Products of tuples of types

module foundation.products-of-tuples-of-types where

open import elementary-number-theory.natural-numbers

open import foundation.tuples-of-types

open import foundation-core.universe-levels

open import univalent-combinatorics.standard-finite-types

Idea

The product of an n-tuple of types is just the dependent product.

Definition

Products of n-tuples of types

product-tuple-types :
  {l : Level} (n : ℕ) → tuple-types l n → UU l
product-tuple-types n A = (i : Fin n) → A i

The projection maps

pr-product-tuple-types :
  {l : Level} {n : ℕ} (A : tuple-types l n) (i : Fin n) →
  product-tuple-types n A → A i
pr-product-tuple-types A i f = f i

{-
equiv-universal-property-product-tuple-types :
  {l : Level} {n : ℕ} (A : tuple-types l (succ-ℕ n)) (i : Fin (succ-ℕ n)) →
  ( product-tuple-types (succ-ℕ n) A) ≃
  ( ( product-tuple-types n {!!}) × A i)
equiv-universal-property-product-tuple-types A i =
  {!!}
  -}

Properties