last rereading?

extensions.tex

@@ -20,7 +20,7 @@ following two type constructors:\\[1.4mm]
 %% \textbf{Types} & t & ::= & \record{\ell_1=t \ldots \ell_n=t}{t}\alt \Undef
 %% \end{array}
 %% \]
-where $\ell$ ranges over an infinite set of labels $\Labels$, $\Undef$
+where $\ell$ ranges over an infinite set of labels $\Labels$ and $\Undef$
 is a special singleton type whose only value is the constant
 $\undefcst$ which is a constant not in $\Any$. The type
 $\record{\ell_1=t_1 \ldots \ell_n=t_n}{t}$ is a \emph{quasi-constant
@@ -34,12 +34,12 @@ following syntactic sugar and that form the \emph{record types} of our language\
 \item $\crecord{\ell_1=t_1, \ldots, \ell_n=t_n}$ for $\record{\ell_1=t_1 \ldots \ell_n=t_n}{\Undef}$ (closed records).
 \item $\orecord{\ell_1=t_1, \ldots, \ell_n=t_n}$ for $\record{\ell_1=t_1 \ldots \ell_n=t_n}{\Any \vee \Undef}$ (open records).
 \end{itemize}
-plus the notation $\mathtt{\ell \eqq t}$ to denote optional fields,
+plus the notation $\mathtt{\ell \eqq} t$ to denote optional fields,
 which corresponds to using in the quasi-constant function notation the
 field $\ell = t \vee \Undef$. 
 
 
-For what concern expressions, we adapt CDuce records to our analysis. In particular records are built starting from the empty record expressions \erecord{} and by adding, updating, or removing fields:\vspace{-1mm}
+For what concerns expressions, we adapt CDuce records to our analysis. In particular records are built starting from the empty record expression \erecord{} and by adding, updating, or removing fields:\vspace{-1mm}
 \[
 \begin{array}{lrcl}
 \textbf{Expr} & e & ::= & \erecord {} \alt \recupd e \ell e \alt \recdel e \ell \alt e.\ell
@@ -62,7 +62,7 @@ To define record type subtyping and record expression type inference we need the
     \proj {\ell'} u \geq \proj {\ell'} t &\text{ otherwise}
     \end{array}\right\}\right\}
 \end{eqnarray}
-Then two record types $t_1$ and $t_2$ are in subtyping relation, $t_1 \leq t_2$, if and only if $\forall \ell \in \Labels. \proj \ell {t_1} \leq \proj \ell {t_2}$. In particular $\orecord{\!\!}$ is the largest record type.
+Then two record types $t_1$ and $t_2$ are in subtyping relation, $t_1 \leq t_2$, if and only if for all $\ell \in \Labels$ we have $\proj \ell {t_1} \leq \proj \ell {t_2}$. In particular $\orecord{\!\!}$ is the largest record type.
 
 Expressions are then typed by the following rules (already in algorithmic form).
 \begin{mathpar}
@@ -86,7 +86,7 @@ Expressions are then typed by the following rules (already in algorithmic form).
         {\Gamma \vdash e.\ell:\proj \ell {t}}
         {e.\ell\not\in\dom\Gamma}\vspace{-2mm}
 \end{mathpar}
-To extend occurrence typing to records we add paths the following values to paths: $\varpi\in\{\ldots,a_\ell,u_\ell^1,u_\ell^2,r_\ell\}^*$, with 
+To extend occurrence typing to records we add the following values to paths: $\varpi\in\{\ldots,a_\ell,u_\ell^1,u_\ell^2,r_\ell\}^*$, with 
 \(e.\ell\downarrow a_\ell.\varpi =\occ{e}\varpi\), 
 \(\recdel e \ell\downarrow r_\ell.\varpi = \occ{e}\varpi\), and
 \(\recupd{e_1}\ell{e_2}\downarrow u_\ell^i.\varpi = \occ{e_i}\varpi\)

refining.tex

@@ -20,7 +20,7 @@ in its body, and use this information to partition the domain of the function
 and to re-type its body. Consider a more involved example
 \begin{alltt}\color{darkblue}\morecompact
   function (x \textcolor{darkred}{: \(\tau\)}) \{
-    (x \(\in\) Real) ? \{ (x \(\in\) Int) ? succ x : sqrt x \} : \{ \(\neg\)x \}
+    (x \(\in\) Real) ? \{ (x \(\in\) Int) ? succ x : sqrt x \} : \{ \(\neg\)x \}  \refstepcounter{equation}                                \mbox{\color{black}\rm(\theequation)}\label{foorefine}
   \}
 \end{alltt}
 When $\tau$ is \code{Real|Bool} (we assume that \code{Int} is a
@@ -139,10 +139,10 @@ domain of $\Gamma$ restricted to variables.
 Simply put, this rule first collects all possible types that are deduced
 for a variable $x$ during typing and then uses them to re-type the body
 of the lambda under this new refined hypothesis for the type of
-$x$. The re-typing ensures that that the type soundness property
+$x$. The re-typing ensures that  the type safety property
 carries over to this new rule.
 
-This system is enough to type our case study for the case $\tau$
+This system is enough to type our case study \eqref{foorefine} for the case $\tau$
 defined as \code{Real|Bool}. Indeed the analysis of the body yields
 $\psi(x)=\{\Int,\Real\setminus\Int\}$ for the branch \code{(x\,$\in$\,Int) ? succ\,x : sqrt\,x} and, since
 \code{$(\Bool\vee\Real)\setminus\Real = \Bool$}, yields
@@ -152,7 +152,7 @@ will be checked for the input types $\Int$, $\Real\setminus\Int$, and
 
 It is not too difficult to generalize this rule when the lambda is
 typed by an intersection type. Indeed, even if the programmer has
-specified a particular intersection type for the function we may want
+specified a particular intersection type for the function, we may want
 to refine each of its arrows. This is exactly what the following rule
 does:\vspace{-1mm}
 \begin{mathpar}

setup.sty

@@ -119,7 +119,7 @@
 \newcommand {\bpl}[1] {\boldsymbol{\pi}_{\boldsymbol 1}(#1)}
 \newcommand {\bpr}[1] {\boldsymbol{\pi}_{\boldsymbol 2}(#1)}
 \newcommand {\proj}[2] {#2.#1}
-\newcommand {\recupd}[3] {\{ #1 \texttt{ with } #2 = #3 \}}
+\newcommand {\recupd}[3] {\texttt{\{} #1 \texttt{ with } #2 = #3 \texttt{\}}}
 \newcommand {\recdel}[2] {#1 \mathtt{ \setminus } #2}
 \DeclareMathOperator{\eqq}{\texttt{=?}}