I have no idea how to implement fresh unification variable generation (my type system is a superset of Damas-Milner, kind of like how the calculus of constructions is a superset of System F-omega) in Coq or Agda. Everywhere I've seen it implemented, it involves mutable state. So, “in principle”, I could fake it using something like the state monad, and plumbing around the collection of unification variables generated so far, but in practice reasoning about this seems like a pain.