# Telescopes

Content created by Fredrik Bakke and Egbert Rijke.

Created on 2023-10-22.

module foundation.telescopes where

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

open import foundation.universe-levels


## Idea

A telescope, or iterated type family, is a list of type families (A₀, A₁, A₂, ..., A_n) consisting of

• a type A₀,
• a type family A₁ : A₀ → 𝒰,
• a type family A₂ : (x₀ : A₀) → A₁ x₀ → 𝒰,
• ...
• a type family A_n : (x₀ : A₀) ... (x_(n-1) : A_(n-1) x₀ ... x_(n-2)) → 𝒰.

We say that a telescope (A₀,...,A_n) has length n+1. In other words, the length of the telescope (A₀,...,A_n) is the length of the (dependent) list (A₀,...,A_n).

We encode the type of telescopes as a family of inductive types

  telescope : (l : Level) → ℕ → UUω


The type of telescopes is a directed tree

  ... → T₃ → T₂ → T₁ → T₀,


where T_n is the type of all telescopes of length n, and the map from T_(n+1) to T_n maps (A₀,...,A_n) to (A₀,...,A_(n-1)). The type of such directed trees can be defined as a coinductive record type, and we will define the tree T of telescopes as a particular element of this tree.

## Definitions

### Telescopes

data
telescope : (l : Level) → ℕ → UUω
where
base-telescope :
{l1 : Level} → UU l1 → telescope l1 0
cons-telescope :
{l1 l2 : Level} {n : ℕ} {X : UU l1} →
(X → telescope l2 n) → telescope (l1 ⊔ l2) (succ-ℕ n)

open telescope public


A very slight reformulation of cons-telescope for convenience:

prepend-telescope :
{l1 l2 : Level} {n : ℕ} →
(A : UU l1) → ({x : A} → telescope l2 n) → telescope (l1 ⊔ l2) (succ-ℕ n)
prepend-telescope A B = cons-telescope {X = A} (λ x → B {x})


### Telescopes at a universe level

One issue with the previous definition of telescopes is that it is impossible to extract any type information from it. At the expense of giving up full universe polymorphism, we can define a notion of telescopes at a universe level that admits such projections. This definition is also compatible with the --level-universe restriction.

data telescope-Level (l : Level) : ℕ → UU (lsuc l)
where
base-telescope-Level :
UU l → telescope-Level l 0
cons-telescope-Level :
{n : ℕ} {X : UU l} →
(X → telescope-Level l n) → telescope-Level l (succ-ℕ n)

open telescope-Level public

telescope-telescope-Level :
{l : Level} {n : ℕ} → telescope-Level l n → telescope l n
telescope-telescope-Level (base-telescope-Level A) =
base-telescope A
telescope-telescope-Level (cons-telescope-Level Γ) =
cons-telescope (λ x → telescope-telescope-Level (Γ x))


### Transformations on telescopes

Given an operation on universes, we can apply it at the base of the telescope.

apply-base-telescope :
{l1 : Level} {n : ℕ}
(P : {l : Level} → UU l → UU l) → telescope l1 n → telescope l1 n
apply-base-telescope P (base-telescope A) = base-telescope (P A)
apply-base-telescope P (cons-telescope A) =
cons-telescope (λ x → apply-base-telescope P (A x))


### Telescopes as instance arguments

To get Agda to infer telescopes, we help it along a little using instance arguments. These are a special kind of implicit argument in Agda that are resolved by the instance resolution algorithm. We register building blocks for this algorithm to use below, i.e. instances. Then Agda will attempt to use those to construct telescopes of the appropriate kind when asked to.

