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🚧 Work on commutativity

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24
src/Category/Instance/AmbientCategory.lagda.md
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@ -114,6 +114,30 @@ module Category.Instance.AmbientCategory where
(distributeˡ⁻¹ ∘ [ ((f ⁂ (g +₁ h)) ∘ (idC ⁂ i₁)) , ((f ⁂ (g +₁ h)) ∘ (idC ⁂ i₂)) ]) ∘ distributeˡ⁻¹ ≈⟨ sym (pullˡ (pullʳ ∘[])) ⟩
(distributeˡ⁻¹ ∘ (f ⁂ (g +₁ h))) ∘ distributeˡ ∘ distributeˡ⁻¹ ≈⟨ sym (introʳ (IsIso.isoʳ isIsoˡ)) ⟩
distributeˡ⁻¹ ∘ (f ⁂ (g +₁ h)) ∎
-- TODO extract distribution proofs of the non inverses
distribute₁' : ∀ {X Y Z U V W} (f : X ⇒ U) (g : Y ⇒ V) (h : Z ⇒ W) → ((g ⁂ f) +₁ (h ⁂ f)) ∘ distributeʳ⁻¹ ≈ distributeʳ⁻¹ ∘ ((g +₁ h) ⁂ f)
distribute₁' f g h =
Iso⇒Mono C (IsIso.iso isIsoʳ) ((g ⁂ f +₁ h ⁂ f) ∘ distributeʳ⁻¹) (distributeʳ⁻¹ ∘ ((g +₁ h) ⁂ f))
(Iso⇒Epi C (IsIso.iso isIsoʳ) (distributeʳ ∘ (g ⁂ f +₁ h ⁂ f) ∘ distributeʳ⁻¹) (distributeʳ ∘ distributeʳ⁻¹ ∘ ((g +₁ h) ⁂ f)) (begin
(distributeʳ ∘ (g ⁂ f +₁ h ⁂ f) ∘ distributeʳ⁻¹) ∘ distributeʳ ≈⟨ pullʳ (cancelʳ (IsIso.isoˡ isIsoʳ)) ⟩
distributeʳ ∘ (g ⁂ f +₁ h ⁂ f) ≈⟨ []∘+₁ ⟩
[ (i₁ ⁂ idC) ∘ (g ⁂ f) , (i₂ ⁂ idC) ∘ (h ⁂ f) ] ≈⟨ []-cong₂ ⁂∘⁂ ⁂∘⁂ ⟩
[ (i₁ ∘ g ⁂ idC ∘ f) , (i₂ ∘ h ⁂ idC ∘ f) ] ≈˘⟨ []-cong₂ (⁂-cong₂ +₁∘i₁ id-comm) (⁂-cong₂ +₁∘i₂ id-comm) ⟩
[ (g +₁ h) ∘ i₁ ⁂ f ∘ idC , (g +₁ h) ∘ i₂ ⁂ f ∘ idC ] ≈˘⟨ []-cong₂ ⁂∘⁂ ⁂∘⁂ ⟩
[ ((g +₁ h) ⁂ f) ∘ (i₁ ⁂ idC) , ((g +₁ h) ⁂ f) ∘ (i₂ ⁂ idC) ] ≈˘⟨ ∘[] ⟩
(((g +₁ h) ⁂ f) ∘ distributeʳ) ≈˘⟨ cancelˡ (IsIso.isoʳ isIsoʳ) ⟩∘⟨refl ⟩
(distributeʳ ∘ distributeʳ⁻¹ ∘ ((g +₁ h) ⁂ f)) ∘ distributeʳ ∎))
dstldstr : ∀ {X Y U V} → (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹ ≈ [ i₁ +₁ i₁ , i₂ +₁ i₂ ] ∘ (distributeʳ⁻¹ +₁ distributeʳ⁻¹) ∘ (distributeˡ⁻¹ {X + Y} {U} {V})
dstldstr = Iso⇒Epi C (IsIso.iso isIsoˡ) ((distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹) ([ i₁ +₁ i₁ , i₂ +₁ i₂ ] ∘ (distributeʳ⁻¹ +₁ distributeʳ⁻¹) ∘ distributeˡ⁻¹) (sym (begin
(([ i₁ +₁ i₁ , i₂ +₁ i₂ ] ∘ (distributeʳ⁻¹ +₁ distributeʳ⁻¹) ∘ distributeˡ⁻¹) ∘ distributeˡ) ≈⟨ pullʳ (cancelʳ (IsIso.isoˡ isIsoˡ)) ⟩
([ i₁ +₁ i₁ , i₂ +₁ i₂ ] ∘ (distributeʳ⁻¹ +₁ distributeʳ⁻¹)) ≈⟨ []∘+₁ ⟩
[ (i₁ +₁ i₁) ∘ distributeʳ⁻¹ , (i₂ +₁ i₂) ∘ distributeʳ⁻¹ ] ≈⟨ {! !} ⟩
{! !} ≈˘⟨ {! !} ⟩
{! !} ≈˘⟨ {! !} ⟩
{! !} ≈˘⟨ {! !} ⟩
{! !} ≈˘⟨ {! !} ⟩
[ ((distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹) ∘ (idC ⁂ i₁) , ((distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹) ∘ (idC ⁂ i₂) ] ≈˘⟨ ∘[] ⟩
(((distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹) ∘ distributeˡ) ∎))
iso-epi-from : ∀ {X Y} → (iso : X ≅ Y) → Epi (_≅_.from iso)
iso-epi-from iso = λ f g eq → begin

131
src/Monad/Instance/Delay/Commutative.lagda.md
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@ -76,41 +76,58 @@ module Monad.Instance.Delay.Commutative {o ℓ e} (ambient : Ambient o ℓ e) (D
(out⁻¹ ∘ out) ∘ σ ∘ (out⁻¹ ⁂ idC) ≈⟨ pullʳ (pullˡ (u-commutes (σ-coalg X Y))) ⟩
out⁻¹ ∘ ((idC +₁ σ) ∘ distributeʳ⁻¹ ∘ (out ⁂ idC)) ∘ (out⁻¹ ⁂ idC) ≈⟨ refl⟩∘⟨ (pullʳ (cancelʳ (⁂∘⁂ ○ ⁂-cong₂ (_≅_.isoʳ out-≅) identity² ○ ⟨⟩-unique id-comm id-comm))) ⟩
out⁻¹ ∘ (idC +₁ σ) ∘ distributeʳ⁻¹ ∎
-- TODO this should be in commutative, it expresses that σ is natural
σ-commute : ∀ {U V W X : Obj} (f : U ⇒ V) (g : W ⇒ X) → σ ∘ (extend (now ∘ f) ⁂ g) ≈ extend (now ∘ (f ⁂ g)) ∘ σ
σ-commute {U} {V} {W} {X} f g = begin
σ ∘ (D₁ f ⁂ g) ≈⟨ pullʳ (pullʳ swap∘⁂) ⟩
D₁ swap ∘ τ ∘ (g ⁂ extend (now ∘ f)) ∘ swap ≈⟨ refl⟩∘⟨ (pullˡ (τ-commute (g , f))) ⟩
D₁ swap ∘ (D₁ (g ⁂ f) ∘ τ) ∘ swap ≈⟨ pullˡ (pullˡ (sym D-homomorphism)) ⟩
(D₁ (swap ∘ (g ⁂ f)) ∘ τ) ∘ swap ≈⟨ ((D-resp-≈ swap∘⁂) ⟩∘⟨refl) ⟩∘⟨refl ⟩
(D₁ ((f ⁂ g) ∘ swap) ∘ τ) ∘ swap ≈⟨ pushˡ D-homomorphism ⟩∘⟨refl ⟩
(D₁ (f ⁂ g) ∘ D₁ swap ∘ τ) ∘ swap ≈⟨ assoc²' ⟩
D₁ (f ⁂ g) ∘ σ ∎
commutes' : ∀ {X Y} → extend σ ∘ τ {D₀ X} {Y} ≈ extend τ ∘ σ
commutes' {X} {Y} = guarded-unique g (extend σ ∘ τ) (extend τ ∘ σ) 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 , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w
guarded : is-guarded g
guarded = [ idC +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w , (begin
(i₁ +₁ idC) ∘ [ idC +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈⟨ pullˡ ∘[] ⟩
[ (i₁ +₁ idC) ∘ (idC +₁ D₁ i₁ ∘ σ) , (i₁ +₁ idC) ∘ i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈⟨ ([]-cong₂ +₁∘+₁ (pullˡ +₁∘i₂)) ⟩∘⟨refl ⟩
[ (i₁ ∘ idC +₁ idC ∘ D₁ i₁ ∘ σ) , (i₂ ∘ idC) ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈⟨ ([]-cong₂ (+₁-cong₂ identityʳ identityˡ) (identityʳ ⟩∘⟨refl)) ⟩∘⟨refl ⟩
[ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈⟨ sym (cancelˡ (_≅_.isoʳ out-≅)) ⟩
out ∘ out⁻¹ ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ∎)
helper₁ : (D₁ distributeʳ⁻¹) ∘ τ ≈ [ D₁ i₁ ∘ τ , D₁ i₂ ∘ τ ] ∘ distributeʳ⁻¹
helper₁ = Iso⇒Epi (IsIso.iso isIsoʳ) ((D₁ distributeʳ⁻¹) ∘ τ) ([ D₁ i₁ ∘ τ , D₁ i₂ ∘ τ ] ∘ distributeʳ⁻¹) (begin
((D₁ distributeʳ⁻¹) ∘ τ) ∘ distributeʳ ≈⟨ ∘[] ⟩
