Ur/Web: doc/manual.tex annotate

annotate doc/manual.tex @ 652:9db6880184d0

Revising manual, through main syntax section

author	Adam Chlipala <adamc@hcoop.net>
date	Thu, 12 Mar 2009 10:16:59 -0400
parents	33500a15b872
children	a0f2b070af01

rev	line source
adamc@524	1 \documentclass{article}
adamc@554	2 \usepackage{fullpage,amsmath,amssymb,proof,url}
adamc@524	3
adamc@524	4 \newcommand{\cd}[1]{\texttt{#1}}
adamc@524	5 \newcommand{\mt}[1]{\mathsf{#1}}
adamc@524	6
adamc@524	7 \newcommand{\rc}{+ \hspace{-.075in} + \;}
adamc@527	8 \newcommand{\rcut}{\; \texttt{--} \;}
adamc@527	9 \newcommand{\rcutM}{\; \texttt{---} \;}
adamc@524	10
adamc@524	11 \begin{document}
adamc@524	12
adamc@524	13 \title{The Ur/Web Manual}
adamc@524	14 \author{Adam Chlipala}
adamc@524	15
adamc@524	16 \maketitle
adamc@524	17
adamc@540	18 \tableofcontents
adamc@540	19
adamc@554	20
adamc@554	21 \section{Introduction}
adamc@554	22
adamc@554	23 \emph{Ur} is a programming language designed to introduce richer type system features into functional programming in the tradition of ML and Haskell. Ur is functional, pure, statically-typed, and strict. Ur supports a powerful kind of \emph{metaprogramming} based on \emph{row types}.
adamc@554	24
adamc@554	25 \emph{Ur/Web} is Ur plus a special standard library and associated rules for parsing and optimization. Ur/Web supports construction of dynamic web applications backed by SQL databases. The signature of the standard library is such that well-typed Ur/Web programs ``don't go wrong'' in a very broad sense. Not only do they not crash during particular page generations, but they also may not:
adamc@554	26
adamc@554	27 \begin{itemize}
adamc@554	28 \item Suffer from any kinds of code-injection attacks
adamc@554	29 \item Return invalid HTML
adamc@554	30 \item Contain dead intra-application links
adamc@554	31 \item Have mismatches between HTML forms and the fields expected by their handlers
adamc@652	32 \item Include client-side code that makes incorrect assumptions about the ``AJAX''-style services that the remote web server provides
adamc@554	33 \item Attempt invalid SQL queries
adamc@652	34 \item Use improper marshaling or unmarshaling in communication with SQL databases or between browsers and web servers
adamc@554	35 \end{itemize}
adamc@554	36
adamc@554	37 This type safety is just the foundation of the Ur/Web methodology. It is also possible to use metaprogramming to build significant application pieces by analysis of type structure. For instance, the demo includes an ML-style functor for building an admin interface for an arbitrary SQL table. The type system guarantees that the admin interface sub-application that comes out will always be free of the above-listed bugs, no matter which well-typed table description is given as input.
adamc@554	38
adamc@652	39 The Ur/Web compiler also produces very efficient object code that does not use garbage collection. These compiled programs will often be even more efficient than what most programmers would bother to write in C. The compiler also generates JavaScript versions of client-side code, with no need to write those parts of applications in a different language.
adamc@554	40
adamc@554	41 \medskip
adamc@554	42
adamc@554	43 The official web site for Ur is:
adamc@554	44 \begin{center}
adamc@554	45 \url{http://www.impredicative.com/ur/}
adamc@554	46 \end{center}
adamc@554	47
adamc@555	48
adamc@555	49 \section{Installation}
adamc@555	50
adamc@555	51 If you are lucky, then the following standard command sequence will suffice for installation, in a directory to which you have unpacked the latest distribution tarball.
adamc@555	52
adamc@555	53 \begin{verbatim}
adamc@555	54 ./configure
adamc@555	55 make
adamc@555	56 sudo make install
adamc@555	57 \end{verbatim}
adamc@555	58
adamc@555	59 Some other packages must be installed for the above to work. At a minimum, you need a standard UNIX shell, with standard UNIX tools like sed and GCC in your execution path; 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	60
adamc@555	61 \begin{verbatim}
adamc@555	62 apt-get install mlton libpq-dev
adamc@555	63 \end{verbatim}
adamc@555	64
adamc@555	65 It is also possible to access the modules of the Ur/Web compiler interactively, within Standard ML of New Jersey. To install the prerequisites in Debian testing:
adamc@555	66
adamc@555	67 \begin{verbatim}
adamc@555	68 apt-get install smlnj libsmlnj-smlnj ml-yacc ml-lpt
adamc@555	69 \end{verbatim}
adamc@555	70
adamc@555	71 To begin an interactive session with the Ur compiler modules, run \texttt{make smlnj}, and then, from within an \texttt{sml} session, run \texttt{CM.make "src/urweb.cm";}. The \texttt{Compiler} module is the main entry point.
adamc@555	72
adamc@555	73 To run an SQL-backed application, you will probably want to install the PostgreSQL server. Version 8.3 or higher is required.
adamc@555	74
adamc@555	75 \begin{verbatim}
adamc@555	76 apt-get install postgresql-8.3
adamc@555	77 \end{verbatim}
adamc@555	78
adamc@555	79 To use the Emacs mode, you must have a modern Emacs installed. We assume that you already know how to do this, if you're in the business of looking for an Emacs mode. The demo generation facility of the compiler will also call out to Emacs to syntax-highlight code, and that process depends on the \texttt{htmlize} module, which can be installed in Debian testing via:
adamc@555	80
adamc@555	81 \begin{verbatim}
adamc@555	82 apt-get install emacs-goodies-el
adamc@555	83 \end{verbatim}
adamc@555	84
adamc@555	85 Even with the right packages installed, configuration and building might fail to work. After you run \texttt{./configure}, you will see the values of some named environment variables printed. You may need to adjust these values to get proper installation for your system. To change a value, store your preferred alternative in the corresponding UNIX environment variable, before running \texttt{./configure}. For instance, here is how to change the list of extra arguments that the Ur/Web compiler will pass to GCC on every invocation.
adamc@555	86
adamc@555	87 \begin{verbatim}
adamc@555	88 GCCARGS=-fnested-functions ./configure
adamc@555	89 \end{verbatim}
adamc@555	90
adamc@555	91 Some OSX users have reported needing to use this particular GCCARGS value.
adamc@555	92
adamc@555	93 The Emacs mode can be set to autoload by adding the following to your \texttt{.emacs} file.
adamc@555	94
adamc@555	95 \begin{verbatim}
adamc@555	96 (add-to-list 'load-path "/usr/local/share/emacs/site-lisp/urweb-mode")
adamc@555	97 (load "urweb-mode-startup")
adamc@555	98 \end{verbatim}
adamc@555	99
adamc@555	100 Change the path in the first line if you chose a different Emacs installation path during configuration.
adamc@555	101
adamc@555	102
adamc@556	103 \section{Command-Line Compiler}
adamc@556	104
adamc@556	105 \subsection{Project Files}
adamc@556	106
adamc@556	107 The basic inputs to the \texttt{urweb} compiler are project files, which have the extension \texttt{.urp}. Here is a sample \texttt{.urp} file.
adamc@556	108
adamc@556	109 \begin{verbatim}
adamc@556	110 database dbname=test
adamc@556	111 sql crud1.sql
adamc@556	112
adamc@556	113 crud
adamc@556	114 crud1
adamc@556	115 \end{verbatim}
adamc@556	116
adamc@556	117 The \texttt{database} line gives the database information string to pass to libpq. In this case, the string only says to connect to a local database named \texttt{test}.
adamc@556	118
adamc@556	119 The \texttt{sql} line asks for an SQL source file to be generated, giving the commands to run to create the tables and sequences that this application expects to find. After building this \texttt{.urp} file, the following commands could be used to initialize the database, assuming that the current UNIX user exists as a Postgres user with database creation privileges:
adamc@556	120
adamc@556	121 \begin{verbatim}
adamc@556	122 createdb test
adamc@556	123 psql -f crud1.sql test
adamc@556	124 \end{verbatim}
adamc@556	125
adamc@556	126 A blank line always separates the named directives from a list of modules to include in the project; if there are no named directives, a blank line must begin the file.
adamc@556	127
adamc@556	128 For each entry \texttt{M} in the module list, the file \texttt{M.urs} is included in the project if it exists, and the file \texttt{M.ur} must exist and is always included.
adamc@556	129
adamc@556	130 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	131
adamc@557	132 \subsection{Building an Application}
adamc@557	133
adamc@557	134 To compile project \texttt{P.urp}, simply run
adamc@557	135 \begin{verbatim}
adamc@557	136 urweb P
adamc@557	137 \end{verbatim}
adamc@558	138 The output executable is a standalone web server. Run it with the command-line argument \texttt{-h} to see which options it takes. If the project file lists a database, the web server will attempt to connect to that database on startup.
adamc@557	139
adamc@557	140 To time how long the different compiler phases run, without generating an executable, run
adamc@557	141 \begin{verbatim}
adamc@557	142 urweb -timing P
adamc@557	143 \end{verbatim}
adamc@557	144
adamc@556	145
adamc@529	146 \section{Ur Syntax}
adamc@529	147
adamc@529	148 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	149
adamc@524	150 \subsection{Lexical Conventions}
adamc@524	151
adamc@524	152 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	153
adamc@524	154 \begin{center}
adamc@524	155 \begin{tabular}{rl}
adamc@524	156 \textbf{\LaTeX} & \textbf{ASCII} \\
adamc@524	157 $\to$ & \cd{->} \\
adamc@652	158 $\longrightarrow$ & \cd{-->} \\
adamc@524	159 $\times$ & \cd{*} \\
adamc@524	160 $\lambda$ & \cd{fn} \\
adamc@524	161 $\Rightarrow$ & \cd{=>} \\
adamc@652	162 $\Longrightarrow$ & \cd{==>} \\
adamc@529	163 $\neq$ & \cd{<>} \\
adamc@529	164 $\leq$ & \cd{<=} \\
adamc@529	165 $\geq$ & \cd{>=} \\
adamc@524	166 \\
adamc@524	167 $x$ & Normal textual identifier, not beginning with an uppercase letter \\
adamc@525	168 $X$ & Normal textual identifier, beginning with an uppercase letter \\
adamc@524	169 \end{tabular}
adamc@524	170 \end{center}
adamc@524	171
adamc@525	172 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	173
adamc@526	174 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	175
adamc@527	176 This version of the manual doesn't include operator precedences; see \texttt{src/urweb.grm} for that.
adamc@527	177
adamc@552	178 \subsection{\label{core}Core Syntax}
adamc@524	179
adamc@524	180 \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	181 $$\begin{array}{rrcll}
adamc@524	182 \textrm{Kinds} & \kappa &::=& \mt{Type} & \textrm{proper types} \\
adamc@525	183 &&& \mt{Unit} & \textrm{the trivial constructor} \\
adamc@525	184 &&& \mt{Name} & \textrm{field names} \\
adamc@525	185 &&& \kappa \to \kappa & \textrm{type-level functions} \\
adamc@525	186 &&& \{\kappa\} & \textrm{type-level records} \\
adamc@525	187 &&& (\kappa\times^+) & \textrm{type-level tuples} \\
adamc@652	188 &&& X & \textrm{variable} \\
adamc@652	189 &&& X \longrightarrow k & \textrm{kind-polymorphic type-level function} \\
adamc@529	190 &&& \_\_ & \textrm{wildcard} \\
adamc@525	191 &&& (\kappa) & \textrm{explicit precedence} \\
adamc@524	192 \end{array}$$
adamc@524	193
adamc@524	194 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	195 $$\begin{array}{rrcll}
adamc@524	196 \textrm{Explicitness} & ? &::=& :: & \textrm{explicit} \\
adamc@558	197 &&& ::: & \textrm{implicit}
adamc@524	198 \end{array}$$
adamc@524	199
adamc@524	200 \emph{Constructors} are the main class of compile-time-only data. They include proper types and are classified by kinds.
adamc@524	201 $$\begin{array}{rrcll}
adamc@524	202 \textrm{Constructors} & c, \tau &::=& (c) :: \kappa & \textrm{kind annotation} \\
adamc@530	203 &&& \hat{x} & \textrm{constructor variable} \\
adamc@524	204 \\
adamc@525	205 &&& \tau \to \tau & \textrm{function type} \\
adamc@525	206 &&& x \; ? \; \kappa \to \tau & \textrm{polymorphic function type} \\
adamc@652	207 &&& X \longrightarrow \tau & \textrm{kind-polymorphic function type} \\
adamc@525	208 &&& \$ c & \textrm{record type} \\
adamc@524	209 \\
adamc@525	210 &&& c \; c & \textrm{type-level function application} \\
adamc@530	211 &&& \lambda x \; :: \; \kappa \Rightarrow c & \textrm{type-level function abstraction} \\
adamc@524	212 \\
adamc@652	213 &&& X \Longrightarrow c & \textrm{type-level kind-polymorphic function abstraction} \\
adamc@652	214 \\
adamc@525	215 &&& () & \textrm{type-level unit} \\
adamc@525	216 &&& \#X & \textrm{field name} \\
adamc@524	217 \\
adamc@525	218 &&& [(c = c)^*] & \textrm{known-length type-level record} \\
adamc@525	219 &&& c \rc c & \textrm{type-level record concatenation} \\
adamc@652	220 &&& \mt{map} & \textrm{type-level record map} \\
adamc@524	221 \\
adamc@558	222 &&& (c,^+) & \textrm{type-level tuple} \\
adamc@525	223 &&& c.n & \textrm{type-level tuple projection ($n \in \mathbb N^+$)} \\
adamc@524	224 \\
adamc@652	225 &&& [c \sim c] \Rightarrow \tau & \textrm{guarded type} \\
adamc@524	226 \\
adamc@529	227 &&& \_ :: \kappa & \textrm{wildcard} \\
adamc@525	228 &&& (c) & \textrm{explicit precedence} \\
adamc@530	229 \\
adamc@530	230 \textrm{Qualified uncapitalized variables} & \hat{x} &::=& x & \textrm{not from a module} \\
adamc@530	231 &&& M.x & \textrm{projection from a module} \\
adamc@525	232 \end{array}$$
adamc@525	233
adamc@525	234 Modules of the module system are described by \emph{signatures}.