In the case of telescopes, this has the unfortunate disadvantage that we can only define instances for fixed length telescopes. We have defined instances of telescopes up to length 18, so although Agda cannot infer a telescope of a general length using this approach, it can infer them up to this given length.

instance-telescope : {l : Level} {n : ℕ} → {{telescope l n}} → telescope l n
instance-telescope {{x}} = x

instance
instance-telescope⁰ : {l : Level} {X : UU l} → telescope l 0
instance-telescope⁰ {X = X} = base-telescope X

instance-telescope¹ :
{ l1 l : Level} {A1 : UU l1} {X : A1 → UU l} → telescope (l1 ⊔ l) 1
instance-telescope¹ {X = X} =
cons-telescope (λ x → instance-telescope⁰ {X = X x})

instance-telescope² :
{ l1 l2 l : Level} {A1 : UU l1} {A2 : A1 → UU l2}
{ X : (x1 : A1) → A2 x1 → UU l} → telescope (l1 ⊔ l2 ⊔ l) 2
instance-telescope² {X = X} =
cons-telescope (λ x → instance-telescope¹ {X = X x})

instance-telescope³ :
{ l1 l2 l3 l : Level}
{ A1 : UU l1} {A2 : A1 → UU l2} {A3 : (x1 : A1) → A2 x1 → UU l3}
{ X : (x1 : A1) (x2 : A2 x1) (x2 : A3 x1 x2) → UU l} →
telescope (l1 ⊔ l2 ⊔ l3 ⊔ l) 3
instance-telescope³ {X = X} =
cons-telescope (λ x → instance-telescope² {X = X x})

instance-telescope⁴ :
{ l1 l2 l3 l4 l : Level}
{ A1 : UU l1} {A2 : A1 → UU l2} {A3 : (x1 : A1) → A2 x1 → UU l3}
{ A4 : (x1 : A1) (x2 : A2 x1) → A3 x1 x2 → UU l4}
{ X : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → UU l} →
telescope (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ l) 4
instance-telescope⁴ {X = X} =
cons-telescope (λ x → instance-telescope³ {X = X x})

instance-telescope⁵ :
{ l1 l2 l3 l4 l5 l : Level}
{ A1 : UU l1} {A2 : A1 → UU l2} {A3 : (x1 : A1) → A2 x1 → UU l3}
{ A4 : (x1 : A1) (x2 : A2 x1) → A3 x1 x2 → UU l4}
{ A5 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → UU l5}
{ X :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) → UU l} →
telescope (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l) 5
instance-telescope⁵ {X = X} =
cons-telescope (λ x → instance-telescope⁴ {X = X x})

instance-telescope⁶ :
{ l1 l2 l3 l4 l5 l6 l : Level}
{ A1 : UU l1} {A2 : A1 → UU l2} {A3 : (x1 : A1) → A2 x1 → UU l3}
{ A4 : (x1 : A1) (x2 : A2 x1) → A3 x1 x2 → UU l4}
{ A5 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → UU l5}
{ A6 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) → UU l6}
{ X :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) → UU l} →
telescope (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l6 ⊔ l) 6
instance-telescope⁶ {X = X} =
cons-telescope (λ x → instance-telescope⁵ {X = X x})

instance-telescope⁷ :
{ l1 l2 l3 l4 l5 l6 l7 l : Level}
{ A1 : UU l1} {A2 : A1 → UU l2} {A3 : (x1 : A1) → A2 x1 → UU l3}
{ A4 : (x1 : A1) (x2 : A2 x1) → A3 x1 x2 → UU l4}
{ A5 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → UU l5}
{ A6 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) → UU l6}
{ A7 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) → UU l7}
{ X :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) → UU l} →
telescope (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l6 ⊔ l7 ⊔ l) 7
instance-telescope⁷ {X = X} =
cons-telescope (λ x → instance-telescope⁶ {X = X x})

instance-telescope⁸ :
{ l1 l2 l3 l4 l5 l6 l7 l8 l : Level}
{ A1 : UU l1} {A2 : A1 → UU l2} {A3 : (x1 : A1) → A2 x1 → UU l3}
{ A4 : (x1 : A1) (x2 : A2 x1) → A3 x1 x2 → UU l4}
{ A5 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → UU l5}
{ A6 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) → UU l6}
{ A7 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) → UU l7}
{ A8 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) → UU l8}
{ X :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) → UU l} →
telescope (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l6 ⊔ l7 ⊔ l8 ⊔ l) 8
instance-telescope⁸ {X = X} =
cons-telescope (λ x → instance-telescope⁷ {X = X x})