[ ((D₁ distributeʳ⁻¹) ∘ τ) ∘ (i₁ ⁂ idC) , ((D₁ distributeʳ⁻¹) ∘ τ) ∘ (i₂ ⁂ idC) ] ≈⟨ []-cong₂ (refl⟩∘⟨ (⁂-cong₂ refl (sym D-identity))) (refl⟩∘⟨ (⁂-cong₂ refl (sym D-identity))) ⟩
[ ((D₁ distributeʳ⁻¹) ∘ τ) ∘ (i₁ ⁂ D₁ idC) , ((D₁ distributeʳ⁻¹) ∘ τ) ∘ (i₂ ⁂ D₁ idC) ] ≈⟨ []-cong₂ (pullʳ (τ-commute (i₁ , idC))) (pullʳ (τ-commute (i₂ , idC))) ⟩
[ (D₁ distributeʳ⁻¹) ∘ D₁ (i₁ ⁂ idC) ∘ τ , (D₁ distributeʳ⁻¹) ∘ D₁ (i₂ ⁂ idC) ∘ τ ] ≈⟨ []-cong₂ (pullˡ (sym D-homomorphism)) (pullˡ (sym D-homomorphism)) ⟩
[ D₁ (distributeʳ⁻¹ ∘ (i₁ ⁂ idC)) ∘ τ , D₁ (distributeʳ⁻¹ ∘ (i₂ ⁂ idC)) ∘ τ ] ≈⟨ []-cong₂ (D-resp-≈ dstr-law₃ ⟩∘⟨refl) ((D-resp-≈ dstr-law₄) ⟩∘⟨refl) ⟩
[ D₁ i₁ ∘ τ , D₁ i₂ ∘ τ ] ≈˘⟨ cancelʳ (IsIso.isoˡ isIsoʳ) ⟩
([ D₁ i₁ ∘ τ , D₁ i₂ ∘ τ ] ∘ distributeʳ⁻¹) ∘ distributeʳ ∎)
guarded = {! !}
-- guarded : is-guarded g
-- guarded = [ idC +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w , (begin
-- (i₁ +₁ idC) ∘ [ idC +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈⟨ pullˡ ∘[] ⟩
-- [ (i₁ +₁ idC) ∘ (idC +₁ D₁ i₁ ∘ σ) , (i₁ +₁ idC) ∘ i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈⟨ ([]-cong₂ +₁∘+₁ (pullˡ +₁∘i₂)) ⟩∘⟨refl ⟩
-- [ (i₁ ∘ idC +₁ idC ∘ D₁ i₁ ∘ σ) , (i₂ ∘ idC) ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈⟨ ([]-cong₂ (+₁-cong₂ identityʳ identityˡ) (identityʳ ⟩∘⟨refl)) ⟩∘⟨refl ⟩
-- [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈⟨ sym (cancelˡ (_≅_.isoʳ out-≅)) ⟩
-- out ∘ out⁻¹ ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ∎)
-- helper₁ : (D₁ distributeʳ⁻¹) ∘ τ ≈ [ D₁ i₁ ∘ τ , D₁ i₂ ∘ τ ] ∘ distributeʳ⁻¹
-- helper₁ = Iso⇒Epi (IsIso.iso isIsoʳ) ((D₁ distributeʳ⁻¹) ∘ τ) ([ D₁ i₁ ∘ τ , D₁ i₂ ∘ τ ] ∘ distributeʳ⁻¹) (begin
-- ((D₁ distributeʳ⁻¹) ∘ τ) ∘ distributeʳ ≈⟨ ∘[] ⟩
-- [ ((D₁ distributeʳ⁻¹) ∘ τ) ∘ (i₁ ⁂ idC) , ((D₁ distributeʳ⁻¹) ∘ τ) ∘ (i₂ ⁂ idC) ] ≈⟨ []-cong₂ (refl⟩∘⟨ (⁂-cong₂ refl (sym D-identity))) (refl⟩∘⟨ (⁂-cong₂ refl (sym D-identity))) ⟩
-- [ ((D₁ distributeʳ⁻¹) ∘ τ) ∘ (i₁ ⁂ D₁ idC) , ((D₁ distributeʳ⁻¹) ∘ τ) ∘ (i₂ ⁂ D₁ idC) ] ≈⟨ []-cong₂ (pullʳ (τ-commute (i₁ , idC))) (pullʳ (τ-commute (i₂ , idC))) ⟩
-- [ (D₁ distributeʳ⁻¹) ∘ D₁ (i₁ ⁂ idC) ∘ τ , (D₁ distributeʳ⁻¹) ∘ D₁ (i₂ ⁂ idC) ∘ τ ] ≈⟨ []-cong₂ (pullˡ (sym D-homomorphism)) (pullˡ (sym D-homomorphism)) ⟩
-- [ D₁ (distributeʳ⁻¹ ∘ (i₁ ⁂ idC)) ∘ τ , D₁ (distributeʳ⁻¹ ∘ (i₂ ⁂ idC)) ∘ τ ] ≈⟨ []-cong₂ (D-resp-≈ dstr-law₃ ⟩∘⟨refl) ((D-resp-≈ dstr-law₄) ⟩∘⟨refl) ⟩
-- [ D₁ i₁ ∘ τ , D₁ i₂ ∘ τ ] ≈˘⟨ cancelʳ (IsIso.isoˡ isIsoʳ) ⟩
