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author	Adam Chlipala <adamc@hcoop.net>
date	Tue, 30 Jun 2009 15:59:41 -0400
parents	19fdeef40ada
children	41971801b62d

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@783	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. To build programs that access SQL databases, you also need libpq, the PostgreSQL client library. As of this writing, in the ``testing'' version of Debian Linux, this command will install the more uncommon of these dependencies:
adamc@555	60
adamc@555	61 \begin{verbatim}
adamc@783	62 apt-get install mlton libmhash-dev libpq-dev
adamc@555	63 \end{verbatim}
adamc@555	64
adamc@555	65 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	66
adamc@555	67 \begin{verbatim}
adamc@555	68 apt-get install smlnj libsmlnj-smlnj ml-yacc ml-lpt
adamc@555	69 \end{verbatim}
adamc@555	70
adamc@555	71 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	72
adamc@555	73 To run an SQL-backed application, you will probably want to install the PostgreSQL server. Version 8.3 or higher is required.
adamc@555	74
adamc@555	75 \begin{verbatim}
adamc@555	76 apt-get install postgresql-8.3
adamc@555	77 \end{verbatim}
adamc@555	78
adamc@555	79 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	80
adamc@555	81 \begin{verbatim}
adamc@555	82 apt-get install emacs-goodies-el
adamc@555	83 \end{verbatim}
adamc@555	84
adamc@555	85 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	86
adamc@555	87 \begin{verbatim}
adamc@555	88 GCCARGS=-fnested-functions ./configure
adamc@555	89 \end{verbatim}
adamc@555	90
adamc@555	91 Some OSX users have reported needing to use this particular GCCARGS value.
adamc@555	92
adamc@555	93 The Emacs mode can be set to autoload by adding the following to your \texttt{.emacs} file.
adamc@555	94
adamc@555	95 \begin{verbatim}
adamc@555	96 (add-to-list 'load-path "/usr/local/share/emacs/site-lisp/urweb-mode")
adamc@555	97 (load "urweb-mode-startup")
adamc@555	98 \end{verbatim}
adamc@555	99
adamc@555	100 Change the path in the first line if you chose a different Emacs installation path during configuration.
adamc@555	101
adamc@555	102
adamc@556	103 \section{Command-Line Compiler}
adamc@556	104
adamc@556	105 \subsection{Project Files}
adamc@556	106
adamc@556	107 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	108
adamc@556	109 \begin{verbatim}
adamc@556	110 database dbname=test
adamc@556	111 sql crud1.sql
adamc@556	112
adamc@556	113 crud
adamc@556	114 crud1
adamc@556	115 \end{verbatim}
adamc@556	116
adamc@556	117 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	118
adamc@556	119 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	120
adamc@556	121 \begin{verbatim}
adamc@556	122 createdb test
adamc@556	123 psql -f crud1.sql test
adamc@556	124 \end{verbatim}
adamc@556	125
adamc@556	126 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	127
adamc@556	128 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	129
adamc@783	130 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	131 \begin{itemize}
adamc@783	132 \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	133 \item \texttt{clientOnly Module.ident} registers an FFI function or transaction that may only be run in client browsers.
adamc@783	134 \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	135 \item \texttt{database DBSTRING} sets the string to pass to libpq to open a database connection.
adamc@783	136 \item \texttt{debug} saves some intermediate C files, which is mostly useful to help in debugging the compiler itself.
adamc@783	137 \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	138 \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	139 \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@783	140 \item \texttt{jsFunc Module.ident=name} gives the JavaScript name of an FFI value.
adamc@783	141 \item \texttt{library FILENAME} parses \texttt{FILENAME.urp} and merges its contents with the rest of the current file's contents.
adamc@783	142 \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	143 \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	144 \item \texttt{prefix PREFIX} sets the prefix included before every URI within the generated application. The default is \texttt{/}.
adamc@783	145 \item \texttt{profile} generates an executable that may be used with gprof.
adamc@783	146 \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	147 \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	148 \item \texttt{serverOnly Module.ident} registers an FFI function or transaction that may only be run on the server.
adamc@783	149 \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	150 \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	151 \end{itemize}
adamc@701	152
adamc@701	153
adamc@557	154 \subsection{Building an Application}
adamc@557	155
adamc@557	156 To compile project \texttt{P.urp}, simply run
adamc@557	157 \begin{verbatim}
adamc@557	158 urweb P
adamc@557	159 \end{verbatim}
adamc@558	160 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	161
adamc@557	162 To time how long the different compiler phases run, without generating an executable, run
adamc@557	163 \begin{verbatim}
adamc@557	164 urweb -timing P
adamc@557	165 \end{verbatim}
adamc@557	166
adamc@556	167
adamc@529	168 \section{Ur Syntax}
adamc@529	169
adamc@784	170 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	171
adamc@524	172 \subsection{Lexical Conventions}
adamc@524	173
adamc@524	174 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	175
adamc@524	176 \begin{center}
adamc@524	177 \begin{tabular}{rl}
adamc@524	178 \textbf{\LaTeX} & \textbf{ASCII} \\
adamc@524	179 $\to$ & \cd{->} \\
adamc@652	180 $\longrightarrow$ & \cd{-->} \\
adamc@524	181 $\times$ & \cd{*} \\
adamc@524	182 $\lambda$ & \cd{fn} \\
adamc@524	183 $\Rightarrow$ & \cd{=>} \\
adamc@652	184 $\Longrightarrow$ & \cd{==>} \\
adamc@529	185 $\neq$ & \cd{<>} \\
adamc@529	186 $\leq$ & \cd{<=} \\
adamc@529	187 $\geq$ & \cd{>=} \\
adamc@524	188 \\
adamc@524	189 $x$ & Normal textual identifier, not beginning with an uppercase letter \\
adamc@525	190 $X$ & Normal textual identifier, beginning with an uppercase letter \\
adamc@524	191 \end{tabular}
adamc@524	192 \end{center}
adamc@524	193
adamc@525	194 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	195
adamc@526	196 We write $\ell$ for literals of the primitive types, for the most part following C conventions. There are $\mt{int}$, $\mt{float}$, and $\mt{string}$ literals.
adamc@526	197
adamc@527	198 This version of the manual doesn't include operator precedences; see \texttt{src/urweb.grm} for that.
adamc@527	199
adamc@552	200 \subsection{\label{core}Core Syntax}
adamc@524	201
adamc@524	202 \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	203 $$\begin{array}{rrcll}
adamc@524	204 \textrm{Kinds} & \kappa &::=& \mt{Type} & \textrm{proper types} \\
adamc@525	205 &&& \mt{Unit} & \textrm{the trivial constructor} \\
adamc@525	206 &&& \mt{Name} & \textrm{field names} \\
adamc@525	207 &&& \kappa \to \kappa & \textrm{type-level functions} \\
adamc@525	208 &&& \{\kappa\} & \textrm{type-level records} \\
adamc@525	209 &&& (\kappa\times^+) & \textrm{type-level tuples} \\
adamc@652	210 &&& X & \textrm{variable} \\
adamc@652	211 &&& X \longrightarrow k & \textrm{kind-polymorphic type-level function} \\
adamc@529	212 &&& \_\_ & \textrm{wildcard} \\
adamc@525	213 &&& (\kappa) & \textrm{explicit precedence} \\
adamc@524	214 \end{array}$$
adamc@524	215
adamc@524	216 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	217 $$\begin{array}{rrcll}
adamc@524	218 \textrm{Explicitness} & ? &::=& :: & \textrm{explicit} \\
adamc@558	219 &&& ::: & \textrm{implicit}
adamc@524	220 \end{array}$$
adamc@524	221
adamc@524	222 \emph{Constructors} are the main class of compile-time-only data. They include proper types and are classified by kinds.
adamc@524	223 $$\begin{array}{rrcll}
adamc@524	224 \textrm{Constructors} & c, \tau &::=& (c) :: \kappa & \textrm{kind annotation} \\
adamc@530	225 &&& \hat{x} & \textrm{constructor variable} \\
adamc@524	226 \\
adamc@525	227 &&& \tau \to \tau & \textrm{function type} \\
adamc@525	228 &&& x \; ? \; \kappa \to \tau & \textrm{polymorphic function type} \\
adamc@652	229 &&& X \longrightarrow \tau & \textrm{kind-polymorphic function type} \\
adamc@525	230 &&& \$ c & \textrm{record type} \\
adamc@524	231 \\
adamc@525	232 &&& c \; c & \textrm{type-level function application} \\
adamc@530	233 &&& \lambda x \; :: \; \kappa \Rightarrow c & \textrm{type-level function abstraction} \\
adamc@524	234 \\
adamc@652	235 &&& X \Longrightarrow c & \textrm{type-level kind-polymorphic function abstraction} \\
adamc@655	236 &&& c [\kappa] & \textrm{type-level kind-polymorphic function application} \\
adamc@652	237 \\
adamc@525	238 &&& () & \textrm{type-level unit} \\
adamc@525	239 &&& \#X & \textrm{field name} \\
adamc@524	240 \\
adamc@525	241 &&& [(c = c)^*] & \textrm{known-length type-level record} \\
adamc@525	242 &&& c \rc c & \textrm{type-level record concatenation} \\
adamc@652	243 &&& \mt{map} & \textrm{type-level record map} \\
adamc@524	244 \\
adamc@558	245 &&& (c,^+) & \textrm{type-level tuple} \\
adamc@525	246 &&& c.n & \textrm{type-level tuple projection ($n \in \mathbb N^+$)} \\
adamc@524	247 \\
adamc@652	248 &&& [c \sim c] \Rightarrow \tau & \textrm{guarded type} \\
adamc@524	249 \\
adamc@529	250 &&& \_ :: \kappa & \textrm{wildcard} \\
adamc@525	251 &&& (c) & \textrm{explicit precedence} \\
adamc@530	252 \\
adamc@530	253 \textrm{Qualified uncapitalized variables} & \hat{x} &::=& x & \textrm{not from a module} \\
adamc@530	254 &&& M.x & \textrm{projection from a module} \\
adamc@525	255 \end{array}$$
adamc@525	256
adamc@655	257 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	258
adamc@525	259 Modules of the module system are described by \emph{signatures}.
adamc@525	260 $$\begin{array}{rrcll}
adamc@525	261 \textrm{Signatures} & S &::=& \mt{sig} \; s^* \; \mt{end} & \textrm{constant} \\
adamc@525	262 &&& X & \textrm{variable} \\
adamc@525	263 &&& \mt{functor}(X : S) : S & \textrm{functor} \\
adamc@529	264 &&& S \; \mt{where} \; \mt{con} \; x = c & \textrm{concretizing an abstract constructor} \\
adamc@525	265 &&& M.X & \textrm{projection from a module} \\
adamc@525	266 \\
adamc@525	267 \textrm{Signature items} & s &::=& \mt{con} \; x :: \kappa & \textrm{abstract constructor} \\
adamc@525	268 &&& \mt{con} \; x :: \kappa = c & \textrm{concrete constructor} \\
adamc@528	269 &&& \mt{datatype} \; x \; x^* = dc\mid^+ & \textrm{algebraic datatype definition} \\
adamc@529	270 &&& \mt{datatype} \; x = \mt{datatype} \; M.x & \textrm{algebraic datatype import} \\
adamc@525	271 &&& \mt{val} \; x : \tau & \textrm{value} \\
adamc@525	272 &&& \mt{structure} \; X : S & \textrm{sub-module} \\
adamc@525	273 &&& \mt{signature} \; X = S & \textrm{sub-signature} \\
adamc@525	274 &&& \mt{include} \; S & \textrm{signature inclusion} \\
adamc@525	275 &&& \mt{constraint} \; c \sim c & \textrm{record disjointness constraint} \\
adamc@654	276 &&& \mt{class} \; x :: \kappa & \textrm{abstract constructor class} \\
adamc@654	277 &&& \mt{class} \; x :: \kappa = c & \textrm{concrete constructor class} \\
adamc@525	278 \\
adamc@525	279 \textrm{Datatype constructors} & dc &::=& X & \textrm{nullary constructor} \\
adamc@525	280 &&& X \; \mt{of} \; \tau & \textrm{unary constructor} \\
adamc@524	281 \end{array}$$
adamc@524	282
adamc@526	283 \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	284 $$\begin{array}{rrcll}
adamc@526	285 \textrm{Patterns} & p &::=& \_ & \textrm{wildcard} \\
adamc@526	286 &&& x & \textrm{variable} \\
adamc@526	287 &&& \ell & \textrm{constant} \\
adamc@526	288 &&& \hat{X} & \textrm{nullary constructor} \\
adamc@526	289 &&& \hat{X} \; p & \textrm{unary constructor} \\
adamc@526	290 &&& \{(x = p,)^*\} & \textrm{rigid record pattern} \\
adamc@526	291 &&& \{(x = p,)^+, \ldots\} & \textrm{flexible record pattern} \\
adamc@852	292 &&& p : \tau & \textrm{type annotation} \\
adamc@527	293 &&& (p) & \textrm{explicit precedence} \\
adamc@526	294 \\
adamc@529	295 \textrm{Qualified capitalized variables} & \hat{X} &::=& X & \textrm{not from a module} \\
adamc@526	296 &&& M.X & \textrm{projection from a module} \\
adamc@526	297 \end{array}$$
adamc@526	298
adamc@527	299 \emph{Expressions} are the main run-time entities, corresponding to both ``expressions'' and ``statements'' in mainstream imperative languages.
adamc@527	300 $$\begin{array}{rrcll}
adamc@527	301 \textrm{Expressions} & e &::=& e : \tau & \textrm{type annotation} \\
adamc@529	302 &&& \hat{x} & \textrm{variable} \\
adamc@529	303 &&& \hat{X} & \textrm{datatype constructor} \\
adamc@527	304 &&& \ell & \textrm{constant} \\
adamc@527	305 \\
adamc@527	306 &&& e \; e & \textrm{function application} \\
adamc@527	307 &&& \lambda x : \tau \Rightarrow e & \textrm{function abstraction} \\
adamc@527	308 &&& e [c] & \textrm{polymorphic function application} \\
adamc@852	309 &&& \lambda [x \; ? \; \kappa] \Rightarrow e & \textrm{polymorphic function abstraction} \\
adamc@655	310 &&& e [\kappa] & \textrm{kind-polymorphic function application} \\
adamc@652	311 &&& X \Longrightarrow e & \textrm{kind-polymorphic function abstraction} \\
adamc@527	312 \\
adamc@527	313 &&& \{(c = e,)^*\} & \textrm{known-length record} \\
adamc@527	314 &&& e.c & \textrm{record field projection} \\
adamc@527	315 &&& e \rc e & \textrm{record concatenation} \\
adamc@527	316 &&& e \rcut c & \textrm{removal of a single record field} \\
adamc@527	317 &&& e \rcutM c & \textrm{removal of multiple record fields} \\
adamc@527	318 \\
adamc@527	319 &&& \mt{let} \; ed^* \; \mt{in} \; e \; \mt{end} & \textrm{local definitions} \\
adamc@527	320 \\
adamc@527	321 &&& \mt{case} \; e \; \mt{of} \; (p \Rightarrow e\|)^+ & \textrm{pattern matching} \\
adamc@527	322 \\
adamc@654	323 &&& \lambda [c \sim c] \Rightarrow e & \textrm{guarded expression abstraction} \\
adamc@654	324 &&& e \; ! & \textrm{guarded expression application} \\
adamc@527	325 \\
adamc@527	326 &&& \_ & \textrm{wildcard} \\
adamc@527	327 &&& (e) & \textrm{explicit precedence} \\
adamc@527	328 \\
adamc@527	329 \textrm{Local declarations} & ed &::=& \cd{val} \; x : \tau = e & \textrm{non-recursive value} \\
adamc@527	330 &&& \cd{val} \; \cd{rec} \; (x : \tau = e \; \cd{and})^+ & \textrm{mutually-recursive values} \\
adamc@527	331 \end{array}$$
adamc@527	332
adamc@655	333 As with constructors, we include both abstraction and application for kind polymorphism, but applications are only inferred internally.
adamc@655	334
adamc@528	335 \emph{Declarations} primarily bring new symbols into context.
adamc@528	336 $$\begin{array}{rrcll}
adamc@528	337 \textrm{Declarations} & d &::=& \mt{con} \; x :: \kappa = c & \textrm{constructor synonym} \\
adamc@528	338 &&& \mt{datatype} \; x \; x^* = dc\mid^+ & \textrm{algebraic datatype definition} \\
adamc@529	339 &&& \mt{datatype} \; x = \mt{datatype} \; M.x & \textrm{algebraic datatype import} \\
adamc@528	340 &&& \mt{val} \; x : \tau = e & \textrm{value} \\
adamc@528	341 &&& \mt{val} \; \cd{rec} \; (x : \tau = e \; \mt{and})^+ & \textrm{mutually-recursive values} \\
adamc@528	342 &&& \mt{structure} \; X : S = M & \textrm{module definition} \\
adamc@528	343 &&& \mt{signature} \; X = S & \textrm{signature definition} \\
adamc@528	344 &&& \mt{open} \; M & \textrm{module inclusion} \\
adamc@528	345 &&& \mt{constraint} \; c \sim c & \textrm{record disjointness constraint} \\
adamc@528	346 &&& \mt{open} \; \mt{constraints} \; M & \textrm{inclusion of just the constraints from a module} \\
adamc@528	347 &&& \mt{table} \; x : c & \textrm{SQL table} \\
adamc@784	348 &&& \mt{view} \; x : c & \textrm{SQL view} \\
adamc@528	349 &&& \mt{sequence} \; x & \textrm{SQL sequence} \\
adamc@535	350 &&& \mt{cookie} \; x : \tau & \textrm{HTTP cookie} \\
adamc@784	351 &&& \mt{style} \; x : \tau & \textrm{CSS class} \\
adamc@654	352 &&& \mt{class} \; x :: \kappa = c & \textrm{concrete constructor class} \\
adamc@528	353 \\
adamc@529	354 \textrm{Modules} & M &::=& \mt{struct} \; d^* \; \mt{end} & \textrm{constant} \\
adamc@529	355 &&& X & \textrm{variable} \\
adamc@529	356 &&& M.X & \textrm{projection} \\
adamc@529	357 &&& M(M) & \textrm{functor application} \\
adamc@529	358 &&& \mt{functor}(X : S) : S = M & \textrm{functor abstraction} \\
adamc@528	359 \end{array}$$
adamc@528	360
adamc@528	361 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	362
adamc@784	363 We omit some extra possibilities in $\mt{table}$ syntax, deferring them to Section \ref{tables}.