instance-telescope⁹ :
{ l1 l2 l3 l4 l5 l6 l7 l8 l9 l : Level}
{ A1 : UU l1} {A2 : A1 → UU l2} {A3 : (x1 : A1) → A2 x1 → UU l3}
{ A4 : (x1 : A1) (x2 : A2 x1) → A3 x1 x2 → UU l4}
{ A5 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → UU l5}
{ A6 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) → UU l6}
{ A7 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) → UU l7}
{ A8 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) → UU l8}
{ A9 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) →
UU l9}
{ X :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) → UU l} →
telescope (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l6 ⊔ l7 ⊔ l8 ⊔ l9 ⊔ l) 9
instance-telescope⁹ {X = X} =
cons-telescope (λ x → instance-telescope⁸ {X = X x})

instance-telescope¹⁰ :
{ l1 l2 l3 l4 l5 l6 l7 l8 l9 l10 l : Level}
{ A1 : UU l1} {A2 : A1 → UU l2} {A3 : (x1 : A1) → A2 x1 → UU l3}
{ A4 : (x1 : A1) (x2 : A2 x1) → A3 x1 x2 → UU l4}
{ A5 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → UU l5}
{ A6 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) → UU l6}
{ A7 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) → UU l7}
{ A8 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) → UU l8}
{ A9 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) →
UU l9}
{ A10 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) →
UU l10}
{ X :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9) →
UU l} →
telescope (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l6 ⊔ l7 ⊔ l8 ⊔ l9 ⊔ l10 ⊔ l) 10
instance-telescope¹⁰ {X = X} =
cons-telescope (λ x → instance-telescope⁹ {X = X x})

instance-telescope¹¹ :
{ l1 l2 l3 l4 l5 l6 l7 l8 l9 l10 l11 l : Level}
{ A1 : UU l1} {A2 : A1 → UU l2} {A3 : (x1 : A1) → A2 x1 → UU l3}
{ A4 : (x1 : A1) (x2 : A2 x1) → A3 x1 x2 → UU l4}
{ A5 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → UU l5}
{ A6 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) → UU l6}
{ A7 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) → UU l7}
{ A8 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) → UU l8}
{ A9 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) →
UU l9}
{ A10 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) →
UU l10}
{ A11 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9) →
UU l11}
{ X :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10) → UU l} →
telescope (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l6 ⊔ l7 ⊔ l8 ⊔ l9 ⊔ l10 ⊔ l11 ⊔ l) 11
instance-telescope¹¹ {X = X} =
cons-telescope (λ x → instance-telescope¹⁰ {X = X x})

instance-telescope¹² :
{ l1 l2 l3 l4 l5 l6 l7 l8 l9 l10 l11 l12 l : Level}
{ A1 : UU l1} {A2 : A1 → UU l2} {A3 : (x1 : A1) → A2 x1 → UU l3}
{ A4 : (x1 : A1) (x2 : A2 x1) → A3 x1 x2 → UU l4}
{ A5 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → UU l5}
{ A6 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) → UU l6}
{ A7 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) → UU l7}
{ A8 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) → UU l8}
{ A9 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) →
UU l9}
{ A10 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) →
UU l10}
{ A11 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9) →
UU l11}
{ A12 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10) → UU l12}
{ X :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(x12 : A12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) → UU l} →
telescope
( l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l6 ⊔ l7 ⊔ l8 ⊔ l9 ⊔ l10 ⊔ l11 ⊔ l12 ⊔ l)
( 12)
instance-telescope¹² {X = X} =
cons-telescope (λ x → instance-telescope¹¹ {X = X x})

