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Leon Vatthauer 2024-02-01 12:51:13 +01:00
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agda/src/Category/Construction/ElgotAlgebras.lagda.md
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@ -366,13 +366,17 @@ module Category.Construction.ElgotAlgebras {o ℓ e} {C : Category o ℓ e} wher
((i₂ ∘ id) ∘ (id ⁂ id)) ∘ ⟨ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ⟩ ≈⟨ (cancelʳ (identityˡ ○ ⟨⟩-unique id-comm id-comm)) ⟩∘⟨refl ⟩
i₂ ∘ ⟨ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ⟩ ∎
Elgot-Algebra-Morphism.h (CanonicalCartesianClosed.curry cccc {A} {B} {C} f) = λg < f >
Elgot-Algebra-Morphism.preserves (CanonicalCartesianClosed.curry cccc {A} {B} {C} f) {X} {g} = begin
λg < f > ∘ (Elgot-Algebra._# A) g ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩
λg ((Elgot-Algebra._# C) ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (((λg < f > +₁ id) ∘ g) ⁂ id))) ∎
Elgot-Algebra-Morphism.preserves (CanonicalCartesianClosed.curry cccc {A} {B} {C} f) {X} {g} = λ-unique′ (begin
eval′ ∘ ((λg < f > ∘ (Elgot-Algebra._# A) g) ⁂ id) ≈⟨ refl⟩∘⟨ ⁂-cong₂ subst refl ⟩
eval′ ∘ ((λg (< f > ∘ (((Elgot-Algebra._# A) g) ⁂ id))) ⁂ id) ≈⟨ β′ ⟩
< f > ∘ (((Elgot-Algebra._# A) g) ⁂ id) ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩
< f > ∘ ⟨ (Elgot-Algebra._# A) ((π₁ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id))) , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id))) ⟩ ≈⟨ Elgot-Algebra-Morphism.preserves f ⟩
(Elgot-Algebra._# C) ((< f > +₁ id) ∘ distributeʳ⁻¹ ∘ (g ⁂ id)) ≈˘⟨ {! !} ⟩
(Elgot-Algebra._# C) ((eval′ +₁ id) ∘ (((λg < f > ⁂ id) +₁ (id ⁂ id)) ∘ distributeʳ⁻¹) ∘ (g ⁂ id)) ≈˘⟨ Elgot-Algebra.#-resp-≈ C (refl⟩∘⟨ (pullˡ (sym (distributeʳ⁻¹-natural id (λg < f >) id)))) ⟩
(Elgot-Algebra._# C) ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ((λg < f > +₁ id) ⁂ id) ∘ (g ⁂ id)) ≈⟨ Elgot-Algebra.#-resp-≈ C (refl⟩∘⟨ (refl⟩∘⟨ (⁂∘⁂ ○ ⁂-cong₂ refl identity²))) ⟩
(Elgot-Algebra._# C) ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (((λg < f > +₁ id) ∘ g) ⁂ id)) ∎)
CanonicalCartesianClosed.eval-comp cccc {A} {B} {C} {f} = {! !}
CanonicalCartesianClosed.curry-unique cccc {A} {B} {C} {f} {g} eq = {! !}
-- record