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align proofs

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Leon Vatthauer 2023-11-20 11:40:04 +01:00
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src/Monad/Instance/K/StrongPreElgot.lagda.md
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@ -192,7 +192,6 @@ isInitialStrongPreElgot = record { ! = !′ ; !-unique = !-unique′ }
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))) ∘ (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) ∘ (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