How can eta-reduction of a well typed function result in a type error?

Question

I was playing around with van Laarhoven lenses and ran into a problem where the type-checker rejects the eta-reduced form of a well-typed function:

{-# LANGUAGE          


        

Answer 1

I'd say that the reason isn't in the η-reduction itself, the problem is that with RankNTypes you lose principal types and type inference.

The problem with type inference with higher-order ranks is when inferring the type of λx.M to obey the rule

     Γ, x:σ |- M:ρ
----------------------
  Γ |- λx:σ.M : σ→ρ

we don't know what type σ we should choose for x. In the case of Hindley-Milner type system, we limit ourselves to type-quantifier-free types for x and the inference is possible, but not with arbitrary ranked types.

So even with RankNTypes, when the compiler encounters a term without explicit type information, it resorts to Hindley-Milner and infers its rank-1 principal type. However, in your case the type you need for getWith id is rank-2 and so compiler can't infer it by itself.

Your explicit case

get lens = getWith id lens

corresponds to the situation where the type of x is already given explicitly λ(x:σ).Mx. The compiler knows the type of lens before type-checking getWith id lens.

In the reduced case

get = getWith id

the compiler has to infer the type of getWidth id on it's own, so it sticks with Hindley-Milner and infers the inadequate rank-1 type.