Ur/Web: doc/manual.tex annotate

annotate doc/manual.tex @ 557:0d3db8d6a586

Building an application

author	Adam Chlipala <adamc@hcoop.net>
date	Tue, 09 Dec 2008 11:57:17 -0500
parents	5703a2ad5221
children	390cba747188

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@557	137
adamc@557	138 To time how long the different compiler phases run, without generating an executable, run
adamc@557	139 \begin{verbatim}
adamc@557	140 urweb -timing P
adamc@557	141 \end{verbatim}
adamc@557	142
adamc@556	143
adamc@529	144 \section{Ur Syntax}
adamc@529	145
adamc@529	146 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	147
adamc@524	148 \subsection{Lexical Conventions}
adamc@524	149
adamc@524	150 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	151
adamc@524	152 \begin{center}
adamc@524	153 \begin{tabular}{rl}
adamc@524	154 \textbf{\LaTeX} & \textbf{ASCII} \\
adamc@524	155 $\to$ & \cd{->} \\
adamc@524	156 $\times$ & \cd{*} \\
adamc@524	157 $\lambda$ & \cd{fn} \\
adamc@524	158 $\Rightarrow$ & \cd{=>} \\
adamc@529	159 $\neq$ & \cd{<>} \\
adamc@529	160 $\leq$ & \cd{<=} \\
adamc@529	161 $\geq$ & \cd{>=} \\
adamc@524	162 \\
adamc@524	163 $x$ & Normal textual identifier, not beginning with an uppercase letter \\
adamc@525	164 $X$ & Normal textual identifier, beginning with an uppercase letter \\
adamc@524	165 \end{tabular}
adamc@524	166 \end{center}
adamc@524	167
adamc@525	168 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	169
adamc@526	170 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	171
adamc@527	172 This version of the manual doesn't include operator precedences; see \texttt{src/urweb.grm} for that.
adamc@527	173
adamc@552	174 \subsection{\label{core}Core Syntax}
adamc@524	175
adamc@524	176 \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	177 $$\begin{array}{rrcll}
adamc@524	178 \textrm{Kinds} & \kappa &::=& \mt{Type} & \textrm{proper types} \\
adamc@525	179 &&& \mt{Unit} & \textrm{the trivial constructor} \\
adamc@525	180 &&& \mt{Name} & \textrm{field names} \\
adamc@525	181 &&& \kappa \to \kappa & \textrm{type-level functions} \\
adamc@525	182 &&& \{\kappa\} & \textrm{type-level records} \\
adamc@525	183 &&& (\kappa\times^+) & \textrm{type-level tuples} \\
adamc@529	184 &&& \_\_ & \textrm{wildcard} \\
adamc@525	185 &&& (\kappa) & \textrm{explicit precedence} \\
adamc@524	186 \end{array}$$
adamc@524	187
adamc@524	188 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	189 $$\begin{array}{rrcll}
adamc@524	190 \textrm{Explicitness} & ? &::=& :: & \textrm{explicit} \\
adamc@525	191 &&& \; ::: & \textrm{implicit}
adamc@524	192 \end{array}$$
adamc@524	193
adamc@524	194 \emph{Constructors} are the main class of compile-time-only data. They include proper types and are classified by kinds.
adamc@524	195 $$\begin{array}{rrcll}
adamc@524	196 \textrm{Constructors} & c, \tau &::=& (c) :: \kappa & \textrm{kind annotation} \\
adamc@530	197 &&& \hat{x} & \textrm{constructor variable} \\
adamc@524	198 \\
adamc@525	199 &&& \tau \to \tau & \textrm{function type} \\
adamc@525	200 &&& x \; ? \; \kappa \to \tau & \textrm{polymorphic function type} \\
adamc@525	201 &&& \$ c & \textrm{record type} \\
adamc@524	202 \\
adamc@525	203 &&& c \; c & \textrm{type-level function application} \\
adamc@530	204 &&& \lambda x \; :: \; \kappa \Rightarrow c & \textrm{type-level function abstraction} \\
adamc@524	205 \\
adamc@525	206 &&& () & \textrm{type-level unit} \\
adamc@525	207 &&& \#X & \textrm{field name} \\
adamc@524	208 \\
adamc@525	209 &&& [(c = c)^*] & \textrm{known-length type-level record} \\
adamc@525	210 &&& c \rc c & \textrm{type-level record concatenation} \\
adamc@525	211 &&& \mt{fold} & \textrm{type-level record fold} \\
adamc@524	212 \\
adamc@525	213 &&& (c^+) & \textrm{type-level tuple} \\
adamc@525	214 &&& c.n & \textrm{type-level tuple projection ($n \in \mathbb N^+$)} \\
adamc@524	215 \\
adamc@525	216 &&& \lambda [c \sim c] \Rightarrow c & \textrm{guarded constructor} \\
adamc@524	217 \\
adamc@529	218 &&& \_ :: \kappa & \textrm{wildcard} \\
adamc@525	219 &&& (c) & \textrm{explicit precedence} \\
adamc@530	220 \\
adamc@530	221 \textrm{Qualified uncapitalized variables} & \hat{x} &::=& x & \textrm{not from a module} \\
adamc@530	222 &&& M.x & \textrm{projection from a module} \\
adamc@525	223 \end{array}$$
adamc@525	224
adamc@525	225 Modules of the module system are described by \emph{signatures}.
adamc@525	226 $$\begin{array}{rrcll}
adamc@525	227 \textrm{Signatures} & S &::=& \mt{sig} \; s^* \; \mt{end} & \textrm{constant} \\
adamc@525	228 &&& X & \textrm{variable} \\
adamc@525	229 &&& \mt{functor}(X : S) : S & \textrm{functor} \\
adamc@529	230 &&& S \; \mt{where} \; \mt{con} \; x = c & \textrm{concretizing an abstract constructor} \\
adamc@525	231 &&& M.X & \textrm{projection from a module} \\
adamc@525	232 \\
adamc@525	233 \textrm{Signature items} & s &::=& \mt{con} \; x :: \kappa & \textrm{abstract constructor} \\
adamc@525	234 &&& \mt{con} \; x :: \kappa = c & \textrm{concrete constructor} \\
adamc@528	235 &&& \mt{datatype} \; x \; x^* = dc\mid^+ & \textrm{algebraic datatype definition} \\
adamc@529	236 &&& \mt{datatype} \; x = \mt{datatype} \; M.x & \textrm{algebraic datatype import} \\
adamc@525	237 &&& \mt{val} \; x : \tau & \textrm{value} \\
adamc@525	238 &&& \mt{structure} \; X : S & \textrm{sub-module} \\
adamc@525	239 &&& \mt{signature} \; X = S & \textrm{sub-signature} \\
adamc@525	240 &&& \mt{include} \; S & \textrm{signature inclusion} \\
adamc@525	241 &&& \mt{constraint} \; c \sim c & \textrm{record disjointness constraint} \\
adamc@525	242 &&& \mt{class} \; x & \textrm{abstract type class} \\
adamc@525	243 &&& \mt{class} \; x = c & \textrm{concrete type class} \\
adamc@525	244 \\
adamc@525	245 \textrm{Datatype constructors} & dc &::=& X & \textrm{nullary constructor} \\
adamc@525	246 &&& X \; \mt{of} \; \tau & \textrm{unary constructor} \\
adamc@524	247 \end{array}$$
adamc@524	248
adamc@526	249 \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	250 $$\begin{array}{rrcll}
adamc@526	251 \textrm{Patterns} & p &::=& \_ & \textrm{wildcard} \\
adamc@526	252 &&& x & \textrm{variable} \\
adamc@526	253 &&& \ell & \textrm{constant} \\
adamc@526	254 &&& \hat{X} & \textrm{nullary constructor} \\
adamc@526	255 &&& \hat{X} \; p & \textrm{unary constructor} \\
adamc@526	256 &&& \{(x = p,)^*\} & \textrm{rigid record pattern} \\
adamc@526	257 &&& \{(x = p,)^+, \ldots\} & \textrm{flexible record pattern} \\
adamc@527	258 &&& (p) & \textrm{explicit precedence} \\
adamc@526	259 \\
adamc@529	260 \textrm{Qualified capitalized variables} & \hat{X} &::=& X & \textrm{not from a module} \\
adamc@526	261 &&& M.X & \textrm{projection from a module} \\
adamc@526	262 \end{array}$$
adamc@526	263
adamc@527	264 \emph{Expressions} are the main run-time entities, corresponding to both ``expressions'' and ``statements'' in mainstream imperative languages.