-- ([ D₁ i₁ ∘ τ , D₁ i₂ ∘ τ ] ∘ distributeʳ⁻¹) ∘ distributeʳ ∎)
fixpoint₁ = {! !}
-- fixpoint₁ = Iso⇒Mono (_≅_.iso out-≅) (extend σ ∘ τ) (out⁻¹ ∘ [ idC +₁ σ , i₂ ∘ [ τ , ▷ ∘ extend σ ∘ τ ] ] ∘ w) (begin
-- out ∘ extend σ ∘ τ ≈⟨ pullˡ (extendlaw σ) ⟩
-- ([ out ∘ σ , i₂ ∘ extend' σ ] ∘ out) ∘ τ ≈⟨ pullʳ (τ-law (D₀ X , Y)) ⟩
-- [ out ∘ σ , i₂ ∘ extend' σ ] ∘ (idC +₁ τ) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ pullˡ []∘+₁ ⟩
-- [ (out ∘ σ) ∘ idC , (i₂ ∘ extend' σ) ∘ τ ] ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ ([]-cong₂ (identityʳ ○ u-commutes (σ-coalg X Y)) assoc) ⟩∘⟨refl ⟩
-- [ (idC +₁ σ) ∘ distributeʳ⁻¹ ∘ (out ⁂ idC) , i₂ ∘ extend' σ ∘ τ ] ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ ⁂-cong₂ (sym (_≅_.isoˡ out-≅)) refl ⟩
-- [ (idC +₁ σ) ∘ distributeʳ⁻¹ ∘ (out ⁂ idC) , i₂ ∘ extend' σ ∘ τ ] ∘ distributeˡ⁻¹ ∘ (out⁻¹ ∘ out ⁂ out) ≈⟨ sym (refl⟩∘⟨ refl⟩∘⟨ (⁂∘⁂ ○ ⁂-cong₂ refl (elimˡ ([]-unique id-comm-sym id-comm-sym)))) ⟩
-- [ (idC +₁ σ) ∘ distributeʳ⁻¹ ∘ (out ⁂ idC) , i₂ ∘ extend' σ ∘ τ ] ∘ distributeˡ⁻¹ ∘ (out⁻¹ ⁂ (idC +₁ idC)) ∘ (out ⁂ out) ≈⟨ refl⟩∘⟨ pullˡ (sym (distribute₁ out⁻¹ idC idC)) ⟩
-- [ (idC +₁ σ) ∘ distributeʳ⁻¹ ∘ (out ⁂ idC) , i₂ ∘ extend' σ ∘ τ ] ∘ (((out⁻¹ ⁂ idC) +₁ (out⁻¹ ⁂ idC)) ∘ distributeˡ⁻¹) ∘ (out ⁂ out) ≈⟨ pullˡ (pullˡ []∘+₁) ⟩
-- ([ ((idC +₁ σ) ∘ distributeʳ⁻¹ ∘ (out ⁂ idC)) ∘ (out⁻¹ ⁂ idC) , (i₂ ∘ extend' σ ∘ τ) ∘ (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) ∘ distributeʳ⁻¹ , i₂ ∘ extend' σ ∘ D₁ (out⁻¹ ⁂ idC) ∘ τ ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ refl (refl⟩∘⟨ (pullˡ (sym k-assoc)) ○ refl⟩∘⟨ ((extend-≈ (pullˡ k-identityʳ)) ⟩∘⟨refl))) ⟩∘⟨refl ⟩
-- [ (idC +₁ σ) ∘ distributeʳ⁻¹ , i₂ ∘ extend' (σ ∘ (out⁻¹ ⁂ idC)) ∘ τ ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ refl (refl⟩∘⟨ ((extend-≈ σ-helper) ⟩∘⟨refl))) ⟩∘⟨refl ⟩
-- [ (idC +₁ σ) ∘ distributeʳ⁻¹ , i₂ ∘ extend' (out⁻¹ ∘ (idC +₁ σ) ∘ distributeʳ⁻¹) ∘ τ ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈˘⟨ ([]-cong₂ refl (refl⟩∘⟨ ((sym k-assoc ○ extend-≈ (pullˡ k-identityʳ) ○ extend-≈ assoc) ⟩∘⟨refl))) ⟩∘⟨refl ⟩
-- [ (idC +₁ σ) ∘ distributeʳ⁻¹ , i₂ ∘ (extend' (out⁻¹ ∘ (idC +₁ σ)) ∘ D₁ distributeʳ⁻¹) ∘ τ ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ refl (refl⟩∘⟨ pullʳ helper₁)) ⟩∘⟨refl ⟩
-- [ (idC +₁ σ) ∘ distributeʳ⁻¹ , i₂ ∘ extend' (out⁻¹ ∘ (idC +₁ σ)) ∘ [ D₁ i₁ ∘ τ , D₁ i₂ ∘ τ ] ∘ distributeʳ⁻¹ ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈˘⟨ pullˡ ([]∘+₁ ○ []-cong₂ refl assoc²') ⟩
