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author	Adam Chlipala <adamc@hcoop.net>
date	Wed, 24 Dec 2008 10:48:31 -0500
parents	af0df56ecc2c
children	9db6880184d0

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@554	32 \item Attempt invalid SQL queries
adamc@554	33 \item Use improper marshaling or unmarshaling in communication with SQL databases
adamc@554	34 \end{itemize}
adamc@554	35
adamc@554	36 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	37
adamc@554	38 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.
adamc@554	39
adamc@554	40 \medskip
adamc@554	41
adamc@554	42 The official web site for Ur is:
adamc@554	43 \begin{center}
adamc@554	44 \url{http://www.impredicative.com/ur/}
adamc@554	45 \end{center}
adamc@554	46
adamc@555	47
adamc@555	48 \section{Installation}
adamc@555	49
adamc@555	50 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	51
adamc@555	52 \begin{verbatim}
adamc@555	53 ./configure
adamc@555	54 make
adamc@555	55 sudo make install
adamc@555	56 \end{verbatim}
adamc@555	57
adamc@555	58 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; and MLton, the whole-program optimizing compiler for Standard ML. 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	59
adamc@555	60 \begin{verbatim}
adamc@555	61 apt-get install mlton libpq-dev
adamc@555	62 \end{verbatim}
adamc@555	63
adamc@555	64 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	65
adamc@555	66 \begin{verbatim}
adamc@555	67 apt-get install smlnj libsmlnj-smlnj ml-yacc ml-lpt
adamc@555	68 \end{verbatim}
adamc@555	69
adamc@555	70 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	71
adamc@555	72 To run an SQL-backed application, you will probably want to install the PostgreSQL server. Version 8.3 or higher is required.
adamc@555	73
adamc@555	74 \begin{verbatim}
adamc@555	75 apt-get install postgresql-8.3
adamc@555	76 \end{verbatim}
adamc@555	77
adamc@555	78 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	79
adamc@555	80 \begin{verbatim}
adamc@555	81 apt-get install emacs-goodies-el
adamc@555	82 \end{verbatim}
adamc@555	83
adamc@555	84 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	85
adamc@555	86 \begin{verbatim}
adamc@555	87 GCCARGS=-fnested-functions ./configure
adamc@555	88 \end{verbatim}
adamc@555	89
adamc@555	90 Some OSX users have reported needing to use this particular GCCARGS value.
adamc@555	91
adamc@555	92 The Emacs mode can be set to autoload by adding the following to your \texttt{.emacs} file.
adamc@555	93
adamc@555	94 \begin{verbatim}
adamc@555	95 (add-to-list 'load-path "/usr/local/share/emacs/site-lisp/urweb-mode")
adamc@555	96 (load "urweb-mode-startup")
adamc@555	97 \end{verbatim}
adamc@555	98
adamc@555	99 Change the path in the first line if you chose a different Emacs installation path during configuration.
adamc@555	100
adamc@555	101
adamc@556	102 \section{Command-Line Compiler}
adamc@556	103
adamc@556	104 \subsection{Project Files}
adamc@556	105
adamc@556	106 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	107
adamc@556	108 \begin{verbatim}
adamc@556	109 database dbname=test
adamc@556	110 sql crud1.sql
adamc@556	111
adamc@556	112 crud
adamc@556	113 crud1
adamc@556	114 \end{verbatim}
adamc@556	115
adamc@556	116 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	117
adamc@556	118 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	119
adamc@556	120 \begin{verbatim}
adamc@556	121 createdb test
adamc@556	122 psql -f crud1.sql test
adamc@556	123 \end{verbatim}
adamc@556	124
adamc@556	125 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	126
adamc@556	127 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	128
adamc@556	129 A few other named directives are supported. \texttt{prefix PREFIX} sets the prefix included before every URI within the generated application; the default is \texttt{/}. \texttt{exe FILENAME} sets the filename to which to write the output executable; the default for file \texttt{P.urp} is \texttt{P.exe}. \texttt{debug} saves some intermediate C files, which is mostly useful to help in debugging the compiler itself. \texttt{profile} generates an executable that may be used with gprof.
adamc@556	130
adamc@557	131 \subsection{Building an Application}
adamc@557	132
adamc@557	133 To compile project \texttt{P.urp}, simply run
adamc@557	134 \begin{verbatim}
adamc@557	135 urweb P
adamc@557	136 \end{verbatim}
adamc@558	137 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	138
adamc@557	139 To time how long the different compiler phases run, without generating an executable, run
adamc@557	140 \begin{verbatim}
adamc@557	141 urweb -timing P
adamc@557	142 \end{verbatim}
adamc@557	143
adamc@556	144
adamc@529	145 \section{Ur Syntax}
adamc@529	146
adamc@529	147 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 tables, sequences, and cookies.
adamc@524	148
adamc@524	149 \subsection{Lexical Conventions}
adamc@524	150
adamc@524	151 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	152
adamc@524	153 \begin{center}
adamc@524	154 \begin{tabular}{rl}
adamc@524	155 \textbf{\LaTeX} & \textbf{ASCII} \\
adamc@524	156 $\to$ & \cd{->} \\
adamc@524	157 $\times$ & \cd{*} \\
adamc@524	158 $\lambda$ & \cd{fn} \\
adamc@524	159 $\Rightarrow$ & \cd{=>} \\
adamc@529	160 $\neq$ & \cd{<>} \\
adamc@529	161 $\leq$ & \cd{<=} \\
adamc@529	162 $\geq$ & \cd{>=} \\
adamc@524	163 \\
adamc@524	164 $x$ & Normal textual identifier, not beginning with an uppercase letter \\
adamc@525	165 $X$ & Normal textual identifier, beginning with an uppercase letter \\
adamc@524	166 \end{tabular}
adamc@524	167 \end{center}
adamc@524	168
adamc@525	169 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	170
adamc@526	171 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	172
adamc@527	173 This version of the manual doesn't include operator precedences; see \texttt{src/urweb.grm} for that.
adamc@527	174
adamc@552	175 \subsection{\label{core}Core Syntax}
adamc@524	176
adamc@524	177 \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	178 $$\begin{array}{rrcll}
adamc@524	179 \textrm{Kinds} & \kappa &::=& \mt{Type} & \textrm{proper types} \\
adamc@525	180 &&& \mt{Unit} & \textrm{the trivial constructor} \\
adamc@525	181 &&& \mt{Name} & \textrm{field names} \\
adamc@525	182 &&& \kappa \to \kappa & \textrm{type-level functions} \\
adamc@525	183 &&& \{\kappa\} & \textrm{type-level records} \\
adamc@525	184 &&& (\kappa\times^+) & \textrm{type-level tuples} \\
adamc@529	185 &&& \_\_ & \textrm{wildcard} \\
adamc@525	186 &&& (\kappa) & \textrm{explicit precedence} \\
adamc@524	187 \end{array}$$
adamc@524	188
adamc@524	189 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	190 $$\begin{array}{rrcll}
adamc@524	191 \textrm{Explicitness} & ? &::=& :: & \textrm{explicit} \\
adamc@558	192 &&& ::: & \textrm{implicit}
adamc@524	193 \end{array}$$
adamc@524	194
adamc@524	195 \emph{Constructors} are the main class of compile-time-only data. They include proper types and are classified by kinds.
adamc@524	196 $$\begin{array}{rrcll}
adamc@524	197 \textrm{Constructors} & c, \tau &::=& (c) :: \kappa & \textrm{kind annotation} \\
adamc@530	198 &&& \hat{x} & \textrm{constructor variable} \\
adamc@524	199 \\
adamc@525	200 &&& \tau \to \tau & \textrm{function type} \\
adamc@525	201 &&& x \; ? \; \kappa \to \tau & \textrm{polymorphic function type} \\
adamc@525	202 &&& \$ c & \textrm{record type} \\
adamc@524	203 \\
adamc@525	204 &&& c \; c & \textrm{type-level function application} \\
adamc@530	205 &&& \lambda x \; :: \; \kappa \Rightarrow c & \textrm{type-level function abstraction} \\
adamc@524	206 \\
adamc@525	207 &&& () & \textrm{type-level unit} \\
adamc@525	208 &&& \#X & \textrm{field name} \\
adamc@524	209 \\
adamc@525	210 &&& [(c = c)^*] & \textrm{known-length type-level record} \\
adamc@525	211 &&& c \rc c & \textrm{type-level record concatenation} \\
adamc@525	212 &&& \mt{fold} & \textrm{type-level record fold} \\
adamc@524	213 \\
adamc@558	214 &&& (c,^+) & \textrm{type-level tuple} \\
adamc@525	215 &&& c.n & \textrm{type-level tuple projection ($n \in \mathbb N^+$)} \\
adamc@524	216 \\
adamc@525	217 &&& \lambda [c \sim c] \Rightarrow c & \textrm{guarded constructor} \\
adamc@524	218 \\
adamc@529	219 &&& \_ :: \kappa & \textrm{wildcard} \\
adamc@525	220 &&& (c) & \textrm{explicit precedence} \\
adamc@530	221 \\
adamc@530	222 \textrm{Qualified uncapitalized variables} & \hat{x} &::=& x & \textrm{not from a module} \\
adamc@530	223 &&& M.x & \textrm{projection from a module} \\
adamc@525	224 \end{array}$$
adamc@525	225
adamc@525	226 Modules of the module system are described by \emph{signatures}.
adamc@525	227 $$\begin{array}{rrcll}
adamc@525	228 \textrm{Signatures} & S &::=& \mt{sig} \; s^* \; \mt{end} & \textrm{constant} \\
adamc@525	229 &&& X & \textrm{variable} \\
adamc@525	230 &&& \mt{functor}(X : S) : S & \textrm{functor} \\
adamc@529	231 &&& S \; \mt{where} \; \mt{con} \; x = c & \textrm{concretizing an abstract constructor} \\
adamc@525	232 &&& M.X & \textrm{projection from a module} \\
adamc@525	233 \\
adamc@525	234 \textrm{Signature items} & s &::=& \mt{con} \; x :: \kappa & \textrm{abstract constructor} \\
adamc@525	235 &&& \mt{con} \; x :: \kappa = c & \textrm{concrete constructor} \\
adamc@528	236 &&& \mt{datatype} \; x \; x^* = dc\mid^+ & \textrm{algebraic datatype definition} \\
adamc@529	237 &&& \mt{datatype} \; x = \mt{datatype} \; M.x & \textrm{algebraic datatype import} \\
adamc@525	238 &&& \mt{val} \; x : \tau & \textrm{value} \\
adamc@525	239 &&& \mt{structure} \; X : S & \textrm{sub-module} \\
adamc@525	240 &&& \mt{signature} \; X = S & \textrm{sub-signature} \\
adamc@525	241 &&& \mt{include} \; S & \textrm{signature inclusion} \\
adamc@525	242 &&& \mt{constraint} \; c \sim c & \textrm{record disjointness constraint} \\
adamc@525	243 &&& \mt{class} \; x & \textrm{abstract type class} \\
adamc@525	244 &&& \mt{class} \; x = c & \textrm{concrete type class} \\
adamc@525	245 \\
adamc@525	246 \textrm{Datatype constructors} & dc &::=& X & \textrm{nullary constructor} \\
adamc@525	247 &&& X \; \mt{of} \; \tau & \textrm{unary constructor} \\
adamc@524	248 \end{array}$$
adamc@524	249
adamc@526	250 \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	251 $$\begin{array}{rrcll}
adamc@526	252 \textrm{Patterns} & p &::=& \_ & \textrm{wildcard} \\
adamc@526	253 &&& x & \textrm{variable} \\
adamc@526	254 &&& \ell & \textrm{constant} \\
adamc@526	255 &&& \hat{X} & \textrm{nullary constructor} \\
adamc@526	256 &&& \hat{X} \; p & \textrm{unary constructor} \\
adamc@526	257 &&& \{(x = p,)^*\} & \textrm{rigid record pattern} \\
adamc@526	258 &&& \{(x = p,)^+, \ldots\} & \textrm{flexible record pattern} \\
adamc@527	259 &&& (p) & \textrm{explicit precedence} \\
adamc@526	260 \\
adamc@529	261 \textrm{Qualified capitalized variables} & \hat{X} &::=& X & \textrm{not from a module} \\
adamc@526	262 &&& M.X & \textrm{projection from a module} \\
adamc@526	263 \end{array}$$
adamc@526	264
adamc@527	265 \emph{Expressions} are the main run-time entities, corresponding to both ``expressions'' and ``statements'' in mainstream imperative languages.
adamc@527	266 $$\begin{array}{rrcll}
adamc@527	267 \textrm{Expressions} & e &::=& e : \tau & \textrm{type annotation} \\
adamc@529	268 &&& \hat{x} & \textrm{variable} \\
adamc@529	269 &&& \hat{X} & \textrm{datatype constructor} \\
adamc@527	270 &&& \ell & \textrm{constant} \\
adamc@527	271 \\
adamc@527	272 &&& e \; e & \textrm{function application} \\
adamc@527	273 &&& \lambda x : \tau \Rightarrow e & \textrm{function abstraction} \\
adamc@527	274 &&& e [c] & \textrm{polymorphic function application} \\
adamc@527	275 &&& \lambda x \; ? \; \kappa \Rightarrow e & \textrm{polymorphic function abstraction} \\
adamc@527	276 \\
adamc@527	277 &&& \{(c = e,)^*\} & \textrm{known-length record} \\
adamc@527	278 &&& e.c & \textrm{record field projection} \\
adamc@527	279 &&& e \rc e & \textrm{record concatenation} \\
adamc@527	280 &&& e \rcut c & \textrm{removal of a single record field} \\
adamc@527	281 &&& e \rcutM c & \textrm{removal of multiple record fields} \\
adamc@527	282 &&& \mt{fold} & \textrm{fold over fields of a type-level record} \\
adamc@527	283 \\
adamc@527	284 &&& \mt{let} \; ed^* \; \mt{in} \; e \; \mt{end} & \textrm{local definitions} \\
adamc@527	285 \\
adamc@527	286 &&& \mt{case} \; e \; \mt{of} \; (p \Rightarrow e\|)^+ & \textrm{pattern matching} \\
adamc@527	287 \\
adamc@527	288 &&& \lambda [c \sim c] \Rightarrow e & \textrm{guarded expression} \\
adamc@527	289 \\
adamc@527	290 &&& \_ & \textrm{wildcard} \\
adamc@527	291 &&& (e) & \textrm{explicit precedence} \\
adamc@527	292 \\
adamc@527	293 \textrm{Local declarations} & ed &::=& \cd{val} \; x : \tau = e & \textrm{non-recursive value} \\
adamc@527	294 &&& \cd{val} \; \cd{rec} \; (x : \tau = e \; \cd{and})^+ & \textrm{mutually-recursive values} \\
adamc@527	295 \end{array}$$
adamc@527	296
adamc@528	297 \emph{Declarations} primarily bring new symbols into context.