adamc@784	364
adamc@529	365 \subsection{Shorthands}
adamc@529	366
adamc@529	367 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	368
adamc@529	369 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	370
adamc@529	371 A record type may be written $\{(c = c,)^\}$, which elaborates to $\$[(c = c,)^]$.
adamc@529	372
adamc@533	373 The notation $[c_1, \ldots, c_n]$ is shorthand for $[c_1 = (), \ldots, c_n = ()]$.
adamc@533	374
adamc@529	375 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	376
adamc@852	377 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	378
adamc@529	379 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	380
adamc@529	381 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	382
adamc@654	383 A signature item or declaration $\mt{class} \; x = \lambda y \Rightarrow c$ may be abbreviated $\mt{class} \; x \; y = c$.
adamc@529	384
adamc@654	385 Handling of implicit and explicit constructor arguments may be tweaked with some prefixes to variable references. An expression $@x$ is a version of $x$ where all implicit constructor arguments have been made explicit. An expression $@@x$ achieves the same effect, additionally halting automatic resolution of type class instances and automatic proving of disjointness constraints. The default is that any prefix of a variable's type consisting only of implicit polymorphism, type class instances, and disjointness obligations is resolved automatically, with the variable treated as having the type that starts after the last implicit element, with suitable unification variables substituted. The same syntax works for variables projected out of modules and for capitalized variables (datatype constructors).
adamc@529	386
adamc@852	387 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	388
adamc@852	389 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	390
adamc@852	391 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	392
adamc@529	393 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	394
adamc@852	395 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	396
adamc@853	397 A declaration $\mt{table} \; x : \{(c = c,)^\}$ is elaborated into $\mt{table} \; x : [(c = c,)^]$.
adamc@529	398
adamc@529	399 The syntax $\mt{where} \; \mt{type}$ is an alternate form of $\mt{where} \; \mt{con}$.
adamc@529	400
adamc@529	401 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	402
adamc@529	403 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	404
adamc@784	405 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	406
adamc@530	407
adamc@530	408 \section{Static Semantics}
adamc@530	409
adamc@530	410 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	411
adamc@530	412 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	413 \begin{itemize}
adamc@655	414 \item $\Gamma \vdash \kappa$ expresses kind well-formedness.
adamc@530	415 \item $\Gamma \vdash c :: \kappa$ assigns a kind to a constructor in a context.
adamc@530	416 \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	417 \item $\Gamma \vdash c \hookrightarrow C$ proves that record constructor $c$ decomposes into set $C$ of field names and record constructors.
adamc@530	418 \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	419 \item $\Gamma \vdash e : \tau$ is a standard typing judgment.
adamc@534	420 \item $\Gamma \vdash p \leadsto \Gamma; \tau$ combines typing of patterns with calculation of which new variables they bind.
adamc@537	421 \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	422 \item $\Gamma \vdash S \equiv S$ is the signature equivalence judgment.
adamc@536	423 \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	424 \item $\Gamma \vdash M : S$ is the module signature checking judgment.
adamc@537	425 \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	426 \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	427 \end{itemize}
adamc@530	428
adamc@655	429
adamc@655	430 \subsection{Kind Well-Formedness}
adamc@655	431
adamc@655	432 $$\infer{\Gamma \vdash \mt{Type}}{}
adamc@655	433 \quad \infer{\Gamma \vdash \mt{Unit}}{}
adamc@655	434 \quad \infer{\Gamma \vdash \mt{Name}}{}
adamc@655	435 \quad \infer{\Gamma \vdash \kappa_1 \to \kappa_2}{
adamc@655	436 \Gamma \vdash \kappa_1
adamc@655	437 & \Gamma \vdash \kappa_2
adamc@655	438 }
adamc@655	439 \quad \infer{\Gamma \vdash \{\kappa\}}{
adamc@655	440 \Gamma \vdash \kappa
adamc@655	441 }
adamc@655	442 \quad \infer{\Gamma \vdash (\kappa_1 \times \ldots \times \kappa_n)}{
adamc@655	443 \forall i: \Gamma \vdash \kappa_i
adamc@655	444 }$$
adamc@655	445
adamc@655	446 $$\infer{\Gamma \vdash X}{
adamc@655	447 X \in \Gamma
adamc@655	448 }
adamc@655	449 \quad \infer{\Gamma \vdash X \longrightarrow \kappa}{
adamc@655	450 \Gamma, X \vdash \kappa
adamc@655	451 }$$
adamc@655	452
adamc@530	453 \subsection{Kinding}
adamc@530	454
adamc@655	455 We write $[X \mapsto \kappa_1]\kappa_2$ for capture-avoiding substitution of $\kappa_1$ for $X$ in $\kappa_2$.
adamc@655	456
adamc@530	457 $$\infer{\Gamma \vdash (c) :: \kappa :: \kappa}{
adamc@530	458 \Gamma \vdash c :: \kappa
adamc@530	459 }
adamc@530	460 \quad \infer{\Gamma \vdash x :: \kappa}{
adamc@530	461 x :: \kappa \in \Gamma
adamc@530	462 }
adamc@530	463 \quad \infer{\Gamma \vdash x :: \kappa}{
adamc@530	464 x :: \kappa = c \in \Gamma
adamc@530	465 }$$
adamc@530	466
adamc@530	467 $$\infer{\Gamma \vdash M.x :: \kappa}{
adamc@537	468 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	469 & \mt{proj}(M, \overline{s}, \mt{con} \; x) = \kappa
adamc@530	470 }
adamc@530	471 \quad \infer{\Gamma \vdash M.x :: \kappa}{
adamc@537	472 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	473 & \mt{proj}(M, \overline{s}, \mt{con} \; x) = (\kappa, c)
adamc@530	474 }$$
adamc@530	475
adamc@530	476 $$\infer{\Gamma \vdash \tau_1 \to \tau_2 :: \mt{Type}}{
adamc@530	477 \Gamma \vdash \tau_1 :: \mt{Type}
adamc@530	478 & \Gamma \vdash \tau_2 :: \mt{Type}
adamc@530	479 }
adamc@530	480 \quad \infer{\Gamma \vdash x \; ? \: \kappa \to \tau :: \mt{Type}}{
adamc@530	481 \Gamma, x :: \kappa \vdash \tau :: \mt{Type}
adamc@530	482 }
adamc@655	483 \quad \infer{\Gamma \vdash X \longrightarrow \tau :: \mt{Type}}{
adamc@655	484 \Gamma, X \vdash \tau :: \mt{Type}
adamc@655	485 }
adamc@530	486 \quad \infer{\Gamma \vdash \$c :: \mt{Type}}{
adamc@530	487 \Gamma \vdash c :: \{\mt{Type}\}
adamc@530	488 }$$
adamc@530	489
adamc@530	490 $$\infer{\Gamma \vdash c_1 \; c_2 :: \kappa_2}{
adamc@530	491 \Gamma \vdash c_1 :: \kappa_1 \to \kappa_2
adamc@530	492 & \Gamma \vdash c_2 :: \kappa_1
adamc@530	493 }
adamc@530	494 \quad \infer{\Gamma \vdash \lambda x \; :: \; \kappa_1 \Rightarrow c :: \kappa_1 \to \kappa_2}{
adamc@530	495 \Gamma, x :: \kappa_1 \vdash c :: \kappa_2
adamc@530	496 }$$
adamc@530	497
adamc@655	498 $$\infer{\Gamma \vdash c[\kappa'] :: [X \mapsto \kappa']\kappa}{
adamc@655	499 \Gamma \vdash c :: X \to \kappa
adamc@655	500 & \Gamma \vdash \kappa'
adamc@655	501 }
adamc@655	502 \quad \infer{\Gamma \vdash X \Longrightarrow c :: X \to \kappa}{
adamc@655	503 \Gamma, X \vdash c :: \kappa
adamc@655	504 }$$
adamc@655	505
adamc@530	506 $$\infer{\Gamma \vdash () :: \mt{Unit}}{}
adamc@530	507 \quad \infer{\Gamma \vdash \#X :: \mt{Name}}{}$$
adamc@530	508
adamc@530	509 $$\infer{\Gamma \vdash [\overline{c_i = c'_i}] :: \{\kappa\}}{
adamc@530	510 \forall i: \Gamma \vdash c_i : \mt{Name}
adamc@530	511 & \Gamma \vdash c'_i :: \kappa
adamc@530	512 & \forall i \neq j: \Gamma \vdash c_i \sim c_j
adamc@530	513 }
adamc@530	514 \quad \infer{\Gamma \vdash c_1 \rc c_2 :: \{\kappa\}}{
adamc@530	515 \Gamma \vdash c_1 :: \{\kappa\}
adamc@530	516 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@530	517 & \Gamma \vdash c_1 \sim c_2
adamc@530	518 }$$
adamc@530	519
adamc@655	520 $$\infer{\Gamma \vdash \mt{map} :: (\kappa_1 \to \kappa_2) \to \{\kappa_1\} \to \{\kappa_2\}}{}$$
adamc@530	521
adamc@573	522 $$\infer{\Gamma \vdash (\overline c) :: (\kappa_1 \times \ldots \times \kappa_n)}{
adamc@573	523 \forall i: \Gamma \vdash c_i :: \kappa_i
adamc@530	524 }
adamc@573	525 \quad \infer{\Gamma \vdash c.i :: \kappa_i}{
adamc@573	526 \Gamma \vdash c :: (\kappa_1 \times \ldots \times \kappa_n)
adamc@530	527 }$$
adamc@530	528
adamc@655	529 $$\infer{\Gamma \vdash \lambda [c_1 \sim c_2] \Rightarrow \tau :: \mt{Type}}{
adamc@655	530 \Gamma \vdash c_1 :: \{\kappa\}
adamc@530	531 & \Gamma \vdash c_2 :: \{\kappa'\}
adamc@655	532 & \Gamma, c_1 \sim c_2 \vdash \tau :: \mt{Type}
adamc@530	533 }$$
adamc@530	534
adamc@531	535 \subsection{Record Disjointness}
adamc@531	536
adamc@531	537 $$\infer{\Gamma \vdash c_1 \sim c_2}{
adamc@558	538 \Gamma \vdash c_1 \hookrightarrow C_1
adamc@558	539 & \Gamma \vdash c_2 \hookrightarrow C_2
adamc@558	540 & \forall c'_1 \in C_1, c'_2 \in C_2: \Gamma \vdash c'_1 \sim c'_2
adamc@531	541 }
adamc@531	542 \quad \infer{\Gamma \vdash X \sim X'}{
adamc@531	543 X \neq X'
adamc@531	544 }$$
adamc@531	545
adamc@531	546 $$\infer{\Gamma \vdash c_1 \sim c_2}{
adamc@531	547 c'_1 \sim c'_2 \in \Gamma
adamc@558	548 & \Gamma \vdash c'_1 \hookrightarrow C_1
adamc@558	549 & \Gamma \vdash c'_2 \hookrightarrow C_2
adamc@558	550 & c_1 \in C_1
adamc@558	551 & c_2 \in C_2
adamc@531	552 }$$
adamc@531	553
adamc@531	554 $$\infer{\Gamma \vdash c \hookrightarrow \{c\}}{}
adamc@531	555 \quad \infer{\Gamma \vdash [\overline{c = c'}] \hookrightarrow \{\overline{c}\}}{}
adamc@531	556 \quad \infer{\Gamma \vdash c_1 \rc c_2 \hookrightarrow C_1 \cup C_2}{
adamc@531	557 \Gamma \vdash c_1 \hookrightarrow C_1
adamc@531	558 & \Gamma \vdash c_2 \hookrightarrow C_2
adamc@531	559 }
adamc@531	560 \quad \infer{\Gamma \vdash c \hookrightarrow C}{
adamc@531	561 \Gamma \vdash c \equiv c'
adamc@531	562 & \Gamma \vdash c' \hookrightarrow C
adamc@531	563 }
adamc@531	564 \quad \infer{\Gamma \vdash \mt{map} \; f \; c \hookrightarrow C}{
adamc@531	565 \Gamma \vdash c \hookrightarrow C
adamc@531	566 }$$
adamc@531	567
adamc@541	568 \subsection{\label{definitional}Definitional Equality}
adamc@532	569
adamc@655	570 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	571
adamc@532	572 $$\infer{\Gamma \vdash c \equiv c}{}
adamc@532	573 \quad \infer{\Gamma \vdash c_1 \equiv c_2}{
adamc@532	574 \Gamma \vdash c_2 \equiv c_1
adamc@532	575 }
adamc@532	576 \quad \infer{\Gamma \vdash c_1 \equiv c_3}{
adamc@532	577 \Gamma \vdash c_1 \equiv c_2
adamc@532	578 & \Gamma \vdash c_2 \equiv c_3
adamc@532	579 }
adamc@532	580 \quad \infer{\Gamma \vdash \mathcal C[c_1] \equiv \mathcal C[c_2]}{
adamc@532	581 \Gamma \vdash c_1 \equiv c_2
adamc@532	582 }$$
adamc@532	583
adamc@532	584 $$\infer{\Gamma \vdash x \equiv c}{
adamc@532	585 x :: \kappa = c \in \Gamma
adamc@532	586 }
adamc@532	587 \quad \infer{\Gamma \vdash M.x \equiv c}{
adamc@537	588 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	589 & \mt{proj}(M, \overline{s}, \mt{con} \; x) = (\kappa, c)
adamc@532	590 }
adamc@532	591 \quad \infer{\Gamma \vdash (\overline c).i \equiv c_i}{}$$
adamc@532	592
adamc@532	593 $$\infer{\Gamma \vdash (\lambda x :: \kappa \Rightarrow c) \; c' \equiv [x \mapsto c'] c}{}
adamc@655	594 \quad \infer{\Gamma \vdash (X \Longrightarrow c) [\kappa] \equiv [X \mapsto \kappa] c}{}$$
adamc@655	595
adamc@655	596 $$\infer{\Gamma \vdash c_1 \rc c_2 \equiv c_2 \rc c_1}{}
adamc@532	597 \quad \infer{\Gamma \vdash c_1 \rc (c_2 \rc c_3) \equiv (c_1 \rc c_2) \rc c_3}{}$$
adamc@532	598
adamc@532	599 $$\infer{\Gamma \vdash [] \rc c \equiv c}{}
adamc@532	600 \quad \infer{\Gamma \vdash [\overline{c_1 = c'_1}] \rc [\overline{c_2 = c'_2}] \equiv [\overline{c_1 = c'_1}, \overline{c_2 = c'_2}]}{}$$
adamc@532	601
adamc@655	602 $$\infer{\Gamma \vdash \mt{map} \; f \; [] \equiv []}{}
adamc@655	603 \quad \infer{\Gamma \vdash \mt{map} \; f \; ([c_1 = c_2] \rc c) \equiv [c_1 = f \; c_2] \rc \mt{map} \; f \; c}{}$$
adamc@532	604
adamc@532	605 $$\infer{\Gamma \vdash \mt{map} \; (\lambda x \Rightarrow x) \; c \equiv c}{}
adamc@655	606 \quad \infer{\Gamma \vdash \mt{map} \; f \; (\mt{map} \; f' \; c)
adamc@655	607 \equiv \mt{map} \; (\lambda x \Rightarrow f \; (f' \; x)) \; c}{}$$
adamc@532	608
adamc@532	609 $$\infer{\Gamma \vdash \mt{map} \; f \; (c_1 \rc c_2) \equiv \mt{map} \; f \; c_1 \rc \mt{map} \; f \; c_2}{}$$
adamc@531	610
adamc@534	611 \subsection{Expression Typing}
adamc@533	612
adamc@533	613 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 $\mt{Basis}.\mt{float}$, and string constants to $\mt{Basis}.\mt{string}$.
