module Monad.Instance.Delay.Commutative {o ℓ e} (ambient : Ambient o ℓ e) (D : DelayM ambient) where
open Ambient ambient
open HomReasoning
open Equiv
open MR C
open M C
open F-Coalgebra-Morphism using () renaming (f to u; commutes to u-commutes)
open import Categories.Morphism.Properties C
open Terminal using (!; !-unique; ⊤)
open DelayM D
open import Monad.Instance.Delay.Strong ambient D
open Functor functor using () renaming (F₁ to D₁; identity to D-identity; homomorphism to D-homomorphism; F-resp-≈ to D-resp-≈)
open RMonad kleisli using (extend; extend-≈) renaming (assoc to k-assoc; identityʳ to k-identityʳ; identityˡ to k-identityˡ)
open Monad monad using (η; μ)
open NaturalTransformation (StrongMonad.strengthen strongMonad) using () renaming (commute to τ-commute)
commutativeMonad : Commutative braided strongMonad
commutativeMonad = record { commutes = λ {X} {Y} → pullˡ (Kleisli⇒Monad⇒Kleisli kleisli _) ○ commutes' ○ pushˡ (sym (Kleisli⇒Monad⇒Kleisli kleisli _)) }
where
We first introduce some helper definitions and lemmas:
open τ-mod
σ : ∀ P → D₀ (proj₁ P) × (proj₂ P) ⇒ D₀ (proj₁ P × proj₂ P)
σ (X , Y) = D₁ swap ∘ (τ (Y , X)) ∘ swap
σ-coalg : ∀ (X Y : Obj) → F-Coalgebra-Morphism {F = (X × Y) +- } (record { A = D₀ X × Y ; α = distributeʳ⁻¹ {Y} {X} {D₀ X} ∘ (out {X} ⁂ idC) }) (record { A = D₀ (X × Y) ; α = out {X × Y} })
σ-coalg X Y = record { f = (σ (X , Y)) ; commutes = begin
out ∘ (σ (X , Y)) ≈⟨ pullˡ (out-law swap) ⟩
((swap +₁ D₁ swap) ∘ out) ∘ (τ (Y , X)) ∘ swap ≈⟨ pullˡ (pullʳ (τ-law (Y , X))) ⟩
((swap +₁ D₁ swap) ∘ (idC +₁ (τ (Y , X))) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out)) ∘ swap ≈⟨ pullʳ (pullʳ (pullʳ (sym swap∘⁂))) ⟩
(swap +₁ D₁ swap) ∘ (idC +₁ (τ (Y , X))) ∘ distributeˡ⁻¹ ∘ swap ∘ (out ⁂ idC) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ distributeˡ⁻¹∘swap ⟩
(swap +₁ D₁ swap) ∘ (idC +₁ (τ (Y , X))) ∘ ((swap +₁ swap) ∘ distributeʳ⁻¹) ∘ (out ⁂ idC) ≈⟨ pullˡ +₁∘+₁ ⟩
(swap ∘ idC +₁ D₁ swap ∘ (τ (Y , X))) ∘ ((swap +₁ swap) ∘ distributeʳ⁻¹) ∘ (out ⁂ idC) ≈⟨ pullˡ (pullˡ +₁∘+₁) ⟩
(((swap ∘ idC) ∘ swap +₁ (D₁ swap ∘ (τ (Y , X))) ∘ swap) ∘ distributeʳ⁻¹) ∘ (out ⁂ idC) ≈⟨ ((+₁-cong₂ (identityʳ ⟩∘⟨refl ○ swap∘swap) assoc) ⟩∘⟨refl) ⟩∘⟨refl ⟩
((idC +₁ D₁ swap ∘ (τ (Y , X)) ∘ swap) ∘ distributeʳ⁻¹) ∘ (out ⁂ idC) ≈⟨ assoc ⟩
(idC +₁ (σ (X , Y))) ∘ distributeʳ⁻¹ ∘ (out ⁂ idC) ∎ }
σ-helper : ∀ {X Y : Obj} → (σ (X , Y)) ∘ (out⁻¹ ⁂ idC) ≈ out⁻¹ ∘ (idC +₁ (σ (X , Y))) ∘ distributeʳ⁻¹
σ-helper {X} {Y} = begin
(σ (X , Y)) ∘ (out⁻¹ ⁂ idC) ≈⟨ introˡ (_≅_.isoˡ out-≅) ⟩
(out⁻¹ ∘ out) ∘ (σ (X , Y)) ∘ (out⁻¹ ⁂ idC) ≈⟨ pullʳ (pullˡ (u-commutes (σ-coalg X Y))) ⟩
out⁻¹ ∘ ((idC +₁ (σ (X , Y))) ∘ distributeʳ⁻¹ ∘ (out ⁂ idC)) ∘ (out⁻¹ ⁂ idC) ≈⟨ refl⟩∘⟨ (pullʳ (cancelʳ (⁂∘⁂ ○ ⁂-cong₂ (_≅_.isoʳ out-≅) identity² ○ ⟨⟩-unique id-comm id-comm))) ⟩
out⁻¹ ∘ (idC +₁ (σ (X , Y))) ∘ distributeʳ⁻¹ ∎
-- this is naturality of σ, i.e. it could be included from Categories.Monad.Strong.Properties but we reiterate the proof to save some imports
σ-commute : ∀ {U V W X : Obj} (f : U ⇒ V) (g : W ⇒ X) → (σ (V , X)) ∘ (extend (now ∘ f) ⁂ g) ≈ extend (now ∘ (f ⁂ g)) ∘ σ (U , W)
σ-commute {U} {V} {W} {X} f g = begin
(σ (V , X)) ∘ (D₁ f ⁂ g) ≈⟨ pullʳ (pullʳ swap∘⁂) ⟩