adamc@527	265 $$\begin{array}{rrcll}
adamc@527	266 \textrm{Expressions} & e &::=& e : \tau & \textrm{type annotation} \\
adamc@529	267 &&& \hat{x} & \textrm{variable} \\
adamc@529	268 &&& \hat{X} & \textrm{datatype constructor} \\
adamc@527	269 &&& \ell & \textrm{constant} \\
adamc@527	270 \\
adamc@527	271 &&& e \; e & \textrm{function application} \\
adamc@527	272 &&& \lambda x : \tau \Rightarrow e & \textrm{function abstraction} \\
adamc@527	273 &&& e [c] & \textrm{polymorphic function application} \\
adamc@527	274 &&& \lambda x \; ? \; \kappa \Rightarrow e & \textrm{polymorphic function abstraction} \\
adamc@527	275 \\
adamc@527	276 &&& \{(c = e,)^*\} & \textrm{known-length record} \\
adamc@527	277 &&& e.c & \textrm{record field projection} \\
adamc@527	278 &&& e \rc e & \textrm{record concatenation} \\
adamc@527	279 &&& e \rcut c & \textrm{removal of a single record field} \\
adamc@527	280 &&& e \rcutM c & \textrm{removal of multiple record fields} \\
adamc@527	281 &&& \mt{fold} & \textrm{fold over fields of a type-level record} \\
adamc@527	282 \\
adamc@527	283 &&& \mt{let} \; ed^* \; \mt{in} \; e \; \mt{end} & \textrm{local definitions} \\
adamc@527	284 \\
adamc@527	285 &&& \mt{case} \; e \; \mt{of} \; (p \Rightarrow e\|)^+ & \textrm{pattern matching} \\
adamc@527	286 \\
adamc@527	287 &&& \lambda [c \sim c] \Rightarrow e & \textrm{guarded expression} \\
adamc@527	288 \\
adamc@527	289 &&& \_ & \textrm{wildcard} \\
adamc@527	290 &&& (e) & \textrm{explicit precedence} \\
adamc@527	291 \\
adamc@527	292 \textrm{Local declarations} & ed &::=& \cd{val} \; x : \tau = e & \textrm{non-recursive value} \\
adamc@527	293 &&& \cd{val} \; \cd{rec} \; (x : \tau = e \; \cd{and})^+ & \textrm{mutually-recursive values} \\
adamc@527	294 \end{array}$$
adamc@527	295
adamc@528	296 \emph{Declarations} primarily bring new symbols into context.
adamc@528	297 $$\begin{array}{rrcll}
adamc@528	298 \textrm{Declarations} & d &::=& \mt{con} \; x :: \kappa = c & \textrm{constructor synonym} \\
adamc@528	299 &&& \mt{datatype} \; x \; x^* = dc\mid^+ & \textrm{algebraic datatype definition} \\
adamc@529	300 &&& \mt{datatype} \; x = \mt{datatype} \; M.x & \textrm{algebraic datatype import} \\
adamc@528	301 &&& \mt{val} \; x : \tau = e & \textrm{value} \\
adamc@528	302 &&& \mt{val} \; \cd{rec} \; (x : \tau = e \; \mt{and})^+ & \textrm{mutually-recursive values} \\
adamc@528	303 &&& \mt{structure} \; X : S = M & \textrm{module definition} \\
adamc@528	304 &&& \mt{signature} \; X = S & \textrm{signature definition} \\
adamc@528	305 &&& \mt{open} \; M & \textrm{module inclusion} \\
adamc@528	306 &&& \mt{constraint} \; c \sim c & \textrm{record disjointness constraint} \\
adamc@528	307 &&& \mt{open} \; \mt{constraints} \; M & \textrm{inclusion of just the constraints from a module} \\
adamc@528	308 &&& \mt{table} \; x : c & \textrm{SQL table} \\
adamc@528	309 &&& \mt{sequence} \; x & \textrm{SQL sequence} \\
adamc@535	310 &&& \mt{cookie} \; x : \tau & \textrm{HTTP cookie} \\
adamc@528	311 &&& \mt{class} \; x = c & \textrm{concrete type class} \\
adamc@528	312 \\
adamc@529	313 \textrm{Modules} & M &::=& \mt{struct} \; d^* \; \mt{end} & \textrm{constant} \\
adamc@529	314 &&& X & \textrm{variable} \\
adamc@529	315 &&& M.X & \textrm{projection} \\
adamc@529	316 &&& M(M) & \textrm{functor application} \\
adamc@529	317 &&& \mt{functor}(X : S) : S = M & \textrm{functor abstraction} \\
adamc@528	318 \end{array}$$
adamc@528	319
adamc@528	320 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	321
adamc@529	322 \subsection{Shorthands}
adamc@529	323
adamc@529	324 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	325
adamc@529	326 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	327
adamc@529	328 A record type may be written $\{(c = c,)^\}$, which elaborates to $\$[(c = c,)^]$.
adamc@529	329
adamc@533	330 The notation $[c_1, \ldots, c_n]$ is shorthand for $[c_1 = (), \ldots, c_n = ()]$.
adamc@533	331
adamc@529	332 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	333
adamc@529	334 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	335
adamc@529	336 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	337
adamc@529	338 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	339
adamc@529	340 A signature item or declaration $\mt{class} \; x = \lambda y :: \mt{Type} \Rightarrow c$ may be abbreviated $\mt{class} \; x \; y = c$.
adamc@529	341
adamc@529	342 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	343
adamc@529	344 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	345
adamc@529	346 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	347
adamc@529	348 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	349
adamc@529	350 A declaration $\mt{table} \; x : \{(c = c,)^\}$ is elaborated into $\mt{table} \; x : [(c = c,)^]$
adamc@529	351
adamc@529	352 The syntax $\mt{where} \; \mt{type}$ is an alternate form of $\mt{where} \; \mt{con}$.
adamc@529	353
adamc@529	354 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	355
adamc@529	356 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	357
adamc@529	358 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	359
adamc@530	360
adamc@530	361 \section{Static Semantics}
adamc@530	362
adamc@530	363 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	364
adamc@530	365 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	366 \begin{itemize}
adamc@530	367 \item $\Gamma \vdash c :: \kappa$ assigns a kind to a constructor in a context.
adamc@530	368 \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	369 \item $\Gamma \vdash c \hookrightarrow C$ proves that record constructor $c$ decomposes into set $C$ of field names and record constructors.
adamc@530	370 \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	371 \item $\Gamma \vdash e : \tau$ is a standard typing judgment.
adamc@534	372 \item $\Gamma \vdash p \leadsto \Gamma; \tau$ combines typing of patterns with calculation of which new variables they bind.
adamc@537	373 \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	374 \item $\Gamma \vdash S \equiv S$ is the signature equivalence judgment.
adamc@536	375 \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	376 \item $\Gamma \vdash M : S$ is the module signature checking judgment.
