typos

algorithm.tex

@@ -19,7 +19,7 @@ rules%
   not.\svvspace{-3.3mm}}
 such as \Rule{Subs} and \Rule{Inter}. We handle this presence
 in the classic way: we define an algorithmic system that tracks the
-miminum type of an expression; this system is obtained from the
+minimum type of an expression; this system is obtained from the
 original system by removing the two structural rules and by
 distributing suitable checks of the subtyping relation in the
 remaining rules. To do that in the presence of set-theoretic types we
@@ -251,7 +251,7 @@ refinement is meaningful only when the function is typed by a union
 type, but also we had to build the expression that causes the
 divergence in quite an \emph{ad hoc} way which makes divergence even
 more unlikely: setting an $n_o$ twice the depth of the syntax tree of
-the outermost type case should be enough to capture all realistic
+the outermost type case should be more than enough to capture all realistic
 cases. For instance, all examples given in Section~\ref{sec:practical}
 can be checked (or found to be ill-typed) with $n_o = 1$.
 \fi
@@ -366,7 +366,7 @@ system is not complete as we discuss in details in the next section.
 \subsubsection{Properties of the algorithmic system}\label{sec:algoprop}
 \rev{%%%
 In what follow we will use $\Gamma\vdashA^{n_o} e:t$ to stress the
-fact that the judgement $\Gamma\vdashA e:t$ is provable in the
+fact that the judgment $\Gamma\vdashA e:t$ is provable in the
 algorithmic system where $\Refinef_{e,t}$ is defined as
 $(\RefineStep{e,t})^{n_o}$; we will omit the index $n_o$---thus keeping
 it implicit---whenever it does not matter in the context.
@@ -390,7 +390,7 @@ be associated to a set of types of the form $\{s\alt
     of $\vdashAts$.
 
 
-Completeness needs a more detailed explaination. The algorithmic
+Completeness needs a more detailed explanation. The algorithmic
 system $\vdashA$
 is not complete w.r.t.\ the language presented in
 Section~\ref{sec:syntax} because it cannot deduce negated arrow

conclusion.tex

@@ -91,6 +91,6 @@ account for polymorphism.
 This needs a whole gamut of non trivial research that we plan to
 develop in the near future building on the work on polymorphic types
 for semantic subtyping~\cite{CX11} and the research on the definition of
-polymophic languages with set-theoretic types
+polymorphic languages with set-theoretic types
 by~\citet{polyduce2,polyduce1,CPN16} and \citet{Pet19phd}.
 \fi

intro.tex

@@ -12,7 +12,7 @@ argument. The annotation specifies that the parameter is either a
 number or a string (the vertical bar denotes a union type).  If this annotation is respected and the function
 is applied to either an integer or a string, then the application
 cannot fail because of a type error (\code{trim()} is a string method of the ECMAScript 5 standard that
-trims whitespaces from the beginning and end of the string) and  both the type-checker of TypeScript and the one of Flow
+trims white-spaces from the beginning and end of the string) and  both the type-checker of TypeScript and the one of Flow
 rightly accept this function. This is possible because both type-checkers
 implement a specific type discipline called \emph{occurrence typing} or \emph{flow
 typing}:\footnote{%

language.tex

@@ -61,7 +61,7 @@ by~\citet{Frisch2008}
 \iflongversion
 and detailed description of the algorithm to
 decide this relation can be found in~\cite{Cas15}. For the reader's
-convenience we succintly recall the definition of the subtyping
+convenience we succinctly recall the definition of the subtyping
 relation in the next subsection but it is possible to skip this
 subsection at first reading and jump directly to Subsection~\ref{sec:syntax}, since to understand
 the rest of the paper

related.tex

@@ -269,8 +269,8 @@ and constructs a more precise \emph{dependent} type. In the dependent
 type, fresh variables are introduced to name sub-expressions and
 record the new constraints.
 This
-process---called \emph{selfification} in the cited works---roughly
-corresponds to a our $\constrf$ and \Refinef{} functions (see Section~\ref{sec:typenv}).
+process---called in the cited works \emph{selfification}---roughly
+corresponds to  our $\constrf$ and \Refinef{} functions (see Section~\ref{sec:typenv}).
 Another approach is the one followed by \citet{RKJ08} which is
 completely based on a program transformation, namely, it consists in putting the term
 in \emph{administrative normal form} (\cite{SF92}). Using a program
@@ -382,13 +382,13 @@ which is a pointer to the environment for the type hypothesis about
 the expression and, as such, it plays the role of our extended type
 environments. Likewise, the \emph{selfification} of \cite{OTMW04}
 and \cite{KF09}, propagates the precise type constraints learned
-during a test. On difference is that with refinement types, the
+during a test. One difference with our approach is that with refinement types the
 information can be
-kept at the level of types types, since dependent types contain terms and can
-introduce variables, while in our approach the mapping is kept in an
-environment.  Although tracking of types for structured expressions thus
-seems thus an aspect commont to different approaches to occurrence types,
-we are confident that even this last non-standard aspect
+kept at the level of types, since dependent types contain terms and can
+introduce variables, while in our approach the mapping is kept separate in a
+type environment.  In summary the tracking of types for structured expressions
+seems an aspect common to different approaches to occurrence types,
+nevertheless we are confident that even this last non-standard aspect
 of our system can be removed and that occurrence typing can be
 explained in a pure standard type-theoretic setting.