adamc@525	235 $$\begin{array}{rrcll}
adamc@525	236 \textrm{Signatures} & S &::=& \mt{sig} \; s^* \; \mt{end} & \textrm{constant} \\
adamc@525	237 &&& X & \textrm{variable} \\
adamc@525	238 &&& \mt{functor}(X : S) : S & \textrm{functor} \\
adamc@529	239 &&& S \; \mt{where} \; \mt{con} \; x = c & \textrm{concretizing an abstract constructor} \\
adamc@525	240 &&& M.X & \textrm{projection from a module} \\
adamc@525	241 \\
adamc@525	242 \textrm{Signature items} & s &::=& \mt{con} \; x :: \kappa & \textrm{abstract constructor} \\
adamc@525	243 &&& \mt{con} \; x :: \kappa = c & \textrm{concrete constructor} \\
adamc@528	244 &&& \mt{datatype} \; x \; x^* = dc\mid^+ & \textrm{algebraic datatype definition} \\
adamc@529	245 &&& \mt{datatype} \; x = \mt{datatype} \; M.x & \textrm{algebraic datatype import} \\
adamc@525	246 &&& \mt{val} \; x : \tau & \textrm{value} \\
adamc@525	247 &&& \mt{structure} \; X : S & \textrm{sub-module} \\
adamc@525	248 &&& \mt{signature} \; X = S & \textrm{sub-signature} \\
adamc@525	249 &&& \mt{include} \; S & \textrm{signature inclusion} \\
adamc@525	250 &&& \mt{constraint} \; c \sim c & \textrm{record disjointness constraint} \\
adamc@525	251 &&& \mt{class} \; x & \textrm{abstract type class} \\
adamc@525	252 &&& \mt{class} \; x = c & \textrm{concrete type class} \\
adamc@525	253 \\
adamc@525	254 \textrm{Datatype constructors} & dc &::=& X & \textrm{nullary constructor} \\
adamc@525	255 &&& X \; \mt{of} \; \tau & \textrm{unary constructor} \\
adamc@524	256 \end{array}$$
adamc@524	257
adamc@526	258 \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	259 $$\begin{array}{rrcll}
adamc@526	260 \textrm{Patterns} & p &::=& \_ & \textrm{wildcard} \\
adamc@526	261 &&& x & \textrm{variable} \\
adamc@526	262 &&& \ell & \textrm{constant} \\
adamc@526	263 &&& \hat{X} & \textrm{nullary constructor} \\
adamc@526	264 &&& \hat{X} \; p & \textrm{unary constructor} \\
adamc@526	265 &&& \{(x = p,)^*\} & \textrm{rigid record pattern} \\
adamc@526	266 &&& \{(x = p,)^+, \ldots\} & \textrm{flexible record pattern} \\
adamc@527	267 &&& (p) & \textrm{explicit precedence} \\
adamc@526	268 \\
adamc@529	269 \textrm{Qualified capitalized variables} & \hat{X} &::=& X & \textrm{not from a module} \\
adamc@526	270 &&& M.X & \textrm{projection from a module} \\
adamc@526	271 \end{array}$$
adamc@526	272
adamc@527	273 \emph{Expressions} are the main run-time entities, corresponding to both ``expressions'' and ``statements'' in mainstream imperative languages.
adamc@527	274 $$\begin{array}{rrcll}
adamc@527	275 \textrm{Expressions} & e &::=& e : \tau & \textrm{type annotation} \\
adamc@529	276 &&& \hat{x} & \textrm{variable} \\
adamc@529	277 &&& \hat{X} & \textrm{datatype constructor} \\
adamc@527	278 &&& \ell & \textrm{constant} \\
adamc@527	279 \\
adamc@527	280 &&& e \; e & \textrm{function application} \\
adamc@527	281 &&& \lambda x : \tau \Rightarrow e & \textrm{function abstraction} \\
adamc@527	282 &&& e [c] & \textrm{polymorphic function application} \\
adamc@527	283 &&& \lambda x \; ? \; \kappa \Rightarrow e & \textrm{polymorphic function abstraction} \\
adamc@652	284 &&& X \Longrightarrow e & \textrm{kind-polymorphic function abstraction} \\
adamc@527	285 \\
adamc@527	286 &&& \{(c = e,)^*\} & \textrm{known-length record} \\
adamc@527	287 &&& e.c & \textrm{record field projection} \\
adamc@527	288 &&& e \rc e & \textrm{record concatenation} \\
adamc@527	289 &&& e \rcut c & \textrm{removal of a single record field} \\
adamc@527	290 &&& e \rcutM c & \textrm{removal of multiple record fields} \\
adamc@527	291 \\
adamc@527	292 &&& \mt{let} \; ed^* \; \mt{in} \; e \; \mt{end} & \textrm{local definitions} \\
adamc@527	293 \\
adamc@527	294 &&& \mt{case} \; e \; \mt{of} \; (p \Rightarrow e\|)^+ & \textrm{pattern matching} \\
adamc@527	295 \\
adamc@527	296 &&& \lambda [c \sim c] \Rightarrow e & \textrm{guarded expression} \\
adamc@527	297 \\
adamc@527	298 &&& \_ & \textrm{wildcard} \\
adamc@527	299 &&& (e) & \textrm{explicit precedence} \\
adamc@527	300 \\
adamc@527	301 \textrm{Local declarations} & ed &::=& \cd{val} \; x : \tau = e & \textrm{non-recursive value} \\
adamc@527	302 &&& \cd{val} \; \cd{rec} \; (x : \tau = e \; \cd{and})^+ & \textrm{mutually-recursive values} \\
adamc@527	303 \end{array}$$
adamc@527	304
adamc@528	305 \emph{Declarations} primarily bring new symbols into context.
adamc@528	306 $$\begin{array}{rrcll}
adamc@528	307 \textrm{Declarations} & d &::=& \mt{con} \; x :: \kappa = c & \textrm{constructor synonym} \\
adamc@528	308 &&& \mt{datatype} \; x \; x^* = dc\mid^+ & \textrm{algebraic datatype definition} \\
adamc@529	309 &&& \mt{datatype} \; x = \mt{datatype} \; M.x & \textrm{algebraic datatype import} \\
adamc@528	310 &&& \mt{val} \; x : \tau = e & \textrm{value} \\
adamc@528	311 &&& \mt{val} \; \cd{rec} \; (x : \tau = e \; \mt{and})^+ & \textrm{mutually-recursive values} \\
adamc@528	312 &&& \mt{structure} \; X : S = M & \textrm{module definition} \\
adamc@528	313 &&& \mt{signature} \; X = S & \textrm{signature definition} \\
adamc@528	314 &&& \mt{open} \; M & \textrm{module inclusion} \\
adamc@528	315 &&& \mt{constraint} \; c \sim c & \textrm{record disjointness constraint} \\
adamc@528	316 &&& \mt{open} \; \mt{constraints} \; M & \textrm{inclusion of just the constraints from a module} \\
adamc@528	317 &&& \mt{table} \; x : c & \textrm{SQL table} \\
adamc@528	318 &&& \mt{sequence} \; x & \textrm{SQL sequence} \\
adamc@535	319 &&& \mt{cookie} \; x : \tau & \textrm{HTTP cookie} \\
adamc@528	320 &&& \mt{class} \; x = c & \textrm{concrete type class} \\
adamc@528	321 \\
adamc@529	322 \textrm{Modules} & M &::=& \mt{struct} \; d^* \; \mt{end} & \textrm{constant} \\
adamc@529	323 &&& X & \textrm{variable} \\
adamc@529	324 &&& M.X & \textrm{projection} \\
adamc@529	325 &&& M(M) & \textrm{functor application} \\
adamc@529	326 &&& \mt{functor}(X : S) : S = M & \textrm{functor abstraction} \\
adamc@528	327 \end{array}$$
adamc@528	328
adamc@528	329 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	330
adamc@529	331 \subsection{Shorthands}
adamc@529	332
adamc@529	333 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	334
adamc@529	335 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	336
adamc@529	337 A record type may be written $\{(c = c,)^\}$, which elaborates to $\$[(c = c,)^]$.
adamc@529	338
adamc@533	339 The notation $[c_1, \ldots, c_n]$ is shorthand for $[c_1 = (), \ldots, c_n = ()]$.
adamc@533	340
adamc@529	341 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	342
adamc@529	343 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	344
adamc@529	345 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	346
adamc@529	347 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	348
adamc@529	349 A signature item or declaration $\mt{class} \; x = \lambda y :: \mt{Type} \Rightarrow c$ may be abbreviated $\mt{class} \; x \; y = c$.
adamc@529	350
adamc@529	351 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	352
adamc@529	353 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	354
adamc@529	355 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	356
adamc@529	357 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	358
adamc@529	359 A declaration $\mt{table} \; x : \{(c = c,)^\}$ is elaborated into $\mt{table} \; x : [(c = c,)^]$
adamc@529	360
adamc@529	361 The syntax $\mt{where} \; \mt{type}$ is an alternate form of $\mt{where} \; \mt{con}$.
adamc@529	362
adamc@529	363 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	364
adamc@529	365 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	366
adamc@529	367 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	368
adamc@530	369
adamc@530	370 \section{Static Semantics}
adamc@530	371
adamc@530	372 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	373
adamc@530	374 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	375 \begin{itemize}
adamc@530	376 \item $\Gamma \vdash c :: \kappa$ assigns a kind to a constructor in a context.
adamc@530	377 \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	378 \item $\Gamma \vdash c \hookrightarrow C$ proves that record constructor $c$ decomposes into set $C$ of field names and record constructors.
adamc@530	379 \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	380 \item $\Gamma \vdash e : \tau$ is a standard typing judgment.
adamc@534	381 \item $\Gamma \vdash p \leadsto \Gamma; \tau$ combines typing of patterns with calculation of which new variables they bind.
adamc@537	382 \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	383 \item $\Gamma \vdash S \equiv S$ is the signature equivalence judgment.
adamc@536	384 \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	385 \item $\Gamma \vdash M : S$ is the module signature checking judgment.
adamc@537	386 \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	387 \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	388 \end{itemize}
adamc@530	389
adamc@530	390 \subsection{Kinding}
adamc@530	391
adamc@530	392 $$\infer{\Gamma \vdash (c) :: \kappa :: \kappa}{
adamc@530	393 \Gamma \vdash c :: \kappa
adamc@530	394 }
adamc@530	395 \quad \infer{\Gamma \vdash x :: \kappa}{
adamc@530	396 x :: \kappa \in \Gamma
adamc@530	397 }
adamc@530	398 \quad \infer{\Gamma \vdash x :: \kappa}{
adamc@530	399 x :: \kappa = c \in \Gamma
adamc@530	400 }$$
adamc@530	401
adamc@530	402 $$\infer{\Gamma \vdash M.x :: \kappa}{
adamc@537	403 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	404 & \mt{proj}(M, \overline{s}, \mt{con} \; x) = \kappa
adamc@530	405 }
adamc@530	406 \quad \infer{\Gamma \vdash M.x :: \kappa}{
adamc@537	407 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	408 & \mt{proj}(M, \overline{s}, \mt{con} \; x) = (\kappa, c)
adamc@530	409 }$$
adamc@530	410
adamc@530	411 $$\infer{\Gamma \vdash \tau_1 \to \tau_2 :: \mt{Type}}{
adamc@530	412 \Gamma \vdash \tau_1 :: \mt{Type}
adamc@530	413 & \Gamma \vdash \tau_2 :: \mt{Type}
adamc@530	414 }
adamc@530	415 \quad \infer{\Gamma \vdash x \; ? \: \kappa \to \tau :: \mt{Type}}{
adamc@530	416 \Gamma, x :: \kappa \vdash \tau :: \mt{Type}
adamc@530	417 }
adamc@530	418 \quad \infer{\Gamma \vdash \$c :: \mt{Type}}{
adamc@530	419 \Gamma \vdash c :: \{\mt{Type}\}
adamc@530	420 }$$
adamc@530	421
adamc@530	422 $$\infer{\Gamma \vdash c_1 \; c_2 :: \kappa_2}{
adamc@530	423 \Gamma \vdash c_1 :: \kappa_1 \to \kappa_2
adamc@530	424 & \Gamma \vdash c_2 :: \kappa_1
adamc@530	425 }
adamc@530	426 \quad \infer{\Gamma \vdash \lambda x \; :: \; \kappa_1 \Rightarrow c :: \kappa_1 \to \kappa_2}{
adamc@530	427 \Gamma, x :: \kappa_1 \vdash c :: \kappa_2
adamc@530	428 }$$
adamc@530	429
adamc@530	430 $$\infer{\Gamma \vdash () :: \mt{Unit}}{}
adamc@530	431 \quad \infer{\Gamma \vdash \#X :: \mt{Name}}{}$$
adamc@530	432
adamc@530	433 $$\infer{\Gamma \vdash [\overline{c_i = c'_i}] :: \{\kappa\}}{
adamc@530	434 \forall i: \Gamma \vdash c_i : \mt{Name}
adamc@530	435 & \Gamma \vdash c'_i :: \kappa
adamc@530	436 & \forall i \neq j: \Gamma \vdash c_i \sim c_j
adamc@530	437 }
adamc@530	438 \quad \infer{\Gamma \vdash c_1 \rc c_2 :: \{\kappa\}}{
adamc@530	439 \Gamma \vdash c_1 :: \{\kappa\}
adamc@530	440 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@530	441 & \Gamma \vdash c_1 \sim c_2
adamc@530	442 }$$
adamc@530	443
adamc@530	444 $$\infer{\Gamma \vdash \mt{fold} :: ((\mt{Name} \to \kappa_1 \to \kappa_2 \to \kappa_2) \to \kappa_2 \to \{\kappa_1\} \to \kappa_2}{}$$