-- [ (idC +₁ σ) , i₂ ∘ extend' (out⁻¹ ∘ (idC +₁ σ)) ∘ [ D₁ i₁ ∘ τ , D₁ i₂ ∘ τ ] ] ∘ (distributeʳ⁻¹ +₁ distributeʳ⁻¹) ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ refl (refl⟩∘⟨ ∘[])) ⟩∘⟨refl ⟩
@ -120,25 +137,69 @@ module Monad.Instance.Delay.Commutative {o ℓ e} (ambient : Ambient o ℓ e) (D
-- [ (idC +₁ σ) , i₂ ∘ [ τ , extend' (▷ ∘ σ) ∘ τ ] ] ∘ (distributeʳ⁻¹ +₁ distributeʳ⁻¹) ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ refl (refl⟩∘⟨ ([]-cong₂ refl ((sym (▷∘extendˡ σ)) ⟩∘⟨refl ○ assoc)))) ⟩∘⟨refl ⟩
-- [ idC +₁ σ , i₂ ∘ [ τ , ▷ ∘ extend σ ∘ τ ] ] ∘ w ≈˘⟨ cancelˡ (_≅_.isoʳ out-≅) ⟩
-- out ∘ out⁻¹ ∘ [ idC +₁ σ , i₂ ∘ [ τ , ▷ ∘ extend σ ∘ τ ] ] ∘ w ∎)
fixpoint₂ = {! !}
fixpoint-eq : ∀ {f : D₀ X × D₀ Y ⇒ D₀ (X × Y)} → f ≈ out⁻¹ ∘ [ idC +₁ σ , i₂ ∘ [ τ , ▷ ∘ f ] ] ∘ w → f ≈ extend [ now , f ] ∘ g
fixpoint-eq {f} fix = begin
f ≈⟨ fix ⟩
out⁻¹ ∘ [ idC +₁ σ , i₂ ∘ [ τ , ▷ ∘ f ] ] ∘ w ≈˘⟨ refl⟩∘⟨ []-cong₂ refl (refl⟩∘⟨ ([]-cong₂ refl (pullʳ inject₂))) ⟩∘⟨refl ⟩
out⁻¹ ∘ [ idC +₁ σ , i₂ ∘ [ τ , (▷ ∘ [ now , f ]) ∘ i₂ ] ] ∘ w ≈˘⟨ refl⟩∘⟨ []-cong₂ refl (sym ∘[] ○ refl⟩∘⟨ []-cong₂ (elimˡ (extend-≈ inject₁ ○ k-identityˡ)) (pullˡ k-identityʳ)) ⟩∘⟨refl ⟩
out⁻¹ ∘ [ idC +₁ σ , [ i₂ ∘ extend ([ now , f ] ∘ i₁) ∘ τ , 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 ] , [ (i₂ ∘ extend [ now , f ]) ∘ D₁ i₁ ∘ τ , (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 ] , (i₂ ∘ extend [ now , f ]) ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈˘⟨ refl⟩∘⟨ (([]-cong₂ []∘+₁ (pullˡ inject₂)) ⟩∘⟨refl) ⟩
out⁻¹ ∘ [ [ [ i₁ , out ∘ f ] , i₂ ∘ extend [ now , f ] ] ∘ (i₁ +₁ D₁ i₁ ∘ σ) , [ [ i₁ , out ∘ f ] , i₂ ∘ extend [ now , f ] ] ∘ i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈˘⟨ refl⟩∘⟨ (pullˡ ∘[]) ⟩
out⁻¹ ∘ [ [ i₁ , out ∘ f ] , i₂ ∘ extend [ now , f ] ] ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈⟨ Iso⇒Mono (_≅_.iso out-≅) (out⁻¹ ∘ [ [ i₁ , out ∘ f ] , i₂ ∘ extend [ now , f ] ] ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w) (extend [ now , f ] ∘ out⁻¹ ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w) helper ⟩
extend [ now , f ] ∘ out⁻¹ ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ∎
where
helper = begin
out ∘ out⁻¹ ∘ [ [ i₁ , out ∘ f ] , i₂ ∘ extend [ now , f ] ] ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈⟨ cancelˡ (_≅_.isoʳ out-≅) ⟩
[ [ i₁ , out ∘ f ] , i₂ ∘ extend [ now , f ] ] ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈˘⟨ ([]-cong₂ (∘[] ○ []-cong₂ unitlaw refl) refl) ⟩∘⟨refl ⟩