adamc@533	614
adamc@533	615 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	616
adamc@533	617 $$\infer{\Gamma \vdash e : \tau : \tau}{
adamc@533	618 \Gamma \vdash e : \tau
adamc@533	619 }
adamc@533	620 \quad \infer{\Gamma \vdash e : \tau}{
adamc@533	621 \Gamma \vdash e : \tau'
adamc@533	622 & \Gamma \vdash \tau' \equiv \tau
adamc@533	623 }
adamc@533	624 \quad \infer{\Gamma \vdash \ell : T(\ell)}{}$$
adamc@533	625
adamc@533	626 $$\infer{\Gamma \vdash x : \mathcal I(\tau)}{
adamc@533	627 x : \tau \in \Gamma
adamc@533	628 }
adamc@533	629 \quad \infer{\Gamma \vdash M.x : \mathcal I(\tau)}{
adamc@537	630 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	631 & \mt{proj}(M, \overline{s}, \mt{val} \; x) = \tau
adamc@533	632 }
adamc@533	633 \quad \infer{\Gamma \vdash X : \mathcal I(\tau)}{
adamc@533	634 X : \tau \in \Gamma
adamc@533	635 }
adamc@533	636 \quad \infer{\Gamma \vdash M.X : \mathcal I(\tau)}{
adamc@537	637 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	638 & \mt{proj}(M, \overline{s}, \mt{val} \; X) = \tau
adamc@533	639 }$$
adamc@533	640
adamc@533	641 $$\infer{\Gamma \vdash e_1 \; e_2 : \tau_2}{
adamc@533	642 \Gamma \vdash e_1 : \tau_1 \to \tau_2
adamc@533	643 & \Gamma \vdash e_2 : \tau_1
adamc@533	644 }
adamc@533	645 \quad \infer{\Gamma \vdash \lambda x : \tau_1 \Rightarrow e : \tau_1 \to \tau_2}{
adamc@533	646 \Gamma, x : \tau_1 \vdash e : \tau_2
adamc@533	647 }$$
adamc@533	648
adamc@533	649 $$\infer{\Gamma \vdash e [c] : [x \mapsto c]\tau}{
adamc@533	650 \Gamma \vdash e : x :: \kappa \to \tau
adamc@533	651 & \Gamma \vdash c :: \kappa
adamc@533	652 }
adamc@852	653 \quad \infer{\Gamma \vdash \lambda [x \; ? \; \kappa] \Rightarrow e : x \; ? \; \kappa \to \tau}{
adamc@533	654 \Gamma, x :: \kappa \vdash e : \tau
adamc@533	655 }$$
adamc@533	656
adamc@655	657 $$\infer{\Gamma \vdash e [\kappa] : [X \mapsto \kappa]\tau}{
adamc@655	658 \Gamma \vdash e : X \longrightarrow \tau
adamc@655	659 & \Gamma \vdash \kappa
adamc@655	660 }
adamc@655	661 \quad \infer{\Gamma \vdash X \Longrightarrow e : X \longrightarrow \tau}{
adamc@655	662 \Gamma, X \vdash e : \tau
adamc@655	663 }$$
adamc@655	664
adamc@533	665 $$\infer{\Gamma \vdash \{\overline{c = e}\} : \{\overline{c : \tau}\}}{
adamc@533	666 \forall i: \Gamma \vdash c_i :: \mt{Name}
adamc@533	667 & \Gamma \vdash e_i : \tau_i
adamc@533	668 & \forall i \neq j: \Gamma \vdash c_i \sim c_j
adamc@533	669 }
adamc@533	670 \quad \infer{\Gamma \vdash e.c : \tau}{
adamc@533	671 \Gamma \vdash e : \$([c = \tau] \rc c')
adamc@533	672 }
adamc@533	673 \quad \infer{\Gamma \vdash e_1 \rc e_2 : \$(c_1 \rc c_2)}{
adamc@533	674 \Gamma \vdash e_1 : \$c_1
adamc@533	675 & \Gamma \vdash e_2 : \$c_2
adamc@573	676 & \Gamma \vdash c_1 \sim c_2
adamc@533	677 }$$
adamc@533	678
adamc@533	679 $$\infer{\Gamma \vdash e \rcut c : \$c'}{
adamc@533	680 \Gamma \vdash e : \$([c = \tau] \rc c')
adamc@533	681 }
adamc@533	682 \quad \infer{\Gamma \vdash e \rcutM c : \$c'}{
adamc@533	683 \Gamma \vdash e : \$(c \rc c')
adamc@533	684 }$$
adamc@533	685
adamc@533	686 $$\infer{\Gamma \vdash \mt{let} \; \overline{ed} \; \mt{in} \; e \; \mt{end} : \tau}{
adamc@533	687 \Gamma \vdash \overline{ed} \leadsto \Gamma'
adamc@533	688 & \Gamma' \vdash e : \tau
adamc@533	689 }
adamc@533	690 \quad \infer{\Gamma \vdash \mt{case} \; e \; \mt{of} \; \overline{p \Rightarrow e} : \tau}{
adamc@533	691 \forall i: \Gamma \vdash p_i \leadsto \Gamma_i, \tau'
adamc@533	692 & \Gamma_i \vdash e_i : \tau
adamc@533	693 }$$
adamc@533	694
adamc@573	695 $$\infer{\Gamma \vdash \lambda [c_1 \sim c_2] \Rightarrow e : \lambda [c_1 \sim c_2] \Rightarrow \tau}{
adamc@533	696 \Gamma \vdash c_1 :: \{\kappa\}
adamc@655	697 & \Gamma \vdash c_2 :: \{\kappa'\}
adamc@533	698 & \Gamma, c_1 \sim c_2 \vdash e : \tau
adamc@662	699 }
adamc@662	700 \quad \infer{\Gamma \vdash e \; ! : \tau}{
adamc@662	701 \Gamma \vdash e : [c_1 \sim c_2] \Rightarrow \tau
adamc@662	702 & \Gamma \vdash c_1 \sim c_2
adamc@533	703 }$$
adamc@533	704
adamc@534	705 \subsection{Pattern Typing}
adamc@534	706
adamc@534	707 $$\infer{\Gamma \vdash \_ \leadsto \Gamma; \tau}{}
adamc@534	708 \quad \infer{\Gamma \vdash x \leadsto \Gamma, x : \tau; \tau}{}
adamc@534	709 \quad \infer{\Gamma \vdash \ell \leadsto \Gamma; T(\ell)}{}$$
adamc@534	710
adamc@534	711 $$\infer{\Gamma \vdash X \leadsto \Gamma; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@534	712 X : \overline{x ::: \mt{Type}} \to \tau \in \Gamma
adamc@534	713 & \textrm{$\tau$ not a function type}
adamc@534	714 }
adamc@534	715 \quad \infer{\Gamma \vdash X \; p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@534	716 X : \overline{x ::: \mt{Type}} \to \tau'' \to \tau \in \Gamma
adamc@534	717 & \Gamma \vdash p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau''
adamc@534	718 }$$
adamc@534	719
adamc@534	720 $$\infer{\Gamma \vdash M.X \leadsto \Gamma; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@537	721 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	722 & \mt{proj}(M, \overline{s}, \mt{val} \; X) = \overline{x ::: \mt{Type}} \to \tau
adamc@534	723 & \textrm{$\tau$ not a function type}
adamc@534	724 }$$
adamc@534	725
adamc@534	726 $$\infer{\Gamma \vdash M.X \; p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@537	727 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	728 & \mt{proj}(M, \overline{s}, \mt{val} \; X) = \overline{x ::: \mt{Type}} \to \tau'' \to \tau
adamc@534	729 & \Gamma \vdash p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau''
adamc@534	730 }$$
adamc@534	731
adamc@534	732 $$\infer{\Gamma \vdash \{\overline{x = p}\} \leadsto \Gamma_n; \{\overline{x = \tau}\}}{
adamc@534	733 \Gamma_0 = \Gamma
adamc@534	734 & \forall i: \Gamma_i \vdash p_i \leadsto \Gamma_{i+1}; \tau_i
adamc@534	735 }
adamc@534	736 \quad \infer{\Gamma \vdash \{\overline{x = p}, \ldots\} \leadsto \Gamma_n; \$([\overline{x = \tau}] \rc c)}{
adamc@534	737 \Gamma_0 = \Gamma
adamc@534	738 & \forall i: \Gamma_i \vdash p_i \leadsto \Gamma_{i+1}; \tau_i
adamc@534	739 }$$
adamc@534	740
adamc@852	741 $$\infer{\Gamma \vdash p : \tau \leadsto \Gamma'; \tau}{
adamc@852	742 \Gamma \vdash p \leadsto \Gamma'; \tau'
adamc@852	743 & \Gamma \vdash \tau' \equiv \tau
adamc@852	744 }$$
adamc@852	745
adamc@535	746 \subsection{Declaration Typing}
adamc@535	747
adamc@535	748 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	749
adamc@655	750 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	751
adamc@558	752 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	753 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	754
adamc@535	755 $$\infer{\Gamma \vdash \cdot \leadsto \Gamma}{}
adamc@535	756 \quad \infer{\Gamma \vdash d, \overline{d} \leadsto \Gamma''}{
adamc@535	757 \Gamma \vdash d \leadsto \Gamma'
adamc@535	758 & \Gamma' \vdash \overline{d} \leadsto \Gamma''
adamc@535	759 }$$
adamc@535	760
adamc@535	761 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa = c \leadsto \Gamma, x :: \kappa = c}{
adamc@535	762 \Gamma \vdash c :: \kappa
adamc@535	763 }
adamc@535	764 \quad \infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leadsto \Gamma'}{
adamc@535	765 \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} \vdash \overline{dc} \leadsto \Gamma'
adamc@535	766 }$$
adamc@535	767
adamc@535	768 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leadsto \Gamma'}{
adamc@537	769 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	770 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@535	771 & \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} = M.z \vdash \overline{dc} \leadsto \Gamma'
adamc@535	772 }$$
adamc@535	773
adamc@535	774 $$\infer{\Gamma \vdash \mt{val} \; x : \tau = e \leadsto \Gamma, x : \tau}{
adamc@535	775 \Gamma \vdash e : \tau
adamc@535	776 }$$
adamc@535	777
adamc@535	778 $$\infer{\Gamma \vdash \mt{val} \; \mt{rec} \; \overline{x : \tau = e} \leadsto \Gamma, \overline{x : \tau}}{
adamc@535	779 \forall i: \Gamma, \overline{x : \tau} \vdash e_i : \tau_i
adamc@535	780 & \textrm{$e_i$ starts with an expression $\lambda$, optionally preceded by constructor and disjointness $\lambda$s}
adamc@535	781 }$$
adamc@535	782
adamc@535	783 $$\infer{\Gamma \vdash \mt{structure} \; X : S = M \leadsto \Gamma, X : S}{
adamc@535	784 \Gamma \vdash M : S
adamc@558	785 & \textrm{ $M$ not a constant or application}
adamc@535	786 }
adamc@558	787 \quad \infer{\Gamma \vdash \mt{structure} \; X : S = M \leadsto \Gamma, X : \mt{selfify}(X, \overline{s})}{
adamc@558	788 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@539	789 }$$
adamc@539	790
adamc@539	791 $$\infer{\Gamma \vdash \mt{signature} \; X = S \leadsto \Gamma, X = S}{
adamc@535	792 \Gamma \vdash S
adamc@535	793 }$$
adamc@535	794
adamc@537	795 $$\infer{\Gamma \vdash \mt{open} \; M \leadsto \Gamma, \mathcal O(M, \overline{s})}{
adamc@537	796 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@535	797 }$$
adamc@535	798
adamc@535	799 $$\infer{\Gamma \vdash \mt{constraint} \; c_1 \sim c_2 \leadsto \Gamma}{
adamc@535	800 \Gamma \vdash c_1 :: \{\kappa\}
adamc@535	801 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@535	802 & \Gamma \vdash c_1 \sim c_2
adamc@535	803 }
adamc@537	804 \quad \infer{\Gamma \vdash \mt{open} \; \mt{constraints} \; M \leadsto \Gamma, \mathcal O_c(M, \overline{s})}{
adamc@537	805 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@535	806 }$$
adamc@535	807
adamc@784	808 $$\infer{\Gamma \vdash \mt{table} \; x : c \leadsto \Gamma, x : \mt{Basis}.\mt{sql\_table} \; c \; []}{
adamc@535	809 \Gamma \vdash c :: \{\mt{Type}\}
adamc@535	810 }
adamc@784	811 \quad \infer{\Gamma \vdash \mt{view} \; x : c \leadsto \Gamma, x : \mt{Basis}.\mt{sql\_view} \; c}{
adamc@784	812 \Gamma \vdash c :: \{\mt{Type}\}
adamc@784	813 }$$
adamc@784	814
adamc@784	815 $$\infer{\Gamma \vdash \mt{sequence} \; x \leadsto \Gamma, x : \mt{Basis}.\mt{sql\_sequence}}{}$$
adamc@535	816
adamc@535	817 $$\infer{\Gamma \vdash \mt{cookie} \; x : \tau \leadsto \Gamma, x : \mt{Basis}.\mt{http\_cookie} \; \tau}{
adamc@535	818 \Gamma \vdash \tau :: \mt{Type}
adamc@784	819 }
adamc@784	820 \quad \infer{\Gamma \vdash \mt{style} \; x \leadsto \Gamma, x : \mt{Basis}.\mt{css\_class}}{}$$
adamc@535	821
adamc@784	822 $$\infer{\Gamma \vdash \mt{class} \; x :: \kappa = c \leadsto \Gamma, x :: \kappa = c}{
adamc@784	823 \Gamma \vdash c :: \kappa
adamc@535	824 }$$
adamc@535	825
adamc@535	826 $$\infer{\overline{y}; x; \Gamma \vdash \cdot \leadsto \Gamma}{}
adamc@535	827 \quad \infer{\overline{y}; x; \Gamma \vdash X \mid \overline{dc} \leadsto \Gamma', X : \overline{y ::: \mt{Type}} \to x \; \overline{y}}{
adamc@535	828 \overline{y}; x; \Gamma \vdash \overline{dc} \leadsto \Gamma'
adamc@535	829 }
adamc@535	830 \quad \infer{\overline{y}; x; \Gamma \vdash X \; \mt{of} \; \tau \mid \overline{dc} \leadsto \Gamma', X : \overline{y ::: \mt{Type}} \to \tau \to x \; \overline{y}}{
adamc@535	831 \overline{y}; x; \Gamma \vdash \overline{dc} \leadsto \Gamma'
adamc@535	832 }$$
adamc@535	833
adamc@537	834 \subsection{Signature Item Typing}
adamc@537	835
adamc@537	836 We appeal to a signature item analogue of the $\mathcal O$ function from the last subsection.
