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532 KiB
module Monad.Instance.Delay.Strong {o ℓ e} (ambient : Ambient o ℓ e) (D : DelayM ambient) where open Ambient ambient open HomReasoning open Equiv open MP C open MR C open M C open F-Coalgebra-Morphism renaming (f to u)
The Delay Monad is strong
module _ where open DelayM D open Functor open RMonad kleisli using (extend; extend-≈) renaming (assoc to k-assoc; identityʳ to k-identityʳ) open Monad monad using (η; μ) -- TODO change 'coinduction' proofs, move the two proofs i.e. f ≈ ! and ! ≈ g to the where clause module τ-mod (P : Category.Obj (CProduct C C)) where private X = proj₁ P Y = proj₂ P open Terminal (coalgebras (X × Y)) τ : X × D₀ Y ⇒ D₀ (X × Y) τ = u (! {A = record { A = X × D₀ Y ; α = distributeˡ⁻¹ ∘ (idC ⁂ out) }}) τ-law : out ∘ τ ≈ (idC +₁ τ) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) τ-law = commutes (! {A = record { A = X × D₀ Y ; α = distributeˡ⁻¹ ∘ (idC ⁂ out) }}) τ-helper : τ ∘ (idC ⁂ out⁻¹) ≈ out⁻¹ ∘ (idC +₁ τ) ∘ distributeˡ⁻¹ τ-helper = begin τ ∘ (idC ⁂ out⁻¹) ≈⟨ introˡ (_≅_.isoˡ out-≅) ⟩ (out⁻¹ ∘ out) ∘ τ ∘ (idC ⁂ out⁻¹) ≈⟨ pullʳ (pullˡ τ-law) ⟩ out⁻¹ ∘ ((idC +₁ τ) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out)) ∘ (idC ⁂ out⁻¹) ≈⟨ refl⟩∘⟨ (assoc ○ refl⟩∘⟨ assoc) ⟩ out⁻¹ ∘ (idC +₁ τ) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ∘ (idC ⁂ out⁻¹) ≈⟨ refl⟩∘⟨ (refl⟩∘⟨ (elimʳ (⁂∘⁂ ○ (⁂-cong₂ identity² (_≅_.isoʳ out-≅)) ○ ((⟨⟩-cong₂ identityˡ identityˡ) ○ ⁂-η)))) ⟩ out⁻¹ ∘ (idC +₁ τ) ∘ distributeˡ⁻¹ ∎ τ-now : τ ∘ (idC ⁂ now) ≈ now τ-now = begin τ ∘ (idC ⁂ now) ≈⟨ refl⟩∘⟨ sym (⁂∘⁂ ○ (⁂-cong₂ identity² refl)) ⟩ τ ∘ (idC ⁂ out⁻¹) ∘ (idC ⁂ i₁) ≈⟨ pullˡ τ-helper ⟩ (out⁻¹ ∘ (idC +₁ τ) ∘ distributeˡ⁻¹) ∘ (idC ⁂ i₁) ≈⟨ pullʳ (pullʳ distributeˡ⁻¹-i₁) ⟩ out⁻¹ ∘ (idC +₁ τ) ∘ i₁ ≈⟨ refl⟩∘⟨ +₁∘i₁ ⟩ out⁻¹ ∘ i₁ ∘ idC ≈⟨ refl⟩∘⟨ identityʳ ⟩ now ∎ ▷∘τ : τ ∘ (idC ⁂ ▷) ≈ ▷ ∘ τ ▷∘τ = begin τ ∘ (idC ⁂ ▷) ≈⟨ refl⟩∘⟨ (sym (⁂∘⁂ ○ ⁂-cong₂ identity² refl)) ⟩ τ ∘ (idC ⁂ out⁻¹) ∘ (idC ⁂ i₂) ≈⟨ pullˡ τ-helper ⟩ (out⁻¹ ∘ (idC +₁ τ) ∘ distributeˡ⁻¹) ∘ (idC ⁂ i₂) ≈⟨ pullʳ (pullʳ distributeˡ⁻¹-i₂) ⟩ out⁻¹ ∘ (idC +₁ τ) ∘ i₂ ≈⟨ refl⟩∘⟨ +₁∘i₂ ⟩ out⁻¹ ∘ i₂ ∘ τ ≈⟨ sym-assoc ⟩ ▷ ∘ τ ∎ τ-unique : (t : X × D₀ Y ⇒ D₀ (X × Y)) → (out ∘ t ≈ (idC +₁ t) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out)) → t ≈ τ τ-unique t t-commutes = sym (!-unique (record { f = t ; commutes = t-commutes })) open τ-mod strength : Strength monoidal monad strength = record { strengthen = ntHelper (record { η = τ ; commute = commute' }) ; identityˡ = identityˡ' -- triangle ; η-comm = λ {A} {B} → τ-now (A , B) ; μ-η-comm = μ-η-comm' -- μ-τ ; strength-assoc = strength-assoc' -- square } where identityˡ' : ∀ {X : Obj} → extend (now ∘ π₂) ∘ τ (Terminal.⊤ terminal , X) ≈ π₂ identityˡ' {X} = begin extend (now ∘ π₂) ∘ τ _ ≈⟨ sym (Terminal.!-unique (coalgebras X) {A = record { A = Terminal.⊤ terminal × D₀ X ; α = (π₂ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) }} (record { f = extend (now ∘ π₂) ∘ τ _ ; commutes = begin out ∘ extend (now ∘ π₂) ∘ τ _ ≈⟨ refl⟩∘⟨ (refl⟩∘⟨ (sym identityʳ)) ⟩ out ∘ extend (now ∘ π₂) ∘ τ _ ∘ idC ≈⟨ pullˡ diag₃ ⟩ ((π₂ +₁ extend (now ∘ π₂)) ∘ out) ∘ τ _ ∘ idC ≈⟨ pullʳ (pullˡ (diag₂ (Terminal.