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src/Monad/Instance/K/StrongPreElgot.lagda.md
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@ -166,35 +166,34 @@ isInitialStrongPreElgot = record { ! = !′ ; !-unique = !-unique′ }
η' X ∎ η' X ∎
α-strength : ∀ {X Y : Obj} → η' (X × Y) ∘ strengthenK.η (X , Y) ≈ strengthen.η (X , Y) ∘ (idC ⁂ η' Y) α-strength : ∀ {X Y : Obj} → η' (X × Y) ∘ strengthenK.η (X , Y) ≈ strengthen.η (X , Y) ∘ (idC ⁂ η' Y)
α-strength {X} {Y} = begin α-strength {X} {Y} = begin
η' (X × Y) ∘ strengthenK.η (X , Y) ≈⟨ IsStableFreeUniformIterationAlgebra.♯-unique (stable Y) (T.η.η (X × Y)) (η' (X × Y) ∘ strengthenK.η (X , Y)) (sym pres₁) pres₃ ⟩ η' (X × Y) ∘ strengthenK.η (X , Y) ≈⟨ IsStableFreeUniformIterationAlgebra.♯-unique (stable Y) (T.η.η (X × Y)) (η' (X × Y) ∘ strengthenK.η (X , Y)) (sym pres₁) pres₃ ⟩
IsStableFreeUniformIterationAlgebra.[ (stable Y) , Functor.₀ elgot-to-uniformF (T-Alg (X × Y)) ]♯ (T.η.η (X × Y)) ≈⟨ sym (IsStableFreeUniformIterationAlgebra.♯-unique (stable Y) (T.η.η (X × Y)) (strengthen.η (X , Y) ∘ (idC ⁂ η' Y)) (sym pres₂) pres₄) ⟩ IsStableFreeUniformIterationAlgebra.[ (stable Y) , Functor.₀ elgot-to-uniformF (T-Alg (X × Y)) ]♯ (T.η.η (X × Y)) ≈⟨ sym (IsStableFreeUniformIterationAlgebra.♯-unique (stable Y) (T.η.η (X × Y)) (strengthen.η (X , Y) ∘ (idC ⁂ η' Y)) (sym pres₂) pres₄) ⟩
strengthen.η (X , Y) ∘ (idC ⁂ η' Y) ∎ strengthen.η (X , Y) ∘ (idC ⁂ η' Y) ∎
where where
pres₁ : (η' (X × Y) ∘ strengthenK.η (X , Y)) ∘ (idC ⁂ ηK.η Y) ≈ T.η.η (X × Y) pres₁ : (η' (X × Y) ∘ strengthenK.η (X , Y)) ∘ (idC ⁂ ηK.η Y) ≈ T.η.η (X × Y)
pres₁ = begin pres₁ = begin
(η' (X × Y) ∘ strengthenK.η (X , Y)) ∘ (idC ⁂ ηK.η Y) ≈⟨ pullʳ (τ-η (X , Y)) ⟩ (η' (X × Y) ∘ strengthenK.η (X , Y)) ∘ (idC ⁂ ηK.η Y) ≈⟨ pullʳ (τ-η (X , Y)) ⟩
η' (X × Y) ∘ ηK.η (X × Y) ≈⟨ α-η ⟩ η' (X × Y) ∘ ηK.η (X × Y) ≈⟨ α-η ⟩
T.η.η (X × Y) ∎ T.η.η (X × Y) ∎
pres₂ : (strengthen.η (X , Y) ∘ (idC ⁂ η' Y)) ∘ (idC ⁂ ηK.η Y) ≈ T.η.η (X × Y) pres₂ : (strengthen.η (X , Y) ∘ (idC ⁂ η' Y)) ∘ (idC ⁂ ηK.η Y) ≈ T.η.η (X × Y)
pres₂ = begin pres₂ = begin
(strengthen.η (X , Y) ∘ (idC ⁂ η' Y)) ∘ (idC ⁂ ηK.η Y) ≈⟨ pullʳ (⁂∘⁂ ○ ⁂-cong₂ identity² α-η) ⟩ (strengthen.η (X , Y) ∘ (idC ⁂ η' Y)) ∘ (idC ⁂ ηK.η Y) ≈⟨ pullʳ (⁂∘⁂ ○ ⁂-cong₂ identity² α-η) ⟩
