@@ -97,7 +97,7 @@ provided to us by occurrence typing and deduce for the function in~\eqref{foo3}

function is applied to a number.

To achieve this, we simply modify the typing rule for functions that we defined

in the previous section to accommodate for gradual typing. Let $\sigma$ and $\tau$ range over \emph{gradual types}, that is the types produced by the grammar in Definition~\ref{def:types} to which we add \dyn{} as basic type (see~\citet{castagna2019gradual} for the definition of the subtyping relation on these types). For every gradual type

$\tau$, define $\tauUp$ as the (non graudal) type obtained from $\tau$ by replacing all

$\tau$, define $\tauUp$ as the (non gradual) type obtained from $\tau$ by replacing all

covariant occurrences of \dyn{} by \Any{} and all contravariant ones by \Empty. The

type $\tauUp$ can be seen as the \emph{maximal} interpretation of $\tau$, that is,

every expression that can safely be cast to $\tau$ is of type $\tauUp$. In