language.tex 21.8 KB
Newer Older
1
\newlength{\sk}
Giuseppe Castagna's avatar
space    
Giuseppe Castagna committed
2
\setlength{\sk}{-1.9pt}
Giuseppe Castagna's avatar
Giuseppe Castagna committed
3
\iflongversion
Giuseppe Castagna's avatar
Giuseppe Castagna committed
4
In this section we formalize the ideas we outlined in the introduction. We start by the definition of types followed by the language and its reduction semantics. The static semantics is the core of our work: we first present a declarative type system that deduces (possibly many) types for well-typed expressions and then the algorithms to decide whether an expression is well typed or not. 
Giuseppe Castagna's avatar
Giuseppe Castagna committed
5
\fi
6
7
8

\subsection{Types}

Giuseppe Castagna's avatar
Giuseppe Castagna committed
9
\begin{definition}[Types]\label{def:types} The set of types \types{} is formed by the terms $t$ coinductively produced by the grammar:\vspace{-1.45mm}
10
11
12
13
14
\[
\begin{array}{lrcl}
\textbf{Types} & t & ::= & b\alt t\to t\alt t\times t\alt t\vee t \alt \neg t \alt \Empty 
\end{array}
\]
Giuseppe Castagna's avatar
Giuseppe Castagna committed
15
16
and that satisfy the following conditions
\begin{itemize}[nosep]
Giuseppe Castagna's avatar
Giuseppe Castagna committed
17
18
\item (regularity) every term has a finite number of different sub-terms;
\item (contractivity) every infinite branch of a term contains an infinite number of occurrences of the
Giuseppe Castagna's avatar
Giuseppe Castagna committed
19
arrow or product type constructors.\vspace{-1mm}
20
21
\end{itemize}
\end{definition}
Giuseppe Castagna's avatar
Giuseppe Castagna committed
22
We use the following abbreviations: $
23
24
    t_1 \land t_2 \eqdef \neg (\neg t_1 \vee \neg t_2)$, 
    $t_ 1 \setminus t_2 \eqdef t_1 \wedge \neg t_2$, $\Any \eqdef \neg \Empty$.
Giuseppe Castagna's avatar
Giuseppe Castagna committed
25
$b$ ranges over basic types
Giuseppe Castagna's avatar
Giuseppe Castagna committed
26
(e.g., \Int, \Bool),
27
28
29
30
31
32
33
34
35
36
$\Empty$ and $\Any$ respectively denote the empty (that types no value)
and top (that types all values) types. Coinduction accounts for
recursive types and the condition on infinite branches bars out
ill-formed types such as 
$t = t \lor t$ (which does not carry any information about the set
denoted by the type) or $t = \neg t$ (which cannot represent any
set). 
It also ensures that the binary relation $\vartriangleright
\,\subseteq\!\types^{2}$ defined by $t_1 \lor t_2 \vartriangleright
t_i$, $t_1 \land t_2 \vartriangleright
Giuseppe Castagna's avatar
Giuseppe Castagna committed
37
t_i$, $\neg t \vartriangleright t$ is Noetherian.
38
This gives an induction principle on $\types$ that we
Giuseppe Castagna's avatar
Giuseppe Castagna committed
39
will use without any further explicit reference to the relation.\footnote{In a nutshell, we can do proofs by induction on the structure of unions and negations---and, thus, intersections---but arrows, products, and basic types are the base cases for the induction.} 
40
41
42
43
44
We refer to $ b $, $\times$, and $ \to $ as \emph{type constructors}
and to $ \lor $, $ \land $, $ \lnot $, and $ \setminus $
as \emph{type connectives}.

The subtyping relation for these types, noted $\leq$, is the one defined
Giuseppe Castagna's avatar
Giuseppe Castagna committed
45
by~\citet{Frisch2008} to which the reader may refer. A detailed description of the algorithm to decide it can be found in~\cite{Cas15}.
