{-# OPTIONS --cubical --safe #-}
module Cubical.Data.Prod.Base where

open import Cubical.Core.Everything

open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Function

-- If × is defined using Σ then transp/hcomp will be compute
-- "negatively", that is, they won't reduce unless we project out the
-- first of second component. This is not always what we want so the
-- default implementation is done using a datatype which computes
-- positively.


private
  variable
     ℓ' : Level

data _×_ (A : Type ) (B : Type ℓ') : Type (ℓ-max  ℓ') where
  _,_ : A  B  A × B

infixr 5 _×_

proj₁ : {A : Type } {B : Type ℓ'}  A × B  A
proj₁ (x , _) = x

proj₂ : {A : Type } {B : Type ℓ'}  A × B  B
proj₂ (_ , x) = x

-- We still export the version using Σ
_×Σ_ : (A : Type ) (B : Type ℓ')  Type (ℓ-max  ℓ')
A ×Σ B = Σ A  _  B)

infixr 5 _×Σ_

private
  variable
    A    : Type 
    B C  : A  Type 

intro : (∀ a  B a)  (∀ a  C a)   a  B a × C a
intro f g a = f a , g a

map : {B : Type } {D : B  Type ℓ'}
    (∀ a  C a)  (∀ b  D b)  (x : A × B)  C (proj₁ x) × D (proj₂ x)
map f g = intro (f  proj₁) (g  proj₂)