D₁ swap ∘ (τ (X , V)) ∘ (g ⁂ extend (now ∘ f)) ∘ swap ≈⟨ refl⟩∘⟨ (pullˡ (τ-commute (g , f))) ⟩
D₁ swap ∘ (D₁ (g ⁂ f) ∘ (τ (W , U))) ∘ swap ≈⟨ pullˡ (pullˡ (sym D-homomorphism)) ⟩
(D₁ (swap ∘ (g ⁂ f)) ∘ (τ (W , U))) ∘ swap ≈⟨ ((D-resp-≈ swap∘⁂) ⟩∘⟨refl) ⟩∘⟨refl ⟩
(D₁ ((f ⁂ g) ∘ swap) ∘ (τ (W , U))) ∘ swap ≈⟨ pushˡ D-homomorphism ⟩∘⟨refl ⟩
(D₁ (f ⁂ g) ∘ D₁ swap ∘ (τ (W , U))) ∘ swap ≈⟨ assoc²' ⟩
D₁ (f ⁂ g) ∘ (σ (U , W)) ∎
We now proof commutativity by stability, i.e. by showing that both sides are solutions to some guarded morphism:
commutes' : ∀ {X Y} → extend (σ (X , Y)) ∘ (τ (D₀ X , Y)) ≈ extend (τ (X , Y)) ∘ (σ (X , D₀ Y))
commutes' {X} {Y} = guarded-unique g (extend (σ (X , Y)) ∘ (τ (D₀ X , Y))) (extend (τ (X , Y)) ∘ (σ (X , D₀ Y))) guarded (fixpoint-eq fixpoint₁) (fixpoint-eq fixpoint₂)
where
w = (distributeʳ⁻¹ +₁ distributeʳ⁻¹) ∘ distributeˡ⁻¹ ∘ (out ⁂ out)
g = out⁻¹ ∘ [ i₁ +₁ D₁ i₁ ∘ (σ (X , Y)) , i₂ ∘ [ D₁ i₁ ∘ (τ (X , Y)) , ▷ ∘ now ∘ i₂ ] ] ∘ w
guarded : is-guarded g
guarded = [ idC +₁ D₁ i₁ ∘ (σ (X , Y)) , i₂ ∘ [ D₁ i₁ ∘ (τ (X , Y)) , ▷ ∘ now ∘ i₂ ] ] ∘ w , (begin
(i₁ +₁ idC) ∘ [ idC +₁ D₁ i₁ ∘ (σ (X , Y)) , i₂ ∘ [ D₁ i₁ ∘ (τ (X , Y)) , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈⟨ pullˡ ∘[] ⟩
[ (i₁ +₁ idC) ∘ (idC +₁ D₁ i₁ ∘ (σ (X , Y))) , (i₁ +₁ idC) ∘ i₂ ∘ [ D₁ i₁ ∘ (τ (X , Y)) , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈⟨ ([]-cong₂ +₁∘+₁ (pullˡ +₁∘i₂)) ⟩∘⟨refl ⟩
[ (i₁ ∘ idC +₁ idC ∘ D₁ i₁ ∘ (σ (X , Y))) , (i₂ ∘ idC) ∘ [ D₁ i₁ ∘ (τ (X , Y)) , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈⟨ ([]-cong₂ (+₁-cong₂ identityʳ identityˡ) (identityʳ ⟩∘⟨refl)) ⟩∘⟨refl ⟩
[ i₁ +₁ D₁ i₁ ∘ (σ (X , Y)) , i₂ ∘ [ D₁ i₁ ∘ (τ (X , Y)) , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈⟨ sym (cancelˡ (_≅_.isoʳ out-≅)) ⟩
out ∘ out⁻¹ ∘ [ i₁ +₁ D₁ i₁ ∘ (σ (X , Y)) , i₂ ∘ [ D₁ i₁ ∘ (τ (X , Y)) , ▷ ∘ now ∘ i₂ ] ] ∘ w ∎)
fixpoint₁ : extend (σ (X , Y)) ∘ (τ (D₀ X , Y)) ≈ out⁻¹ ∘ [ idC +₁ (σ (X , Y)) , i₂ ∘ [ (τ (X , Y)) , ▷ ∘ extend (σ (X , Y)) ∘ (τ (D₀ X , Y)) ] ] ∘ w
fixpoint₁ = Iso⇒Mono (_≅_.iso out-≅) (extend (σ (X , Y)) ∘ (τ (D₀ X , Y))) (out⁻¹ ∘ [ idC +₁ (σ (X , Y)) , i₂ ∘ [ (τ (X , Y)) , ▷ ∘ extend (σ (X , Y)) ∘ (τ (D₀ X , Y)) ] ] ∘ w) (begin
out ∘ extend (σ (X , Y)) ∘ (τ (D₀ X , Y)) ≈⟨ pullˡ (extendlaw (σ (X , Y))) ⟩
([ out ∘ (σ (X , Y)) , i₂ ∘ extend' (σ (X , Y)) ] ∘ out) ∘ (τ (D₀ X , Y)) ≈⟨ pullʳ (τ-law (D₀ X , Y)) ⟩
[ out ∘ (σ (X , Y)) , i₂ ∘ extend' (σ (X , Y)) ] ∘ (idC +₁ (τ (D₀ X , Y))) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ pullˡ []∘+₁ ⟩
[ (out ∘ (σ (X , Y))) ∘ idC , (i₂ ∘ extend' (σ (X , Y))) ∘ (τ (D₀ X , Y)) ] ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ ([]-cong₂ (identityʳ ○ u-commutes (σ-coalg X Y)) assoc) ⟩∘⟨refl ⟩
[ (idC +₁ (σ (X , Y))) ∘ distributeʳ⁻¹ ∘ (out ⁂ idC) , i₂ ∘ extend' (σ (X , Y)) ∘ (τ (D₀ X , Y)) ] ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ ⁂-cong₂ (sym (_≅_.isoˡ out-≅)) refl ⟩
[ (idC +₁ (σ (X , Y))) ∘ distributeʳ⁻¹ ∘ (out ⁂ idC) , i₂ ∘ extend' (σ (X , Y)) ∘ (τ (D₀ X , Y)) ] ∘ distributeˡ⁻¹ ∘ (out⁻¹ ∘ out ⁂ out) ≈⟨ sym (refl⟩∘⟨ refl⟩∘⟨ (⁂∘⁂ ○ ⁂-cong₂ refl (elimˡ ([]-unique id-comm-sym id-comm-sym)))) ⟩