adamc@537	377 \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	378 \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	379 \end{itemize}
adamc@530	380
adamc@530	381 \subsection{Kinding}
adamc@530	382
adamc@530	383 $$\infer{\Gamma \vdash (c) :: \kappa :: \kappa}{
adamc@530	384 \Gamma \vdash c :: \kappa
adamc@530	385 }
adamc@530	386 \quad \infer{\Gamma \vdash x :: \kappa}{
adamc@530	387 x :: \kappa \in \Gamma
adamc@530	388 }
adamc@530	389 \quad \infer{\Gamma \vdash x :: \kappa}{
adamc@530	390 x :: \kappa = c \in \Gamma
adamc@530	391 }$$
adamc@530	392
adamc@530	393 $$\infer{\Gamma \vdash M.x :: \kappa}{
adamc@537	394 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	395 & \mt{proj}(M, \overline{s}, \mt{con} \; x) = \kappa
adamc@530	396 }
adamc@530	397 \quad \infer{\Gamma \vdash M.x :: \kappa}{
adamc@537	398 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	399 & \mt{proj}(M, \overline{s}, \mt{con} \; x) = (\kappa, c)
adamc@530	400 }$$
adamc@530	401
adamc@530	402 $$\infer{\Gamma \vdash \tau_1 \to \tau_2 :: \mt{Type}}{
adamc@530	403 \Gamma \vdash \tau_1 :: \mt{Type}
adamc@530	404 & \Gamma \vdash \tau_2 :: \mt{Type}
adamc@530	405 }
adamc@530	406 \quad \infer{\Gamma \vdash x \; ? \: \kappa \to \tau :: \mt{Type}}{
adamc@530	407 \Gamma, x :: \kappa \vdash \tau :: \mt{Type}
adamc@530	408 }
adamc@530	409 \quad \infer{\Gamma \vdash \$c :: \mt{Type}}{
adamc@530	410 \Gamma \vdash c :: \{\mt{Type}\}
adamc@530	411 }$$
adamc@530	412
adamc@530	413 $$\infer{\Gamma \vdash c_1 \; c_2 :: \kappa_2}{
adamc@530	414 \Gamma \vdash c_1 :: \kappa_1 \to \kappa_2
adamc@530	415 & \Gamma \vdash c_2 :: \kappa_1
adamc@530	416 }
adamc@530	417 \quad \infer{\Gamma \vdash \lambda x \; :: \; \kappa_1 \Rightarrow c :: \kappa_1 \to \kappa_2}{
adamc@530	418 \Gamma, x :: \kappa_1 \vdash c :: \kappa_2
adamc@530	419 }$$
adamc@530	420
adamc@530	421 $$\infer{\Gamma \vdash () :: \mt{Unit}}{}
adamc@530	422 \quad \infer{\Gamma \vdash \#X :: \mt{Name}}{}$$
adamc@530	423
adamc@530	424 $$\infer{\Gamma \vdash [\overline{c_i = c'_i}] :: \{\kappa\}}{
adamc@530	425 \forall i: \Gamma \vdash c_i : \mt{Name}
adamc@530	426 & \Gamma \vdash c'_i :: \kappa
adamc@530	427 & \forall i \neq j: \Gamma \vdash c_i \sim c_j
adamc@530	428 }
adamc@530	429 \quad \infer{\Gamma \vdash c_1 \rc c_2 :: \{\kappa\}}{
adamc@530	430 \Gamma \vdash c_1 :: \{\kappa\}
adamc@530	431 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@530	432 & \Gamma \vdash c_1 \sim c_2
adamc@530	433 }$$
adamc@530	434
adamc@530	435 $$\infer{\Gamma \vdash \mt{fold} :: ((\mt{Name} \to \kappa_1 \to \kappa_2 \to \kappa_2) \to \kappa_2 \to \{\kappa_1\} \to \kappa_2}{}$$
adamc@530	436
adamc@530	437 $$\infer{\Gamma \vdash (\overline c) :: (k_1 \times \ldots \times k_n)}{
adamc@530	438 \forall i: \Gamma \vdash c_i :: k_i
adamc@530	439 }
adamc@530	440 \quad \infer{\Gamma \vdash c.i :: k_i}{
adamc@530	441 \Gamma \vdash c :: (k_1 \times \ldots \times k_n)
adamc@530	442 }$$
adamc@530	443
adamc@530	444 $$\infer{\Gamma \vdash \lambda [c_1 \sim c_2] \Rightarrow c :: \kappa}{
adamc@530	445 \Gamma \vdash c_1 :: \{\kappa'\}
adamc@530	446 & \Gamma \vdash c_2 :: \{\kappa'\}
adamc@530	447 & \Gamma, c_1 \sim c_2 \vdash c :: \kappa
adamc@530	448 }$$
adamc@530	449
adamc@531	450 \subsection{Record Disjointness}
adamc@531	451
adamc@531	452 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	453
adamc@531	454 $$\infer{\Gamma \vdash c_1 \sim c_2}{
adamc@531	455 \Gamma \vdash c_1 \hookrightarrow c'_1
adamc@531	456 & \Gamma \vdash c_2 \hookrightarrow c'_2
adamc@531	457 & \forall c''_1 \in c'_1, c''_2 \in c'_2: \Gamma \vdash c''_1 \sim c''_2
adamc@531	458 }
adamc@531	459 \quad \infer{\Gamma \vdash X \sim X'}{
adamc@531	460 X \neq X'
adamc@531	461 }$$
adamc@531	462
adamc@531	463 $$\infer{\Gamma \vdash c_1 \sim c_2}{
adamc@531	464 c'_1 \sim c'_2 \in \Gamma
adamc@531	465 & \Gamma \vdash c'_1 \hookrightarrow c''_1
adamc@531	466 & \Gamma \vdash c'_2 \hookrightarrow c''_2
adamc@531	467 & c_1 \in c''_1
adamc@531	468 & c_2 \in c''_2
adamc@531	469 }$$
adamc@531	470
adamc@531	471 $$\infer{\Gamma \vdash c \hookrightarrow \{c\}}{}
adamc@531	472 \quad \infer{\Gamma \vdash [\overline{c = c'}] \hookrightarrow \{\overline{c}\}}{}
adamc@531	473 \quad \infer{\Gamma \vdash c_1 \rc c_2 \hookrightarrow C_1 \cup C_2}{
adamc@531	474 \Gamma \vdash c_1 \hookrightarrow C_1
adamc@531	475 & \Gamma \vdash c_2 \hookrightarrow C_2
adamc@531	476 }
adamc@531	477 \quad \infer{\Gamma \vdash c \hookrightarrow C}{
adamc@531	478 \Gamma \vdash c \equiv c'
adamc@531	479 & \Gamma \vdash c' \hookrightarrow C
adamc@531	480 }
adamc@531	481 \quad \infer{\Gamma \vdash \mt{map} \; f \; c \hookrightarrow C}{
adamc@531	482 \Gamma \vdash c \hookrightarrow C
adamc@531	483 }$$
adamc@531	484
adamc@541	485 \subsection{\label{definitional}Definitional Equality}
adamc@532	486
adamc@532	487 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	488
adamc@532	489 $$\infer{\Gamma \vdash c \equiv c}{}
adamc@532	490 \quad \infer{\Gamma \vdash c_1 \equiv c_2}{
adamc@532	491 \Gamma \vdash c_2 \equiv c_1
adamc@532	492 }
adamc@532	493 \quad \infer{\Gamma \vdash c_1 \equiv c_3}{
adamc@532	494 \Gamma \vdash c_1 \equiv c_2
adamc@532	495 & \Gamma \vdash c_2 \equiv c_3
adamc@532	496 }
adamc@532	497 \quad \infer{\Gamma \vdash \mathcal C[c_1] \equiv \mathcal C[c_2]}{
adamc@532	498 \Gamma \vdash c_1 \equiv c_2
adamc@532	499 }$$
adamc@532	500
adamc@532	501 $$\infer{\Gamma \vdash x \equiv c}{
adamc@532	502 x :: \kappa = c \in \Gamma
adamc@532	503 }
adamc@532	504 \quad \infer{\Gamma \vdash M.x \equiv c}{
adamc@537	505 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	506 & \mt{proj}(M, \overline{s}, \mt{con} \; x) = (\kappa, c)
adamc@532	507 }
adamc@532	508 \quad \infer{\Gamma \vdash (\overline c).i \equiv c_i}{}$$
adamc@532	509
adamc@532	510 $$\infer{\Gamma \vdash (\lambda x :: \kappa \Rightarrow c) \; c' \equiv [x \mapsto c'] c}{}
adamc@532	511 \quad \infer{\Gamma \vdash c_1 \rc c_2 \equiv c_2 \rc c_1}{}
