```------------------------------------------------------------------------
-- The Agda standard library
--
-- Code for converting Vec A n → B to and from n-ary functions
------------------------------------------------------------------------

{-# OPTIONS --cubical-compatible --safe #-}

module Data.Vec.N-ary where

open import Axiom.Extensionality.Propositional using (Extensionality)
open import Function.Bundles using (_↔_; Inverse; mk↔′)
open import Data.Nat.Base hiding (_⊔_)
open import Data.Product.Base as Prod using (∃; _,_)
open import Data.Vec.Base using (Vec; []; _∷_; head; tail)
open import Function.Base using (_∘_; id; flip; constᵣ)
open import Function.Bundles using (_⇔_; mk⇔)
open import Level using (Level; _⊔_)
open import Relation.Binary.Core using (REL)
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)

private
variable
a b c ℓ ℓ₁ ℓ₂ : Level
A : Set a
B : Set b
C : Set c

------------------------------------------------------------------------
-- N-ary functions

N-ary-level : Level → Level → ℕ → Level
N-ary-level ℓ₁ ℓ₂ zero    = ℓ₂
N-ary-level ℓ₁ ℓ₂ (suc n) = ℓ₁ ⊔ N-ary-level ℓ₁ ℓ₂ n

N-ary : ∀ (n : ℕ) → Set ℓ₁ → Set ℓ₂ → Set (N-ary-level ℓ₁ ℓ₂ n)
N-ary zero    A B = B
N-ary (suc n) A B = A → N-ary n A B

------------------------------------------------------------------------
-- Conversion

curryⁿ : ∀ {n} → (Vec A n → B) → N-ary n A B
curryⁿ {n = zero}  f = f []
curryⁿ {n = suc n} f = λ x → curryⁿ (f ∘ _∷_ x)

infix -1 _\$ⁿ_

_\$ⁿ_ : ∀ {n} → N-ary n A B → (Vec A n → B)
f \$ⁿ []       = f
f \$ⁿ (x ∷ xs) = f x \$ⁿ xs

------------------------------------------------------------------------
-- Quantifiers

module _ {A : Set a} where

-- Universal quantifier.

∀ⁿ : ∀ n → N-ary n A (Set ℓ) → Set (N-ary-level a ℓ n)
∀ⁿ zero    P = P
∀ⁿ (suc n) P = ∀ x → ∀ⁿ n (P x)

-- Universal quantifier with implicit (hidden) arguments.

∀ⁿʰ : ∀ n → N-ary n A (Set ℓ) → Set (N-ary-level a ℓ n)
∀ⁿʰ zero    P = P
∀ⁿʰ (suc n) P = ∀ {x} → ∀ⁿʰ n (P x)

-- Existential quantifier.

∃ⁿ : ∀ n → N-ary n A (Set ℓ) → Set (N-ary-level a ℓ n)
∃ⁿ zero    P = P
∃ⁿ (suc n) P = ∃ λ x → ∃ⁿ n (P x)

------------------------------------------------------------------------
-- N-ary function equality

Eq : ∀ {A : Set a} {B : Set b} {C : Set c} n →
REL B C ℓ → REL (N-ary n A B) (N-ary n A C) (N-ary-level a ℓ n)
Eq n _∼_ f g = ∀ⁿ n (curryⁿ {n = n} λ xs → (f \$ⁿ xs) ∼ (g \$ⁿ xs))

-- A variant where all the arguments are implicit (hidden).

Eqʰ : ∀ {A : Set a} {B : Set b} {C : Set c} n →
REL B C ℓ → REL (N-ary n A B) (N-ary n A C) (N-ary-level a ℓ n)
Eqʰ n _∼_ f g = ∀ⁿʰ n (curryⁿ {n = n} λ xs → (f \$ⁿ xs) ∼ (g \$ⁿ xs))

------------------------------------------------------------------------
-- Some lemmas

-- The functions curryⁿ and _\$ⁿ_ are inverses.

left-inverse : ∀ {n} (f : Vec A n → B) →
∀ xs → (curryⁿ f \$ⁿ xs) ≡ f xs
left-inverse f []       = refl
left-inverse f (x ∷ xs) = left-inverse (f ∘ _∷_ x) xs

right-inverse : ∀ n (f : N-ary n A B) →
Eq n _≡_ (curryⁿ (_\$ⁿ_ {n = n} f)) f
right-inverse zero    f = refl
right-inverse (suc n) f = λ x → right-inverse n (f x)

-- ∀ⁿ can be expressed in an "uncurried" way.

uncurry-∀ⁿ : ∀ n {P : N-ary n A (Set ℓ)} →
∀ⁿ n P ⇔ (∀ (xs : Vec A n) → P \$ⁿ xs)
uncurry-∀ⁿ {a} {A} {ℓ} n = mk⇔ (⇒ n) (⇐ n)
where
⇒ : ∀ n {P : N-ary n A (Set ℓ)} →
∀ⁿ n P → (∀ (xs : Vec A n) → P \$ⁿ xs)
⇒ zero    p []       = p
⇒ (suc n) p (x ∷ xs) = ⇒ n (p x) xs

⇐ : ∀ n {P : N-ary n A (Set ℓ)} →
(∀ (xs : Vec A n) → P \$ⁿ xs) → ∀ⁿ n P
⇐ zero    p = p []
⇐ (suc n) p = λ x → ⇐ n (p ∘ _∷_ x)

-- ∃ⁿ can be expressed in an "uncurried" way.