[ (idC +₁ (σ (X , Y))) ∘ distributeʳ⁻¹ ∘ (out ⁂ idC) , i₂ ∘ extend' (σ (X , Y)) ∘ (τ (D₀ X , Y)) ] ∘ distributeˡ⁻¹ ∘ (out⁻¹ ⁂ (idC +₁ idC)) ∘ (out ⁂ out) ≈⟨ refl⟩∘⟨ pullˡ (sym (distributeˡ⁻¹-natural out⁻¹ idC idC)) ⟩
[ (idC +₁ (σ (X , Y))) ∘ distributeʳ⁻¹ ∘ (out ⁂ idC) , i₂ ∘ extend' (σ (X , Y)) ∘ (τ (D₀ X , Y)) ] ∘ (((out⁻¹ ⁂ idC) +₁ (out⁻¹ ⁂ idC)) ∘ distributeˡ⁻¹) ∘ (out ⁂ out) ≈⟨ pullˡ (pullˡ []∘+₁) ⟩
([ ((idC +₁ (σ (X , Y))) ∘ distributeʳ⁻¹ ∘ (out ⁂ idC)) ∘ (out⁻¹ ⁂ idC) , (i₂ ∘ extend' (σ (X , Y)) ∘ (τ (D₀ X , Y))) ∘ (out⁻¹ ⁂ idC) ] ∘ distributeˡ⁻¹) ∘ (out ⁂ out) ≈⟨ assoc ○ ([]-cong₂ (pullʳ (cancelʳ (⁂∘⁂ ○ ⁂-cong₂ (_≅_.isoʳ out-≅) identity² ○ ⟨⟩-unique id-comm id-comm))) (refl⟩∘⟨ (⁂-cong₂ refl (sym D-identity)) ○ (pullʳ (pullʳ (τ-commute (out⁻¹ , idC)))))) ⟩∘⟨refl ⟩
[ (idC +₁ (σ (X , Y))) ∘ distributeʳ⁻¹ , i₂ ∘ extend' (σ (X , Y)) ∘ D₁ (out⁻¹ ⁂ idC) ∘ (τ (X + D₀ X , Y)) ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ refl (refl⟩∘⟨ (pullˡ (sym k-assoc)) ○ refl⟩∘⟨ ((extend-≈ (pullˡ k-identityʳ)) ⟩∘⟨refl))) ⟩∘⟨refl ⟩
[ (idC +₁ (σ (X , Y))) ∘ distributeʳ⁻¹ , i₂ ∘ extend' (σ (X , Y) ∘ (out⁻¹ ⁂ idC)) ∘ (τ (X + D₀ X , Y)) ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ refl (refl⟩∘⟨ ((extend-≈ σ-helper) ⟩∘⟨refl))) ⟩∘⟨refl ⟩
[ (idC +₁ (σ (X , Y))) ∘ distributeʳ⁻¹ , i₂ ∘ extend' (out⁻¹ ∘ (idC +₁ (σ (X , Y))) ∘ distributeʳ⁻¹) ∘ (τ (X + D₀ X , Y)) ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈˘⟨ ([]-cong₂ refl (refl⟩∘⟨ ((sym k-assoc ○ extend-≈ (pullˡ k-identityʳ) ○ extend-≈ assoc) ⟩∘⟨refl))) ⟩∘⟨refl ⟩
[ (idC +₁ (σ (X , Y))) ∘ distributeʳ⁻¹ , i₂ ∘ (extend' (out⁻¹ ∘ (idC +₁ (σ (X , Y)))) ∘ D₁ distributeʳ⁻¹) ∘ (τ (X + D₀ X , Y)) ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ refl (refl⟩∘⟨ pullʳ helper₁)) ⟩∘⟨refl ⟩
[ (idC +₁ (σ (X , Y))) ∘ distributeʳ⁻¹ , i₂ ∘ extend' (out⁻¹ ∘ (idC +₁ (σ (X , Y)))) ∘ [ D₁ i₁ ∘ (τ (X , Y)) , D₁ i₂ ∘ (τ (D₀ X , Y)) ] ∘ distributeʳ⁻¹ ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈˘⟨ pullˡ ([]∘+₁ ○ []-cong₂ refl assoc²') ⟩
[ (idC +₁ (σ (X , Y))) , i₂ ∘ extend' (out⁻¹ ∘ (idC +₁ (σ (X , Y)))) ∘ [ D₁ i₁ ∘ (τ (X , Y)) , D₁ i₂ ∘ (τ (D₀ X , Y)) ] ] ∘ (distributeʳ⁻¹ +₁ distributeʳ⁻¹) ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ refl (refl⟩∘⟨ ∘[])) ⟩∘⟨refl ⟩
[ (idC +₁ (σ (X , Y))) , i₂ ∘ [ extend' (out⁻¹ ∘ (idC +₁ (σ (X , Y)))) ∘ D₁ i₁ ∘ (τ (X , Y)) , extend' (out⁻¹ ∘ (idC +₁ (σ (X , Y)))) ∘ D₁ i₂ ∘ (τ (D₀ X , Y)) ] ] ∘ (distributeʳ⁻¹ +₁ distributeʳ⁻¹) ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ refl (refl⟩∘⟨ ([]-cong₂ (pullˡ ((sym k-assoc) ○ extend-≈ (pullˡ k-identityʳ))) (pullˡ ((sym k-assoc) ○ extend-≈ (pullˡ k-identityʳ)))))) ⟩∘⟨refl ⟩
[ (idC +₁ (σ (X , Y))) , i₂ ∘ [ extend' ((out⁻¹ ∘ (idC +₁ (σ (X , Y)))) ∘ i₁) ∘ (τ (X , Y)) , extend' ((out⁻¹ ∘ (idC +₁ (σ (X , Y)))) ∘ i₂) ∘ (τ (D₀ X , Y)) ] ] ∘ (distributeʳ⁻¹ +₁ distributeʳ⁻¹) ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ refl (refl⟩∘⟨ ([]-cong₂ ((extend-≈ (pullʳ +₁∘i₁)) ⟩∘⟨refl) ((extend-≈ (pullʳ +₁∘i₂)) ⟩∘⟨refl)))) ⟩∘⟨refl ⟩
[ (idC +₁ (σ (X , Y))) , i₂ ∘ [ extend' (out⁻¹ ∘ i₁ ∘ idC) ∘ (τ (X , Y)) , extend' (out⁻¹ ∘ i₂ ∘ (σ (X , Y))) ∘ (τ (D₀ X , Y)) ] ] ∘ (distributeʳ⁻¹ +₁ distributeʳ⁻¹) ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ refl (refl⟩∘⟨ ([]-cong₂ (elimˡ ((extend-≈ (refl⟩∘⟨ identityʳ)) ○ k-identityˡ)) ((extend-≈ sym-assoc) ⟩∘⟨refl)))) ⟩∘⟨refl ⟩
