I hope you can help me confirm my understanding of the order of kinds and rank of types as reported in [1,2]. I have been validating the mapping functions in Section 2 against the GHC 7.4.2 type checker by explicitly annotating the universal quantifiers as described in [1]. 1. Just to baseline my understanding, given the following data type data U a = UC (forall a. a) -- :k U, U :: * -> * -- order of kind = 1 -- :t UC, UC :: (forall a1. a1) -> U a -- rank of type = 2 am I correct that the order of the kind is 1 and the rank of the type constructor is 2? See page 3 of [1] for counting the order and also page 3 of [2] for counting rank. 2. Assuming the answer to #1 above is yes, am I also correct that the order of Fix is 2 and the rank of In is also 2 and the annotation to In does not affect its rank? newtype Fix f = In (forall f. f (Fix f)) -- :k Fix, Fix :: (* -> *) -> * -- order of kind = 2 -- :t In, In :: (forall (f1 :: * -> *). f1 (Fix f1)) -> Fix f -- rank of type still = 2 3. Are the explicit annotations I added to the following fixpoint operator correct? Hinze indicates this is a third order kind and it seems that although the annotation on HIn has a higher order, the rank of the type hasn't changed, correct? newtype HFix h a = HIn (forall h. (forall a. h (HFix h) a)) -- :k HFix, HFix :: ((* -> *) -> * -> *) -> * -> * -- per hinze order of kind = 3 -- :t HIn, HIn :: (forall (h1 :: (* -> *) -> * -> *) a1. h1 (HFix h1) a1) -> HFix h a -- still rank of type = 2 because there's one quantifier to the left of the function arrow. If all that all checks out, I have a few follow up questions: 4. Hinze reports that mapHFix is Rank3, but the following type checks with the Rank2Type pragma and if I remove the explicit universal quantifiers in HIn. Put the quantifiers back in an it does not type check. mapHFix maph mapa (HIn v) = HIn (maph (mapHFix maph) mapa v) 5. Also, the inferred signature for mapHFix differs from the one stated by Hinze. It's this ... mapHFix :: ((t2 -> HFix t t1 -> HFix h a) -> t2 -> t (HFix t) t1 -> h (HFix h) a) -> t2 -> HFix t t1 -> HFix h a What is the rank of this type signature? The type variables are not in the same order as Hinze's and I was able to just wrap a (forall t2 t t1 h a.) around the whole thing and it type checked. Why are they different? 6. What is the correct type signature for mapHFix? The correct type signature should support changing the pragma from Rank2Types to RankNTypes, then today's 7.4.2 type checker will succeed where it could not in 1999 when the paper was written, correct? Thanks so much for helping me confirm my understanding of these issues! -- dude 1. Polytypic Values Posess Polykinded Types. (Hinze, 1999) 2. A Direct Algorithm for Type Inference in the Rank 2 Fragment of the Second Order Lambda Calculus. (Kfoury, 1993)