1665 \mt{val} \; \mt{sql\_order\_by\_Nil} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{exps} :: \{\mt{Type}\} \to \mt{sql\_order\_by} \; \mt{tables} \; \mt{exps} \\
adamc@544 1666 \mt{val} \; \mt{sql\_order\_by\_Cons} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544 1667 \hspace{.1in} \to \mt{sql\_exp} \; \mt{tables} \; [] \; \mt{exps} \; \mt{t} \to \mt{sql\_direction} \to \mt{sql\_order\_by} \; \mt{tables} \; \mt{exps} \to \mt{sql\_order\_by} \; \mt{tables} \; \mt{exps} \\
adamc@544 1668 \\
adamc@544 1669 \mt{type} \; \mt{sql\_limit} \\
adamc@544 1670 \mt{val} \; \mt{sql\_no\_limit} : \mt{sql\_limit} \\
adamc@544 1671 \mt{val} \; \mt{sql\_limit} : \mt{int} \to \mt{sql\_limit} \\
adamc@544 1672 \\
adamc@544 1673 \mt{type} \; \mt{sql\_offset} \\
adamc@544 1674 \mt{val} \; \mt{sql\_no\_offset} : \mt{sql\_offset} \\
adamc@544 1675 \mt{val} \; \mt{sql\_offset} : \mt{int} \to \mt{sql\_offset}
adamc@544 1676 \end{array}$$
adamc@544 1677
adamc@545 1678
adamc@545 1679 \subsubsection{DML}
adamc@545 1680
adamc@545 1681 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 1682
adamc@545 1683 $$\begin{array}{l}
adamc@545 1684 \mt{type} \; \mt{dml} \\
adamc@545 1685 \mt{val} \; \mt{dml} : \mt{dml} \to \mt{transaction} \; \mt{unit}
adamc@545 1686 \end{array}$$
adamc@545 1687
adamc@545 1688 Properly-typed records may be used to form $\mt{INSERT}$ commands.
adamc@545 1689 $$\begin{array}{l}
adamc@545 1690 \mt{val} \; \mt{insert} : \mt{fields} ::: \{\mt{Type}\} \to \mt{sql\_table} \; \mt{fields} \\
adamc@658 1691 \hspace{.1in} \to \$(\mt{map} \; (\mt{sql\_exp} \; [] \; [] \; []) \; \mt{fields}) \to \mt{dml}
adamc@545 1692 \end{array}$$
adamc@545 1693
adamc@545 1694 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 1695 $$\begin{array}{l}
adamc@545 1696 \mt{val} \; \mt{update} : \mt{unchanged} ::: \{\mt{Type}\} \to \mt{changed} :: \{\mt{Type}\} \to \lambda [\mt{changed} \sim \mt{unchanged}] \\
adamc@658 1697 \hspace{.1in} \Rightarrow \$(\mt{map} \; (\mt{sql\_exp} \; [\mt{T} = \mt{changed} \rc \mt{unchanged}] \; [] \; []) \; \mt{changed}) \\
adamc@545 1698 \hspace{.1in} \to \mt{sql\_table} \; (\mt{changed} \rc \mt{unchanged}) \to \mt{sql\_exp} \; [\mt{T} = \mt{changed} \rc \mt{unchanged}] \; [] \; [] \; \mt{bool} \to \mt{dml}
adamc@545 1699 \end{array}$$
adamc@545 1700
adamc@545 1701 A $\mt{DELETE}$ command is formed from a table and a $\mt{WHERE}$ clause.