[ (idC +₁ (σ (X , Y))) , i₂ ∘ [ (τ (X , Y)) , extend' (▷ ∘ (σ (X , Y))) ∘ (τ (D₀ X , Y)) ] ] ∘ (distributeʳ⁻¹ +₁ distributeʳ⁻¹) ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ refl (refl⟩∘⟨ ([]-cong₂ refl ((sym (▷∘extendˡ (σ (X , Y)))) ⟩∘⟨refl ○ assoc)))) ⟩∘⟨refl ⟩
[ idC +₁ (σ (X , Y)) , i₂ ∘ [ (τ (X , Y)) , ▷ ∘ extend (σ (X , Y)) ∘ (τ (D₀ X , Y)) ] ] ∘ w ≈˘⟨ cancelˡ (_≅_.isoʳ out-≅) ⟩
out ∘ out⁻¹ ∘ [ idC +₁ (σ (X , Y)) , i₂ ∘ [ (τ (X , Y)) , ▷ ∘ extend (σ (X , Y)) ∘ (τ (D₀ X , Y)) ] ] ∘ w ∎)
where
helper₁ : (D₁ distributeʳ⁻¹) ∘ (τ (X + D₀ X , Y)) ≈ [ D₁ i₁ ∘ (τ (X , Y)) , D₁ i₂ ∘ (τ (D₀ X , Y)) ] ∘ distributeʳ⁻¹
helper₁ = Iso⇒Epi (IsIso.iso isIsoʳ) ((D₁ distributeʳ⁻¹) ∘ (τ (X + D₀ X , Y))) ([ D₁ i₁ ∘ (τ (X , Y)) , D₁ i₂ ∘ (τ (D₀ X , Y)) ] ∘ distributeʳ⁻¹) (begin
((D₁ distributeʳ⁻¹) ∘ (τ (X + D₀ X , Y))) ∘ distributeʳ ≈⟨ ∘[] ⟩
[ ((D₁ distributeʳ⁻¹) ∘ (τ (X + D₀ X , Y))) ∘ (i₁ ⁂ idC) , ((D₁ distributeʳ⁻¹) ∘ (τ (X + D₀ X , Y))) ∘ (i₂ ⁂ idC) ] ≈⟨ []-cong₂ (refl⟩∘⟨ (⁂-cong₂ refl (sym D-identity))) (refl⟩∘⟨ (⁂-cong₂ refl (sym D-identity))) ⟩
[ ((D₁ distributeʳ⁻¹) ∘ (τ (X + D₀ X , Y))) ∘ (i₁ ⁂ D₁ idC) , ((D₁ distributeʳ⁻¹) ∘ (τ (X + D₀ X , Y))) ∘ (i₂ ⁂ D₁ idC) ] ≈⟨ []-cong₂ (pullʳ (τ-commute (i₁ , idC))) (pullʳ (τ-commute (i₂ , idC))) ⟩
[ (D₁ distributeʳ⁻¹) ∘ D₁ (i₁ ⁂ idC) ∘ (τ (X , Y)) , (D₁ distributeʳ⁻¹) ∘ D₁ (i₂ ⁂ idC) ∘ (τ (D₀ X , Y)) ] ≈⟨ []-cong₂ (pullˡ (sym D-homomorphism)) (pullˡ (sym D-homomorphism)) ⟩
[ D₁ (distributeʳ⁻¹ ∘ (i₁ ⁂ idC)) ∘ (τ (X , Y)) , D₁ (distributeʳ⁻¹ ∘ (i₂ ⁂ idC)) ∘ (τ (D₀ X , Y)) ] ≈⟨ []-cong₂ (D-resp-≈ distributeʳ⁻¹-i₁ ⟩∘⟨refl) ((D-resp-≈ distributeʳ⁻¹-i₂) ⟩∘⟨refl) ⟩
[ D₁ i₁ ∘ (τ (X , Y)) , D₁ i₂ ∘ (τ (D₀ X , Y)) ] ≈˘⟨ cancelʳ (IsIso.isoˡ isIsoʳ) ⟩
([ D₁ i₁ ∘ (τ (X , Y)) , D₁ i₂ ∘ (τ (D₀ X , Y)) ] ∘ distributeʳ⁻¹) ∘ distributeʳ ∎)
fixpoint₂ : extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ≈ out⁻¹ ∘ [ idC +₁ (σ (X , Y)) , i₂ ∘ [ (τ (X , Y)) , ▷ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ] ] ∘ w
fixpoint₂ = Iso⇒Mono ((_≅_.iso out-≅)) (extend (τ (X , Y)) ∘ (σ (X , D₀ Y))) (out⁻¹ ∘ [ idC +₁ (σ (X , Y)) , i₂ ∘ [ (τ (X , Y)) , ▷ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ] ] ∘ w) (begin
out ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ≈⟨ pullˡ (extendlaw (τ (X , Y))) ⟩
([ out ∘ (τ (X , Y)) , i₂ ∘ extend (τ (X , Y)) ] ∘ out) ∘ (σ (X , D₀ Y)) ≈⟨ pullʳ (u-commutes (σ-coalg X (D₀ Y))) ⟩
[ out ∘ (τ (X , Y)) , i₂ ∘ extend (τ (X , Y)) ] ∘ (idC +₁ (σ (X , D₀ Y))) ∘ distributeʳ⁻¹ ∘ (out ⁂ idC) ≈⟨ pullˡ []∘+₁ ⟩
[ (out ∘ (τ (X , Y))) ∘ idC , (i₂ ∘ extend (τ (X , Y))) ∘ (σ (X , D₀ Y)) ] ∘ distributeʳ⁻¹ ∘ (out ⁂ idC) ≈⟨ ([]-cong₂ (identityʳ ○ τ-law (X , Y)) assoc) ⟩∘⟨refl ⟩
[ (idC +₁ (τ (X , Y))) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ] ∘ distributeʳ⁻¹ ∘ (out ⁂ idC) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ sym (⁂∘⁂ ○ ⁂-cong₂ (elimˡ ([]-unique id-comm-sym id-comm-sym)) (_≅_.isoˡ out-≅)) ⟩
[ (idC +₁ (τ (X , Y))) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ] ∘ distributeʳ⁻¹ ∘ ((idC +₁ idC) ⁂ out⁻¹) ∘ (out ⁂ out) ≈⟨ refl⟩∘⟨ (pullˡ (sym (distributeʳ⁻¹-natural out⁻¹ idC idC))) ⟩