adamc@532	512 \quad \infer{\Gamma \vdash c_1 \rc (c_2 \rc c_3) \equiv (c_1 \rc c_2) \rc c_3}{}$$
adamc@532	513
adamc@532	514 $$\infer{\Gamma \vdash [] \rc c \equiv c}{}
adamc@532	515 \quad \infer{\Gamma \vdash [\overline{c_1 = c'_1}] \rc [\overline{c_2 = c'_2}] \equiv [\overline{c_1 = c'_1}, \overline{c_2 = c'_2}]}{}$$
adamc@532	516
adamc@532	517 $$\infer{\Gamma \vdash \lambda [c_1 \sim c_2] \Rightarrow c \equiv c}{
adamc@532	518 \Gamma \vdash c_1 \sim c_2
adamc@532	519 }
adamc@532	520 \quad \infer{\Gamma \vdash \mt{fold} \; f \; i \; [] \equiv i}{}
adamc@532	521 \quad \infer{\Gamma \vdash \mt{fold} \; f \; i \; ([c_1 = c_2] \rc c) \equiv f \; c_1 \; c_2 \; (\mt{fold} \; f \; i \; c)}{}$$
adamc@532	522
adamc@532	523 $$\infer{\Gamma \vdash \mt{map} \; (\lambda x \Rightarrow x) \; c \equiv c}{}
adamc@532	524 \quad \infer{\Gamma \vdash \mt{fold} \; f \; i \; (\mt{map} \; f' \; c)
adamc@532	525 \equiv \mt{fold} \; (\lambda (x_1 :: \mt{Name}) (x_2 :: \kappa) \Rightarrow f \; x_1 \; (f' \; x_2)) \; i \; c}{}$$
adamc@532	526
adamc@532	527 $$\infer{\Gamma \vdash \mt{map} \; f \; (c_1 \rc c_2) \equiv \mt{map} \; f \; c_1 \rc \mt{map} \; f \; c_2}{}$$
adamc@531	528
adamc@534	529 \subsection{Expression Typing}
adamc@533	530
adamc@533	531 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	532
adamc@533	533 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	534
adamc@533	535 $$\infer{\Gamma \vdash e : \tau : \tau}{
adamc@533	536 \Gamma \vdash e : \tau
adamc@533	537 }
adamc@533	538 \quad \infer{\Gamma \vdash e : \tau}{
adamc@533	539 \Gamma \vdash e : \tau'
adamc@533	540 & \Gamma \vdash \tau' \equiv \tau
adamc@533	541 }
adamc@533	542 \quad \infer{\Gamma \vdash \ell : T(\ell)}{}$$
adamc@533	543
adamc@533	544 $$\infer{\Gamma \vdash x : \mathcal I(\tau)}{
adamc@533	545 x : \tau \in \Gamma
adamc@533	546 }
adamc@533	547 \quad \infer{\Gamma \vdash M.x : \mathcal I(\tau)}{
adamc@537	548 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	549 & \mt{proj}(M, \overline{s}, \mt{val} \; x) = \tau
adamc@533	550 }
adamc@533	551 \quad \infer{\Gamma \vdash X : \mathcal I(\tau)}{
adamc@533	552 X : \tau \in \Gamma
adamc@533	553 }
adamc@533	554 \quad \infer{\Gamma \vdash M.X : \mathcal I(\tau)}{
adamc@537	555 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	556 & \mt{proj}(M, \overline{s}, \mt{val} \; X) = \tau
adamc@533	557 }$$
adamc@533	558
adamc@533	559 $$\infer{\Gamma \vdash e_1 \; e_2 : \tau_2}{
adamc@533	560 \Gamma \vdash e_1 : \tau_1 \to \tau_2
adamc@533	561 & \Gamma \vdash e_2 : \tau_1
adamc@533	562 }
adamc@533	563 \quad \infer{\Gamma \vdash \lambda x : \tau_1 \Rightarrow e : \tau_1 \to \tau_2}{
adamc@533	564 \Gamma, x : \tau_1 \vdash e : \tau_2
adamc@533	565 }$$
adamc@533	566
adamc@533	567 $$\infer{\Gamma \vdash e [c] : [x \mapsto c]\tau}{
adamc@533	568 \Gamma \vdash e : x :: \kappa \to \tau
adamc@533	569 & \Gamma \vdash c :: \kappa
adamc@533	570 }
adamc@533	571 \quad \infer{\Gamma \vdash \lambda x \; ? \; \kappa \Rightarrow e : x \; ? \; \kappa \to \tau}{
adamc@533	572 \Gamma, x :: \kappa \vdash e : \tau
adamc@533	573 }$$
adamc@533	574
adamc@533	575 $$\infer{\Gamma \vdash \{\overline{c = e}\} : \{\overline{c : \tau}\}}{
adamc@533	576 \forall i: \Gamma \vdash c_i :: \mt{Name}
adamc@533	577 & \Gamma \vdash e_i : \tau_i
adamc@533	578 & \forall i \neq j: \Gamma \vdash c_i \sim c_j
adamc@533	579 }
adamc@533	580 \quad \infer{\Gamma \vdash e.c : \tau}{
adamc@533	581 \Gamma \vdash e : \$([c = \tau] \rc c')
adamc@533	582 }
adamc@533	583 \quad \infer{\Gamma \vdash e_1 \rc e_2 : \$(c_1 \rc c_2)}{
adamc@533	584 \Gamma \vdash e_1 : \$c_1
adamc@533	585 & \Gamma \vdash e_2 : \$c_2
adamc@533	586 }$$
adamc@533	587
adamc@533	588 $$\infer{\Gamma \vdash e \rcut c : \$c'}{
adamc@533	589 \Gamma \vdash e : \$([c = \tau] \rc c')
adamc@533	590 }
adamc@533	591 \quad \infer{\Gamma \vdash e \rcutM c : \$c'}{
adamc@533	592 \Gamma \vdash e : \$(c \rc c')
adamc@533	593 }$$
adamc@533	594
adamc@533	595 $$\infer{\Gamma \vdash \mt{fold} : \begin{array}{c}
adamc@533	596 x_1 :: (\{\kappa\} \to \tau)
adamc@533	597 \to (x_2 :: \mt{Name} \to x_3 :: \kappa \to x_4 :: \{\kappa\} \to \lambda [[x_2] \sim x_4]
adamc@533	598 \Rightarrow x_1 \; x_4 \to x_1 \; ([x_2 = x_3] \rc x_4)) \\
adamc@533	599 \to x_1 \; [] \to x_5 :: \{\kappa\} \to x_1 \; x_5
adamc@533	600 \end{array}}{}$$
adamc@533	601
adamc@533	602 $$\infer{\Gamma \vdash \mt{let} \; \overline{ed} \; \mt{in} \; e \; \mt{end} : \tau}{
adamc@533	603 \Gamma \vdash \overline{ed} \leadsto \Gamma'
adamc@533	604 & \Gamma' \vdash e : \tau
adamc@533	605 }
adamc@533	606 \quad \infer{\Gamma \vdash \mt{case} \; e \; \mt{of} \; \overline{p \Rightarrow e} : \tau}{
adamc@533	607 \forall i: \Gamma \vdash p_i \leadsto \Gamma_i, \tau'
adamc@533	608 & \Gamma_i \vdash e_i : \tau
adamc@533	609 }$$
adamc@533	610
adamc@533	611 $$\infer{\Gamma \vdash [c_1 \sim c_2] \Rightarrow e : [c_1 \sim c_2] \Rightarrow \tau}{
adamc@533	612 \Gamma \vdash c_1 :: \{\kappa\}
adamc@533	613 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@533	614 & \Gamma, c_1 \sim c_2 \vdash e : \tau
adamc@533	615 }$$
adamc@533	616
adamc@534	617 \subsection{Pattern Typing}
adamc@534	618
adamc@534	619 $$\infer{\Gamma \vdash \_ \leadsto \Gamma; \tau}{}
adamc@534	620 \quad \infer{\Gamma \vdash x \leadsto \Gamma, x : \tau; \tau}{}
adamc@534	621 \quad \infer{\Gamma \vdash \ell \leadsto \Gamma; T(\ell)}{}$$
adamc@534	622
adamc@534	623 $$\infer{\Gamma \vdash X \leadsto \Gamma; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@534	624 X : \overline{x ::: \mt{Type}} \to \tau \in \Gamma
adamc@534	625 & \textrm{$\tau$ not a function type}
adamc@534	626 }
adamc@534	627 \quad \infer{\Gamma \vdash X \; p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@534	628 X : \overline{x ::: \mt{Type}} \to \tau'' \to \tau \in \Gamma
adamc@534	629 & \Gamma \vdash p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau''
adamc@534	630 }$$
adamc@534	631