adamc@528	298 $$\begin{array}{rrcll}
adamc@528	299 \textrm{Declarations} & d &::=& \mt{con} \; x :: \kappa = c & \textrm{constructor synonym} \\
adamc@528	300 &&& \mt{datatype} \; x \; x^* = dc\mid^+ & \textrm{algebraic datatype definition} \\
adamc@529	301 &&& \mt{datatype} \; x = \mt{datatype} \; M.x & \textrm{algebraic datatype import} \\
adamc@528	302 &&& \mt{val} \; x : \tau = e & \textrm{value} \\
adamc@528	303 &&& \mt{val} \; \cd{rec} \; (x : \tau = e \; \mt{and})^+ & \textrm{mutually-recursive values} \\
adamc@528	304 &&& \mt{structure} \; X : S = M & \textrm{module definition} \\
adamc@528	305 &&& \mt{signature} \; X = S & \textrm{signature definition} \\
adamc@528	306 &&& \mt{open} \; M & \textrm{module inclusion} \\
adamc@528	307 &&& \mt{constraint} \; c \sim c & \textrm{record disjointness constraint} \\
adamc@528	308 &&& \mt{open} \; \mt{constraints} \; M & \textrm{inclusion of just the constraints from a module} \\
adamc@528	309 &&& \mt{table} \; x : c & \textrm{SQL table} \\
adamc@528	310 &&& \mt{sequence} \; x & \textrm{SQL sequence} \\
adamc@535	311 &&& \mt{cookie} \; x : \tau & \textrm{HTTP cookie} \\
adamc@528	312 &&& \mt{class} \; x = c & \textrm{concrete type class} \\
adamc@528	313 \\
adamc@529	314 \textrm{Modules} & M &::=& \mt{struct} \; d^* \; \mt{end} & \textrm{constant} \\
adamc@529	315 &&& X & \textrm{variable} \\
adamc@529	316 &&& M.X & \textrm{projection} \\
adamc@529	317 &&& M(M) & \textrm{functor application} \\
adamc@529	318 &&& \mt{functor}(X : S) : S = M & \textrm{functor abstraction} \\
adamc@528	319 \end{array}$$
adamc@528	320
adamc@528	321 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	322
adamc@529	323 \subsection{Shorthands}
adamc@529	324
adamc@529	325 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	326
adamc@529	327 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	328
adamc@529	329 A record type may be written $\{(c = c,)^\}$, which elaborates to $\$[(c = c,)^]$.
adamc@529	330
adamc@533	331 The notation $[c_1, \ldots, c_n]$ is shorthand for $[c_1 = (), \ldots, c_n = ()]$.
adamc@533	332
adamc@529	333 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	334
adamc@529	335 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 $b ::= x \mid (x :: \kappa) \mid [c \sim c]$ 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	336
adamc@529	337 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	338
adamc@529	339 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	340
adamc@529	341 A signature item or declaration $\mt{class} \; x = \lambda y :: \mt{Type} \Rightarrow c$ may be abbreviated $\mt{class} \; x \; y = c$.
adamc@529	342
adamc@529	343 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. The same syntax works for variables projected out of modules and for capitalized variables (datatype constructors).
adamc@529	344
adamc@529	345 At the expression level, an analogue is available of the composite $\lambda$ form for constructors. We define the language of binders as $b ::= x \mid (x : \tau) \mid (x \; ? \; \kappa) \mid [c \sim c]$. A lone variable $x$ as a binder stands for an expression variable of unspecified type.
adamc@529	346
adamc@529	347 A $\mt{val}$ or $\mt{val} \; \mt{rec}$ declaration may include expression binders before the equal sign, following the binder grammar from the last paragraph. Such declarations are elaborated into versions that add additional $\lambda$s to the fronts of the righthand sides, as appropriate. The keyword $\mt{fun}$ is a synonym for $\mt{val} \; \mt{rec}$.
adamc@529	348
adamc@529	349 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	350
adamc@529	351 A declaration $\mt{table} \; x : \{(c = c,)^\}$ is elaborated into $\mt{table} \; x : [(c = c,)^]$
adamc@529	352
adamc@529	353 The syntax $\mt{where} \; \mt{type}$ is an alternate form of $\mt{where} \; \mt{con}$.
adamc@529	354
adamc@529	355 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	356
adamc@529	357 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	358
adamc@529	359 A signature item $\mt{table} \; x : c$ is shorthand for $\mt{val} \; x : \mt{Basis}.\mt{sql\_table} \; c$. $\mt{sequence} \; x$ is short for $\mt{val} \; x : \mt{Basis}.\mt{sql\_sequence}$, and $\mt{cookie} \; x : \tau$ is shorthand for $\mt{val} \; x : \mt{Basis}.\mt{http\_cookie} \; \tau$.
adamc@529	360
adamc@530	361
adamc@530	362 \section{Static Semantics}
adamc@530	363
adamc@530	364 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	365
adamc@530	366 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	367 \begin{itemize}
adamc@530	368 \item $\Gamma \vdash c :: \kappa$ assigns a kind to a constructor in a context.
adamc@530	369 \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	370 \item $\Gamma \vdash c \hookrightarrow C$ proves that record constructor $c$ decomposes into set $C$ of field names and record constructors.
adamc@530	371 \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	372 \item $\Gamma \vdash e : \tau$ is a standard typing judgment.
adamc@534	373 \item $\Gamma \vdash p \leadsto \Gamma; \tau$ combines typing of patterns with calculation of which new variables they bind.
adamc@537	374 \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	375 \item $\Gamma \vdash S \equiv S$ is the signature equivalence judgment.
adamc@536	376 \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	377 \item $\Gamma \vdash M : S$ is the module signature checking judgment.
adamc@537	378 \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	379 \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	380 \end{itemize}
adamc@530	381
adamc@530	382 \subsection{Kinding}
adamc@530	383
adamc@530	384 $$\infer{\Gamma \vdash (c) :: \kappa :: \kappa}{
adamc@530	385 \Gamma \vdash c :: \kappa
adamc@530	386 }
adamc@530	387 \quad \infer{\Gamma \vdash x :: \kappa}{
adamc@530	388 x :: \kappa \in \Gamma
adamc@530	389 }
adamc@530	390 \quad \infer{\Gamma \vdash x :: \kappa}{
adamc@530	391 x :: \kappa = c \in \Gamma
adamc@530	392 }$$
adamc@530	393
adamc@530	394 $$\infer{\Gamma \vdash M.x :: \kappa}{
adamc@537	395 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	396 & \mt{proj}(M, \overline{s}, \mt{con} \; x) = \kappa
adamc@530	397 }
adamc@530	398 \quad \infer{\Gamma \vdash M.x :: \kappa}{
adamc@537	399 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	400 & \mt{proj}(M, \overline{s}, \mt{con} \; x) = (\kappa, c)
adamc@530	401 }$$
adamc@530	402
adamc@530	403 $$\infer{\Gamma \vdash \tau_1 \to \tau_2 :: \mt{Type}}{
adamc@530	404 \Gamma \vdash \tau_1 :: \mt{Type}
adamc@530	405 & \Gamma \vdash \tau_2 :: \mt{Type}
adamc@530	406 }
adamc@530	407 \quad \infer{\Gamma \vdash x \; ? \: \kappa \to \tau :: \mt{Type}}{
adamc@530	408 \Gamma, x :: \kappa \vdash \tau :: \mt{Type}
adamc@530	409 }
adamc@530	410 \quad \infer{\Gamma \vdash \$c :: \mt{Type}}{
adamc@530	411 \Gamma \vdash c :: \{\mt{Type}\}
adamc@530	412 }$$
adamc@530	413
adamc@530	414 $$\infer{\Gamma \vdash c_1 \; c_2 :: \kappa_2}{
adamc@530	415 \Gamma \vdash c_1 :: \kappa_1 \to \kappa_2
adamc@530	416 & \Gamma \vdash c_2 :: \kappa_1
adamc@530	417 }
adamc@530	418 \quad \infer{\Gamma \vdash \lambda x \; :: \; \kappa_1 \Rightarrow c :: \kappa_1 \to \kappa_2}{
adamc@530	419 \Gamma, x :: \kappa_1 \vdash c :: \kappa_2
adamc@530	420 }$$
adamc@530	421
adamc@530	422 $$\infer{\Gamma \vdash () :: \mt{Unit}}{}
adamc@530	423 \quad \infer{\Gamma \vdash \#X :: \mt{Name}}{}$$
adamc@530	424
adamc@530	425 $$\infer{\Gamma \vdash [\overline{c_i = c'_i}] :: \{\kappa\}}{
adamc@530	426 \forall i: \Gamma \vdash c_i : \mt{Name}
adamc@530	427 & \Gamma \vdash c'_i :: \kappa
adamc@530	428 & \forall i \neq j: \Gamma \vdash c_i \sim c_j
adamc@530	429 }
adamc@530	430 \quad \infer{\Gamma \vdash c_1 \rc c_2 :: \{\kappa\}}{
adamc@530	431 \Gamma \vdash c_1 :: \{\kappa\}
adamc@530	432 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@530	433 & \Gamma \vdash c_1 \sim c_2
adamc@530	434 }$$
adamc@530	435
adamc@530	436 $$\infer{\Gamma \vdash \mt{fold} :: ((\mt{Name} \to \kappa_1 \to \kappa_2 \to \kappa_2) \to \kappa_2 \to \{\kappa_1\} \to \kappa_2}{}$$
adamc@530	437
adamc@573	438 $$\infer{\Gamma \vdash (\overline c) :: (\kappa_1 \times \ldots \times \kappa_n)}{
adamc@573	439 \forall i: \Gamma \vdash c_i :: \kappa_i
adamc@530	440 }
adamc@573	441 \quad \infer{\Gamma \vdash c.i :: \kappa_i}{
adamc@573	442 \Gamma \vdash c :: (\kappa_1 \times \ldots \times \kappa_n)
adamc@530	443 }$$
adamc@530	444
adamc@530	445 $$\infer{\Gamma \vdash \lambda [c_1 \sim c_2] \Rightarrow c :: \kappa}{
adamc@530	446 \Gamma \vdash c_1 :: \{\kappa'\}
adamc@530	447 & \Gamma \vdash c_2 :: \{\kappa'\}
adamc@530	448 & \Gamma, c_1 \sim c_2 \vdash c :: \kappa
adamc@530	449 }$$
adamc@530	450
adamc@531	451 \subsection{Record Disjointness}
adamc@531	452
adamc@531	453 We will use a keyword $\mt{map}$ as a shorthand, such that, for $f$ of kind $\kappa \to \kappa'$, $\mt{map} \; f$ stands for $\mt{fold} \; (\lambda (x_1 :: \mt{Name}) (x_2 :: \kappa) (x_3 :: \{\kappa'\}) \Rightarrow [x_1 = f \; x_2] \rc x_3) \; []$.
adamc@531	454
adamc@531	455 $$\infer{\Gamma \vdash c_1 \sim c_2}{
adamc@558	456 \Gamma \vdash c_1 \hookrightarrow C_1
adamc@558	457 & \Gamma \vdash c_2 \hookrightarrow C_2
adamc@558	458 & \forall c'_1 \in C_1, c'_2 \in C_2: \Gamma \vdash c'_1 \sim c'_2
adamc@531	459 }
adamc@531	460 \quad \infer{\Gamma \vdash X \sim X'}{
adamc@531	461 X \neq X'
adamc@531	462 }$$
adamc@531	463
adamc@531	464 $$\infer{\Gamma \vdash c_1 \sim c_2}{
adamc@531	465 c'_1 \sim c'_2 \in \Gamma
adamc@558	466 & \Gamma \vdash c'_1 \hookrightarrow C_1
adamc@558	467 & \Gamma \vdash c'_2 \hookrightarrow C_2
adamc@558	468 & c_1 \in C_1
adamc@558	469 & c_2 \in C_2
adamc@531	470 }$$
adamc@531	471
adamc@531	472 $$\infer{\Gamma \vdash c \hookrightarrow \{c\}}{}
adamc@531	473 \quad \infer{\Gamma \vdash [\overline{c = c'}] \hookrightarrow \{\overline{c}\}}{}
adamc@531	474 \quad \infer{\Gamma \vdash c_1 \rc c_2 \hookrightarrow C_1 \cup C_2}{
adamc@531	475 \Gamma \vdash c_1 \hookrightarrow C_1
adamc@531	476 & \Gamma \vdash c_2 \hookrightarrow C_2
adamc@531	477 }
adamc@531	478 \quad \infer{\Gamma \vdash c \hookrightarrow C}{
adamc@531	479 \Gamma \vdash c \equiv c'
adamc@531	480 & \Gamma \vdash c' \hookrightarrow C
adamc@531	481 }
adamc@531	482 \quad \infer{\Gamma \vdash \mt{map} \; f \; c \hookrightarrow C}{
adamc@531	483 \Gamma \vdash c \hookrightarrow C
adamc@531	484 }$$
adamc@531	485
adamc@541	486 \subsection{\label{definitional}Definitional Equality}
adamc@532	487
adamc@532	488 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 $c_2$.