⊤ terminal , X))) ⟩ (π₂ +₁ extend (now ∘ π₂)) ∘ ((idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out)) ∘ idC ≈⟨ refl⟩∘⟨ (pullʳ (assoc ○ sym diag₁)) ⟩ (π₂ +₁ extend (now ∘ π₂)) ∘ (idC +₁ τ _) ∘ (⟨ Terminal.! terminal , idC ⟩ +₁ idC) ∘ (π₂ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ (pullˡ (+₁∘+₁ ○ +₁-cong₂ (_≅_.isoˡ ⊤×A≅A) identity² ○ CC.+-unique ((identity² ⟩∘⟨refl) ○ id-comm-sym) ((identity² ⟩∘⟨refl) ○ id-comm-sym)) ) ⟩ (π₂ +₁ extend (now ∘ π₂)) ∘ (idC +₁ τ _) ∘ (idC ∘ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ pullˡ +₁∘+₁ ⟩ (π₂ ∘ idC +₁ extend (now ∘ π₂) ∘ τ _) ∘ (idC ∘ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ (+₁-cong₂ id-comm refl) ⟩∘⟨refl ⟩ (idC ∘ π₂ +₁ extend (now ∘ π₂) ∘ τ _) ∘ (idC ∘ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈˘⟨ pullˡ +₁∘+₁ ⟩ (idC +₁ extend (now ∘ π₂)) ∘ (π₂ +₁ τ _) ∘ (idC ∘ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ refl⟩∘⟨ (refl⟩∘⟨ (elimˡ identity²)) ⟩ (idC +₁ extend (now ∘ π₂)) ∘ (π₂ +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈˘⟨ refl⟩∘⟨ ((+₁-cong₂ identityˡ identityʳ) ⟩∘⟨refl) ⟩ (idC +₁ extend (now ∘ π₂)) ∘ (idC ∘ π₂ +₁ τ _ ∘ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈˘⟨ pullʳ (pullˡ +₁∘+₁) ⟩ ((idC +₁ extend (now ∘ π₂)) ∘ (idC +₁ τ _)) ∘ (π₂ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈˘⟨ (homomorphism (X +-)) ⟩∘⟨refl ⟩ (idC +₁ extend (now ∘ π₂) ∘ τ _) ∘ (π₂ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ∎ })) ⟩ u (Terminal.! (coalgebras X)) ≈⟨ Terminal.!-unique (coalgebras X) {A = record { A = Terminal.⊤ terminal × D₀ X ; α = (π₂ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) }} (record { f = π₂ ; commutes = begin out ∘ π₂ ≈˘⟨ π₂∘⁂ ⟩ π₂ ∘ (idC ⁂ out) ≈˘⟨ pullˡ distributeˡ⁻¹-π₂ ⟩ (π₂ +₁ π₂) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈˘⟨ (+₁-cong₂ identityˡ identityʳ) ⟩∘⟨refl ⟩ (idC ∘ π₂ +₁ π₂ ∘ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈˘⟨ pullˡ +₁∘+₁ ⟩ (idC +₁ π₂) ∘ (π₂ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ∎ }) ⟩ π₂ ∎ where diag₁ : (⟨ Terminal.! terminal , idC ⟩ +₁ idC) ∘ (π₂ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈ distributeˡ⁻¹ {Terminal.⊤ terminal} {X} {D₀ X} ∘ (idC ⁂ out) ∘ idC diag₁ = begin (⟨ Terminal.! terminal , idC ⟩ +₁ idC) ∘ (π₂ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ pullˡ +₁∘+₁ ⟩ (⟨ Terminal.! terminal , idC ⟩ ∘ π₂ +₁ idC ∘ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ +₁-cong₂ (_≅_.isoˡ ⊤×A≅A) identity² ⟩∘⟨ (refl⟩∘⟨ sym identityʳ) ⟩ (idC +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ∘ idC ≈⟨ elimˡ (CC.+-unique id-comm-sym id-comm-sym) ⟩ distributeˡ⁻¹ ∘ (idC ⁂ out) ∘ idC ∎ diag₂ = τ-law diag₃ : out {X} ∘ extend (now ∘ π₂ {A = Terminal.⊤ terminal} {B = X}) ≈ (π₂ +₁ extend (now ∘ π₂)) ∘ out diag₃ = out-law π₂ μ-η-comm' : ∀ {X Y} → extend idC ∘ (extend (now ∘ τ _)) ∘ τ _ ≈ τ (X , Y) ∘ (idC ⁂ extend idC) μ-η-comm' {X} {Y} = begin extend idC ∘ (extend (now ∘ τ _)) ∘ τ _ ≈⟨ sym (Terminal.!