46
For this presentation it suffices to consider that
Giuseppe Castagna's avatar
Giuseppe Castagna committed
47
types are interpreted as sets of \emph{values} ({i.e., either
Victor Lanvin's avatar
Fixes    
Victor Lanvin committed
48
constants, $\lambda$-abstractions, or pairs of values: see
49
Section~\ref{sec:syntax} right below) that have that type, and that subtyping is set
Giuseppe Castagna's avatar
Giuseppe Castagna committed
50
containment (i.e., a type $s$ is a subtype of a type $t$ if and only if $t$
51
52
contains all the values of type $s$). In particular, $s\to t$
contains all $\lambda$-abstractions that when applied to a value of
53
type $s$, if their computation terminates, then they return a result of
Giuseppe Castagna's avatar
Giuseppe Castagna committed
54
type $t$ (e.g., $\Empty\to\Any$ is the set of all
55
56
57
58
functions\footnote{\label{allfunctions}Actually, for every type $t$,
all types of the form $\Empty{\to}t$ are equivalent and each of them
denotes the set of all functions.} and $\Any\to\Empty$ is the set
of functions that diverge on every argument). Type connectives
Giuseppe Castagna's avatar
Giuseppe Castagna committed
59
60
(i.e., union, intersection, negation) are interpreted as the
corresponding set-theoretic operators (e.g.,~$s\vee t$ is the
61
union of the values of the two types). We use $\simeq$ to denote the
Giuseppe Castagna's avatar
Giuseppe Castagna committed
62
symmetric closure of $\leq$: thus $s\simeq t$ (read, $s$ is equivalent to $t$) means that $s$ and $t$ denote the same set of values and, as such, they are semantically the same type.
63
64

\subsection{Syntax}\label{sec:syntax}
Giuseppe Castagna's avatar
Giuseppe Castagna committed
65
The expressions $e$ and values $v$ of our language are inductively generated by the following grammars:\vspace{-1mm}
66
\begin{equation}\label{expressions}
67
\begin{array}{lrclr}  
Giuseppe Castagna's avatar
Giuseppe Castagna committed
68
69
  \textbf{Expr} &e &::=& c\alt x\alt ee\alt\lambda^{\wedge_{i\in I}s_i\to t_i} x.e\alt \pi_j e\alt(e,e)\alt\tcase{e}{t}{e}{e}\\[.3mm]
  \textbf{Values} &v &::=& c\alt\lambda^{\wedge_{i\in I}s_i\to t_i} x.e\alt (v,v)\\[-1mm]
70
71
\end{array}
\end{equation}
Giuseppe Castagna's avatar
wording    
Giuseppe Castagna committed
72
for $j=1,2$. In~\eqref{expressions}, $c$ ranges over constants
Giuseppe Castagna's avatar
Giuseppe Castagna committed
73
(e.g., \texttt{true}, \texttt{false}, \texttt{1}, \texttt{2},
74
75
...) which are values of basic types (we use $\basic{c}$ to denote the
basic type of the constant $c$); $x$ ranges over variables; $(e,e)$
Kim Nguyễn's avatar
typos.    
Kim Nguyễn committed
76
denotes pairs and $\pi_i e$ their projections; $\tcase{e}{t}{e_1}{e_2}$
77
78
79
80
81
82
83
84
85
86
denotes the type-case expression that evaluates either $e_1$ or $e_2$
according to whether the value returned by $e$ (if any) is of type $t$
or not; $\lambda^{\wedge_{i\in I}s_i\to t_i} x.e$ is a value of type
$\wedge_{i\in I}s_i\to t_i$ and denotes the function of parameter $x$
and body $e$. An expression has an intersection type if and only if it
has all the types that compose the intersection. Therefore,
intuitively, $\lambda^{\wedge_{i\in I}s_i\to t_i} x.e$ is a well-typed
value if for all $i{\in} I$ the hypothesis that $x$ is of type $s_i$
implies that the body $e$ has type $t_i$, that is to say, it is well
typed if $\lambda^{\wedge_{i\in I}s_i\to t_i} x.e$ has type $s_i\to
Giuseppe Castagna's avatar
typos    
Giuseppe Castagna committed
87
t_i$ for all $i\in I$. Every value is associated to a most specific type (mst): the mst of $c$ is $\basic c$; the mst of
88
 $\lambda^{\wedge_{i\in I}s_i\to t_i} x.e$ is $\wedge_{i\in I}s_i\to t_i$; and, inductively,
Giuseppe Castagna's avatar
Giuseppe Castagna committed
89
90
the mst of a pair of values is the product of the mst's of the
values. We write $v\in t$ if the most specific type of $v$ is a subtype of $t$ (see Appendix~\ref{app:typeschemes} for the formal definition of $v\in t$ which  deals with some corner cases for negated arrow types).
