Modern programming language type systems help programmers write correct software, and furthermore helps them write the software they actually intended to write. We show how expressive types can be used to encode dimension and units of measure information, which can be used to avoid dimensional mistakes and guide software construction, and how types can even help to generate code automatically, which eliminates a whole class of bugs.

We investigate containers and polynomial functors in Quantitative Type Theory, and give initial algebra semantics of inductive data types in the presence of linearity. We show that reasoning by induction is supported, and equivalent to initiality, also in the linear setting.