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Still working on delay strength, finished strength-assoc
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2 changed files with 86 additions and 36 deletions
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@ -66,8 +66,21 @@ module Category.Instance.AmbientCategory where
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distributeʳ⁻¹ : ∀ {A B C : Obj} → (B + C) × A ⇒ B × A + C × A
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distributeʳ⁻¹ = IsIso.inv isIsoʳ
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module M = M'
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module MR = MR'
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distribute₁ : ∀ {X Y Z U V W} (f : X ⇒ U) (g : Y ⇒ V) (h : Z ⇒ W) → ((f ⁂ g) +₁ (f ⁂ h)) ∘ distributeˡ⁻¹ ≈ distributeˡ⁻¹ ∘ (f ⁂ (g +₁ h))
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distribute₁ f g h = begin
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((f ⁂ g) +₁ (f ⁂ h)) ∘ distributeˡ⁻¹ ≈⟨ introˡ (IsIso.isoˡ isIsoˡ) ⟩
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(distributeˡ⁻¹ ∘ distributeˡ) ∘ ((f ⁂ g) +₁ (f ⁂ h)) ∘ distributeˡ⁻¹ ≈⟨ pullˡ (pullʳ []∘+₁) ⟩
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(distributeˡ⁻¹ ∘ [(idC ⁂ i₁) ∘ (f ⁂ g) , (idC ⁂ i₂) ∘ (f ⁂ h)]) ∘ distributeˡ⁻¹ ≈⟨ (refl⟩∘⟨ ([]-cong₂ ⁂∘⁂ ⁂∘⁂)) ⟩∘⟨refl ⟩
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(distributeˡ⁻¹ ∘ [ idC ∘ f ⁂ i₁ ∘ g , idC ∘ f ⁂ i₂ ∘ h ]) ∘ distributeˡ⁻¹ ≈⟨ sym ((refl⟩∘⟨ ([]-cong₂ (⁂-cong₂ id-comm +₁∘i₁) (⁂-cong₂ id-comm +₁∘i₂))) ⟩∘⟨refl) ⟩
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(distributeˡ⁻¹ ∘ [ f ∘ idC ⁂ (g +₁ h) ∘ i₁ , f ∘ idC ⁂ (g +₁ h) ∘ i₂ ]) ∘ distributeˡ⁻¹ ≈˘⟨ (refl⟩∘⟨ ([]-cong₂ ⁂∘⁂ ⁂∘⁂)) ⟩∘⟨refl ⟩
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(distributeˡ⁻¹ ∘ [ ((f ⁂ (g +₁ h)) ∘ (idC ⁂ i₁)) , ((f ⁂ (g +₁ h)) ∘ (idC ⁂ i₂)) ]) ∘ distributeˡ⁻¹ ≈⟨ sym (pullˡ (pullʳ ∘[])) ⟩
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(distributeˡ⁻¹ ∘ (f ⁂ (g +₁ h))) ∘ distributeˡ ∘ distributeˡ⁻¹ ≈⟨ sym (introʳ (IsIso.isoʳ isIsoˡ)) ⟩
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distributeˡ⁻¹ ∘ (f ⁂ (g +₁ h)) ∎
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where
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open MR C
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open HomReasoning
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open Equiv
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```
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@ -305,23 +305,15 @@ Next we will show that the delay monad is strong, by giving a natural transforma
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{ η = τ
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; commute = commute' })
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; identityˡ = identityˡ' -- triangle
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; η-comm = begin -- η
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; η-comm = begin -- η-τ
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τ _ ∘ (idC ⁂ now) ≈⟨ refl⟩∘⟨ (⁂-cong₂ (sym identity²) refl ○ sym ⁂∘⁂) ⟩
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τ _ ∘ (idC ⁂ out⁻¹) ∘ (idC ⁂ i₁) ≈⟨ pullˡ (τ-helper _) ⟩
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(out⁻¹ ∘ (idC +₁ τ _) ∘ distributeˡ⁻¹) ∘ (idC ⁂ i₁) ≈⟨ pullʳ (pullʳ dstr-law) ⟩
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(out⁻¹ ∘ (idC +₁ τ _) ∘ distributeˡ⁻¹) ∘ (idC ⁂ i₁) ≈⟨ pullʳ (pullʳ dstr-law₁) ⟩
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out⁻¹ ∘ (idC +₁ τ _) ∘ i₁ ≈⟨ refl⟩∘⟨ +₁∘i₁ ⟩
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out⁻¹ ∘ i₁ ∘ idC ≈⟨ refl⟩∘⟨ identityʳ ⟩
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now ∎
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; μ-η-comm = μ-η-comm' -- μ-η
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; strength-assoc = λ {X} {Y} {Z} → begin
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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