adamc@530	445
adamc@573	446 $$\infer{\Gamma \vdash (\overline c) :: (\kappa_1 \times \ldots \times \kappa_n)}{
adamc@573	447 \forall i: \Gamma \vdash c_i :: \kappa_i
adamc@530	448 }
adamc@573	449 \quad \infer{\Gamma \vdash c.i :: \kappa_i}{
adamc@573	450 \Gamma \vdash c :: (\kappa_1 \times \ldots \times \kappa_n)
adamc@530	451 }$$
adamc@530	452
adamc@530	453 $$\infer{\Gamma \vdash \lambda [c_1 \sim c_2] \Rightarrow c :: \kappa}{
adamc@530	454 \Gamma \vdash c_1 :: \{\kappa'\}
adamc@530	455 & \Gamma \vdash c_2 :: \{\kappa'\}
adamc@530	456 & \Gamma, c_1 \sim c_2 \vdash c :: \kappa
adamc@530	457 }$$
adamc@530	458
adamc@531	459 \subsection{Record Disjointness}
adamc@531	460
adamc@531	461 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	462
adamc@531	463 $$\infer{\Gamma \vdash c_1 \sim c_2}{
adamc@558	464 \Gamma \vdash c_1 \hookrightarrow C_1
adamc@558	465 & \Gamma \vdash c_2 \hookrightarrow C_2
adamc@558	466 & \forall c'_1 \in C_1, c'_2 \in C_2: \Gamma \vdash c'_1 \sim c'_2
adamc@531	467 }
adamc@531	468 \quad \infer{\Gamma \vdash X \sim X'}{
adamc@531	469 X \neq X'
adamc@531	470 }$$
adamc@531	471
adamc@531	472 $$\infer{\Gamma \vdash c_1 \sim c_2}{
adamc@531	473 c'_1 \sim c'_2 \in \Gamma
adamc@558	474 & \Gamma \vdash c'_1 \hookrightarrow C_1
adamc@558	475 & \Gamma \vdash c'_2 \hookrightarrow C_2
adamc@558	476 & c_1 \in C_1
adamc@558	477 & c_2 \in C_2
adamc@531	478 }$$
adamc@531	479
adamc@531	480 $$\infer{\Gamma \vdash c \hookrightarrow \{c\}}{}
adamc@531	481 \quad \infer{\Gamma \vdash [\overline{c = c'}] \hookrightarrow \{\overline{c}\}}{}
adamc@531	482 \quad \infer{\Gamma \vdash c_1 \rc c_2 \hookrightarrow C_1 \cup C_2}{
adamc@531	483 \Gamma \vdash c_1 \hookrightarrow C_1
adamc@531	484 & \Gamma \vdash c_2 \hookrightarrow C_2
adamc@531	485 }
adamc@531	486 \quad \infer{\Gamma \vdash c \hookrightarrow C}{
adamc@531	487 \Gamma \vdash c \equiv c'
adamc@531	488 & \Gamma \vdash c' \hookrightarrow C
adamc@531	489 }
adamc@531	490 \quad \infer{\Gamma \vdash \mt{map} \; f \; c \hookrightarrow C}{
adamc@531	491 \Gamma \vdash c \hookrightarrow C
adamc@531	492 }$$
adamc@531	493
adamc@541	494 \subsection{\label{definitional}Definitional Equality}
adamc@532	495
adamc@532	496 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	497
adamc@532	498 $$\infer{\Gamma \vdash c \equiv c}{}
adamc@532	499 \quad \infer{\Gamma \vdash c_1 \equiv c_2}{
adamc@532	500 \Gamma \vdash c_2 \equiv c_1
adamc@532	501 }
adamc@532	502 \quad \infer{\Gamma \vdash c_1 \equiv c_3}{
adamc@532	503 \Gamma \vdash c_1 \equiv c_2
adamc@532	504 & \Gamma \vdash c_2 \equiv c_3
adamc@532	505 }
adamc@532	506 \quad \infer{\Gamma \vdash \mathcal C[c_1] \equiv \mathcal C[c_2]}{
adamc@532	507 \Gamma \vdash c_1 \equiv c_2
adamc@532	508 }$$
adamc@532	509
adamc@532	510 $$\infer{\Gamma \vdash x \equiv c}{
adamc@532	511 x :: \kappa = c \in \Gamma
adamc@532	512 }
adamc@532	513 \quad \infer{\Gamma \vdash M.x \equiv c}{
adamc@537	514 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	515 & \mt{proj}(M, \overline{s}, \mt{con} \; x) = (\kappa, c)
adamc@532	516 }
adamc@532	517 \quad \infer{\Gamma \vdash (\overline c).i \equiv c_i}{}$$
adamc@532	518
adamc@532	519 $$\infer{\Gamma \vdash (\lambda x :: \kappa \Rightarrow c) \; c' \equiv [x \mapsto c'] c}{}
adamc@532	520 \quad \infer{\Gamma \vdash c_1 \rc c_2 \equiv c_2 \rc c_1}{}
adamc@532	521 \quad \infer{\Gamma \vdash c_1 \rc (c_2 \rc c_3) \equiv (c_1 \rc c_2) \rc c_3}{}$$
adamc@532	522
adamc@532	523 $$\infer{\Gamma \vdash [] \rc c \equiv c}{}
adamc@532	524 \quad \infer{\Gamma \vdash [\overline{c_1 = c'_1}] \rc [\overline{c_2 = c'_2}] \equiv [\overline{c_1 = c'_1}, \overline{c_2 = c'_2}]}{}$$
adamc@532	525
adamc@532	526 $$\infer{\Gamma \vdash \lambda [c_1 \sim c_2] \Rightarrow c \equiv c}{
adamc@532	527 \Gamma \vdash c_1 \sim c_2
adamc@532	528 }
adamc@532	529 \quad \infer{\Gamma \vdash \mt{fold} \; f \; i \; [] \equiv i}{}
adamc@532	530 \quad \infer{\Gamma \vdash \mt{fold} \; f \; i \; ([c_1 = c_2] \rc c) \equiv f \; c_1 \; c_2 \; (\mt{fold} \; f \; i \; c)}{}$$
adamc@532	531
adamc@532	532 $$\infer{\Gamma \vdash \mt{map} \; (\lambda x \Rightarrow x) \; c \equiv c}{}
adamc@532	533 \quad \infer{\Gamma \vdash \mt{fold} \; f \; i \; (\mt{map} \; f' \; c)
adamc@532	534 \equiv \mt{fold} \; (\lambda (x_1 :: \mt{Name}) (x_2 :: \kappa) \Rightarrow f \; x_1 \; (f' \; x_2)) \; i \; c}{}$$
adamc@532	535
adamc@532	536 $$\infer{\Gamma \vdash \mt{map} \; f \; (c_1 \rc c_2) \equiv \mt{map} \; f \; c_1 \rc \mt{map} \; f \; c_2}{}$$
adamc@531	537
adamc@534	538 \subsection{Expression Typing}
adamc@533	539
adamc@533	540 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	541
adamc@533	542 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	543
adamc@533	544 $$\infer{\Gamma \vdash e : \tau : \tau}{
adamc@533	545 \Gamma \vdash e : \tau
adamc@533	546 }
adamc@533	547 \quad \infer{\Gamma \vdash e : \tau}{
adamc@533	548 \Gamma \vdash e : \tau'
adamc@533	549 & \Gamma \vdash \tau' \equiv \tau
adamc@533	550 }
adamc@533	551 \quad \infer{\Gamma \vdash \ell : T(\ell)}{}$$
adamc@533	552
adamc@533	553 $$\infer{\Gamma \vdash x : \mathcal I(\tau)}{
adamc@533	554 x : \tau \in \Gamma
adamc@533	555 }
adamc@533	556 \quad \infer{\Gamma \vdash M.x : \mathcal I(\tau)}{
adamc@537	557 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	558 & \mt{proj}(M, \overline{s}, \mt{val} \; x) = \tau
adamc@533	559 }
adamc@533	560 \quad \infer{\Gamma \vdash X : \mathcal I(\tau)}{
adamc@533	561 X : \tau \in \Gamma
adamc@533	562 }
adamc@533	563 \quad \infer{\Gamma \vdash M.X : \mathcal I(\tau)}{
adamc@537	564 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	565 & \mt{proj}(M, \overline{s}, \mt{val} \; X) = \tau
adamc@533	566 }$$
adamc@533	567
adamc@533	568 $$\infer{\Gamma \vdash e_1 \; e_2 : \tau_2}{
adamc@533	569 \Gamma \vdash e_1 : \tau_1 \to \tau_2
adamc@533	570 & \Gamma \vdash e_2 : \tau_1
adamc@533	571 }
adamc@533	572 \quad \infer{\Gamma \vdash \lambda x : \tau_1 \Rightarrow e : \tau_1 \to \tau_2}{
adamc@533	573 \Gamma, x : \tau_1 \vdash e : \tau_2
adamc@533	574 }$$
adamc@533	575
adamc@533	576 $$\infer{\Gamma \vdash e [c] : [x \mapsto c]\tau}{
adamc@533	577 \Gamma \vdash e : x :: \kappa \to \tau
adamc@533	578 & \Gamma \vdash c :: \kappa
adamc@533	579 }
adamc@533	580 \quad \infer{\Gamma \vdash \lambda x \; ? \; \kappa \Rightarrow e : x \; ? \; \kappa \to \tau}{
adamc@533	581 \Gamma, x :: \kappa \vdash e : \tau
adamc@533	582 }$$
adamc@533	583
adamc@533	584 $$\infer{\Gamma \vdash \{\overline{c = e}\} : \{\overline{c : \tau}\}}{
adamc@533	585 \forall i: \Gamma \vdash c_i :: \mt{Name}
adamc@533	586 & \Gamma \vdash e_i : \tau_i
adamc@533	587 & \forall i \neq j: \Gamma \vdash c_i \sim c_j
adamc@533	588 }
adamc@533	589 \quad \infer{\Gamma \vdash e.c : \tau}{
adamc@533	590 \Gamma \vdash e : \$([c = \tau] \rc c')
adamc@533	591 }
adamc@533	592 \quad \infer{\Gamma \vdash e_1 \rc e_2 : \$(c_1 \rc c_2)}{
adamc@533	593 \Gamma \vdash e_1 : \$c_1
adamc@533	594 & \Gamma \vdash e_2 : \$c_2
adamc@573	595 & \Gamma \vdash c_1 \sim c_2
adamc@533	596 }$$
adamc@533	597
adamc@533	598 $$\infer{\Gamma \vdash e \rcut c : \$c'}{
adamc@533	599 \Gamma \vdash e : \$([c = \tau] \rc c')
adamc@533	600 }
adamc@533	601 \quad \infer{\Gamma \vdash e \rcutM c : \$c'}{
adamc@533	602 \Gamma \vdash e : \$(c \rc c')
adamc@533	603 }$$
adamc@533	604
adamc@533	605 $$\infer{\Gamma \vdash \mt{fold} : \begin{array}{c}
adamc@533	606 x_1 :: (\{\kappa\} \to \tau)
adamc@533	607 \to (x_2 :: \mt{Name} \to x_3 :: \kappa \to x_4 :: \{\kappa\} \to \lambda [[x_2] \sim x_4]
adamc@533	608 \Rightarrow x_1 \; x_4 \to x_1 \; ([x_2 = x_3] \rc x_4)) \\
adamc@533	609 \to x_1 \; [] \to x_5 :: \{\kappa\} \to x_1 \; x_5
adamc@533	610 \end{array}}{}$$
adamc@533	611
adamc@533	612 $$\infer{\Gamma \vdash \mt{let} \; \overline{ed} \; \mt{in} \; e \; \mt{end} : \tau}{
adamc@533	613 \Gamma \vdash \overline{ed} \leadsto \Gamma'
adamc@533	614 & \Gamma' \vdash e : \tau
adamc@533	615 }
adamc@533	616 \quad \infer{\Gamma \vdash \mt{case} \; e \; \mt{of} \; \overline{p \Rightarrow e} : \tau}{
adamc@533	617 \forall i: \Gamma \vdash p_i \leadsto \Gamma_i, \tau'
adamc@533	618 & \Gamma_i \vdash e_i : \tau
adamc@533	619 }$$
adamc@533	620
adamc@573	621 $$\infer{\Gamma \vdash \lambda [c_1 \sim c_2] \Rightarrow e : \lambda [c_1 \sim c_2] \Rightarrow \tau}{
adamc@533	622 \Gamma \vdash c_1 :: \{\kappa\}
adamc@533	623 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@533	624 & \Gamma, c_1 \sim c_2 \vdash e : \tau
adamc@533	625 }$$
adamc@533	626
adamc@534	627 \subsection{Pattern Typing}
adamc@534	628
adamc@534	629 $$\infer{\Gamma \vdash \_ \leadsto \Gamma; \tau}{}
adamc@534	630 \quad \infer{\Gamma \vdash x \leadsto \Gamma, x : \tau; \tau}{}
adamc@534	631 \quad \infer{\Gamma \vdash \ell \leadsto \Gamma; T(\ell)}{}$$
adamc@534	632
adamc@534	633 $$\infer{\Gamma \vdash X \leadsto \Gamma; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@534	634 X : \overline{x ::: \mt{Type}} \to \tau \in \Gamma
adamc@534	635 & \textrm{$\tau$ not a function type}
adamc@534	636 }
adamc@534	637 \quad \infer{\Gamma \vdash X \; p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@534	638 X : \overline{x ::: \mt{Type}} \to \tau'' \to \tau \in \Gamma
adamc@534	639 & \Gamma \vdash p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau''
adamc@534	640 }$$
adamc@534	641
adamc@534	642 $$\infer{\Gamma \vdash M.X \leadsto \Gamma; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@537	643 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	644 & \mt{proj}(M, \overline{s}, \mt{val} \; X) = \overline{x ::: \mt{Type}} \to \tau
adamc@534	645 & \textrm{$\tau$ not a function type}
adamc@534	646 }$$
adamc@534	647
adamc@534	648 $$\infer{\Gamma \vdash M.X \; p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau}{
adamc@537	649 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	650 & \mt{proj}(M, \overline{s}, \mt{val} \; X) = \overline{x ::: \mt{Type}} \to \tau'' \to \tau
adamc@534	651 & \Gamma \vdash p \leadsto \Gamma'; \overline{[x_i \mapsto \tau'_i]}\tau''
adamc@534	652 }$$
adamc@534	653
adamc@534	654 $$\infer{\Gamma \vdash \{\overline{x = p}\} \leadsto \Gamma_n; \{\overline{x = \tau}\}}{
adamc@534	655 \Gamma_0 = \Gamma
adamc@534	656 & \forall i: \Gamma_i \vdash p_i \leadsto \Gamma_{i+1}; \tau_i
adamc@534	657 }
adamc@534	658 \quad \infer{\Gamma \vdash \{\overline{x = p}, \ldots\} \leadsto \Gamma_n; \$([\overline{x = \tau}] \rc c)}{
adamc@534	659 \Gamma_0 = \Gamma
adamc@534	660 & \forall i: \Gamma_i \vdash p_i \leadsto \Gamma_{i+1}; \tau_i
adamc@534	661 }$$
adamc@534	662
adamc@535	663 \subsection{Declaration Typing}
adamc@535	664
adamc@535	665 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	666
adamc@541	667 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	668
adamc@558	669 We presuppose the existence of a function $\mathcal O$, where $\mathcal O(M, \overline{s})$ implements the $\mt{open}$ declaration by producing a context with the appropriate entry for each available component of module $M$ with signature items $\overline{s}$. Where possible, $\mathcal O$ uses ``transparent'' entries (e.g., an abstract type $M.x$ is mapped to $x :: \mt{Type} = M.x$), so that the relationship with $M$ is maintained. A related function $\mathcal O_c$ builds a context containing the disjointness constraints found in $\overline s$.