adamc@537	837
adamc@537	838 $$\infer{\Gamma \vdash \cdot \leadsto \Gamma}{}
adamc@537	839 \quad \infer{\Gamma \vdash s, \overline{s} \leadsto \Gamma''}{
adamc@537	840 \Gamma \vdash s \leadsto \Gamma'
adamc@537	841 & \Gamma' \vdash \overline{s} \leadsto \Gamma''
adamc@537	842 }$$
adamc@537	843
adamc@537	844 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa \leadsto \Gamma, x :: \kappa}{}
adamc@537	845 \quad \infer{\Gamma \vdash \mt{con} \; x :: \kappa = c \leadsto \Gamma, x :: \kappa = c}{
adamc@537	846 \Gamma \vdash c :: \kappa
adamc@537	847 }
adamc@537	848 \quad \infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leadsto \Gamma'}{
adamc@537	849 \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} \vdash \overline{dc} \leadsto \Gamma'
adamc@537	850 }$$
adamc@537	851
adamc@537	852 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leadsto \Gamma'}{
adamc@537	853 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	854 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@537	855 & \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} = M.z \vdash \overline{dc} \leadsto \Gamma'
adamc@537	856 }$$
adamc@537	857
adamc@537	858 $$\infer{\Gamma \vdash \mt{val} \; x : \tau \leadsto \Gamma, x : \tau}{
adamc@537	859 \Gamma \vdash \tau :: \mt{Type}
adamc@537	860 }$$
adamc@537	861
adamc@537	862 $$\infer{\Gamma \vdash \mt{structure} \; X : S \leadsto \Gamma, X : S}{
adamc@537	863 \Gamma \vdash S
adamc@537	864 }
adamc@537	865 \quad \infer{\Gamma \vdash \mt{signature} \; X = S \leadsto \Gamma, X = S}{
adamc@537	866 \Gamma \vdash S
adamc@537	867 }$$
adamc@537	868
adamc@537	869 $$\infer{\Gamma \vdash \mt{include} \; S \leadsto \Gamma, \mathcal O(\overline{s})}{
adamc@537	870 \Gamma \vdash S
adamc@537	871 & \Gamma \vdash S \equiv \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	872 }$$
adamc@537	873
adamc@537	874 $$\infer{\Gamma \vdash \mt{constraint} \; c_1 \sim c_2 \leadsto \Gamma, c_1 \sim c_2}{
adamc@537	875 \Gamma \vdash c_1 :: \{\kappa\}
adamc@537	876 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@537	877 }$$
adamc@537	878
adamc@784	879 $$\infer{\Gamma \vdash \mt{class} \; x :: \kappa = c \leadsto \Gamma, x :: \kappa = c}{
adamc@784	880 \Gamma \vdash c :: \kappa
adamc@537	881 }
adamc@784	882 \quad \infer{\Gamma \vdash \mt{class} \; x :: \kappa \leadsto \Gamma, x :: \kappa}{}$$
adamc@537	883
adamc@536	884 \subsection{Signature Compatibility}
adamc@536	885
adamc@558	886 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	887
adamc@537	888 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	889
adamc@536	890 $$\infer{\Gamma \vdash S \equiv S}{}
adamc@536	891 \quad \infer{\Gamma \vdash S_1 \equiv S_2}{
adamc@536	892 \Gamma \vdash S_2 \equiv S_1
adamc@536	893 }
adamc@536	894 \quad \infer{\Gamma \vdash X \equiv S}{
adamc@536	895 X = S \in \Gamma
adamc@536	896 }
adamc@536	897 \quad \infer{\Gamma \vdash M.X \equiv S}{
adamc@537	898 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	899 & \mt{proj}(M, \overline{s}, \mt{signature} \; X) = S
adamc@536	900 }$$
adamc@536	901
adamc@536	902 $$\infer{\Gamma \vdash S \; \mt{where} \; \mt{con} \; x = c \equiv \mt{sig} \; \overline{s^1} \; \mt{con} \; x :: \kappa = c \; \overline{s_2} \; \mt{end}}{
adamc@536	903 \Gamma \vdash S \equiv \mt{sig} \; \overline{s^1} \; \mt{con} \; x :: \kappa \; \overline{s_2} \; \mt{end}
adamc@536	904 & \Gamma \vdash c :: \kappa
adamc@537	905 }
adamc@537	906 \quad \infer{\Gamma \vdash \mt{sig} \; \overline{s^1} \; \mt{include} \; S \; \overline{s^2} \; \mt{end} \equiv \mt{sig} \; \overline{s^1} \; \overline{s} \; \overline{s^2} \; \mt{end}}{
adamc@537	907 \Gamma \vdash S \equiv \mt{sig} \; \overline{s} \; \mt{end}
adamc@536	908 }$$
adamc@536	909
adamc@536	910 $$\infer{\Gamma \vdash S_1 \leq S_2}{
adamc@536	911 \Gamma \vdash S_1 \equiv S_2
adamc@536	912 }
adamc@536	913 \quad \infer{\Gamma \vdash \mt{sig} \; \overline{s} \; \mt{end} \leq \mt{sig} \; \mt{end}}{}
adamc@537	914 \quad \infer{\Gamma \vdash \mt{sig} \; \overline{s} \; \mt{end} \leq \mt{sig} \; s' \; \overline{s'} \; \mt{end}}{
adamc@537	915 \Gamma \vdash \overline{s} \leq s'
adamc@537	916 & \Gamma \vdash s' \leadsto \Gamma'
adamc@537	917 & \Gamma' \vdash \mt{sig} \; \overline{s} \; \mt{end} \leq \mt{sig} \; \overline{s'} \; \mt{end}
adamc@537	918 }$$
adamc@537	919
adamc@537	920 $$\infer{\Gamma \vdash s \; \overline{s} \leq s'}{
adamc@537	921 \Gamma \vdash s \leq s'
adamc@537	922 }
adamc@537	923 \quad \infer{\Gamma \vdash s \; \overline{s} \leq s'}{
adamc@537	924 \Gamma \vdash s \leadsto \Gamma'
adamc@537	925 & \Gamma' \vdash \overline{s} \leq s'
adamc@536	926 }$$
adamc@536	927
adamc@536	928 $$\infer{\Gamma \vdash \mt{functor} (X : S_1) : S_2 \leq \mt{functor} (X : S'_1) : S'_2}{
adamc@536	929 \Gamma \vdash S'_1 \leq S_1
adamc@536	930 & \Gamma, X : S'_1 \vdash S_2 \leq S'_2
adamc@536	931 }$$
adamc@536	932
adamc@537	933 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa \leq \mt{con} \; x :: \kappa}{}
adamc@537	934 \quad \infer{\Gamma \vdash \mt{con} \; x :: \kappa = c \leq \mt{con} \; x :: \kappa}{}
adamc@558	935 \quad \infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leq \mt{con} \; x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type}}{}$$
adamc@537	936
adamc@537	937 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leq \mt{con} \; x :: \mt{Type}^{\mt{len}(y)} \to \mt{Type}}{
adamc@537	938 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	939 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@537	940 }$$
adamc@537	941
adamc@784	942 $$\infer{\Gamma \vdash \mt{class} \; x :: \kappa \leq \mt{con} \; x :: \kappa}{}
adamc@784	943 \quad \infer{\Gamma \vdash \mt{class} \; x :: \kappa = c \leq \mt{con} \; x :: \kappa}{}$$
adamc@537	944
adamc@537	945 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa = c_1 \leq \mt{con} \; x :: \mt{\kappa} = c_2}{
adamc@537	946 \Gamma \vdash c_1 \equiv c_2
adamc@537	947 }
adamc@784	948 \quad \infer{\Gamma \vdash \mt{class} \; x :: \kappa = c_1 \leq \mt{con} \; x :: \kappa = c_2}{
adamc@537	949 \Gamma \vdash c_1 \equiv c_2
adamc@537	950 }$$
adamc@537	951
adamc@537	952 $$\infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leq \mt{datatype} \; x \; \overline{y} = \overline{dc'}}{
adamc@537	953 \Gamma, \overline{y :: \mt{Type}} \vdash \overline{dc} \leq \overline{dc'}
adamc@537	954 }$$
adamc@537	955
adamc@537	956 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leq \mt{datatype} \; x \; \overline{y} = \overline{dc'}}{
adamc@537	957 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	958 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@537	959 & \Gamma, \overline{y :: \mt{Type}} \vdash \overline{dc} \leq \overline{dc'}
adamc@537	960 }$$
adamc@537	961
adamc@537	962 $$\infer{\Gamma \vdash \cdot \leq \cdot}{}
adamc@537	963 \quad \infer{\Gamma \vdash X; \overline{dc} \leq X; \overline{dc'}}{
adamc@537	964 \Gamma \vdash \overline{dc} \leq \overline{dc'}
adamc@537	965 }
adamc@537	966 \quad \infer{\Gamma \vdash X \; \mt{of} \; \tau_1; \overline{dc} \leq X \; \mt{of} \; \tau_2; \overline{dc'}}{
adamc@537	967 \Gamma \vdash \tau_1 \equiv \tau_2
adamc@537	968 & \Gamma \vdash \overline{dc} \leq \overline{dc'}
adamc@537	969 }$$
adamc@537	970
adamc@537	971 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leq \mt{datatype} \; x = \mt{datatype} \; M'.z'}{
adamc@537	972 \Gamma \vdash M.z \equiv M'.z'
adamc@537	973 }$$
adamc@537	974
adamc@537	975 $$\infer{\Gamma \vdash \mt{val} \; x : \tau_1 \leq \mt{val} \; x : \tau_2}{
adamc@537	976 \Gamma \vdash \tau_1 \equiv \tau_2
adamc@537	977 }
adamc@537	978 \quad \infer{\Gamma \vdash \mt{structure} \; X : S_1 \leq \mt{structure} \; X : S_2}{
adamc@537	979 \Gamma \vdash S_1 \leq S_2
adamc@537	980 }
adamc@537	981 \quad \infer{\Gamma \vdash \mt{signature} \; X = S_1 \leq \mt{signature} \; X = S_2}{
adamc@537	982 \Gamma \vdash S_1 \leq S_2
adamc@537	983 & \Gamma \vdash S_2 \leq S_1
adamc@537	984 }$$
adamc@537	985
adamc@537	986 $$\infer{\Gamma \vdash \mt{constraint} \; c_1 \sim c_2 \leq \mt{constraint} \; c'_1 \sim c'_2}{
adamc@537	987 \Gamma \vdash c_1 \equiv c'_1
adamc@537	988 & \Gamma \vdash c_2 \equiv c'_2
adamc@537	989 }$$
adamc@537	990
adamc@655	991 $$\infer{\Gamma \vdash \mt{class} \; x :: \kappa \leq \mt{class} \; x :: \kappa}{}
adamc@655	992 \quad \infer{\Gamma \vdash \mt{class} \; x :: \kappa = c \leq \mt{class} \; x :: \kappa}{}
adamc@655	993 \quad \infer{\Gamma \vdash \mt{class} \; x :: \kappa = c_1 \leq \mt{class} \; x :: \kappa = c_2}{
adamc@537	994 \Gamma \vdash c_1 \equiv c_2
adamc@537	995 }$$
adamc@537	996
adamc@538	997 \subsection{Module Typing}
adamc@538	998
adamc@538	999 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	1000
adamc@538	1001 $$\infer{\Gamma \vdash M : S}{
adamc@538	1002 \Gamma \vdash M : S'
adamc@538	1003 & \Gamma \vdash S' \leq S
adamc@538	1004 }
adamc@538	1005 \quad \infer{\Gamma \vdash \mt{struct} \; \overline{d} \; \mt{end} : \mt{sig} \; \mt{sigOf}(\overline{d}) \; \mt{end}}{
adamc@538	1006 \Gamma \vdash \overline{d} \leadsto \Gamma'
adamc@538	1007 }
adamc@538	1008 \quad \infer{\Gamma \vdash X : S}{
adamc@538	1009 X : S \in \Gamma
adamc@538	1010 }$$
adamc@538	1011
adamc@538	1012 $$\infer{\Gamma \vdash M.X : S}{
adamc@538	1013 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@538	1014 & \mt{proj}(M, \overline{s}, \mt{structure} \; X) = S
adamc@538	1015 }$$
adamc@538	1016
adamc@538	1017 $$\infer{\Gamma \vdash M_1(M_2) : [X \mapsto M_2]S_2}{
adamc@538	1018 \Gamma \vdash M_1 : \mt{functor}(X : S_1) : S_2
adamc@538	1019 & \Gamma \vdash M_2 : S_1
adamc@538	1020 }
adamc@538	1021 \quad \infer{\Gamma \vdash \mt{functor} (X : S_1) : S_2 = M : \mt{functor} (X : S_1) : S_2}{
adamc@538	1022 \Gamma \vdash S_1
adamc@538	1023 & \Gamma, X : S_1 \vdash S_2
adamc@538	1024 & \Gamma, X : S_1 \vdash M : S_2
adamc@538	1025 }$$
adamc@538	1026
adamc@538	1027 \begin{eqnarray*}
adamc@538	1028 \mt{sigOf}(\cdot) &=& \cdot \\
adamc@538	1029 \mt{sigOf}(s \; \overline{s'}) &=& \mt{sigOf}(s) \; \mt{sigOf}(\overline{s'}) \\
adamc@538	1030 \\
adamc@538	1031 \mt{sigOf}(\mt{con} \; x :: \kappa = c) &=& \mt{con} \; x :: \kappa = c \\
adamc@538	1032 \mt{sigOf}(\mt{datatype} \; x \; \overline{y} = \overline{dc}) &=& \mt{datatype} \; x \; \overline{y} = \overline{dc} \\
adamc@538	1033 \mt{sigOf}(\mt{datatype} \; x = \mt{datatype} \; M.z) &=& \mt{datatype} \; x = \mt{datatype} \; M.z \\
adamc@538	1034 \mt{sigOf}(\mt{val} \; x : \tau = e) &=& \mt{val} \; x : \tau \\
adamc@538	1035 \mt{sigOf}(\mt{val} \; \mt{rec} \; \overline{x : \tau = e}) &=& \overline{\mt{val} \; x : \tau} \\
adamc@538	1036 \mt{sigOf}(\mt{structure} \; X : S = M) &=& \mt{structure} \; X : S \\
adamc@538	1037 \mt{sigOf}(\mt{signature} \; X = S) &=& \mt{signature} \; X = S \\
adamc@538	1038 \mt{sigOf}(\mt{open} \; M) &=& \mt{include} \; S \textrm{ (where $\Gamma \vdash M : S$)} \\
adamc@538	1039 \mt{sigOf}(\mt{constraint} \; c_1 \sim c_2) &=& \mt{constraint} \; c_1 \sim c_2 \\
adamc@538	1040 \mt{sigOf}(\mt{open} \; \mt{constraints} \; M) &=& \cdot \\
adamc@538	1041 \mt{sigOf}(\mt{table} \; x : c) &=& \mt{table} \; x : c \\
adamc@784	1042 \mt{sigOf}(\mt{view} \; x : c) &=& \mt{view} \; x : c \\
adamc@538	1043 \mt{sigOf}(\mt{sequence} \; x) &=& \mt{sequence} \; x \\
adamc@538	1044 \mt{sigOf}(\mt{cookie} \; x : \tau) &=& \mt{cookie} \; x : \tau \\
adamc@784	1045 \mt{sigOf}(\mt{style} \; x) &=& \mt{style} \; x \\
adamc@655	1046 \mt{sigOf}(\mt{class} \; x :: \kappa = c) &=& \mt{class} \; x :: \kappa = c \\
adamc@538	1047 \end{eqnarray*}
adamc@539	1048 \begin{eqnarray*}
adamc@539	1049 \mt{selfify}(M, \cdot) &=& \cdot \\
adamc@558	1050 \mt{selfify}(M, s \; \overline{s'}) &=& \mt{selfify}(M, s) \; \mt{selfify}(M, \overline{s'}) \\
adamc@539	1051 \\
adamc@539	1052 \mt{selfify}(M, \mt{con} \; x :: \kappa) &=& \mt{con} \; x :: \kappa = M.x \\
adamc@539	1053 \mt{selfify}(M, \mt{con} \; x :: \kappa = c) &=& \mt{con} \; x :: \kappa = c \\
adamc@539	1054 \mt{selfify}(M, \mt{datatype} \; x \; \overline{y} = \overline{dc}) &=& \mt{datatype} \; x \; \overline{y} = \mt{datatype} \; M.x \\
adamc@539	1055 \mt{selfify}(M, \mt{datatype} \; x = \mt{datatype} \; M'.z) &=& \mt{datatype} \; x = \mt{datatype} \; M'.z \\
adamc@539	1056 \mt{selfify}(M, \mt{val} \; x : \tau) &=& \mt{val} \; x : \tau \\
adamc@539	1057 \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	1058 \mt{selfify}(M, \mt{signature} \; X = S) &=& \mt{signature} \; X = S \\
adamc@539	1059 \mt{selfify}(M, \mt{include} \; S) &=& \mt{include} \; S \\
adamc@539	1060 \mt{selfify}(M, \mt{constraint} \; c_1 \sim c_2) &=& \mt{constraint} \; c_1 \sim c_2 \\
adamc@655	1061 \mt{selfify}(M, \mt{class} \; x :: \kappa) &=& \mt{class} \; x :: \kappa = M.x \\
adamc@655	1062 \mt{selfify}(M, \mt{class} \; x :: \kappa = c) &=& \mt{class} \; x :: \kappa = c \\
adamc@539	1063 \end{eqnarray*}
adamc@539	1064
adamc@540	1065 \subsection{Module Projection}
adamc@540	1066
adamc@540	1067 \begin{eqnarray*}
adamc@540	1068 \mt{proj}(M, \mt{con} \; x :: \kappa \; \overline{s}, \mt{con} \; x) &=& \kappa \\
adamc@540	1069 \mt{proj}(M, \mt{con} \; x :: \kappa = c \; \overline{s}, \mt{con} \; x) &=& (\kappa, c) \\
adamc@540	1070 \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	1071 \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	1072 && \textrm{and $\mt{proj}(M', \overline{s'}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})$)} \\
adamc@655	1073 \mt{proj}(M, \mt{class} \; x :: \kappa \; \overline{s}, \mt{con} \; x) &=& \kappa \to \mt{Type} \\
adamc@655	1074 \mt{proj}(M, \mt{class} \; x :: \kappa = c \; \overline{s}, \mt{con} \; x) &=& (\kappa \to \mt{Type}, c) \\
adamc@540	1075 \\
adamc@540	1076 \mt{proj}(M, \mt{datatype} \; x \; \overline{y} = \overline{dc} \; \overline{s}, \mt{datatype} \; x) &=& (\overline{y}, \overline{dc}) \\
adamc@540	1077 \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	1078 \\
adamc@540	1079 \mt{proj}(M, \mt{val} \; x : \tau \; \overline{s}, \mt{val} \; x) &=& \tau \\
adamc@540	1080 \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	1081 \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	1082 \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	1083 && \textrm{and $\mt{proj}(M', \overline{s'}, \mt{datatype} \; z = (\overline{y}, \overline{dc})$ and $X \in \overline{dc}$)} \\
adamc@540	1084 \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	1085 && \textrm{and $\mt{proj}(M', \overline{s'}, \mt{datatype} \; z = (\overline{y}, \overline{dc})$ and $X \; \mt{of} \; \tau \in \overline{dc}$)} \\
adamc@540	1086 \\
adamc@540	1087 \mt{proj}(M, \mt{structure} \; X : S \; \overline{s}, \mt{structure} \; X) &=& S \\
adamc@540	1088 \\
adamc@540	1089 \mt{proj}(M, \mt{signature} \; X = S \; \overline{s}, \mt{signature} \; X) &=& S \\
adamc@540	1090 \\
adamc@540	1091 \mt{proj}(M, \mt{con} \; x :: \kappa \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@540	1092 \mt{proj}(M, \mt{con} \; x :: \kappa = c \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@540	1093 \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	1094 \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	1095 \mt{proj}(M, \mt{val} \; x : \tau \; \overline{s}, V) &=& \mt{proj}(M, \overline{s}, V) \\
adamc@540	1096 \mt{proj}(M, \mt{structure} \; X : S \; \overline{s}, V) &=& [X \mapsto M.X]\mt{proj}(M, \overline{s}, V) \\
adamc@540	1097 \mt{proj}(M, \mt{signature} \; X = S \; \overline{s}, V) &=& [X \mapsto M.X]\mt{proj}(M, \overline{s}, V) \\
adamc@540	1098 \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	1099 \mt{proj}(M, \mt{constraint} \; c_1 \sim c_2 \; \overline{s}, V) &=& \mt{proj}(M, \overline{s}, V) \\
adamc@655	1100 \mt{proj}(M, \mt{class} \; x :: \kappa \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@655	1101 \mt{proj}(M, \mt{class} \; x :: \kappa = c \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@540	1102 \end{eqnarray*}
adamc@540	1103
adamc@541	1104
adamc@541	1105 \section{Type Inference}
adamc@541	1106
adamc@541	1107 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	1108
adamc@541	1109 \subsection{Basic Unification}
adamc@541	1110
adamc@560	1111 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	1112
adamc@656	1113 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	1114
adamc@541	1115 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	1116
adamc@541	1117 \subsection{Unifying Record Types}
adamc@541	1118
adamc@570	1119 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	1120
adamc@656	1121 \subsection{\label{typeclasses}Constructor Classes}
adamc@541	1122
adamc@784	1123 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	1124
adamc@784	1125 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	1126
adamc@656	1127 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	1128
adamc@656	1129 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	1130
adamc@541	1131 \subsection{Reverse-Engineering Record Types}
adamc@541	1132
adamc@656	1133 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	1134
adamc@541	1135 \subsection{Implicit Arguments in Functor Applications}
adamc@541	1136
adamc@656	1137 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	1138
adamc@541	1139
adamc@542	1140 \section{The Ur Standard Library}
adamc@542	1141
adamc@542	1142 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	1143
adamc@542	1144 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	1145
adamc@542	1146 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	1147 $$\begin{array}{l}
adamc@542	1148 \mt{type} \; \mt{int} \\
adamc@542	1149 \mt{type} \; \mt{float} \\
adamc@542	1150 \mt{type} \; \mt{string} \\
adamc@542	1151 \mt{type} \; \mt{time} \\
adamc@785	1152 \mt{type} \; \mt{blob} \\
adamc@542	1153 \\
adamc@542	1154 \mt{type} \; \mt{unit} = \{\} \\
adamc@542	1155 \\
adamc@542	1156 \mt{datatype} \; \mt{bool} = \mt{False} \mid \mt{True} \\
adamc@542	1157 \\
adamc@785	1158 \mt{datatype} \; \mt{option} \; \mt{t} = \mt{None} \mid \mt{Some} \; \mt{of} \; \mt{t} \\
adamc@785	1159 \\
adamc@785	1160 \mt{datatype} \; \mt{list} \; \mt{t} = \mt{Nil} \mid \mt{Cons} \; \mt{of} \; \mt{t} \times \mt{list} \; \mt{t}
adamc@542	1161 \end{array}$$
adamc@542	1162
adamc@785	1163 The only unusual element of this list is the $\mt{blob}$ type, which stands for binary sequences.
