I tend to reason about these kind higher kind programs using Agda.
The problem here is that you want to pattern match on *
(Set
in Agda), violate parametericity, as mentioned in the comment. That is not good, so you cannot just do it. You have to provide witness. I.e. following is not possible
P : Set → Set → Set
P Unit b = b
P a b = a × b
You can overcome the limitiation by using aux type:
P : Aux → Set → Set
P auxunit b = b
P (auxpair a) b = a × b
Or in Haskell:
data Aux x a = AuxUnit x | AuxPair x a
type family P (x :: Aux * *) :: * where
P (AuxUnit x) = x
P (AuxPair x a) = (x, a)
But doing so you'll have problems expressing T
, as you need to pattern match on its parameter again, to select right Aux
constructor.
"Simple" solution, is to express T a ~ Integer
when a ~ ()
, and T a ~ (Integer, a)
directly:
module fmap where
record Unit : Set where
constructor tt
data ⊥ : Set where
data Nat : Set where
zero : Nat
suc : Nat → Nat
data _≡_ {ℓ} {a : Set ℓ} : a → a → Set ℓ where
refl : {x : a} → x ≡ x
¬_ : ∀ {ℓ} → Set ℓ → Set ℓ
¬ x = x → ⊥
-- GADTs
data T : Set → Set1 where
tunit : Nat → T Unit
tpair : (a : Set) → ¬ (a ≡ Unit) → a → T a
test : T Unit → Nat
test (tunit x) = x
test (tpair .Unit contra _) with contra refl
test (tpair .Unit contra x) | ()
You can try to encode this in Haskell to.
You can express it using e.g. 'idiomatic' Haskell type inequality
I'll leave the Haskell version as an exercise :)
Hmm or did you meant by data T a = T Integer (P (T a) a)
:
T () ~ Integer × (P (T ()) ())
~ Integer × (T ())
~ Integer × Integer × ... -- infinite list of integers?
-- a /= ()
T a ~ Integer × (P (T a) a)
~ Integer × (T a × a) ~ Integer × T a × a
~ Integer × Integer × ... × a × a
Those are easier to encode directly as well.