91
92
93



94
\subsection{Dynamic semantics}\label{sec:opsem}
95

Giuseppe Castagna's avatar
Giuseppe Castagna committed
96
The dynamic semantics is defined as a classic left-to-right call-by-value reduction for a $\lambda$-calculus with pairs, enriched with specific rules for type-cases. We have the following  notions of reduction:\vspace{-1.2mm}
97
98
\[
\begin{array}{rcll}
Kim Nguyễn's avatar
Kim Nguyễn committed
99
  (\lambda^{\wedge_{i\in I}s_i\to t_i} x.e)\,v &\reduces& e\subst x v\\[-.4mm]
Giuseppe Castagna's avatar
Giuseppe Castagna committed
100
101
  \pi_i(v_1,v_2) &\reduces& v_i & i=1,2\\[-.4mm]
  \tcase{v}{t}{e_1}{e_2} &\reduces& e_1 &v\in t\\[-.4mm] 
Giuseppe Castagna's avatar
Giuseppe Castagna committed
102
  \tcase{v}{t}{e_1}{e_2} &\reduces& e_2 &v\not\in t\\[-1.3mm]
103
104
\end{array}
\]
Giuseppe Castagna's avatar
Giuseppe Castagna committed
105
Contextual reductions are
Giuseppe Castagna's avatar
Giuseppe Castagna committed
106
107
108
defined by the following evaluation contexts:\\[1mm]
\centerline{\(
%\[
109
\Cx[] ::= [\,]\alt \Cx e\alt v\Cx \alt (\Cx,e)\alt (v,\Cx)\alt \pi_i\Cx\alt \tcase{\Cx}tee
Giuseppe Castagna's avatar
Giuseppe Castagna committed
110
111
%\]
\)}\\[1mm]
112
113
114
As usual we denote by $\Cx[e]$ the term obtained by replacing $e$ for
the hole in the context $\Cx$ and we have that $e\reduces e'$ implies
$\Cx[e]\reduces\Cx[e']$.
115

Giuseppe Castagna's avatar
Giuseppe Castagna committed
116
\subsection{Static semantics}\label{sec:static}
117

118
While the syntax and reduction semantics are, on the whole, pretty
Giuseppe Castagna's avatar
bla    
Giuseppe Castagna committed
119
standard, for what concerns the type system we will have to introduce several
120
unconventional features that we anticipated in
Giuseppe Castagna's avatar
Giuseppe Castagna committed
121
Section~\ref{sec:challenges} and are at the core of our work. Let
122
us start with the standard part, that is the typing of the functional
Giuseppe Castagna's avatar
Giuseppe Castagna committed
123
core and the use of subtyping, given by the following typing rules:\vspace{-1mm}
124
125
126
127
128
129
130
131
\begin{mathpar}
  \Infer[Const]
      { }
      {\Gamma\vdash c:\basic{c}}
      { }
  \quad
  \Infer[App]
      {
132
        \Gamma \vdash e_1: \arrow {t_1}{t_2}\quad
133
134
135
136
137
138
        \Gamma \vdash e_2: t_1
      }
      { \Gamma \vdash {e_1}{e_2}: t_2 }
      { }
  \quad
  \Infer[Abs+]
139
      {{\scriptstyle\forall i\in I}\quad\Gamma,x:s_i\vdash e:t_i}
140
141
142
143
144
145
146
147
148
149
150
151
      {
      \Gamma\vdash\lambda^{\wedge_{i\in I}\arrow {s_i} {t_i}}x.e:\textstyle \bigwedge_{i\in I}\arrow {s_i} {t_i}
      }
      { }
%      \Infer[If]
%            {\Gamma\vdash e:t_0\\
%            %t_0\not\leq \neg t \Rightarrow
%            \Gamma \cvdash + e t e_1:t'\\
%            %t_0\not\leq t \Rightarrow
%            \Gamma \cvdash - e t e_2:t'}
%            {\Gamma\vdash \ite {e} t {e_1}{e_2}: t'}
%            { }
Giuseppe Castagna's avatar
Giuseppe Castagna committed
152
153
\end{mathpar}
\begin{mathpar}\vspace{-4mm}\\ 
Giuseppe Castagna's avatar
Giuseppe Castagna committed
154
      \Infer[Sel]
155
156
157
158
159
160
161
162
163
164
165
166
167
  {\Gamma \vdash e:\pair{t_1}{t_2}}
  {\Gamma \vdash \pi_i e:t_i}
  { }
  \qquad
  \Infer[Pair]
  {\Gamma \vdash e_1:t_1 \and \Gamma \vdash e_2:t_2}
  {\Gamma \vdash (e_1,e_2):\pair {t_1} {t_2}}
  { }
  \qquad
    \Infer[Subs]
      { \Gamma \vdash e:t\\t\leq t' }
      { \Gamma \vdash e: t' }
      { }
Giuseppe Castagna's avatar
Giuseppe Castagna committed
168
  \qquad\vspace{-3mm}
169
\end{mathpar}
Giuseppe Castagna's avatar
Giuseppe Castagna committed
170
These rules are quite standard and do not need any particular explanation besides those already given in Section~\ref{sec:syntax}. Just notice subtyping is embedded in the system by the classic \Rule{Subs} subsumption rule. Next we focus on the unconventional aspects of our system, from the simplest to the hardest.