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out ∘ extend (now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ τ _ ≈⟨ pullˡ (extendlaw (now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩)) ⟩
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{! !} ≈⟨ {! !} ⟩
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{! !} ≈⟨ {! !} ⟩
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{! !} ≈⟨ {! !} ⟩
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(idC +₁ extend (now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ τ _) ∘ {! !} ∎ })) ⟩
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u (Terminal.! (coalgebras (X × Y × Z)) {A = record { A = (X × Y) × D₀ Z ; α = {! !} }}) ≈⟨ {! !} ⟩
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τ _ ∘ (idC ⁂ τ _) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ∎ -- square
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; μ-η-comm = μ-η-comm' -- μ-τ
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; strength-assoc = strength-assoc' -- square
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}
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where
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open import Agda.Builtin.Sigma
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@ -330,8 +322,10 @@ Next we will show that the delay monad is strong, by giving a natural transforma
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out ∘ extend (now ∘ f) ≈⟨ extendlaw (now ∘ f) ⟩
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[ out ∘ now ∘ f , i₂ ∘ extend (now ∘ f) ] ∘ out ≈⟨ ([]-cong₂ (pullˡ unitlaw) refl) ⟩∘⟨refl ⟩
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(f +₁ extend (now ∘ f)) ∘ out ∎
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dstr-law : ∀ {A B C} → distributeˡ⁻¹ {A} {B} {C} ∘ (idC ⁂ i₁) ≈ i₁
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dstr-law = (refl⟩∘⟨ (sym inject₁)) ○ (cancelˡ (IsIso.isoˡ isIsoˡ))
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dstr-law₁ : ∀ {A B C} → distributeˡ⁻¹ {A} {B} {C} ∘ (idC ⁂ i₁) ≈ i₁
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dstr-law₁ = (refl⟩∘⟨ (sym inject₁)) ○ (cancelˡ (IsIso.isoˡ isIsoˡ))
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dstr-law₂ : ∀ {A B C} → distributeˡ⁻¹ {A} {B} {C} ∘ (idC ⁂ i₂) ≈ i₂
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dstr-law₂ = (refl⟩∘⟨ (sym inject₂)) ○ (cancelˡ (IsIso.isoˡ isIsoˡ))
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module _ (P : Category.Obj (CProduct C C)) where
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X = fst P
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@ -428,18 +422,71 @@ Next we will show that the delay monad is strong, by giving a natural transforma
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[ (idC +₁ now) ∘ i₁ , i₂ ] ≈⟨ []-cong₂ (+₁∘i₁ ○ identityʳ) refl ⟩
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[ i₁ , i₂ ] ≈⟨ +-η ⟩
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idC ∎
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diag₁ : ((idC ⁂ now) +₁ τ _) ∘ distributeˡ⁻¹ {X} {Y} {D₀ (D₀ Y)} ∘ ( idC ⁂ [ (idC +₁ now) ∘ out , i₂ ] ∘ out) ≈ out ∘ τ _
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diag₁ = begin
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((idC ⁂ now) +₁ τ _) ∘ distributeˡ⁻¹ {X} {Y} {D₀ (D₀ Y)} ∘ ( idC ⁂ [ (idC +₁ now) ∘ out , i₂ ] ∘ out) ≈⟨ {! !} ⟩
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(idC +₁ τ _) ∘ ((idC ⁂ now) +₁ idC) ∘ distributeˡ⁻¹ {X} {Y} {D₀ (D₀ Y)} ∘ ( idC ⁂ [ (idC +₁ now) ∘ out , i₂ ] ∘ out) ≈⟨ {! !} ⟩
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(idC +₁ τ _) ∘ distributeˡ⁻¹ {X} {D₀ Y} {D₀ (D₀ Y)} ∘ ((idC ⁂ (now +₁ idC))) ∘ ( idC ⁂ [ (idC +₁ now) ∘ out , i₂ ] ∘ out) ≈⟨ {! !} ⟩
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(idC +₁ τ _) ∘ distributeˡ⁻¹ {X} {D₀ Y} {D₀ (D₀ Y)} ∘ ( idC ⁂ [ (now +₁ idC) ∘ (idC +₁ now) ∘ out , (now +₁ idC) ∘ i₂ ] ∘ out) ≈⟨ {! !} ⟩