adamc@545 1702 $$\begin{array}{l}
adamc@545 1703 \mt{val} \; \mt{delete} : \mt{fields} ::: \{\mt{Type}\} \to \mt{sql\_table} \; \mt{fields} \to \mt{sql\_exp} \; [\mt{T} = \mt{fields}] \; [] \; [] \; \mt{bool} \to \mt{dml}
adamc@545 1704 \end{array}$$
adamc@545 1705
adamc@546 1706 \subsubsection{Sequences}
adamc@546 1707
adamc@546 1708 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 1709
adamc@546 1710 $$\begin{array}{l}
adamc@546 1711 \mt{type} \; \mt{sql\_sequence} \\
adamc@1085 1712 \mt{val} \; \mt{nextval} : \mt{sql\_sequence} \to \mt{transaction} \; \mt{int} \\
adamc@1085 1713 \mt{val} \; \mt{setval} : \mt{sql\_sequence} \to \mt{int} \to \mt{transaction} \; \mt{unit}
adamc@546 1714 \end{array}$$
adamc@546 1715
adamc@546 1716
adamc@547 1717 \subsection{XML}
adamc@547 1718
adamc@547 1719 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 1720
adamc@547 1721 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 1722 $$\begin{array}{l}
adamc@547 1723 \mt{con} \; \mt{xml} :: \{\mt{Unit}\} \to \{\mt{Type}\} \to \{\mt{Type}\} \to \mt{Type}
adamc@547 1724 \end{array}$$
adamc@547 1725
adamc@547 1726 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 1727 $$\begin{array}{l}
adamc@547 1728 \mt{con} \; \mt{tag} :: \{\mt{Type}\} \to \{\mt{Unit}\} \to \{\mt{Unit}\} \to \{\mt{Type}\} \to \{\mt{Type}\} \to \mt{Type}
adamc@547 1729 \end{array}$$
adamc@547 1730
adamc@547 1731 Literal text may be injected into XML as ``CDATA.''
adamc@547 1732 $$\begin{array}{l}
adamc@547 1733 \mt{val} \; \mt{cdata} : \mt{ctx} ::: \{\mt{Unit}\} \to \mt{use} ::: \{\mt{Type}\} \to \mt{string} \to \mt{xml} \; \mt{ctx} \; \mt{use} \; []
adamc@547 1734 \end{array}$$
adamc@547 1735
adamc@547 1736 There is a function for producing an XML tree with a particular tag at its root.
adamc@547 1737 $$\begin{array}{l}
adamc@547 1738 \mt{val} \; \mt{tag} : \mt{attrsGiven} ::: \{\mt{Type}\} \to \mt{attrsAbsent} ::: \{\mt{Type}\} \to \mt{ctxOuter} ::: \{\mt{Unit}\} \to \mt{ctxInner} ::: \{\mt{Unit}\} \\
adamc@547 1739 \hspace{.1in} \to \mt{useOuter} ::: \{\mt{Type}\} \to \mt{useInner} ::: \{\mt{Type}\} \to \mt{bindOuter} ::: \{\mt{Type}\} \to \mt{bindInner} ::: \{\mt{Type}\} \\
adamc@787 1740 \hspace{.1in} \to \lambda [\mt{attrsGiven} \sim \mt{attrsAbsent}] \; [\mt{useOuter} \sim \mt{useInner}] \; [\mt{bindOuter} \sim \mt{bindInner}] \\
adamc@787 1741 \hspace{.1in} \Rightarrow \mt{option} \; \mt{css\_class} \\
adamc@787 1742 \hspace{.1in} \to \$\mt{attrsGiven} \\
adamc@547 1743 \hspace{.1in} \to \mt{tag} \; (\mt{attrsGiven} \rc \mt{attrsAbsent}) \; \mt{ctxOuter} \; \mt{ctxInner} \; \mt{useOuter} \; \mt{bindOuter} \\
adamc@547 1744 \hspace{.1in} \to \mt{xml} \; \mt{ctxInner} \; \mt{useInner} \; \mt{bindInner} \to \mt{xml} \; \mt{ctxOuter} \; (\mt{useOuter} \rc \mt{useInner}) \; (\mt{bindOuter} \rc \mt{bindInner})
adamc@547 1745 \end{array}$$
adamc@787 1746 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 1747
adamc@547 1748 Two XML fragments may be concatenated.
adamc@547 1749 $$\begin{array}{l}
adamc@547 1750 \mt{val} \; \mt{join} : \mt{ctx} ::: \{\mt{Unit}\} \to \mt{use_1} ::: \{\mt{Type}\} \to \mt{bind_1} ::: \{\mt{Type}\} \to \mt{bind_2} ::: \{\mt{Type}\} \\
adamc@547 1751 \hspace{.1in} \to \lambda [\mt{use_1} \sim \mt{bind_1}] \; [\mt{bind_1} \sim \mt{bind_2}] \\
adamc@547 1752 \hspace{.1in} \Rightarrow \mt{xml} \; \mt{ctx} \; \mt{use_1} \; \mt{bind_1} \to \mt{xml} \; \mt{ctx} \; (\mt{use_1} \rc \mt{bind_1}) \; \mt{bind_2} \to \mt{xml} \; \mt{ctx} \; \mt{use_1} \; (\mt{bind_1} \rc \mt{bind_2})
adamc@547 1753 \end{array}$$
adamc@547 1754
adamc@547 1755 Finally, any XML fragment may be updated to ``claim'' to use more form fields than it does.