171

Giuseppe Castagna's avatar
Giuseppe Castagna committed
172
The first unconventional aspect is that, as explained in
173
Section~\ref{sec:challenges}, our type assumptions are about
Giuseppe Castagna's avatar
Giuseppe Castagna committed
174
expressions. Therefore, in our rules the type environments, ranged over
175
by $\Gamma$, map \emph{expressions}---rather than just variables---into
Giuseppe Castagna's avatar
Giuseppe Castagna committed
176
types. This explains why the classic typing rule for variables is replaced by a more general \Rule{Env} rule defined below:\vspace{-1mm}
177
178
179
180
181
182
\begin{mathpar}
  \Infer[Env]
      { }
      { \Gamma \vdash e: \Gamma(e) }
      { e\in\dom\Gamma }
  \qquad
183
  \Infer[Inter]
184
185
      { \Gamma \vdash e:t_1\\\Gamma \vdash e:t_2 }
      { \Gamma \vdash e: t_1 \wedge t_2 }
Giuseppe Castagna's avatar
Giuseppe Castagna committed
186
      { }\vspace{-3mm}
187
\end{mathpar}
188
The \Rule{Env} rule is coupled with the standard intersection introduction rule \Rule{Inter}
Giuseppe Castagna's avatar
Giuseppe Castagna committed
189
which allows us to deduce for a complex expression the intersection of
190
191
192
193
the types recorded by the occurrence typing analysis in the
environment $\Gamma$ with the static type deduced for the same
expression by using the other typing rules. This same intersection
rule is also used to infer the second unconventional aspect of our
194
system, that is, the fact that $\lambda$-abstractions can have negated
Giuseppe Castagna's avatar
Giuseppe Castagna committed
195
arrow types, as long as these negated types do not make the type deduced for the function empty:\vspace{-.5mm}
196
197
\begin{mathpar}
  \Infer[Abs-]
Mickael Laurent's avatar
Mickael Laurent committed
198
199
    {\Gamma \vdash \lambda^{\wedge_{i\in I}\arrow {s_i} {t_i}}x.e:t}
    { \Gamma \vdash\lambda^{\wedge_{i\in I}\arrow {s_i} {t_i}}x.e:\neg(t_1\to t_2)  }
Giuseppe Castagna's avatar
Giuseppe Castagna committed
200
    { ((\wedge_{i\in I}\arrow {s_i} {t_i})\wedge\neg(t_1\to t_2))\not\simeq\Empty }\vspace{-1mm}
201
\end{mathpar}
Giuseppe Castagna's avatar
Giuseppe Castagna committed
202
203
204
205
206
207
208
%\beppe{I have doubt: is this safe or should we play it safer and
%  deduce $t\wedge\neg(t_1\to t_2)$? In other terms is is possible to
%  deduce two separate negation of arrow types that when intersected
%  with the interface are non empty, but by intersecting everything
%  makes the type empty? It should be safe since otherwise intersection
%  would not be admissible in semantic subtyping (see Theorem 6.15 in
%  JACM), but I think we should doube ckeck it.}
209
As explained in Section~\ref{sec:challenges}, we need to be able to
210
211
212
deduce for, say, the function $\lambda^{\Int\to\Int} x.x$ a type such
as $(\Int\to\Int)\wedge\neg(\Bool\to\Bool)$ (in particular, if this is
the term $e$ in equation \eqref{bistwo} we need to deduce for it the
Giuseppe Castagna's avatar
Giuseppe Castagna committed
213
214
type $(\Int\to t)\wedge\neg(\Int\to\neg\Bool)$, that is,
$(\Int\to t)\setminus(\Int\to\neg\Bool)$ ). But the sole rule \Rule{Abs+}
215
216
above does not allow us to deduce  negations of
arrows for abstractions: the rule \Rule{Abs-} makes this possible. As an aside, note that this kind
Giuseppe Castagna's avatar
Giuseppe Castagna committed
217
of deduction was already present in the system by~\citet{Frisch2008}
Kim Nguyễn's avatar
typo.    