adamc@534	632 $$\infer{\Gamma \vdash M.X \leadsto \Gamma; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@537	633 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	634 & \mt{proj}(M, \overline{s}, \mt{val} \; X) = \overline{x ::: \mt{Type}} \to \tau
adamc@534	635 & \textrm{$\tau$ not a function type}
adamc@534	636 }$$
adamc@534	637
adamc@534	638 $$\infer{\Gamma \vdash M.X \; p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@537	639 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	640 & \mt{proj}(M, \overline{s}, \mt{val} \; X) = \overline{x ::: \mt{Type}} \to \tau'' \to \tau
adamc@534	641 & \Gamma \vdash p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau''
adamc@534	642 }$$
adamc@534	643
adamc@534	644 $$\infer{\Gamma \vdash \{\overline{x = p}\} \leadsto \Gamma_n; \{\overline{x = \tau}\}}{
adamc@534	645 \Gamma_0 = \Gamma
adamc@534	646 & \forall i: \Gamma_i \vdash p_i \leadsto \Gamma_{i+1}; \tau_i
adamc@534	647 }
adamc@534	648 \quad \infer{\Gamma \vdash \{\overline{x = p}, \ldots\} \leadsto \Gamma_n; \$([\overline{x = \tau}] \rc c)}{
adamc@534	649 \Gamma_0 = \Gamma
adamc@534	650 & \forall i: \Gamma_i \vdash p_i \leadsto \Gamma_{i+1}; \tau_i
adamc@534	651 }$$
adamc@534	652
adamc@535	653 \subsection{Declaration Typing}
adamc@535	654
adamc@535	655 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	656
adamc@541	657 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	658
adamc@537	659 We presuppose the existence of a function $\mathcal O$, where $\mathcal(M, \overline{s})$ 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 $S$.
adamc@537	660
adamc@537	661 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	662
adamc@535	663 $$\infer{\Gamma \vdash \cdot \leadsto \Gamma}{}
adamc@535	664 \quad \infer{\Gamma \vdash d, \overline{d} \leadsto \Gamma''}{
adamc@535	665 \Gamma \vdash d \leadsto \Gamma'
adamc@535	666 & \Gamma' \vdash \overline{d} \leadsto \Gamma''
adamc@535	667 }$$
adamc@535	668
adamc@535	669 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa = c \leadsto \Gamma, x :: \kappa = c}{
adamc@535	670 \Gamma \vdash c :: \kappa
adamc@535	671 }
adamc@535	672 \quad \infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leadsto \Gamma'}{
adamc@535	673 \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} \vdash \overline{dc} \leadsto \Gamma'
adamc@535	674 }$$
adamc@535	675
adamc@535	676 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leadsto \Gamma'}{
adamc@537	677 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	678 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@535	679 & \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} = M.z \vdash \overline{dc} \leadsto \Gamma'
adamc@535	680 }$$
adamc@535	681
adamc@535	682 $$\infer{\Gamma \vdash \mt{val} \; x : \tau = e \leadsto \Gamma, x : \tau}{
adamc@535	683 \Gamma \vdash e : \tau
adamc@535	684 }$$
adamc@535	685
adamc@535	686 $$\infer{\Gamma \vdash \mt{val} \; \mt{rec} \; \overline{x : \tau = e} \leadsto \Gamma, \overline{x : \tau}}{
adamc@535	687 \forall i: \Gamma, \overline{x : \tau} \vdash e_i : \tau_i
adamc@535	688 & \textrm{$e_i$ starts with an expression $\lambda$, optionally preceded by constructor and disjointness $\lambda$s}
adamc@535	689 }$$
adamc@535	690
adamc@535	691 $$\infer{\Gamma \vdash \mt{structure} \; X : S = M \leadsto \Gamma, X : S}{
adamc@535	692 \Gamma \vdash M : S
adamc@539	693 & \textrm{ ($M$ not a $\mt{struct} \; \ldots \; \mt{end}$)}
adamc@535	694 }
adamc@539	695 \quad \infer{\Gamma \vdash \mt{structure} \; X : S = \mt{struct} \; \overline{d} \; \mt{end} \leadsto \Gamma, X : \mt{selfify}(X, \overline{s})}{
adamc@539	696 \Gamma \vdash \mt{struct} \; \overline{d} \; \mt{end} : \mt{sig} \; \overline{s} \; \mt{end}
adamc@539	697 }$$
adamc@539	698
adamc@539	699 $$\infer{\Gamma \vdash \mt{signature} \; X = S \leadsto \Gamma, X = S}{
adamc@535	700 \Gamma \vdash S
adamc@535	701 }$$
adamc@535	702
adamc@537	703 $$\infer{\Gamma \vdash \mt{open} \; M \leadsto \Gamma, \mathcal O(M, \overline{s})}{
adamc@537	704 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@535	705 }$$
adamc@535	706
adamc@535	707 $$\infer{\Gamma \vdash \mt{constraint} \; c_1 \sim c_2 \leadsto \Gamma}{
adamc@535	708 \Gamma \vdash c_1 :: \{\kappa\}
adamc@535	709 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@535	710 & \Gamma \vdash c_1 \sim c_2
adamc@535	711 }
adamc@537	712 \quad \infer{\Gamma \vdash \mt{open} \; \mt{constraints} \; M \leadsto \Gamma, \mathcal O_c(M, \overline{s})}{
adamc@537	713 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@535	714 }$$
adamc@535	715
adamc@535	716 $$\infer{\Gamma \vdash \mt{table} \; x : c \leadsto \Gamma, x : \mt{Basis}.\mt{sql\_table} \; c}{
adamc@535	717 \Gamma \vdash c :: \{\mt{Type}\}
adamc@535	718 }
adamc@535	719 \quad \infer{\Gamma \vdash \mt{sequence} \; x \leadsto \Gamma, x : \mt{Basis}.\mt{sql\_sequence}}{}$$
adamc@535	720
adamc@535	721 $$\infer{\Gamma \vdash \mt{cookie} \; x : \tau \leadsto \Gamma, x : \mt{Basis}.\mt{http\_cookie} \; \tau}{
adamc@535	722 \Gamma \vdash \tau :: \mt{Type}
adamc@535	723 }$$
adamc@535	724
adamc@535	725 $$\infer{\Gamma \vdash \mt{class} \; x = c \leadsto \Gamma, x :: \mt{Type} \to \mt{Type} = c}{
adamc@535	726 \Gamma \vdash c :: \mt{Type} \to \mt{Type}
adamc@535	727 }$$
adamc@535	728
adamc@535	729 $$\infer{\overline{y}; x; \Gamma \vdash \cdot \leadsto \Gamma}{}
adamc@535	730 \quad \infer{\overline{y}; x; \Gamma \vdash X \mid \overline{dc} \leadsto \Gamma', X : \overline{y ::: \mt{Type}} \to x \; \overline{y}}{
adamc@535	731 \overline{y}; x; \Gamma \vdash \overline{dc} \leadsto \Gamma'
adamc@535	732 }
adamc@535	733 \quad \infer{\overline{y}; x; \Gamma \vdash X \; \mt{of} \; \tau \mid \overline{dc} \leadsto \Gamma', X : \overline{y ::: \mt{Type}} \to \tau \to x \; \overline{y}}{
adamc@535	734 \overline{y}; x; \Gamma \vdash \overline{dc} \leadsto \Gamma'
adamc@535	735 }$$
adamc@535	736
adamc@537	737 \subsection{Signature Item Typing}
adamc@537	738
adamc@537	739 We appeal to a signature item analogue of the $\mathcal O$ function from the last subsection.
