Here’s a quick reference for the state of the art on homotopy groups? of spheres in HoTT. Everything listed here is also discussed on the page on Formalized Homotopy Theory.
Guillaume Brunerie has proved that there exists an such that is . Given a computational interpretation, we could run this proof and check that is 2. Added June 2016: Brunerie now has a proof that , using cohomology calculations and a Gysin sequence argument.
Calculuation of
Peter LeFanu Lumsdaine has constructed the Hopf fibration as a dependent type. Lots of people around know the construction, but I don’t know anywhere it’s written up. Here’s some Agda code with it in it.
Guillaume Brunerie’s proof that the total space of the Hopf fibration is , together with , imply this by a long-exact-sequence argument.
Mike Shulman’s proof by contractibility of total space of universal cover (HoTT blog).
Dan Licata’s encode/decode-style proof (HoTT blog). A paper mostly about the encode/decode-style proof, but also describing the relationship between the two.