Kim Nguyễn committed
218
though in that system this presence was motivated by the semantics of types rather than, as in our case,
219
220
221
222
223
by the soundness of the type system.

Rules \Rule{Abs+} and \Rule{Abs-} are not enough to deduce for
$\lambda$-abstractions all the types we wish. In particular, these
rules alone are not enough to type general overloaded functions. For
Giuseppe Castagna's avatar
Giuseppe Castagna committed
224
instance, consider this simple example of a function that applied to an
225
integer returns its successor and applied to anything else returns
Giuseppe Castagna's avatar
Giuseppe Castagna committed
226
227
228
\textsf{true}:\\[1mm]
\centerline{\(
%\[
229
\lambda^{(\Int\to\Int)\wedge(\neg\Int\to\Bool)} x\,.\,\tcase{x}{\Int}{x+1}{\textsf{true}}
Giuseppe Castagna's avatar
Giuseppe Castagna committed
230
231
%\]
\)}\\[1mm]
232
Clearly, the expression above is well typed, but the rule \Rule{Abs+} alone
233
is not enough to type it. In particular, according to \Rule{Abs+} we
234
have to prove that under the hypothesis that $x$ is of type $\Int$ the expression
235
$(\tcase{x}{\Int}{x+1}{\textsf{true}})$ is of type $\Int$, too.  That is, that under the
Giuseppe Castagna's avatar
Giuseppe Castagna committed
236
hypothesis that $x$ has type $\Int\wedge\Int$ (we apply occurrence
237
typing) the expression $x+1$ is of type \Int{} (which holds) and that under the
238
hypothesis that $x$ has type $\Int\setminus\Int$, that is $\Empty$
239
240
(we apply once more occurrence typing), \textsf{true} is of type \Int{}
(which \emph{does not} hold). The problem is that we are trying to type the
241
second case of a type-case even if we know that there is no chance that, when $x$ is bound to an integer,
242
243
that case will be ever selected. The fact that it is never selected is witnessed
by the presence of a type hypothesis with  $\Empty$ type. To
244
avoid this problem (and type the term above) we add the rule
245
246
247
\Rule{Efq} (\emph{ex falso quodlibet}) that allows the system to deduce any type
for an expression that will never be selected, that is, for an
expression whose type environment contains an empty assumption:
248
249
250
251
\begin{mathpar}
  \Infer[Efq]
  { }
  { \Gamma, (e:\Empty) \vdash e': t }
Giuseppe Castagna's avatar
Giuseppe Castagna committed
252
  { }\vspace{-3mm}
253
\end{mathpar}
254
255
Once more, this kind of deduction was already present in the system
by~\citet{Frisch2008} to type full fledged overloaded functions,
Giuseppe Castagna's avatar
Giuseppe Castagna committed
256
257
though it was embedded in the typing rule for the type-case.\pagebreak
Here we
Giuseppe Castagna's avatar
Giuseppe Castagna committed
258
need the rule \Rule{Efq}, which is more general, to ensure the
Giuseppe Castagna's avatar
Giuseppe Castagna committed
259
260
property of subject reduction.