adamc@547 1756 $$\begin{array}{l}
adamc@547 1757 \mt{val} \; \mt{useMore} : \mt{ctx} ::: \{\mt{Unit}\} \to \mt{use_1} ::: \{\mt{Type}\} \to \mt{use_2} ::: \{\mt{Type}\} \to \mt{bind} ::: \{\mt{Type}\} \to \lambda [\mt{use_1} \sim \mt{use_2}] \\
adamc@547 1758 \hspace{.1in} \Rightarrow \mt{xml} \; \mt{ctx} \; \mt{use_1} \; \mt{bind} \to \mt{xml} \; \mt{ctx} \; (\mt{use_1} \rc \mt{use_2}) \; \mt{bind}
adamc@547 1759 \end{array}$$
adamc@547 1760
adamc@547 1761 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 1762
adamc@547 1763 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 1764 $$\begin{array}{l}
adamc@547 1765 \mt{val} \; \mt{error} : \mt{t} ::: \mt{Type} \to \mt{xml} \; [\mt{Body}] \; [] \; [] \to \mt{t}
adamc@547 1766 \end{array}$$
adamc@547 1767
adamc@549 1768
adamc@701 1769 \subsection{Client-Side Programming}
adamc@659 1770
adamc@701 1771 Ur/Web supports running code on web browsers, via automatic compilation to JavaScript.
adamc@701 1772
adamc@701 1773 \subsubsection{The Basics}
adamc@701 1774
adamc@701 1775 Clients can open alert dialog boxes, in the usual annoying JavaScript way.
adamc@701 1776 $$\begin{array}{l}
adamc@701 1777 \mt{val} \; \mt{alert} : \mt{string} \to \mt{transaction} \; \mt{unit}
adamc@701 1778 \end{array}$$
adamc@701 1779
adamc@701 1780 Any transaction may be run in a new thread with the $\mt{spawn}$ function.
adamc@701 1781 $$\begin{array}{l}
adamc@701 1782 \mt{val} \; \mt{spawn} : \mt{transaction} \; \mt{unit} \to \mt{transaction} \; \mt{unit}
adamc@701 1783 \end{array}$$
adamc@701 1784
adamc@701 1785 The current thread can be paused for at least a specified number of milliseconds.
adamc@701 1786 $$\begin{array}{l}
adamc@701 1787 \mt{val} \; \mt{sleep} : \mt{int} \to \mt{transaction} \; \mt{unit}
adamc@701 1788 \end{array}$$
adamc@701 1789
adamc@787 1790 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 1791 $$\begin{array}{l}
adamc@787 1792 \mt{val} \; \mt{onError} : (\mt{xbody} \to \mt{transaction} \; \mt{unit}) \to \mt{transaction} \; \mt{unit} \\
adamc@787 1793 \mt{val} \; \mt{onFail} : (\mt{string} \to \mt{transaction} \; \mt{unit}) \to \mt{transaction} \; \mt{unit} \\
adamc@787 1794 \mt{val} \; \mt{onConnectFail} : \mt{transaction} \; \mt{unit} \to \mt{transaction} \; \mt{unit} \\
adamc@787 1795 \mt{val} \; \mt{onDisconnect} : \mt{transaction} \; \mt{unit} \to \mt{transaction} \; \mt{unit} \\
adamc@787 1796 \mt{val} \; \mt{onServerError} : (\mt{string} \to \mt{transaction} \; \mt{unit}) \to \mt{transaction} \; \mt{unit}
adamc@787 1797 \end{array}$$
adamc@787 1798
adamc@701 1799 \subsubsection{Functional-Reactive Page Generation}
adamc@701 1800
adamc@701 1801 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 1802
adamc@659 1803 $$\begin{array}{l}
adamc@659 1804 \mt{con} \; \mt{source} :: \mt{Type} \to \mt{Type} \\
adamc@659 1805 \mt{val} \; \mt{source} : \mt{t} ::: \mt{Type} \to \mt{t} \to \mt{transaction} \; (\mt{source} \; \mt{t}) \\
adamc@659 1806 \mt{val} \; \mt{set} : \mt{t} ::: \mt{Type} \to \mt{source} \; \mt{t} \to \mt{t} \to \mt{transaction} \; \mt{unit} \\
adamc@659 1807 \mt{val} \; \mt{get} : \mt{t} ::: \mt{Type} \to \mt{source} \; \mt{t} \to \mt{transaction} \; \mt{t}
adamc@659 1808 \end{array}$$
adamc@659 1809
adamc@659 1810 Pure functions over sources are represented in a monad of \emph{signals}.