adamc@532	489
adamc@532	490 $$\infer{\Gamma \vdash c \equiv c}{}
adamc@532	491 \quad \infer{\Gamma \vdash c_1 \equiv c_2}{
adamc@532	492 \Gamma \vdash c_2 \equiv c_1
adamc@532	493 }
adamc@532	494 \quad \infer{\Gamma \vdash c_1 \equiv c_3}{
adamc@532	495 \Gamma \vdash c_1 \equiv c_2
adamc@532	496 & \Gamma \vdash c_2 \equiv c_3
adamc@532	497 }
adamc@532	498 \quad \infer{\Gamma \vdash \mathcal C[c_1] \equiv \mathcal C[c_2]}{
adamc@532	499 \Gamma \vdash c_1 \equiv c_2
adamc@532	500 }$$
adamc@532	501
adamc@532	502 $$\infer{\Gamma \vdash x \equiv c}{
adamc@532	503 x :: \kappa = c \in \Gamma
adamc@532	504 }
adamc@532	505 \quad \infer{\Gamma \vdash M.x \equiv c}{
adamc@537	506 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	507 & \mt{proj}(M, \overline{s}, \mt{con} \; x) = (\kappa, c)
adamc@532	508 }
adamc@532	509 \quad \infer{\Gamma \vdash (\overline c).i \equiv c_i}{}$$
adamc@532	510
adamc@532	511 $$\infer{\Gamma \vdash (\lambda x :: \kappa \Rightarrow c) \; c' \equiv [x \mapsto c'] c}{}
adamc@532	512 \quad \infer{\Gamma \vdash c_1 \rc c_2 \equiv c_2 \rc c_1}{}
adamc@532	513 \quad \infer{\Gamma \vdash c_1 \rc (c_2 \rc c_3) \equiv (c_1 \rc c_2) \rc c_3}{}$$
adamc@532	514
adamc@532	515 $$\infer{\Gamma \vdash [] \rc c \equiv c}{}
adamc@532	516 \quad \infer{\Gamma \vdash [\overline{c_1 = c'_1}] \rc [\overline{c_2 = c'_2}] \equiv [\overline{c_1 = c'_1}, \overline{c_2 = c'_2}]}{}$$
adamc@532	517
adamc@532	518 $$\infer{\Gamma \vdash \lambda [c_1 \sim c_2] \Rightarrow c \equiv c}{
adamc@532	519 \Gamma \vdash c_1 \sim c_2
adamc@532	520 }
adamc@532	521 \quad \infer{\Gamma \vdash \mt{fold} \; f \; i \; [] \equiv i}{}
adamc@532	522 \quad \infer{\Gamma \vdash \mt{fold} \; f \; i \; ([c_1 = c_2] \rc c) \equiv f \; c_1 \; c_2 \; (\mt{fold} \; f \; i \; c)}{}$$
adamc@532	523
adamc@532	524 $$\infer{\Gamma \vdash \mt{map} \; (\lambda x \Rightarrow x) \; c \equiv c}{}
adamc@532	525 \quad \infer{\Gamma \vdash \mt{fold} \; f \; i \; (\mt{map} \; f' \; c)
adamc@532	526 \equiv \mt{fold} \; (\lambda (x_1 :: \mt{Name}) (x_2 :: \kappa) \Rightarrow f \; x_1 \; (f' \; x_2)) \; i \; c}{}$$
adamc@532	527
adamc@532	528 $$\infer{\Gamma \vdash \mt{map} \; f \; (c_1 \rc c_2) \equiv \mt{map} \; f \; c_1 \rc \mt{map} \; f \; c_2}{}$$
adamc@531	529
adamc@534	530 \subsection{Expression Typing}
adamc@533	531
adamc@533	532 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	533
adamc@533	534 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	535
adamc@533	536 $$\infer{\Gamma \vdash e : \tau : \tau}{
adamc@533	537 \Gamma \vdash e : \tau
adamc@533	538 }
adamc@533	539 \quad \infer{\Gamma \vdash e : \tau}{
adamc@533	540 \Gamma \vdash e : \tau'
adamc@533	541 & \Gamma \vdash \tau' \equiv \tau
adamc@533	542 }
adamc@533	543 \quad \infer{\Gamma \vdash \ell : T(\ell)}{}$$
adamc@533	544
adamc@533	545 $$\infer{\Gamma \vdash x : \mathcal I(\tau)}{
adamc@533	546 x : \tau \in \Gamma
adamc@533	547 }
adamc@533	548 \quad \infer{\Gamma \vdash M.x : \mathcal I(\tau)}{
adamc@537	549 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	550 & \mt{proj}(M, \overline{s}, \mt{val} \; x) = \tau
adamc@533	551 }
adamc@533	552 \quad \infer{\Gamma \vdash X : \mathcal I(\tau)}{
adamc@533	553 X : \tau \in \Gamma
adamc@533	554 }
adamc@533	555 \quad \infer{\Gamma \vdash M.X : \mathcal I(\tau)}{
adamc@537	556 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	557 & \mt{proj}(M, \overline{s}, \mt{val} \; X) = \tau
adamc@533	558 }$$
adamc@533	559
adamc@533	560 $$\infer{\Gamma \vdash e_1 \; e_2 : \tau_2}{
adamc@533	561 \Gamma \vdash e_1 : \tau_1 \to \tau_2
adamc@533	562 & \Gamma \vdash e_2 : \tau_1
adamc@533	563 }
adamc@533	564 \quad \infer{\Gamma \vdash \lambda x : \tau_1 \Rightarrow e : \tau_1 \to \tau_2}{
adamc@533	565 \Gamma, x : \tau_1 \vdash e : \tau_2
adamc@533	566 }$$
adamc@533	567
adamc@533	568 $$\infer{\Gamma \vdash e [c] : [x \mapsto c]\tau}{
adamc@533	569 \Gamma \vdash e : x :: \kappa \to \tau
adamc@533	570 & \Gamma \vdash c :: \kappa
adamc@533	571 }
adamc@533	572 \quad \infer{\Gamma \vdash \lambda x \; ? \; \kappa \Rightarrow e : x \; ? \; \kappa \to \tau}{
adamc@533	573 \Gamma, x :: \kappa \vdash e : \tau
adamc@533	574 }$$
adamc@533	575
adamc@533	576 $$\infer{\Gamma \vdash \{\overline{c = e}\} : \{\overline{c : \tau}\}}{
adamc@533	577 \forall i: \Gamma \vdash c_i :: \mt{Name}
adamc@533	578 & \Gamma \vdash e_i : \tau_i
adamc@533	579 & \forall i \neq j: \Gamma \vdash c_i \sim c_j
adamc@533	580 }
adamc@533	581 \quad \infer{\Gamma \vdash e.c : \tau}{
adamc@533	582 \Gamma \vdash e : \$([c = \tau] \rc c')
adamc@533	583 }
adamc@533	584 \quad \infer{\Gamma \vdash e_1 \rc e_2 : \$(c_1 \rc c_2)}{
adamc@533	585 \Gamma \vdash e_1 : \$c_1
adamc@533	586 & \Gamma \vdash e_2 : \$c_2
adamc@573	587 & \Gamma \vdash c_1 \sim c_2
adamc@533	588 }$$
adamc@533	589
adamc@533	590 $$\infer{\Gamma \vdash e \rcut c : \$c'}{
adamc@533	591 \Gamma \vdash e : \$([c = \tau] \rc c')
adamc@533	592 }
adamc@533	593 \quad \infer{\Gamma \vdash e \rcutM c : \$c'}{
adamc@533	594 \Gamma \vdash e : \$(c \rc c')
adamc@533	595 }$$
adamc@533	596
adamc@533	597 $$\infer{\Gamma \vdash \mt{fold} : \begin{array}{c}
adamc@533	598 x_1 :: (\{\kappa\} \to \tau)
adamc@533	599 \to (x_2 :: \mt{Name} \to x_3 :: \kappa \to x_4 :: \{\kappa\} \to \lambda [[x_2] \sim x_4]
adamc@533	600 \Rightarrow x_1 \; x_4 \to x_1 \; ([x_2 = x_3] \rc x_4)) \\
adamc@533	601 \to x_1 \; [] \to x_5 :: \{\kappa\} \to x_1 \; x_5
adamc@533	602 \end{array}}{}$$
adamc@533	603
adamc@533	604 $$\infer{\Gamma \vdash \mt{let} \; \overline{ed} \; \mt{in} \; e \; \mt{end} : \tau}{
adamc@533	605 \Gamma \vdash \overline{ed} \leadsto \Gamma'
adamc@533	606 & \Gamma' \vdash e : \tau
adamc@533	607 }
adamc@533	608 \quad \infer{\Gamma \vdash \mt{case} \; e \; \mt{of} \; \overline{p \Rightarrow e} : \tau}{
adamc@533	609 \forall i: \Gamma \vdash p_i \leadsto \Gamma_i, \tau'
adamc@533	610 & \Gamma_i \vdash e_i : \tau
adamc@533	611 }$$
adamc@533	612
adamc@573	613 $$\infer{\Gamma \vdash \lambda [c_1 \sim c_2] \Rightarrow e : \lambda [c_1 \sim c_2] \Rightarrow \tau}{
adamc@533	614 \Gamma \vdash c_1 :: \{\kappa\}
adamc@533	615 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@533	616 & \Gamma, c_1 \sim c_2 \vdash e : \tau
adamc@533	617 }$$
adamc@533	618
adamc@534	619 \subsection{Pattern Typing}
adamc@534	620
adamc@534	621 $$\infer{\Gamma \vdash \_ \leadsto \Gamma; \tau}{}
adamc@534	622 \quad \infer{\Gamma \vdash x \leadsto \Gamma, x : \tau; \tau}{}
adamc@534	623 \quad \infer{\Gamma \vdash \ell \leadsto \Gamma; T(\ell)}{}$$
adamc@534	624
adamc@534	625 $$\infer{\Gamma \vdash X \leadsto \Gamma; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@534	626 X : \overline{x ::: \mt{Type}} \to \tau \in \Gamma
adamc@534	627 & \textrm{$\tau$ not a function type}
adamc@534	628 }
adamc@534	629 \quad \infer{\Gamma \vdash X \; p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@534	630 X : \overline{x ::: \mt{Type}} \to \tau'' \to \tau \in \Gamma
adamc@534	631 & \Gamma \vdash p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau''
adamc@534	632 }$$
adamc@534	633
adamc@534	634 $$\infer{\Gamma \vdash M.X \leadsto \Gamma; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@537	635 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	636 & \mt{proj}(M, \overline{s}, \mt{val} \; X) = \overline{x ::: \mt{Type}} \to \tau
adamc@534	637 & \textrm{$\tau$ not a function type}
adamc@534	638 }$$
adamc@534	639
adamc@534	640 $$\infer{\Gamma \vdash M.X \; p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@537	641 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	642 & \mt{proj}(M, \overline{s}, \mt{val} \; X) = \overline{x ::: \mt{Type}} \to \tau'' \to \tau
adamc@534	643 & \Gamma \vdash p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau''
adamc@534	644 }$$
adamc@534	645
adamc@534	646 $$\infer{\Gamma \vdash \{\overline{x = p}\} \leadsto \Gamma_n; \{\overline{x = \tau}\}}{
adamc@534	647 \Gamma_0 = \Gamma
adamc@534	648 & \forall i: \Gamma_i \vdash p_i \leadsto \Gamma_{i+1}; \tau_i
adamc@534	649 }
adamc@534	650 \quad \infer{\Gamma \vdash \{\overline{x = p}, \ldots\} \leadsto \Gamma_n; \$([\overline{x = \tau}] \rc c)}{
adamc@534	651 \Gamma_0 = \Gamma
adamc@534	652 & \forall i: \Gamma_i \vdash p_i \leadsto \Gamma_{i+1}; \tau_i
adamc@534	653 }$$
adamc@534	654
adamc@535	655 \subsection{Declaration Typing}
adamc@535	656
adamc@535	657 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	658
adamc@541	659 This is the first judgment where we deal with type 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 type classes influence type inference.
