{-# OPTIONS --guardedness --safe #-}
module Cubical.Codata.M.AsLimit.M.Base where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Equiv using (_≃_)
open import Cubical.Foundations.Function using (_∘_)
open import Cubical.Data.Unit
open import Cubical.Data.Prod
open import Cubical.Data.Nat as ℕ using (ℕ ; suc ; _+_ )
open import Cubical.Data.Sigma hiding (_×_)
open import Cubical.Data.Nat.Algebra
open import Cubical.Foundations.Transport
open import Cubical.Foundations.Univalence
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Function
open import Cubical.Foundations.Path
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.GroupoidLaws
open import Cubical.Data.Sum
open import Cubical.Codata.M.AsLimit.helper
open import Cubical.Codata.M.AsLimit.Container
open import Cubical.Codata.M.AsLimit.Coalg.Base
open Iso
private
limit-collapse : ∀ {ℓ} {S : Container ℓ} (X : ℕ → Type ℓ) (l : (n : ℕ) → X n → X (suc n)) → (x₀ : X 0) → ∀ (n : ℕ) → X n
limit-collapse X l x₀ 0 = x₀
limit-collapse {S = S} X l x₀ (suc n) = l n (limit-collapse {S = S} X l x₀ n)
lemma11-Iso :
∀ {ℓ} {S : Container ℓ} (X : ℕ → Type ℓ) (l : (n : ℕ) → X n → X (suc n))
→ Iso (Σ[ x ∈ ((n : ℕ) → X n)] ((n : ℕ) → x (suc n) ≡ l n (x n)))
(X 0)
fun (lemma11-Iso X l) (x , y) = x 0
inv (lemma11-Iso {S = S} X l) x₀ = limit-collapse {S = S} X l x₀ , (λ n → refl {x = limit-collapse {S = S} X l x₀ (suc n)})
rightInv (lemma11-Iso X l) _ = refl
leftInv (lemma11-Iso {ℓ = ℓ} {S = S} X l) (x , y) i =
let temp = χ-prop (x 0) (fst (inv (lemma11-Iso {S = S} X l) (fun (lemma11-Iso {S = S} X l) (x , y))) , refl , (λ n → refl {x = limit-collapse {S = S} X l (x 0) (suc n)})) (x , refl , y)
in temp i .fst , proj₂ (temp i .snd)
where
open AlgebraPropositionality
open NatSection
X-fiber-over-ℕ : (x₀ : X 0) -> NatFiber NatAlgebraℕ ℓ
X-fiber-over-ℕ x₀ = record { Fiber = X ; fib-zero = x₀ ; fib-suc = λ {n : ℕ} xₙ → l n xₙ }
X-section : (x₀ : X 0) → (z : Σ[ x ∈ ((n : ℕ) → X n)] (x 0 ≡ x₀) × (∀ n → (x (suc n)) ≡ l n (x n))) -> NatSection (X-fiber-over-ℕ x₀)
X-section = λ x₀ z → record { section = fst z ; sec-comm-zero = proj₁ (snd z) ; sec-comm-suc = proj₂ (snd z) }
Z-is-Section : (x₀ : X 0) →
Iso (Σ[ x ∈ ((n : ℕ) → X n)] (x 0 ≡ x₀) × (∀ n → (x (suc n)) ≡ l n (x n)))
(NatSection (X-fiber-over-ℕ x₀))
fun (Z-is-Section x₀) (x , (z , y)) = record { section = x ; sec-comm-zero = z ; sec-comm-suc = y }
inv (Z-is-Section x₀) x = NatSection.section x , (sec-comm-zero x , sec-comm-suc x)
rightInv (Z-is-Section x₀) _ = refl
leftInv (Z-is-Section x₀) (x , (z , y)) = refl