adamc@659 1811
adamc@659 1812 $$\begin{array}{l}
adamc@659 1813 \mt{con} \; \mt{signal} :: \mt{Type} \to \mt{Type} \\
adamc@659 1814 \mt{val} \; \mt{signal\_monad} : \mt{monad} \; \mt{signal} \\
adamc@659 1815 \mt{val} \; \mt{signal} : \mt{t} ::: \mt{Type} \to \mt{source} \; \mt{t} \to \mt{signal} \; \mt{t}
adamc@659 1816 \end{array}$$
adamc@659 1817
adamc@659 1818 A reactive portion of an HTML page is injected with a $\mt{dyn}$ tag, which has a signal-valued attribute $\mt{Signal}$.
adamc@659 1819
adamc@659 1820 $$\begin{array}{l}
adamc@701 1821 \mt{val} \; \mt{dyn} : \mt{use} ::: \{\mt{Type}\} \to \mt{bind} ::: \{\mt{Type}\} \to \mt{unit} \\
adamc@701 1822 \hspace{.1in} \to \mt{tag} \; [\mt{Signal} = \mt{signal} \; (\mt{xml} \; \mt{body} \; \mt{use} \; \mt{bind})] \; \mt{body} \; [] \; \mt{use} \; \mt{bind}
adamc@659 1823 \end{array}$$
adamc@659 1824
adamc@701 1825 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 1826
adamc@914 1827 \subsubsection{Remote Procedure Calls}
adamc@914 1828
adamc@914 1829 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 1830
adamc@914 1831 $$\begin{array}{l}
adamc@914 1832 \mt{val} \; \mt{rpc} : \mt{t} ::: \mt{Type} \to \mt{transaction} \; \mt{t} \to \mt{transaction} \; \mt{t}
adamc@914 1833 \end{array}$$
adamc@914 1834
adamc@701 1835 \subsubsection{Asynchronous Message-Passing}
adamc@701 1836
adamc@701 1837 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 1838
adamc@701 1839 $$\begin{array}{l}
adamc@701 1840 \mt{type} \; \mt{client} \\
adamc@701 1841 \mt{val} \; \mt{self} : \mt{transaction} \; \mt{client}
adamc@701 1842 \end{array}$$
adamc@701 1843
adamc@701 1844 \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 1845
adamc@701 1846 $$\begin{array}{l}
adamc@701 1847 \mt{con} \; \mt{channel} :: \mt{Type} \to \mt{Type} \\
adamc@701 1848 \mt{val} \; \mt{channel} : \mt{t} ::: \mt{Type} \to \mt{transaction} \; (\mt{channel} \; \mt{t}) \\
adamc@701 1849 \mt{val} \; \mt{send} : \mt{t} ::: \mt{Type} \to \mt{channel} \; \mt{t} \to \mt{t} \to \mt{transaction} \; \mt{unit} \\
adamc@701 1850 \mt{val} \; \mt{recv} : \mt{t} ::: \mt{Type} \to \mt{channel} \; \mt{t} \to \mt{transaction} \; \mt{t}
adamc@701 1851 \end{array}$$
adamc@701 1852
adamc@701 1853 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 1854
adamc@701 1855 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 1856
adamc@659 1857
adamc@549 1858 \section{Ur/Web Syntax Extensions}
adamc@549 1859
adamc@549 1860 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 1861
adamc@549 1862 \subsection{SQL}
adamc@549 1863
adamc@786 1864 \subsubsection{\label{tables}Table Declarations}
adamc@786 1865
adamc@788 1866 $\mt{table}$ declarations may include constraints, via these grammar rules.
