Anyone here use Qi, Mercury, or Agda? I’m interested in what people think of those languages.
Qi is interesting but the development community is much smaller than its competitors (mainly Haskell). You probably want to look into Shen, not Qi. Qi development has mostly stalled.
(As far as I know. I've only worked through Pierce's book.)
That's where you are wrong. Partial functions introduce the bottom value and then you can 'prove' arbitrary incorrect things.