strengthen.η (X , Y) ∘ (idC ⁂ T.η.η Y) ≈⟨ SM.η-comm ⟩ strengthen.η (X , Y) ∘ (idC ⁂ T.η.η Y) ≈⟨ SM.η-comm ⟩
T.η.η (X × Y) ∎ T.η.η (X × Y) ∎
pres₃ : ∀ {Z : Obj} (h : Z ⇒ K.₀ Y + Z) → (η' (X × Y) ∘ strengthenK.η (X , Y)) ∘ (idC ⁂ h #K) ≈ ((η' (X × Y) ∘ strengthenK.η (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #T pres₃ : ∀ {Z : Obj} (h : Z ⇒ K.₀ Y + Z) → (η' (X × Y) ∘ strengthenK.η (X , Y)) ∘ (idC ⁂ h #K) ≈ ((η' (X × Y) ∘ strengthenK.η (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #T
pres₃ {Z} h = begin pres₃ {Z} h = begin
(η' (X × Y) ∘ strengthenK.η (X , Y)) ∘ (idC ⁂ h #K) ≈⟨ pullʳ (τ-comm h) ⟩ (η' (X × Y) ∘ strengthenK.η (X , Y)) ∘ (idC ⁂ h #K) ≈⟨ pullʳ (τ-comm h) ⟩
η' (X × Y) ∘ ((τ (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #K ≈⟨ η'-preserves ((τ (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) ⟩ η' (X × Y) ∘ ((τ (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #K ≈⟨ η'-preserves ((τ (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) ⟩
((η' (X × Y) +₁ idC) ∘ (strengthenK.η (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #T ≈⟨ #-resp-≈ (StrongPreElgotMonad.elgotalgebras A) (pullˡ (+₁∘+₁ ○ +₁-cong₂ refl identity²)) ⟩ ((η' (X × Y) +₁ idC) ∘ (strengthenK.η (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #T ≈⟨ #-resp-≈ (StrongPreElgotMonad.elgotalgebras A) (pullˡ (+₁∘+₁ ○ +₁-cong₂ refl identity²)) ⟩
((η' (X × Y) ∘ strengthenK.η (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #T ∎ ((η' (X × Y) ∘ strengthenK.η (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #T ∎
pres₄ : ∀ {Z : Obj} (h : Z ⇒ K.₀ Y + Z) → (strengthen.η (X , Y) ∘ (idC ⁂ η' Y)) ∘ (idC ⁂ h #K) ≈ ((strengthen.η (X , Y) ∘ (idC ⁂ η' Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #T pres₄ : ∀ {Z : Obj} (h : Z ⇒ K.₀ Y + Z) → (strengthen.η (X , Y) ∘ (idC ⁂ η' Y)) ∘ (idC ⁂ h #K) ≈ ((strengthen.η (X , Y) ∘ (idC ⁂ η' Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #T
pres₄ {Z} h = begin pres₄ {Z} h = begin
(strengthen.η (X , Y) ∘ (idC ⁂ η' Y)) ∘ (idC ⁂ h #K) ≈⟨ pullʳ (⁂∘⁂ ○ ⁂-cong₂ identity² (η'-preserves h)) ⟩ (strengthen.η (X , Y) ∘ (idC ⁂ η' Y)) ∘ (idC ⁂ h #K) ≈⟨ pullʳ (⁂∘⁂ ○ ⁂-cong₂ identity² (η'-preserves h)) ⟩