%\beppe{Example?}
261

Giuseppe Castagna's avatar
Giuseppe Castagna committed
262
Finally, there remains one last rule in our type system, the one that
263
implements occurrence typing, that is, the rule for the
Giuseppe Castagna's avatar
Giuseppe Castagna committed
264
type-case:\vspace{-1mm}
265
266
267
268
\begin{mathpar}
    \Infer[Case]
        {\Gamma\vdash e:t_0\\
        %t_0\not\leq \neg t \Rightarrow
269
        \Gamma \evdash e t \Gamma_1 \\ \Gamma_1 \vdash e_1:t'\\
270
        %t_0\not\leq t \Rightarrow
271
        \Gamma \evdash e {\neg t} \Gamma_2 \\ \Gamma_2 \vdash e_2:t'}
272
        {\Gamma\vdash \tcase {e} t {e_1}{e_2}: t'}
Giuseppe Castagna's avatar
Giuseppe Castagna committed
273
        { }\vspace{-3mm}
274
\end{mathpar}
275
276
277
278
279
280
281
282
283
284
285
286
The rule \Rule{Case} checks whether the expression $e$, whose type is
being tested, is well-typed and then performs the occurrence typing
analysis that produces the environments $\Gamma_i$'s under whose
hypothesis the expressions $e_i$'s are typed. The production of these
environments is represented by the judgments $\Gamma \evdash e
{(\neg)t} \Gamma_i$. The intuition is that when $\Gamma \evdash e t
\Gamma_1$ is provable then $\Gamma_1$ is a version of $\Gamma$
extended with type hypotheses for all expressions occurring in $e$,
type hypotheses that can be deduced assuming that the test $e\in t$
succeeds. Likewise, $\Gamma \evdash e {\neg t} \Gamma_2$ (notice the negation on $t$) extends
$\Gamma$ with the hypothesis deduced assuming that $e\in\neg t$, that
is, for when the test $e\in t$ fails.
287

288
All it remains to do is to show how to deduce judgments of the form
Giuseppe Castagna's avatar
Giuseppe Castagna committed
289
$\Gamma \evdash e t \Gamma'$. For that we first define how
Giuseppe Castagna's avatar
Giuseppe Castagna committed
290
291
to denote occurrences of an expression. These are identified by paths in the
syntax tree of the expressions, that is, by possibly empty strings of
292
293
294
295
296
297
characters denoting directions starting from the root of the tree (we
use $\epsilon$ for the empty string/path, which corresponds to the
root of the tree).

Let $e$ be an expression and $\varpi\in\{0,1,l,r,f,s\}^*$ a
\emph{path}; we denote $\occ e\varpi$ the occurrence of $e$ reached by
Kim Nguyễn's avatar
Kim Nguyễn committed
298
the path $\varpi$, that is (for $i=0,1$, and undefined otherwise)\vspace{-.4mm}
Giuseppe Castagna's avatar
Giuseppe Castagna committed
299
300
301
302
303
304
305
306
307
308
309
310
311
%% \[
%% \begin{array}{l}
%% \begin{array}{r@{\downarrow}l@{\quad=\quad}l}
%% e&\epsilon & e\\
%% e_0e_1& i.\varpi & \occ{e_i}\varpi\qquad i=0,1\\
%% (e_0,e_1)& l.\varpi & \occ{e_0}\varpi\\
%% (e_0,e_1)& r.\varpi & \occ{e_1}\varpi\\
%% \pi_1 e& f.\varpi & \occ{e}\varpi\\
%% \pi_2 e& s.\varpi & \occ{e}\varpi\\
%% \end{array}\\
%% \text{undefined otherwise}
%% \end{array}
%% \]
312
\[
Giuseppe Castagna's avatar
Giuseppe Castagna committed
313
\begin{array}{r@{\downarrow}l@{\quad=\quad}lr@{\downarrow}l@{\quad=\quad}lr@{\downarrow}l@{\quad=\quad}l}
Kim Nguyễn's avatar
Kim Nguyễn committed
314
315
e&\epsilon & e & (e_1,e_2)& l.\varpi & \occ{e_1}\varpi &\pi_1 e& f.\varpi & \occ{e}\varpi\\
e_0\,e_1& i.\varpi & \occ{e_i}\varpi \quad\qquad& (e_1,e_2)& r.\varpi & \occ{e_2}\varpi \quad\qquad&
Giuseppe Castagna's avatar
spaces    
Giuseppe Castagna committed
316
\pi_2 e& s.\varpi & \occ{e}\varpi\\[-.4mm]
317
318
319
320
\end{array}
\]
To ease our analysis we used different directions for each kind of
term. So we have $0$ and $1$ for the function and argument of an
321
application, $l$ and $r$ for the $l$eft and $r$ight expressions forming a pair,
Giuseppe Castagna's avatar
Giuseppe Castagna committed
322
and $f$ and $s$ for the argument of a $f$irst or of a $s$econd projection. Note also that we do not consider occurrences
Giuseppe Castagna's avatar
Giuseppe Castagna committed
323
under $\lambda$'s (since their type is frozen in their annotations) and type-cases (since they reset the analysis).