adamc@537	670 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	671
adamc@535	672 $$\infer{\Gamma \vdash \cdot \leadsto \Gamma}{}
adamc@535	673 \quad \infer{\Gamma \vdash d, \overline{d} \leadsto \Gamma''}{
adamc@535	674 \Gamma \vdash d \leadsto \Gamma'
adamc@535	675 & \Gamma' \vdash \overline{d} \leadsto \Gamma''
adamc@535	676 }$$
adamc@535	677
adamc@535	678 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa = c \leadsto \Gamma, x :: \kappa = c}{
adamc@535	679 \Gamma \vdash c :: \kappa
adamc@535	680 }
adamc@535	681 \quad \infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leadsto \Gamma'}{
adamc@535	682 \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} \vdash \overline{dc} \leadsto \Gamma'
adamc@535	683 }$$
adamc@535	684
adamc@535	685 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leadsto \Gamma'}{
adamc@537	686 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	687 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@535	688 & \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} = M.z \vdash \overline{dc} \leadsto \Gamma'
adamc@535	689 }$$
adamc@535	690
adamc@535	691 $$\infer{\Gamma \vdash \mt{val} \; x : \tau = e \leadsto \Gamma, x : \tau}{
adamc@535	692 \Gamma \vdash e : \tau
adamc@535	693 }$$
adamc@535	694
adamc@535	695 $$\infer{\Gamma \vdash \mt{val} \; \mt{rec} \; \overline{x : \tau = e} \leadsto \Gamma, \overline{x : \tau}}{
adamc@535	696 \forall i: \Gamma, \overline{x : \tau} \vdash e_i : \tau_i
adamc@535	697 & \textrm{$e_i$ starts with an expression $\lambda$, optionally preceded by constructor and disjointness $\lambda$s}
adamc@535	698 }$$
adamc@535	699
adamc@535	700 $$\infer{\Gamma \vdash \mt{structure} \; X : S = M \leadsto \Gamma, X : S}{
adamc@535	701 \Gamma \vdash M : S
adamc@558	702 & \textrm{ $M$ not a constant or application}
adamc@535	703 }
adamc@558	704 \quad \infer{\Gamma \vdash \mt{structure} \; X : S = M \leadsto \Gamma, X : \mt{selfify}(X, \overline{s})}{
adamc@558	705 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@539	706 }$$
adamc@539	707
adamc@539	708 $$\infer{\Gamma \vdash \mt{signature} \; X = S \leadsto \Gamma, X = S}{
adamc@535	709 \Gamma \vdash S
adamc@535	710 }$$
adamc@535	711
adamc@537	712 $$\infer{\Gamma \vdash \mt{open} \; M \leadsto \Gamma, \mathcal O(M, \overline{s})}{
adamc@537	713 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@535	714 }$$
adamc@535	715
adamc@535	716 $$\infer{\Gamma \vdash \mt{constraint} \; c_1 \sim c_2 \leadsto \Gamma}{
adamc@535	717 \Gamma \vdash c_1 :: \{\kappa\}
adamc@535	718 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@535	719 & \Gamma \vdash c_1 \sim c_2
adamc@535	720 }
adamc@537	721 \quad \infer{\Gamma \vdash \mt{open} \; \mt{constraints} \; M \leadsto \Gamma, \mathcal O_c(M, \overline{s})}{
adamc@537	722 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@535	723 }$$
adamc@535	724
adamc@535	725 $$\infer{\Gamma \vdash \mt{table} \; x : c \leadsto \Gamma, x : \mt{Basis}.\mt{sql\_table} \; c}{
adamc@535	726 \Gamma \vdash c :: \{\mt{Type}\}
adamc@535	727 }
adamc@535	728 \quad \infer{\Gamma \vdash \mt{sequence} \; x \leadsto \Gamma, x : \mt{Basis}.\mt{sql\_sequence}}{}$$
adamc@535	729
adamc@535	730 $$\infer{\Gamma \vdash \mt{cookie} \; x : \tau \leadsto \Gamma, x : \mt{Basis}.\mt{http\_cookie} \; \tau}{
adamc@535	731 \Gamma \vdash \tau :: \mt{Type}
adamc@535	732 }$$
adamc@535	733
adamc@535	734 $$\infer{\Gamma \vdash \mt{class} \; x = c \leadsto \Gamma, x :: \mt{Type} \to \mt{Type} = c}{
adamc@535	735 \Gamma \vdash c :: \mt{Type} \to \mt{Type}
adamc@535	736 }$$
adamc@535	737
adamc@535	738 $$\infer{\overline{y}; x; \Gamma \vdash \cdot \leadsto \Gamma}{}
adamc@535	739 \quad \infer{\overline{y}; x; \Gamma \vdash X \mid \overline{dc} \leadsto \Gamma', X : \overline{y ::: \mt{Type}} \to x \; \overline{y}}{
adamc@535	740 \overline{y}; x; \Gamma \vdash \overline{dc} \leadsto \Gamma'
adamc@535	741 }
adamc@535	742 \quad \infer{\overline{y}; x; \Gamma \vdash X \; \mt{of} \; \tau \mid \overline{dc} \leadsto \Gamma', X : \overline{y ::: \mt{Type}} \to \tau \to x \; \overline{y}}{
adamc@535	743 \overline{y}; x; \Gamma \vdash \overline{dc} \leadsto \Gamma'
adamc@535	744 }$$
adamc@535	745
adamc@537	746 \subsection{Signature Item Typing}
adamc@537	747
adamc@537	748 We appeal to a signature item analogue of the $\mathcal O$ function from the last subsection.
adamc@537	749
adamc@537	750 $$\infer{\Gamma \vdash \cdot \leadsto \Gamma}{}
adamc@537	751 \quad \infer{\Gamma \vdash s, \overline{s} \leadsto \Gamma''}{
adamc@537	752 \Gamma \vdash s \leadsto \Gamma'
adamc@537	753 & \Gamma' \vdash \overline{s} \leadsto \Gamma''
adamc@537	754 }$$
adamc@537	755
adamc@537	756 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa \leadsto \Gamma, x :: \kappa}{}
adamc@537	757 \quad \infer{\Gamma \vdash \mt{con} \; x :: \kappa = c \leadsto \Gamma, x :: \kappa = c}{
adamc@537	758 \Gamma \vdash c :: \kappa
adamc@537	759 }
adamc@537	760 \quad \infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leadsto \Gamma'}{
adamc@537	761 \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} \vdash \overline{dc} \leadsto \Gamma'
adamc@537	762 }$$
adamc@537	763
adamc@537	764 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leadsto \Gamma'}{
adamc@537	765 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	766 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@537	767 & \overline{y}; x; \Gamma, x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type} = M.z \vdash \overline{dc} \leadsto \Gamma'
adamc@537	768 }$$
adamc@537	769
adamc@537	770 $$\infer{\Gamma \vdash \mt{val} \; x : \tau \leadsto \Gamma, x : \tau}{
adamc@537	771 \Gamma \vdash \tau :: \mt{Type}
adamc@537	772 }$$
adamc@537	773
adamc@537	774 $$\infer{\Gamma \vdash \mt{structure} \; X : S \leadsto \Gamma, X : S}{
adamc@537	775 \Gamma \vdash S
adamc@537	776 }
adamc@537	777 \quad \infer{\Gamma \vdash \mt{signature} \; X = S \leadsto \Gamma, X = S}{
adamc@537	778 \Gamma \vdash S
adamc@537	779 }$$
adamc@537	780
adamc@537	781 $$\infer{\Gamma \vdash \mt{include} \; S \leadsto \Gamma, \mathcal O(\overline{s})}{
adamc@537	782 \Gamma \vdash S
adamc@537	783 & \Gamma \vdash S \equiv \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	784 }$$
adamc@537	785
adamc@537	786 $$\infer{\Gamma \vdash \mt{constraint} \; c_1 \sim c_2 \leadsto \Gamma, c_1 \sim c_2}{
adamc@537	787 \Gamma \vdash c_1 :: \{\kappa\}
adamc@537	788 & \Gamma \vdash c_2 :: \{\kappa\}
adamc@537	789 }$$
adamc@537	790
adamc@537	791 $$\infer{\Gamma \vdash \mt{class} \; x = c \leadsto \Gamma, x :: \mt{Type} \to \mt{Type} = c}{
adamc@537	792 \Gamma \vdash c :: \mt{Type} \to \mt{Type}
adamc@537	793 }
adamc@537	794 \quad \infer{\Gamma \vdash \mt{class} \; x \leadsto \Gamma, x :: \mt{Type} \to \mt{Type}}{}$$
adamc@537	795
adamc@536	796 \subsection{Signature Compatibility}
adamc@536	797
adamc@558	798 To simplify the judgments in this section, we assume that all signatures are alpha-varied as necessary to avoid including multiple bindings for the same identifier. This is in addition to the usual alpha-variation of locally-bound variables.
adamc@537	799
adamc@537	800 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	801
adamc@536	802 $$\infer{\Gamma \vdash S \equiv S}{}
adamc@536	803 \quad \infer{\Gamma \vdash S_1 \equiv S_2}{
adamc@536	804 \Gamma \vdash S_2 \equiv S_1
adamc@536	805 }
adamc@536	806 \quad \infer{\Gamma \vdash X \equiv S}{
adamc@536	807 X = S \in \Gamma
adamc@536	808 }
adamc@536	809 \quad \infer{\Gamma \vdash M.X \equiv S}{
adamc@537	810 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	811 & \mt{proj}(M, \overline{s}, \mt{signature} \; X) = S
adamc@536	812 }$$
adamc@536	813
adamc@536	814 $$\infer{\Gamma \vdash S \; \mt{where} \; \mt{con} \; x = c \equiv \mt{sig} \; \overline{s^1} \; \mt{con} \; x :: \kappa = c \; \overline{s_2} \; \mt{end}}{
adamc@536	815 \Gamma \vdash S \equiv \mt{sig} \; \overline{s^1} \; \mt{con} \; x :: \kappa \; \overline{s_2} \; \mt{end}
adamc@536	816 & \Gamma \vdash c :: \kappa
adamc@537	817 }
adamc@537	818 \quad \infer{\Gamma \vdash \mt{sig} \; \overline{s^1} \; \mt{include} \; S \; \overline{s^2} \; \mt{end} \equiv \mt{sig} \; \overline{s^1} \; \overline{s} \; \overline{s^2} \; \mt{end}}{
adamc@537	819 \Gamma \vdash S \equiv \mt{sig} \; \overline{s} \; \mt{end}
adamc@536	820 }$$
adamc@536	821
adamc@536	822 $$\infer{\Gamma \vdash S_1 \leq S_2}{
adamc@536	823 \Gamma \vdash S_1 \equiv S_2
adamc@536	824 }
adamc@536	825 \quad \infer{\Gamma \vdash \mt{sig} \; \overline{s} \; \mt{end} \leq \mt{sig} \; \mt{end}}{}
adamc@537	826 \quad \infer{\Gamma \vdash \mt{sig} \; \overline{s} \; \mt{end} \leq \mt{sig} \; s' \; \overline{s'} \; \mt{end}}{
adamc@537	827 \Gamma \vdash \overline{s} \leq s'
adamc@537	828 & \Gamma \vdash s' \leadsto \Gamma'
adamc@537	829 & \Gamma' \vdash \mt{sig} \; \overline{s} \; \mt{end} \leq \mt{sig} \; \overline{s'} \; \mt{end}
adamc@537	830 }$$
adamc@537	831
adamc@537	832 $$\infer{\Gamma \vdash s \; \overline{s} \leq s'}{
adamc@537	833 \Gamma \vdash s \leq s'
adamc@537	834 }
adamc@537	835 \quad \infer{\Gamma \vdash s \; \overline{s} \leq s'}{
adamc@537	836 \Gamma \vdash s \leadsto \Gamma'
adamc@537	837 & \Gamma' \vdash \overline{s} \leq s'
adamc@536	838 }$$
adamc@536	839
adamc@536	840 $$\infer{\Gamma \vdash \mt{functor} (X : S_1) : S_2 \leq \mt{functor} (X : S'_1) : S'_2}{
adamc@536	841 \Gamma \vdash S'_1 \leq S_1
adamc@536	842 & \Gamma, X : S'_1 \vdash S_2 \leq S'_2
adamc@536	843 }$$
adamc@536	844
adamc@537	845 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa \leq \mt{con} \; x :: \kappa}{}
adamc@537	846 \quad \infer{\Gamma \vdash \mt{con} \; x :: \kappa = c \leq \mt{con} \; x :: \kappa}{}
adamc@558	847 \quad \infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leq \mt{con} \; x :: \mt{Type}^{\mt{len}(\overline y)} \to \mt{Type}}{}$$
adamc@537	848
adamc@537	849 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leq \mt{con} \; x :: \mt{Type}^{\mt{len}(y)} \to \mt{Type}}{
adamc@537	850 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	851 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@537	852 }$$
adamc@537	853
adamc@537	854 $$\infer{\Gamma \vdash \mt{class} \; x \leq \mt{con} \; x :: \mt{Type} \to \mt{Type}}{}
adamc@537	855 \quad \infer{\Gamma \vdash \mt{class} \; x = c \leq \mt{con} \; x :: \mt{Type} \to \mt{Type}}{}$$
adamc@537	856
adamc@537	857 $$\infer{\Gamma \vdash \mt{con} \; x :: \kappa = c_1 \leq \mt{con} \; x :: \mt{\kappa} = c_2}{
adamc@537	858 \Gamma \vdash c_1 \equiv c_2
adamc@537	859 }
adamc@537	860 \quad \infer{\Gamma \vdash \mt{class} \; x = c_1 \leq \mt{con} \; x :: \mt{Type} \to \mt{Type} = c_2}{
adamc@537	861 \Gamma \vdash c_1 \equiv c_2
adamc@537	862 }$$
adamc@537	863
adamc@537	864 $$\infer{\Gamma \vdash \mt{datatype} \; x \; \overline{y} = \overline{dc} \leq \mt{datatype} \; x \; \overline{y} = \overline{dc'}}{
adamc@537	865 \Gamma, \overline{y :: \mt{Type}} \vdash \overline{dc} \leq \overline{dc'}
adamc@537	866 }$$
adamc@537	867
adamc@537	868 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leq \mt{datatype} \; x \; \overline{y} = \overline{dc'}}{
