# Iterated dependent pair types

Content created by Fredrik Bakke and Egbert Rijke.

Created on 2023-10-22.

module foundation.iterated-dependent-pair-types where

open import foundation.telescopes public

Imports
open import elementary-number-theory.natural-numbers

open import foundation.dependent-pair-types
open import foundation.universe-levels

open import foundation-core.cartesian-product-types
open import foundation-core.contractible-types


## Idea

Iterated dependent pair types are defined by iteratively applying the dependent pair operator Σ. More formally, iterated-Σ is defined as an operation telescope l n → UU l from the type of telescopes to the universe of types of universe level l. For example, the iterated dependent pair type of the telescope

  A₀ : 𝒰 l₀
A₁ : A₀ → 𝒰 l₁
A₂ : (x₀ : A₀) → A₁ x₀ → 𝒰 l₂
A₃ : (x₀ : A₀) (x₁ : A₁ x₀) → A₂ x₀ x₁ → 𝒰 l₃


is the dependent pair type

  Σ A₀ (λ x₀ → Σ (A₁ x₀) (λ x₁ → Σ (A₂ x₀ x₁) (A₃ x₀ x₁)))


of universe level l₀ ⊔ l₁ ⊔ l₂ ⊔ l₃.

## Definitions

### Iterated dependent products of iterated type families

iterated-Σ : {l : Level} {n : ℕ} → telescope l n → UU l
iterated-Σ (base-telescope A) = A
iterated-Σ (cons-telescope {X = X} A) = Σ X (λ x → iterated-Σ (A x))


### Iterated elements of iterated type families

data
iterated-element : {l : Level} {n : ℕ} → telescope l n → UUω
where
base-iterated-element :
{l1 : Level} {A : UU l1} → A → iterated-element (base-telescope A)
cons-iterated-element :
{l1 l2 : Level} {n : ℕ} {X : UU l1} {Y : X → telescope l2 n} →
(x : X) → iterated-element (Y x) → iterated-element (cons-telescope Y)


### Iterated pairing

iterated-pair :
{l : Level} {n : ℕ} {A : telescope l n} →
iterated-element A → iterated-Σ A
iterated-pair (base-iterated-element x) = x
pr1 (iterated-pair (cons-iterated-element y a)) = y
pr2 (iterated-pair (cons-iterated-element y a)) = iterated-pair a


## Properties

### Contractiblity of iterated Σ-types

is-contr-Σ-telescope : {l : Level} {n : ℕ} → telescope l n → UU l
is-contr-Σ-telescope (base-telescope A) = is-contr A
is-contr-Σ-telescope (cons-telescope {X = X} A) =
(is-contr X) × (Σ X (λ x → is-contr-Σ-telescope (A x)))

is-contr-iterated-Σ :
{l : Level} (n : ℕ) {{A : telescope l n}} →
is-contr-Σ-telescope A → is-contr (iterated-Σ A)
is-contr-iterated-Σ .0 {{base-telescope A}} is-contr-A = is-contr-A
is-contr-iterated-Σ ._ {{cons-telescope A}} (is-contr-X , x , H) =
is-contr-Σ is-contr-X x (is-contr-iterated-Σ _ {{A x}} H)


### Contractiblity of iterated Σ-types without choice of contracting center

is-contr-Σ-telescope' : {l : Level} {n : ℕ} → telescope l n → UU l
is-contr-Σ-telescope' (base-telescope A) = is-contr A
is-contr-Σ-telescope' (cons-telescope {X = X} A) =
(is-contr X) × ((x : X) → is-contr-Σ-telescope' (A x))

is-contr-iterated-Σ' :
{l : Level} (n : ℕ) {{A : telescope l n}} →
is-contr-Σ-telescope' A → is-contr (iterated-Σ A)
is-contr-iterated-Σ' .0 {{base-telescope A}} is-contr-A = is-contr-A
is-contr-iterated-Σ' ._ {{cons-telescope A}} (is-contr-X , H) =
is-contr-Σ' is-contr-X (λ x → is-contr-iterated-Σ' _ {{A x}} (H x))