[ out ∘ [ now , f ] , i₂ ∘ extend [ now , f ] ] ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈˘⟨ pullʳ (cancelˡ (_≅_.isoʳ out-≅)) ⟩
([ out ∘ [ now , f ] , i₂ ∘ extend [ now , f ] ] ∘ out) ∘ out⁻¹ ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈˘⟨ pullˡ (extendlaw [ now , f ]) ⟩
out ∘ extend [ now , f ] ∘ out⁻¹ ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ∎
helper₂ : (D₁ distributeˡ⁻¹) ∘ σ ≈ [ D₁ i₁ ∘ σ , D₁ i₂ ∘ σ ] ∘ distributeˡ⁻¹
helper₂ = Iso⇒Epi (IsIso.iso isIsoˡ) ((D₁ distributeˡ⁻¹) ∘ σ) ([ D₁ i₁ ∘ σ , D₁ i₂ ∘ σ ] ∘ distributeˡ⁻¹) (begin
((D₁ distributeˡ⁻¹) ∘ σ) ∘ distributeˡ ≈⟨ ∘[] ⟩
[ ((D₁ distributeˡ⁻¹) ∘ σ) ∘ (idC ⁂ i₁) , ((D₁ distributeˡ⁻¹) ∘ σ) ∘ (idC ⁂ i₂) ] ≈⟨ []-cong₂ (refl⟩∘⟨ ⁂-cong₂ (sym D-identity) refl) (refl⟩∘⟨ ⁂-cong₂ (sym D-identity) refl) ⟩
[ ((D₁ distributeˡ⁻¹) ∘ σ) ∘ (D₁ idC ⁂ i₁) , ((D₁ distributeˡ⁻¹) ∘ σ) ∘ (D₁ idC ⁂ i₂) ] ≈⟨ []-cong₂ (pullʳ (σ-commute idC i₁)) (pullʳ (σ-commute idC i₂)) ⟩
[ (D₁ distributeˡ⁻¹) ∘ D₁ (idC ⁂ i₁) ∘ σ , (D₁ distributeˡ⁻¹) ∘ D₁ (idC ⁂ i₂) ∘ σ ] ≈⟨ []-cong₂ (pullˡ (sym D-homomorphism)) (pullˡ (sym D-homomorphism)) ⟩
[ D₁ (distributeˡ⁻¹ ∘ (idC ⁂ i₁)) ∘ σ , D₁ (distributeˡ⁻¹ ∘ (idC ⁂ i₂)) ∘ σ ] ≈⟨ []-cong₂ (D-resp-≈ dstr-law₁ ⟩∘⟨refl) (D-resp-≈ dstr-law₂ ⟩∘⟨refl) ⟩
[ D₁ i₁ ∘ σ , D₁ i₂ ∘ σ ] ≈˘⟨ cancelʳ (IsIso.isoˡ isIsoˡ) ⟩
([ D₁ i₁ ∘ σ , D₁ i₂ ∘ σ ] ∘ distributeˡ⁻¹) ∘ distributeˡ ∎)
fixpoint₂ = Iso⇒Mono ((_≅_.iso out-≅)) (extend τ ∘ σ) (out⁻¹ ∘ [ idC +₁ σ , i₂ ∘ [ τ , ▷ ∘ extend τ ∘ σ ] ] ∘ w) (begin
out ∘ extend τ ∘ σ ≈⟨ pullˡ (extendlaw τ) ⟩
([ out ∘ τ , i₂ ∘ extend τ ] ∘ out) ∘ σ ≈⟨ pullʳ (u-commutes (σ-coalg X (D₀ Y))) ⟩
[ out ∘ τ , i₂ ∘ extend τ ] ∘ (idC +₁ σ) ∘ distributeʳ⁻¹ ∘ (out ⁂ idC) ≈⟨ pullˡ []∘+₁ ⟩
[ (out ∘ τ) ∘ idC , (i₂ ∘ extend τ) ∘ σ ] ∘ distributeʳ⁻¹ ∘ (out ⁂ idC) ≈⟨ ([]-cong₂ (identityʳ ○ τ-law (X , Y)) assoc) ⟩∘⟨refl ⟩
[ (idC +₁ τ) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ∘ extend τ ∘ σ ] ∘ distributeʳ⁻¹ ∘ (out ⁂ idC) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ sym (⁂∘⁂ ○ ⁂-cong₂ (elimˡ ([]-unique id-comm-sym id-comm-sym)) (_≅_.isoˡ out-≅)) ⟩
[ (idC +₁ τ) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ∘ extend τ ∘ σ ] ∘ distributeʳ⁻¹ ∘ ((idC +₁ idC) ⁂ out⁻¹) ∘ (out ⁂ out) ≈⟨ refl⟩∘⟨ (pullˡ (sym (distribute₁' out⁻¹ idC idC))) ⟩
[ (idC +₁ τ) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ∘ extend τ ∘ σ ] ∘ (((idC ⁂ out⁻¹) +₁ (idC ⁂ out⁻¹)) ∘ distributeʳ⁻¹) ∘ (out ⁂ out) ≈⟨ pullˡ (pullˡ []∘+₁) ⟩