adamc@537	869 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@537	870 & \mt{proj}(M, \overline{s}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})
adamc@537	871 & \Gamma, \overline{y :: \mt{Type}} \vdash \overline{dc} \leq \overline{dc'}
adamc@537	872 }$$
adamc@537	873
adamc@537	874 $$\infer{\Gamma \vdash \cdot \leq \cdot}{}
adamc@537	875 \quad \infer{\Gamma \vdash X; \overline{dc} \leq X; \overline{dc'}}{
adamc@537	876 \Gamma \vdash \overline{dc} \leq \overline{dc'}
adamc@537	877 }
adamc@537	878 \quad \infer{\Gamma \vdash X \; \mt{of} \; \tau_1; \overline{dc} \leq X \; \mt{of} \; \tau_2; \overline{dc'}}{
adamc@537	879 \Gamma \vdash \tau_1 \equiv \tau_2
adamc@537	880 & \Gamma \vdash \overline{dc} \leq \overline{dc'}
adamc@537	881 }$$
adamc@537	882
adamc@537	883 $$\infer{\Gamma \vdash \mt{datatype} \; x = \mt{datatype} \; M.z \leq \mt{datatype} \; x = \mt{datatype} \; M'.z'}{
adamc@537	884 \Gamma \vdash M.z \equiv M'.z'
adamc@537	885 }$$
adamc@537	886
adamc@537	887 $$\infer{\Gamma \vdash \mt{val} \; x : \tau_1 \leq \mt{val} \; x : \tau_2}{
adamc@537	888 \Gamma \vdash \tau_1 \equiv \tau_2
adamc@537	889 }
adamc@537	890 \quad \infer{\Gamma \vdash \mt{structure} \; X : S_1 \leq \mt{structure} \; X : S_2}{
adamc@537	891 \Gamma \vdash S_1 \leq S_2
adamc@537	892 }
adamc@537	893 \quad \infer{\Gamma \vdash \mt{signature} \; X = S_1 \leq \mt{signature} \; X = S_2}{
adamc@537	894 \Gamma \vdash S_1 \leq S_2
adamc@537	895 & \Gamma \vdash S_2 \leq S_1
adamc@537	896 }$$
adamc@537	897
adamc@537	898 $$\infer{\Gamma \vdash \mt{constraint} \; c_1 \sim c_2 \leq \mt{constraint} \; c'_1 \sim c'_2}{
adamc@537	899 \Gamma \vdash c_1 \equiv c'_1
adamc@537	900 & \Gamma \vdash c_2 \equiv c'_2
adamc@537	901 }$$
adamc@537	902
adamc@537	903 $$\infer{\Gamma \vdash \mt{class} \; x \leq \mt{class} \; x}{}
adamc@537	904 \quad \infer{\Gamma \vdash \mt{class} \; x = c \leq \mt{class} \; x}{}
adamc@537	905 \quad \infer{\Gamma \vdash \mt{class} \; x = c_1 \leq \mt{class} \; x = c_2}{
adamc@537	906 \Gamma \vdash c_1 \equiv c_2
adamc@537	907 }$$
adamc@537	908
adamc@538	909 \subsection{Module Typing}
adamc@538	910
adamc@538	911 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	912
adamc@538	913 $$\infer{\Gamma \vdash M : S}{
adamc@538	914 \Gamma \vdash M : S'
adamc@538	915 & \Gamma \vdash S' \leq S
adamc@538	916 }
adamc@538	917 \quad \infer{\Gamma \vdash \mt{struct} \; \overline{d} \; \mt{end} : \mt{sig} \; \mt{sigOf}(\overline{d}) \; \mt{end}}{
adamc@538	918 \Gamma \vdash \overline{d} \leadsto \Gamma'
adamc@538	919 }
adamc@538	920 \quad \infer{\Gamma \vdash X : S}{
adamc@538	921 X : S \in \Gamma
adamc@538	922 }$$
adamc@538	923
adamc@538	924 $$\infer{\Gamma \vdash M.X : S}{
adamc@538	925 \Gamma \vdash M : \mt{sig} \; \overline{s} \; \mt{end}
adamc@538	926 & \mt{proj}(M, \overline{s}, \mt{structure} \; X) = S
adamc@538	927 }$$
adamc@538	928
adamc@538	929 $$\infer{\Gamma \vdash M_1(M_2) : [X \mapsto M_2]S_2}{
adamc@538	930 \Gamma \vdash M_1 : \mt{functor}(X : S_1) : S_2
adamc@538	931 & \Gamma \vdash M_2 : S_1
adamc@538	932 }
adamc@538	933 \quad \infer{\Gamma \vdash \mt{functor} (X : S_1) : S_2 = M : \mt{functor} (X : S_1) : S_2}{
adamc@538	934 \Gamma \vdash S_1
adamc@538	935 & \Gamma, X : S_1 \vdash S_2
adamc@538	936 & \Gamma, X : S_1 \vdash M : S_2
adamc@538	937 }$$
adamc@538	938
adamc@538	939 \begin{eqnarray*}
adamc@538	940 \mt{sigOf}(\cdot) &=& \cdot \\
adamc@538	941 \mt{sigOf}(s \; \overline{s'}) &=& \mt{sigOf}(s) \; \mt{sigOf}(\overline{s'}) \\
adamc@538	942 \\
adamc@538	943 \mt{sigOf}(\mt{con} \; x :: \kappa = c) &=& \mt{con} \; x :: \kappa = c \\
adamc@538	944 \mt{sigOf}(\mt{datatype} \; x \; \overline{y} = \overline{dc}) &=& \mt{datatype} \; x \; \overline{y} = \overline{dc} \\
adamc@538	945 \mt{sigOf}(\mt{datatype} \; x = \mt{datatype} \; M.z) &=& \mt{datatype} \; x = \mt{datatype} \; M.z \\
adamc@538	946 \mt{sigOf}(\mt{val} \; x : \tau = e) &=& \mt{val} \; x : \tau \\
adamc@538	947 \mt{sigOf}(\mt{val} \; \mt{rec} \; \overline{x : \tau = e}) &=& \overline{\mt{val} \; x : \tau} \\
adamc@538	948 \mt{sigOf}(\mt{structure} \; X : S = M) &=& \mt{structure} \; X : S \\
adamc@538	949 \mt{sigOf}(\mt{signature} \; X = S) &=& \mt{signature} \; X = S \\
adamc@538	950 \mt{sigOf}(\mt{open} \; M) &=& \mt{include} \; S \textrm{ (where $\Gamma \vdash M : S$)} \\
adamc@538	951 \mt{sigOf}(\mt{constraint} \; c_1 \sim c_2) &=& \mt{constraint} \; c_1 \sim c_2 \\
adamc@538	952 \mt{sigOf}(\mt{open} \; \mt{constraints} \; M) &=& \cdot \\
adamc@538	953 \mt{sigOf}(\mt{table} \; x : c) &=& \mt{table} \; x : c \\
adamc@538	954 \mt{sigOf}(\mt{sequence} \; x) &=& \mt{sequence} \; x \\
adamc@538	955 \mt{sigOf}(\mt{cookie} \; x : \tau) &=& \mt{cookie} \; x : \tau \\
adamc@538	956 \mt{sigOf}(\mt{class} \; x = c) &=& \mt{class} \; x = c \\
adamc@538	957 \end{eqnarray*}
adamc@539	958 \begin{eqnarray*}
adamc@539	959 \mt{selfify}(M, \cdot) &=& \cdot \\
adamc@558	960 \mt{selfify}(M, s \; \overline{s'}) &=& \mt{selfify}(M, s) \; \mt{selfify}(M, \overline{s'}) \\
adamc@539	961 \\
adamc@539	962 \mt{selfify}(M, \mt{con} \; x :: \kappa) &=& \mt{con} \; x :: \kappa = M.x \\
adamc@539	963 \mt{selfify}(M, \mt{con} \; x :: \kappa = c) &=& \mt{con} \; x :: \kappa = c \\
adamc@539	964 \mt{selfify}(M, \mt{datatype} \; x \; \overline{y} = \overline{dc}) &=& \mt{datatype} \; x \; \overline{y} = \mt{datatype} \; M.x \\
adamc@539	965 \mt{selfify}(M, \mt{datatype} \; x = \mt{datatype} \; M'.z) &=& \mt{datatype} \; x = \mt{datatype} \; M'.z \\
adamc@539	966 \mt{selfify}(M, \mt{val} \; x : \tau) &=& \mt{val} \; x : \tau \\
adamc@539	967 \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	968 \mt{selfify}(M, \mt{signature} \; X = S) &=& \mt{signature} \; X = S \\
adamc@539	969 \mt{selfify}(M, \mt{include} \; S) &=& \mt{include} \; S \\
adamc@539	970 \mt{selfify}(M, \mt{constraint} \; c_1 \sim c_2) &=& \mt{constraint} \; c_1 \sim c_2 \\
adamc@539	971 \mt{selfify}(M, \mt{class} \; x) &=& \mt{class} \; x = M.x \\
adamc@539	972 \mt{selfify}(M, \mt{class} \; x = c) &=& \mt{class} \; x = c \\
adamc@539	973 \end{eqnarray*}
adamc@539	974
adamc@540	975 \subsection{Module Projection}
adamc@540	976
adamc@540	977 \begin{eqnarray*}
adamc@540	978 \mt{proj}(M, \mt{con} \; x :: \kappa \; \overline{s}, \mt{con} \; x) &=& \kappa \\
adamc@540	979 \mt{proj}(M, \mt{con} \; x :: \kappa = c \; \overline{s}, \mt{con} \; x) &=& (\kappa, c) \\
adamc@540	980 \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	981 \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	982 && \textrm{and $\mt{proj}(M', \overline{s'}, \mt{datatype} \; z) = (\overline{y}, \overline{dc})$)} \\
adamc@540	983 \mt{proj}(M, \mt{class} \; x \; \overline{s}, \mt{con} \; x) &=& \mt{Type} \to \mt{Type} \\
adamc@540	984 \mt{proj}(M, \mt{class} \; x = c \; \overline{s}, \mt{con} \; x) &=& (\mt{Type} \to \mt{Type}, c) \\
adamc@540	985 \\
adamc@540	986 \mt{proj}(M, \mt{datatype} \; x \; \overline{y} = \overline{dc} \; \overline{s}, \mt{datatype} \; x) &=& (\overline{y}, \overline{dc}) \\
adamc@540	987 \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	988 \\
adamc@540	989 \mt{proj}(M, \mt{val} \; x : \tau \; \overline{s}, \mt{val} \; x) &=& \tau \\
adamc@540	990 \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	991 \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	992 \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	993 && \textrm{and $\mt{proj}(M', \overline{s'}, \mt{datatype} \; z = (\overline{y}, \overline{dc})$ and $X \in \overline{dc}$)} \\
adamc@540	994 \mt{proj}(M, \mt{datatype} \; x = \mt{datatype} \; M'.z, \mt{val} \; X) &=& \overline{y ::: \mt{Type}} \to \tau \to M.x \; \overline y \textrm{ (where $\Gamma \vdash M' : \mt{sig} \; \overline{s'} \; \mt{end}$} \\
adamc@558	995 && \textrm{and $\mt{proj}(M', \overline{s'}, \mt{datatype} \; z = (\overline{y}, \overline{dc})$ and $X \; \mt{of} \; \tau \in \overline{dc}$)} \\
adamc@540	996 \\
adamc@540	997 \mt{proj}(M, \mt{structure} \; X : S \; \overline{s}, \mt{structure} \; X) &=& S \\
adamc@540	998 \\
adamc@540	999 \mt{proj}(M, \mt{signature} \; X = S \; \overline{s}, \mt{signature} \; X) &=& S \\
adamc@540	1000 \\
adamc@540	1001 \mt{proj}(M, \mt{con} \; x :: \kappa \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@540	1002 \mt{proj}(M, \mt{con} \; x :: \kappa = c \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@540	1003 \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	1004 \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	1005 \mt{proj}(M, \mt{val} \; x : \tau \; \overline{s}, V) &=& \mt{proj}(M, \overline{s}, V) \\
adamc@540	1006 \mt{proj}(M, \mt{structure} \; X : S \; \overline{s}, V) &=& [X \mapsto M.X]\mt{proj}(M, \overline{s}, V) \\
adamc@540	1007 \mt{proj}(M, \mt{signature} \; X = S \; \overline{s}, V) &=& [X \mapsto M.X]\mt{proj}(M, \overline{s}, V) \\
adamc@540	1008 \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	1009 \mt{proj}(M, \mt{constraint} \; c_1 \sim c_2 \; \overline{s}, V) &=& \mt{proj}(M, \overline{s}, V) \\
adamc@540	1010 \mt{proj}(M, \mt{class} \; x \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@540	1011 \mt{proj}(M, \mt{class} \; x = c \; \overline{s}, V) &=& [x \mapsto M.x]\mt{proj}(M, \overline{s}, V) \\
adamc@540	1012 \end{eqnarray*}
adamc@540	1013
adamc@541	1014
adamc@541	1015 \section{Type Inference}
adamc@541	1016
adamc@541	1017 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	1018
adamc@541	1019 \subsection{Basic Unification}
adamc@541	1020
adamc@560	1021 Type-checkers for languages based on the Hindley-Milner type discipline, like ML and Haskell, take advantage of \emph{principal typing} properties, making complete type inference relatively straightforward. Inference algorithms are traditionally implemented using type unification variables, at various points asserting equalities between types, in the process discovering the values of type variables. The Ur/Web compiler uses the same basic strategy, but the complexity of the type system rules out easy completeness.
adamc@541	1022
adamc@560	1023 Type-checking can require evaluating recursive functional programs, thanks to the type-level $\mt{fold}$ operator. When a unification variable appears in such a type, the next step of computation can be undetermined. The value of that variable might be determined later, but this would be ``too late'' for the unification problems generated at the first occurrence. This is the essential source of incompleteness.
adamc@541	1024
adamc@541	1025 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	1026
adamc@541	1027 \subsection{Unifying Record Types}
adamc@541	1028
adamc@570	1029 The type inference engine tries to take advantage of the algebraic rules governing type-level records, as shown in Section \ref{definitional}. When two constructors of record kind are unified, they are reduced to normal forms, with like terms crossed off from each normal form until, hopefully, nothing remains. This cannot be complete, with the inclusion of unification variables. The type-checker can help you understand what goes wrong when the process fails, as it outputs the unmatched remainders of the two normal forms.
