There is RedPRL, there is also cubical Agda. I think cubical Agda is the best developed so far. It lacks automation. But that is not fundamental, it is thanks to the philosophies of the people involved.
Lean is not weak, it just commits to something called Uniqueness of Identity Proofs (UIP). This is inconsistent with Univalence, a property that cubical type theories have. Necessarily, neither system can express "all of math." So it goes.
Leo chose UIP to get a novel way to get really powerful automation of equality proofs. Leo is very, very good at this; this is his whole shindig. But he had to hack around things a bit in order to get what he wanted. The paper by Bas and his student shows that in cubical, you don't have to hack around things; this automation of equality proofs arises in the natural way.
The gap is not fundamental, and it irritates me when it is attributed to the choice of UIP. The reason Lean is so useful is because the people working on it are good at building automation. The people working on cubical are much more interested in foundations. We need people who are interested in automating cubical; once we get that, the proof assistant that arises from it will be better than anything we have ever seen.
Do you have any introductory material on cubical type theory, suitable for the average mathematician? The type theory in Lean feels dead simple to me, but all the online resources on cubical type theory feel impenetrable.
Lean is not weak, it just commits to something called Uniqueness of Identity Proofs (UIP). This is inconsistent with Univalence, a property that cubical type theories have. Necessarily, neither system can express "all of math." So it goes.
Leo chose UIP to get a novel way to get really powerful automation of equality proofs. Leo is very, very good at this; this is his whole shindig. But he had to hack around things a bit in order to get what he wanted. The paper by Bas and his student shows that in cubical, you don't have to hack around things; this automation of equality proofs arises in the natural way.
The gap is not fundamental, and it irritates me when it is attributed to the choice of UIP. The reason Lean is so useful is because the people working on it are good at building automation. The people working on cubical are much more interested in foundations. We need people who are interested in automating cubical; once we get that, the proof assistant that arises from it will be better than anything we have ever seen.