adamc@537	740
adamc@537	741 $$\infer{\Gamma \vdash \cdot \leadsto \Gamma}{}
adamc@537	742 \quad \infer{\Gamma \vdash s, \overline{s} \leadsto \Gamma''}{
adamc@537	743 \Gamma \vdash s \leadsto \Gamma'
adamc@537	744 & \Gamma' \vdash \overline{s} \leadsto \Gamma''
adamc@537	745 }$$
adamc@537	746
adamc@537	747 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa \leadsto \Gamma, x :: \kappa}{}
adamc@537	748 \quad \infer{\Gamma \vdash \mt{con} \; x :: \kappa = c \leadsto \Gamma, x :: \kappa = c}{
adamc@537	749 \Gamma \vdash c :: \kappa
adamc@537	750 }
adamc@537	751 \quad \infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leadsto \Gamma'}{
adamc@537	752 \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} \vdash \overline{dc} \leadsto \Gamma'
adamc@537	753 }$$
adamc@537	754
adamc@537	755 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leadsto \Gamma'}{
adamc@537	756 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	757 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@537	758 & \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} = M.z \vdash \overline{dc} \leadsto \Gamma'
adamc@537	759 }$$
adamc@537	760
adamc@537	761 $$\infer{\Gamma \vdash \mt{val} \; x : \tau \leadsto \Gamma, x : \tau}{
adamc@537	762 \Gamma \vdash \tau :: \mt{Type}
adamc@537	763 }$$
adamc@537	764
adamc@537	765 $$\infer{\Gamma \vdash \mt{structure} \; X : S \leadsto \Gamma, X : S}{
adamc@537	766 \Gamma \vdash S
adamc@537	767 }
adamc@537	768 \quad \infer{\Gamma \vdash \mt{signature} \; X = S \leadsto \Gamma, X = S}{
adamc@537	769 \Gamma \vdash S
adamc@537	770 }$$
adamc@537	771
adamc@537	772 $$\infer{\Gamma \vdash \mt{include} \; S \leadsto \Gamma, \mathcal O(\overline{s})}{
adamc@537	773 \Gamma \vdash S
adamc@537	774 & \Gamma \vdash S \equiv \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	775 }$$
adamc@537	776
adamc@537	777 $$\infer{\Gamma \vdash \mt{constraint} \; c_1 \sim c_2 \leadsto \Gamma, c_1 \sim c_2}{
adamc@537	778 \Gamma \vdash c_1 :: \{\kappa\}
adamc@537	779 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@537	780 }$$
adamc@537	781
adamc@537	782 $$\infer{\Gamma \vdash \mt{class} \; x = c \leadsto \Gamma, x :: \mt{Type} \to \mt{Type} = c}{
adamc@537	783 \Gamma \vdash c :: \mt{Type} \to \mt{Type}
adamc@537	784 }
adamc@537	785 \quad \infer{\Gamma \vdash \mt{class} \; x \leadsto \Gamma, x :: \mt{Type} \to \mt{Type}}{}$$
adamc@537	786
adamc@536	787 \subsection{Signature Compatibility}
adamc@536	788
adamc@537	789 To simplify the judgments in this section, we assume that all signatures are alpha-varied as necessary to avoid including mmultiple bindings for the same identifier. This is in addition to the usual alpha-variation of locally-bound variables.
adamc@537	790
adamc@537	791 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	792
adamc@536	793 $$\infer{\Gamma \vdash S \equiv S}{}
adamc@536	794 \quad \infer{\Gamma \vdash S_1 \equiv S_2}{
adamc@536	795 \Gamma \vdash S_2 \equiv S_1
adamc@536	796 }
adamc@536	797 \quad \infer{\Gamma \vdash X \equiv S}{
adamc@536	798 X = S \in \Gamma
adamc@536	799 }
adamc@536	800 \quad \infer{\Gamma \vdash M.X \equiv S}{
adamc@537	801 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	802 & \mt{proj}(M, \overline{s}, \mt{signature} \; X) = S
adamc@536	803 }$$
adamc@536	804
adamc@536	805 $$\infer{\Gamma \vdash S \; \mt{where} \; \mt{con} \; x = c \equiv \mt{sig} \; \overline{s^1} \; \mt{con} \; x :: \kappa = c \; \overline{s_2} \; \mt{end}}{
adamc@536	806 \Gamma \vdash S \equiv \mt{sig} \; \overline{s^1} \; \mt{con} \; x :: \kappa \; \overline{s_2} \; \mt{end}
adamc@536	807 & \Gamma \vdash c :: \kappa
adamc@537	808 }
adamc@537	809 \quad \infer{\Gamma \vdash \mt{sig} \; \overline{s^1} \; \mt{include} \; S \; \overline{s^2} \; \mt{end} \equiv \mt{sig} \; \overline{s^1} \; \overline{s} \; \overline{s^2} \; \mt{end}}{
adamc@537	810 \Gamma \vdash S \equiv \mt{sig} \; \overline{s} \; \mt{end}
adamc@536	811 }$$
adamc@536	812
adamc@536	813 $$\infer{\Gamma \vdash S_1 \leq S_2}{
adamc@536	814 \Gamma \vdash S_1 \equiv S_2
adamc@536	815 }
adamc@536	816 \quad \infer{\Gamma \vdash \mt{sig} \; \overline{s} \; \mt{end} \leq \mt{sig} \; \mt{end}}{}
adamc@537	817 \quad \infer{\Gamma \vdash \mt{sig} \; \overline{s} \; \mt{end} \leq \mt{sig} \; s' \; \overline{s'} \; \mt{end}}{
adamc@537	818 \Gamma \vdash \overline{s} \leq s'
adamc@537	819 & \Gamma \vdash s' \leadsto \Gamma'
adamc@537	820 & \Gamma' \vdash \mt{sig} \; \overline{s} \; \mt{end} \leq \mt{sig} \; \overline{s'} \; \mt{end}
adamc@537	821 }$$
adamc@537	822
adamc@537	823 $$\infer{\Gamma \vdash s \; \overline{s} \leq s'}{
adamc@537	824 \Gamma \vdash s \leq s'
adamc@537	825 }
adamc@537	826 \quad \infer{\Gamma \vdash s \; \overline{s} \leq s'}{
adamc@537	827 \Gamma \vdash s \leadsto \Gamma'
adamc@537	828 & \Gamma' \vdash \overline{s} \leq s'
adamc@536	829 }$$
adamc@536	830
adamc@536	831 $$\infer{\Gamma \vdash \mt{functor} (X : S_1) : S_2 \leq \mt{functor} (X : S'_1) : S'_2}{
adamc@536	832 \Gamma \vdash S'_1 \leq S_1
adamc@536	833 & \Gamma, X : S'_1 \vdash S_2 \leq S'_2
adamc@536	834 }$$
adamc@536	835
adamc@537	836 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa \leq \mt{con} \; x :: \kappa}{}
adamc@537	837 \quad \infer{\Gamma \vdash \mt{con} \; x :: \kappa = c \leq \mt{con} \; x :: \kappa}{}
adamc@537	838 \quad \infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leq \mt{con} \; x :: \mt{Type}}{}$$
adamc@537	839
adamc@537	840 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leq \mt{con} \; x :: \mt{Type}^{\mt{len}(y)} \to \mt{Type}}{
adamc@537	841 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	842 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@537	843 }$$
adamc@537	844
adamc@537	845 $$\infer{\Gamma \vdash \mt{class} \; x \leq \mt{con} \; x :: \mt{Type} \to \mt{Type}}{}
adamc@537	846 \quad \infer{\Gamma \vdash \mt{class} \; x = c \leq \mt{con} \; x :: \mt{Type} \to \mt{Type}}{}$$
adamc@537	847
adamc@537	848 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa = c_1 \leq \mt{con} \; x :: \mt{\kappa} = c_2}{