adamc@541	1030
adamc@541	1031 \subsection{\label{typeclasses}Type Classes}
adamc@541	1032
adamc@541	1033 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	1034
adamc@541	1035 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	1036
adamc@541	1037 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	1038
adamc@541	1039 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	1040
adamc@541	1041 \subsection{Reverse-Engineering Record Types}
adamc@541	1042
adamc@541	1043 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	1044
adamc@541	1045 \subsection{Implicit Arguments in Functor Applications}
adamc@541	1046
adamc@541	1047 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	1048
adamc@541	1049
adamc@542	1050 \section{The Ur Standard Library}
adamc@542	1051
adamc@542	1052 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	1053
adamc@542	1054 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	1055
adamc@542	1056 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	1057 $$\begin{array}{l}
adamc@542	1058 \mt{type} \; \mt{int} \\
adamc@542	1059 \mt{type} \; \mt{float} \\
adamc@542	1060 \mt{type} \; \mt{string} \\
adamc@542	1061 \mt{type} \; \mt{time} \\
adamc@542	1062 \\
adamc@542	1063 \mt{type} \; \mt{unit} = \{\} \\
adamc@542	1064 \\
adamc@542	1065 \mt{datatype} \; \mt{bool} = \mt{False} \mid \mt{True} \\
adamc@542	1066 \\
adamc@542	1067 \mt{datatype} \; \mt{option} \; \mt{t} = \mt{None} \mid \mt{Some} \; \mt{of} \; \mt{t}
adamc@542	1068 \end{array}$$
adamc@542	1069
adamc@542	1070
adamc@542	1071 \section{The Ur/Web Standard Library}
adamc@542	1072
adamc@542	1073 \subsection{Transactions}
adamc@542	1074
adamc@542	1075 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	1076 $$\begin{array}{l}
adamc@542	1077 \mt{con} \; \mt{transaction} :: \mt{Type} \to \mt{Type} \\
adamc@542	1078 \\
adamc@542	1079 \mt{val} \; \mt{return} : \mt{t} ::: \mt{Type} \to \mt{t} \to \mt{transaction} \; \mt{t} \\
adamc@542	1080 \mt{val} \; \mt{bind} : \mt{t_1} ::: \mt{Type} \to \mt{t_2} ::: \mt{Type} \to \mt{transaction} \; \mt{t_1} \to (\mt{t_1} \to \mt{transaction} \; \mt{t_2}) \to \mt{transaction} \; \mt{t_2}
adamc@542	1081 \end{array}$$
adamc@542	1082
adamc@542	1083 \subsection{HTTP}
adamc@542	1084
adamc@542	1085 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	1086 $$\begin{array}{l}
adamc@542	1087 \mt{val} \; \mt{requestHeader} : \mt{string} \to \mt{transaction} \; (\mt{option} \; \mt{string}) \\
adamc@542	1088 \\
adamc@542	1089 \mt{con} \; \mt{http\_cookie} :: \mt{Type} \to \mt{Type} \\
adamc@542	1090 \mt{val} \; \mt{getCookie} : \mt{t} ::: \mt{Type} \to \mt{http\_cookie} \; \mt{t} \to \mt{transaction} \; (\mt{option} \; \mt{t}) \\
adamc@542	1091 \mt{val} \; \mt{setCookie} : \mt{t} ::: \mt{Type} \to \mt{http\_cookie} \; \mt{t} \to \mt{t} \to \mt{transaction} \; \mt{unit}
adamc@542	1092 \end{array}$$
adamc@542	1093
adamc@543	1094 \subsection{SQL}
adamc@543	1095
adamc@543	1096 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	1097 $$\begin{array}{l}
adamc@543	1098 \mt{con} \; \mt{sql\_table} :: \{\mt{Type}\} \to \mt{Type}
adamc@543	1099 \end{array}$$
adamc@543	1100
adamc@543	1101 \subsubsection{Queries}
adamc@543	1102
adamc@543	1103 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	1104 $$\begin{array}{l}
adamc@543	1105 \mt{con} \; \mt{sql\_query} :: \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@543	1106 \mt{val} \; \mt{sql\_query} : \mt{tables} ::: \{\{\mt{Type}\}\} \\
adamc@543	1107 \hspace{.1in} \to \mt{selectedFields} ::: \{\{\mt{Type}\}\} \\
adamc@543	1108 \hspace{.1in} \to \mt{selectedExps} ::: \{\mt{Type}\} \\
adamc@543	1109 \hspace{.1in} \to \{\mt{Rows} : \mt{sql\_query1} \; \mt{tables} \; \mt{selectedFields} \; \mt{selectedExps}, \\
adamc@543	1110 \hspace{.2in} \mt{OrderBy} : \mt{sql\_order\_by} \; \mt{tables} \; \mt{selectedExps}, \\
adamc@543	1111 \hspace{.2in} \mt{Limit} : \mt{sql\_limit}, \\
adamc@543	1112 \hspace{.2in} \mt{Offset} : \mt{sql\_offset}\} \\
adamc@543	1113 \hspace{.1in} \to \mt{sql\_query} \; \mt{selectedFields} \; \mt{selectedExps}
adamc@543	1114 \end{array}$$
adamc@543	1115
adamc@545	1116 Queries are used by folding over their results inside transactions.
adamc@545	1117 $$\begin{array}{l}
adamc@545	1118 \mt{val} \; \mt{query} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \lambda [\mt{tables} \sim \mt{exps}] \Rightarrow \mt{state} ::: \mt{Type} \to \mt{sql\_query} \; \mt{tables} \; \mt{exps} \\
adamc@545	1119 \hspace{.1in} \to (\$(\mt{exps} \rc \mt{fold} \; (\lambda \mt{nm} \; (\mt{fields} :: \{\mt{Type}\}) \; \mt{acc} \; [[\mt{nm}] \sim \mt{acc}] \Rightarrow [\mt{nm} = \$\mt{fields}] \rc \mt{acc}) \; [] \; \mt{tables}) \\
adamc@545	1120 \hspace{.2in} \to \mt{state} \to \mt{transaction} \; \mt{state}) \\
adamc@545	1121 \hspace{.1in} \to \mt{state} \to \mt{transaction} \; \mt{state}
adamc@545	1122 \end{array}$$
adamc@545	1123
adamc@543	1124 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	1125 $$\begin{array}{l}
adamc@543	1126 \mt{con} \; \mt{sql\_query1} :: \{\{\mt{Type}\}\} \to \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@543	1127 \\
adamc@543	1128 \mt{type} \; \mt{sql\_relop} \\
adamc@543	1129 \mt{val} \; \mt{sql\_union} : \mt{sql\_relop} \\
adamc@543	1130 \mt{val} \; \mt{sql\_intersect} : \mt{sql\_relop} \\
adamc@543	1131 \mt{val} \; \mt{sql\_except} : \mt{sql\_relop} \\
adamc@543	1132 \mt{val} \; \mt{sql\_relop} : \mt{tables1} ::: \{\{\mt{Type}\}\} \\
adamc@543	1133 \hspace{.1in} \to \mt{tables2} ::: \{\{\mt{Type}\}\} \\
adamc@543	1134 \hspace{.1in} \to \mt{selectedFields} ::: \{\{\mt{Type}\}\} \\
adamc@543	1135 \hspace{.1in} \to \mt{selectedExps} ::: \{\mt{Type}\} \\
adamc@543	1136 \hspace{.1in} \to \mt{sql\_relop} \\
adamc@543	1137 \hspace{.1in} \to \mt{sql\_query1} \; \mt{tables1} \; \mt{selectedFields} \; \mt{selectedExps} \\
adamc@543	1138 \hspace{.1in} \to \mt{sql\_query1} \; \mt{tables2} \; \mt{selectedFields} \; \mt{selectedExps} \\
adamc@543	1139 \hspace{.1in} \to \mt{sql\_query1} \; \mt{selectedFields} \; \mt{selectedFields} \; \mt{selectedExps}
adamc@543	1140 \end{array}$$
adamc@543	1141
adamc@543	1142 $$\begin{array}{l}
adamc@543	1143 \mt{val} \; \mt{sql\_query1} : \mt{tables} ::: \{\{\mt{Type}\}\} \\
adamc@543	1144 \hspace{.1in} \to \mt{grouped} ::: \{\{\mt{Type}\}\} \\
adamc@543	1145 \hspace{.1in} \to \mt{selectedFields} ::: \{\{\mt{Type}\}\} \\
adamc@543	1146 \hspace{.1in} \to \mt{selectedExps} ::: \{\mt{Type}\} \\
adamc@543	1147 \hspace{.1in} \to \{\mt{From} : \$(\mt{fold} \; (\lambda \mt{nm} \; (\mt{fields} :: \{\mt{Type}\}) \; \mt{acc} \; [[\mt{nm}] \sim \mt{acc}] \Rightarrow [\mt{nm} = \mt{sql\_table} \; \mt{fields}] \rc \mt{acc}) \; [] \; \mt{tables}), \\
adamc@543	1148 \hspace{.2in} \mt{Where} : \mt{sql\_exp} \; \mt{tables} \; [] \; [] \; \mt{bool}, \\
adamc@543	1149 \hspace{.2in} \mt{GroupBy} : \mt{sql\_subset} \; \mt{tables} \; \mt{grouped}, \\
adamc@543	1150 \hspace{.2in} \mt{Having} : \mt{sql\_exp} \; \mt{grouped} \; \mt{tables} \; [] \; \mt{bool}, \\
adamc@543	1151 \hspace{.2in} \mt{SelectFields} : \mt{sql\_subset} \; \mt{grouped} \; \mt{selectedFields}, \\
adamc@543	1152 \hspace{.2in} \mt {SelectExps} : \$(\mt{fold} \; (\lambda \mt{nm} \; (\mt{t} :: \mt{Type}) \; \mt{acc} \; [[\mt{nm}] \sim \mt{acc}] \Rightarrow [\mt{nm} = \mt{sql\_exp} \; \mt{grouped} \; \mt{tables} \; [] \; \mt{t}] \rc \mt{acc}) \; [] \; \mt{selectedExps}) \} \\
adamc@543	1153 \hspace{.1in} \to \mt{sql\_query1} \; \mt{tables} \; \mt{selectedFields} \; \mt{selectedExps}
adamc@543	1154 \end{array}$$
adamc@543	1155
adamc@543	1156 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	1157 $$\begin{array}{l}
adamc@543	1158 \mt{con} \; \mt{sql\_subset} :: \{\{\mt{Type}\}\} \to \{\{\mt{Type}\}\} \to \mt{Type} \\
adamc@543	1159 \mt{val} \; \mt{sql\_subset} : \mt{keep\_drop} :: \{(\{\mt{Type}\} \times \{\mt{Type}\})\} \\
adamc@543	1160 \hspace{.1in} \to \mt{sql\_subset} \\
adamc@543	1161 \hspace{.2in} (\mt{fold} \; (\lambda \mt{nm} \; (\mt{fields} :: (\{\mt{Type}\} \times \{\mt{Type}\})) \; \mt{acc} \; [[\mt{nm}] \sim \mt{acc}] \; [\mt{fields}.1 \sim \mt{fields}.2] \Rightarrow \\
adamc@543	1162 \hspace{.3in} [\mt{nm} = \mt{fields}.1 \rc \mt{fields}.2] \rc \mt{acc}) \; [] \; \mt{keep\_drop}) \\
adamc@543	1163 \hspace{.2in} (\mt{fold} \; (\lambda \mt{nm} \; (\mt{fields} :: (\{\mt{Type}\} \times \{\mt{Type}\})) \; \mt{acc} \; [[\mt{nm}] \sim \mt{acc}] \Rightarrow [\mt{nm} = \mt{fields}.1] \rc \mt{acc}) \; [] \; \mt{keep\_drop}) \\
adamc@543	1164 \mt{val} \; \mt{sql\_subset\_all} : \mt{tables} :: \{\{\mt{Type}\}\} \to \mt{sql\_subset} \; \mt{tables} \; \mt{tables}
adamc@543	1165 \end{array}$$
adamc@543	1166
adamc@560	1167 SQL expressions are used in several places, including $\mt{SELECT}$, $\mt{WHERE}$, $\mt{HAVING}$, and $\mt{ORDER} \; \mt{BY}$ clauses. They reify a fragment of the standard SQL expression language, while making it possible to inject ``native'' Ur values in some places. The arguments to the $\mt{sql\_exp}$ type family respectively give the unrestricted-availability table fields, the table fields that may only be used in arguments to aggregate functions, the available selected expressions, and the type of the expression.
adamc@543	1168 $$\begin{array}{l}
adamc@543	1169 \mt{con} \; \mt{sql\_exp} :: \{\{\mt{Type}\}\} \to \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \to \mt{Type}
adamc@543	1170 \end{array}$$
adamc@543	1171
adamc@543	1172 Any field in scope may be converted to an expression.
adamc@543	1173 $$\begin{array}{l}
adamc@543	1174 \mt{val} \; \mt{sql\_field} : \mt{otherTabs} ::: \{\{\mt{Type}\}\} \to \mt{otherFields} ::: \{\mt{Type}\} \\
adamc@543	1175 \hspace{.1in} \to \mt{fieldType} ::: \mt{Type} \to \mt{agg} ::: \{\{\mt{Type}\}\} \\
adamc@543	1176 \hspace{.1in} \to \mt{exps} ::: \{\mt{Type}\} \\
adamc@543	1177 \hspace{.1in} \to \mt{tab} :: \mt{Name} \to \mt{field} :: \mt{Name} \\
adamc@543	1178 \hspace{.1in} \to \mt{sql\_exp} \; ([\mt{tab} = [\mt{field} = \mt{fieldType}] \rc \mt{otherFields}] \rc \mt{otherTabs}) \; \mt{agg} \; \mt{exps} \; \mt{fieldType}
adamc@543	1179 \end{array}$$
adamc@543	1180
adamc@544	1181 There is an analogous function for referencing named expressions.
adamc@544	1182 $$\begin{array}{l}
adamc@544	1183 \mt{val} \; \mt{sql\_exp} : \mt{tabs} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{t} ::: \mt{Type} \to \mt{rest} ::: \{\mt{Type}\} \to \mt{nm} :: \mt{Name} \\
adamc@544	1184 \hspace{.1in} \to \mt{sql\_exp} \; \mt{tabs} \; \mt{agg} \; ([\mt{nm} = \mt{t}] \rc \mt{rest}) \; \mt{t}
adamc@544	1185 \end{array}$$
adamc@544	1186
adamc@544	1187 Ur values of appropriate types may be injected into SQL expressions.
adamc@544	1188 $$\begin{array}{l}
adamc@544	1189 \mt{class} \; \mt{sql\_injectable} \\
adamc@544	1190 \mt{val} \; \mt{sql\_bool} : \mt{sql\_injectable} \; \mt{bool} \\
adamc@544	1191 \mt{val} \; \mt{sql\_int} : \mt{sql\_injectable} \; \mt{int} \\
adamc@544	1192 \mt{val} \; \mt{sql\_float} : \mt{sql\_injectable} \; \mt{float} \\
adamc@544	1193 \mt{val} \; \mt{sql\_string} : \mt{sql\_injectable} \; \mt{string} \\
adamc@544	1194 \mt{val} \; \mt{sql\_time} : \mt{sql\_injectable} \; \mt{time} \\
adamc@544	1195 \mt{val} \; \mt{sql\_option\_bool} : \mt{sql\_injectable} \; (\mt{option} \; \mt{bool}) \\
adamc@544	1196 \mt{val} \; \mt{sql\_option\_int} : \mt{sql\_injectable} \; (\mt{option} \; \mt{int}) \\
adamc@544	1197 \mt{val} \; \mt{sql\_option\_float} : \mt{sql\_injectable} \; (\mt{option} \; \mt{float}) \\
adamc@544	1198 \mt{val} \; \mt{sql\_option\_string} : \mt{sql\_injectable} \; (\mt{option} \; \mt{string}) \\
adamc@544	1199 \mt{val} \; \mt{sql\_option\_time} : \mt{sql\_injectable} \; (\mt{option} \; \mt{time}) \\
adamc@544	1200 \mt{val} \; \mt{sql\_inject} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \to \mt{sql\_injectable} \; \mt{t} \\
adamc@544	1201 \hspace{.1in} \to \mt{t} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{t}
adamc@544	1202 \end{array}$$
adamc@544	1203
adamc@544	1204 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	1205 $$\begin{array}{l}
adamc@544	1206 \mt{val} \; \mt{sql\_is\_null} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544	1207 \hspace{.1in} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; (\mt{option} \; \mt{t}) \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{bool}
adamc@544	1208 \end{array}$$
adamc@544	1209
adamc@559	1210 We have generic nullary, unary, and binary operators.