instance-telescope¹³ :
{ l1 l2 l3 l4 l5 l6 l7 l8 l9 l10 l11 l12 l13 l : Level}
{ A1 : UU l1} {A2 : A1 → UU l2} {A3 : (x1 : A1) → A2 x1 → UU l3}
{ A4 : (x1 : A1) (x2 : A2 x1) → A3 x1 x2 → UU l4}
{ A5 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → UU l5}
{ A6 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) → UU l6}
{ A7 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) → UU l7}
{ A8 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) → UU l8}
{ A9 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) →
UU l9}
{ A10 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) →
UU l10}
{ A11 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9) →
UU l11}
{ A12 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10) → UU l12}
{ A13 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(x12 : A12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) → UU l13}
{ X :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(x12 : A12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11)
(x13 : A13 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) → UU l} →
telescope
( l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l6 ⊔ l7 ⊔ l8 ⊔ l9 ⊔ l10 ⊔ l11 ⊔ l12 ⊔ l13 ⊔ l)
( 13)
instance-telescope¹³ {X = X} =
cons-telescope (λ x → instance-telescope¹² {X = X x})

instance-telescope¹⁴ :
{ l1 l2 l3 l4 l5 l6 l7 l8 l9 l10 l11 l12 l13 l14 l : Level}
{ A1 : UU l1} {A2 : A1 → UU l2} {A3 : (x1 : A1) → A2 x1 → UU l3}
{ A4 : (x1 : A1) (x2 : A2 x1) → A3 x1 x2 → UU l4}
{ A5 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → UU l5}
{ A6 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) → UU l6}
{ A7 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) → UU l7}
{ A8 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) → UU l8}
{ A9 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) →
UU l9}
{ A10 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) →
UU l10}
{ A11 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9) →
UU l11}
{ A12 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10) → UU l12}
{ A13 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(x12 : A12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) → UU l13}
{ A14 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(x12 : A12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11)
(x13 : A13 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) → UU l14}
{ X :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(x12 : A12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11)
(x13 : A13 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12)
(x14 : A14 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13) → UU l} →
telescope
( l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l6 ⊔ l7 ⊔ l8 ⊔ l9 ⊔ l10 ⊔ l11 ⊔ l12 ⊔ l13 ⊔
l14 ⊔ l)
( 14)
instance-telescope¹⁴ {X = X} =
cons-telescope (λ x → instance-telescope¹³ {X = X x})

instance-telescope¹⁵ :
{ l1 l2 l3 l4 l5 l6 l7 l8 l9 l10 l11 l12 l13 l14 l15 l : Level}
{ A1 : UU l1} {A2 : A1 → UU l2} {A3 : (x1 : A1) → A2 x1 → UU l3}
{ A4 : (x1 : A1) (x2 : A2 x1) → A3 x1 x2 → UU l4}
{ A5 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → UU l5}
{ A6 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) → UU l6}
{ A7 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) → UU l7}
{ A8 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) → UU l8}
{ A9 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) →
UU l9}
{ A10 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) →
UU l10}
{ A11 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9) →
UU l11}
{ A12 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10) → UU l12}
{ A13 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(x12 : A12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) → UU l13}
{ A14 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(x12 : A12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11)
(x13 : A13 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) → UU l14}
{ A15 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(x12 : A12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11)
(x13 : A13 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12)
(x14 : A14 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13) → UU l15}
{ X :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(x12 : A12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11)
(x13 : A13 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12)
(x14 : A14 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13)
(x15 : A15 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14) → UU l} →
telescope
( l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l6 ⊔ l7 ⊔ l8 ⊔ l9 ⊔ l10 ⊔ l11 ⊔ l12 ⊔ l13 ⊔
l14 ⊔ l15 ⊔ l)
( 15)
instance-telescope¹⁵ {X = X} =
cons-telescope (λ x → instance-telescope¹⁴ {X = X x})