adamc@785	1164
adamc@657	1165 Another important generic Ur element comes at the beginning of \texttt{top.urs}.
adamc@657	1166
adamc@657	1167 $$\begin{array}{l}
adamc@657	1168 \mt{con} \; \mt{folder} :: \mt{K} \longrightarrow \{\mt{K}\} \to \mt{Type} \\
adamc@657	1169 \\
adamc@657	1170 \mt{val} \; \mt{fold} : \mt{K} \longrightarrow \mt{tf} :: (\{\mt{K}\} \to \mt{Type}) \\
adamc@657	1171 \hspace{.1in} \to (\mt{nm} :: \mt{Name} \to \mt{v} :: \mt{K} \to \mt{r} :: \{\mt{K}\} \to [[\mt{nm}] \sim \mt{r}] \Rightarrow \\
adamc@657	1172 \hspace{.2in} \mt{tf} \; \mt{r} \to \mt{tf} \; ([\mt{nm} = \mt{v}] \rc \mt{r})) \\
adamc@657	1173 \hspace{.1in} \to \mt{tf} \; [] \\
adamc@657	1174 \hspace{.1in} \to \mt{r} :: \{\mt{K}\} \to \mt{folder} \; \mt{r} \to \mt{tf} \; \mt{r}
adamc@657	1175 \end{array}$$
adamc@657	1176
adamc@657	1177 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	1178
adamc@664	1179 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	1180
adamc@542	1181
adamc@542	1182 \section{The Ur/Web Standard Library}
adamc@542	1183
adamc@658	1184 \subsection{Monads}
adamc@658	1185
adamc@658	1186 The Ur Basis defines the monad constructor class from Haskell.
adamc@658	1187
adamc@658	1188 $$\begin{array}{l}
adamc@658	1189 \mt{class} \; \mt{monad} :: \mt{Type} \to \mt{Type} \\
adamc@658	1190 \mt{val} \; \mt{return} : \mt{m} ::: (\mt{Type} \to \mt{Type}) \to \mt{t} ::: \mt{Type} \\
adamc@658	1191 \hspace{.1in} \to \mt{monad} \; \mt{m} \\
adamc@658	1192 \hspace{.1in} \to \mt{t} \to \mt{m} \; \mt{t} \\
adamc@658	1193 \mt{val} \; \mt{bind} : \mt{m} ::: (\mt{Type} \to \mt{Type}) \to \mt{t1} ::: \mt{Type} \to \mt{t2} ::: \mt{Type} \\
adamc@658	1194 \hspace{.1in} \to \mt{monad} \; \mt{m} \\
adamc@658	1195 \hspace{.1in} \to \mt{m} \; \mt{t1} \to (\mt{t1} \to \mt{m} \; \mt{t2}) \\
adamc@658	1196 \hspace{.1in} \to \mt{m} \; \mt{t2}
adamc@658	1197 \end{array}$$
adamc@658	1198
adamc@542	1199 \subsection{Transactions}
adamc@542	1200
adamc@542	1201 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	1202 $$\begin{array}{l}
adamc@542	1203 \mt{con} \; \mt{transaction} :: \mt{Type} \to \mt{Type} \\
adamc@658	1204 \mt{val} \; \mt{transaction\_monad} : \mt{monad} \; \mt{transaction}
adamc@542	1205 \end{array}$$
adamc@542	1206
adamc@542	1207 \subsection{HTTP}
adamc@542	1208
adamc@542	1209 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	1210 $$\begin{array}{l}
adamc@786	1211 \mt{val} \; \mt{requestHeader} : \mt{string} \to \mt{transaction} \; (\mt{option} \; \mt{string}) \\
adamc@786	1212 \\
adamc@786	1213 \mt{con} \; \mt{http\_cookie} :: \mt{Type} \to \mt{Type} \\
adamc@786	1214 \mt{val} \; \mt{getCookie} : \mt{t} ::: \mt{Type} \to \mt{http\_cookie} \; \mt{t} \to \mt{transaction} \; (\mt{option} \; \mt{t}) \\
adamc@786	1215 \mt{val} \; \mt{setCookie} : \mt{t} ::: \mt{Type} \to \mt{http\_cookie} \; \mt{t} \to \mt{t} \to \mt{transaction} \; \mt{unit}
adamc@786	1216 \end{array}$$
adamc@786	1217
adamc@786	1218 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	1219 $$\begin{array}{l}
adamc@786	1220 \mt{type} \; \mt{url} \\
adamc@786	1221 \mt{val} \; \mt{bless} : \mt{string} \to \mt{url} \\
adamc@786	1222 \mt{val} \; \mt{checkUrl} : \mt{string} \to \mt{option} \; \mt{url}
adamc@786	1223 \end{array}$$
adamc@786	1224 $\mt{bless}$ raises a runtime error if the string passed to it fails the URL policy.
adamc@786	1225
adamc@786	1226 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	1227 $$\begin{array}{l}
adamc@786	1228 \mt{type} \; \mt{file} \\
adamc@786	1229 \mt{val} \; \mt{fileName} : \mt{file} \to \mt{option} \; \mt{string} \\
adamc@786	1230 \mt{val} \; \mt{fileMimeType} : \mt{file} \to \mt{string} \\
adamc@786	1231 \mt{val} \; \mt{fileData} : \mt{file} \to \mt{blob}
adamc@786	1232 \end{array}$$
adamc@786	1233
adamc@786	1234 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	1235 $$\begin{array}{l}
adamc@786	1236 \mt{type} \; \mt{mimeType} \\
adamc@786	1237 \mt{val} \; \mt{blessMime} : \mt{string} \to \mt{mimeType} \\
adamc@786	1238 \mt{val} \; \mt{checkMime} : \mt{string} \to \mt{option} \; \mt{mimeType} \\
adamc@786	1239 \mt{val} \; \mt{returnBlob} : \mt{t} ::: \mt{Type} \to \mt{blob} \to \mt{mimeType} \to \mt{transaction} \; \mt{t}
adamc@542	1240 \end{array}$$
adamc@542	1241
adamc@543	1242 \subsection{SQL}
adamc@543	1243
adamc@543	1244 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	1245 $$\begin{array}{l}
adamc@785	1246 \mt{con} \; \mt{sql\_table} :: \{\mt{Type}\} \to \{\{\mt{Unit}\}\} \to \mt{Type}
adamc@785	1247 \end{array}$$
adamc@785	1248 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	1249
adamc@785	1250 We also have the simpler type family of SQL views, which have no keys.
adamc@785	1251 $$\begin{array}{l}
adamc@785	1252 \mt{con} \; \mt{sql\_view} :: \{\mt{Type}\} \to \mt{Type}
adamc@543	1253 \end{array}$$
adamc@543	1254
adamc@785	1255 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	1256 $$\begin{array}{l}
adamc@785	1257 \mt{class} \; \mt{fieldsOf} :: \mt{Type} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@785	1258 \mt{val} \; \mt{fieldsOf\_table} : \mt{fs} ::: \{\mt{Type}\} \to \mt{keys} ::: \{\{\mt{Unit}\}\} \to \mt{fieldsOf} \; (\mt{sql\_table} \; \mt{fs} \; \mt{keys}) \; \mt{fs} \\
adamc@785	1259 \mt{val} \; \mt{fieldsOf\_view} : \mt{fs} ::: \{\mt{Type}\} \to \mt{fieldsOf} \; (\mt{sql\_view} \; \mt{fs}) \; \mt{fs}
adamc@785	1260 \end{array}$$
adamc@785	1261
adamc@785	1262 \subsubsection{Table Constraints}
adamc@785	1263
adamc@785	1264 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	1265
adamc@785	1266 $$\begin{array}{l}
adamc@785	1267 \mt{con} \; \mt{primary\_key} :: \{\mt{Type}\} \to \{\{\mt{Unit}\}\} \to \mt{Type} \\
adamc@785	1268 \mt{val} \; \mt{no\_primary\_key} : \mt{fs} ::: \{\mt{Type}\} \to \mt{primary\_key} \; \mt{fs} \; [] \\
adamc@785	1269 \mt{val} \; \mt{primary\_key} : \mt{rest} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \to \mt{key1} :: \mt{Name} \to \mt{keys} :: \{\mt{Type}\} \\
adamc@785	1270 \hspace{.1in} \to [[\mt{key1}] \sim \mt{keys}] \Rightarrow [[\mt{key1} = \mt{t}] \rc \mt{keys} \sim \mt{rest}] \\
adamc@785	1271 \hspace{.1in} \Rightarrow \$([\mt{key1} = \mt{sql\_injectable\_prim} \; \mt{t}] \rc \mt{map} \; \mt{sql\_injectable\_prim} \; \mt{keys}) \\
adamc@785	1272 \hspace{.1in} \to \mt{primary\_key} \; ([\mt{key1} = \mt{t}] \rc \mt{keys} \rc \mt{rest}) \; [\mt{Pkey} = [\mt{key1}] \rc \mt{map} \; (\lambda \_ \Rightarrow ()) \; \mt{keys}]
adamc@785	1273 \end{array}$$
adamc@785	1274 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	1275
adamc@785	1276 A type family stands for sets of named constraints of the remaining varieties.
adamc@785	1277 $$\begin{array}{l}
adamc@785	1278 \mt{con} \; \mt{sql\_constraints} :: \{\mt{Type}\} \to \{\{\mt{Unit}\}\} \to \mt{Type}
adamc@785	1279 \end{array}$$
adamc@785	1280 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	1281
adamc@785	1282 There is a type family of individual, unnamed constraints.
adamc@785	1283 $$\begin{array}{l}
adamc@785	1284 \mt{con} \; \mt{sql\_constraint} :: \{\mt{Type}\} \to \{\mt{Unit}\} \to \mt{Type}
adamc@785	1285 \end{array}$$
adamc@785	1286 The first argument is the same as above, and the second argument gives the key columns for just this constraint.
adamc@785	1287
adamc@785	1288 We have operations for assembling constraints into constraint sets.
adamc@785	1289 $$\begin{array}{l}
adamc@785	1290 \mt{val} \; \mt{no\_constraint} : \mt{fs} ::: \{\mt{Type}\} \to \mt{sql\_constraints} \; \mt{fs} \; [] \\
adamc@785	1291 \mt{val} \; \mt{one\_constraint} : \mt{fs} ::: \{\mt{Type}\} \to \mt{unique} ::: \{\mt{Unit}\} \to \mt{name} :: \mt{Name} \\
adamc@785	1292 \hspace{.1in} \to \mt{sql\_constraint} \; \mt{fs} \; \mt{unique} \to \mt{sql\_constraints} \; \mt{fs} \; [\mt{name} = \mt{unique}] \\
adamc@785	1293 \mt{val} \; \mt{join\_constraints} : \mt{fs} ::: \{\mt{Type}\} \to \mt{uniques1} ::: \{\{\mt{Unit}\}\} \to \mt{uniques2} ::: \{\{\mt{Unit}\}\} \to [\mt{uniques1} \sim \mt{uniques2}] \\
adamc@785	1294 \hspace{.1in} \Rightarrow \mt{sql\_constraints} \; \mt{fs} \; \mt{uniques1} \to \mt{sql\_constraints} \; \mt{fs} \; \mt{uniques2} \to \mt{sql\_constraints} \; \mt{fs} \; (\mt{uniques1} \rc \mt{uniques2})
adamc@785	1295 \end{array}$$
adamc@785	1296
adamc@785	1297 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	1298 $$\begin{array}{l}
adamc@785	1299 \mt{val} \; \mt{unique} : \mt{rest} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \to \mt{unique1} :: \mt{Name} \to \mt{unique} :: \{\mt{Type}\} \\
adamc@785	1300 \hspace{.1in} \to [[\mt{unique1}] \sim \mt{unique}] \Rightarrow [[\mt{unique1} = \mt{t}] \rc \mt{unique} \sim \mt{rest}] \\
adamc@785	1301 \hspace{.1in} \Rightarrow \mt{sql\_constraint} \; ([\mt{unique1} = \mt{t}] \rc \mt{unique} \rc \mt{rest}) \; ([\mt{unique1}] \rc \mt{map} \; (\lambda \_ \Rightarrow ()) \; \mt{unique})
adamc@785	1302 \end{array}$$
adamc@785	1303
adamc@785	1304 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	1305 $$\begin{array}{l}
adamc@785	1306 \mt{class} \; \mt{linkable} :: \mt{Type} \to \mt{Type} \to \mt{Type} \\
adamc@785	1307 \mt{val} \; \mt{linkable\_same} : \mt{t} ::: \mt{Type} \to \mt{linkable} \; \mt{t} \; \mt{t} \\
adamc@785	1308 \mt{val} \; \mt{linkable\_from\_nullable} : \mt{t} ::: \mt{Type} \to \mt{linkable} \; (\mt{option} \; \mt{t}) \; \mt{t} \\
adamc@785	1309 \mt{val} \; \mt{linkable\_to\_nullable} : \mt{t} ::: \mt{Type} \to \mt{linkable} \; \mt{t} \; (\mt{option} \; \mt{t})
adamc@785	1310 \end{array}$$
adamc@785	1311
adamc@785	1312 The $\mt{matching}$ type family uses $\mt{linkable}$ to define when two keys match up type-wise.
adamc@785	1313 $$\begin{array}{l}
adamc@785	1314 \mt{con} \; \mt{matching} :: \{\mt{Type}\} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@785	1315 \mt{val} \; \mt{mat\_nil} : \mt{matching} \; [] \; [] \\
adamc@785	1316 \mt{val} \; \mt{mat\_cons} : \mt{t1} ::: \mt{Type} \to \mt{rest1} ::: \{\mt{Type}\} \to \mt{t2} ::: \mt{Type} \to \mt{rest2} ::: \{\mt{Type}\} \to \mt{nm1} :: \mt{Name} \to \mt{nm2} :: \mt{Name} \\
adamc@785	1317 \hspace{.1in} \to [[\mt{nm1}] \sim \mt{rest1}] \Rightarrow [[\mt{nm2}] \sim \mt{rest2}] \Rightarrow \mt{linkable} \; \mt{t1} \; \mt{t2} \to \mt{matching} \; \mt{rest1} \; \mt{rest2} \\
adamc@785	1318 \hspace{.1in} \to \mt{matching} \; ([\mt{nm1} = \mt{t1}] \rc \mt{rest1}) \; ([\mt{nm2} = \mt{t2}] \rc \mt{rest2})
adamc@785	1319 \end{array}$$
adamc@785	1320
adamc@785	1321 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	1322 $$\begin{array}{l}
adamc@785	1323 \mt{con} \; \mt{propagation\_mode} :: \{\mt{Type}\} \to \mt{Type} \\
adamc@785	1324 \mt{val} \; \mt{restrict} : \mt{fs} ::: \{\mt{Type}\} \to \mt{propagation\_mode} \; \mt{fs} \\
adamc@785	1325 \mt{val} \; \mt{cascade} : \mt{fs} ::: \{\mt{Type}\} \to \mt{propagation\_mode} \; \mt{fs} \\
adamc@785	1326 \mt{val} \; \mt{no\_action} : \mt{fs} ::: \{\mt{Type}\} \to \mt{propagation\_mode} \; \mt{fs} \\
adamc@785	1327 \mt{val} \; \mt{set\_null} : \mt{fs} ::: \{\mt{Type}\} \to \mt{propagation\_mode} \; (\mt{map} \; \mt{option} \; \mt{fs})
adamc@785	1328 \end{array}$$
adamc@785	1329
adamc@785	1330 Finally, we put these ingredient together to define the \texttt{FOREIGN KEY} constraint function.