adamc@544	1211 $$\begin{array}{l}
adamc@544	1212 \mt{con} \; \mt{sql\_nfunc} :: \mt{Type} \to \mt{Type} \\
adamc@544	1213 \mt{val} \; \mt{sql\_current\_timestamp} : \mt{sql\_nfunc} \; \mt{time} \\
adamc@544	1214 \mt{val} \; \mt{sql\_nfunc} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544	1215 \hspace{.1in} \to \mt{sql\_nfunc} \; \mt{t} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{t} \\\end{array}$$
adamc@544	1216
adamc@544	1217 $$\begin{array}{l}
adamc@544	1218 \mt{con} \; \mt{sql\_unary} :: \mt{Type} \to \mt{Type} \to \mt{Type} \\
adamc@544	1219 \mt{val} \; \mt{sql\_not} : \mt{sql\_unary} \; \mt{bool} \; \mt{bool} \\
adamc@544	1220 \mt{val} \; \mt{sql\_unary} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{arg} ::: \mt{Type} \to \mt{res} ::: \mt{Type} \\
adamc@544	1221 \hspace{.1in} \to \mt{sql\_unary} \; \mt{arg} \; \mt{res} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{arg} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{res} \\
adamc@544	1222 \end{array}$$
adamc@544	1223
adamc@544	1224 $$\begin{array}{l}
adamc@544	1225 \mt{con} \; \mt{sql\_binary} :: \mt{Type} \to \mt{Type} \to \mt{Type} \to \mt{Type} \\
adamc@544	1226 \mt{val} \; \mt{sql\_and} : \mt{sql\_binary} \; \mt{bool} \; \mt{bool} \; \mt{bool} \\
adamc@544	1227 \mt{val} \; \mt{sql\_or} : \mt{sql\_binary} \; \mt{bool} \; \mt{bool} \; \mt{bool} \\
adamc@544	1228 \mt{val} \; \mt{sql\_binary} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{arg_1} ::: \mt{Type} \to \mt{arg_2} ::: \mt{Type} \to \mt{res} ::: \mt{Type} \\
adamc@544	1229 \hspace{.1in} \to \mt{sql\_binary} \; \mt{arg_1} \; \mt{arg_2} \; \mt{res} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{arg_1} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{arg_2} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{res}
adamc@544	1230 \end{array}$$
adamc@544	1231
adamc@544	1232 $$\begin{array}{l}
adamc@559	1233 \mt{class} \; \mt{sql\_arith} \\
adamc@559	1234 \mt{val} \; \mt{sql\_int\_arith} : \mt{sql\_arith} \; \mt{int} \\
adamc@559	1235 \mt{val} \; \mt{sql\_float\_arith} : \mt{sql\_arith} \; \mt{float} \\
adamc@559	1236 \mt{val} \; \mt{sql\_neg} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_unary} \; \mt{t} \; \mt{t} \\
adamc@559	1237 \mt{val} \; \mt{sql\_plus} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559	1238 \mt{val} \; \mt{sql\_minus} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559	1239 \mt{val} \; \mt{sql\_times} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559	1240 \mt{val} \; \mt{sql\_div} : \mt{t} ::: \mt{Type} \to \mt{sql\_arith} \; \mt{t} \to \mt{sql\_binary} \; \mt{t} \; \mt{t} \; \mt{t} \\
adamc@559	1241 \mt{val} \; \mt{sql\_mod} : \mt{sql\_binary} \; \mt{int} \; \mt{int} \; \mt{int}
adamc@559	1242 \end{array}$$
adamc@544	1243
adamc@544	1244 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	1245
adamc@544	1246 $$\begin{array}{l}
adamc@544	1247 \mt{val} \; \mt{sql\_count} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{int}
adamc@544	1248 \end{array}$$
adamc@544	1249
adamc@544	1250 $$\begin{array}{l}
adamc@544	1251 \mt{con} \; \mt{sql\_aggregate} :: \mt{Type} \to \mt{Type} \\
adamc@544	1252 \mt{val} \; \mt{sql\_aggregate} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{agg} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544	1253 \hspace{.1in} \to \mt{sql\_aggregate} \; \mt{t} \to \mt{sql\_exp} \; \mt{agg} \; \mt{agg} \; \mt{exps} \; \mt{t} \to \mt{sql\_exp} \; \mt{tables} \; \mt{agg} \; \mt{exps} \; \mt{t}
adamc@544	1254 \end{array}$$
adamc@544	1255
adamc@544	1256 $$\begin{array}{l}
adamc@544	1257 \mt{class} \; \mt{sql\_summable} \\
adamc@544	1258 \mt{val} \; \mt{sql\_summable\_int} : \mt{sql\_summable} \; \mt{int} \\
adamc@544	1259 \mt{val} \; \mt{sql\_summable\_float} : \mt{sql\_summable} \; \mt{float} \\
adamc@544	1260 \mt{val} \; \mt{sql\_avg} : \mt{t} ::: \mt{Type} \to \mt{sql\_summable} \; \mt{t} \to \mt{sql\_aggregate} \; \mt{t} \\
adamc@544	1261 \mt{val} \; \mt{sql\_sum} : \mt{t} ::: \mt{Type} \to \mt{sql\_summable} \mt{t} \to \mt{sql\_aggregate} \; \mt{t}
adamc@544	1262 \end{array}$$
adamc@544	1263
adamc@544	1264 $$\begin{array}{l}
adamc@544	1265 \mt{class} \; \mt{sql\_maxable} \\
adamc@544	1266 \mt{val} \; \mt{sql\_maxable\_int} : \mt{sql\_maxable} \; \mt{int} \\
adamc@544	1267 \mt{val} \; \mt{sql\_maxable\_float} : \mt{sql\_maxable} \; \mt{float} \\
adamc@544	1268 \mt{val} \; \mt{sql\_maxable\_string} : \mt{sql\_maxable} \; \mt{string} \\
adamc@544	1269 \mt{val} \; \mt{sql\_maxable\_time} : \mt{sql\_maxable} \; \mt{time} \\
adamc@544	1270 \mt{val} \; \mt{sql\_max} : \mt{t} ::: \mt{Type} \to \mt{sql\_maxable} \; \mt{t} \to \mt{sql\_aggregate} \; \mt{t} \\
adamc@544	1271 \mt{val} \; \mt{sql\_min} : \mt{t} ::: \mt{Type} \to \mt{sql\_maxable} \; \mt{t} \to \mt{sql\_aggregate} \; \mt{t}
adamc@544	1272 \end{array}$$
adamc@544	1273
adamc@544	1274 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	1275 $$\begin{array}{l}
adamc@544	1276 \mt{type} \; \mt{sql\_direction} \\
adamc@544	1277 \mt{val} \; \mt{sql\_asc} : \mt{sql\_direction} \\
adamc@544	1278 \mt{val} \; \mt{sql\_desc} : \mt{sql\_direction} \\
adamc@544	1279 \\
adamc@544	1280 \mt{con} \; \mt{sql\_order\_by} :: \{\{\mt{Type}\}\} \to \{\mt{Type}\} \to \mt{Type} \\
adamc@544	1281 \mt{val} \; \mt{sql\_order\_by\_Nil} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{exps} :: \{\mt{Type}\} \to \mt{sql\_order\_by} \; \mt{tables} \; \mt{exps} \\
adamc@544	1282 \mt{val} \; \mt{sql\_order\_by\_Cons} : \mt{tables} ::: \{\{\mt{Type}\}\} \to \mt{exps} ::: \{\mt{Type}\} \to \mt{t} ::: \mt{Type} \\
adamc@544	1283 \hspace{.1in} \to \mt{sql\_exp} \; \mt{tables} \; [] \; \mt{exps} \; \mt{t} \to \mt{sql\_direction} \to \mt{sql\_order\_by} \; \mt{tables} \; \mt{exps} \to \mt{sql\_order\_by} \; \mt{tables} \; \mt{exps} \\
adamc@544	1284 \\
adamc@544	1285 \mt{type} \; \mt{sql\_limit} \\
adamc@544	1286 \mt{val} \; \mt{sql\_no\_limit} : \mt{sql\_limit} \\
adamc@544	1287 \mt{val} \; \mt{sql\_limit} : \mt{int} \to \mt{sql\_limit} \\
adamc@544	1288 \\
adamc@544	1289 \mt{type} \; \mt{sql\_offset} \\
adamc@544	1290 \mt{val} \; \mt{sql\_no\_offset} : \mt{sql\_offset} \\
adamc@544	1291 \mt{val} \; \mt{sql\_offset} : \mt{int} \to \mt{sql\_offset}
adamc@544	1292 \end{array}$$
adamc@544	1293
adamc@545	1294
adamc@545	1295 \subsubsection{DML}
adamc@545	1296
adamc@545	1297 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	1298
adamc@545	1299 $$\begin{array}{l}
adamc@545	1300 \mt{type} \; \mt{dml} \\
adamc@545	1301 \mt{val} \; \mt{dml} : \mt{dml} \to \mt{transaction} \; \mt{unit}
adamc@545	1302 \end{array}$$
adamc@545	1303
adamc@545	1304 Properly-typed records may be used to form $\mt{INSERT}$ commands.
adamc@545	1305 $$\begin{array}{l}
adamc@545	1306 \mt{val} \; \mt{insert} : \mt{fields} ::: \{\mt{Type}\} \to \mt{sql\_table} \; \mt{fields} \\
adamc@545	1307 \hspace{.1in} \to \$(\mt{fold} \; (\lambda \mt{nm} \; (\mt{t} :: \mt{Type}) \; \mt{acc} \; [[\mt{nm}] \sim \mt{acc}] \Rightarrow [\mt{nm} = \mt{sql\_exp} \; [] \; [] \; [] \; \mt{t}] \rc \mt{acc}) \; [] \; \mt{fields}) \to \mt{dml}
adamc@545	1308 \end{array}$$
adamc@545	1309
adamc@545	1310 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	1311 $$\begin{array}{l}
adamc@545	1312 \mt{val} \; \mt{update} : \mt{unchanged} ::: \{\mt{Type}\} \to \mt{changed} :: \{\mt{Type}\} \to \lambda [\mt{changed} \sim \mt{unchanged}] \\
adamc@545	1313 \hspace{.1in} \Rightarrow \$(\mt{fold} \; (\lambda \mt{nm} \; (\mt{t} :: \mt{Type}) \; \mt{acc} \; [[\mt{nm}] \sim \mt{acc}] \Rightarrow [\mt{nm} = \mt{sql\_exp} \; [\mt{T} = \mt{changed} \rc \mt{unchanged}] \; [] \; [] \; \mt{t}] \rc \mt{acc}) \; [] \; \mt{changed}) \\
adamc@545	1314 \hspace{.1in} \to \mt{sql\_table} \; (\mt{changed} \rc \mt{unchanged}) \to \mt{sql\_exp} \; [\mt{T} = \mt{changed} \rc \mt{unchanged}] \; [] \; [] \; \mt{bool} \to \mt{dml}
adamc@545	1315 \end{array}$$
adamc@545	1316
adamc@545	1317 A $\mt{DELETE}$ command is formed from a table and a $\mt{WHERE}$ clause.
adamc@545	1318 $$\begin{array}{l}
adamc@545	1319 \mt{val} \; \mt{delete} : \mt{fields} ::: \{\mt{Type}\} \to \mt{sql\_table} \; \mt{fields} \to \mt{sql\_exp} \; [\mt{T} = \mt{fields}] \; [] \; [] \; \mt{bool} \to \mt{dml}
adamc@545	1320 \end{array}$$
adamc@545	1321
adamc@546	1322 \subsubsection{Sequences}
adamc@546	1323
adamc@546	1324 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	1325
adamc@546	1326 $$\begin{array}{l}
adamc@546	1327 \mt{type} \; \mt{sql\_sequence} \\
adamc@546	1328 \mt{val} \; \mt{nextval} : \mt{sql\_sequence} \to \mt{transaction} \; \mt{int}
adamc@546	1329 \end{array}$$
adamc@546	1330
adamc@546	1331
adamc@547	1332 \subsection{XML}
adamc@547	1333
adamc@547	1334 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	1335
adamc@547	1336 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	1337 $$\begin{array}{l}
adamc@547	1338 \mt{con} \; \mt{xml} :: \{\mt{Unit}\} \to \{\mt{Type}\} \to \{\mt{Type}\} \to \mt{Type}
adamc@547	1339 \end{array}$$
adamc@547	1340
adamc@547	1341 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	1342 $$\begin{array}{l}
adamc@547	1343 \mt{con} \; \mt{tag} :: \{\mt{Type}\} \to \{\mt{Unit}\} \to \{\mt{Unit}\} \to \{\mt{Type}\} \to \{\mt{Type}\} \to \mt{Type}
adamc@547	1344 \end{array}$$
adamc@547	1345
adamc@547	1346 Literal text may be injected into XML as ``CDATA.''
adamc@547	1347 $$\begin{array}{l}
adamc@547	1348 \mt{val} \; \mt{cdata} : \mt{ctx} ::: \{\mt{Unit}\} \to \mt{use} ::: \{\mt{Type}\} \to \mt{string} \to \mt{xml} \; \mt{ctx} \; \mt{use} \; []
adamc@547	1349 \end{array}$$
adamc@547	1350
adamc@547	1351 There is a function for producing an XML tree with a particular tag at its root.
adamc@547	1352 $$\begin{array}{l}
adamc@547	1353 \mt{val} \; \mt{tag} : \mt{attrsGiven} ::: \{\mt{Type}\} \to \mt{attrsAbsent} ::: \{\mt{Type}\} \to \mt{ctxOuter} ::: \{\mt{Unit}\} \to \mt{ctxInner} ::: \{\mt{Unit}\} \\
adamc@547	1354 \hspace{.1in} \to \mt{useOuter} ::: \{\mt{Type}\} \to \mt{useInner} ::: \{\mt{Type}\} \to \mt{bindOuter} ::: \{\mt{Type}\} \to \mt{bindInner} ::: \{\mt{Type}\} \\
adamc@547	1355 \hspace{.1in} \to \lambda [\mt{attrsGiven} \sim \mt{attrsAbsent}] \; [\mt{useOuter} \sim \mt{useInner}] \; [\mt{bindOuter} \sim \mt{bindInner}] \Rightarrow \$\mt{attrsGiven} \\
adamc@547	1356 \hspace{.1in} \to \mt{tag} \; (\mt{attrsGiven} \rc \mt{attrsAbsent}) \; \mt{ctxOuter} \; \mt{ctxInner} \; \mt{useOuter} \; \mt{bindOuter} \\
adamc@547	1357 \hspace{.1in} \to \mt{xml} \; \mt{ctxInner} \; \mt{useInner} \; \mt{bindInner} \to \mt{xml} \; \mt{ctxOuter} \; (\mt{useOuter} \rc \mt{useInner}) \; (\mt{bindOuter} \rc \mt{bindInner})
adamc@547	1358 \end{array}$$
adamc@547	1359
adamc@547	1360 Two XML fragments may be concatenated.