Giuseppe Castagna's avatar
space    
Giuseppe Castagna committed
324
325
%
The judgments  $\Gamma \evdash e t \Gamma'$ are then deduced by the following two rules:\vspace{-1mm} \begin{mathpar}
326
327
328
329
330
331
332
333
334
335
336
%        \Infer[Base]
%            { \Gamma \vdash e':t' }
%            { \Gamma \cvdash p e t e':t' }
%            { }
%            \qquad
%        \Infer[Path]
%            { \pvdash \Gamma p e t \varpi:t_1 \\ \Gamma,(\occ e \varpi:t_1) \cvdash p e t e':t_2 }
%            { \Gamma \cvdash p e t e':t_2 }
%            { }
    \Infer[Base]
      { }
337
      { \Gamma \evdash e t \Gamma }
338
339
340
      { }
    \qquad
    \Infer[Path]
341
342
      { \pvdash {\Gamma'} e t \varpi:t' \\ \Gamma \evdash e t \Gamma' }
      { \Gamma \evdash e t \Gamma',(\occ e \varpi:t') }
Giuseppe Castagna's avatar
space    
Giuseppe Castagna committed
343
      { }\vspace{-1.5mm}
344
\end{mathpar}
345
346
347
348
These rules describe how to produce by occurrence typing the type
environments while checking that an expression $e$ has type $t$. They state that $(i)$ we can
deduce from $\Gamma$ all the hypothesis already in $\Gamma$ (rule
\Rule{Base}) and that $(ii)$ if we can deduce a given type $t'$ for a particular
Giuseppe Castagna's avatar
Giuseppe Castagna committed
349
occurrence $\varpi$ of the expression $e$ being checked, then we can add this
350
hypothesis to the produced type environment (rule \Rule{Path}). The rule
351
\Rule{Path} uses a (last) auxiliary judgement $\pvdash {\Gamma}  e t
Giuseppe Castagna's avatar
Giuseppe Castagna committed
352
\varpi:t'$ to deduce the type $t'$ of the occurrence $\occ e \varpi$ when
353
354
355
356
357
checking $e$ against $t$ under the hypotheses $\Gamma$. This rule \Rule{Path} is subtler than it may appear at
first sight, insofar as the deduction of the type for $\varpi$ may already use
some hypothesis on $\occ e \varpi$ (in $\Gamma'$) and, from an
algorithmic viewpoint, this will imply the computation of a fix-point
(see Section~\ref{sec:typenv}). The last ingredient for our type system is the deduction of the
358
judgements of the form $\pvdash {\Gamma}  e t \varpi:t'$ where
359
$\varpi$ is a path to an expression occurring in $e$. This is given by the following set
360
361
362
of rules.
\begin{mathpar}
    \Infer[PSubs]
363
364
        { \pvdash \Gamma e t \varpi:t_1 \\ t_1\leq t_2 }
        { \pvdash \Gamma e t \varpi:t_2 }
365
        { }
366
        \quad
367
    \Infer[PInter]
368
369
        { \pvdash \Gamma e t \varpi:t_1 \\ \pvdash \Gamma e t \varpi:t_2 }
        { \pvdash \Gamma e t \varpi:t_1\land t_2 }
370
        { }
371
        \quad
372
373
    \Infer[PTypeof]
        { \Gamma \vdash \occ e \varpi:t' }
374
        { \pvdash \Gamma e t \varpi:t' }
375
        { }
376
\vspace{-1.2mm}\\
377
    \Infer[PEps]
378
        { }
379
        { \pvdash \Gamma e t \epsilon:t }
380
381
382
        { }
        \qquad
    \Infer[PAppR]
383
384
        { \pvdash \Gamma e t \varpi.0:\arrow{t_1}{t_2} \\ \pvdash \Gamma e t \varpi:t_2'}
        { \pvdash \Gamma e t \varpi.1:\neg t_1 }
385
        { t_2\land t_2' \simeq \Empty  }
Giuseppe Castagna's avatar
Giuseppe Castagna committed
386
\ifsubmission\vspace{-1.2mm}\\\else\end{mathpar}\begin{mathpar}\fi
387
    \Infer[PAppL]
388
389
        { \pvdash \Gamma e t \varpi.1:t_1 \\ \pvdash \Gamma e t \varpi:t_2 }
        { \pvdash \Gamma e t \varpi.0:\neg (\arrow {t_1} {\neg t_2}) }
390
391