[ (idC +₁ (τ (X , Y))) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ] ∘ (((idC ⁂ out⁻¹) +₁ (idC ⁂ out⁻¹)) ∘ distributeʳ⁻¹) ∘ (out ⁂ out) ≈⟨ pullˡ (pullˡ []∘+₁) ⟩
([ ((idC +₁ (τ (X , Y))) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out)) ∘ (idC ⁂ out⁻¹) , (i₂ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y))) ∘ (idC ⁂ out⁻¹) ] ∘ distributeʳ⁻¹) ∘ (out ⁂ out) ≈⟨ assoc ○ ([]-cong₂ (pullʳ (cancelʳ (⁂∘⁂ ○ ⁂-cong₂ identity² (_≅_.isoʳ out-≅) ○ ⟨⟩-unique id-comm id-comm))) (refl⟩∘⟨ (⁂-cong₂ (sym D-identity) refl))) ⟩∘⟨refl ⟩
[ (idC +₁ (τ (X , Y))) ∘ distributeˡ⁻¹ , (i₂ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y))) ∘ (D₁ idC ⁂ out⁻¹) ] ∘ distributeʳ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ refl (pullʳ (pullʳ (σ-commute idC out⁻¹)))) ⟩∘⟨refl ⟩
[ (idC +₁ (τ (X , Y))) ∘ distributeˡ⁻¹ , i₂ ∘ extend (τ (X , Y)) ∘ D₁ (idC ⁂ out⁻¹) ∘ (σ (X , Y + D₀ Y)) ] ∘ distributeʳ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ refl (refl⟩∘⟨ (pullˡ ((sym k-assoc) ○ extend-≈ (pullˡ k-identityʳ))))) ⟩∘⟨refl ⟩
[ (idC +₁ (τ (X , Y))) ∘ distributeˡ⁻¹ , i₂ ∘ extend (τ (X , Y) ∘ (idC ⁂ out⁻¹)) ∘ (σ (X , Y + D₀ Y)) ] ∘ distributeʳ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ refl (refl⟩∘⟨ ((extend-≈ (τ-helper (X , Y))) ⟩∘⟨refl))) ⟩∘⟨refl ⟩
[ (idC +₁ (τ (X , Y))) ∘ distributeˡ⁻¹ , i₂ ∘ extend (out⁻¹ ∘ (idC +₁ (τ (X , Y))) ∘ distributeˡ⁻¹) ∘ (σ (X , Y + D₀ Y)) ] ∘ distributeʳ⁻¹ ∘ (out ⁂ out) ≈˘⟨ ([]-cong₂ refl (refl⟩∘⟨ pullˡ ((sym k-assoc) ○ (extend-≈ (pullˡ k-identityʳ) ○ extend-≈ assoc)))) ⟩∘⟨refl ⟩
[ (idC +₁ (τ (X , Y))) ∘ distributeˡ⁻¹ , i₂ ∘ extend (out⁻¹ ∘ (idC +₁ (τ (X , Y)))) ∘ D₁ distributeˡ⁻¹ ∘ (σ (X , Y + D₀ Y)) ] ∘ distributeʳ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ refl ((refl⟩∘⟨ (refl⟩∘⟨ helper₂)) ○ sym assoc²')) ⟩∘⟨refl ⟩
[ (idC +₁ (τ (X , Y))) ∘ distributeˡ⁻¹ , (i₂ ∘ extend (out⁻¹ ∘ (idC +₁ (τ (X , Y)))) ∘ [ D₁ i₁ ∘ (σ (X , Y)) , D₁ i₂ ∘ (σ (X , D₀ Y)) ]) ∘ distributeˡ⁻¹ ] ∘ distributeʳ⁻¹ ∘ (out ⁂ out) ≈˘⟨ []∘+₁ ⟩∘⟨refl ⟩
([ (idC +₁ (τ (X , Y))) , i₂ ∘ extend (out⁻¹ ∘ (idC +₁ (τ (X , Y)))) ∘ [ D₁ i₁ ∘ (σ (X , Y)) , D₁ i₂ ∘ (σ (X , D₀ Y)) ] ] ∘ (distributeˡ⁻¹ +₁ distributeˡ⁻¹)) ∘ distributeʳ⁻¹ ∘ (out ⁂ out) ≈⟨ assoc ○ ([]-cong₂ refl (refl⟩∘⟨ ∘[])) ⟩∘⟨refl ⟩
[ (idC +₁ (τ (X , Y))) , i₂ ∘ [ extend (out⁻¹ ∘ (idC +₁ (τ (X , Y)))) ∘ D₁ i₁ ∘ (σ (X , Y)) , extend (out⁻¹ ∘ (idC +₁ (τ (X , Y)))) ∘ D₁ i₂ ∘ (σ (X , D₀ Y)) ] ] ∘ (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ refl (refl⟩∘⟨ ([]-cong₂ (pullˡ (sym k-assoc ○ extend-≈ (pullˡ k-identityʳ))) (pullˡ (sym k-assoc ○ extend-≈ (pullˡ k-identityʳ)))))) ⟩∘⟨refl ⟩
[ (idC +₁ (τ (X , Y))) , i₂ ∘ [ extend ((out⁻¹ ∘ (idC +₁ (τ (X , Y)))) ∘ i₁) ∘ (σ (X , Y)) , extend ((out⁻¹ ∘ (idC +₁ (τ (X , Y)))) ∘ i₂) ∘ (σ (X , D₀ Y)) ] ] ∘ (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹ ∘ (out ⁂ out) ≈⟨ []-cong₂ refl (refl⟩∘⟨ []-cong₂ (extend-≈ (pullʳ inject₁) ⟩∘⟨refl) (extend-≈ (pullʳ inject₂) ⟩∘⟨refl)) ⟩∘⟨refl ⟩
[ (idC +₁ (τ (X , Y))) , i₂ ∘ [ extend (out⁻¹ ∘ i₁ ∘ idC) ∘ (σ (X , Y)) , extend (out⁻¹ ∘ i₂ ∘ (τ (X , Y))) ∘ (σ (X , D₀ Y)) ] ] ∘ (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹ ∘ (out ⁂ out) ≈⟨ []-cong₂ refl (refl⟩∘⟨ []-cong₂ (elimˡ (extend-≈ (refl⟩∘⟨ identityʳ) ○ k-identityˡ)) (extend-≈ sym-assoc ⟩∘⟨refl ○ sym (pullˡ (▷∘extendˡ (τ (X , Y)))))) ⟩∘⟨refl ⟩