χ-prop' : (x₀ : X 0) → isProp (NatSection (X-fiber-over-ℕ x₀))
χ-prop' x₀ a b = SectionProp.S≡T isNatInductiveℕ (X-section x₀ (inv (Z-is-Section x₀) a)) (X-section x₀ (inv (Z-is-Section x₀) b))
χ-prop : (x₀ : X 0) → isProp (Σ[ x ∈ ((n : ℕ) → X n) ] (x 0 ≡ x₀) × (∀ n → (x (suc n)) ≡ l n (x n)))
χ-prop x₀ = subst isProp (sym (isoToPath (Z-is-Section x₀))) (χ-prop' x₀)
shift-iso : ∀ {ℓ} (S : Container ℓ) -> Iso (P₀ S (M S)) (M S)
shift-iso S@(A , B) =
P₀ S (M S)
Iso⟨ Σ-cong-iso-snd
(λ x → iso (λ f → (λ n z → f z .fst n) , λ n i a → f a .snd n i)
(λ (u , q) z → (λ n → u n z) , λ n i → q n i z)
(λ _ → refl)
(λ _ → refl)) ⟩
(Σ[ a ∈ A ] (Σ[ u ∈ ((n : ℕ) → B a → X (sequence S) n) ] ((n : ℕ) → π (sequence S) ∘ (u (suc n)) ≡ u n)))
Iso⟨ invIso α-iso-step-5-Iso ⟩
(Σ[ a ∈ (Σ[ a ∈ ((n : ℕ) → A) ] ((n : ℕ) → a (suc n) ≡ a n)) ]
Σ[ u ∈ ((n : ℕ) → B (a .fst n) → X (sequence S) n) ]
((n : ℕ) → PathP (λ x → B (a .snd n x) → X (sequence S) n)
(π (sequence S) ∘ u (suc n))
(u n)))
Iso⟨ α-iso-step-1-4-Iso-lem-12 ⟩
M S ∎Iso
where
α-iso-step-5-Iso-helper0 :
∀ (a : (ℕ -> A))
→ (p : (n : ℕ) → a (suc n) ≡ a n)
→ (n : ℕ)
→ a n ≡ a 0
α-iso-step-5-Iso-helper0 a p 0 = refl
α-iso-step-5-Iso-helper0 a p (suc n) = p n ∙ α-iso-step-5-Iso-helper0 a p n
α-iso-step-5-Iso-helper1-Iso :
∀ (a : ℕ -> A)
→ (p : (n : ℕ) → a (suc n) ≡ a n)
→ (u : (n : ℕ) → B (a n) → Wₙ S n)
→ (n : ℕ)
→ (PathP (λ x → B (p n x) → Wₙ S n) (πₙ S ∘ u (suc n)) (u n)) ≡
(πₙ S ∘ (subst (\k -> B k → Wₙ S (suc n)) (α-iso-step-5-Iso-helper0 a p (suc n))) (u (suc n))
≡ subst (λ k → B k → Wₙ S n) (α-iso-step-5-Iso-helper0 a p n) (u n))
α-iso-step-5-Iso-helper1-Iso a p u n =
PathP (λ x → B (p n x) → Wₙ S n) (πₙ S ∘ u (suc n)) (u n)
≡⟨ PathP≡Path (λ x → B (p n x) → Wₙ S n) (πₙ S ∘ u (suc n)) (u n) ⟩
subst (λ k → B k → Wₙ S n) (p n) (πₙ S ∘ u (suc n)) ≡ (u n)
≡⟨ (λ i → transp (λ j → B (α-iso-step-5-Iso-helper0 a p n (i ∧ j)) → Wₙ S n) (~ i) (subst (λ k → B k → Wₙ S n) (p n) (πₙ S ∘ u (suc n)))
≡ transp (λ j → B (α-iso-step-5-Iso-helper0 a p n (i ∧ j)) → Wₙ S n) (~ i) (u n)) ⟩
subst (λ k → B k → Wₙ S n) (α-iso-step-5-Iso-helper0 a p n) (subst (λ k → B k → Wₙ S n) (p n) (πₙ S ∘ u (suc n))) ≡
subst (λ k → B k → Wₙ S n) (α-iso-step-5-Iso-helper0 a p n) (u n)
≡⟨ (λ i → sym (substComposite (λ k → B k → Wₙ S n) (p n) (α-iso-step-5-Iso-helper0 a p n) (πₙ S ∘ u (suc n))) i
≡ subst (λ k → B k → Wₙ S n) (α-iso-step-5-Iso-helper0 a p n) (u n)) ⟩
subst (λ k → B k → Wₙ S n) (α-iso-step-5-Iso-helper0 a p (suc n)) (πₙ S ∘ u (suc n)) ≡
subst (λ k → B k → Wₙ S n) (α-iso-step-5-Iso-helper0 a p n) (u n)
≡⟨ (λ i → substCommSlice (λ k → B k → Wₙ S (suc n)) (λ k → B k → Wₙ S n)