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(idC +₁ τ _) ∘ distributeˡ⁻¹ {X} {D₀ Y} {D₀ (D₀ Y)} ∘ ( idC ⁂ [ (now +₁ now) ∘ out , i₂ ] ∘ out) ≈⟨ {! !} ⟩
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(idC +₁ τ _) ∘ distributeˡ⁻¹ {X} {D₀ Y} {D₀ (D₀ Y)} ∘ ( idC ⁂ [ out ∘ now , i₂ ] ∘ out) ≈⟨ {! !} ⟩
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(idC +₁ τ _) ∘ distributeˡ⁻¹ {X} {D₀ Y} {D₀ (D₀ Y)} ∘ ( idC ⁂ out) ≈⟨ sym (τ-law (X , A (Terminal.⊤ (coalgebras Y)))) ⟩
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out ∘ τ _ ∎
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-- diag₁ : ((idC ⁂ now) +₁ τ _) ∘ distributeˡ⁻¹ {X} {Y} {D₀ (D₀ Y)} ∘ ( idC ⁂ [ (idC +₁ now) ∘ out , i₂ ] ∘ out) ≈ out ∘ τ _
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-- diag₁ = begin
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-- ((idC ⁂ now) +₁ τ _) ∘ distributeˡ⁻¹ {X} {Y} {D₀ (D₀ Y)} ∘ ( idC ⁂ [ (idC +₁ now) ∘ out , i₂ ] ∘ out) ≈⟨ {! !} ⟩
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-- (idC +₁ τ _) ∘ ((idC ⁂ now) +₁ idC) ∘ distributeˡ⁻¹ {X} {Y} {D₀ (D₀ Y)} ∘ ( idC ⁂ [ (idC +₁ now) ∘ out , i₂ ] ∘ out) ≈⟨ {! !} ⟩
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-- (idC +₁ τ _) ∘ distributeˡ⁻¹ {X} {D₀ Y} {D₀ (D₀ Y)} ∘ ((idC ⁂ (now +₁ idC))) ∘ ( idC ⁂ [ (idC +₁ now) ∘ out , i₂ ] ∘ out) ≈⟨ {! !} ⟩
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-- (idC +₁ τ _) ∘ distributeˡ⁻¹ {X} {D₀ Y} {D₀ (D₀ Y)} ∘ ( idC ⁂ [ (now +₁ idC) ∘ (idC +₁ now) ∘ out , (now +₁ idC) ∘ i₂ ] ∘ out) ≈⟨ {! !} ⟩
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-- (idC +₁ τ _) ∘ distributeˡ⁻¹ {X} {D₀ Y} {D₀ (D₀ Y)} ∘ ( idC ⁂ [ (now +₁ now) ∘ out , i₂ ] ∘ out) ≈⟨ {! !} ⟩
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-- (idC +₁ τ _) ∘ distributeˡ⁻¹ {X} {D₀ Y} {D₀ (D₀ Y)} ∘ ( idC ⁂ [ out ∘ now , i₂ ] ∘ out) ≈⟨ {! !} ⟩
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-- (idC +₁ τ _) ∘ distributeˡ⁻¹ {X} {D₀ Y} {D₀ (D₀ Y)} ∘ ( idC ⁂ out) ≈⟨ sym (τ-law (X , A (Terminal.⊤ (coalgebras Y)))) ⟩
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-- out ∘ τ _ ∎
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diag₂ = τ-law
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diag₃ = out-law {Y = D₀ (X × Y)} (extend idC)
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strength-assoc' : ∀ {X Y Z} → extend (now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ τ ((X × Y), Z) ≈ τ _ ∘ (idC ⁂ τ _) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩
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strength-assoc' {X} {Y} {Z} = begin
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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
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out ∘ extend (now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ τ _ ≈⟨ pullˡ (extendlaw (now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩)) ⟩
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([ out ∘ now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ , i₂ ∘ extend (now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ] ∘ out) ∘ τ _ ≈⟨ pullʳ (τ-law (X × Y , Z)) ⟩
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[ 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 ⟩
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(⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ +₁ extend (now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩)) ∘ (idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ pullˡ +₁∘+₁ ⟩
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(⟨ 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 ⟩
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(idC ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ +₁ (extend (now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ τ _) ∘ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ sym (pullˡ +₁∘+₁) ⟩
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(idC +₁ extend (now ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ τ _) ∘ (⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ∎ })) ⟩
