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Thanks for pointing this out! I was vaguely aware that GHC used some form of evidence of type equality, but I didn't know any specifics. However, skimming through the presentation you linked, it seems that System FC is used mainly during compilation, by the GHC intermediate language. In my experiments, however, I focused on type systems with a form of type inference, which System F (and its extensions) lacks. There are ways of type inference for GADTs, but none of them is complete.
I've recently been looking around at System FC, which is what is used in haskell. I don't know if you've heard of it but it's interesting.
https://www.youtube.com/watch?v=2IZQx7WNOMs
https://www.cs.uoregon.edu/research/summerschool/summer13/lectures/FC_in_GHC_July13.pdf
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