Schematic Affine Recursion, Oh My!

Iowa Type Theory Commute

To solve the problem raised in the last episode, I propose schematic affine recursion. We saw that affine lambda calculus (where lambda-bound variables are used at most once) plus structural recursion does not enforce termination, even if you restrict the recursor so that the function to be iterated is closed when you reduce ("closed at reduction"). You have to restrict it so that recursion terms are disallowed entirely unless the function to be iterated is closed ("closed at construction"). But this prevents higher-order functions like map, which need to repeat a computation involving a variable f to be mapped over the elements of a list. The solution is to allow schematic definition of terms, using schema variables ranging over closed terms.