adamc@535	660
adamc@558	661 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	662 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	663
adamc@535	664 $$\infer{\Gamma \vdash \cdot \leadsto \Gamma}{}
adamc@535	665 \quad \infer{\Gamma \vdash d, \overline{d} \leadsto \Gamma''}{
adamc@535	666 \Gamma \vdash d \leadsto \Gamma'
adamc@535	667 & \Gamma' \vdash \overline{d} \leadsto \Gamma''
adamc@535	668 }$$
adamc@535	669
adamc@535	670 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa = c \leadsto \Gamma, x :: \kappa = c}{
adamc@535	671 \Gamma \vdash c :: \kappa
adamc@535	672 }
adamc@535	673 \quad \infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leadsto \Gamma'}{
adamc@535	674 \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} \vdash \overline{dc} \leadsto \Gamma'
adamc@535	675 }$$
adamc@535	676
adamc@535	677 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leadsto \Gamma'}{
adamc@537	678 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	679 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@535	680 & \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} = M.z \vdash \overline{dc} \leadsto \Gamma'
adamc@535	681 }$$
adamc@535	682
adamc@535	683 $$\infer{\Gamma \vdash \mt{val} \; x : \tau = e \leadsto \Gamma, x : \tau}{
adamc@535	684 \Gamma \vdash e : \tau
adamc@535	685 }$$
adamc@535	686
adamc@535	687 $$\infer{\Gamma \vdash \mt{val} \; \mt{rec} \; \overline{x : \tau = e} \leadsto \Gamma, \overline{x : \tau}}{
adamc@535	688 \forall i: \Gamma, \overline{x : \tau} \vdash e_i : \tau_i
adamc@535	689 & \textrm{$e_i$ starts with an expression $\lambda$, optionally preceded by constructor and disjointness $\lambda$s}
adamc@535	690 }$$
adamc@535	691
adamc@535	692 $$\infer{\Gamma \vdash \mt{structure} \; X : S = M \leadsto \Gamma, X : S}{
adamc@535	693 \Gamma \vdash M : S
adamc@558	694 & \textrm{ $M$ not a constant or application}
adamc@535	695 }
adamc@558	696 \quad \infer{\Gamma \vdash \mt{structure} \; X : S = M \leadsto \Gamma, X : \mt{selfify}(X, \overline{s})}{
adamc@558	697 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@539	698 }$$
adamc@539	699
adamc@539	700 $$\infer{\Gamma \vdash \mt{signature} \; X = S \leadsto \Gamma, X = S}{
adamc@535	701 \Gamma \vdash S
adamc@535	702 }$$
adamc@535	703
adamc@537	704 $$\infer{\Gamma \vdash \mt{open} \; M \leadsto \Gamma, \mathcal O(M, \overline{s})}{
adamc@537	705 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@535	706 }$$
adamc@535	707
adamc@535	708 $$\infer{\Gamma \vdash \mt{constraint} \; c_1 \sim c_2 \leadsto \Gamma}{
adamc@535	709 \Gamma \vdash c_1 :: \{\kappa\}
adamc@535	710 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@535	711 & \Gamma \vdash c_1 \sim c_2
adamc@535	712 }
adamc@537	713 \quad \infer{\Gamma \vdash \mt{open} \; \mt{constraints} \; M \leadsto \Gamma, \mathcal O_c(M, \overline{s})}{
adamc@537	714 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@535	715 }$$
adamc@535	716
adamc@535	717 $$\infer{\Gamma \vdash \mt{table} \; x : c \leadsto \Gamma, x : \mt{Basis}.\mt{sql\_table} \; c}{
adamc@535	718 \Gamma \vdash c :: \{\mt{Type}\}
adamc@535	719 }
adamc@535	720 \quad \infer{\Gamma \vdash \mt{sequence} \; x \leadsto \Gamma, x : \mt{Basis}.\mt{sql\_sequence}}{}$$
adamc@535	721
adamc@535	722 $$\infer{\Gamma \vdash \mt{cookie} \; x : \tau \leadsto \Gamma, x : \mt{Basis}.\mt{http\_cookie} \; \tau}{
adamc@535	723 \Gamma \vdash \tau :: \mt{Type}
adamc@535	724 }$$
adamc@535	725
adamc@535	726 $$\infer{\Gamma \vdash \mt{class} \; x = c \leadsto \Gamma, x :: \mt{Type} \to \mt{Type} = c}{
adamc@535	727 \Gamma \vdash c :: \mt{Type} \to \mt{Type}
adamc@535	728 }$$
adamc@535	729
adamc@535	730 $$\infer{\overline{y}; x; \Gamma \vdash \cdot \leadsto \Gamma}{}
adamc@535	731 \quad \infer{\overline{y}; x; \Gamma \vdash X \mid \overline{dc} \leadsto \Gamma', X : \overline{y ::: \mt{Type}} \to x \; \overline{y}}{
adamc@535	732 \overline{y}; x; \Gamma \vdash \overline{dc} \leadsto \Gamma'
adamc@535	733 }
adamc@535	734 \quad \infer{\overline{y}; x; \Gamma \vdash X \; \mt{of} \; \tau \mid \overline{dc} \leadsto \Gamma', X : \overline{y ::: \mt{Type}} \to \tau \to x \; \overline{y}}{
adamc@535	735 \overline{y}; x; \Gamma \vdash \overline{dc} \leadsto \Gamma'
adamc@535	736 }$$
adamc@535	737
adamc@537	738 \subsection{Signature Item Typing}
adamc@537	739
adamc@537	740 We appeal to a signature item analogue of the $\mathcal O$ function from the last subsection.
adamc@537	741
adamc@537	742 $$\infer{\Gamma \vdash \cdot \leadsto \Gamma}{}
adamc@537	743 \quad \infer{\Gamma \vdash s, \overline{s} \leadsto \Gamma''}{
adamc@537	744 \Gamma \vdash s \leadsto \Gamma'
adamc@537	745 & \Gamma' \vdash \overline{s} \leadsto \Gamma''
adamc@537	746 }$$
adamc@537	747
adamc@537	748 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa \leadsto \Gamma, x :: \kappa}{}
adamc@537	749 \quad \infer{\Gamma \vdash \mt{con} \; x :: \kappa = c \leadsto \Gamma, x :: \kappa = c}{
adamc@537	750 \Gamma \vdash c :: \kappa
adamc@537	751 }
adamc@537	752 \quad \infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leadsto \Gamma'}{
adamc@537	753 \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} \vdash \overline{dc} \leadsto \Gamma'
adamc@537	754 }$$
adamc@537	755
adamc@537	756 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leadsto \Gamma'}{
adamc@537	757 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	758 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@537	759 & \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} = M.z \vdash \overline{dc} \leadsto \Gamma'
adamc@537	760 }$$
adamc@537	761
adamc@537	762 $$\infer{\Gamma \vdash \mt{val} \; x : \tau \leadsto \Gamma, x : \tau}{
adamc@537	763 \Gamma \vdash \tau :: \mt{Type}
adamc@537	764 }$$
adamc@537	765
adamc@537	766 $$\infer{\Gamma \vdash \mt{structure} \; X : S \leadsto \Gamma, X : S}{
adamc@537	767 \Gamma \vdash S
adamc@537	768 }
adamc@537	769 \quad \infer{\Gamma \vdash \mt{signature} \; X = S \leadsto \Gamma, X = S}{
adamc@537	770 \Gamma \vdash S
adamc@537	771 }$$
adamc@537	772
adamc@537	773 $$\infer{\Gamma \vdash \mt{include} \; S \leadsto \Gamma, \mathcal O(\overline{s})}{
adamc@537	774 \Gamma \vdash S
adamc@537	775 & \Gamma \vdash S \equiv \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	776 }$$
adamc@537	777
adamc@537	778 $$\infer{\Gamma \vdash \mt{constraint} \; c_1 \sim c_2 \leadsto \Gamma, c_1 \sim c_2}{
adamc@537	779 \Gamma \vdash c_1 :: \{\kappa\}
adamc@537	780 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@537	781 }$$
adamc@537	782
adamc@537	783 $$\infer{\Gamma \vdash \mt{class} \; x = c \leadsto \Gamma, x :: \mt{Type} \to \mt{Type} = c}{
adamc@537	784 \Gamma \vdash c :: \mt{Type} \to \mt{Type}
adamc@537	785 }
adamc@537	786 \quad \infer{\Gamma \vdash \mt{class} \; x \leadsto \Gamma, x :: \mt{Type} \to \mt{Type}}{}$$
adamc@537	787
adamc@536	788 \subsection{Signature Compatibility}
adamc@536	789
adamc@558	790 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	791
adamc@537	792 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	793
adamc@536	794 $$\infer{\Gamma \vdash S \equiv S}{}
adamc@536	795 \quad \infer{\Gamma \vdash S_1 \equiv S_2}{
adamc@536	796 \Gamma \vdash S_2 \equiv S_1
adamc@536	797 }
adamc@536	798 \quad \infer{\Gamma \vdash X \equiv S}{
adamc@536	799 X = S \in \Gamma
adamc@536	800 }
adamc@536	801 \quad \infer{\Gamma \vdash M.X \equiv S}{
adamc@537	802 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	803 & \mt{proj}(M, \overline{s}, \mt{signature} \; X) = S
adamc@536	804 }$$
adamc@536	805
adamc@536	806 $$\infer{\Gamma \vdash S \; \mt{where} \; \mt{con} \; x = c \equiv \mt{sig} \; \overline{s^1} \; \mt{con} \; x :: \kappa = c \; \overline{s_2} \; \mt{end}}{
adamc@536	807 \Gamma \vdash S \equiv \mt{sig} \; \overline{s^1} \; \mt{con} \; x :: \kappa \; \overline{s_2} \; \mt{end}
adamc@536	808 & \Gamma \vdash c :: \kappa
adamc@537	809 }
adamc@537	810 \quad \infer{\Gamma \vdash \mt{sig} \; \overline{s^1} \; \mt{include} \; S \; \overline{s^2} \; \mt{end} \equiv \mt{sig} \; \overline{s^1} \; \overline{s} \; \overline{s^2} \; \mt{end}}{
adamc@537	811 \Gamma \vdash S \equiv \mt{sig} \; \overline{s} \; \mt{end}
adamc@536	812 }$$
adamc@536	813
adamc@536	814 $$\infer{\Gamma \vdash S_1 \leq S_2}{
adamc@536	815 \Gamma \vdash S_1 \equiv S_2
adamc@536	816 }
adamc@536	817 \quad \infer{\Gamma \vdash \mt{sig} \; \overline{s} \; \mt{end} \leq \mt{sig} \; \mt{end}}{}
adamc@537	818 \quad \infer{\Gamma \vdash \mt{sig} \; \overline{s} \; \mt{end} \leq \mt{sig} \; s' \; \overline{s'} \; \mt{end}}{
adamc@537	819 \Gamma \vdash \overline{s} \leq s'
adamc@537	820 & \Gamma \vdash s' \leadsto \Gamma'
adamc@537	821 & \Gamma' \vdash \mt{sig} \; \overline{s} \; \mt{end} \leq \mt{sig} \; \overline{s'} \; \mt{end}
adamc@537	822 }$$
adamc@537	823
adamc@537	824 $$\infer{\Gamma \vdash s \; \overline{s} \leq s'}{
adamc@537	825 \Gamma \vdash s \leq s'
adamc@537	826 }
adamc@537	827 \quad \infer{\Gamma \vdash s \; \overline{s} \leq s'}{
adamc@537	828 \Gamma \vdash s \leadsto \Gamma'
adamc@537	829 & \Gamma' \vdash \overline{s} \leq s'
adamc@536	830 }$$
adamc@536	831
adamc@536	832 $$\infer{\Gamma \vdash \mt{functor} (X : S_1) : S_2 \leq \mt{functor} (X : S'_1) : S'_2}{
adamc@536	833 \Gamma \vdash S'_1 \leq S_1
adamc@536	834 & \Gamma, X : S'_1 \vdash S_2 \leq S'_2
adamc@536	835 }$$
adamc@536	836
adamc@537	837 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa \leq \mt{con} \; x :: \kappa}{}
adamc@537	838 \quad \infer{\Gamma \vdash \mt{con} \; x :: \kappa = c \leq \mt{con} \; x :: \kappa}{}
adamc@558	839 \quad \infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leq \mt{con} \; x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type}}{}$$
adamc@537	840
adamc@537	841 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leq \mt{con} \; x :: \mt{Type}^{\mt{len}(y)} \to \mt{Type}}{
adamc@537	842 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	843 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@537	844 }$$
adamc@537	845
adamc@537	846 $$\infer{\Gamma \vdash \mt{class} \; x \leq \mt{con} \; x :: \mt{Type} \to \mt{Type}}{}
adamc@537	847 \quad \infer{\Gamma \vdash \mt{class} \; x = c \leq \mt{con} \; x :: \mt{Type} \to \mt{Type}}{}$$
adamc@537	848
adamc@537	849 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa = c_1 \leq \mt{con} \; x :: \mt{\kappa} = c_2}{
adamc@537	850 \Gamma \vdash c_1 \equiv c_2
adamc@537	851 }
adamc@537	852 \quad \infer{\Gamma \vdash \mt{class} \; x = c_1 \leq \mt{con} \; x :: \mt{Type} \to \mt{Type} = c_2}{
adamc@537	853 \Gamma \vdash c_1 \equiv c_2
adamc@537	854 }$$
adamc@537	855
adamc@537	856 $$\infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leq \mt{datatype} \; x \; \overline{y} = \overline{dc'}}{
adamc@537	857 \Gamma, \overline{y :: \mt{Type}} \vdash \overline{dc} \leq \overline{dc'}
adamc@537	858 }$$
adamc@537	859
adamc@537	860 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leq \mt{datatype} \; x \; \overline{y} = \overline{dc'}}{
adamc@537	861 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	862 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@537	863 & \Gamma, \overline{y :: \mt{Type}} \vdash \overline{dc} \leq \overline{dc'}
adamc@537	864 }$$
adamc@537	865
adamc@537	866 $$\infer{\Gamma \vdash \cdot \leq \cdot}{}
adamc@537	867 \quad \infer{\Gamma \vdash X; \overline{dc} \leq X; \overline{dc'}}{
adamc@537	868 \Gamma \vdash \overline{dc} \leq \overline{dc'}
adamc@537	869 }
adamc@537	870 \quad \infer{\Gamma \vdash X \; \mt{of} \; \tau_1; \overline{dc} \leq X \; \mt{of} \; \tau_2; \overline{dc'}}{
adamc@537	871 \Gamma \vdash \tau_1 \equiv \tau_2
adamc@537	872 & \Gamma \vdash \overline{dc} \leq \overline{dc'}
adamc@537	873 }$$
adamc@537	874
adamc@537	875 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leq \mt{datatype} \; x = \mt{datatype} \; M'.z'}{
adamc@537	876 \Gamma \vdash M.z \equiv M'.z'
adamc@537	877 }$$
adamc@537	878
adamc@537	879 $$\infer{\Gamma \vdash \mt{val} \; x : \tau_1 \leq \mt{val} \; x : \tau_2}{
adamc@537	880 \Gamma \vdash \tau_1 \equiv \tau_2
adamc@537	881 }
adamc@537	882 \quad \infer{\Gamma \vdash \mt{structure} \; X : S_1 \leq \mt{structure} \; X : S_2}{
adamc@537	883 \Gamma \vdash S_1 \leq S_2
adamc@537	884 }
adamc@537	885 \quad \infer{\Gamma \vdash \mt{signature} \; X = S_1 \leq \mt{signature} \; X = S_2}{
adamc@537	886 \Gamma \vdash S_1 \leq S_2
adamc@537	887 & \Gamma \vdash S_2 \leq S_1
adamc@537	888 }$$
adamc@537	889
adamc@537	890 $$\infer{\Gamma \vdash \mt{constraint} \; c_1 \sim c_2 \leq \mt{constraint} \; c'_1 \sim c'_2}{
adamc@537	891 \Gamma \vdash c_1 \equiv c'_1
adamc@537	892 & \Gamma \vdash c_2 \equiv c'_2
adamc@537	893 }$$
adamc@537	894
adamc@537	895 $$\infer{\Gamma \vdash \mt{class} \; x \leq \mt{class} \; x}{}
adamc@537	896 \quad \infer{\Gamma \vdash \mt{class} \; x = c \leq \mt{class} \; x}{}
adamc@537	897 \quad \infer{\Gamma \vdash \mt{class} \; x = c_1 \leq \mt{class} \; x = c_2}{
adamc@537	898 \Gamma \vdash c_1 \equiv c_2
adamc@537	899 }$$
adamc@537	900
adamc@538	901 \subsection{Module Typing}
