Theory and Applications of Type-Theoretic Universes

Application of universes: staged metaprogramming

Let $Ty_{0} : Set$ and $Tm_{0} : Ty_{0} \to Set$ be a universe with products and unit type. Then if $N : Set$ is a meta-level type of natural numbers, we can define

Vec : N \to Ty_{0} \to Ty_{0} Vec 0_:\equiv ⊤ Vec (S n) A :\equiv prod A (Vec n A) map : {n : N, A B : Ty_{0}} \to (Tm_{0} A \to Tm_{0} B) \to Tm_{0} (Vec n A) \to Tm_{0} (Vec n B) map {0}__:\equiv tt map {S n} f v :\equiv pair (f a) (map f a s) where (a, a s) :\equiv proj v

Loop unrolling for compile-time-known $n$

Note that $Ty_{0}$ need not have dependent types or even function types for this to work. See [Kiselyov 2014Kiselyov, O. 2014. The Design and Implementation of BER MetaOCaml: System Description. In: Functional and Logic Programming. Springer International Publishing, 86–102., Kovács 2022Kovács, A. 2022. Staged compilation with two-level type theory. Proceedings of the ACM on Programming Languages 6, 540–569. http://dx.doi.org/10.1145/3547641.]