[ (idC +₁ (τ (X , Y))) , i₂ ∘ [ (σ (X , Y)) , ▷ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ] ] ∘ (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹ ∘ (out ⁂ out) ≈⟨ helper₃ ⟩
[ idC +₁ (σ (X , Y)) , i₂ ∘ [ (τ (X , Y)) , ▷ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ] ] ∘ w ≈˘⟨ cancelˡ (_≅_.isoʳ out-≅) ⟩
out ∘ out⁻¹ ∘ [ idC +₁ (σ (X , Y)) , i₂ ∘ [ (τ (X , Y)) , ▷ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ] ] ∘ w ∎)
where
helper₂ : (D₁ distributeˡ⁻¹) ∘ (σ (X , Y + D₀ Y)) ≈ [ D₁ i₁ ∘ (σ (X , Y)) , D₁ i₂ ∘ (σ (X , D₀ Y)) ] ∘ distributeˡ⁻¹
helper₂ = Iso⇒Epi (IsIso.iso isIsoˡ) ((D₁ distributeˡ⁻¹) ∘ (σ (X , Y + D₀ Y))) ([ D₁ i₁ ∘ (σ (X , Y)) , D₁ i₂ ∘ (σ (X , D₀ Y)) ] ∘ distributeˡ⁻¹) (begin
((D₁ distributeˡ⁻¹) ∘ (σ (X , Y + D₀ Y))) ∘ distributeˡ ≈⟨ ∘[] ⟩
[ ((D₁ distributeˡ⁻¹) ∘ (σ (X , Y + D₀ Y))) ∘ (idC ⁂ i₁) , ((D₁ distributeˡ⁻¹) ∘ (σ (X , Y + D₀ Y))) ∘ (idC ⁂ i₂) ] ≈⟨ []-cong₂ (refl⟩∘⟨ ⁂-cong₂ (sym D-identity) refl) (refl⟩∘⟨ ⁂-cong₂ (sym D-identity) refl) ⟩
[ ((D₁ distributeˡ⁻¹) ∘ (σ (X , Y + D₀ Y))) ∘ (D₁ idC ⁂ i₁) , ((D₁ distributeˡ⁻¹) ∘ (σ (X , Y + D₀ Y))) ∘ (D₁ idC ⁂ i₂) ] ≈⟨ []-cong₂ (pullʳ (σ-commute idC i₁)) (pullʳ (σ-commute idC i₂)) ⟩
[ (D₁ distributeˡ⁻¹) ∘ D₁ (idC ⁂ i₁) ∘ (σ (X , Y)) , (D₁ distributeˡ⁻¹) ∘ D₁ (idC ⁂ i₂) ∘ (σ (X , D₀ Y)) ] ≈⟨ []-cong₂ (pullˡ (sym D-homomorphism)) (pullˡ (sym D-homomorphism)) ⟩
[ D₁ (distributeˡ⁻¹ ∘ (idC ⁂ i₁)) ∘ (σ (X , Y)) , D₁ (distributeˡ⁻¹ ∘ (idC ⁂ i₂)) ∘ (σ (X , D₀ Y)) ] ≈⟨ []-cong₂ (D-resp-≈ distributeˡ⁻¹-i₁ ⟩∘⟨refl) (D-resp-≈ distributeˡ⁻¹-i₂ ⟩∘⟨refl) ⟩
[ D₁ i₁ ∘ (σ (X , Y)) , D₁ i₂ ∘ (σ (X , D₀ Y)) ] ≈˘⟨ cancelʳ (IsIso.isoˡ isIsoˡ) ⟩
([ D₁ i₁ ∘ (σ (X , Y)) , D₁ i₂ ∘ (σ (X , D₀ Y)) ] ∘ distributeˡ⁻¹) ∘ distributeˡ ∎)
helper₃ : [ idC +₁ (τ (X , Y)) , i₂ ∘ [ D₁ swap ∘ (τ (Y , X)) ∘ swap , ▷ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ] ] ∘ (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹ ∘ (out {X} ⁂ out {Y}) ≈ [ idC +₁ (σ (X , Y)) , i₂ ∘ [ (τ (X , Y)) , ▷ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ] ] ∘ w
helper₃ = begin
[ idC +₁ (τ (X , Y)) , i₂ ∘ [ D₁ swap ∘ (τ (Y , X)) ∘ swap , ▷ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ] ] ∘ (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹ ∘ (out {X} ⁂ out {Y}) ≈⟨ refl⟩∘⟨ helper ⟩
[ idC +₁ (τ (X , Y)) , i₂ ∘ [ D₁ swap ∘ (τ (Y , X)) ∘ swap , ▷ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ] ] ∘ [ [ i₁ ∘ i₁ , i₂ ∘ i₁ ] , [ (i₁ ∘ i₂) , (i₂ ∘ i₂) ] ] ∘ w ≈⟨ pullˡ ∘[] ⟩
[ [ idC +₁ (τ (X , Y)) , i₂ ∘ [ D₁ swap ∘ (τ (Y , X)) ∘ swap , ▷ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ] ] ∘ [ i₁ ∘ i₁ , i₂ ∘ i₁ ]
, [ idC +₁ (τ (X , Y)) , i₂ ∘ [ D₁ swap ∘ (τ (Y , X)) ∘ swap , ▷ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ] ] ∘ [ (i₁ ∘ i₂) , (i₂ ∘ i₂) ] ] ∘ w ≈⟨ ([]-cong₂ ∘[] ∘[]) ⟩∘⟨refl ⟩
[ [ [ idC +₁ (τ (X , Y)) , i₂ ∘ [ D₁ swap ∘ (τ (Y , X)) ∘ swap , ▷ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ] ] ∘ i₁ ∘ i₁ , [ idC +₁ (τ (X , Y)) , i₂ ∘ [ D₁ swap ∘ (τ (Y , X)) ∘ swap , ▷ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ] ] ∘ i₂ ∘ i₁ ]