([ ((idC +₁ τ) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out)) ∘ (idC ⁂ out⁻¹) , (i₂ ∘ extend τ ∘ σ) ∘ (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) ∘ distributeˡ⁻¹ , (i₂ ∘ extend τ ∘ σ) ∘ (D₁ idC ⁂ out⁻¹) ] ∘ distributeʳ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ refl (pullʳ (pullʳ (σ-commute idC out⁻¹)))) ⟩∘⟨refl ⟩
[ (idC +₁ τ) ∘ distributeˡ⁻¹ , i₂ ∘ extend τ ∘ D₁ (idC ⁂ out⁻¹) ∘ σ ] ∘ distributeʳ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ refl (refl⟩∘⟨ (pullˡ ((sym k-assoc) ○ extend-≈ (pullˡ k-identityʳ))))) ⟩∘⟨refl ⟩
[ (idC +₁ τ) ∘ distributeˡ⁻¹ , i₂ ∘ extend (τ ∘ (idC ⁂ out⁻¹)) ∘ σ ] ∘ distributeʳ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ refl (refl⟩∘⟨ ((extend-≈ (τ-helper (X , Y))) ⟩∘⟨refl))) ⟩∘⟨refl ⟩
[ (idC +₁ τ) ∘ distributeˡ⁻¹ , i₂ ∘ extend (out⁻¹ ∘ (idC +₁ τ) ∘ distributeˡ⁻¹) ∘ σ ] ∘ distributeʳ⁻¹ ∘ (out ⁂ out) ≈˘⟨ ([]-cong₂ refl (refl⟩∘⟨ pullˡ ((sym k-assoc) ○ (extend-≈ (pullˡ k-identityʳ) ○ extend-≈ assoc)))) ⟩∘⟨refl ⟩
[ (idC +₁ τ) ∘ distributeˡ⁻¹ , i₂ ∘ extend (out⁻¹ ∘ (idC +₁ τ)) ∘ D₁ distributeˡ⁻¹ ∘ σ ] ∘ distributeʳ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ refl ((refl⟩∘⟨ (refl⟩∘⟨ helper₂)) ○ sym assoc²')) ⟩∘⟨refl ⟩
[ (idC +₁ τ) ∘ distributeˡ⁻¹ , (i₂ ∘ extend (out⁻¹ ∘ (idC +₁ τ)) ∘ [ D₁ i₁ ∘ σ , D₁ i₂ ∘ σ ]) ∘ distributeˡ⁻¹ ] ∘ distributeʳ⁻¹ ∘ (out ⁂ out) ≈˘⟨ []∘+₁ ⟩∘⟨refl ⟩
([ (idC +₁ τ) , i₂ ∘ extend (out⁻¹ ∘ (idC +₁ τ)) ∘ [ D₁ i₁ ∘ σ , D₁ i₂ ∘ σ ] ] ∘ (distributeˡ⁻¹ +₁ distributeˡ⁻¹)) ∘ distributeʳ⁻¹ ∘ (out ⁂ out) ≈⟨ assoc ○ ([]-cong₂ refl (refl⟩∘⟨ ∘[])) ⟩∘⟨refl ⟩
[ (idC +₁ τ) , i₂ ∘ [ extend (out⁻¹ ∘ (idC +₁ τ)) ∘ D₁ i₁ ∘ σ , extend (out⁻¹ ∘ (idC +₁ τ)) ∘ D₁ i₂ ∘ σ ] ] ∘ (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) , i₂ ∘ [ extend ((out⁻¹ ∘ (idC +₁ τ)) ∘ i₁) ∘ σ , extend ((out⁻¹ ∘ (idC +₁ τ)) ∘ i₂) ∘ σ ] ] ∘ (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) , i₂ ∘ [ extend (out⁻¹ ∘ i₁ ∘ idC) ∘ σ , extend (out⁻¹ ∘ i₂ ∘ τ) ∘ σ ] ] ∘ (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)))) ⟩∘⟨refl ⟩
[ (idC +₁ τ) , i₂ ∘ [ σ , ▷ ∘ extend τ ∘ σ ] ] ∘ (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹ ∘ (out ⁂ out) ≈⟨ {! !} ⟩
[ (idC +₁ τ) , i₂ ∘ [ D₁ swap ∘ τ , ▷ ∘ extend τ ∘ σ ] ] ∘ (idC +₁ (swap +₁ idC)) ∘ (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹ ∘ (out ⁂ out) ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩
-- TODO proof D₁ swap ∘ τ ≈ σ ∘ swap , follows by definition!!