instance-telescope¹⁶ :
{ l1 l2 l3 l4 l5 l6 l7 l8 l9 l10 l11 l12 l13 l14 l15 l16 l : Level}
{ A1 : UU l1} {A2 : A1 → UU l2} {A3 : (x1 : A1) → A2 x1 → UU l3}
{ A4 : (x1 : A1) (x2 : A2 x1) → A3 x1 x2 → UU l4}
{ A5 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → UU l5}
{ A6 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) → UU l6}
{ A7 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) → UU l7}
{ A8 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) → UU l8}
{ A9 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) →
UU l9}
{ A10 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) →
UU l10}
{ A11 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9) →
UU l11}
{ A12 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10) → UU l12}
{ A13 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(x12 : A12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) → UU l13}
{ A14 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(x12 : A12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11)
(x13 : A13 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) → UU l14}
{ A15 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(x12 : A12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11)
(x13 : A13 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12)
(x14 : A14 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13) → UU l15}
{ A16 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(x12 : A12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11)
(x13 : A13 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12)
(x14 : A14 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13)
(x15 : A15 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14) → UU l16}
{ X :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(x12 : A12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11)
(x13 : A13 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12)
(x14 : A14 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13)
(x15 : A15 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14)
(x16 : A16 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15) → UU l} →
telescope
( l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l6 ⊔ l7 ⊔ l8 ⊔ l9 ⊔ l10 ⊔ l11 ⊔ l12 ⊔ l13 ⊔
l14 ⊔ l15 ⊔ l16 ⊔ l)
( 16)
instance-telescope¹⁶ {X = X} =
cons-telescope (λ x → instance-telescope¹⁵ {X = X x})

instance-telescope¹⁷ :
{ l1 l2 l3 l4 l5 l6 l7 l8 l9 l10 l11 l12 l13 l14 l15 l16 l17 l : Level}
{ A1 : UU l1} {A2 : A1 → UU l2} {A3 : (x1 : A1) → A2 x1 → UU l3}
{ A4 : (x1 : A1) (x2 : A2 x1) → A3 x1 x2 → UU l4}
{ A5 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → UU l5}
{ A6 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) → UU l6}
{ A7 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) → UU l7}
{ A8 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) → UU l8}
{ A9 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) →
UU l9}
{ A10 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) →
UU l10}
{ A11 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9) →
UU l11}
{ A12 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10) → UU l12}
{ A13 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(x12 : A12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) → UU l13}
{ A14 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(x12 : A12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11)
(x13 : A13 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) → UU l14}
{ A15 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(x12 : A12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11)
(x13 : A13 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12)
(x14 : A14 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13) → UU l15}
{ A16 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(x12 : A12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11)
(x13 : A13 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12)
(x14 : A14 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13)
(x15 : A15 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14) → UU l16}
{ A17 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(x12 : A12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11)
(x13 : A13 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12)
(x14 : A14 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13)
(x15 : A15 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14)
(x16 : A16 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15) → UU l17}
{ X :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(x11 : A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(x12 : A12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11)
(x13 : A13 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12)
(x14 : A14 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13)
(x15 : A15 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14)
(x16 : A16 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15)
(x17 : A17 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16) →
UU l} →
telescope
( l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l6 ⊔ l7 ⊔ l8 ⊔ l9 ⊔ l10 ⊔ l11 ⊔ l12 ⊔ l13 ⊔
l14 ⊔ l15 ⊔ l16 ⊔ l17 ⊔ l)
( 17)
instance-telescope¹⁷ {X = X} =
cons-telescope (λ x → instance-telescope¹⁶ {X = X x})

instance-telescope¹⁸ :
{ l1 l2 l3 l4 l5 l6 l7 l8 l9 l10 l11 l12 l13 l14 l15 l16 l17 l18 l : Level}
{ A1 : UU l1} {A2 : A1 → UU l2} {A3 : (x1 : A1) → A2 x1 → UU l3}
{ A4 : (x1 : A1) (x2 : A2 x1) → A3 x1 x2 → UU l4}
{ A5 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → UU l5}
{ A6 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) → UU l6}
{ A7 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) → UU l7}
{ A8 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) → UU l8}
{ A9 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) →
UU l9}
{ A10 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7)
(x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) →
UU l10}
{ A11 :
(x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3)
(x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5)
(x7 : A7 x1 x2 x3 x4 x5 x6) (