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Category：Languages

So I was asked whether these 3 type expressions where equivalent in Haskell:

`τ1 = (a -> a) -> (a -> a -> a) τ2 = a -> a -> ((a -> a) -> a) τ3 = a -> a -> (a -> (a -> a)) `

if I take away the parenthesis I'm left with this

`τ1 = (a -> a) -> a -> a -> a τ2 = a -> a -> (a -> a) -> a τ3 = a -> a -> a -> a -> a `

So it's obvious to me that they are all different from each other. However, according to the question, these two answers are wrong:

`τ1 !≡ τ2 !≡ τ3 !≡ τ1 τ1 !≡ τ2 ≡ τ3 `

So I'm a bit confused right here, what would be the right answer and why?

Indeed, they are all distinct for the reason you mention.

We can even ask GHC to confirm it. (Below, I chose `a ~ Int`

to obtain a closed type.)

`> import Data.Type.Equality > type T1 a = (a -> a) -> (a -> a -> a) > type T2 a = a -> a -> ((a -> a) -> a) > type T3 a = a -> a -> (a -> (a -> a)) > :kind! T1 Int == T2 Int T1 Int == T2 Int :: Bool = 'False > :kind! T1 Int == T3 Int T1 Int == T3 Int :: Bool = 'False > :kind! T2 Int == T3 Int T2 Int == T3 Int :: Bool = 'False `