```------------------------------------------------------------------------
-- Products
------------------------------------------------------------------------

{-# OPTIONS --universe-polymorphism #-}

module Data.Product where

open import Data.Function
open import Level
open import Relation.Nullary.Core

infixr 4 _,_
infix  4 ,_
infixr 2 _×_ _-×-_ _-,-_

------------------------------------------------------------------------
-- Definition

data Σ {a b} (A : Set a) (B : A → Set b) : Set (a ⊔ b) where
_,_ : (x : A) (y : B x) → Σ A B

∃ : ∀ {a b} {A : Set a} → (A → Set b) → Set (a ⊔ b)
∃ = Σ _

∄ : ∀ {a b} {A : Set a} → (A → Set b) → Set (a ⊔ b)
∄ P = ¬ ∃ P

∃₂ : ∀ {a b c} {A : Set a} {B : A → Set b}
(C : (x : A) → B x → Set c) → Set (a ⊔ b ⊔ c)
∃₂ C = ∃ λ a → ∃ λ b → C a b

_×_ : ∀ {a b} (A : Set a) (B : Set b) → Set (a ⊔ b)
A × B = Σ A (λ _ → B)

------------------------------------------------------------------------
-- Unique existence

-- Parametrised on the underlying equality.

∃! : ∀ {a b ℓ} {A : Set a} →
(A → A → Set ℓ) → (A → Set b) → Set (a ⊔ b ⊔ ℓ)
∃! _≈_ B = ∃ λ x → B x × (∀ {y} → B y → x ≈ y)

------------------------------------------------------------------------
-- Functions

-- Sometimes the first component can be inferred.

,_ : ∀ {a b} {A : Set a} {B : A → Set b} {x} → B x → ∃ B
, y = _ , y

proj₁ : ∀ {a b} {A : Set a} {B : A → Set b} → Σ A B → A
proj₁ (x , y) = x

proj₂ : ∀ {a b} {A : Set a} {B : A → Set b} →
(p : Σ A B) → B (proj₁ p)
proj₂ (x , y) = y

<_,_> : ∀ {a b c} {A : Set a} {B : A → Set b} {C : ∀ {x} → B x → Set c}
(f : (x : A) → B x) → ((x : A) → C (f x)) →
((x : A) → Σ (B x) C)
< f , g > x = (f x , g x)

map : ∀ {a b p q}
{A : Set a} {B : Set b} {P : A → Set p} {Q : B → Set q} →
(f : A → B) → (∀ {x} → P x → Q (f x)) →
Σ A P → Σ B Q
map f g = < f ∘ proj₁ , g ∘ proj₂ >

zip : ∀ {a b c p q r}
{A : Set a} {B : Set b} {C : Set c}
{P : A → Set p} {Q : B → Set q} {R : C → Set r} →
(_∙_ : A → B → C) →
(∀ {x y} → P x → Q y → R (x ∙ y)) →
Σ A P → Σ B Q → Σ C R
zip _∙_ _∘_ p₁ p₂ = (proj₁ p₁ ∙ proj₁ p₂ , proj₂ p₁ ∘ proj₂ p₂)

swap : ∀ {a b} {A : Set a} {B : Set b} → A × B → B × A
swap = < proj₂ , proj₁ >

_-×-_ : ∀ {a b i j} {A : Set a} {B : Set b} →
(A → B → Set i) → (A → B → Set j) → (A → B → Set _)
f -×- g = f -[ _×_ ]- g

_-,-_ : ∀ {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} →
(A → B → C) → (A → B → D) → (A → B → C × D)
f -,- g = f -[ _,_ ]- g

curry : ∀ {a b c} {A : Set a} {B : A → Set b} {C : Σ A B → Set c} →
((p : Σ A B) → C p) →
((x : A) → (y : B x) → C (x , y))
curry f x y = f (x , y)

uncurry : ∀ {a b c} {A : Set a} {B : A → Set b} {C : Σ A B → Set c} →
((x : A) → (y : B x) → C (x , y)) →
((p : Σ A B) → C p)
uncurry f (p₁ , p₂) = f p₁ p₂
```