, [ [ idC +₁ (τ (X , Y)) , i₂ ∘ [ D₁ swap ∘ (τ (Y , X)) ∘ swap , ▷ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ] ] ∘ (i₁ ∘ i₂)
, [ idC +₁ (τ (X , Y)) , i₂ ∘ [ D₁ swap ∘ (τ (Y , X)) ∘ swap , ▷ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ] ] ∘ (i₂ ∘ i₂) ] ] ∘ w ≈⟨ ([]-cong₂ ([]-cong₂ (pullˡ inject₁) (pullˡ inject₂ ○ assoc)) ([]-cong₂ (pullˡ inject₁) (pullˡ inject₂ ○ assoc))) ⟩∘⟨refl ⟩
[ [ (idC +₁ (τ (X , Y))) ∘ i₁ , i₂ ∘ [ D₁ swap ∘ (τ (Y , X)) ∘ swap , ▷ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ] ∘ i₁ ] , [ (idC +₁ (τ (X , Y))) ∘ i₂ , i₂ ∘ [ D₁ swap ∘ (τ (Y , X)) ∘ swap , ▷ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ] ∘ i₂ ] ] ∘ w ≈⟨ ([]-cong₂ ([]-cong₂ (+₁∘i₁ ○ identityʳ) (refl⟩∘⟨ inject₁)) ([]-cong₂ +₁∘i₂ (refl⟩∘⟨ inject₂))) ⟩∘⟨refl ⟩
[ [ i₁ , i₂ ∘ (σ (X , Y)) ] , [ i₂ ∘ (τ (X , Y)) , i₂ ∘ ▷ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ] ] ∘ w ≈˘⟨ ([]-cong₂ ([]-cong₂ identityʳ refl) ∘[]) ⟩∘⟨refl ⟩
[ idC +₁ (σ (X , Y)) , i₂ ∘ [ (τ (X , Y)) , ▷ ∘ extend (τ (X , Y)) ∘ (σ (X , D₀ Y)) ] ] ∘ w ∎
where
helper : (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹ ∘ (out {X} ⁂ out {Y}) ≈ [ [ i₁ ∘ i₁ , i₂ ∘ i₁ ] , [ (i₁ ∘ i₂) , (i₂ ∘ i₂) ] ] ∘ w
helper = sym (begin
[ [ i₁ ∘ i₁ , i₂ ∘ i₁ ] , [ (i₁ ∘ i₂) , (i₂ ∘ i₂) ] ] ∘ w ≈⟨ pullˡ []∘+₁ ⟩
[ (i₁ +₁ i₁) ∘ distributeʳ⁻¹ , (i₂ +₁ i₂) ∘ distributeʳ⁻¹ ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ ((+₁-cong₂ (sym distributeˡ⁻¹-i₁) (sym distributeˡ⁻¹-i₁)) ⟩∘⟨refl) ((+₁-cong₂ (sym distributeˡ⁻¹-i₂) (sym distributeˡ⁻¹-i₂)) ⟩∘⟨refl)) ⟩∘⟨refl ⟩
[ (distributeˡ⁻¹ ∘ (idC ⁂ i₁) +₁ distributeˡ⁻¹ ∘ (idC ⁂ i₁)) ∘ distributeʳ⁻¹
, (distributeˡ⁻¹ ∘ (idC ⁂ i₂) +₁ distributeˡ⁻¹ ∘ (idC ⁂ i₂)) ∘ distributeʳ⁻¹ ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ sym (([]-cong₂ (pullˡ +₁∘+₁) (pullˡ +₁∘+₁)) ⟩∘⟨refl) ⟩
[ (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ ((idC ⁂ i₁) +₁ (idC ⁂ i₁)) ∘ distributeʳ⁻¹
, (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ ((idC ⁂ i₂) +₁ (idC ⁂ i₂)) ∘ distributeʳ⁻¹ ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ sym (pullˡ ∘[]) ⟩
(distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ [ ((idC ⁂ i₁) +₁ (idC ⁂ i₁)) ∘ distributeʳ⁻¹ , ((idC ⁂ i₂) +₁ (idC ⁂ i₂)) ∘ distributeʳ⁻¹ ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ refl⟩∘⟨ []-cong₂ (distributeʳ⁻¹-natural i₁ idC idC) (distributeʳ⁻¹-natural i₂ idC idC) ⟩∘⟨refl ⟩
(distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ [ distributeʳ⁻¹ ∘ ((idC +₁ idC) ⁂ i₁) , distributeʳ⁻¹ ∘ ((idC +₁ idC) ⁂ i₂) ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ refl⟩∘⟨ (sym (pullˡ ∘[])) ⟩
(distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹ ∘ [ ((idC +₁ idC) ⁂ i₁) , ((idC +₁ idC) ⁂ i₂) ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ ([]-cong₂ (⁂-cong₂ ([]-unique id-comm-sym id-comm-sym) refl) (⁂-cong₂ ([]-unique id-comm-sym id-comm-sym) refl)) ⟩∘⟨refl ⟩
(distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹ ∘ [ (idC ⁂ i₁) , (idC ⁂ i₂) ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ cancelˡ (IsIso.isoʳ isIsoˡ) ⟩
(distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹ ∘ (out ⁂ out) ∎)