adamc@538	902
adamc@538	903 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	904
adamc@538	905 $$\infer{\Gamma \vdash M : S}{
adamc@538	906 \Gamma \vdash M : S'
adamc@538	907 & \Gamma \vdash S' \leq S
adamc@538	908 }
adamc@538	909 \quad \infer{\Gamma \vdash \mt{struct} \; \overline{d} \; \mt{end} : \mt{sig} \; \mt{sigOf}(\overline{d}) \; \mt{end}}{
adamc@538	910 \Gamma \vdash \overline{d} \leadsto \Gamma'
adamc@538	911 }
adamc@538	912 \quad \infer{\Gamma \vdash X : S}{
adamc@538	913 X : S \in \Gamma
adamc@538	914 }$$
adamc@538	915
adamc@538	916 $$\infer{\Gamma \vdash M.X : S}{
adamc@538	917 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@538	918 & \mt{proj}(M, \overline{s}, \mt{structure} \; X) = S
adamc@538	919 }$$
adamc@538	920
adamc@538	921 $$\infer{\Gamma \vdash M_1(M_2) : [X \mapsto M_2]S_2}{
adamc@538	922 \Gamma \vdash M_1 : \mt{functor}(X : S_1) : S_2
adamc@538	923 & \Gamma \vdash M_2 : S_1
adamc@538	924 }
adamc@538	925 \quad \infer{\Gamma \vdash \mt{functor} (X : S_1) : S_2 = M : \mt{functor} (X : S_1) : S_2}{
adamc@538	926 \Gamma \vdash S_1
adamc@538	927 & \Gamma, X : S_1 \vdash S_2
adamc@538	928 & \Gamma, X : S_1 \vdash M : S_2
adamc@538	929 }$$
adamc@538	930
adamc@538	931 \begin{eqnarray*}
adamc@538	932 \mt{sigOf}(\cdot) &=& \cdot \\
adamc@538	933 \mt{sigOf}(s \; \overline{s'}) &=& \mt{sigOf}(s) \; \mt{sigOf}(\overline{s'}) \\
adamc@538	934 \\
adamc@538	935 \mt{sigOf}(\mt{con} \; x :: \kappa = c) &=& \mt{con} \; x :: \kappa = c \\
adamc@538	936 \mt{sigOf}(\mt{datatype} \; x \; \overline{y} = \overline{dc}) &=& \mt{datatype} \; x \; \overline{y} = \overline{dc} \\
adamc@538	937 \mt{sigOf}(\mt{datatype} \; x = \mt{datatype} \; M.z) &=& \mt{datatype} \; x = \mt{datatype} \; M.z \\
adamc@538	938 \mt{sigOf}(\mt{val} \; x : \tau = e) &=& \mt{val} \; x : \tau \\
adamc@538	939 \mt{sigOf}(\mt{val} \; \mt{rec} \; \overline{x : \tau = e}) &=& \overline{\mt{val} \; x : \tau} \\
adamc@538	940 \mt{sigOf}(\mt{structure} \; X : S = M) &=& \mt{structure} \; X : S \\
adamc@538	941 \mt{sigOf}(\mt{signature} \; X = S) &=& \mt{signature} \; X = S \\
adamc@538	942 \mt{sigOf}(\mt{open} \; M) &=& \mt{include} \; S \textrm{ (where $\Gamma \vdash M : S$)} \\
adamc@538	943 \mt{sigOf}(\mt{constraint} \; c_1 \sim c_2) &=& \mt{constraint} \; c_1 \sim c_2 \\
adamc@538	944 \mt{sigOf}(\mt{open} \; \mt{constraints} \; M) &=& \cdot \\
adamc@538	945 \mt{sigOf}(\mt{table} \; x : c) &=& \mt{table} \; x : c \\
adamc@538	946 \mt{sigOf}(\mt{sequence} \; x) &=& \mt{sequence} \; x \\
adamc@538	947 \mt{sigOf}(\mt{cookie} \; x : \tau) &=& \mt{cookie} \; x : \tau \\
adamc@538	948 \mt{sigOf}(\mt{class} \; x = c) &=& \mt{class} \; x = c \\
adamc@538	949 \end{eqnarray*}
adamc@539	950 \begin{eqnarray*}
adamc@539	951 \mt{selfify}(M, \cdot) &=& \cdot \\
adamc@558	952 \mt{selfify}(M, s \; \overline{s'}) &=& \mt{selfify}(M, s) \; \mt{selfify}(M, \overline{s'}) \\
adamc@539	953 \\
adamc@539	954 \mt{selfify}(M, \mt{con} \; x :: \kappa) &=& \mt{con} \; x :: \kappa = M.x \\
adamc@539	955 \mt{selfify}(M, \mt{con} \; x :: \kappa = c) &=& \mt{con} \; x :: \kappa = c \\
adamc@539	956 \mt{selfify}(M, \mt{datatype} \; x \; \overline{y} = \overline{dc}) &=& \mt{datatype} \; x \; \overline{y} = \mt{datatype} \; M.x \\
adamc@539	957 \mt{selfify}(M, \mt{datatype} \; x = \mt{datatype} \; M'.z) &=& \mt{datatype} \; x = \mt{datatype} \; M'.z \\
adamc@539	958 \mt{selfify}(M, \mt{val} \; x : \tau) &=& \mt{val} \; x : \tau \\
adamc@539	959 \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	960 \mt{selfify}(M, \mt{signature} \; X = S) &=& \mt{signature} \; X = S \\
adamc@539	961 \mt{selfify}(M, \mt{include} \; S) &=& \mt{include} \; S \\
adamc@539	962 \mt{selfify}(M, \mt{constraint} \; c_1 \sim c_2) &=& \mt{constraint} \; c_1 \sim c_2 \\
adamc@539	963 \mt{selfify}(M, \mt{class} \; x) &=& \mt{class} \; x = M.x \\
adamc@539	964 \mt{selfify}(M, \mt{class} \; x = c) &=& \mt{class} \; x = c \\
adamc@539	965 \end{eqnarray*}
adamc@539	966
adamc@540	967 \subsection{Module Projection}
adamc@540	968
adamc@540	969 \begin{eqnarray*}
adamc@540	970 \mt{proj}(M, \mt{con} \; x :: \kappa \; \overline{s}, \mt{con} \; x) &=& \kappa \\
adamc@540	971 \mt{proj}(M, \mt{con} \; x :: \kappa = c \; \overline{s}, \mt{con} \; x) &=& (\kappa, c) \\
adamc@540	972 \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	973 \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	974 && \textrm{and $\mt{proj}(M', \overline{s'}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})$)} \\
adamc@540	975 \mt{proj}(M, \mt{class} \; x \; \overline{s}, \mt{con} \; x) &=& \mt{Type} \to \mt{Type} \\
adamc@540	976 \mt{proj}(M, \mt{class} \; x = c \; \overline{s}, \mt{con} \; x) &=& (\mt{Type} \to \mt{Type}, c) \\
adamc@540	977 \\
adamc@540	978 \mt{proj}(M, \mt{datatype} \; x \; \overline{y} = \overline{dc} \; \overline{s}, \mt{datatype} \; x) &=& (\overline{y}, \overline{dc}) \\
adamc@540	979 \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	980 \\
adamc@540	981 \mt{proj}(M, \mt{val} \; x : \tau \; \overline{s}, \mt{val} \; x) &=& \tau \\
adamc@540	982 \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	983 \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	984 \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	985 && \textrm{and $\mt{proj}(M', \overline{s'}, \mt{datatype} \; z = (\overline{y}, \overline{dc})$ and $X \in \overline{dc}$)} \\
adamc@540	986 \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	987 && \textrm{and $\mt{proj}(M', \overline{s'}, \mt{datatype} \; z = (\overline{y}, \overline{dc})$ and $X \; \mt{of} \; \tau \in \overline{dc}$)} \\
adamc@540	988 \\
adamc@540	989 \mt{proj}(M, \mt{structure} \; X : S \; \overline{s}, \mt{structure} \; X) &=& S \\
adamc@540	990 \\
adamc@540	991 \mt{proj}(M, \mt{signature} \; X = S \; \overline{s}, \mt{signature} \; X) &=& S \\
adamc@540	992 \\
adamc@540	993 \mt{proj}(M, \mt{con} \; x :: \kappa \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@540	994 \mt{proj}(M, \mt{con} \; x :: \kappa = c \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@540	995 \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	996 \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	997 \mt{proj}(M, \mt{val} \; x : \tau \; \overline{s}, V) &=& \mt{proj}(M, \overline{s}, V) \\
adamc@540	998 \mt{proj}(M, \mt{structure} \; X : S \; \overline{s}, V) &=& [X \mapsto M.X]\mt{proj}(M, \overline{s}, V) \\
adamc@540	999 \mt{proj}(M, \mt{signature} \; X = S \; \overline{s}, V) &=& [X \mapsto M.X]\mt{proj}(M, \overline{s}, V) \\
adamc@540	1000 \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	1001 \mt{proj}(M, \mt{constraint} \; c_1 \sim c_2 \; \overline{s}, V) &=& \mt{proj}(M, \overline{s}, V) \\
adamc@540	1002 \mt{proj}(M, \mt{class} \; x \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@540	1003 \mt{proj}(M, \mt{class} \; x = c \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@540	1004 \end{eqnarray*}
adamc@540	1005
adamc@541	1006
adamc@541	1007 \section{Type Inference}
adamc@541	1008
adamc@541	1009 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	1010
adamc@541	1011 \subsection{Basic Unification}
adamc@541	1012
adamc@560	1013 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	1014
adamc@560	1015 Type-checking can require evaluating recursive functional programs, thanks to the type-level $\mt{fold}$ 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	1016
adamc@541	1017 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	1018
adamc@541	1019 \subsection{Unifying Record Types}
adamc@541	1020
adamc@570	1021 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	1022
adamc@541	1023 \subsection{\label{typeclasses}Type Classes}
adamc@541	1024
adamc@541	1025 Ur includes a type class facility inspired by Haskell's. The current version is very rudimentary, only supporting instances for particular types built up from abstract types and datatypes and type-level application.
adamc@541	1026
adamc@541	1027 Type classes are integrated with the module system. A type class is just a constructor of kind $\mt{Type} \to \mt{Type}$. By marking such a constructor $c$ as a type class, the programmer instructs the type inference engine to, in each scope, record all values of types $c \; \tau$ 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	1028
adamc@541	1029 The ``dictionary encoding'' often used in Haskell implementations is made explicit in Ur. Type 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	1030
adamc@541	1031 Just as for types, type classes may be exported from modules, and they may be exported as concrete or abstract. Concrete type classes have their ``real'' definitions exposed, so that client code may add new instances freely. Abstract type 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	1032
adamc@541	1033 \subsection{Reverse-Engineering Record Types}
adamc@541	1034
adamc@541	1035 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 folded over, yielding known results. Often, as in the case of $\mt{map}$-like folds, the base and recursive cases of a fold produce constructors with different top-level structure. Thus, if the result of the fold is known, examining its top-level structure reveals whether the record being folded over is empty or not. If it's 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	1036
adamc@541	1037 \subsection{Implicit Arguments in Functor Applications}
adamc@541	1038
adamc@541	1039 Constructor, constraint, and type 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	1040
adamc@541	1041
adamc@542	1042 \section{The Ur Standard Library}
adamc@542	1043
adamc@542	1044 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	1045
adamc@542	1046 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	1047
adamc@542	1048 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	1049 $$\begin{array}{l}
adamc@542	1050 \mt{type} \; \mt{int} \\
adamc@542	1051 \mt{type} \; \mt{float} \\
adamc@542	1052 \mt{type} \; \mt{string} \\
adamc@542	1053 \mt{type} \; \mt{time} \\
adamc@542	1054 \\
adamc@542	1055 \mt{type} \; \mt{unit} = \{\} \\
adamc@542	1056 \\
adamc@542	1057 \mt{datatype} \; \mt{bool} = \mt{False} \mid \mt{True} \\
adamc@542	1058 \\
adamc@542	1059 \mt{datatype} \; \mt{option} \; \mt{t} = \mt{None} \mid \mt{Some} \; \mt{of} \; \mt{t}
adamc@542	1060 \end{array}$$
adamc@542	1061
adamc@542	1062
adamc@542	1063 \section{The Ur/Web Standard Library}
adamc@542	1064
adamc@542	1065 \subsection{Transactions}
adamc@542	1066
adamc@542	1067 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	1068 $$\begin{array}{l}
adamc@542	1069 \mt{con} \; \mt{transaction} :: \mt{Type} \to \mt{Type} \\
adamc@542	1070 \\
adamc@542	1071 \mt{val} \; \mt{return} : \mt{t} ::: \mt{Type} \to \mt{t} \to \mt{transaction} \; \mt{t} \\
adamc@542	1072 \mt{val} \; \mt{bind} : \mt{t_1} ::: \mt{Type} \to \mt{t_2} ::: \mt{Type} \to \mt{transaction} \; \mt{t_1} \to (\mt{t_1} \to \mt{transaction} \; \mt{t_2}) \to \mt{transaction} \; \mt{t_2}
adamc@542	1073 \end{array}$$
adamc@542	1074
adamc@542	1075 \subsection{HTTP}
adamc@542	1076
adamc@542	1077 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	1078 $$\begin{array}{l}
adamc@542	1079 \mt{val} \; \mt{requestHeader} : \mt{string} \to \mt{transaction} \; (\mt{option} \; \mt{string}) \\
adamc@542	1080 \\
adamc@542	1081 \mt{con} \; \mt{http\_cookie} :: \mt{Type} \to \mt{Type} \\
adamc@542	1082 \mt{val} \; \mt{getCookie} : \mt{t} ::: \mt{Type} \to \mt{http\_cookie} \; \mt{t} \to \mt{transaction} \; (\mt{option} \; \mt{t}) \\
adamc@542	1083 \mt{val} \; \mt{setCookie} : \mt{t} ::: \mt{Type} \to \mt{http\_cookie} \; \mt{t} \to \mt{t} \to \mt{transaction} \; \mt{unit}
adamc@542	1084 \end{array}$$
adamc@542	1085
adamc@543	1086 \subsection{SQL}
adamc@543	1087
adamc@543	1088 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	1089 $$\begin{array}{l}
adamc@543	1090 \mt{con} \; \mt{sql\_table} :: \{\mt{Type}\} \to \mt{Type}
adamc@543	1091 \end{array}$$
adamc@543	1092
adamc@543	1093 \subsubsection{Queries}
adamc@543	1094
adamc@543	1095 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	1096 $$\begin{array}{l}
adamc@543	1097 \mt{con} \; \mt{sql\_query} :: \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@543	1098 \mt{val} \; \mt{sql\_query} : \mt{tables} ::: \{\{\mt{Type}\}\} \\
adamc@543	1099 \hspace{.1in} \to \mt{selectedFields} ::: \{\{\mt{Type}\}\} \\
adamc@543	1100 \hspace{.1in} \to \mt{selectedExps} ::: \{\mt{Type}\} \\
adamc@543	1101 \hspace{.1in} \to \{\mt{Rows} : \mt{sql\_query1} \; \mt{tables} \; \mt{selectedFields} \; \mt{selectedExps}, \\
adamc@543	1102 \hspace{.2in} \mt{OrderBy} : \mt{sql\_order\_by} \; \mt{tables} \; \mt{selectedExps}, \\
adamc@543	1103 \hspace{.2in} \mt{Limit} : \mt{sql\_limit}, \\
adamc@543	1104 \hspace{.2in} \mt{Offset} : \mt{sql\_offset}\} \\
adamc@543	1105 \hspace{.1in} \to \mt{sql\_query} \; \mt{selectedFields} \; \mt{selectedExps}
adamc@543	1106 \end{array}$$
adamc@543	1107
adamc@545	1108 Queries are used by folding over their results inside transactions.
