Cartesian product types

Content created by Egbert Rijke, Fredrik Bakke and Jonathan Prieto-Cubides.

Created on 2022-01-26.
Last modified on 2024-02-06.

module foundation.cartesian-product-types where

open import foundation-core.cartesian-product-types public
open import foundation.dependent-pair-types
open import foundation.subuniverses
open import foundation.universe-levels

open import foundation-core.identity-types
open import foundation-core.transport-along-identifications


Transport in a family of cartesian products

tr-product :
  {l1 l2 : Level} {A : UU l1} {a0 a1 : A}
  (B C : A  UU l2) (p : a0  a1) (u : B a0 × C a0) 
  (tr  a  B a × C a) p u)  (pair (tr B p (pr1 u)) (tr C p (pr2 u)))
tr-product B C refl u = refl

Subuniverses closed under cartesian product types

is-closed-under-products-subuniverses :
  {l1 l2 l3 l4 l5 : Level} (P : subuniverse l1 l2) (Q : subuniverse l3 l4)
  (R : subuniverse (l1  l3) l5)  UU (lsuc l1  l2  lsuc l3  l4  l5)
is-closed-under-products-subuniverses {l1} {l2} {l3} P Q R =
  {X : UU l1} {Y : UU l3} 
  is-in-subuniverse P X  is-in-subuniverse Q Y  is-in-subuniverse R (X × Y)

is-closed-under-products-subuniverse :
  {l1 l2 : Level} (P : subuniverse l1 l2)  UU (lsuc l1  l2)
is-closed-under-products-subuniverse P =
  is-closed-under-products-subuniverses P P P

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