adamc@785	1331 $$\begin{array}{l}
adamc@785	1332 \mt{val} \; \mt{foreign\_key} : \mt{mine1} ::: \mt{Name} \to \mt{t} ::: \mt{Type} \to \mt{mine} ::: \{\mt{Type}\} \to \mt{munused} ::: \{\mt{Type}\} \to \mt{foreign} ::: \{\mt{Type}\} \\
adamc@785	1333 \hspace{.1in} \to \mt{funused} ::: \{\mt{Type}\} \to \mt{nm} ::: \mt{Name} \to \mt{uniques} ::: \{\{\mt{Unit}\}\} \\
adamc@785	1334 \hspace{.1in} \to [[\mt{mine1}] \sim \mt{mine}] \Rightarrow [[\mt{mine1} = \mt{t}] \rc \mt{mine} \sim \mt{munused}] \Rightarrow [\mt{foreign} \sim \mt{funused}] \Rightarrow [[\mt{nm}] \sim \mt{uniques}] \\
adamc@785	1335 \hspace{.1in} \Rightarrow \mt{matching} \; ([\mt{mine1} = \mt{t}] \rc \mt{mine}) \; \mt{foreign} \\
adamc@785	1336 \hspace{.1in} \to \mt{sql\_table} \; (\mt{foreign} \rc \mt{funused}) \; ([\mt{nm} = \mt{map} \; (\lambda \_ \Rightarrow ()) \; \mt{foreign}] \rc \mt{uniques}) \\
adamc@785	1337 \hspace{.1in} \to \{\mt{OnDelete} : \mt{propagation\_mode} \; ([\mt{mine1} = \mt{t}] \rc \mt{mine}), \\
adamc@785	1338 \hspace{.2in} \mt{OnUpdate} : \mt{propagation\_mode} \; ([\mt{mine1} = \mt{t}] \rc \mt{mine})\} \\
adamc@785	1339 \hspace{.1in} \to \mt{sql\_constraint} \; ([\mt{mine1} = \mt{t}] \rc \mt{mine} \rc \mt{munused}) \; []
adamc@785	1340 \end{array}$$
adamc@785	1341
adamc@785	1342 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	1343 $$\begin{array}{l}
adamc@785	1344 \mt{val} \; \mt{check} : \mt{fs} ::: \{\mt{Type}\} \to \mt{sql\_exp} \; [] \; [] \; \mt{fs} \; \mt{bool} \to \mt{sql\_constraint} \; \mt{fs} \; []
adamc@785	1345 \end{array}$$
adamc@785	1346
adamc@785	1347 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	1348
adamc@784	1349
adamc@543	1350 \subsubsection{Queries}
adamc@543	1351
adamc@543	1352 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	1353 $$\begin{array}{l}
adamc@543	1354 \mt{con} \; \mt{sql\_query} :: \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@543	1355 \mt{val} \; \mt{sql\_query} : \mt{tables} ::: \{\{\mt{Type}\}\} \\
adamc@543	1356 \hspace{.1in} \to \mt{selectedFields} ::: \{\{\mt{Type}\}\} \\
adamc@543	1357 \hspace{.1in} \to \mt{selectedExps} ::: \{\mt{Type}\} \\
adamc@543	1358 \hspace{.1in} \to \{\mt{Rows} : \mt{sql\_query1} \; \mt{tables} \; \mt{selectedFields} \; \mt{selectedExps}, \\
adamc@543	1359 \hspace{.2in} \mt{OrderBy} : \mt{sql\_order\_by} \; \mt{tables} \; \mt{selectedExps}, \\
adamc@543	1360 \hspace{.2in} \mt{Limit} : \mt{sql\_limit}, \\
adamc@543	1361 \hspace{.2in} \mt{Offset} : \mt{sql\_offset}\} \\
adamc@543	1362 \hspace{.1in} \to \mt{sql\_query} \; \mt{selectedFields} \; \mt{selectedExps}
adamc@543	1363 \end{array}$$
adamc@543	1364
adamc@545	1365 Queries are used by folding over their results inside transactions.
adamc@545	1366 $$\begin{array}{l}
adamc@545	1367 \mt{val} \; \mt{query} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \lambda [\mt{tables} \sim \mt{exps}] \Rightarrow \mt{state} ::: \mt{Type} \to \mt{sql\_query} \; \mt{tables} \; \mt{exps} \\
adamc@658	1368 \hspace{.1in} \to (\$(\mt{exps} \rc \mt{map} \; (\lambda \mt{fields} :: \{\mt{Type}\} \Rightarrow \$\mt{fields}) \; \mt{tables}) \\
adamc@545	1369 \hspace{.2in} \to \mt{state} \to \mt{transaction} \; \mt{state}) \\
adamc@545	1370 \hspace{.1in} \to \mt{state} \to \mt{transaction} \; \mt{state}
adamc@545	1371 \end{array}$$
adamc@545	1372
adamc@543	1373 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	1374 $$\begin{array}{l}
adamc@543	1375 \mt{con} \; \mt{sql\_query1} :: \{\{\mt{Type}\}\} \to \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@543	1376 \\
adamc@543	1377 \mt{type} \; \mt{sql\_relop} \\
adamc@543	1378 \mt{val} \; \mt{sql\_union} : \mt{sql\_relop} \\
adamc@543	1379 \mt{val} \; \mt{sql\_intersect} : \mt{sql\_relop} \\
adamc@543	1380 \mt{val} \; \mt{sql\_except} : \mt{sql\_relop} \\
adamc@543	1381 \mt{val} \; \mt{sql\_relop} : \mt{tables1} ::: \{\{\mt{Type}\}\} \\
adamc@543	1382 \hspace{.1in} \to \mt{tables2} ::: \{\{\mt{Type}\}\} \\
adamc@543	1383 \hspace{.1in} \to \mt{selectedFields} ::: \{\{\mt{Type}\}\} \\
adamc@543	1384 \hspace{.1in} \to \mt{selectedExps} ::: \{\mt{Type}\} \\
adamc@543	1385 \hspace{.1in} \to \mt{sql\_relop} \\
adamc@543	1386 \hspace{.1in} \to \mt{sql\_query1} \; \mt{tables1} \; \mt{selectedFields} \; \mt{selectedExps} \\
adamc@543	1387 \hspace{.1in} \to \mt{sql\_query1} \; \mt{tables2} \; \mt{selectedFields} \; \mt{selectedExps} \\
adamc@543	1388 \hspace{.1in} \to \mt{sql\_query1} \; \mt{selectedFields} \; \mt{selectedFields} \; \mt{selectedExps}
adamc@543	1389 \end{array}$$
adamc@543	1390
adamc@543	1391 $$\begin{array}{l}
adamc@543	1392 \mt{val} \; \mt{sql\_query1} : \mt{tables} ::: \{\{\mt{Type}\}\} \\
adamc@543	1393 \hspace{.1in} \to \mt{grouped} ::: \{\{\mt{Type}\}\} \\
adamc@543	1394 \hspace{.1in} \to \mt{selectedFields} ::: \{\{\mt{Type}\}\} \\
adamc@543	1395 \hspace{.1in} \to \mt{selectedExps} ::: \{\mt{Type}\} \\
adamc@786	1396 \hspace{.1in} \to \{\mt{From} : \mt{sql\_from\_items} \; \mt{tables}, \\
adamc@543	1397 \hspace{.2in} \mt{Where} : \mt{sql\_exp} \; \mt{tables} \; [] \; [] \; \mt{bool}, \\
adamc@543	1398 \hspace{.2in} \mt{GroupBy} : \mt{sql\_subset} \; \mt{tables} \; \mt{grouped}, \\
adamc@543	1399 \hspace{.2in} \mt{Having} : \mt{sql\_exp} \; \mt{grouped} \; \mt{tables} \; [] \; \mt{bool}, \\
adamc@543	1400 \hspace{.2in} \mt{SelectFields} : \mt{sql\_subset} \; \mt{grouped} \; \mt{selectedFields}, \\
adamc@658	1401 \hspace{.2in} \mt {SelectExps} : \$(\mt{map} \; (\mt{sql\_exp} \; \mt{grouped} \; \mt{tables} \; []) \; \mt{selectedExps}) \} \\
adamc@543	1402 \hspace{.1in} \to \mt{sql\_query1} \; \mt{tables} \; \mt{selectedFields} \; \mt{selectedExps}
adamc@543	1403 \end{array}$$
adamc@543	1404
adamc@543	1405 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	1406 $$\begin{array}{l}
adamc@543	1407 \mt{con} \; \mt{sql\_subset} :: \{\{\mt{Type}\}\} \to \{\{\mt{Type}\}\} \to \mt{Type} \\
adamc@543	1408 \mt{val} \; \mt{sql\_subset} : \mt{keep\_drop} :: \{(\{\mt{Type}\} \times \{\mt{Type}\})\} \\
adamc@543	1409 \hspace{.1in} \to \mt{sql\_subset} \\
adamc@658	1410 \hspace{.2in} (\mt{map} \; (\lambda \mt{fields} :: (\{\mt{Type}\} \times \{\mt{Type}\}) \Rightarrow \mt{fields}.1 \rc \mt{fields}.2)\; \mt{keep\_drop}) \\
adamc@658	1411 \hspace{.2in} (\mt{map} \; (\lambda \mt{fields} :: (\{\mt{Type}\} \times \{\mt{Type}\}) \Rightarrow \mt{fields}.1) \; \mt{keep\_drop}) \\
adamc@543	1412 \mt{val} \; \mt{sql\_subset\_all} : \mt{tables} :: \{\{\mt{Type}\}\} \to \mt{sql\_subset} \; \mt{tables} \; \mt{tables}
adamc@543	1413 \end{array}$$
adamc@543	1414
adamc@560	1415 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	1416 $$\begin{array}{l}
adamc@543	1417 \mt{con} \; \mt{sql\_exp} :: \{\{\mt{Type}\}\} \to \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \to \mt{Type}
adamc@543	1418 \end{array}$$
adamc@543	1419
adamc@543	1420 Any field in scope may be converted to an expression.
adamc@543	1421 $$\begin{array}{l}
adamc@543	1422 \mt{val} \; \mt{sql\_field} : \mt{otherTabs} ::: \{\{\mt{Type}\}\} \to \mt{otherFields} ::: \{\mt{Type}\} \\
adamc@543	1423 \hspace{.1in} \to \mt{fieldType} ::: \mt{Type} \to \mt{agg} ::: \{\{\mt{Type}\}\} \\
adamc@543	1424 \hspace{.1in} \to \mt{exps} ::: \{\mt{Type}\} \\
adamc@543	1425 \hspace{.1in} \to \mt{tab} :: \mt{Name} \to \mt{field} :: \mt{Name} \\
adamc@543	1426 \hspace{.1in} \to \mt{sql\_exp} \; ([\mt{tab} = [\mt{field} = \mt{fieldType}] \rc \mt{otherFields}] \rc \mt{otherTabs}) \; \mt{agg} \; \mt{exps} \; \mt{fieldType}
adamc@543	1427 \end{array}$$
adamc@543	1428
adamc@544	1429 There is an analogous function for referencing named expressions.
adamc@544	1430 $$\begin{array}{l}
adamc@544	1431 \mt{val} \; \mt{sql\_exp} : \mt{tabs} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{t} ::: \mt{Type} \to \mt{rest} ::: \{\mt{Type}\} \to \mt{nm} :: \mt{Name} \\
adamc@544	1432 \hspace{.1in} \to \mt{sql\_exp} \; \mt{tabs} \; \mt{agg} \; ([\mt{nm} = \mt{t}] \rc \mt{rest}) \; \mt{t}
adamc@544	1433 \end{array}$$
adamc@544	1434
adamc@544	1435 Ur values of appropriate types may be injected into SQL expressions.
adamc@544	1436 $$\begin{array}{l}
adamc@786	1437 \mt{class} \; \mt{sql\_injectable\_prim} \\
adamc@786	1438 \mt{val} \; \mt{sql\_bool} : \mt{sql\_injectable\_prim} \; \mt{bool} \\
adamc@786	1439 \mt{val} \; \mt{sql\_int} : \mt{sql\_injectable\_prim} \; \mt{int} \\
adamc@786	1440 \mt{val} \; \mt{sql\_float} : \mt{sql\_injectable\_prim} \; \mt{float} \\
adamc@786	1441 \mt{val} \; \mt{sql\_string} : \mt{sql\_injectable\_prim} \; \mt{string} \\
adamc@786	1442 \mt{val} \; \mt{sql\_time} : \mt{sql\_injectable\_prim} \; \mt{time} \\
adamc@786	1443 \mt{val} \; \mt{sql\_blob} : \mt{sql\_injectable\_prim} \; \mt{blob} \\
adamc@786	1444 \mt{val} \; \mt{sql\_channel} : \mt{t} ::: \mt{Type} \to \mt{sql\_injectable\_prim} \; (\mt{channel} \; \mt{t}) \\
adamc@786	1445 \mt{val} \; \mt{sql\_client} : \mt{sql\_injectable\_prim} \; \mt{client} \\
adamc@786	1446 \\
adamc@544	1447 \mt{class} \; \mt{sql\_injectable} \\
adamc@786	1448 \mt{val} \; \mt{sql\_prim} : \mt{t} ::: \mt{Type} \to \mt{sql\_injectable\_prim} \; \mt{t} \to \mt{sql\_injectable} \; \mt{t} \\
adamc@786	1449 \mt{val} \; \mt{sql\_option\_prim} : \mt{t} ::: \mt{Type} \to \mt{sql\_injectable\_prim} \; \mt{t} \to \mt{sql\_injectable} \; (\mt{option} \; \mt{t}) \\
adamc@786	1450 \\
adamc@544	1451 \mt{val} \; \mt{sql\_inject} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \to \mt{sql\_injectable} \; \mt{t} \\
adamc@544	1452 \hspace{.1in} \to \mt{t} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{t}
adamc@544	1453 \end{array}$$
adamc@544	1454
adamc@544	1455 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	1456 $$\begin{array}{l}
adamc@544	1457 \mt{val} \; \mt{sql\_is\_null} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544	1458 \hspace{.1in} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; (\mt{option} \; \mt{t}) \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{bool}
adamc@544	1459 \end{array}$$
adamc@544	1460
adamc@559	1461 We have generic nullary, unary, and binary operators.
adamc@544	1462 $$\begin{array}{l}
adamc@544	1463 \mt{con} \; \mt{sql\_nfunc} :: \mt{Type} \to \mt{Type} \\
adamc@544	1464 \mt{val} \; \mt{sql\_current\_timestamp} : \mt{sql\_nfunc} \; \mt{time} \\
adamc@544	1465 \mt{val} \; \mt{sql\_nfunc} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544	1466 \hspace{.1in} \to \mt{sql\_nfunc} \; \mt{t} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{t} \\\end{array}$$
adamc@544	1467
adamc@544	1468 $$\begin{array}{l}
adamc@544	1469 \mt{con} \; \mt{sql\_unary} :: \mt{Type} \to \mt{Type} \to \mt{Type} \\
adamc@544	1470 \mt{val} \; \mt{sql\_not} : \mt{sql\_unary} \; \mt{bool} \; \mt{bool} \\
adamc@544	1471 \mt{val} \; \mt{sql\_unary} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{arg} ::: \mt{Type} \to \mt{res} ::: \mt{Type} \\
adamc@544	1472 \hspace{.1in} \to \mt{sql\_unary} \; \mt{arg} \; \mt{res} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{arg} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{res} \\
adamc@544	1473 \end{array}$$
adamc@544	1474
adamc@544	1475 $$\begin{array}{l}
adamc@544	1476 \mt{con} \; \mt{sql\_binary} :: \mt{Type} \to \mt{Type} \to \mt{Type} \to \mt{Type} \\
adamc@544	1477 \mt{val} \; \mt{sql\_and} : \mt{sql\_binary} \; \mt{bool} \; \mt{bool} \; \mt{bool} \\
adamc@544	1478 \mt{val} \; \mt{sql\_or} : \mt{sql\_binary} \; \mt{bool} \; \mt{bool} \; \mt{bool} \\
adamc@544	1479 \mt{val} \; \mt{sql\_binary} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{arg_1} ::: \mt{Type} \to \mt{arg_2} ::: \mt{Type} \to \mt{res} ::: \mt{Type} \\
adamc@544	1480 \hspace{.1in} \to \mt{sql\_binary} \; \mt{arg_1} \; \mt{arg_2} \; \mt{res} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{arg_1} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{arg_2} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{res}
adamc@544	1481 \end{array}$$
adamc@544	1482
adamc@544	1483 $$\begin{array}{l}
adamc@559	1484 \mt{class} \; \mt{sql\_arith} \\
adamc@559	1485 \mt{val} \; \mt{sql\_int\_arith} : \mt{sql\_arith} \; \mt{int} \\
adamc@559	1486 \mt{val} \; \mt{sql\_float\_arith} : \mt{sql\_arith} \; \mt{float} \\
adamc@559	1487 \mt{val} \; \mt{sql\_neg} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_unary} \; \mt{t} \; \mt{t} \\
adamc@559	1488 \mt{val} \; \mt{sql\_plus} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559	1489 \mt{val} \; \mt{sql\_minus} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559	1490 \mt{val} \; \mt{sql\_times} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559	1491 \mt{val} \; \mt{sql\_div} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559	1492 \mt{val} \; \mt{sql\_mod} : \mt{sql\_binary} \; \mt{int} \; \mt{int} \; \mt{int}
adamc@559	1493 \end{array}$$
adamc@544	1494
adamc@656	1495 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	1496 $$\begin{array}{l}
adamc@544	1497 \mt{val} \; \mt{sql\_count} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{int}
adamc@544	1498 \end{array}$$
adamc@544	1499
adamc@544	1500 $$\begin{array}{l}
adamc@544	1501 \mt{con} \; \mt{sql\_aggregate} :: \mt{Type} \to \mt{Type} \\
adamc@544	1502 \mt{val} \; \mt{sql\_aggregate} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544	1503 \hspace{.1in} \to \mt{sql\_aggregate} \; \mt{t} \to \mt{sql\_exp} \; \mt{agg} \; \mt{agg} \; \mt{exps} \; \mt{t} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{t}
adamc@544	1504 \end{array}$$
adamc@544	1505
adamc@544	1506 $$\begin{array}{l}
adamc@544	1507 \mt{class} \; \mt{sql\_summable} \\
adamc@544	1508 \mt{val} \; \mt{sql\_summable\_int} : \mt{sql\_summable} \; \mt{int} \\
adamc@544	1509 \mt{val} \; \mt{sql\_summable\_float} : \mt{sql\_summable} \; \mt{float} \\
adamc@544	1510 \mt{val} \; \mt{sql\_avg} : \mt{t} ::: \mt{Type} \to \mt{sql\_summable} \; \mt{t} \to \mt{sql\_aggregate} \; \mt{t} \\
adamc@544	1511 \mt{val} \; \mt{sql\_sum} : \mt{t} ::: \mt{Type} \to \mt{sql\_summable} \mt{t} \to \mt{sql\_aggregate} \; \mt{t}
adamc@544	1512 \end{array}$$
adamc@544	1513
adamc@544	1514 $$\begin{array}{l}
adamc@544	1515 \mt{class} \; \mt{sql\_maxable} \\
adamc@544	1516 \mt{val} \; \mt{sql\_maxable\_int} : \mt{sql\_maxable} \; \mt{int} \\
adamc@544	1517 \mt{val} \; \mt{sql\_maxable\_float} : \mt{sql\_maxable} \; \mt{float} \\
adamc@544	1518 \mt{val} \; \mt{sql\_maxable\_string} : \mt{sql\_maxable} \; \mt{string} \\
adamc@544	1519 \mt{val} \; \mt{sql\_maxable\_time} : \mt{sql\_maxable} \; \mt{time} \\
adamc@544	1520 \mt{val} \; \mt{sql\_max} : \mt{t} ::: \mt{Type} \to \mt{sql\_maxable} \; \mt{t} \to \mt{sql\_aggregate} \; \mt{t} \\
adamc@544	1521 \mt{val} \; \mt{sql\_min} : \mt{t} ::: \mt{Type} \to \mt{sql\_maxable} \; \mt{t} \to \mt{sql\_aggregate} \; \mt{t}
adamc@544	1522 \end{array}$$
adamc@544	1523
adamc@786	1524 \texttt{FROM} clauses are specified using a type family.
