Working with the simple types over a base type of natural numbers (including product types), we consider the question of when a type $\sigma$ is encodable as a definable retract of $\tau$: that is, when there are $\lambda$-terms $e:\sigma\rightarrow\tau$ and $d:\tau\rightarrow\sigma$ with $d \circ e = id$. In general, the answer to this question may vary according to both the choice of $\lambda$-calculus and the notion of equality considered; however, we shall show that the encodability… Expand