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 ... ... @@ -28,7 +28,7 @@ recursive) systems: one is the system to infer the type environments, whose judgements are of the form $\Gamma\vdash e\triangleright \{\Gamma_i\}_{i\in I}$; the other is the type inference system we presented in Section~\ref{sec:static} which we modify/extend to exploit the inference of $\Gamma\vdash e\triangleright exploit the results of the inference of$\Gamma\vdash e\triangleright \{\Gamma_i\}_{i\in I}$. For what concerns the type inference system we have to modify the rule ... ...  ... ... @@ -57,8 +57,7 @@ as \emph{type connectives}. The subtyping relation for these types, noted$\leq$, is the one defined by~\citet{Frisch2008} to which the reader may refer for the formal definition (see~\citet{types18post} for a simpler alternative definition that we recall in Appendix~\ref{sec:subtyping} for the definition (we recall it in Appendix~\ref{sec:subtyping} for the reader's convenience). A detailed description of the algorithm to decide this relation can be found in~\cite{Cas15}. For this presentation it suffices to consider that ... ...  ... ... @@ -74,11 +74,10 @@ t_i$, $t_1 \land t_2 \vartriangleright t_i$, $\neg t \vartriangleright t$ is Noetherian. This gives an induction principle\footnote{In a nutshell, we can do proofs and give definitions 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.} on $\types$ that we use combined with structural induction on $\Domain$ to give the following definition (due to~\citet{types18post} and equivalent to but simpler than the definition in~\cite{Frisch2008}), use combined with structural induction on $\Domain$ to give the following definition, which validates these equalities. \fi%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{definition}[Set-theoretic interpretation of types~\cite{types18post}]\label{def:interpretation-of-types} \begin{definition}[Set-theoretic interpretation of types~\cite{Frisch2008}]\label{def:interpretation-of-types} We define a binary predicate $(d : t)$ (the element $d$ belongs to the type $t$''), where $d \in \Domain$ and $t \in \types$, ... ...
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