uncurry-∃ⁿ : ∀ n {P : N-ary n A (Set ℓ)} →
∃ⁿ n P ⇔ (∃ λ (xs : Vec A n) → P \$ⁿ xs)
uncurry-∃ⁿ {a} {A} {ℓ} n = mk⇔ (⇒ n) (⇐ n)
where
⇒ : ∀ n {P : N-ary n A (Set ℓ)} →
∃ⁿ n P → (∃ λ (xs : Vec A n) → P \$ⁿ xs)
⇒ zero    p       = ([] , p)
⇒ (suc n) (x , p) = Prod.map (_∷_ x) id (⇒ n p)

⇐ : ∀ n {P : N-ary n A (Set ℓ)} →
(∃ λ (xs : Vec A n) → P \$ⁿ xs) → ∃ⁿ n P
⇐ zero    ([] , p)     = p
⇐ (suc n) (x ∷ xs , p) = (x , ⇐ n (xs , p))

-- Conversion preserves equality.

module _ (_∼_ : REL B C ℓ) where

curryⁿ-cong : ∀ {n} (f : Vec A n → B) (g : Vec A n → C) →
(∀ xs → f xs ∼ g xs) →
Eq n _∼_ (curryⁿ f) (curryⁿ g)
curryⁿ-cong {n = zero}  f g hyp = hyp []
curryⁿ-cong {n = suc n} f g hyp = λ x →
curryⁿ-cong (f ∘ _∷_ x) (g ∘ _∷_ x) (λ xs → hyp (x ∷ xs))

curryⁿ-cong⁻¹ : ∀ {n} (f : Vec A n → B) (g : Vec A n → C) →
Eq n _∼_ (curryⁿ f) (curryⁿ g) →
∀ xs → f xs ∼ g xs
curryⁿ-cong⁻¹ f g hyp []       = hyp
curryⁿ-cong⁻¹ f g hyp (x ∷ xs) =
curryⁿ-cong⁻¹ (f ∘ _∷_ x) (g ∘ _∷_ x) (hyp x) xs

appⁿ-cong : ∀ {n} (f : N-ary n A B) (g : N-ary n A C) →
Eq n _∼_ f g →
(xs : Vec A n) → (f \$ⁿ xs) ∼ (g \$ⁿ xs)
appⁿ-cong f g hyp []       = hyp
appⁿ-cong f g hyp (x ∷ xs) = appⁿ-cong (f x) (g x) (hyp x) xs

appⁿ-cong⁻¹ : ∀ {n} (f : N-ary n A B) (g : N-ary n A C) →
((xs : Vec A n) → (f \$ⁿ xs) ∼ (g \$ⁿ xs)) →
Eq n _∼_ f g
appⁿ-cong⁻¹ {n = zero}  f g hyp = hyp []
appⁿ-cong⁻¹ {n = suc n} f g hyp = λ x →
appⁿ-cong⁻¹ (f x) (g x) (λ xs → hyp (x ∷ xs))

-- Eq and Eqʰ are equivalent.

Eq-to-Eqʰ : ∀ n (_∼_ : REL B C ℓ) {f : N-ary n A B} {g : N-ary n A C} →
Eq n _∼_ f g → Eqʰ n _∼_ f g
Eq-to-Eqʰ zero    _∼_ eq = eq
Eq-to-Eqʰ (suc n) _∼_ eq = Eq-to-Eqʰ n _∼_ (eq _)

Eqʰ-to-Eq : ∀ n (_∼_ : REL B C ℓ) {f : N-ary n A B} {g : N-ary n A C} →
Eqʰ n _∼_ f g → Eq n _∼_ f g
Eqʰ-to-Eq zero    _∼_ eq = eq
Eqʰ-to-Eq (suc n) _∼_ eq = λ _ → Eqʰ-to-Eq n _∼_ eq

module _ (ext : ∀ {a b} → Extensionality a b) where

Vec↔N-ary : ∀ n → (Vec A n → B) ↔ N-ary n A B
Vec↔N-ary zero = mk↔′ (λ vxs → vxs []) (flip constᵣ) (λ _ → refl)
(λ vxs → ext λ where [] → refl)
Vec↔N-ary (suc n) = let open Inverse (Vec↔N-ary n) in
mk↔′ (λ vxs x → to λ xs → vxs (x ∷ xs))
(λ any xs → from (any (head xs)) (tail xs))
(λ any → ext λ x → strictlyInverseˡ _)
(λ vxs → ext λ where (x ∷ xs) → cong (λ f → f xs) (strictlyInverseʳ (λ ys → vxs (x ∷ ys))))
```