adamc@537	849 \Gamma \vdash c_1 \equiv c_2
adamc@537	850 }
adamc@537	851 \quad \infer{\Gamma \vdash \mt{class} \; x = c_1 \leq \mt{con} \; x :: \mt{Type} \to \mt{Type} = c_2}{
adamc@537	852 \Gamma \vdash c_1 \equiv c_2
adamc@537	853 }$$
adamc@537	854
adamc@537	855 $$\infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leq \mt{datatype} \; x \; \overline{y} = \overline{dc'}}{
adamc@537	856 \Gamma, \overline{y :: \mt{Type}} \vdash \overline{dc} \leq \overline{dc'}
adamc@537	857 }$$
adamc@537	858
adamc@537	859 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leq \mt{datatype} \; x \; \overline{y} = \overline{dc'}}{
adamc@537	860 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	861 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@537	862 & \Gamma, \overline{y :: \mt{Type}} \vdash \overline{dc} \leq \overline{dc'}
adamc@537	863 }$$
adamc@537	864
adamc@537	865 $$\infer{\Gamma \vdash \cdot \leq \cdot}{}
adamc@537	866 \quad \infer{\Gamma \vdash X; \overline{dc} \leq X; \overline{dc'}}{
adamc@537	867 \Gamma \vdash \overline{dc} \leq \overline{dc'}
adamc@537	868 }
adamc@537	869 \quad \infer{\Gamma \vdash X \; \mt{of} \; \tau_1; \overline{dc} \leq X \; \mt{of} \; \tau_2; \overline{dc'}}{
adamc@537	870 \Gamma \vdash \tau_1 \equiv \tau_2
adamc@537	871 & \Gamma \vdash \overline{dc} \leq \overline{dc'}
adamc@537	872 }$$
adamc@537	873
adamc@537	874 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leq \mt{datatype} \; x = \mt{datatype} \; M'.z'}{
adamc@537	875 \Gamma \vdash M.z \equiv M'.z'
adamc@537	876 }$$
adamc@537	877
adamc@537	878 $$\infer{\Gamma \vdash \mt{val} \; x : \tau_1 \leq \mt{val} \; x : \tau_2}{
adamc@537	879 \Gamma \vdash \tau_1 \equiv \tau_2
adamc@537	880 }
adamc@537	881 \quad \infer{\Gamma \vdash \mt{structure} \; X : S_1 \leq \mt{structure} \; X : S_2}{
adamc@537	882 \Gamma \vdash S_1 \leq S_2
adamc@537	883 }
adamc@537	884 \quad \infer{\Gamma \vdash \mt{signature} \; X = S_1 \leq \mt{signature} \; X = S_2}{
adamc@537	885 \Gamma \vdash S_1 \leq S_2
adamc@537	886 & \Gamma \vdash S_2 \leq S_1
adamc@537	887 }$$
adamc@537	888
adamc@537	889 $$\infer{\Gamma \vdash \mt{constraint} \; c_1 \sim c_2 \leq \mt{constraint} \; c'_1 \sim c'_2}{
adamc@537	890 \Gamma \vdash c_1 \equiv c'_1
adamc@537	891 & \Gamma \vdash c_2 \equiv c'_2
adamc@537	892 }$$
adamc@537	893
adamc@537	894 $$\infer{\Gamma \vdash \mt{class} \; x \leq \mt{class} \; x}{}
adamc@537	895 \quad \infer{\Gamma \vdash \mt{class} \; x = c \leq \mt{class} \; x}{}
adamc@537	896 \quad \infer{\Gamma \vdash \mt{class} \; x = c_1 \leq \mt{class} \; x = c_2}{
adamc@537	897 \Gamma \vdash c_1 \equiv c_2
adamc@537	898 }$$
adamc@537	899
adamc@538	900 \subsection{Module Typing}
adamc@538	901
adamc@538	902 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	903
adamc@538	904 $$\infer{\Gamma \vdash M : S}{
adamc@538	905 \Gamma \vdash M : S'
adamc@538	906 & \Gamma \vdash S' \leq S
adamc@538	907 }
adamc@538	908 \quad \infer{\Gamma \vdash \mt{struct} \; \overline{d} \; \mt{end} : \mt{sig} \; \mt{sigOf}(\overline{d}) \; \mt{end}}{
adamc@538	909 \Gamma \vdash \overline{d} \leadsto \Gamma'
adamc@538	910 }
adamc@538	911 \quad \infer{\Gamma \vdash X : S}{
adamc@538	912 X : S \in \Gamma
adamc@538	913 }$$
adamc@538	914
adamc@538	915 $$\infer{\Gamma \vdash M.X : S}{
adamc@538	916 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@538	917 & \mt{proj}(M, \overline{s}, \mt{structure} \; X) = S
adamc@538	918 }$$
adamc@538	919
adamc@538	920 $$\infer{\Gamma \vdash M_1(M_2) : [X \mapsto M_2]S_2}{
adamc@538	921 \Gamma \vdash M_1 : \mt{functor}(X : S_1) : S_2
adamc@538	922 & \Gamma \vdash M_2 : S_1
adamc@538	923 }
adamc@538	924 \quad \infer{\Gamma \vdash \mt{functor} (X : S_1) : S_2 = M : \mt{functor} (X : S_1) : S_2}{
adamc@538	925 \Gamma \vdash S_1
adamc@538	926 & \Gamma, X : S_1 \vdash S_2
adamc@538	927 & \Gamma, X : S_1 \vdash M : S_2
adamc@538	928 }$$
adamc@538	929
adamc@538	930 \begin{eqnarray*}
adamc@538	931 \mt{sigOf}(\cdot) &=& \cdot \\
adamc@538	932 \mt{sigOf}(s \; \overline{s'}) &=& \mt{sigOf}(s) \; \mt{sigOf}(\overline{s'}) \\
adamc@538	933 \\
adamc@538	934 \mt{sigOf}(\mt{con} \; x :: \kappa = c) &=& \mt{con} \; x :: \kappa = c \\
adamc@538	935 \mt{sigOf}(\mt{datatype} \; x \; \overline{y} = \overline{dc}) &=& \mt{datatype} \; x \; \overline{y} = \overline{dc} \\
adamc@538	936 \mt{sigOf}(\mt{datatype} \; x = \mt{datatype} \; M.z) &=& \mt{datatype} \; x = \mt{datatype} \; M.z \\
adamc@538	937 \mt{sigOf}(\mt{val} \; x : \tau = e) &=& \mt{val} \; x : \tau \\
adamc@538	938 \mt{sigOf}(\mt{val} \; \mt{rec} \; \overline{x : \tau = e}) &=& \overline{\mt{val} \; x : \tau} \\
adamc@538	939 \mt{sigOf}(\mt{structure} \; X : S = M) &=& \mt{structure} \; X : S \\
adamc@538	940 \mt{sigOf}(\mt{signature} \; X = S) &=& \mt{signature} \; X = S \\
adamc@538	941 \mt{sigOf}(\mt{open} \; M) &=& \mt{include} \; S \textrm{ (where $\Gamma \vdash M : S$)} \\
adamc@538	942 \mt{sigOf}(\mt{constraint} \; c_1 \sim c_2) &=& \mt{constraint} \; c_1 \sim c_2 \\
adamc@538	943 \mt{sigOf}(\mt{open} \; \mt{constraints} \; M) &=& \cdot \\
adamc@538	944 \mt{sigOf}(\mt{table} \; x : c) &=& \mt{table} \; x : c \\
adamc@538	945 \mt{sigOf}(\mt{sequence} \; x) &=& \mt{sequence} \; x \\
adamc@538	946 \mt{sigOf}(\mt{cookie} \; x : \tau) &=& \mt{cookie} \; x : \tau \\
adamc@538	947 \mt{sigOf}(\mt{class} \; x = c) &=& \mt{class} \; x = c \\
adamc@538	948 \end{eqnarray*}
adamc@538	949
adamc@539	950 \begin{eqnarray*}
adamc@539	951 \mt{selfify}(M, \cdot) &=& \cdot \\
adamc@539	952 \mt{selfify}(M, s \; \overline{s'}) &=& \mt{selfify}(M, \sigma, 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@540	987 && \textrm{and $\mt{proj}(M', \overline{s'}, \mt{datatype} \; z = (\overline{y}, \overline{dc})$ and $X : \tau \in \overline{dc}$)} \\
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@541	1013 Type-checkers for languages based on the Hindly-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@541	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 incompletness.