adamc@788 1867 $$\begin{array}{rrcll}
adamc@788 1868 \textrm{Declarations} & d &::=& \mt{table} \; x : c \; [pk[,]] \; cts \\
adamc@788 1869 \textrm{Primary key constraints} & pk &::=& \mt{PRIMARY} \; \mt{KEY} \; K \\
adamc@788 1870 \textrm{Keys} & K &::=& f \mid (f, (f,)^+) \\
adamc@788 1871 \textrm{Constraint sets} & cts &::=& \mt{CONSTRAINT} f \; ct \mid cts, cts \mid \{\{e\}\} \\
adamc@788 1872 \textrm{Constraints} & ct &::=& \mt{UNIQUE} \; K \mid \mt{CHECK} \; E \\
adamc@788 1873 &&& \mid \mt{FOREIGN} \; \mt{KEY} \; K \; \mt{REFERENCES} \; F \; (K) \; [\mt{ON} \; \mt{DELETE} \; pr] \; [\mt{ON} \; \mt{UPDATE} \; pr] \\
adamc@788 1874 \textrm{Foreign tables} & F &::=& x \mid \{\{e\}\} \\
adamc@788 1875 \textrm{Propagation modes} & pr &::=& \mt{NO} \; \mt{ACTION} \mid \mt{RESTRICT} \mid \mt{CASCADE} \mid \mt{SET} \; \mt{NULL}
adamc@788 1876 \end{array}$$
adamc@788 1877
adamc@788 1878 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 1879
adamc@788 1880
adamc@549 1881 \subsubsection{Queries}
adamc@549 1882
adamc@550 1883 Queries $Q$ are added to the rules for expressions $e$.
adamc@550 1884
adamc@549 1885 $$\begin{array}{rrcll}
adamc@550 1886 \textrm{Queries} & Q &::=& (q \; [\mt{ORDER} \; \mt{BY} \; (E \; [o],)^+] \; [\mt{LIMIT} \; N] \; [\mt{OFFSET} \; N]) \\
adamc@1085 1887 \textrm{Pre-queries} & q &::=& \mt{SELECT} \; [\mt{DISTINCT}] \; P \; \mt{FROM} \; F,^+ \; [\mt{WHERE} \; E] \; [\mt{GROUP} \; \mt{BY} \; p,^+] \; [\mt{HAVING} \; E] \\
adamc@1085 1888 &&& \mid q \; R \; q \mid \{\{\{e\}\}\} \\
adamc@549 1889 \textrm{Relational operators} & R &::=& \mt{UNION} \mid \mt{INTERSECT} \mid \mt{EXCEPT}
adamc@549 1890 \end{array}$$
adamc@549 1891
adamc@549 1892 $$\begin{array}{rrcll}
adamc@549 1893 \textrm{Projections} & P &::=& \ast & \textrm{all columns} \\
adamc@549 1894 &&& p,^+ & \textrm{particular columns} \\
adamc@549 1895 \textrm{Pre-projections} & p &::=& t.f & \textrm{one column from a table} \\
adamc@558 1896 &&& t.\{\{c\}\} & \textrm{a record of columns from a table (of kind $\{\mt{Type}\}$)} \\
adamc@549 1897 \textrm{Table names} & t &::=& x & \textrm{constant table name (automatically capitalized)} \\
adamc@549 1898 &&& X & \textrm{constant table name} \\
adamc@549 1899 &&& \{\{c\}\} & \textrm{computed table name (of kind $\mt{Name}$)} \\
adamc@549 1900 \textrm{Column names} & f &::=& X & \textrm{constant column name} \\
adamc@549 1901 &&& \{c\} & \textrm{computed column name (of kind $\mt{Name}$)} \\
adamc@549 1902 \textrm{Tables} & T &::=& x & \textrm{table variable, named locally by its own capitalization} \\
adamc@549 1903 &&& x \; \mt{AS} \; t & \textrm{table variable, with local name} \\
adamc@549 1904 &&& \{\{e\}\} \; \mt{AS} \; t & \textrm{computed table expression, with local name} \\
adamc@1085 1905 \textrm{$\mt{FROM}$ items} & F &::=& T \mid \{\{e\}\} \mid F \; J \; \mt{JOIN} \; F \; \mt{ON} \; E \\
adamc@1085 1906 &&& \mid F \; \mt{CROSS} \; \mt{JOIN} \ F \\
adamc@1085 1907 \textrm{Joins} & J &::=& [\mt{INNER}] \\
adamc@1085 1908 &&& \mid [\mt{LEFT} \mid \mt{RIGHT} \mid \mt{FULL}] \; [\mt{OUTER}] \\
adamc@549 1909 \textrm{SQL expressions} & E &::=& p & \textrm{column references} \\
adamc@549 1910 &&& X & \textrm{named expression references} \\
adamc@549 1911 &&& \{\{e\}\} & \textrm{injected native Ur expressions} \\
adamc@549 1912 &&& \{e\} & \textrm{computed expressions, probably using $\mt{sql\_exp}$ directly} \\
adamc@549 1913 &&& \mt{TRUE} \mid \mt{FALSE} & \textrm{boolean constants} \\
adamc@549 1914 &&& \ell & \textrm{primitive type literals} \\
adamc@549 1915 &&& \mt{NULL} & \textrm{null value (injection of $\mt{None}$)} \\
adamc@549 1916 &&& E \; \mt{IS} \; \mt{NULL} & \textrm{nullness test} \\
adamc@549 1917 &&& n & \textrm{nullary operators} \\
adamc@549 1918 &&& u \; E & \textrm{unary operators} \\
