Dependent telescopes

Content created by Egbert Rijke and Fredrik Bakke.

Created on 2023-10-22.
Last modified on 2023-10-22.

module foundation.dependent-telescopes where
Imports 
open import elementary-number-theory.multiplication-natural-numbers
open import elementary-number-theory.natural-numbers

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

Idea

A dependent telescope over a telescope A of length n is a dependent list of dependent types over each of the entries in A. For example, a dependent telescope over the telescope

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

consists of

  B₀ : A₀ → 𝒰 k₀
  B₁ : (x₀ : A₀) (x₁ : A₁ x₀) → B₀ x₀ → UU k₁
  B₂ : (x₀ : A₀) (x₁ : A₁ x₀) (x₂ : A₂ x₀ x₁) (y₀ : B x₀) → B₁ x₀ → UU k₂

Definitions

Dependent telescopes

data
  dependent-telescope :
    {l : Level} (k : Level) → {n : ℕ} → telescope l n → UUω
  where
  base-dependent-telescope :
    {l1 k1 : Level} {A : UU l1} → (A → UU k1) →
    dependent-telescope k1 (base-telescope A)
  cons-dependent-telescope :
    {l1 l2 k1 k2 : Level} {n : ℕ} {X : UU l1} {A : X → telescope l2 n}
    {Y : X → UU k1} (B : (x : X) → Y x → dependent-telescope k2 (A x)) →
    dependent-telescope (k1 ⊔ k2) (cons-telescope A)

Expansion of telescopes

An expansion of a telescope A by a dependent telescope B over it is a new telescope of the same length as A, constructed by taking dependent pairs componentwise.

expand-telescope :
  {l1 l2 : Level} {n : ℕ} {A : telescope l1 n} →
  dependent-telescope l2 A → telescope (l1 ⊔ l2) n
expand-telescope (base-dependent-telescope Y) =
  base-telescope (Σ _ Y)
expand-telescope (cons-dependent-telescope B) =
  cons-telescope (λ x → expand-telescope (B (pr1 x) (pr2 x)))

Interleaving telescopes

Given a telescope A of length n and a dependent telescope B over it, we can define the interleaved telescope whose length is 2n.

interleave-telescope :
  {l1 l2 : Level} {n : ℕ} {A : telescope l1 n} →
  dependent-telescope l2 A → telescope (l1 ⊔ l2) (succ-ℕ (n *ℕ 2))
interleave-telescope (base-dependent-telescope A) =
  cons-telescope (λ x → base-telescope (A x))
interleave-telescope (cons-dependent-telescope B) =
  cons-telescope (λ x → cons-telescope λ y → interleave-telescope (B x y))

Mapping telescopes

Given a telescope A and a dependent telescope B over it, we can define the mapping telescope by taking dependent function types componentwise.

telescope-Π :
  {l1 l2 : Level} {n : ℕ} {A : telescope l1 n} →
  dependent-telescope l2 A → telescope (l1 ⊔ l2) n
telescope-Π (base-dependent-telescope Y) =
  base-telescope ((x : _) → Y x)
telescope-Π (cons-dependent-telescope B) =
  cons-telescope (λ x → telescope-Π (B (pr1 x) (pr2 x)))

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