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@541	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 reduce 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@543	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-availablity 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@544	1202 We have generic nullary, unary, and binary operators, as well as comparison 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 $$\begin{array}{l}
adamc@544	1225 \mt{type} \; \mt{sql\_comparison} \\
adamc@544	1226 \mt{val} \; \mt{sql\_eq} : \mt{sql\_comparison} \\
adamc@544	1227 \mt{val} \; \mt{sql\_ne} : \mt{sql\_comparison} \\
adamc@544	1228 \mt{val} \; \mt{sql\_lt} : \mt{sql\_comparison} \\
adamc@544	1229 \mt{val} \; \mt{sql\_le} : \mt{sql\_comparison} \\
adamc@544	1230 \mt{val} \; \mt{sql\_gt} : \mt{sql\_comparison} \\
adamc@544	1231 \mt{val} \; \mt{sql\_ge} : \mt{sql\_comparison} \\
adamc@544	1232 \mt{val} \; \mt{sql\_comparison} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544	1233 \hspace{.1in} \to \mt{sql\_comparison} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{t} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{t} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{bool}
adamc@544	1234 \end{array}$$
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@549	1394 &&& t.\{\{c\}\} & \textrm{a record of colums 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 $$\begin{array}{rrcll}
adamc@551	1445 \textrm{XML fragments} & L &::=& \texttt{<xml/>} \mid \texttt{<xml>}l^*\texttt{</xml>} \\
adamc@551	1446 \textrm{XML pieces} & l &::=& \textrm{text} & \textrm{cdata} \\
adamc@551	1447 &&& \texttt{<}g\texttt{/>} & \textrm{tag with no children} \\
adamc@551	1448 &&& \texttt{<}g\texttt{>}l^*\texttt{</}x\texttt{>} & \textrm{tag with children} \\
adamc@551	1449 \textrm{Tag} & g &::=& h \; (x = v)^* \\
adamc@551	1450 \textrm{Tag head} & h &::=& x & \textrm{tag name} \\
adamc@551	1451 &&& h\{c\} & \textrm{constructor parameter} \\
adamc@551	1452 \textrm{Attribute value} & v &::=& \ell & \textrm{literal value} \\
adamc@551	1453 &&& \{e\} & \textrm{computed value} \\
adamc@551	1454 \end{array}$$
adamc@551	1455
adamc@552	1456
adamc@553	1457 \section{The Structure of Web Applications}
adamc@553	1458
adamc@553	1459 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	1460
adamc@553	1461 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	1462
adamc@553	1463 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	1464
adamc@553	1465 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 appeared in the source code.
adamc@553	1466
adamc@553	1467
adamc@552	1468 \section{Compiler Phases}
adamc@552	1469
adamc@552	1470 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	1471
adamc@552	1472 In this section, we step through the main phases of compilation, noting what consequences each phase has for effective programming.
adamc@552	1473
adamc@552	1474 \subsection{Parse}
adamc@552	1475
adamc@552	1476 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	1477
adamc@552	1478 \subsection{Elaborate}
adamc@552	1479
adamc@552	1480 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	1481
adamc@552	1482 \subsection{Unnest}
adamc@552	1483
adamc@552	1484 Named local function definitions are moved to the top level, to avoid the need to generate closures.
adamc@552	1485
adamc@552	1486 \subsection{Corify}
adamc@552	1487
adamc@552	1488 Module system features are compiled away, through inlining of functor definitions at application sites. Afterward, most abstraction boundaries are broken, facilitating optimization.
adamc@552	1489
adamc@552	1490 \subsection{Especialize}
adamc@552	1491
adamc@552	1492 Functions are specialized to particular argument patterns. This is an important trick for avoiding the need to maintain any closures at runtime.
adamc@552	1493
adamc@552	1494 \subsection{Untangle}
adamc@552	1495
adamc@552	1496 Remove unnecessary mutual recursion, splitting recursive groups into strongly-connected components.
adamc@552	1497
adamc@552	1498 \subsection{Shake}
adamc@552	1499
adamc@552	1500 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	1501
adamc@553	1502 \subsection{\label{tag}Tag}
adamc@552	1503
adamc@552	1504 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	1505
adamc@552	1506 \subsection{Reduce}
adamc@552	1507
adamc@552	1508 Apply definitional equality rules to simplify the program as much as possible. This effectively includes inlining of every non-recursive definition.
adamc@552	1509
adamc@552	1510 \subsection{Unpoly}
adamc@552	1511
adamc@552	1512 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	1513
adamc@552	1514 \subsection{Specialize}
adamc@552	1515
adamc@552	1516 Replace uses of parametrized 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	1517
adamc@552	1518 \subsection{Shake}
adamc@552	1519
adamc@552	1520 Here the compiler repeats the earlier shake phase.
adamc@552	1521
adamc@552	1522 \subsection{Monoize}
adamc@552	1523
adamc@552	1524 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	1525
adamc@552	1526 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	1527 \subsection{MonoOpt}
adamc@552	1528
adamc@552	1529 Simple algebraic laws are applied to simplify the program, focusing especially on efficient imperative generation of HTML pages.
adamc@552	1530
adamc@552	1531 \subsection{MonoUntangle}
adamc@552	1532
adamc@552	1533 Unnecessary mutual recursion is broken up again.
adamc@552	1534
adamc@552	1535 \subsection{MonoReduce}
adamc@552	1536
adamc@552	1537 Equivalents of the definitional equality rules are applied to simplify programs, with inlining again playing a major role.
adamc@552	1538
adamc@552	1539 \subsection{MonoShake, MonoOpt}
adamc@552	1540
adamc@552	1541 Unneeded declarations are removed, and basic optimizations are repeated.
adamc@552	1542
adamc@552	1543 \subsection{Fuse}
adamc@552	1544
adamc@552	1545 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	1546
adamc@552	1547 \subsection{MonoUntangle, MonoShake}
adamc@552	1548
adamc@552	1549 Fuse often creates more opportunities to remove spurious mutual recursion.
adamc@552	1550
adamc@552	1551 \subsection{Pathcheck}
adamc@552	1552
adamc@552	1553 The compiler checks that no link or action name has been used more than once.
adamc@552	1554
adamc@552	1555 \subsection{Cjrize}
adamc@552	1556
adamc@552	1557 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	1558
adamc@552	1559 \subsection{C Compilation and Linking}
adamc@552	1560
adamc@552	1561 The output of the last phase is pretty-printed as C source code and passed to GCC.
adamc@552	1562
adamc@552	1563
adamc@524	1564 \end{document}

Mercurial > urweb

annotate doc/manual.tex @ 557:0d3db8d6a586