Dear Haskell-café, I don't know if this is the best forum to ask questions about the OutsideIn(X) paper that lies below type inference in GHC. But given that this is the place where I know many GHC devs look at, I'll just try :) My question related about the proof of soundness and principality, specifically Lemma 7.2 (to be found in page 67). In that lemma, it's stated that QQ and \phi' Q_q ||- \phi Q_w <-> \phi' Q_w'. I'm trying to recover the proof (which is omitted in the text), but I stumble upon a wall when trying to work out what happens in the case an axiom is applied. In particular, I'm playing with an example where QQ (the set of axioms) = { forall. C a => D a } (where C and D are one-parameter type classes) Q_q = { } Q_w = { D Int } Thus, if I apply the rule DINSTW (to be found in page 65), I get a new Q_w' = { C Int } Now, if the lemma 7.2 is true, it should be the case that (1) QQ ||- C Int <-> D Int which in particular means that I have the two implications (2) { forall. C a => D a, C Int } ||- D Int (3) { forall. C a => D a, D Int } ||- C Int (2) follows easily by applying the AXIOM rule of ||- (as shown in page 54). However, I don't see how to make (3) work :( I think that understanding this example will be key for my understanding of the whole system. Anybody could point to the error in my reasoning or to places where I could find more information? Thanks in advance.