adamc@545	1109 $$\begin{array}{l}
adamc@545	1110 \mt{val} \; \mt{query} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \lambda [\mt{tables} \sim \mt{exps}] \Rightarrow \mt{state} ::: \mt{Type} \to \mt{sql\_query} \; \mt{tables} \; \mt{exps} \\
adamc@545	1111 \hspace{.1in} \to (\$(\mt{exps} \rc \mt{fold} \; (\lambda \mt{nm} \; (\mt{fields} :: \{\mt{Type}\}) \; \mt{acc} \; [[\mt{nm}] \sim \mt{acc}] \Rightarrow [\mt{nm} = \$\mt{fields}] \rc \mt{acc}) \; [] \; \mt{tables}) \\
adamc@545	1112 \hspace{.2in} \to \mt{state} \to \mt{transaction} \; \mt{state}) \\
adamc@545	1113 \hspace{.1in} \to \mt{state} \to \mt{transaction} \; \mt{state}
adamc@545	1114 \end{array}$$
adamc@545	1115
adamc@543	1116 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	1117 $$\begin{array}{l}
adamc@543	1118 \mt{con} \; \mt{sql\_query1} :: \{\{\mt{Type}\}\} \to \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@543	1119 \\
adamc@543	1120 \mt{type} \; \mt{sql\_relop} \\
adamc@543	1121 \mt{val} \; \mt{sql\_union} : \mt{sql\_relop} \\
adamc@543	1122 \mt{val} \; \mt{sql\_intersect} : \mt{sql\_relop} \\
adamc@543	1123 \mt{val} \; \mt{sql\_except} : \mt{sql\_relop} \\
adamc@543	1124 \mt{val} \; \mt{sql\_relop} : \mt{tables1} ::: \{\{\mt{Type}\}\} \\
adamc@543	1125 \hspace{.1in} \to \mt{tables2} ::: \{\{\mt{Type}\}\} \\
adamc@543	1126 \hspace{.1in} \to \mt{selectedFields} ::: \{\{\mt{Type}\}\} \\
adamc@543	1127 \hspace{.1in} \to \mt{selectedExps} ::: \{\mt{Type}\} \\
adamc@543	1128 \hspace{.1in} \to \mt{sql\_relop} \\
adamc@543	1129 \hspace{.1in} \to \mt{sql\_query1} \; \mt{tables1} \; \mt{selectedFields} \; \mt{selectedExps} \\
adamc@543	1130 \hspace{.1in} \to \mt{sql\_query1} \; \mt{tables2} \; \mt{selectedFields} \; \mt{selectedExps} \\
adamc@543	1131 \hspace{.1in} \to \mt{sql\_query1} \; \mt{selectedFields} \; \mt{selectedFields} \; \mt{selectedExps}
adamc@543	1132 \end{array}$$
adamc@543	1133
adamc@543	1134 $$\begin{array}{l}
adamc@543	1135 \mt{val} \; \mt{sql\_query1} : \mt{tables} ::: \{\{\mt{Type}\}\} \\
adamc@543	1136 \hspace{.1in} \to \mt{grouped} ::: \{\{\mt{Type}\}\} \\
adamc@543	1137 \hspace{.1in} \to \mt{selectedFields} ::: \{\{\mt{Type}\}\} \\
adamc@543	1138 \hspace{.1in} \to \mt{selectedExps} ::: \{\mt{Type}\} \\
adamc@543	1139 \hspace{.1in} \to \{\mt{From} : \$(\mt{fold} \; (\lambda \mt{nm} \; (\mt{fields} :: \{\mt{Type}\}) \; \mt{acc} \; [[\mt{nm}] \sim \mt{acc}] \Rightarrow [\mt{nm} = \mt{sql\_table} \; \mt{fields}] \rc \mt{acc}) \; [] \; \mt{tables}), \\
adamc@543	1140 \hspace{.2in} \mt{Where} : \mt{sql\_exp} \; \mt{tables} \; [] \; [] \; \mt{bool}, \\
adamc@543	1141 \hspace{.2in} \mt{GroupBy} : \mt{sql\_subset} \; \mt{tables} \; \mt{grouped}, \\
adamc@543	1142 \hspace{.2in} \mt{Having} : \mt{sql\_exp} \; \mt{grouped} \; \mt{tables} \; [] \; \mt{bool}, \\
adamc@543	1143 \hspace{.2in} \mt{SelectFields} : \mt{sql\_subset} \; \mt{grouped} \; \mt{selectedFields}, \\
adamc@543	1144 \hspace{.2in} \mt {SelectExps} : \$(\mt{fold} \; (\lambda \mt{nm} \; (\mt{t} :: \mt{Type}) \; \mt{acc} \; [[\mt{nm}] \sim \mt{acc}] \Rightarrow [\mt{nm} = \mt{sql\_exp} \; \mt{grouped} \; \mt{tables} \; [] \; \mt{t}] \rc \mt{acc}) \; [] \; \mt{selectedExps}) \} \\
adamc@543	1145 \hspace{.1in} \to \mt{sql\_query1} \; \mt{tables} \; \mt{selectedFields} \; \mt{selectedExps}
adamc@543	1146 \end{array}$$
adamc@543	1147
adamc@543	1148 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	1149 $$\begin{array}{l}
adamc@543	1150 \mt{con} \; \mt{sql\_subset} :: \{\{\mt{Type}\}\} \to \{\{\mt{Type}\}\} \to \mt{Type} \\
adamc@543	1151 \mt{val} \; \mt{sql\_subset} : \mt{keep\_drop} :: \{(\{\mt{Type}\} \times \{\mt{Type}\})\} \\
adamc@543	1152 \hspace{.1in} \to \mt{sql\_subset} \\
adamc@543	1153 \hspace{.2in} (\mt{fold} \; (\lambda \mt{nm} \; (\mt{fields} :: (\{\mt{Type}\} \times \{\mt{Type}\})) \; \mt{acc} \; [[\mt{nm}] \sim \mt{acc}] \; [\mt{fields}.1 \sim \mt{fields}.2] \Rightarrow \\
adamc@543	1154 \hspace{.3in} [\mt{nm} = \mt{fields}.1 \rc \mt{fields}.2] \rc \mt{acc}) \; [] \; \mt{keep\_drop}) \\
adamc@543	1155 \hspace{.2in} (\mt{fold} \; (\lambda \mt{nm} \; (\mt{fields} :: (\{\mt{Type}\} \times \{\mt{Type}\})) \; \mt{acc} \; [[\mt{nm}] \sim \mt{acc}] \Rightarrow [\mt{nm} = \mt{fields}.1] \rc \mt{acc}) \; [] \; \mt{keep\_drop}) \\
adamc@543	1156 \mt{val} \; \mt{sql\_subset\_all} : \mt{tables} :: \{\{\mt{Type}\}\} \to \mt{sql\_subset} \; \mt{tables} \; \mt{tables}
adamc@543	1157 \end{array}$$
adamc@543	1158
adamc@560	1159 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	1160 $$\begin{array}{l}
adamc@543	1161 \mt{con} \; \mt{sql\_exp} :: \{\{\mt{Type}\}\} \to \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \to \mt{Type}
adamc@543	1162 \end{array}$$
adamc@543	1163
adamc@543	1164 Any field in scope may be converted to an expression.
adamc@543	1165 $$\begin{array}{l}
adamc@543	1166 \mt{val} \; \mt{sql\_field} : \mt{otherTabs} ::: \{\{\mt{Type}\}\} \to \mt{otherFields} ::: \{\mt{Type}\} \\
adamc@543	1167 \hspace{.1in} \to \mt{fieldType} ::: \mt{Type} \to \mt{agg} ::: \{\{\mt{Type}\}\} \\
adamc@543	1168 \hspace{.1in} \to \mt{exps} ::: \{\mt{Type}\} \\
adamc@543	1169 \hspace{.1in} \to \mt{tab} :: \mt{Name} \to \mt{field} :: \mt{Name} \\
adamc@543	1170 \hspace{.1in} \to \mt{sql\_exp} \; ([\mt{tab} = [\mt{field} = \mt{fieldType}] \rc \mt{otherFields}] \rc \mt{otherTabs}) \; \mt{agg} \; \mt{exps} \; \mt{fieldType}
adamc@543	1171 \end{array}$$
adamc@543	1172
adamc@544	1173 There is an analogous function for referencing named expressions.
adamc@544	1174 $$\begin{array}{l}
adamc@544	1175 \mt{val} \; \mt{sql\_exp} : \mt{tabs} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{t} ::: \mt{Type} \to \mt{rest} ::: \{\mt{Type}\} \to \mt{nm} :: \mt{Name} \\
adamc@544	1176 \hspace{.1in} \to \mt{sql\_exp} \; \mt{tabs} \; \mt{agg} \; ([\mt{nm} = \mt{t}] \rc \mt{rest}) \; \mt{t}
adamc@544	1177 \end{array}$$
adamc@544	1178
adamc@544	1179 Ur values of appropriate types may be injected into SQL expressions.
adamc@544	1180 $$\begin{array}{l}
adamc@544	1181 \mt{class} \; \mt{sql\_injectable} \\
adamc@544	1182 \mt{val} \; \mt{sql\_bool} : \mt{sql\_injectable} \; \mt{bool} \\
adamc@544	1183 \mt{val} \; \mt{sql\_int} : \mt{sql\_injectable} \; \mt{int} \\
adamc@544	1184 \mt{val} \; \mt{sql\_float} : \mt{sql\_injectable} \; \mt{float} \\
adamc@544	1185 \mt{val} \; \mt{sql\_string} : \mt{sql\_injectable} \; \mt{string} \\
adamc@544	1186 \mt{val} \; \mt{sql\_time} : \mt{sql\_injectable} \; \mt{time} \\
adamc@544	1187 \mt{val} \; \mt{sql\_option\_bool} : \mt{sql\_injectable} \; (\mt{option} \; \mt{bool}) \\
adamc@544	1188 \mt{val} \; \mt{sql\_option\_int} : \mt{sql\_injectable} \; (\mt{option} \; \mt{int}) \\
adamc@544	1189 \mt{val} \; \mt{sql\_option\_float} : \mt{sql\_injectable} \; (\mt{option} \; \mt{float}) \\
adamc@544	1190 \mt{val} \; \mt{sql\_option\_string} : \mt{sql\_injectable} \; (\mt{option} \; \mt{string}) \\
adamc@544	1191 \mt{val} \; \mt{sql\_option\_time} : \mt{sql\_injectable} \; (\mt{option} \; \mt{time}) \\
adamc@544	1192 \mt{val} \; \mt{sql\_inject} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \to \mt{sql\_injectable} \; \mt{t} \\
adamc@544	1193 \hspace{.1in} \to \mt{t} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{t}
adamc@544	1194 \end{array}$$
adamc@544	1195
adamc@544	1196 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	1197 $$\begin{array}{l}
adamc@544	1198 \mt{val} \; \mt{sql\_is\_null} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544	1199 \hspace{.1in} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; (\mt{option} \; \mt{t}) \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{bool}
adamc@544	1200 \end{array}$$
adamc@544	1201
adamc@559	1202 We have generic nullary, unary, and binary operators.