adamc@549 1919 &&& E \; b \; E & \textrm{binary operators} \\
adamc@549 1920 &&& \mt{COUNT}(\ast) & \textrm{count number of rows} \\
adamc@549 1921 &&& a(E) & \textrm{other aggregate function} \\
adamc@549 1922 &&& (E) & \textrm{explicit precedence} \\
adamc@549 1923 \textrm{Nullary operators} & n &::=& \mt{CURRENT\_TIMESTAMP} \\
adamc@549 1924 \textrm{Unary operators} & u &::=& \mt{NOT} \\
adamc@549 1925 \textrm{Binary operators} & b &::=& \mt{AND} \mid \mt{OR} \mid \neq \mid < \mid \leq \mid > \mid \geq \\
adamc@549 1926 \textrm{Aggregate functions} & a &::=& \mt{AVG} \mid \mt{SUM} \mid \mt{MIN} \mid \mt{MAX} \\
adamc@550 1927 \textrm{Directions} & o &::=& \mt{ASC} \mid \mt{DESC} \\
adamc@549 1928 \textrm{SQL integer} & N &::=& n \mid \{e\} \\
adamc@549 1929 \end{array}$$
adamc@549 1930
adamc@1085 1931 Additionally, an SQL expression may be inserted into normal Ur code with the syntax $(\mt{SQL} \; E)$ or $(\mt{WHERE} \; E)$. Similar shorthands exist for other nonterminals, with the prefix $\mt{FROM}$ for $\mt{FROM}$ items and $\mt{SELECT1}$ for pre-queries.
adamc@549 1932
adamc@550 1933 \subsubsection{DML}
adamc@550 1934
adamc@550 1935 DML commands $D$ are added to the rules for expressions $e$.
adamc@550 1936
adamc@550 1937 $$\begin{array}{rrcll}
adamc@550 1938 \textrm{Commands} & D &::=& (\mt{INSERT} \; \mt{INTO} \; T^E \; (f,^+) \; \mt{VALUES} \; (E,^+)) \\
adamc@550 1939 &&& (\mt{UPDATE} \; T^E \; \mt{SET} \; (f = E,)^+ \; \mt{WHERE} \; E) \\
adamc@550 1940 &&& (\mt{DELETE} \; \mt{FROM} \; T^E \; \mt{WHERE} \; E) \\
adamc@550 1941 \textrm{Table expressions} & T^E &::=& x \mid \{\{e\}\}
adamc@550 1942 \end{array}$$
adamc@550 1943
adamc@550 1944 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 1945
adamc@551 1946 \subsection{XML}
adamc@551 1947
adamc@551 1948 XML fragments $L$ are added to the rules for expressions $e$.
adamc@551 1949
adamc@551 1950 $$\begin{array}{rrcll}
adamc@551 1951 \textrm{XML fragments} & L &::=& \texttt{<xml/>} \mid \texttt{<xml>}l^*\texttt{</xml>} \\
adamc@551 1952 \textrm{XML pieces} & l &::=& \textrm{text} & \textrm{cdata} \\
adamc@551 1953 &&& \texttt{<}g\texttt{/>} & \textrm{tag with no children} \\
adamc@551 1954 &&& \texttt{<}g\texttt{>}l^*\texttt{</}x\texttt{>} & \textrm{tag with children} \\
adamc@559 1955 &&& \{e\} & \textrm{computed XML fragment} \\
adamc@559 1956 &&& \{[e]\} & \textrm{injection of an Ur expression, via the $\mt{Top}.\mt{txt}$ function} \\
adamc@551 1957 \textrm{Tag} & g &::=& h \; (x = v)^* \\
adamc@551 1958 \textrm{Tag head} & h &::=& x & \textrm{tag name} \\
adamc@551 1959 &&& h\{c\} & \textrm{constructor parameter} \\
adamc@551 1960 \textrm{Attribute value} & v &::=& \ell & \textrm{literal value} \\
adamc@551 1961 &&& \{e\} & \textrm{computed value} \\
adamc@551 1962 \end{array}$$
adamc@551 1963
adamc@552 1964
adamc@553 1965 \section{The Structure of Web Applications}
adamc@553 1966
adamc@553 1967 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 1968
adamc@553 1969 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 1970
adamc@553 1971 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 1972
adamc@558 1973 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 1974
adamc@660 1975 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 1976
adamc@789 1977 \medskip
adamc@789 1978
adamc@789 1979 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 1980
adamc@789 1981 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 1982
adamc@553 1983
adamc@897 1984 \section{The Foreign Function Interface}
adamc@897 1985
adamc@897 1986 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 1987
adamc@897 1988 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 1989
adamc@897 1990 \begin{itemize}
adamc@897 1991 \item \texttt{clientOnly Module.ident} registers a value as being allowed only in client-side code.