fixpoint-eq : ∀ {f : D₀ X × D₀ Y ⇒ D₀ (X × Y)} → f ≈ out⁻¹ ∘ [ idC +₁ (σ (X , Y)) , i₂ ∘ [ (τ (X , Y)) , ▷ ∘ f ] ] ∘ w → f ≈ extend [ now , f ] ∘ g
fixpoint-eq {f} fix = begin
f ≈⟨ fix ⟩
out⁻¹ ∘ [ idC +₁ (σ (X , Y)) , i₂ ∘ [ (τ (X , Y)) , ▷ ∘ f ] ] ∘ w ≈˘⟨ refl⟩∘⟨ []-cong₂ refl (refl⟩∘⟨ ([]-cong₂ refl (pullʳ inject₂))) ⟩∘⟨refl ⟩
out⁻¹ ∘ [ idC +₁ (σ (X , Y)) , i₂ ∘ [ (τ (X , Y)) , (▷ ∘ [ now , f ]) ∘ i₂ ] ] ∘ w ≈˘⟨ refl⟩∘⟨ []-cong₂ refl (sym ∘[] ○ refl⟩∘⟨ []-cong₂ (elimˡ (extend-≈ inject₁ ○ k-identityˡ)) (pullˡ k-identityʳ)) ⟩∘⟨refl ⟩
out⁻¹ ∘ [ idC +₁ (σ (X , Y)) , [ i₂ ∘ extend ([ now , f ] ∘ i₁) ∘ (τ (X , Y)) , i₂ ∘ extend (▷ ∘ [ now , f ]) ∘ now ∘ i₂ ] ] ∘ w ≈˘⟨ refl⟩∘⟨ []-cong₂ ([]-cong₂ (sym identityʳ) (refl⟩∘⟨ (elimˡ ((extend-≈ inject₁) ○ k-identityˡ)))) ([]-cong₂ (pullʳ (pullˡ ((sym k-assoc) ○ extend-≈ (pullˡ k-identityʳ)))) (pullʳ (pullˡ (▷∘extendʳ [ now , f ])))) ⟩∘⟨refl ⟩
out⁻¹ ∘ [ [ i₁ , i₂ ∘ extend ([ now , f ] ∘ i₁) ∘ (σ (X , Y)) ] , [ (i₂ ∘ extend [ now , f ]) ∘ D₁ i₁ ∘ (τ (X , Y)) , (i₂ ∘ extend [ now , f ]) ∘ ▷ ∘ now ∘ i₂ ] ] ∘ w ≈˘⟨ refl⟩∘⟨ []-cong₂ ([]-cong₂ inject₁ (pullʳ (pullˡ ((sym k-assoc) ○ extend-≈ (pullˡ k-identityʳ))))) ∘[] ⟩∘⟨refl ⟩
out⁻¹ ∘ [ [ [ i₁ , out ∘ f ] ∘ i₁ , (i₂ ∘ extend [ now , f ]) ∘ D₁ i₁ ∘ (σ (X , Y)) ] , (i₂ ∘ extend [ now , f ]) ∘ [ D₁ i₁ ∘ (τ (X , Y)) , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈˘⟨ refl⟩∘⟨ (([]-cong₂ []∘+₁ (pullˡ inject₂)) ⟩∘⟨refl) ⟩
out⁻¹ ∘ [ [ [ i₁ , out ∘ f ] , i₂ ∘ extend [ now , f ] ] ∘ (i₁ +₁ D₁ i₁ ∘ (σ (X , Y))) , [ [ i₁ , out ∘ f ] , i₂ ∘ extend [ now , f ] ] ∘ i₂ ∘ [ D₁ i₁ ∘ (τ (X , Y)) , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈˘⟨ refl⟩∘⟨ (pullˡ ∘[]) ⟩
out⁻¹ ∘ [ [ i₁ , out ∘ f ] , i₂ ∘ extend [ now , f ] ] ∘ [ i₁ +₁ D₁ i₁ ∘ (σ (X , Y)) , i₂ ∘ [ D₁ i₁ ∘ (τ (X , Y)) , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈⟨ Iso⇒Mono (_≅_.iso out-≅) (out⁻¹ ∘ [ [ i₁ , out ∘ f ] , i₂ ∘ extend [ now , f ] ] ∘ [ i₁ +₁ D₁ i₁ ∘ (σ (X , Y)) , i₂ ∘ [ D₁ i₁ ∘ (τ (X , Y)) , ▷ ∘ now ∘ i₂ ] ] ∘ w) (extend [ now , f ] ∘ out⁻¹ ∘ [ i₁ +₁ D₁ i₁ ∘ (σ (X , Y)) , i₂ ∘ [ D₁ i₁ ∘ (τ (X , Y)) , ▷ ∘ now ∘ i₂ ] ] ∘ w) helper ⟩
extend [ now , f ] ∘ out⁻¹ ∘ [ i₁ +₁ D₁ i₁ ∘ (σ (X , Y)) , i₂ ∘ [ D₁ i₁ ∘ (τ (X , Y)) , ▷ ∘ now ∘ i₂ ] ] ∘ w ∎
where
helper = begin
out ∘ out⁻¹ ∘ [ [ i₁ , out ∘ f ] , i₂ ∘ extend [ now , f ] ] ∘ [ i₁ +₁ D₁ i₁ ∘ (σ (X , Y)) , i₂ ∘ [ D₁ i₁ ∘ (τ (X , Y)) , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈⟨ cancelˡ (_≅_.isoʳ out-≅) ⟩
[ [ i₁ , out ∘ f ] , i₂ ∘ extend [ now , f ] ] ∘ [ i₁ +₁ D₁ i₁ ∘ (σ (X , Y)) , i₂ ∘ [ D₁ i₁ ∘ (τ (X , Y)) , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈˘⟨ ([]-cong₂ (∘[] ○ []-cong₂ unitlaw refl) refl) ⟩∘⟨refl ⟩
[ out ∘ [ now , f ] , i₂ ∘ extend [ now , f ] ] ∘ [ i₁ +₁ D₁ i₁ ∘ (σ (X , Y)) , i₂ ∘ [ D₁ i₁ ∘ (τ (X , Y)) , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈˘⟨ pullʳ (cancelˡ (_≅_.isoʳ out-≅)) ⟩
([ out ∘ [ now , f ] , i₂ ∘ extend [ now , f ] ] ∘ out) ∘ out⁻¹ ∘ [ i₁ +₁ D₁ i₁ ∘ (σ (X , Y)) , i₂ ∘ [ D₁ i₁ ∘ (τ (X , Y)) , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈˘⟨ pullˡ (extendlaw [ now , f ]) ⟩
out ∘ extend [ now , f ] ∘ out⁻¹ ∘ [ i₁ +₁ D₁ i₁ ∘ (σ (X , Y)) , i₂ ∘ [ D₁ i₁ ∘ (τ (X , Y)) , ▷ ∘ now ∘ i₂ ] ] ∘ w ∎