module Data.Product.NAry where
n-Ary products🔗
This is an alternative definition of the type of n-ary products,
which, unlike Vec, is
defined by recursion on the length.
Vecₓ : ∀ {ℓ} → Type ℓ → Nat → Type ℓ Vecₓ A 0 = Lift _ ⊤ Vecₓ A 1 = A Vecₓ A (suc (suc n)) = A × Vecₓ A (suc n)
The point of having the type Vecₓ around is that its
inhabitants can be written using the usual tuple syntax, so that we can
define a From-product
“type class” for those type constructors (possibly indexed by the
length) which can be built as a sequence of arguments.
_ : Vecₓ Nat 3 _ = 1 , 2 , 3 record From-product {ℓ} (A : Type ℓ) (T : Nat → Type ℓ) : Type ℓ where field from-prod : ∀ n → Vecₓ A n → T n
Shuffling the arguments to from-prod slightly, we
arrive at the “list syntax” construction:
[_] : ∀ {ℓ} {A : Type ℓ} {T : Nat → Type ℓ} {n} ⦃ fp : From-product A T ⦄ → Vecₓ A n → T n [_] ⦃ fp = fp ⦄ = fp .From-product.from-prod _