Commit 0b36acea authored by Giuseppe Castagna's avatar Giuseppe Castagna
Browse files

spacing

parent f2e04704
......@@ -253,7 +253,12 @@ cases.
\subsubsection{Algorithmic typing rules}\label{sec:algorules}
We now have all the notions we need for our typing algorithm, which is defined by the following rules.
We now have all the notions we need for our typing algorithm%
\iflongversion%
, which is defined by the following rules.
\else%
:
\fi
\begin{mathpar}
\Infer[Efq\Aa]
{ }
......
......@@ -197,7 +197,7 @@ arrow types, as long as these negated types do not make the type deduced for the
\Infer[Abs-]
{\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) }
{ ((\wedge_{i\in I}\arrow {s_i} {t_i})\wedge\neg(t_1\to t_2))\not\simeq\Empty }\vspace{-1mm}
{ ((\wedge_{i\in I}\arrow {s_i} {t_i})\wedge\neg(t_1\to t_2))\not\simeq\Empty }\vspace{-1.2mm}
\end{mathpar}
%\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
......@@ -228,7 +228,7 @@ integer returns its successor and applied to anything else returns
%\[
\lambda^{(\Int\to\Int)\wedge(\neg\Int\to\Bool)} x\,.\,\tcase{x}{\Int}{x+1}{\textsf{true}}
%\]
\)}\\[1mm]
\)}\\[.6mm]
Clearly, the expression above is well typed, but the rule \Rule{Abs+} alone
is not enough to type it. In particular, according to \Rule{Abs+} we
have to prove that under the hypothesis that $x$ is of type $\Int$ the expression
......@@ -253,7 +253,7 @@ expression whose type environment contains an empty assumption:
\end{mathpar}
Once more, this kind of deduction was already present in the system
by~\citet{Frisch2008} to type full fledged overloaded functions,
though it was embedded in the typing rule for the type-case.\pagebreak
though it was embedded in the typing rule for the type-case.~\pagebreak
Here we
need the rule \Rule{Efq}, which is more general, to ensure the
property of subject reduction.
......
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