strengthen.η (X , Y) ∘ (idC ⁂ ((η' Y +₁ idC) ∘ h) #T) ≈⟨ StrongPreElgotMonad.strengthen-preserves A ((η' Y +₁ idC) ∘ h) ⟩ strengthen.η (X , Y) ∘ (idC ⁂ ((η' Y +₁ idC) ∘ h) #T) ≈⟨ StrongPreElgotMonad.strengthen-preserves A ((η' Y +₁ idC) ∘ h) ⟩
((strengthen.η (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ (η' Y +₁ idC) ∘ h)) #T ≈⟨ sym (#-resp-≈ (StrongPreElgotMonad.elgotalgebras A) (refl⟩∘⟨ (pullʳ (⁂∘⁂ ○ ⁂-cong₂ identity² refl)))) ⟩ ((strengthen.η (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ (η' Y +₁ idC) ∘ h)) #T ≈⟨ sym (#-resp-≈ (StrongPreElgotMonad.elgotalgebras A) (refl⟩∘⟨ (pullʳ (⁂∘⁂ ○ ⁂-cong₂ identity² refl)))) ⟩
(((strengthen.η (X , Y) +₁ idC) ∘ (distributeˡ⁻¹ ∘ (idC ⁂ (η' Y +₁ idC))) ∘ (idC ⁂ h)) #T) ≈⟨ sym (#-resp-≈ (StrongPreElgotMonad.elgotalgebras A) (refl⟩∘⟨ (pullˡ ((+₁-cong₂ refl (sym (⟨⟩-unique id-comm id-comm))) ⟩∘⟨refl ○ distribute₁ idC (η' Y) idC)))) ⟩ (((strengthen.η (X , Y) +₁ idC) ∘ (distributeˡ⁻¹ ∘ (idC ⁂ (η' Y +₁ idC))) ∘ (idC ⁂ h)) #T) ≈⟨ sym (#-resp-≈ (StrongPreElgotMonad.elgotalgebras A) (refl⟩∘⟨ (pullˡ ((+₁-cong₂ refl (sym (⟨⟩-unique id-comm id-comm))) ⟩∘⟨refl ○ distribute₁ idC (η' Y) idC)))) ⟩
-- ((strengthen.η (X , Y) +₁ idC) ∘ ((idC ⁂ η' Y) +₁ (idC ⁂ idC)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #T ≈⟨ {! !} ⟩ ((strengthen.η (X , Y) +₁ idC) ∘ ((idC ⁂ η' Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #T ≈⟨ #-resp-≈ (StrongPreElgotMonad.elgotalgebras A) (pullˡ (+₁∘+₁ ○ +₁-cong₂ refl identity²)) ⟩
((strengthen.η (X , Y) +₁ idC) ∘ ((idC ⁂ η' Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #T ≈⟨ #-resp-≈ (StrongPreElgotMonad.elgotalgebras A) (pullˡ (+₁∘+₁ ○ +₁-cong₂ refl identity²)) ⟩ ((strengthen.η (X , Y) ∘ (idC ⁂ η' Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #T ∎
((strengthen.η (X , Y) ∘ (idC ⁂ η' Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #T ∎
!-unique′ : ∀ {A : StrongPreElgotMonad} (f : StrongPreElgotMonad-Morphism strongPreElgot A) → StrongPreElgotMonad-Morphism.α (!′ {A = A}) ≃ StrongPreElgotMonad-Morphism.α f !-unique′ : ∀ {A : StrongPreElgotMonad} (f : StrongPreElgotMonad-Morphism strongPreElgot A) → StrongPreElgotMonad-Morphism.α (!′ {A = A}) ≃ StrongPreElgotMonad-Morphism.α f
!-unique′ {A} f {X} = sym (FreeObject.*-uniq !-unique′ {A} f {X} = sym (FreeObject.*-uniq
(freeElgot X) (freeElgot X)