-- ... wrong lead, brings σ back..
{-
extend τ ∘ σ
≈ extend τ ∘ D₁ swap ∘ τ ∘ swap
≈ extend (τ ∘ swap) ∘ τ ∘ swap
-}
{! !} ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩
[ idC +₁ D₁ swap ∘ τ , i₂ ∘ [ τ , ▷ ∘ extend τ ∘ σ ] ] ∘ ((idC +₁ swap) +₁ idC) ∘ w ≈⟨ {! !} ⟩
[ idC +₁ σ , i₂ ∘ [ τ , ▷ ∘ extend τ ∘ σ ] ] ∘ w ≈˘⟨ cancelˡ (_≅_.isoʳ out-≅) ⟩
out ∘ out⁻¹ ∘ [ idC +₁ σ , i₂ ∘ [ τ , ▷ ∘ extend τ ∘ σ ] ] ∘ w ∎)
fixpoint-eq = {! !}
-- fixpoint-eq : ∀ {f : D₀ X × D₀ Y ⇒ D₀ (X × Y)} → f ≈ out⁻¹ ∘ [ idC +₁ σ , i₂ ∘ [ τ , ▷ ∘ f ] ] ∘ w → f ≈ extend [ now , f ] ∘ g
-- fixpoint-eq {f} fix = begin
-- f ≈⟨ fix ⟩
-- out⁻¹ ∘ [ idC +₁ σ , i₂ ∘ [ τ , ▷ ∘ f ] ] ∘ w ≈˘⟨ refl⟩∘⟨ []-cong₂ refl (refl⟩∘⟨ ([]-cong₂ refl (pullʳ inject₂))) ⟩∘⟨refl ⟩
-- out⁻¹ ∘ [ idC +₁ σ , i₂ ∘ [ τ , (▷ ∘ [ now , f ]) ∘ i₂ ] ] ∘ w ≈˘⟨ refl⟩∘⟨ []-cong₂ refl (sym ∘[] ○ refl⟩∘⟨ []-cong₂ (elimˡ (extend-≈ inject₁ ○ k-identityˡ)) (pullˡ k-identityʳ)) ⟩∘⟨refl ⟩
-- out⁻¹ ∘ [ idC +₁ σ , [ i₂ ∘ extend ([ now , f ] ∘ i₁) ∘ τ , 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 ] , [ (i₂ ∘ extend [ now , f ]) ∘ D₁ i₁ ∘ τ , (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 ] , (i₂ ∘ extend [ now , f ]) ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈˘⟨ refl⟩∘⟨ (([]-cong₂ []∘+₁ (pullˡ inject₂)) ⟩∘⟨refl) ⟩
-- out⁻¹ ∘ [ [ [ i₁ , out ∘ f ] , i₂ ∘ extend [ now , f ] ] ∘ (i₁ +₁ D₁ i₁ ∘ σ) , [ [ i₁ , out ∘ f ] , i₂ ∘ extend [ now , f ] ] ∘ i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈˘⟨ refl⟩∘⟨ (pullˡ ∘[]) ⟩
-- out⁻¹ ∘ [ [ i₁ , out ∘ f ] , i₂ ∘ extend [ now , f ] ] ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈⟨ Iso⇒Mono (_≅_.iso out-≅) (out⁻¹ ∘ [ [ i₁ , out ∘ f ] , i₂ ∘ extend [ now , f ] ] ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w) (extend [ now , f ] ∘ out⁻¹ ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w) helper ⟩
-- extend [ now , f ] ∘ out⁻¹ ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ∎
-- where
-- helper = begin
-- out ∘ out⁻¹ ∘ [ [ i₁ , out ∘ f ] , i₂ ∘ extend [ now , f ] ] ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈⟨ cancelˡ (_≅_.isoʳ out-≅) ⟩
-- [ [ i₁ , out ∘ f ] , i₂ ∘ extend [ now , f ] ] ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈˘⟨ ([]-cong₂ (∘[] ○ []-cong₂ unitlaw refl) refl) ⟩∘⟨refl ⟩
-- [ out ∘ [ now , f ] , i₂ ∘ extend [ now , f ] ] ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈˘⟨ pullʳ (cancelˡ (_≅_.isoʳ out-≅)) ⟩
-- ([ out ∘ [ now , f ] , i₂ ∘ extend [ now , f ] ] ∘ out) ∘ out⁻¹ ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈˘⟨ pullˡ (extendlaw [ now , f ]) ⟩
-- out ∘ extend [ now , f ] ∘ out⁻¹ ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ∎
{-
TODO there's an error in the paper, at the end of the proof of proposition two:
the last line of the 3 line calulation 'f = ....'