-unique (coalgebras (X × Y)) (record { f = extend idC ∘ (extend (now ∘ τ _)) ∘ τ _ ; commutes = begin out ∘ extend idC ∘ extend (now ∘ τ (X , Y)) ∘ τ _ ≈⟨ refl⟩∘⟨ (pullˡ id*∘Dτ) ⟩ out ∘ extend (τ (X , Y)) ∘ τ _ ≈⟨ square ⟩ [ out ∘ τ _ , i₂ ∘ extend (τ _) ] ∘ (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ sym-assoc ○ sym-assoc ○ (assoc ○ tri) ⟩∘⟨refl ⟩ ((idC +₁ extend (τ _) ∘ τ _) ∘ [ (idC +₁ (idC ⁂ now)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ] ∘ distributeˡ⁻¹) ∘ (idC ⁂ out) ≈⟨ assoc ○ refl⟩∘⟨ assoc ⟩ (idC +₁ (extend (τ _) ∘ τ _)) ∘ [ (idC +₁ (idC ⁂ now)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ] ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ (+₁-cong₂ refl (sym (pullˡ id*∘Dτ))) ⟩∘⟨refl ⟩ (idC +₁ (extend idC ∘ extend (now ∘ τ (X , Y)) ∘ τ _)) ∘ [ (idC +₁ (idC ⁂ now)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ] ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ∎ })) ⟩ u (Terminal.! (coalgebras (X × Y)) {A = record { A = X × D₀ (D₀ Y) ; α = [ (idC +₁ (idC ⁂ now)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ] ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) }}) ≈⟨ Terminal.!-unique (coalgebras (X × Y)) (record { f = τ _ ∘ (idC ⁂ extend idC) ; commutes = begin out ∘ τ _ ∘ (idC ⁂ extend idC) ≈⟨ pullˡ (τ-law (X , Y)) ⟩ ((idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out)) ∘ (idC ⁂ extend idC) ≈⟨ pullʳ (pullʳ (⁂∘⁂ ○ ⁂-cong₂ identity² refl)) ⟩ (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out ∘ extend idC) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ ⁂-cong₂ refl (extendlaw idC) ⟩ (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ [ out ∘ idC , i₂ ∘ extend idC ] ∘ out) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ ⁂-cong₂ refl (([]-cong₂ identityʳ refl) ⟩∘⟨refl) ⟩ (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ [ out , i₂ ∘ extend idC ] ∘ out) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ sym (⁂∘⁂ ○ ⁂-cong₂ identity² refl) ⟩ (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ [ out , i₂ ∘ extend idC ]) ∘ (idC ⁂ out) ≈⟨ sym-assoc ○ pullˡ (assoc ○ tri₂) ⟩ ((idC +₁ τ _ ∘ (idC ⁂ extend idC)) ∘ [ (idC +₁ (idC ⁂ now)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ] ∘ distributeˡ⁻¹) ∘ (idC ⁂ out) ≈⟨ assoc ○ refl⟩∘⟨ assoc ⟩ (idC +₁ τ _ ∘ (idC ⁂ extend idC)) ∘ [ (idC +₁ (idC ⁂ now)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ] ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ∎ }) ⟩ τ _ ∘ (idC ⁂ extend idC) ∎ where -- diagram: https://q.uiver.app/#q=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 square : out ∘ extend (τ (X , Y)) ∘ τ _ ≈ [ out ∘ τ _ , i₂ ∘ extend (τ _) ] ∘ (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) square = begin out ∘ extend (τ (X , Y)) ∘ τ _ ≈⟨ pullˡ (extendlaw (τ (X , Y))) ⟩ ([ out ∘ τ (X , Y) , i₂ ∘ extend (τ (X , Y)) ] ∘ out) ∘ τ _ ≈⟨ pullʳ (τ-law _) ⟩ [ out ∘ τ _ , i₂ ∘ extend (τ _) ] ∘ (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ∎ tri : [ out ∘ τ _ , i₂ ∘ extend (τ (X , Y)) ] ∘ (idC +₁ τ _) ∘ distributeˡ⁻¹ ≈ (idC +₁ extend (τ _) ∘ τ _) ∘ [ (idC +₁ (idC ⁂ now)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ] ∘ distributeˡ⁻¹ tri = begin [ out ∘ τ _ , i₂ ∘ extend (τ (X , Y)) ] ∘ (idC +₁ τ _) ∘ distributeˡ⁻¹ ≈⟨ pullˡ []∘+₁ ⟩ [ (out ∘ τ _) ∘ idC , (i₂ ∘ extend (τ (X , Y))) ∘ τ _ ] ∘ distributeˡ⁻¹ ≈⟨ ([]-cong₂ (identityʳ ○ τ-law (X , Y)) assoc) ⟩∘⟨refl ⟩ [ (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ∘ extend (τ (X , Y)) ∘ τ _ ] ∘ distributeˡ⁻¹ ≈⟨ sym (([]-cong₂ ((+₁-cong₂ refl k-identityʳ) ⟩∘⟨refl) refl) ⟩∘⟨refl) ⟩ [ (idC +₁ extend (τ _) ∘ now) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ∘ extend (τ _) ∘ τ _ ] ∘ distributeˡ⁻¹ ≈⟨ sym (([]-cong₂ ((+₁-cong₂ identity² (pullʳ (τ-now (X , D₀ Y)))) ⟩∘⟨refl) refl) ⟩∘⟨refl) ⟩ [ (idC ∘ idC +₁ (extend (τ _) ∘ τ _) ∘ (idC ⁂ now)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ∘ extend (τ _) ∘ τ _ ] ∘ distributeˡ⁻¹ ≈⟨ sym (([]-cong₂ (pullˡ +₁∘+₁) +₁∘i₂) ⟩∘⟨refl) ⟩ [ (idC +₁ extend (τ _) ∘ τ _) ∘ (idC +₁ (idC ⁂ now)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , (idC +₁ extend (τ _) ∘ τ _) ∘ i₂ ] ∘ distributeˡ⁻¹ ≈⟨ sym (pullˡ ∘[]) ⟩ (idC +₁ extend (τ _) ∘ τ _) ∘ [ (idC +₁ (idC ⁂ now)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ] ∘ distributeˡ⁻¹ ∎ id*∘Dτ : extend idC ∘ extend (now ∘ τ _) ≈ extend (τ _) id*∘Dτ = begin extend idC ∘ extend (now ∘ τ _) ≈⟨ sym k-assoc ⟩ extend (extend idC ∘ now ∘ τ _) ≈⟨ extend-≈ (pullˡ k-identityʳ) ⟩ extend (idC ∘ τ _) ≈⟨ extend-≈ identityˡ ⟩ extend (τ _) ∎ tri₂' : (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ [ out , i₂ ∘ extend idC ]) ∘ [ (idC ⁂ i₁) , (idC ⁂ i₂) ] ≈ (idC +₁ τ _ ∘ (idC ⁂ extend idC)) ∘ [ (idC +₁ (idC ⁂ now)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ] ∘ distributeˡ⁻¹ ∘ distributeˡ tri₂' = begin (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ [ out , i₂ ∘ extend idC ]) ∘ [ (idC ⁂ i₁) , (idC ⁂ i₂) ] ≈⟨ refl⟩∘⟨ (refl⟩∘⟨ ∘[]) ⟩ (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ [ (idC ⁂ [ out , i₂ ∘ extend idC ]) ∘ (idC ⁂ i₁) , (idC ⁂ [ out , i₂ ∘ extend idC ]) ∘ (idC ⁂ i₂) ] ≈⟨ refl⟩∘⟨ refl⟩∘⟨ []-cong₂ ⁂∘⁂ ⁂∘⁂ ⟩ (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ [ (idC ∘ idC ⁂ [ out , i₂ ∘ extend idC ] ∘ i₁) , (idC ∘ idC ⁂ [ out , i₂ ∘ extend idC ] ∘ i₂) ] ≈⟨ refl⟩∘⟨ refl⟩∘⟨ []-cong₂ (⁂-cong₂ identity² inject₁) (⁂-cong₂ identity² inject₂) ⟩ (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ [ idC ⁂ out , idC ⁂ i₂ ∘ extend idC ] ≈⟨ refl⟩∘⟨ ∘[] ⟩ (idC +₁ τ _) ∘ [ distributeˡ⁻¹ ∘ (idC ⁂ out) , distributeˡ⁻¹ ∘ (idC ⁂ i₂ ∘ extend idC) ] ≈⟨ ∘[] ⟩ [ (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ i₂ ∘ extend idC) ] ≈⟨ sym ([]-cong₂ refl (pullʳ (pullʳ (⁂∘⁂ ○ ⁂-cong₂ identity² refl)))) ⟩ [ (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , ((idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ i₂)) ∘ (idC ⁂ extend idC) ] ≈⟨ []-cong₂ refl ((refl⟩∘⟨ distributeˡ⁻¹-i₂) ⟩∘⟨refl) ⟩ [ (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , ((idC +₁ τ _) ∘ i₂) ∘ (idC ⁂ extend idC) ] ≈⟨ []-cong₂ refl (pushˡ +₁∘i₂) ⟩ [ (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ∘ τ _ ∘ (idC ⁂ extend idC) ] ≈⟨ []-cong₂ ((+₁-cong₂ refl (introʳ (⟨⟩-unique id-comm (id-comm ○ (sym k-identityʳ) ⟩∘⟨refl)))) ⟩∘⟨refl) refl ⟩ [ (idC +₁ τ _ ∘ (idC ⁂ extend idC ∘ now)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ∘ τ _ ∘ (idC ⁂ extend idC) ] ≈⟨ []-cong₂ ((sym (+₁-cong₂ refl (refl⟩∘⟨ (⁂∘⁂ ○ ⁂-cong₂ identity² refl)))) ⟩∘⟨refl) refl ⟩ [ (idC +₁ τ _ ∘ (idC ⁂ extend idC) ∘ (idC ⁂ now)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ∘ τ _ ∘ (idC ⁂ extend idC) ] ≈⟨ sym ([]-cong₂ (pullˡ (+₁∘+₁ ○ +₁-cong₂ identity² assoc)) +₁∘i₂) ⟩ [ (idC +₁ τ _ ∘ (idC ⁂ extend idC)) ∘ (idC +₁ (idC ⁂ now)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , (idC +₁ τ _ ∘ (idC ⁂ extend idC)) ∘ i₂ ] ≈⟨ sym ∘[] ⟩ (idC +₁ τ _ ∘ (idC ⁂ extend idC)) ∘ [ (idC +₁ (idC ⁂ now)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ] ≈⟨ sym (refl⟩∘⟨ (elimʳ (IsIso.isoˡ isIsoˡ))) ⟩ (idC +₁ τ _ ∘ (idC ⁂ extend idC)) ∘ [ (idC +₁ (idC ⁂ now)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ] ∘ distributeˡ⁻¹ ∘ distributeˡ ∎ tri₂ : (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ [ out , i₂ ∘ extend idC ]) ≈ (idC +₁ τ _ ∘ (idC ⁂ extend idC)) ∘ [ (idC +₁ (idC ⁂ now)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ] ∘ distributeˡ⁻¹ tri₂ = Iso⇒Epi (IsIso.iso isIsoˡ) ((idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ [ out , i₂ ∘ extend idC ])) ((idC +₁ τ _ ∘ (idC ⁂ extend idC)) ∘ [ (idC +₁ (idC ⁂ now)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) , i₂ ] ∘ distributeˡ⁻¹) (assoc ○ refl⟩∘⟨ assoc ○ tri₂' ○ sym-assoc ○ sym-assoc ○ assoc ⟩∘⟨refl) strength-assoc' : ∀ {X Y Z} → extend (now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ τ ((X × Y), Z) ≈ τ _ ∘ (idC ⁂ τ _) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ strength-assoc' {X} {Y} {Z} = begin extend (now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ τ _ ≈⟨ sym (Terminal.!-unique (coalgebras (X × Y × Z)) (record { f = extend (now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ τ _ ; commutes = begin out ∘ extend (now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ τ _ ≈⟨ pullˡ (extendlaw (now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩)) ⟩ ([ out ∘ now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ , i₂ ∘ extend (now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ] ∘ out) ∘ τ _ ≈⟨ pullʳ (τ-law (X × Y , Z)) ⟩ [ out ∘ now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ , i₂ ∘ extend (now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ] ∘ (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ ([]-cong₂ (pullˡ unitlaw) refl) ⟩∘⟨refl ⟩ (⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ +₁ extend (now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩)) ∘ (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ pullˡ +₁∘+₁ ⟩ (⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ∘ idC +₁ extend (now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ (+₁-cong₂ id-comm (sym identityʳ)) ⟩∘⟨refl ⟩ (idC ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ +₁ (extend (now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ τ _) ∘ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ sym (pullˡ +₁∘+₁) ⟩ (idC +₁ extend (now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ τ _) ∘ (⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ∎ })) ⟩ u (Terminal.! (coalgebras (X × Y × Z)) {A = record { A = (X × Y) × D₀ Z ; α = (⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) }}) ≈⟨ Terminal.!-unique (coalgebras (X × Y × Z)) (record { f = τ _ ∘ (idC ⁂ τ _) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ; commutes = begin out ∘ τ _ ∘ (idC ⁂ τ _) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ≈⟨ pullˡ (τ-law (X , Y × Z)) ⟩ ((idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out)) ∘ (idC ⁂ τ _) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ≈⟨ pullʳ (pullˡ (pullʳ ⁂∘⁂)) ⟩ (idC +₁ τ _) ∘ (distributeˡ⁻¹ ∘ (idC ∘ idC ⁂ out ∘ τ _)) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ≈⟨ refl⟩∘⟨ assoc ⟩ (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ∘ idC ⁂ out ∘ τ _) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ≈˘⟨ refl⟩∘⟨ (refl⟩∘⟨ (⁂-cong₂ identityʳ (assoc ○ sym (τ-law (Y , Z)))) ⟩∘⟨refl) ⟩ (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ ((idC ∘ idC) ∘ idC ⁂ ((idC +₁ τ _) ∘ distributeˡ⁻¹) ∘ (idC ⁂ out)) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ≈⟨ sym (refl⟩∘⟨ (pullʳ (pullˡ ⁂∘⁂ ○ pullˡ ⁂∘⁂))) ⟩ (idC +₁ τ _) ∘ (distributeˡ⁻¹ ∘ (idC ⁂ (idC +₁ τ _))) ∘ (idC ⁂ distributeˡ⁻¹) ∘ (idC ⁂ (idC ⁂ out)) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ≈⟨ refl⟩∘⟨ (refl⟩∘⟨ (refl⟩∘⟨ helper₁)) ⟩ (idC +₁ τ _) ∘ (distributeˡ⁻¹ ∘ (idC ⁂ (idC +₁ τ _))) ∘ (idC ⁂ distributeˡ⁻¹) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ∘ (idC ⁂ out) ≈⟨ refl⟩∘⟨ (helper₂ ⟩∘⟨refl) ⟩ (idC +₁ τ _) ∘ ((idC +₁ (idC ⁂ τ (Y , Z))) ∘ distributeˡ⁻¹) ∘ (idC ⁂ distributeˡ⁻¹) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ∘ (idC ⁂ out) ≈⟨ refl⟩∘⟨ (assoc ○ refl⟩∘⟨ sym assoc²') ⟩ (idC +₁ τ _) ∘ (idC +₁ (idC ⁂ τ (Y , Z))) ∘ (distributeˡ⁻¹ ∘ (idC ⁂ distributeˡ⁻¹) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ (idC ⁂ out) ≈⟨ pullˡ (+₁∘+₁ ○ +₁-cong₂ identity² refl) ⟩ (idC +₁ τ _ ∘ (idC ⁂ τ (Y , Z))) ∘ (distributeˡ⁻¹ ∘ (idC ⁂ distributeˡ⁻¹) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ (idC ⁂ out) ≈⟨ refl⟩∘⟨ (helper₃ ⟩∘⟨refl) ⟩ (idC +₁ τ _ ∘ (idC ⁂ τ (Y , Z))) ∘ ((⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ +₁ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ distributeˡ⁻¹) ∘ (idC ⁂ out) ≈⟨ assoc²'' ⟩ ((idC +₁ τ _ ∘ (idC ⁂ τ _)) ∘ (⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ +₁ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ sym ((+₁-cong₂ refl (identityʳ ○ sym-assoc) ○ sym +₁∘+₁) ⟩∘⟨refl) ⟩ (idC ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ +₁ (τ _ ∘ (idC ⁂ τ _) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ sym (pullˡ +₁∘+₁) ⟩ (idC +₁ τ _ ∘ (idC ⁂ τ _) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ (⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ∎ }) ⟩ τ _ ∘ (idC ⁂ τ _) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ∎ where helper₁ : (idC ⁂ (idC ⁂ out)) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ≈ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ∘ (idC {X × Y} ⁂ out {Z}) helper₁ = begin (idC ⁂ (idC ⁂ out)) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ≈⟨ ⁂∘⟨⟩ ⟩ ⟨ idC ∘ π₁ ∘ π₁ , (idC ⁂ out) ∘ ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ≈⟨ ⟨⟩-cong₂ identityˡ ⁂∘⟨⟩ ⟩ ⟨ π₁ ∘ π₁ , ⟨ idC ∘ π₂ ∘ π₁ , out ∘ π₂ ⟩ ⟩ ≈⟨ ⟨⟩-cong₂ refl (⟨⟩-cong₂ identityˡ refl) ⟩ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , out ∘ π₂ ⟩ ⟩ ≈⟨ sym (⟨⟩-cong₂ refl (⟨⟩-cong₂ (refl⟩∘⟨ identityˡ) refl)) ⟩ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ idC ∘ π₁ , out ∘ π₂ ⟩ ⟩ ≈⟨ sym (⟨⟩-cong₂ (refl⟩∘⟨ identityˡ) (⟨⟩-cong₂ (pullʳ π₁∘⁂) π₂∘⁂)) ⟩ ⟨ π₁ ∘ idC ∘ π₁ , ⟨ (π₂ ∘ π₁) ∘ (idC {X × Y} ⁂ out {Z}) , π₂ ∘ (idC {X × Y} ⁂ out {Z}) ⟩ ⟩ ≈⟨ sym (⟨⟩-cong₂ (pullʳ project₁) ⟨⟩∘) ⟩ ⟨ (π₁ ∘ π₁) ∘ (idC {X × Y} ⁂ out {Z}) , ⟨ π₂ ∘ π₁ , π₂ ⟩ ∘ (idC {X × Y} ⁂ out {Z}) ⟩ ≈⟨ sym ⟨⟩∘ ⟩ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ∘ (idC {X × Y} ⁂ out {Z}) ∎ helper₂ : distributeˡ⁻¹ ∘ (idC ⁂ (idC +₁ τ _)) ≈ (idC +₁ (idC ⁂ τ (Y , Z))) ∘ distributeˡ⁻¹ helper₂ = sym (distributeˡ⁻¹-natural idC idC (τ _)) ○ (+₁-cong₂ (⟨⟩-unique id-comm id-comm) refl) ⟩∘⟨refl helper₃ : distributeˡ⁻¹ ∘ (idC ⁂ distributeˡ⁻¹) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ≈ (⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ +₁ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ distributeˡ⁻¹ helper₃ = sym (begin (⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ +₁ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ distributeˡ⁻¹ ≈⟨ introˡ (IsIso.isoˡ isIsoˡ) ⟩ (distributeˡ⁻¹ ∘ distributeˡ) ∘ (⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ +₁ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ distributeˡ⁻¹ ≈⟨ pullʳ (pullˡ []∘+₁) ⟩ distributeˡ⁻¹ ∘ [ (idC ⁂ i₁) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ , (idC ⁂ i₂) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ] ∘ distributeˡ⁻¹ ≈⟨ refl⟩∘⟨ (([]-cong₂ ⁂∘⟨⟩ ⁂∘⟨⟩) ⟩∘⟨refl) ⟩ distributeˡ⁻¹ ∘ [ ⟨ idC ∘ π₁ ∘ π₁ , i₁ ∘ ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ , ⟨ idC ∘ π₁ ∘ π₁ , i₂ ∘ ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ] ∘ distributeˡ⁻¹ ≈⟨ refl⟩∘⟨ []-cong₂ (⟨⟩-cong₂ identityˡ (pushˡ (sym distributeˡ⁻¹-i₁))) (⟨⟩-cong₂ identityˡ ((pushˡ (sym distributeˡ⁻¹-i₂)))) ⟩∘⟨refl ⟩ distributeˡ⁻¹ ∘ [ ⟨ π₁ ∘ π₁ , distributeˡ⁻¹ ∘ (idC ⁂ i₁) ∘ ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ , ⟨ π₁ ∘ π₁ , distributeˡ⁻¹ ∘ (idC ⁂ i₂) ∘ ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ] ∘ distributeˡ⁻¹ ≈⟨ sym (refl⟩∘⟨ []-cong₂ (⟨⟩-cong₂ refl (refl⟩∘⟨ (⟨⟩-cong₂ (refl⟩∘⟨ identityˡ ○ sym identityˡ) refl ○ sym ⁂∘⟨⟩))) (⟨⟩-cong₂ refl (refl⟩∘⟨ (⟨⟩-cong₂ (refl⟩∘⟨ identityˡ ○ sym identityˡ) refl ○ sym ⁂∘⟨⟩))) ⟩∘⟨refl) ⟩ distributeˡ⁻¹ ∘ [ ⟨ π₁ ∘ π₁ , distributeˡ⁻¹ ∘ ⟨ π₂ ∘ idC ∘ π₁ , i₁ ∘ π₂ ⟩ ⟩ , ⟨ π₁ ∘ π₁ , distributeˡ⁻¹ ∘ ⟨ π₂ ∘ idC ∘ π₁ , i₂ ∘ π₂ ⟩ ⟩ ] ∘ distributeˡ⁻¹ ≈⟨ sym (refl⟩∘⟨ []-cong₂ (⟨⟩-cong₂ (identityˡ ○ refl⟩∘⟨ identityˡ) (refl⟩∘⟨ (⟨⟩-cong₂ (pullʳ π₁∘⁂) π₂∘⁂))) (⟨⟩-cong₂ (identityˡ ○ refl⟩∘⟨ identityˡ) (refl⟩∘⟨ (⟨⟩-cong₂ (pullʳ π₁∘⁂) π₂∘⁂))) ⟩∘⟨refl) ⟩ distributeˡ⁻¹ ∘ [ ⟨ idC ∘ π₁ ∘ idC ∘ π₁ , distributeˡ⁻¹ ∘ ⟨ (π₂ ∘ π₁) ∘ (idC ⁂ i₁) , π₂ ∘ (idC ⁂ i₁) ⟩ ⟩ , ⟨ idC ∘ π₁ ∘ idC ∘ π₁ , distributeˡ⁻¹ ∘ ⟨ (π₂ ∘ π₁) ∘ (idC ⁂ i₂) , π₂ ∘ (idC ⁂ i₂) ⟩ ⟩ ] ∘ distributeˡ⁻¹ ≈⟨ sym (refl⟩∘⟨ []-cong₂ (⟨⟩-cong₂ (pullʳ (pullʳ π₁∘⁂)) (pullʳ ⟨⟩∘)) (⟨⟩-cong₂ (pullʳ (pullʳ π₁∘⁂)) (pullʳ ⟨⟩∘)) ⟩∘⟨refl) ⟩ distributeˡ⁻¹ ∘ [ ⟨ (idC ∘ π₁ ∘ π₁) ∘ (idC ⁂ i₁) , (distributeˡ⁻¹ ∘ ⟨ π₂ ∘ π₁ , π₂ ⟩) ∘ (idC ⁂ i₁) ⟩ , ⟨ (idC ∘ π₁ ∘ π₁) ∘ (idC ⁂ i₂) , (distributeˡ⁻¹ ∘ ⟨ π₂ ∘ π₁ , π₂ ⟩) ∘ (idC ⁂ i₂) ⟩ ] ∘ distributeˡ⁻¹ ≈⟨ sym (refl⟩∘⟨ []-cong₂ ⟨⟩∘ ⟨⟩∘ ⟩∘⟨refl) ⟩ distributeˡ⁻¹ ∘ [ (⟨ idC ∘ π₁ ∘ π₁ , distributeˡ⁻¹ ∘ ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ∘ (idC ⁂ i₁)) , (⟨ idC ∘ π₁ ∘ π₁ , distributeˡ⁻¹ ∘ ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ∘ (idC ⁂ i₂)) ] ∘ distributeˡ⁻¹ ≈˘⟨ pullʳ (pullˡ ∘[]) ⟩ (distributeˡ⁻¹ ∘ ⟨ idC ∘ π₁ ∘ π₁ , distributeˡ⁻¹ ∘ ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ distributeˡ ∘ distributeˡ⁻¹ ≈⟨ sym (introʳ (IsIso.isoʳ isIsoˡ)) ⟩ distributeˡ⁻¹ ∘ ⟨ idC ∘ π₁ ∘ π₁ , distributeˡ⁻¹ ∘ ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ≈⟨ sym (refl⟩∘⟨ ⁂∘⟨⟩) ⟩ distributeˡ⁻¹ ∘ (idC ⁂ distributeˡ⁻¹) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ∎) -- {- ⁂ ⊤ ○ -- Diagram for identityˡ': -- https://q.uiver.app/#q=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 -- -} commute' : ∀ {P₁ : Category.Obj (CProduct C C)} {P₂ : Category.Obj (CProduct C C)} (fg : _[_,_] (CProduct C C) P₁ P₂) → τ P₂ ∘ ((proj₁ fg) ⁂ extend (now ∘ (proj₂ fg))) ≈ extend (now ∘ ((proj₁ fg) ⁂ (proj₂ fg))) ∘ τ P₁ commute' {(U , V)} {(W , X)} (f , g) = begin τ _ ∘ (f ⁂ extend (now ∘ g)) ≈⟨ sym (!-unique (record { f = τ _ ∘ (f ⁂ extend (now ∘ g)) ; commutes = begin out ∘ τ (W , X) ∘ (f ⁂ extend (now ∘ g)) ≈⟨ pullˡ (τ-law (W , X)) ⟩ ((idC +₁ τ (W , X)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out)) ∘ (f ⁂ extend (now ∘ g)) ≈⟨ pullʳ (pullʳ ⁂∘⁂) ⟩ (idC +₁ τ (W , X)) ∘ distributeˡ⁻¹ ∘ (idC ∘ f ⁂ out ∘ extend (now ∘ g)) ≈⟨ refl⟩∘⟨ (refl⟩∘⟨ (⁂-cong₂ identityˡ (extendlaw (now ∘ g)))) ⟩ (idC +₁ τ (W , X)) ∘ distributeˡ⁻¹ ∘ (f ⁂ [ out ∘ now ∘ g , i₂ ∘ extend (now ∘ g) ] ∘ out) ≈⟨ refl⟩∘⟨ (refl⟩∘⟨ (⁂-cong₂ refl (([]-cong₂ (pullˡ unitlaw) refl) ⟩∘⟨refl))) ⟩ (idC +₁ τ (W , X)) ∘ distributeˡ⁻¹ ∘ (f ⁂ [ i₁ ∘ g , i₂ ∘ extend (now ∘ g) ] ∘ out) ≈⟨ refl⟩∘⟨ (refl⟩∘⟨ (⁂-cong₂ (sym identityʳ) refl)) ⟩ (idC +₁ τ (W , X)) ∘ distributeˡ⁻¹ ∘ (f ∘ idC ⁂ (g +₁ extend (now ∘ g)) ∘ out) ≈⟨ sym (pullʳ (pullʳ ⁂∘⁂)) ⟩ ((idC +₁ τ (W , X)) ∘ distributeˡ⁻¹ ∘ (f ⁂ (g +₁ extend (now ∘ g)))) ∘ (idC ⁂ out) ≈⟨ (refl⟩∘⟨ (sym (distributeˡ⁻¹-natural f g (extend (now ∘ g))))) ⟩∘⟨refl ⟩ ((idC +₁ τ (W , X)) ∘ ((f ⁂ g) +₁ (f ⁂ extend (now ∘ g))) ∘ distributeˡ⁻¹) ∘ (idC ⁂ out) ≈⟨ (pullˡ +₁∘+₁) ⟩∘⟨refl ⟩ ((idC ∘ (f ⁂ g) +₁ τ (W , X) ∘ (f ⁂ extend (now ∘ g))) ∘ distributeˡ⁻¹) ∘ (idC ⁂ out) ≈⟨ sym (((+₁-cong₂ refl identityʳ) ⟩∘⟨refl) ⟩∘⟨refl) ⟩ ((idC ∘ (f ⁂ g) +₁ (τ (W , X) ∘ (f ⁂ extend (now ∘ g))) ∘ idC) ∘ distributeˡ⁻¹) ∘ (idC ⁂ out) ≈⟨ sym (pullˡ (pullˡ +₁∘+₁)) ⟩ (idC +₁ (τ (W , X) ∘ (f ⁂ extend (now ∘ g)))) ∘ ((f ⁂ g +₁ idC) ∘ distributeˡ⁻¹) ∘ (idC ⁂ out) ∎ })) ⟩ u ! ≈⟨ !-unique (record { f = extend (now ∘ (f ⁂ g)) ∘ τ _ ; commutes = begin out ∘ extend (now ∘ (f ⁂ g)) ∘ τ _ ≈⟨ pullˡ (extendlaw (now ∘ (f ⁂ g))) ⟩ ([ out ∘ now ∘ (f ⁂ g) , i₂ ∘ extend (now ∘ (f ⁂ g)) ] ∘ out) ∘ τ (U , V) ≈⟨ (([]-cong₂ (pullˡ unitlaw) refl) ⟩∘⟨refl) ⟩∘⟨refl ⟩ ([ i₁ ∘ (f ⁂ g) , i₂ ∘ extend (now ∘ (f ⁂ g)) ] ∘ out) ∘ τ (U , V) ≈⟨ pullʳ (τ-law (U , V)) ⟩ ((f ⁂ g) +₁ (extend (now ∘ (f ⁂ g)))) ∘ (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ sym-assoc ○ sym-assoc ⟩ ((((f ⁂ g) +₁ (extend (now ∘ (f ⁂ g)))) ∘ (idC +₁ τ _)) ∘ distributeˡ⁻¹) ∘ (idC ⁂ out) ≈⟨ (+₁∘+₁ ⟩∘⟨refl) ⟩∘⟨refl ⟩ ((((f ⁂ g) ∘ idC) +₁ (extend (now ∘ (f ⁂ g)) ∘ τ (U , V))) ∘ distributeˡ⁻¹) ∘ (idC ⁂ out) ≈⟨ sym (((+₁-cong₂ id-comm-sym identityʳ) ⟩∘⟨refl) ⟩∘⟨refl) ⟩ (((idC ∘ (f ⁂ g)) +₁ ((extend (now ∘ (f ⁂ g)) ∘ τ (U , V)) ∘ idC)) ∘ distributeˡ⁻¹) ∘ (idC ⁂ out) ≈⟨ sym (pullˡ (pullˡ +₁∘+₁)) ⟩ (idC +₁ (extend (now ∘ (f ⁂ g)) ∘ τ (U , V))) ∘ (((f ⁂ g) +₁ idC) ∘ distributeˡ⁻¹) ∘ (idC ⁂ out) ∎ }) ⟩ extend (now ∘ (f ⁂ g)) ∘ τ _ ∎ where open Terminal (coalgebras (W × X)) alg' : F-Coalgebra ((W × X) +-) alg' = record { A = U × D₀ V ; α = ((f ⁂ g) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) } strongMonad : StrongMonad monoidal strongMonad = record { M = monad ; strength = strength }