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}