adamc@544	1203 $$\begin{array}{l}
adamc@544	1204 \mt{con} \; \mt{sql\_nfunc} :: \mt{Type} \to \mt{Type} \\
adamc@544	1205 \mt{val} \; \mt{sql\_current\_timestamp} : \mt{sql\_nfunc} \; \mt{time} \\
adamc@544	1206 \mt{val} \; \mt{sql\_nfunc} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544	1207 \hspace{.1in} \to \mt{sql\_nfunc} \; \mt{t} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{t} \\\end{array}$$
adamc@544	1208
adamc@544	1209 $$\begin{array}{l}
adamc@544	1210 \mt{con} \; \mt{sql\_unary} :: \mt{Type} \to \mt{Type} \to \mt{Type} \\
adamc@544	1211 \mt{val} \; \mt{sql\_not} : \mt{sql\_unary} \; \mt{bool} \; \mt{bool} \\
adamc@544	1212 \mt{val} \; \mt{sql\_unary} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{arg} ::: \mt{Type} \to \mt{res} ::: \mt{Type} \\
adamc@544	1213 \hspace{.1in} \to \mt{sql\_unary} \; \mt{arg} \; \mt{res} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{arg} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{res} \\
adamc@544	1214 \end{array}$$
adamc@544	1215
adamc@544	1216 $$\begin{array}{l}
adamc@544	1217 \mt{con} \; \mt{sql\_binary} :: \mt{Type} \to \mt{Type} \to \mt{Type} \to \mt{Type} \\
adamc@544	1218 \mt{val} \; \mt{sql\_and} : \mt{sql\_binary} \; \mt{bool} \; \mt{bool} \; \mt{bool} \\
adamc@544	1219 \mt{val} \; \mt{sql\_or} : \mt{sql\_binary} \; \mt{bool} \; \mt{bool} \; \mt{bool} \\
adamc@544	1220 \mt{val} \; \mt{sql\_binary} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{arg_1} ::: \mt{Type} \to \mt{arg_2} ::: \mt{Type} \to \mt{res} ::: \mt{Type} \\
adamc@544	1221 \hspace{.1in} \to \mt{sql\_binary} \; \mt{arg_1} \; \mt{arg_2} \; \mt{res} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{arg_1} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{arg_2} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{res}
adamc@544	1222 \end{array}$$
adamc@544	1223
adamc@544	1224 $$\begin{array}{l}
adamc@559	1225 \mt{class} \; \mt{sql\_arith} \\
adamc@559	1226 \mt{val} \; \mt{sql\_int\_arith} : \mt{sql\_arith} \; \mt{int} \\
adamc@559	1227 \mt{val} \; \mt{sql\_float\_arith} : \mt{sql\_arith} \; \mt{float} \\
adamc@559	1228 \mt{val} \; \mt{sql\_neg} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_unary} \; \mt{t} \; \mt{t} \\
adamc@559	1229 \mt{val} \; \mt{sql\_plus} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559	1230 \mt{val} \; \mt{sql\_minus} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559	1231 \mt{val} \; \mt{sql\_times} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559	1232 \mt{val} \; \mt{sql\_div} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559	1233 \mt{val} \; \mt{sql\_mod} : \mt{sql\_binary} \; \mt{int} \; \mt{int} \; \mt{int}
adamc@559	1234 \end{array}$$
adamc@544	1235
adamc@544	1236 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 type 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	1237
adamc@544	1238 $$\begin{array}{l}
adamc@544	1239 \mt{val} \; \mt{sql\_count} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{int}
adamc@544	1240 \end{array}$$
adamc@544	1241
adamc@544	1242 $$\begin{array}{l}
adamc@544	1243 \mt{con} \; \mt{sql\_aggregate} :: \mt{Type} \to \mt{Type} \\
adamc@544	1244 \mt{val} \; \mt{sql\_aggregate} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544	1245 \hspace{.1in} \to \mt{sql\_aggregate} \; \mt{t} \to \mt{sql\_exp} \; \mt{agg} \; \mt{agg} \; \mt{exps} \; \mt{t} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{t}
adamc@544	1246 \end{array}$$
adamc@544	1247
adamc@544	1248 $$\begin{array}{l}
adamc@544	1249 \mt{class} \; \mt{sql\_summable} \\
adamc@544	1250 \mt{val} \; \mt{sql\_summable\_int} : \mt{sql\_summable} \; \mt{int} \\
adamc@544	1251 \mt{val} \; \mt{sql\_summable\_float} : \mt{sql\_summable} \; \mt{float} \\
adamc@544	1252 \mt{val} \; \mt{sql\_avg} : \mt{t} ::: \mt{Type} \to \mt{sql\_summable} \; \mt{t} \to \mt{sql\_aggregate} \; \mt{t} \\
adamc@544	1253 \mt{val} \; \mt{sql\_sum} : \mt{t} ::: \mt{Type} \to \mt{sql\_summable} \mt{t} \to \mt{sql\_aggregate} \; \mt{t}
adamc@544	1254 \end{array}$$
adamc@544	1255
adamc@544	1256 $$\begin{array}{l}
adamc@544	1257 \mt{class} \; \mt{sql\_maxable} \\
adamc@544	1258 \mt{val} \; \mt{sql\_maxable\_int} : \mt{sql\_maxable} \; \mt{int} \\
adamc@544	1259 \mt{val} \; \mt{sql\_maxable\_float} : \mt{sql\_maxable} \; \mt{float} \\
adamc@544	1260 \mt{val} \; \mt{sql\_maxable\_string} : \mt{sql\_maxable} \; \mt{string} \\
adamc@544	1261 \mt{val} \; \mt{sql\_maxable\_time} : \mt{sql\_maxable} \; \mt{time} \\
adamc@544	1262 \mt{val} \; \mt{sql\_max} : \mt{t} ::: \mt{Type} \to \mt{sql\_maxable} \; \mt{t} \to \mt{sql\_aggregate} \; \mt{t} \\
adamc@544	1263 \mt{val} \; \mt{sql\_min} : \mt{t} ::: \mt{Type} \to \mt{sql\_maxable} \; \mt{t} \to \mt{sql\_aggregate} \; \mt{t}
adamc@544	1264 \end{array}$$
adamc@544	1265
adamc@544	1266 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	1267 $$\begin{array}{l}
adamc@544	1268 \mt{type} \; \mt{sql\_direction} \\
adamc@544	1269 \mt{val} \; \mt{sql\_asc} : \mt{sql\_direction} \\
adamc@544	1270 \mt{val} \; \mt{sql\_desc} : \mt{sql\_direction} \\
adamc@544	1271 \\
adamc@544	1272 \mt{con} \; \mt{sql\_order\_by} :: \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@544	1273 \mt{val} \; \mt{sql\_order\_by\_Nil} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{exps} :: \{\mt{Type}\} \to \mt{sql\_order\_by} \; \mt{tables} \; \mt{exps} \\
adamc@544	1274 \mt{val} \; \mt{sql\_order\_by\_Cons} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544	1275 \hspace{.1in} \to \mt{sql\_exp} \; \mt{tables} \; [] \; \mt{exps} \; \mt{t} \to \mt{sql\_direction} \to \mt{sql\_order\_by} \; \mt{tables} \; \mt{exps} \to \mt{sql\_order\_by} \; \mt{tables} \; \mt{exps} \\
adamc@544	1276 \\
adamc@544	1277 \mt{type} \; \mt{sql\_limit} \\
adamc@544	1278 \mt{val} \; \mt{sql\_no\_limit} : \mt{sql\_limit} \\
adamc@544	1279 \mt{val} \; \mt{sql\_limit} : \mt{int} \to \mt{sql\_limit} \\
adamc@544	1280 \\
adamc@544	1281 \mt{type} \; \mt{sql\_offset} \\
adamc@544	1282 \mt{val} \; \mt{sql\_no\_offset} : \mt{sql\_offset} \\
adamc@544	1283 \mt{val} \; \mt{sql\_offset} : \mt{int} \to \mt{sql\_offset}
adamc@544	1284 \end{array}$$
adamc@544	1285
adamc@545	1286
adamc@545	1287 \subsubsection{DML}
adamc@545	1288
adamc@545	1289 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	1290
adamc@545	1291 $$\begin{array}{l}
adamc@545	1292 \mt{type} \; \mt{dml} \\
adamc@545	1293 \mt{val} \; \mt{dml} : \mt{dml} \to \mt{transaction} \; \mt{unit}
adamc@545	1294 \end{array}$$
adamc@545	1295
adamc@545	1296 Properly-typed records may be used to form $\mt{INSERT}$ commands.
adamc@545	1297 $$\begin{array}{l}
adamc@545	1298 \mt{val} \; \mt{insert} : \mt{fields} ::: \{\mt{Type}\} \to \mt{sql\_table} \; \mt{fields} \\
adamc@545	1299 \hspace{.1in} \to \$(\mt{fold} \; (\lambda \mt{nm} \; (\mt{t} :: \mt{Type}) \; \mt{acc} \; [[\mt{nm}] \sim \mt{acc}] \Rightarrow [\mt{nm} = \mt{sql\_exp} \; [] \; [] \; [] \; \mt{t}] \rc \mt{acc}) \; [] \; \mt{fields}) \to \mt{dml}
adamc@545	1300 \end{array}$$
adamc@545	1301
adamc@545	1302 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	1303 $$\begin{array}{l}
adamc@545	1304 \mt{val} \; \mt{update} : \mt{unchanged} ::: \{\mt{Type}\} \to \mt{changed} :: \{\mt{Type}\} \to \lambda [\mt{changed} \sim \mt{unchanged}] \\
adamc@545	1305 \hspace{.1in} \Rightarrow \$(\mt{fold} \; (\lambda \mt{nm} \; (\mt{t} :: \mt{Type}) \; \mt{acc} \; [[\mt{nm}] \sim \mt{acc}] \Rightarrow [\mt{nm} = \mt{sql\_exp} \; [\mt{T} = \mt{changed} \rc \mt{unchanged}] \; [] \; [] \; \mt{t}] \rc \mt{acc}) \; [] \; \mt{changed}) \\
adamc@545	1306 \hspace{.1in} \to \mt{sql\_table} \; (\mt{changed} \rc \mt{unchanged}) \to \mt{sql\_exp} \; [\mt{T} = \mt{changed} \rc \mt{unchanged}] \; [] \; [] \; \mt{bool} \to \mt{dml}
adamc@545	1307 \end{array}$$
adamc@545	1308
adamc@545	1309 A $\mt{DELETE}$ command is formed from a table and a $\mt{WHERE}$ clause.
adamc@545	1310 $$\begin{array}{l}
adamc@545	1311 \mt{val} \; \mt{delete} : \mt{fields} ::: \{\mt{Type}\} \to \mt{sql\_table} \; \mt{fields} \to \mt{sql\_exp} \; [\mt{T} = \mt{fields}] \; [] \; [] \; \mt{bool} \to \mt{dml}
adamc@545	1312 \end{array}$$
adamc@545	1313
adamc@546	1314 \subsubsection{Sequences}
adamc@546	1315
adamc@546	1316 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	1317
adamc@546	1318 $$\begin{array}{l}
adamc@546	1319 \mt{type} \; \mt{sql\_sequence} \\
adamc@546	1320 \mt{val} \; \mt{nextval} : \mt{sql\_sequence} \to \mt{transaction} \; \mt{int}
adamc@546	1321 \end{array}$$
adamc@546	1322
adamc@546	1323
adamc@547	1324 \subsection{XML}
adamc@547	1325
adamc@547	1326 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	1327
adamc@547	1328 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	1329 $$\begin{array}{l}
adamc@547	1330 \mt{con} \; \mt{xml} :: \{\mt{Unit}\} \to \{\mt{Type}\} \to \{\mt{Type}\} \to \mt{Type}
adamc@547	1331 \end{array}$$
adamc@547	1332
adamc@547	1333 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	1334 $$\begin{array}{l}
adamc@547	1335 \mt{con} \; \mt{tag} :: \{\mt{Type}\} \to \{\mt{Unit}\} \to \{\mt{Unit}\} \to \{\mt{Type}\} \to \{\mt{Type}\} \to \mt{Type}
adamc@547	1336 \end{array}$$
adamc@547	1337
adamc@547	1338 Literal text may be injected into XML as ``CDATA.''
adamc@547	1339 $$\begin{array}{l}
adamc@547	1340 \mt{val} \; \mt{cdata} : \mt{ctx} ::: \{\mt{Unit}\} \to \mt{use} ::: \{\mt{Type}\} \to \mt{string} \to \mt{xml} \; \mt{ctx} \; \mt{use} \; []
adamc@547	1341 \end{array}$$
adamc@547	1342
adamc@547	1343 There is a function for producing an XML tree with a particular tag at its root.
adamc@547	1344 $$\begin{array}{l}
adamc@547	1345 \mt{val} \; \mt{tag} : \mt{attrsGiven} ::: \{\mt{Type}\} \to \mt{attrsAbsent} ::: \{\mt{Type}\} \to \mt{ctxOuter} ::: \{\mt{Unit}\} \to \mt{ctxInner} ::: \{\mt{Unit}\} \\
adamc@547	1346 \hspace{.1in} \to \mt{useOuter} ::: \{\mt{Type}\} \to \mt{useInner} ::: \{\mt{Type}\} \to \mt{bindOuter} ::: \{\mt{Type}\} \to \mt{bindInner} ::: \{\mt{Type}\} \\
adamc@547	1347 \hspace{.1in} \to \lambda [\mt{attrsGiven} \sim \mt{attrsAbsent}] \; [\mt{useOuter} \sim \mt{useInner}] \; [\mt{bindOuter} \sim \mt{bindInner}] \Rightarrow \$\mt{attrsGiven} \\
adamc@547	1348 \hspace{.1in} \to \mt{tag} \; (\mt{attrsGiven} \rc \mt{attrsAbsent}) \; \mt{ctxOuter} \; \mt{ctxInner} \; \mt{useOuter} \; \mt{bindOuter} \\
adamc@547	1349 \hspace{.1in} \to \mt{xml} \; \mt{ctxInner} \; \mt{useInner} \; \mt{bindInner} \to \mt{xml} \; \mt{ctxOuter} \; (\mt{useOuter} \rc \mt{useInner}) \; (\mt{bindOuter} \rc \mt{bindInner})
adamc@547	1350 \end{array}$$
adamc@547	1351
adamc@547	1352 Two XML fragments may be concatenated.
