If you look at the Isabelle (https://isabelle.in.tum.de/overview.html https://isabelle.in.tum.de/overview.html) and Coq (https://coq.inria.fr https://coq.inria.

If you look at the Isabelle (https://isabelle.in.tum.de/overview.html https://isabelle.in.tum.de/overview.html) and Coq (https://coq.inria.fr https://coq.inria.fr) proof assistants, you can see that enumerating all possible expressions F and automatically verify if there is a proof of length less than n, has not turned out to be a fruitful approach. These tools do not even particularly try to do that ... ;-)