adamc@897 1992 \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 1993 \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 1994 \item \texttt{ffi FILE.urs} names the file giving your library's signature. You can include multiple such files in a single \texttt{.urp} file, and each file \texttt{mod.urp} defines an FFI module \texttt{Mod}.
adamc@1099 1995 \item \texttt{include FILE} requests inclusion of a C header file.
adamc@897 1996 \item \texttt{jsFunc Module.ident=name} gives a mapping from an Ur name for a value to a JavaScript name.
adamc@897 1997 \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 1998 \item \texttt{script URL} requests inclusion of a JavaScript source file within application HTML.
adamc@897 1999 \item \texttt{serverOnly Module.ident} registers a value as being allowed only in server-side code.
adamc@897 2000 \end{itemize}
adamc@897 2001
adamc@897 2002 \subsection{Writing C FFI Code}
adamc@897 2003
adamc@897 2004 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 2005
adamc@897 2006 \begin{itemize}
adamc@897 2007 \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 2008 \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 2009 \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 2010 \end{itemize}
adamc@897 2011
adamc@897 2012 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 2013 \begin{itemize}
adamc@897 2014 \item \begin{verbatim}
adamc@897 2015 void uw_error(uw_context, failure_kind, const char *fmt, ...);
adamc@897 2016 \end{verbatim}
adamc@897 2017 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 2018
adamc@897 2019 \item \begin{verbatim}
adamc@897 2020 void uw_push_cleanup(uw_context, void (*func)(void *), void *arg);
adamc@897 2021 void uw_pop_cleanup(uw_context);
adamc@897 2022 \end{verbatim}
adamc@897 2023 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 2024
adamc@897 2025 \item \begin{verbatim}
adamc@897 2026 void *uw_malloc(uw_context, size_t);
adamc@897 2027 \end{verbatim}
adamc@897 2028 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 2029
adamc@897 2030 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 2031
adamc@897 2032 \item \begin{verbatim}
adamc@897 2033 typedef void (*uw_callback)(void *);
adamc@897 2034 void uw_register_transactional(uw_context, void *data, uw_callback commit,
adamc@897 2035 uw_callback rollback, uw_callback free);
adamc@897 2036 \end{verbatim}
adamc@897 2037 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 2038
adamc@1085 2039 Any of the callbacks may be \texttt{NULL}. To accommodate some stubbornly non-transactional real-world actions like sending an e-mail message, Ur/Web treats \texttt{NULL} \texttt{rollback} callbacks specially. When a transaction commits, all \texttt{commit} actions that have non-\texttt{NULL} rollback actions are tried before any \texttt{commit} actions that have \texttt{NULL} rollback actions. Thus, if a single execution uses only one non-transactional action, and if that action never fails partway through its execution while still causing an observable side effect, then Ur/Web can maintain the transactional abstraction.
adamc@1085 2040
adamc@1085 2041 \item \begin{verbatim}
adamc@1085 2042 void *uw_get_global(uw_context, char *name);
adamc@1085 2043 void uw_set_global(uw_context, char *name, void *data, uw_callback free);
adamc@1085 2044 \end{verbatim}
adamc@1085 2045 Different FFI-based extensions may want to associate their own pieces of data with contexts. The global interface provides a way of doing that, where each extension must come up with its own unique key. The \texttt{free} argument to \texttt{uw\_set\_global()} explains how to deallocate the saved data.
adamc@1085 2046
adamc@897 2047 \end{itemize}
adamc@897 2048
adamc@897 2049 \subsection{Writing JavaScript FFI Code}
adamc@897 2050
adamc@897 2051 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 2052
adamc@897 2053 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 2054
adamc@897 2055 \begin{itemize}
adamc@897 2056 \item Integers, floats, strings, characters, and booleans are represented in the usual JavaScript way.