adamc@786	1525 $$\begin{array}{l}
adamc@786	1526 \mt{con} \; \mt{sql\_from\_items} :: \{\{\mt{Type}\}\} \to \mt{Type} \\
adamc@786	1527 \mt{val} \; \mt{sql\_from\_table} : \mt{t} ::: \mt{Type} \to \mt{fs} ::: \{\mt{Type}\} \to \mt{fieldsOf} \; \mt{t} \; \mt{fs} \to \mt{name} :: \mt{Name} \to \mt{t} \to \mt{sql\_from\_items} \; [\mt{name} = \mt{fs}] \\
adamc@786	1528 \mt{val} \; \mt{sql\_from\_comma} : \mt{tabs1} ::: \{\{\mt{Type}\}\} \to \mt{tabs2} ::: \{\{\mt{Type}\}\} \to [\mt{tabs1} \sim \mt{tabs2}] \\
adamc@786	1529 \hspace{.1in} \Rightarrow \mt{sql\_from\_items} \; \mt{tabs1} \to \mt{sql\_from\_items} \; \mt{tabs2} \\
adamc@786	1530 \hspace{.1in} \to \mt{sql\_from\_items} \; (\mt{tabs1} \rc \mt{tabs2}) \\
adamc@786	1531 \mt{val} \; \mt{sql\_inner\_join} : \mt{tabs1} ::: \{\{\mt{Type}\}\} \to \mt{tabs2} ::: \{\{\mt{Type}\}\} \to [\mt{tabs1} \sim \mt{tabs2}] \\
adamc@786	1532 \hspace{.1in} \Rightarrow \mt{sql\_from\_items} \; \mt{tabs1} \to \mt{sql\_from\_items} \; \mt{tabs2} \\
adamc@786	1533 \hspace{.1in} \to \mt{sql\_exp} \; (\mt{tabs1} \rc \mt{tabs2}) \; [] \; [] \; \mt{bool} \\
adamc@786	1534 \hspace{.1in} \to \mt{sql\_from\_items} \; (\mt{tabs1} \rc \mt{tabs2})
adamc@786	1535 \end{array}$$
adamc@786	1536
adamc@786	1537 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	1538 $$\begin{array}{l}
adamc@786	1539 \mt{class} \; \mt{nullify} :: \mt{Type} \to \mt{Type} \to \mt{Type} \\
adamc@786	1540 \mt{val} \; \mt{nullify\_option} : \mt{t} ::: \mt{Type} \to \mt{nullify} \; (\mt{option} \; \mt{t}) \; (\mt{option} \; \mt{t}) \\
adamc@786	1541 \mt{val} \; \mt{nullify\_prim} : \mt{t} ::: \mt{Type} \to \mt{sql\_injectable\_prim} \; \mt{t} \to \mt{nullify} \; \mt{t} \; (\mt{option} \; \mt{t})
adamc@786	1542 \end{array}$$
adamc@786	1543
adamc@786	1544 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	1545
adamc@786	1546 $$\begin{array}{l}
adamc@786	1547 \mt{val} \; \mt{sql\_left\_join} : \mt{tabs1} ::: \{\{\mt{Type}\}\} \to \mt{tabs2} ::: \{\{(\mt{Type} \times \mt{Type})\}\} \to [\mt{tabs1} \sim \mt{tabs2}] \\
adamc@786	1548 \hspace{.1in} \Rightarrow \$(\mt{map} \; (\lambda \mt{r} \Rightarrow \$(\mt{map} \; (\lambda \mt{p} :: (\mt{Type} \times \mt{Type}) \Rightarrow \mt{nullify} \; \mt{p}.1 \; \mt{p}.2) \; \mt{r})) \; \mt{tabs2}) \\
adamc@786	1549 \hspace{.1in} \to \mt{sql\_from\_items} \; \mt{tabs1} \to \mt{sql\_from\_items} \; (\mt{map} \; (\mt{map} \; (\lambda \mt{p} :: (\mt{Type} \times \mt{Type}) \Rightarrow \mt{p}.1)) \; \mt{tabs2}) \\
adamc@786	1550 \hspace{.1in} \to \mt{sql\_exp} \; (\mt{tabs1} \rc \mt{map} \; (\mt{map} \; (\lambda \mt{p} :: (\mt{Type} \times \mt{Type}) \Rightarrow \mt{p}.1)) \; \mt{tabs2}) \; [] \; [] \; \mt{bool} \\
adamc@786	1551 \hspace{.1in} \to \mt{sql\_from\_items} \; (\mt{tabs1} \rc \mt{map} \; (\mt{map} \; (\lambda \mt{p} :: (\mt{Type} \times \mt{Type}) \Rightarrow \mt{p}.2)) \; \mt{tabs2})
adamc@786	1552 \end{array}$$
adamc@786	1553
adamc@544	1554 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	1555 $$\begin{array}{l}
adamc@544	1556 \mt{type} \; \mt{sql\_direction} \\
adamc@544	1557 \mt{val} \; \mt{sql\_asc} : \mt{sql\_direction} \\
adamc@544	1558 \mt{val} \; \mt{sql\_desc} : \mt{sql\_direction} \\
adamc@544	1559 \\
adamc@544	1560 \mt{con} \; \mt{sql\_order\_by} :: \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@544	1561 \mt{val} \; \mt{sql\_order\_by\_Nil} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{exps} :: \{\mt{Type}\} \to \mt{sql\_order\_by} \; \mt{tables} \; \mt{exps} \\
adamc@544	1562 \mt{val} \; \mt{sql\_order\_by\_Cons} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544	1563 \hspace{.1in} \to \mt{sql\_exp} \; \mt{tables} \; [] \; \mt{exps} \; \mt{t} \to \mt{sql\_direction} \to \mt{sql\_order\_by} \; \mt{tables} \; \mt{exps} \to \mt{sql\_order\_by} \; \mt{tables} \; \mt{exps} \\
adamc@544	1564 \\
adamc@544	1565 \mt{type} \; \mt{sql\_limit} \\
adamc@544	1566 \mt{val} \; \mt{sql\_no\_limit} : \mt{sql\_limit} \\
adamc@544	1567 \mt{val} \; \mt{sql\_limit} : \mt{int} \to \mt{sql\_limit} \\
adamc@544	1568 \\
adamc@544	1569 \mt{type} \; \mt{sql\_offset} \\
adamc@544	1570 \mt{val} \; \mt{sql\_no\_offset} : \mt{sql\_offset} \\
adamc@544	1571 \mt{val} \; \mt{sql\_offset} : \mt{int} \to \mt{sql\_offset}
adamc@544	1572 \end{array}$$
adamc@544	1573
adamc@545	1574
adamc@545	1575 \subsubsection{DML}
adamc@545	1576
adamc@545	1577 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	1578
adamc@545	1579 $$\begin{array}{l}
adamc@545	1580 \mt{type} \; \mt{dml} \\
adamc@545	1581 \mt{val} \; \mt{dml} : \mt{dml} \to \mt{transaction} \; \mt{unit}
adamc@545	1582 \end{array}$$
adamc@545	1583
adamc@545	1584 Properly-typed records may be used to form $\mt{INSERT}$ commands.
adamc@545	1585 $$\begin{array}{l}
adamc@545	1586 \mt{val} \; \mt{insert} : \mt{fields} ::: \{\mt{Type}\} \to \mt{sql\_table} \; \mt{fields} \\
adamc@658	1587 \hspace{.1in} \to \$(\mt{map} \; (\mt{sql\_exp} \; [] \; [] \; []) \; \mt{fields}) \to \mt{dml}
adamc@545	1588 \end{array}$$
adamc@545	1589
adamc@545	1590 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	1591 $$\begin{array}{l}
adamc@545	1592 \mt{val} \; \mt{update} : \mt{unchanged} ::: \{\mt{Type}\} \to \mt{changed} :: \{\mt{Type}\} \to \lambda [\mt{changed} \sim \mt{unchanged}] \\
adamc@658	1593 \hspace{.1in} \Rightarrow \$(\mt{map} \; (\mt{sql\_exp} \; [\mt{T} = \mt{changed} \rc \mt{unchanged}] \; [] \; []) \; \mt{changed}) \\
adamc@545	1594 \hspace{.1in} \to \mt{sql\_table} \; (\mt{changed} \rc \mt{unchanged}) \to \mt{sql\_exp} \; [\mt{T} = \mt{changed} \rc \mt{unchanged}] \; [] \; [] \; \mt{bool} \to \mt{dml}
adamc@545	1595 \end{array}$$
adamc@545	1596
adamc@545	1597 A $\mt{DELETE}$ command is formed from a table and a $\mt{WHERE}$ clause.
adamc@545	1598 $$\begin{array}{l}
adamc@545	1599 \mt{val} \; \mt{delete} : \mt{fields} ::: \{\mt{Type}\} \to \mt{sql\_table} \; \mt{fields} \to \mt{sql\_exp} \; [\mt{T} = \mt{fields}] \; [] \; [] \; \mt{bool} \to \mt{dml}
adamc@545	1600 \end{array}$$
adamc@545	1601
adamc@546	1602 \subsubsection{Sequences}
adamc@546	1603
adamc@546	1604 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	1605
adamc@546	1606 $$\begin{array}{l}
adamc@546	1607 \mt{type} \; \mt{sql\_sequence} \\
adamc@546	1608 \mt{val} \; \mt{nextval} : \mt{sql\_sequence} \to \mt{transaction} \; \mt{int}
adamc@546	1609 \end{array}$$
adamc@546	1610
adamc@546	1611
adamc@547	1612 \subsection{XML}
adamc@547	1613
adamc@547	1614 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	1615
adamc@547	1616 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	1617 $$\begin{array}{l}
adamc@547	1618 \mt{con} \; \mt{xml} :: \{\mt{Unit}\} \to \{\mt{Type}\} \to \{\mt{Type}\} \to \mt{Type}
adamc@547	1619 \end{array}$$
adamc@547	1620
adamc@547	1621 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	1622 $$\begin{array}{l}
adamc@547	1623 \mt{con} \; \mt{tag} :: \{\mt{Type}\} \to \{\mt{Unit}\} \to \{\mt{Unit}\} \to \{\mt{Type}\} \to \{\mt{Type}\} \to \mt{Type}
adamc@547	1624 \end{array}$$
adamc@547	1625
adamc@547	1626 Literal text may be injected into XML as ``CDATA.''
adamc@547	1627 $$\begin{array}{l}
adamc@547	1628 \mt{val} \; \mt{cdata} : \mt{ctx} ::: \{\mt{Unit}\} \to \mt{use} ::: \{\mt{Type}\} \to \mt{string} \to \mt{xml} \; \mt{ctx} \; \mt{use} \; []
adamc@547	1629 \end{array}$$
adamc@547	1630
adamc@547	1631 There is a function for producing an XML tree with a particular tag at its root.
adamc@547	1632 $$\begin{array}{l}
adamc@547	1633 \mt{val} \; \mt{tag} : \mt{attrsGiven} ::: \{\mt{Type}\} \to \mt{attrsAbsent} ::: \{\mt{Type}\} \to \mt{ctxOuter} ::: \{\mt{Unit}\} \to \mt{ctxInner} ::: \{\mt{Unit}\} \\
adamc@547	1634 \hspace{.1in} \to \mt{useOuter} ::: \{\mt{Type}\} \to \mt{useInner} ::: \{\mt{Type}\} \to \mt{bindOuter} ::: \{\mt{Type}\} \to \mt{bindInner} ::: \{\mt{Type}\} \\
adamc@787	1635 \hspace{.1in} \to \lambda [\mt{attrsGiven} \sim \mt{attrsAbsent}] \; [\mt{useOuter} \sim \mt{useInner}] \; [\mt{bindOuter} \sim \mt{bindInner}] \\
adamc@787	1636 \hspace{.1in} \Rightarrow \mt{option} \; \mt{css\_class} \\
adamc@787	1637 \hspace{.1in} \to \$\mt{attrsGiven} \\
adamc@547	1638 \hspace{.1in} \to \mt{tag} \; (\mt{attrsGiven} \rc \mt{attrsAbsent}) \; \mt{ctxOuter} \; \mt{ctxInner} \; \mt{useOuter} \; \mt{bindOuter} \\
adamc@547	1639 \hspace{.1in} \to \mt{xml} \; \mt{ctxInner} \; \mt{useInner} \; \mt{bindInner} \to \mt{xml} \; \mt{ctxOuter} \; (\mt{useOuter} \rc \mt{useInner}) \; (\mt{bindOuter} \rc \mt{bindInner})
adamc@547	1640 \end{array}$$
adamc@787	1641 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	1642
adamc@547	1643 Two XML fragments may be concatenated.
adamc@547	1644 $$\begin{array}{l}
adamc@547	1645 \mt{val} \; \mt{join} : \mt{ctx} ::: \{\mt{Unit}\} \to \mt{use_1} ::: \{\mt{Type}\} \to \mt{bind_1} ::: \{\mt{Type}\} \to \mt{bind_2} ::: \{\mt{Type}\} \\
adamc@547	1646 \hspace{.1in} \to \lambda [\mt{use_1} \sim \mt{bind_1}] \; [\mt{bind_1} \sim \mt{bind_2}] \\
adamc@547	1647 \hspace{.1in} \Rightarrow \mt{xml} \; \mt{ctx} \; \mt{use_1} \; \mt{bind_1} \to \mt{xml} \; \mt{ctx} \; (\mt{use_1} \rc \mt{bind_1}) \; \mt{bind_2} \to \mt{xml} \; \mt{ctx} \; \mt{use_1} \; (\mt{bind_1} \rc \mt{bind_2})
adamc@547	1648 \end{array}$$
adamc@547	1649
adamc@547	1650 Finally, any XML fragment may be updated to ``claim'' to use more form fields than it does.
adamc@547	1651 $$\begin{array}{l}
adamc@547	1652 \mt{val} \; \mt{useMore} : \mt{ctx} ::: \{\mt{Unit}\} \to \mt{use_1} ::: \{\mt{Type}\} \to \mt{use_2} ::: \{\mt{Type}\} \to \mt{bind} ::: \{\mt{Type}\} \to \lambda [\mt{use_1} \sim \mt{use_2}] \\
adamc@547	1653 \hspace{.1in} \Rightarrow \mt{xml} \; \mt{ctx} \; \mt{use_1} \; \mt{bind} \to \mt{xml} \; \mt{ctx} \; (\mt{use_1} \rc \mt{use_2}) \; \mt{bind}
adamc@547	1654 \end{array}$$
adamc@547	1655
adamc@547	1656 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	1657
adamc@547	1658 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	1659 $$\begin{array}{l}
adamc@547	1660 \mt{val} \; \mt{error} : \mt{t} ::: \mt{Type} \to \mt{xml} \; [\mt{Body}] \; [] \; [] \to \mt{t}
adamc@547	1661 \end{array}$$
adamc@547	1662
adamc@549	1663
adamc@701	1664 \subsection{Client-Side Programming}
adamc@659	1665
adamc@701	1666 Ur/Web supports running code on web browsers, via automatic compilation to JavaScript.
adamc@701	1667
adamc@701	1668 \subsubsection{The Basics}
adamc@701	1669
adamc@701	1670 Clients can open alert dialog boxes, in the usual annoying JavaScript way.
adamc@701	1671 $$\begin{array}{l}
adamc@701	1672 \mt{val} \; \mt{alert} : \mt{string} \to \mt{transaction} \; \mt{unit}
adamc@701	1673 \end{array}$$
adamc@701	1674
adamc@701	1675 Any transaction may be run in a new thread with the $\mt{spawn}$ function.
adamc@701	1676 $$\begin{array}{l}
adamc@701	1677 \mt{val} \; \mt{spawn} : \mt{transaction} \; \mt{unit} \to \mt{transaction} \; \mt{unit}
adamc@701	1678 \end{array}$$
adamc@701	1679
adamc@701	1680 The current thread can be paused for at least a specified number of milliseconds.