adamc@547	1361 $$\begin{array}{l}
adamc@547	1362 \mt{val} \; \mt{join} : \mt{ctx} ::: \{\mt{Unit}\} \to \mt{use_1} ::: \{\mt{Type}\} \to \mt{bind_1} ::: \{\mt{Type}\} \to \mt{bind_2} ::: \{\mt{Type}\} \\
adamc@547	1363 \hspace{.1in} \to \lambda [\mt{use_1} \sim \mt{bind_1}] \; [\mt{bind_1} \sim \mt{bind_2}] \\
adamc@547	1364 \hspace{.1in} \Rightarrow \mt{xml} \; \mt{ctx} \; \mt{use_1} \; \mt{bind_1} \to \mt{xml} \; \mt{ctx} \; (\mt{use_1} \rc \mt{bind_1}) \; \mt{bind_2} \to \mt{xml} \; \mt{ctx} \; \mt{use_1} \; (\mt{bind_1} \rc \mt{bind_2})
adamc@547	1365 \end{array}$$
adamc@547	1366
adamc@547	1367 Finally, any XML fragment may be updated to ``claim'' to use more form fields than it does.
adamc@547	1368 $$\begin{array}{l}
adamc@547	1369 \mt{val} \; \mt{useMore} : \mt{ctx} ::: \{\mt{Unit}\} \to \mt{use_1} ::: \{\mt{Type}\} \to \mt{use_2} ::: \{\mt{Type}\} \to \mt{bind} ::: \{\mt{Type}\} \to \lambda [\mt{use_1} \sim \mt{use_2}] \\
adamc@547	1370 \hspace{.1in} \Rightarrow \mt{xml} \; \mt{ctx} \; \mt{use_1} \; \mt{bind} \to \mt{xml} \; \mt{ctx} \; (\mt{use_1} \rc \mt{use_2}) \; \mt{bind}
adamc@547	1371 \end{array}$$
adamc@547	1372
adamc@547	1373 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	1374
adamc@547	1375 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	1376 $$\begin{array}{l}
adamc@547	1377 \mt{val} \; \mt{error} : \mt{t} ::: \mt{Type} \to \mt{xml} \; [\mt{Body}] \; [] \; [] \to \mt{t}
adamc@547	1378 \end{array}$$
adamc@547	1379
adamc@549	1380
adamc@549	1381 \section{Ur/Web Syntax Extensions}
adamc@549	1382
adamc@549	1383 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	1384
adamc@549	1385 \subsection{SQL}
adamc@549	1386
adamc@549	1387 \subsubsection{Queries}
adamc@549	1388
adamc@550	1389 Queries $Q$ are added to the rules for expressions $e$.
adamc@550	1390
adamc@549	1391 $$\begin{array}{rrcll}
adamc@550	1392 \textrm{Queries} & Q &::=& (q \; [\mt{ORDER} \; \mt{BY} \; (E \; [o],)^+] \; [\mt{LIMIT} \; N] \; [\mt{OFFSET} \; N]) \\
adamc@549	1393 \textrm{Pre-queries} & q &::=& \mt{SELECT} \; P \; \mt{FROM} \; T,^+ \; [\mt{WHERE} \; E] \; [\mt{GROUP} \; \mt{BY} \; p,^+] \; [\mt{HAVING} \; E] \\
adamc@549	1394 &&& \mid q \; R \; q \\
adamc@549	1395 \textrm{Relational operators} & R &::=& \mt{UNION} \mid \mt{INTERSECT} \mid \mt{EXCEPT}
adamc@549	1396 \end{array}$$
adamc@549	1397
adamc@549	1398 $$\begin{array}{rrcll}
adamc@549	1399 \textrm{Projections} & P &::=& \ast & \textrm{all columns} \\
adamc@549	1400 &&& p,^+ & \textrm{particular columns} \\
adamc@549	1401 \textrm{Pre-projections} & p &::=& t.f & \textrm{one column from a table} \\
adamc@558	1402 &&& t.\{\{c\}\} & \textrm{a record of columns from a table (of kind $\{\mt{Type}\}$)} \\
adamc@549	1403 \textrm{Table names} & t &::=& x & \textrm{constant table name (automatically capitalized)} \\
adamc@549	1404 &&& X & \textrm{constant table name} \\
adamc@549	1405 &&& \{\{c\}\} & \textrm{computed table name (of kind $\mt{Name}$)} \\
adamc@549	1406 \textrm{Column names} & f &::=& X & \textrm{constant column name} \\
adamc@549	1407 &&& \{c\} & \textrm{computed column name (of kind $\mt{Name}$)} \\
adamc@549	1408 \textrm{Tables} & T &::=& x & \textrm{table variable, named locally by its own capitalization} \\
adamc@549	1409 &&& x \; \mt{AS} \; t & \textrm{table variable, with local name} \\
adamc@549	1410 &&& \{\{e\}\} \; \mt{AS} \; t & \textrm{computed table expression, with local name} \\
adamc@549	1411 \textrm{SQL expressions} & E &::=& p & \textrm{column references} \\
adamc@549	1412 &&& X & \textrm{named expression references} \\
adamc@549	1413 &&& \{\{e\}\} & \textrm{injected native Ur expressions} \\
adamc@549	1414 &&& \{e\} & \textrm{computed expressions, probably using $\mt{sql\_exp}$ directly} \\
adamc@549	1415 &&& \mt{TRUE} \mid \mt{FALSE} & \textrm{boolean constants} \\
adamc@549	1416 &&& \ell & \textrm{primitive type literals} \\
adamc@549	1417 &&& \mt{NULL} & \textrm{null value (injection of $\mt{None}$)} \\
adamc@549	1418 &&& E \; \mt{IS} \; \mt{NULL} & \textrm{nullness test} \\
adamc@549	1419 &&& n & \textrm{nullary operators} \\
adamc@549	1420 &&& u \; E & \textrm{unary operators} \\
adamc@549	1421 &&& E \; b \; E & \textrm{binary operators} \\
adamc@549	1422 &&& \mt{COUNT}(\ast) & \textrm{count number of rows} \\
adamc@549	1423 &&& a(E) & \textrm{other aggregate function} \\
adamc@549	1424 &&& (E) & \textrm{explicit precedence} \\
adamc@549	1425 \textrm{Nullary operators} & n &::=& \mt{CURRENT\_TIMESTAMP} \\
adamc@549	1426 \textrm{Unary operators} & u &::=& \mt{NOT} \\
adamc@549	1427 \textrm{Binary operators} & b &::=& \mt{AND} \mid \mt{OR} \mid \neq \mid < \mid \leq \mid > \mid \geq \\
adamc@549	1428 \textrm{Aggregate functions} & a &::=& \mt{AVG} \mid \mt{SUM} \mid \mt{MIN} \mid \mt{MAX} \\
adamc@550	1429 \textrm{Directions} & o &::=& \mt{ASC} \mid \mt{DESC} \\
adamc@549	1430 \textrm{SQL integer} & N &::=& n \mid \{e\} \\
adamc@549	1431 \end{array}$$
adamc@549	1432
adamc@549	1433 Additionally, an SQL expression may be inserted into normal Ur code with the syntax $(\mt{SQL} \; E)$ or $(\mt{WHERE} \; E)$.
adamc@549	1434
adamc@550	1435 \subsubsection{DML}
adamc@550	1436
adamc@550	1437 DML commands $D$ are added to the rules for expressions $e$.
adamc@550	1438
adamc@550	1439 $$\begin{array}{rrcll}
adamc@550	1440 \textrm{Commands} & D &::=& (\mt{INSERT} \; \mt{INTO} \; T^E \; (f,^+) \; \mt{VALUES} \; (E,^+)) \\
adamc@550	1441 &&& (\mt{UPDATE} \; T^E \; \mt{SET} \; (f = E,)^+ \; \mt{WHERE} \; E) \\
adamc@550	1442 &&& (\mt{DELETE} \; \mt{FROM} \; T^E \; \mt{WHERE} \; E) \\
adamc@550	1443 \textrm{Table expressions} & T^E &::=& x \mid \{\{e\}\}
adamc@550	1444 \end{array}$$
adamc@550	1445
adamc@550	1446 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	1447
adamc@551	1448 \subsection{XML}
adamc@551	1449
adamc@551	1450 XML fragments $L$ are added to the rules for expressions $e$.
adamc@551	1451
adamc@551	1452 $$\begin{array}{rrcll}
adamc@551	1453 \textrm{XML fragments} & L &::=& \texttt{<xml/>} \mid \texttt{<xml>}l^*\texttt{</xml>} \\
adamc@551	1454 \textrm{XML pieces} & l &::=& \textrm{text} & \textrm{cdata} \\
adamc@551	1455 &&& \texttt{<}g\texttt{/>} & \textrm{tag with no children} \\
adamc@551	1456 &&& \texttt{<}g\texttt{>}l^*\texttt{</}x\texttt{>} & \textrm{tag with children} \\
adamc@559	1457 &&& \{e\} & \textrm{computed XML fragment} \\
adamc@559	1458 &&& \{[e]\} & \textrm{injection of an Ur expression, via the $\mt{Top}.\mt{txt}$ function} \\
adamc@551	1459 \textrm{Tag} & g &::=& h \; (x = v)^* \\
adamc@551	1460 \textrm{Tag head} & h &::=& x & \textrm{tag name} \\
adamc@551	1461 &&& h\{c\} & \textrm{constructor parameter} \\
adamc@551	1462 \textrm{Attribute value} & v &::=& \ell & \textrm{literal value} \\
adamc@551	1463 &&& \{e\} & \textrm{computed value} \\
adamc@551	1464 \end{array}$$
adamc@551	1465
adamc@552	1466
adamc@553	1467 \section{The Structure of Web Applications}
adamc@553	1468
adamc@553	1469 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	1470
adamc@553	1471 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	1472
adamc@553	1473 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	1474
adamc@558	1475 For both links and actions, direct arguments and local variables mentioned implicitly via closures are automatically included in serialized form in URLs, in the order in which they appear in the source code.
adamc@553	1476
adamc@553	1477
adamc@552	1478 \section{Compiler Phases}
adamc@552	1479
adamc@552	1480 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	1481
adamc@552	1482 In this section, we step through the main phases of compilation, noting what consequences each phase has for effective programming.
adamc@552	1483
adamc@552	1484 \subsection{Parse}
adamc@552	1485
adamc@552	1486 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	1487
adamc@552	1488 \subsection{Elaborate}
adamc@552	1489
adamc@552	1490 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	1491
adamc@552	1492 \subsection{Unnest}
adamc@552	1493
adamc@552	1494 Named local function definitions are moved to the top level, to avoid the need to generate closures.
adamc@552	1495
adamc@552	1496 \subsection{Corify}
adamc@552	1497
adamc@552	1498 Module system features are compiled away, through inlining of functor definitions at application sites. Afterward, most abstraction boundaries are broken, facilitating optimization.
adamc@552	1499
adamc@552	1500 \subsection{Especialize}
adamc@552	1501
adamc@552	1502 Functions are specialized to particular argument patterns. This is an important trick for avoiding the need to maintain any closures at runtime.
adamc@552	1503
adamc@552	1504 \subsection{Untangle}
adamc@552	1505
adamc@552	1506 Remove unnecessary mutual recursion, splitting recursive groups into strongly-connected components.
adamc@552	1507
adamc@552	1508 \subsection{Shake}
adamc@552	1509
adamc@552	1510 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	1511
adamc@553	1512 \subsection{\label{tag}Tag}
adamc@552	1513
adamc@552	1514 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	1515
adamc@552	1516 \subsection{Reduce}
adamc@552	1517
adamc@552	1518 Apply definitional equality rules to simplify the program as much as possible. This effectively includes inlining of every non-recursive definition.
adamc@552	1519
adamc@552	1520 \subsection{Unpoly}
adamc@552	1521
adamc@552	1522 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	1523
adamc@552	1524 \subsection{Specialize}
adamc@552	1525
adamc@558	1526 Replace uses of parameterized datatypes with versions specialized to specific parameters. As for Unpoly, this phase will not be effective enough in the presence of polymorphic recursion or other fancy uses of impredicative polymorphism.
adamc@552	1527
adamc@552	1528 \subsection{Shake}
adamc@552	1529
adamc@558	1530 Here the compiler repeats the earlier Shake phase.
adamc@552	1531
adamc@552	1532 \subsection{Monoize}
adamc@552	1533
adamc@552	1534 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	1535
adamc@552	1536 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	1537 \subsection{MonoOpt}
adamc@552	1538
adamc@552	1539 Simple algebraic laws are applied to simplify the program, focusing especially on efficient imperative generation of HTML pages.
adamc@552	1540
adamc@552	1541 \subsection{MonoUntangle}
adamc@552	1542
adamc@552	1543 Unnecessary mutual recursion is broken up again.
adamc@552	1544
adamc@552	1545 \subsection{MonoReduce}
adamc@552	1546
adamc@552	1547 Equivalents of the definitional equality rules are applied to simplify programs, with inlining again playing a major role.
adamc@552	1548
adamc@552	1549 \subsection{MonoShake, MonoOpt}
adamc@552	1550
adamc@552	1551 Unneeded declarations are removed, and basic optimizations are repeated.
adamc@552	1552
adamc@552	1553 \subsection{Fuse}
adamc@552	1554
adamc@552	1555 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	1556
adamc@552	1557 \subsection{MonoUntangle, MonoShake}
adamc@552	1558
adamc@552	1559 Fuse often creates more opportunities to remove spurious mutual recursion.
adamc@552	1560
adamc@552	1561 \subsection{Pathcheck}
adamc@552	1562
adamc@552	1563 The compiler checks that no link or action name has been used more than once.
adamc@552	1564
adamc@552	1565 \subsection{Cjrize}
adamc@552	1566
adamc@552	1567 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	1568
adamc@552	1569 \subsection{C Compilation and Linking}
adamc@552	1570
adamc@552	1571 The output of the last phase is pretty-printed as C source code and passed to GCC.
adamc@552	1572
adamc@552	1573
adamc@524	1574 \end{document}

Mercurial > urweb

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