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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
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out ∘ τ _ ∘ (idC ⁂ τ _) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ≈⟨ pullˡ (τ-law (X , Y × Z)) ⟩
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((idC +₁ τ _) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out)) ∘ (idC ⁂ τ _) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ≈⟨ pullʳ (pullˡ (pullʳ ⁂∘⁂)) ⟩
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(idC +₁ τ _) ∘ (distributeˡ⁻¹ ∘ (idC ∘ idC ⁂ out ∘ τ _)) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ≈⟨ refl⟩∘⟨ assoc ⟩
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(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) ⟩
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(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 ⁂∘⁂))) ⟩
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(idC +₁ τ _) ∘ (distributeˡ⁻¹ ∘ (idC ⁂ (idC +₁ τ _))) ∘ (idC ⁂ distributeˡ⁻¹) ∘ (idC ⁂ (idC ⁂ out)) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ≈⟨ refl⟩∘⟨ (refl⟩∘⟨ (refl⟩∘⟨ helper₁)) ⟩
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(idC +₁ τ _) ∘ (distributeˡ⁻¹ ∘ (idC ⁂ (idC +₁ τ _))) ∘ (idC ⁂ distributeˡ⁻¹) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ∘ (idC ⁂ out) ≈⟨ refl⟩∘⟨ (helper₂ ⟩∘⟨refl) ⟩
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(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') ⟩
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(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) ⟩
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(idC +₁ τ _ ∘ (idC ⁂ τ (Y , Z))) ∘ (distributeˡ⁻¹ ∘ (idC ⁂ distributeˡ⁻¹) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ (idC ⁂ out) ≈⟨ refl⟩∘⟨ (helper₃ ⟩∘⟨refl) ⟩
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(idC +₁ τ _ ∘ (idC ⁂ τ (Y , Z))) ∘ ((⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ +₁ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ distributeˡ⁻¹) ∘ (idC ⁂ out) ≈⟨ assoc²'' ⟩
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((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) ⟩
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(idC ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ +₁ (τ _ ∘ (idC ⁂ τ _) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ≈⟨ sym (pullˡ +₁∘+₁) ⟩
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(idC +₁ τ _ ∘ (idC ⁂ τ _) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ (⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ out) ∎ }) ⟩
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τ _ ∘ (idC ⁂ τ _) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ∎
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where
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helper₁ : (idC ⁂ (idC ⁂ out)) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ≈ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ∘ (idC {X × Y} ⁂ out {Z})
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helper₁ = begin
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(idC ⁂ (idC ⁂ out)) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ≈⟨ ⁂∘⟨⟩ ⟩
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⟨ idC ∘ π₁ ∘ π₁ , (idC ⁂ out) ∘ ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ≈⟨ ⟨⟩-cong₂ identityˡ ⁂∘⟨⟩ ⟩
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⟨ π₁ ∘ π₁ , ⟨ idC ∘ π₂ ∘ π₁ , out ∘ π₂ ⟩ ⟩ ≈⟨ ⟨⟩-cong₂ refl (⟨⟩-cong₂ identityˡ refl) ⟩
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⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , out ∘ π₂ ⟩ ⟩ ≈⟨ sym (⟨⟩-cong₂ refl (⟨⟩-cong₂ (refl⟩∘⟨ identityˡ) refl)) ⟩
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⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ idC ∘ π₁ , out ∘ π₂ ⟩ ⟩ ≈⟨ sym (⟨⟩-cong₂ (refl⟩∘⟨ identityˡ) (⟨⟩-cong₂ (pullʳ π₁∘⁂) π₂∘⁂)) ⟩
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⟨ π₁ ∘ idC ∘ π₁ , ⟨ (π₂ ∘ π₁) ∘ (idC {X × Y} ⁂ out {Z}) , π₂ ∘ (idC {X × Y} ⁂ out {Z}) ⟩ ⟩ ≈⟨ sym (⟨⟩-cong₂ (pullʳ project₁) ⟨⟩∘) ⟩
