@@ -57,17 +58,26 @@ that assigns to each basic type the set of constants of that type, so

We cannot take the equations above

directly as an inductive definition of $\TypeInter{}$

because types are not defined inductively but coinductively.

\iflongversion%%%%%%%%%%%%%%%%%%%

However, recall that the contractivity condition of

Definition~\ref{def:types} ensures that the binary relation $\vartriangleright

\,\subseteq\!\types{\times}\types$ defined by $t_1\lor t_2\vartriangleright

t_i$, $t_1\land t_2\vartriangleright t_i$, $\neg t \vartriangleright t$ is Noetherian which gives an induction principle 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}),

which validates these equalities.

\else

Notice however that the contractivity condition of

Definition~\ref{def:types} ensures that the binary relation $\vartriangleright

\,\subseteq\!\types^{2}$ defined by $t_1\lor t_2\vartriangleright

\,\subseteq\!\types{\times}\types$ defined by $t_1\lor t_2\vartriangleright

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

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}),

which validates these equalities.

\fi%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{definition}[Set-theoretic interpretation of types~\cite{types18post}]\label{def:interpretation-of-types}