adamc@547	1353 $$\begin{array}{l}
adamc@547	1354 \mt{val} \; \mt{join} : \mt{ctx} ::: \{\mt{Unit}\} \to \mt{use_1} ::: \{\mt{Type}\} \to \mt{bind_1} ::: \{\mt{Type}\} \to \mt{bind_2} ::: \{\mt{Type}\} \\
adamc@547	1355 \hspace{.1in} \to \lambda [\mt{use_1} \sim \mt{bind_1}] \; [\mt{bind_1} \sim \mt{bind_2}] \\
adamc@547	1356 \hspace{.1in} \Rightarrow \mt{xml} \; \mt{ctx} \; \mt{use_1} \; \mt{bind_1} \to \mt{xml} \; \mt{ctx} \; (\mt{use_1} \rc \mt{bind_1}) \; \mt{bind_2} \to \mt{xml} \; \mt{ctx} \; \mt{use_1} \; (\mt{bind_1} \rc \mt{bind_2})
adamc@547	1357 \end{array}$$
adamc@547	1358
adamc@547	1359 Finally, any XML fragment may be updated to ``claim'' to use more form fields than it does.
adamc@547	1360 $$\begin{array}{l}
adamc@547	1361 \mt{val} \; \mt{useMore} : \mt{ctx} ::: \{\mt{Unit}\} \to \mt{use_1} ::: \{\mt{Type}\} \to \mt{use_2} ::: \{\mt{Type}\} \to \mt{bind} ::: \{\mt{Type}\} \to \lambda [\mt{use_1} \sim \mt{use_2}] \\
adamc@547	1362 \hspace{.1in} \Rightarrow \mt{xml} \; \mt{ctx} \; \mt{use_1} \; \mt{bind} \to \mt{xml} \; \mt{ctx} \; (\mt{use_1} \rc \mt{use_2}) \; \mt{bind}
adamc@547	1363 \end{array}$$
adamc@547	1364
adamc@547	1365 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	1366
adamc@547	1367 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	1368 $$\begin{array}{l}
adamc@547	1369 \mt{val} \; \mt{error} : \mt{t} ::: \mt{Type} \to \mt{xml} \; [\mt{Body}] \; [] \; [] \to \mt{t}
adamc@547	1370 \end{array}$$
adamc@547	1371
adamc@549	1372
adamc@549	1373 \section{Ur/Web Syntax Extensions}
adamc@549	1374
adamc@549	1375 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	1376
adamc@549	1377 \subsection{SQL}
adamc@549	1378
adamc@549	1379 \subsubsection{Queries}
adamc@549	1380
adamc@550	1381 Queries $Q$ are added to the rules for expressions $e$.
adamc@550	1382
adamc@549	1383 $$\begin{array}{rrcll}
adamc@550	1384 \textrm{Queries} & Q &::=& (q \; [\mt{ORDER} \; \mt{BY} \; (E \; [o],)^+] \; [\mt{LIMIT} \; N] \; [\mt{OFFSET} \; N]) \\
adamc@549	1385 \textrm{Pre-queries} & q &::=& \mt{SELECT} \; P \; \mt{FROM} \; T,^+ \; [\mt{WHERE} \; E] \; [\mt{GROUP} \; \mt{BY} \; p,^+] \; [\mt{HAVING} \; E] \\
adamc@549	1386 &&& \mid q \; R \; q \\
adamc@549	1387 \textrm{Relational operators} & R &::=& \mt{UNION} \mid \mt{INTERSECT} \mid \mt{EXCEPT}
adamc@549	1388 \end{array}$$
adamc@549	1389
adamc@549	1390 $$\begin{array}{rrcll}
adamc@549	1391 \textrm{Projections} & P &::=& \ast & \textrm{all columns} \\
adamc@549	1392 &&& p,^+ & \textrm{particular columns} \\
adamc@549	1393 \textrm{Pre-projections} & p &::=& t.f & \textrm{one column from a table} \\
adamc@558	1394 &&& t.\{\{c\}\} & \textrm{a record of columns from a table (of kind $\{\mt{Type}\}$)} \\
adamc@549	1395 \textrm{Table names} & t &::=& x & \textrm{constant table name (automatically capitalized)} \\
adamc@549	1396 &&& X & \textrm{constant table name} \\
adamc@549	1397 &&& \{\{c\}\} & \textrm{computed table name (of kind $\mt{Name}$)} \\
adamc@549	1398 \textrm{Column names} & f &::=& X & \textrm{constant column name} \\
adamc@549	1399 &&& \{c\} & \textrm{computed column name (of kind $\mt{Name}$)} \\
adamc@549	1400 \textrm{Tables} & T &::=& x & \textrm{table variable, named locally by its own capitalization} \\
adamc@549	1401 &&& x \; \mt{AS} \; t & \textrm{table variable, with local name} \\
adamc@549	1402 &&& \{\{e\}\} \; \mt{AS} \; t & \textrm{computed table expression, with local name} \\
adamc@549	1403 \textrm{SQL expressions} & E &::=& p & \textrm{column references} \\
adamc@549	1404 &&& X & \textrm{named expression references} \\
adamc@549	1405 &&& \{\{e\}\} & \textrm{injected native Ur expressions} \\
adamc@549	1406 &&& \{e\} & \textrm{computed expressions, probably using $\mt{sql\_exp}$ directly} \\
adamc@549	1407 &&& \mt{TRUE} \mid \mt{FALSE} & \textrm{boolean constants} \\
adamc@549	1408 &&& \ell & \textrm{primitive type literals} \\
adamc@549	1409 &&& \mt{NULL} & \textrm{null value (injection of $\mt{None}$)} \\
adamc@549	1410 &&& E \; \mt{IS} \; \mt{NULL} & \textrm{nullness test} \\
adamc@549	1411 &&& n & \textrm{nullary operators} \\
adamc@549	1412 &&& u \; E & \textrm{unary operators} \\
adamc@549	1413 &&& E \; b \; E & \textrm{binary operators} \\
adamc@549	1414 &&& \mt{COUNT}(\ast) & \textrm{count number of rows} \\
adamc@549	1415 &&& a(E) & \textrm{other aggregate function} \\
adamc@549	1416 &&& (E) & \textrm{explicit precedence} \\
adamc@549	1417 \textrm{Nullary operators} & n &::=& \mt{CURRENT\_TIMESTAMP} \\
adamc@549	1418 \textrm{Unary operators} & u &::=& \mt{NOT} \\
adamc@549	1419 \textrm{Binary operators} & b &::=& \mt{AND} \mid \mt{OR} \mid \neq \mid < \mid \leq \mid > \mid \geq \\
adamc@549	1420 \textrm{Aggregate functions} & a &::=& \mt{AVG} \mid \mt{SUM} \mid \mt{MIN} \mid \mt{MAX} \\
adamc@550	1421 \textrm{Directions} & o &::=& \mt{ASC} \mid \mt{DESC} \\
adamc@549	1422 \textrm{SQL integer} & N &::=& n \mid \{e\} \\
adamc@549	1423 \end{array}$$
adamc@549	1424
adamc@549	1425 Additionally, an SQL expression may be inserted into normal Ur code with the syntax $(\mt{SQL} \; E)$ or $(\mt{WHERE} \; E)$.
adamc@549	1426
adamc@550	1427 \subsubsection{DML}
adamc@550	1428
adamc@550	1429 DML commands $D$ are added to the rules for expressions $e$.
adamc@550	1430
adamc@550	1431 $$\begin{array}{rrcll}
adamc@550	1432 \textrm{Commands} & D &::=& (\mt{INSERT} \; \mt{INTO} \; T^E \; (f,^+) \; \mt{VALUES} \; (E,^+)) \\
adamc@550	1433 &&& (\mt{UPDATE} \; T^E \; \mt{SET} \; (f = E,)^+ \; \mt{WHERE} \; E) \\
adamc@550	1434 &&& (\mt{DELETE} \; \mt{FROM} \; T^E \; \mt{WHERE} \; E) \\
adamc@550	1435 \textrm{Table expressions} & T^E &::=& x \mid \{\{e\}\}
adamc@550	1436 \end{array}$$
adamc@550	1437
adamc@550	1438 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	1439
adamc@551	1440 \subsection{XML}
adamc@551	1441
adamc@551	1442 XML fragments $L$ are added to the rules for expressions $e$.
adamc@551	1443
adamc@551	1444 $$\begin{array}{rrcll}
adamc@551	1445 \textrm{XML fragments} & L &::=& \texttt{<xml/>} \mid \texttt{<xml>}l^*\texttt{</xml>} \\
adamc@551	1446 \textrm{XML pieces} & l &::=& \textrm{text} & \textrm{cdata} \\
adamc@551	1447 &&& \texttt{<}g\texttt{/>} & \textrm{tag with no children} \\
adamc@551	1448 &&& \texttt{<}g\texttt{>}l^*\texttt{</}x\texttt{>} & \textrm{tag with children} \\
adamc@559	1449 &&& \{e\} & \textrm{computed XML fragment} \\
adamc@559	1450 &&& \{[e]\} & \textrm{injection of an Ur expression, via the $\mt{Top}.\mt{txt}$ function} \\
adamc@551	1451 \textrm{Tag} & g &::=& h \; (x = v)^* \\
adamc@551	1452 \textrm{Tag head} & h &::=& x & \textrm{tag name} \\
adamc@551	1453 &&& h\{c\} & \textrm{constructor parameter} \\
adamc@551	1454 \textrm{Attribute value} & v &::=& \ell & \textrm{literal value} \\
adamc@551	1455 &&& \{e\} & \textrm{computed value} \\
adamc@551	1456 \end{array}$$
adamc@551	1457
adamc@552	1458
adamc@553	1459 \section{The Structure of Web Applications}
adamc@553	1460
adamc@553	1461 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	1462
adamc@553	1463 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	1464
adamc@553	1465 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	1466
adamc@558	1467 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	1468
adamc@553	1469
adamc@552	1470 \section{Compiler Phases}
adamc@552	1471
adamc@552	1472 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	1473
adamc@552	1474 In this section, we step through the main phases of compilation, noting what consequences each phase has for effective programming.
adamc@552	1475
adamc@552	1476 \subsection{Parse}
adamc@552	1477
adamc@552	1478 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	1479
adamc@552	1480 \subsection{Elaborate}
adamc@552	1481
adamc@552	1482 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	1483
adamc@552	1484 \subsection{Unnest}
adamc@552	1485
adamc@552	1486 Named local function definitions are moved to the top level, to avoid the need to generate closures.
adamc@552	1487
adamc@552	1488 \subsection{Corify}
adamc@552	1489
adamc@552	1490 Module system features are compiled away, through inlining of functor definitions at application sites. Afterward, most abstraction boundaries are broken, facilitating optimization.
adamc@552	1491
adamc@552	1492 \subsection{Especialize}
adamc@552	1493
adamc@552	1494 Functions are specialized to particular argument patterns. This is an important trick for avoiding the need to maintain any closures at runtime.
adamc@552	1495
adamc@552	1496 \subsection{Untangle}
adamc@552	1497
adamc@552	1498 Remove unnecessary mutual recursion, splitting recursive groups into strongly-connected components.
adamc@552	1499
adamc@552	1500 \subsection{Shake}
adamc@552	1501
adamc@552	1502 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	1503
adamc@553	1504 \subsection{\label{tag}Tag}
adamc@552	1505
adamc@552	1506 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	1507
adamc@552	1508 \subsection{Reduce}
adamc@552	1509
adamc@552	1510 Apply definitional equality rules to simplify the program as much as possible. This effectively includes inlining of every non-recursive definition.
adamc@552	1511
adamc@552	1512 \subsection{Unpoly}
adamc@552	1513
adamc@552	1514 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	1515
adamc@552	1516 \subsection{Specialize}
adamc@552	1517
adamc@558	1518 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	1519
adamc@552	1520 \subsection{Shake}
adamc@552	1521
adamc@558	1522 Here the compiler repeats the earlier Shake phase.
adamc@552	1523
adamc@552	1524 \subsection{Monoize}
adamc@552	1525
adamc@552	1526 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	1527
adamc@552	1528 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@552	1529 \subsection{MonoOpt}
adamc@552	1530
adamc@552	1531 Simple algebraic laws are applied to simplify the program, focusing especially on efficient imperative generation of HTML pages.
adamc@552	1532
adamc@552	1533 \subsection{MonoUntangle}
adamc@552	1534
adamc@552	1535 Unnecessary mutual recursion is broken up again.
adamc@552	1536
adamc@552	1537 \subsection{MonoReduce}
adamc@552	1538
adamc@552	1539 Equivalents of the definitional equality rules are applied to simplify programs, with inlining again playing a major role.
adamc@552	1540
adamc@552	1541 \subsection{MonoShake, MonoOpt}
adamc@552	1542
adamc@552	1543 Unneeded declarations are removed, and basic optimizations are repeated.
adamc@552	1544
adamc@552	1545 \subsection{Fuse}
adamc@552	1546
adamc@552	1547 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	1548
adamc@552	1549 \subsection{MonoUntangle, MonoShake}
adamc@552	1550
adamc@552	1551 Fuse often creates more opportunities to remove spurious mutual recursion.
adamc@552	1552
adamc@552	1553 \subsection{Pathcheck}
adamc@552	1554
adamc@552	1555 The compiler checks that no link or action name has been used more than once.
adamc@552	1556
adamc@552	1557 \subsection{Cjrize}
adamc@552	1558
adamc@552	1559 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	1560
adamc@552	1561 \subsection{C Compilation and Linking}
adamc@552	1562
adamc@552	1563 The output of the last phase is pretty-printed as C source code and passed to GCC.
adamc@552	1564
adamc@552	1565
adamc@524	1566 \end{document}

Mercurial > urweb

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