392
        { }
        \qquad
    \Infer[PPairL]
393
394
        { \pvdash \Gamma e t \varpi:\pair{t_1}{t_2} }
        { \pvdash \Gamma e t \varpi.l:t_1 }
395
        { }
Giuseppe Castagna's avatar
space    
Giuseppe Castagna committed
396
\vspace{-1.2mm}\\
397
    \Infer[PPairR]
398
399
        { \pvdash \Gamma e t \varpi:\pair{t_1}{t_2} }
        { \pvdash \Gamma e t \varpi.r:t_2 }
400
401
402
        { }
        \qquad
    \Infer[PFst]
403
404
        { \pvdash \Gamma e t \varpi:t' }
        { \pvdash \Gamma e t \varpi.f:\pair {t'} \Any }
405
406
407
        { }
        \qquad
    \Infer[PSnd]
408
409
        { \pvdash \Gamma e t \varpi:t' }
        { \pvdash \Gamma e t \varpi.s:\pair \Any {t'} }
Giuseppe Castagna's avatar
space    
Giuseppe Castagna committed
410
        { }\vspace{-1.9mm}
411
\end{mathpar}
412
413
414
These rules implement the analysis described in
Section~\ref{sec:ideas} for functions and extend it to products.  Let
us comment each rule in detail. \Rule{PSubs} is just subsumption for
415
the deduction $\vdashp$. The rule \Rule{PInter} combined with
416
417
\Rule{PTypeof} allows the system to deduce for an occurrence $\varpi$
the intersection of the static type of $\occ e \varpi$ (deduced by
418
\Rule{PTypeof}) with the type deduced for $\varpi$ by the other $\vdashp$ rules. The
Giuseppe Castagna's avatar
Giuseppe Castagna committed
419
rule \Rule{PEps} is the starting point of the analysis: if we are assuming that the test $e\in t$ succeeds, then we can assume that $e$ (i.e.,
420
$\occ e\epsilon$) has type $t$ (recall that assuming that the test $e\in t$ fails corresponds to having $\neg t$ at the index of the turnstyle).
Giuseppe Castagna's avatar
Giuseppe Castagna committed
421
The rule \Rule{PAppR} implements occurrence typing for
422
423
424
the arguments of applications, since it states that if a function maps
arguments of type $t_1$ in results of type $t_2$ and an application of
this function yields results (in $t'_2$) that cannot be in $t_2$
Giuseppe Castagna's avatar
Giuseppe Castagna committed
425
(since $t_2\land t_2' \simeq \Empty$), then the argument of this application cannot be of type $t_1$. \Rule{PAppL} performs the
426
427
428
429
occurrence typing analysis for the function part of an application,
since it states that if an application has type $t_2$ and the argument
of this application has type $t_1$, then the function in this
application cannot have type $t_1\to\neg t_2$. Rules \Rule{PPair\_}
430
are straightforward since they state that the $i$-th projection of a pair
Giuseppe Castagna's avatar
Giuseppe Castagna committed
431
that is of type $\pair{t_1}{t_2}$ must be of type $t_i$. So are the last two
432
433
434
435
436
rules that essentially state that if $\pi_1 e$ (respectively, $\pi_2
e$) is of type $t'$, then the type of $e$ must be of the form
$\pair{t'}\Any$ (respectively, $\pair\Any{t'}$).

This concludes the presentation of our type system, which satisfies
Giuseppe Castagna's avatar
Giuseppe Castagna committed
437
438
439
the property of safety, deduced, as customary, from the properties
of progress and subject reduction (\emph{cf.} Appendix~\ref{app:soundness}).
\begin{theorem}[type safety]
Giuseppe Castagna's avatar
Giuseppe Castagna committed
440
441
442
For every expression $e$ such that $\varnothing\vdash e:t$ either  $e$
diverges or there
exists a value $v$ of type $t$ such that $e\reduces^* v$.
443
\end{theorem}
444
445
446
447
448