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⟨ (π₁ ∘ π₁) ∘ (idC {X × Y} ⁂ out {Z}) , ⟨ π₂ ∘ π₁ , π₂ ⟩ ∘ (idC {X × Y} ⁂ out {Z}) ⟩ ≈⟨ sym ⟨⟩∘ ⟩
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⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ∘ (idC {X × Y} ⁂ out {Z}) ∎
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helper₂ : distributeˡ⁻¹ ∘ (idC ⁂ (idC +₁ τ _)) ≈ (idC +₁ (idC ⁂ τ (Y , Z))) ∘ distributeˡ⁻¹
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helper₂ = sym (distribute₁ idC idC (τ _)) ○ (+₁-cong₂ (⟨⟩-unique id-comm id-comm) refl) ⟩∘⟨refl
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helper₃ : distributeˡ⁻¹ ∘ (idC ⁂ distributeˡ⁻¹) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ≈ (⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ +₁ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ distributeˡ⁻¹
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helper₃ = sym (begin
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(⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ +₁ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ distributeˡ⁻¹ ≈⟨ introˡ (IsIso.isoˡ isIsoˡ) ⟩
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(distributeˡ⁻¹ ∘ distributeˡ) ∘ (⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ +₁ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩) ∘ distributeˡ⁻¹ ≈⟨ pullʳ (pullˡ []∘+₁) ⟩
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distributeˡ⁻¹ ∘ [ (idC ⁂ i₁) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ , (idC ⁂ i₂) ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ] ∘ distributeˡ⁻¹ ≈⟨ refl⟩∘⟨ (([]-cong₂ ⁂∘⟨⟩ ⁂∘⟨⟩) ⟩∘⟨refl) ⟩
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distributeˡ⁻¹ ∘ [ ⟨ idC ∘ π₁ ∘ π₁ , i₁ ∘ ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ , ⟨ idC ∘ π₁ ∘ π₁ , i₂ ∘ ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ] ∘ distributeˡ⁻¹ ≈⟨ refl⟩∘⟨ []-cong₂ (⟨⟩-cong₂ identityˡ (pushˡ (sym dstr-law₁))) (⟨⟩-cong₂ identityˡ ((pushˡ (sym dstr-law₂)))) ⟩∘⟨refl ⟩
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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) ⟩
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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) ⟩
|
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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) ⟩
|
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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) ⟩
|
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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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
|
||||
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|
@ -474,16 +521,6 @@ https://q.uiver.app/#q=WzAsOSxbMCwyLCIxIFxcdGltZXMgREIiXSxbMiwyLCIxIFxcdGltZXMgK
|
|||
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) }
|
||||
distribute₁ : ∀ {X Y Z U V W} (f : X ⇒ U) (g : Y ⇒ V) (h : Z ⇒ W) → ((f ⁂ g) +₁ (f ⁂ h)) ∘ distributeˡ⁻¹ ≈ distributeˡ⁻¹ ∘ (f ⁂ (g +₁ h))
|
||||
distribute₁ f g h = begin
|
||||
((f ⁂ g) +₁ (f ⁂ h)) ∘ distributeˡ⁻¹ ≈⟨ introˡ (IsIso.isoˡ isIsoˡ) ⟩
|
||||
(distributeˡ⁻¹ ∘ distributeˡ) ∘ ((f ⁂ g) +₁ (f ⁂ h)) ∘ distributeˡ⁻¹ ≈⟨ pullˡ (pullʳ []∘+₁) ⟩
|
||||
(distributeˡ⁻¹ ∘ [(idC ⁂ i₁) ∘ (f ⁂ g) , (idC ⁂ i₂) ∘ (f ⁂ h)]) ∘ distributeˡ⁻¹ ≈⟨ (refl⟩∘⟨ ([]-cong₂ ⁂∘⁂ ⁂∘⁂)) ⟩∘⟨refl ⟩
|
||||
(distributeˡ⁻¹ ∘ [ idC ∘ f ⁂ i₁ ∘ g , idC ∘ f ⁂ i₂ ∘ h ]) ∘ distributeˡ⁻¹ ≈⟨ sym ((refl⟩∘⟨ ([]-cong₂ (⁂-cong₂ id-comm +₁∘i₁) (⁂-cong₂ id-comm +₁∘i₂))) ⟩∘⟨refl) ⟩
|
||||
(distributeˡ⁻¹ ∘ [ f ∘ idC ⁂ (g +₁ h) ∘ i₁ , f ∘ idC ⁂ (g +₁ h) ∘ i₂ ]) ∘ distributeˡ⁻¹ ≈˘⟨ (refl⟩∘⟨ ([]-cong₂ ⁂∘⁂ ⁂∘⁂)) ⟩∘⟨refl ⟩
|
||||
(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)) ⟩
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distributeˡ⁻¹ ∘ (f ⁂ (g +₁ h)) ∎
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-- ⁂ ○ ˡ ʳ
|
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strongMonad : StrongMonad monoidal
|
||||
strongMonad = record { M = monad ; strength = strength }
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Reference in a new issue