(λ a x x₁ → (πₙ S) (x x₁))
(α-iso-step-5-Iso-helper0 a p (suc n)) (u (suc n)) i
≡ subst (λ k → B k → Wₙ S n) (α-iso-step-5-Iso-helper0 a p n) (u n)) ⟩
πₙ S ∘ subst (λ k → B k → Wₙ S (suc n)) (α-iso-step-5-Iso-helper0 a p (suc n)) (u (suc n))
≡
subst (λ k → B k → Wₙ S n) (α-iso-step-5-Iso-helper0 a p n) (u n) ∎
α-iso-step-5-Iso :
Iso
(Σ[ a ∈ (Σ[ a ∈ ((n : ℕ) → A) ] ((n : ℕ) → a (suc n) ≡ a n)) ]
Σ[ u ∈ ((n : ℕ) → B (a .fst n) → X (sequence S) n) ]
((n : ℕ) → PathP (λ x → B (a .snd n x) → X (sequence S) n)
(π (sequence S) ∘ u (suc n))
(u n)))
(Σ[ a ∈ A ] (Σ[ u ∈ ((n : ℕ) → B a → X (sequence S) n) ] ((n : ℕ) → π (sequence S) ∘ (u (suc n)) ≡ u n)))
α-iso-step-5-Iso =
Σ-cong-iso (lemma11-Iso {S = S} (λ _ → A) (λ _ x → x)) (λ a,p →
Σ-cong-iso (pathToIso (cong (λ k → (n : ℕ) → k n) (funExt λ n → cong (λ k → B k → Wₙ S n) (α-iso-step-5-Iso-helper0 (a,p .fst) (a,p .snd) n)))) λ u →
pathToIso (cong (λ k → (n : ℕ) → k n) (funExt λ n → α-iso-step-5-Iso-helper1-Iso (a,p .fst) (a,p .snd) u n)))
α-iso-step-1-4-Iso-lem-12 :
Iso (Σ[ a ∈ (Σ[ a ∈ ((n : ℕ) → A)] ((n : ℕ) → a (suc n) ≡ a n)) ]
(Σ[ u ∈ ((n : ℕ) → B (a .fst n) → X (sequence S) n) ]
((n : ℕ) → PathP (λ x → B (a .snd n x) → X (sequence S) n)
(π (sequence S) ∘ u (suc n))
(u n))))
(limit-of-chain (sequence S))
fun α-iso-step-1-4-Iso-lem-12 (a , b) = (λ { 0 → lift tt ; (suc n) → (a .fst n) , (b .fst n)}) , λ { 0 → refl {x = lift tt} ; (suc m) i → a .snd m i , b .snd m i }
inv α-iso-step-1-4-Iso-lem-12 x = ((λ n → (x .fst) (suc n) .fst) , λ n i → (x .snd) (suc n) i .fst) , (λ n → (x .fst) (suc n) .snd) , λ n i → (x .snd) (suc n) i .snd
fst (rightInv α-iso-step-1-4-Iso-lem-12 (b , c) i) 0 = lift tt
fst (rightInv α-iso-step-1-4-Iso-lem-12 (b , c) i) (suc n) = refl i
snd (rightInv α-iso-step-1-4-Iso-lem-12 (b , c) i) 0 = refl
snd (rightInv α-iso-step-1-4-Iso-lem-12 (b , c) i) (suc n) = c (suc n)
leftInv α-iso-step-1-4-Iso-lem-12 (a , b) = refl
shift : ∀ {ℓ} (S : Container ℓ) -> P₀ S (M S) ≡ M S
shift S = isoToPath (shift-iso S)
in-fun : ∀ {ℓ} {S : Container ℓ} -> P₀ S (M S) -> M S
in-fun {S = S} = fun (shift-iso S)
out-fun : ∀ {ℓ} {S : Container ℓ} -> M S -> P₀ S (M S)
out-fun {S = S} = inv (shift-iso S)
lift-to-M : ∀ {ℓ} {A : Type ℓ} {S : Container ℓ}
→ (x : (n : ℕ) -> A → X (sequence S) n)
→ ((n : ℕ) → (a : A) → π (sequence S) (x (suc n) a) ≡ x n a)
→ (A → M S)
lift-to-M x p a = (λ n → x n a) , λ n i → p n a i
lift-direct-M : ∀ {ℓ} {S : Container ℓ}
→ (x : (n : ℕ) → X (sequence S) n)
→ ((n : ℕ) → π (sequence S) (x (suc n)) ≡ x n)
→ M S
lift-direct-M x p = x , p