adamc@701	1681 $$\begin{array}{l}
adamc@701	1682 \mt{val} \; \mt{sleep} : \mt{int} \to \mt{transaction} \; \mt{unit}
adamc@701	1683 \end{array}$$
adamc@701	1684
adamc@787	1685 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	1686 $$\begin{array}{l}
adamc@787	1687 \mt{val} \; \mt{onError} : (\mt{xbody} \to \mt{transaction} \; \mt{unit}) \to \mt{transaction} \; \mt{unit} \\
adamc@787	1688 \mt{val} \; \mt{onFail} : (\mt{string} \to \mt{transaction} \; \mt{unit}) \to \mt{transaction} \; \mt{unit} \\
adamc@787	1689 \mt{val} \; \mt{onConnectFail} : \mt{transaction} \; \mt{unit} \to \mt{transaction} \; \mt{unit} \\
adamc@787	1690 \mt{val} \; \mt{onDisconnect} : \mt{transaction} \; \mt{unit} \to \mt{transaction} \; \mt{unit} \\
adamc@787	1691 \mt{val} \; \mt{onServerError} : (\mt{string} \to \mt{transaction} \; \mt{unit}) \to \mt{transaction} \; \mt{unit}
adamc@787	1692 \end{array}$$
adamc@787	1693
adamc@701	1694 \subsubsection{Functional-Reactive Page Generation}
adamc@701	1695
adamc@701	1696 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	1697
adamc@659	1698 $$\begin{array}{l}
adamc@659	1699 \mt{con} \; \mt{source} :: \mt{Type} \to \mt{Type} \\
adamc@659	1700 \mt{val} \; \mt{source} : \mt{t} ::: \mt{Type} \to \mt{t} \to \mt{transaction} \; (\mt{source} \; \mt{t}) \\
adamc@659	1701 \mt{val} \; \mt{set} : \mt{t} ::: \mt{Type} \to \mt{source} \; \mt{t} \to \mt{t} \to \mt{transaction} \; \mt{unit} \\
adamc@659	1702 \mt{val} \; \mt{get} : \mt{t} ::: \mt{Type} \to \mt{source} \; \mt{t} \to \mt{transaction} \; \mt{t}
adamc@659	1703 \end{array}$$
adamc@659	1704
adamc@659	1705 Pure functions over sources are represented in a monad of \emph{signals}.
adamc@659	1706
adamc@659	1707 $$\begin{array}{l}
adamc@659	1708 \mt{con} \; \mt{signal} :: \mt{Type} \to \mt{Type} \\
adamc@659	1709 \mt{val} \; \mt{signal\_monad} : \mt{monad} \; \mt{signal} \\
adamc@659	1710 \mt{val} \; \mt{signal} : \mt{t} ::: \mt{Type} \to \mt{source} \; \mt{t} \to \mt{signal} \; \mt{t}
adamc@659	1711 \end{array}$$
adamc@659	1712
adamc@659	1713 A reactive portion of an HTML page is injected with a $\mt{dyn}$ tag, which has a signal-valued attribute $\mt{Signal}$.
adamc@659	1714
adamc@659	1715 $$\begin{array}{l}
adamc@701	1716 \mt{val} \; \mt{dyn} : \mt{use} ::: \{\mt{Type}\} \to \mt{bind} ::: \{\mt{Type}\} \to \mt{unit} \\
adamc@701	1717 \hspace{.1in} \to \mt{tag} \; [\mt{Signal} = \mt{signal} \; (\mt{xml} \; \mt{body} \; \mt{use} \; \mt{bind})] \; \mt{body} \; [] \; \mt{use} \; \mt{bind}
adamc@659	1718 \end{array}$$
adamc@659	1719
adamc@701	1720 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	1721
adamc@701	1722 \subsubsection{Asynchronous Message-Passing}
adamc@701	1723
adamc@701	1724 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	1725
adamc@701	1726 $$\begin{array}{l}
adamc@701	1727 \mt{type} \; \mt{client} \\
adamc@701	1728 \mt{val} \; \mt{self} : \mt{transaction} \; \mt{client}
adamc@701	1729 \end{array}$$
adamc@701	1730
adamc@701	1731 \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	1732
adamc@701	1733 $$\begin{array}{l}
adamc@701	1734 \mt{con} \; \mt{channel} :: \mt{Type} \to \mt{Type} \\
adamc@701	1735 \mt{val} \; \mt{channel} : \mt{t} ::: \mt{Type} \to \mt{transaction} \; (\mt{channel} \; \mt{t}) \\
adamc@701	1736 \mt{val} \; \mt{send} : \mt{t} ::: \mt{Type} \to \mt{channel} \; \mt{t} \to \mt{t} \to \mt{transaction} \; \mt{unit} \\
adamc@701	1737 \mt{val} \; \mt{recv} : \mt{t} ::: \mt{Type} \to \mt{channel} \; \mt{t} \to \mt{transaction} \; \mt{t}
adamc@701	1738 \end{array}$$
adamc@701	1739
adamc@701	1740 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	1741
adamc@701	1742 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	1743
adamc@659	1744
adamc@549	1745 \section{Ur/Web Syntax Extensions}
adamc@549	1746
adamc@549	1747 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	1748
adamc@549	1749 \subsection{SQL}
adamc@549	1750
adamc@786	1751 \subsubsection{\label{tables}Table Declarations}
adamc@786	1752
adamc@788	1753 $\mt{table}$ declarations may include constraints, via these grammar rules.
adamc@788	1754 $$\begin{array}{rrcll}
adamc@788	1755 \textrm{Declarations} & d &::=& \mt{table} \; x : c \; [pk[,]] \; cts \\
adamc@788	1756 \textrm{Primary key constraints} & pk &::=& \mt{PRIMARY} \; \mt{KEY} \; K \\
adamc@788	1757 \textrm{Keys} & K &::=& f \mid (f, (f,)^+) \\
adamc@788	1758 \textrm{Constraint sets} & cts &::=& \mt{CONSTRAINT} f \; ct \mid cts, cts \mid \{\{e\}\} \\
adamc@788	1759 \textrm{Constraints} & ct &::=& \mt{UNIQUE} \; K \mid \mt{CHECK} \; E \\
adamc@788	1760 &&& \mid \mt{FOREIGN} \; \mt{KEY} \; K \; \mt{REFERENCES} \; F \; (K) \; [\mt{ON} \; \mt{DELETE} \; pr] \; [\mt{ON} \; \mt{UPDATE} \; pr] \\
adamc@788	1761 \textrm{Foreign tables} & F &::=& x \mid \{\{e\}\} \\
adamc@788	1762 \textrm{Propagation modes} & pr &::=& \mt{NO} \; \mt{ACTION} \mid \mt{RESTRICT} \mid \mt{CASCADE} \mid \mt{SET} \; \mt{NULL}
adamc@788	1763 \end{array}$$
adamc@788	1764
adamc@788	1765 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	1766
adamc@788	1767
adamc@549	1768 \subsubsection{Queries}
adamc@549	1769
adamc@550	1770 Queries $Q$ are added to the rules for expressions $e$.
adamc@550	1771
adamc@549	1772 $$\begin{array}{rrcll}
adamc@550	1773 \textrm{Queries} & Q &::=& (q \; [\mt{ORDER} \; \mt{BY} \; (E \; [o],)^+] \; [\mt{LIMIT} \; N] \; [\mt{OFFSET} \; N]) \\
adamc@549	1774 \textrm{Pre-queries} & q &::=& \mt{SELECT} \; P \; \mt{FROM} \; T,^+ \; [\mt{WHERE} \; E] \; [\mt{GROUP} \; \mt{BY} \; p,^+] \; [\mt{HAVING} \; E] \\
adamc@549	1775 &&& \mid q \; R \; q \\
adamc@549	1776 \textrm{Relational operators} & R &::=& \mt{UNION} \mid \mt{INTERSECT} \mid \mt{EXCEPT}
adamc@549	1777 \end{array}$$
adamc@549	1778
adamc@549	1779 $$\begin{array}{rrcll}
adamc@549	1780 \textrm{Projections} & P &::=& \ast & \textrm{all columns} \\
adamc@549	1781 &&& p,^+ & \textrm{particular columns} \\
adamc@549	1782 \textrm{Pre-projections} & p &::=& t.f & \textrm{one column from a table} \\
adamc@558	1783 &&& t.\{\{c\}\} & \textrm{a record of columns from a table (of kind $\{\mt{Type}\}$)} \\
adamc@549	1784 \textrm{Table names} & t &::=& x & \textrm{constant table name (automatically capitalized)} \\
adamc@549	1785 &&& X & \textrm{constant table name} \\
adamc@549	1786 &&& \{\{c\}\} & \textrm{computed table name (of kind $\mt{Name}$)} \\
adamc@549	1787 \textrm{Column names} & f &::=& X & \textrm{constant column name} \\
adamc@549	1788 &&& \{c\} & \textrm{computed column name (of kind $\mt{Name}$)} \\
adamc@549	1789 \textrm{Tables} & T &::=& x & \textrm{table variable, named locally by its own capitalization} \\
adamc@549	1790 &&& x \; \mt{AS} \; t & \textrm{table variable, with local name} \\
adamc@549	1791 &&& \{\{e\}\} \; \mt{AS} \; t & \textrm{computed table expression, with local name} \\
adamc@549	1792 \textrm{SQL expressions} & E &::=& p & \textrm{column references} \\
adamc@549	1793 &&& X & \textrm{named expression references} \\
adamc@549	1794 &&& \{\{e\}\} & \textrm{injected native Ur expressions} \\
adamc@549	1795 &&& \{e\} & \textrm{computed expressions, probably using $\mt{sql\_exp}$ directly} \\
adamc@549	1796 &&& \mt{TRUE} \mid \mt{FALSE} & \textrm{boolean constants} \\
adamc@549	1797 &&& \ell & \textrm{primitive type literals} \\
adamc@549	1798 &&& \mt{NULL} & \textrm{null value (injection of $\mt{None}$)} \\
adamc@549	1799 &&& E \; \mt{IS} \; \mt{NULL} & \textrm{nullness test} \\
adamc@549	1800 &&& n & \textrm{nullary operators} \\
adamc@549	1801 &&& u \; E & \textrm{unary operators} \\
adamc@549	1802 &&& E \; b \; E & \textrm{binary operators} \\
adamc@549	1803 &&& \mt{COUNT}(\ast) & \textrm{count number of rows} \\
adamc@549	1804 &&& a(E) & \textrm{other aggregate function} \\
adamc@549	1805 &&& (E) & \textrm{explicit precedence} \\
adamc@549	1806 \textrm{Nullary operators} & n &::=& \mt{CURRENT\_TIMESTAMP} \\
adamc@549	1807 \textrm{Unary operators} & u &::=& \mt{NOT} \\
adamc@549	1808 \textrm{Binary operators} & b &::=& \mt{AND} \mid \mt{OR} \mid \neq \mid < \mid \leq \mid > \mid \geq \\
adamc@549	1809 \textrm{Aggregate functions} & a &::=& \mt{AVG} \mid \mt{SUM} \mid \mt{MIN} \mid \mt{MAX} \\
adamc@550	1810 \textrm{Directions} & o &::=& \mt{ASC} \mid \mt{DESC} \\
adamc@549	1811 \textrm{SQL integer} & N &::=& n \mid \{e\} \\
adamc@549	1812 \end{array}$$
adamc@549	1813
adamc@549	1814 Additionally, an SQL expression may be inserted into normal Ur code with the syntax $(\mt{SQL} \; E)$ or $(\mt{WHERE} \; E)$.
adamc@549	1815
adamc@550	1816 \subsubsection{DML}
adamc@550	1817
adamc@550	1818 DML commands $D$ are added to the rules for expressions $e$.
adamc@550	1819
adamc@550	1820 $$\begin{array}{rrcll}
adamc@550	1821 \textrm{Commands} & D &::=& (\mt{INSERT} \; \mt{INTO} \; T^E \; (f,^+) \; \mt{VALUES} \; (E,^+)) \\
adamc@550	1822 &&& (\mt{UPDATE} \; T^E \; \mt{SET} \; (f = E,)^+ \; \mt{WHERE} \; E) \\
adamc@550	1823 &&& (\mt{DELETE} \; \mt{FROM} \; T^E \; \mt{WHERE} \; E) \\
adamc@550	1824 \textrm{Table expressions} & T^E &::=& x \mid \{\{e\}\}
adamc@550	1825 \end{array}$$
adamc@550	1826
adamc@550	1827 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	1828
adamc@551	1829 \subsection{XML}
adamc@551	1830
adamc@551	1831 XML fragments $L$ are added to the rules for expressions $e$.
adamc@551	1832
adamc@551	1833 $$\begin{array}{rrcll}
adamc@551	1834 \textrm{XML fragments} & L &::=& \texttt{<xml/>} \mid \texttt{<xml>}l^*\texttt{</xml>} \\
adamc@551	1835 \textrm{XML pieces} & l &::=& \textrm{text} & \textrm{cdata} \\
adamc@551	1836 &&& \texttt{<}g\texttt{/>} & \textrm{tag with no children} \\
adamc@551	1837 &&& \texttt{<}g\texttt{>}l^*\texttt{</}x\texttt{>} & \textrm{tag with children} \\
adamc@559	1838 &&& \{e\} & \textrm{computed XML fragment} \\
adamc@559	1839 &&& \{[e]\} & \textrm{injection of an Ur expression, via the $\mt{Top}.\mt{txt}$ function} \\
adamc@551	1840 \textrm{Tag} & g &::=& h \; (x = v)^* \\
adamc@551	1841 \textrm{Tag head} & h &::=& x & \textrm{tag name} \\
adamc@551	1842 &&& h\{c\} & \textrm{constructor parameter} \\
adamc@551	1843 \textrm{Attribute value} & v &::=& \ell & \textrm{literal value} \\
adamc@551	1844 &&& \{e\} & \textrm{computed value} \\
adamc@551	1845 \end{array}$$
adamc@551	1846
adamc@552	1847
adamc@553	1848 \section{The Structure of Web Applications}
adamc@553	1849
adamc@553	1850 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	1851
adamc@553	1852 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	1853
adamc@553	1854 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	1855
adamc@558	1856 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	1857
adamc@660	1858 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	1859
adamc@789	1860 \medskip
adamc@789	1861
adamc@789	1862 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	1863
adamc@789	1864 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	1865
adamc@553	1866
adamc@552	1867 \section{Compiler Phases}
adamc@552	1868
adamc@552	1869 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	1870
adamc@552	1871 In this section, we step through the main phases of compilation, noting what consequences each phase has for effective programming.
adamc@552	1872
adamc@552	1873 \subsection{Parse}
adamc@552	1874
adamc@552	1875 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	1876
adamc@552	1877 \subsection{Elaborate}
adamc@552	1878
adamc@552	1879 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	1880
adamc@552	1881 \subsection{Unnest}
adamc@552	1882
adamc@552	1883 Named local function definitions are moved to the top level, to avoid the need to generate closures.
adamc@552	1884
adamc@552	1885 \subsection{Corify}
adamc@552	1886
adamc@552	1887 Module system features are compiled away, through inlining of functor definitions at application sites. Afterward, most abstraction boundaries are broken, facilitating optimization.
adamc@552	1888
adamc@552	1889 \subsection{Especialize}
adamc@552	1890
adamc@552	1891 Functions are specialized to particular argument patterns. This is an important trick for avoiding the need to maintain any closures at runtime.
adamc@552	1892
adamc@552	1893 \subsection{Untangle}
adamc@552	1894
adamc@552	1895 Remove unnecessary mutual recursion, splitting recursive groups into strongly-connected components.
adamc@552	1896
adamc@552	1897 \subsection{Shake}
adamc@552	1898
adamc@552	1899 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	1900
adamc@661	1901 \subsection{Rpcify}
adamc@661	1902
adamc@661	1903 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	1904
adamc@661	1905 \subsection{Untangle, Shake}
adamc@661	1906
adamc@661	1907 Repeat these simplifications.
adamc@661	1908
adamc@553	1909 \subsection{\label{tag}Tag}
adamc@552	1910
adamc@552	1911 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	1912
adamc@552	1913 \subsection{Reduce}
adamc@552	1914
adamc@552	1915 Apply definitional equality rules to simplify the program as much as possible. This effectively includes inlining of every non-recursive definition.
adamc@552	1916
adamc@552	1917 \subsection{Unpoly}
adamc@552	1918
adamc@552	1919 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	1920
adamc@552	1921 \subsection{Specialize}
adamc@552	1922
adamc@558	1923 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	1924
adamc@552	1925 \subsection{Shake}
adamc@552	1926
adamc@558	1927 Here the compiler repeats the earlier Shake phase.
adamc@552	1928
adamc@552	1929 \subsection{Monoize}
adamc@552	1930
adamc@552	1931 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	1932
adamc@552	1933 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	1934
adamc@552	1935 \subsection{MonoOpt}
adamc@552	1936
adamc@552	1937 Simple algebraic laws are applied to simplify the program, focusing especially on efficient imperative generation of HTML pages.
adamc@552	1938
adamc@552	1939 \subsection{MonoUntangle}
adamc@552	1940
adamc@552	1941 Unnecessary mutual recursion is broken up again.
adamc@552	1942
adamc@552	1943 \subsection{MonoReduce}
adamc@552	1944
adamc@552	1945 Equivalents of the definitional equality rules are applied to simplify programs, with inlining again playing a major role.
adamc@552	1946
adamc@552	1947 \subsection{MonoShake, MonoOpt}
adamc@552	1948
adamc@552	1949 Unneeded declarations are removed, and basic optimizations are repeated.
adamc@552	1950
adamc@552	1951 \subsection{Fuse}
adamc@552	1952
adamc@552	1953 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	1954
adamc@552	1955 \subsection{MonoUntangle, MonoShake}
adamc@552	1956
adamc@552	1957 Fuse often creates more opportunities to remove spurious mutual recursion.
adamc@552	1958
adamc@552	1959 \subsection{Pathcheck}
adamc@552	1960
adamc@552	1961 The compiler checks that no link or action name has been used more than once.
adamc@552	1962
adamc@552	1963 \subsection{Cjrize}
adamc@552	1964
adamc@552	1965 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	1966
adamc@552	1967 \subsection{C Compilation and Linking}
adamc@552	1968
adamc@552	1969 The output of the last phase is pretty-printed as C source code and passed to GCC.
adamc@552	1970
adamc@552	1971
adamc@524	1972 \end{document}

Mercurial > urweb

annotate doc/manual.tex @ 871:3ae6b655ced0