adamc@985 2057 \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 2058 \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 2059 \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 2060 \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 2061 \end{itemize}
adamc@897 2062
adamc@897 2063 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 2064
adamc@897 2065
adamc@552 2066 \section{Compiler Phases}
adamc@552 2067
adamc@552 2068 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 2069
adamc@552 2070 In this section, we step through the main phases of compilation, noting what consequences each phase has for effective programming.
adamc@552 2071
adamc@552 2072 \subsection{Parse}
adamc@552 2073
adamc@552 2074 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 2075
adamc@552 2076 \subsection{Elaborate}
adamc@552 2077
adamc@552 2078 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 2079
adamc@552 2080 \subsection{Unnest}
adamc@552 2081
adamc@552 2082 Named local function definitions are moved to the top level, to avoid the need to generate closures.
adamc@552 2083
adamc@552 2084 \subsection{Corify}
adamc@552 2085
adamc@552 2086 Module system features are compiled away, through inlining of functor definitions at application sites. Afterward, most abstraction boundaries are broken, facilitating optimization.
adamc@552 2087
adamc@552 2088 \subsection{Especialize}
adamc@552 2089
adamc@552 2090 Functions are specialized to particular argument patterns. This is an important trick for avoiding the need to maintain any closures at runtime.
adamc@552 2091
adamc@552 2092 \subsection{Untangle}
adamc@552 2093
adamc@552 2094 Remove unnecessary mutual recursion, splitting recursive groups into strongly-connected components.
adamc@552 2095
adamc@552 2096 \subsection{Shake}
adamc@552 2097
adamc@552 2098 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 2099
adamc@661 2100 \subsection{Rpcify}
adamc@661 2101
adamc@661 2102 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 2103
adamc@661 2104 \subsection{Untangle, Shake}
adamc@661 2105
adamc@661 2106 Repeat these simplifications.
adamc@661 2107
adamc@553 2108 \subsection{\label{tag}Tag}
adamc@552 2109
adamc@552 2110 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 2111
adamc@552 2112 \subsection{Reduce}
adamc@552 2113
adamc@552 2114 Apply definitional equality rules to simplify the program as much as possible. This effectively includes inlining of every non-recursive definition.
adamc@552 2115
adamc@552 2116 \subsection{Unpoly}
adamc@552 2117
adamc@552 2118 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 2119
adamc@552 2120 \subsection{Specialize}
adamc@552 2121
adamc@558 2122 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 2123
adamc@552 2124 \subsection{Shake}
adamc@552 2125
adamc@558 2126 Here the compiler repeats the earlier Shake phase.
adamc@552 2127
adamc@552 2128 \subsection{Monoize}
adamc@552 2129
adamc@552 2130 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 2131
adamc@552 2132 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 2133
adamc@552 2134 \subsection{MonoOpt}
adamc@552 2135
adamc@552 2136 Simple algebraic laws are applied to simplify the program, focusing especially on efficient imperative generation of HTML pages.
adamc@552 2137
adamc@552 2138 \subsection{MonoUntangle}
adamc@552 2139
adamc@552 2140 Unnecessary mutual recursion is broken up again.
adamc@552 2141
adamc@552 2142 \subsection{MonoReduce}
adamc@552 2143
adamc@552 2144 Equivalents of the definitional equality rules are applied to simplify programs, with inlining again playing a major role.
adamc@552 2145
adamc@552 2146 \subsection{MonoShake, MonoOpt}
adamc@552 2147
adamc@552 2148 Unneeded declarations are removed, and basic optimizations are repeated.
adamc@552 2149
adamc@552 2150 \subsection{Fuse}
adamc@552 2151
adamc@552 2152 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 2153
adamc@552 2154 \subsection{MonoUntangle, MonoShake}
adamc@552 2155
adamc@552 2156 Fuse often creates more opportunities to remove spurious mutual recursion.
adamc@552 2157
adamc@552 2158 \subsection{Pathcheck}
adamc@552 2159
adamc@552 2160 The compiler checks that no link or action name has been used more than once.
adamc@552 2161
adamc@552 2162 \subsection{Cjrize}
adamc@552 2163
adamc@552 2164 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 2165
adamc@552 2166 \subsection{C Compilation and Linking}
adamc@552 2167
adamc@552 2168 The output of the last phase is pretty-printed as C source code and passed to GCC.
adamc@552 2169
adamc@552 2170
adamc@524 2171 \end{document}