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Finish proof of distributivity

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Frederik Hanghøj Iversen 2018-03-01 20:47:36 +01:00
parent 2ceb027f7a
commit c4e3625746
1 changed files with 44 additions and 15 deletions
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59
src/Cat/Category/Monad.agda
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@ -77,26 +77,55 @@ module Monoidal {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
isDistributive {X} {Y} {Z} g f = sym done isDistributive {X} {Y} {Z} g f = sym done
where where
module R² = Functor F[ R ∘ R ] module R² = Functor F[ R ∘ R ]
postulate distrib : ∀ {A B C D} {a : Arrow C D} {b : Arrow B C} {c : Arrow A B}
distrib : ∀ {A B C D} {a : Arrow C D} {b : Arrow B C} {c : Arrow A B} → R.func→ (a ∘ b ∘ c)
→ R.func→ (a ∘ b ∘ c) ≡ R.func→ a ∘ R.func→ b ∘ R.func→ c
≡ R.func→ a ∘ R.func→ b ∘ R.func→ c distrib {a = a} {b} {c} = begin
comm : ∀ {A B C D E} R.func→ (a ∘ b ∘ c) ≡⟨ distr ⟩
→ {a : Arrow D E} {b : Arrow C D} {c : Arrow B C} {d : Arrow A B} R.func→ (a ∘ b) ∘ R.func→ c ≡⟨ cong (_∘ _) distr ⟩
→ a ∘ (b ∘ c ∘ d) ≡ a ∘ b ∘ c ∘ d R.func→ a ∘ R.func→ b ∘ R.func→ c ∎
where
distr = R.isDistributive
comm : ∀ {A B C D E}
→ {a : Arrow D E} {b : Arrow C D} {c : Arrow B C} {d : Arrow A B}
→ a ∘ (b ∘ c ∘ d) ≡ a ∘ b ∘ c ∘ d
comm {a = a} {b} {c} {d} = begin
a ∘ (b ∘ c ∘ d) ≡⟨⟩
a ∘ ((b ∘ c) ∘ d) ≡⟨ cong (_∘_ a) (sym asc) ⟩
a ∘ (b ∘ (c ∘ d)) ≡⟨ asc ⟩
(a ∘ b) ∘ (c ∘ d) ≡⟨ asc ⟩
((a ∘ b) ∘ c) ∘ d ≡⟨⟩
a ∘ b ∘ c ∘ d ∎
where
asc = ℂ.isAssociative
lemmm : μ Z ∘ R.func→ (μ Z) ≡ μ Z ∘ μ (R.func* Z) lemmm : μ Z ∘ R.func→ (μ Z) ≡ μ Z ∘ μ (R.func* Z)
lemmm = isAssociative lemmm = isAssociative
lem4 : μ (R.func* Z) ∘ R².func→ g ≡ R.func→ g ∘ μ Y lem4 : μ (R.func* Z) ∘ R².func→ g ≡ R.func→ g ∘ μ Y
lem4 = μNat g lem4 = μNat g
done = begin done = begin
μ Z ∘ R.func→ (μ Z ∘ R.func→ g ∘ f) ≡⟨ cong (λ φ → μ Z ∘ φ) distrib ⟩ μ Z ∘ R.func→ (μ Z ∘ R.func→ g ∘ f)
μ Z ∘ (R.func→ (μ Z) ∘ R.func→ (R.func→ g) ∘ R.func→ f) ≡⟨⟩ ≡⟨ cong (λ φ → μ Z ∘ φ) distrib ⟩
μ Z ∘ (R.func→ (μ Z) ∘ R².func→ g ∘ R.func→ f) ≡⟨ {!!} ⟩ -- ●-solver? μ Z ∘ (R.func→ (μ Z) ∘ R.func→ (R.func→ g) ∘ R.func→ f)
(μ Z ∘ R.func→ (μ Z)) ∘ (R².func→ g ∘ R.func→ f) ≡⟨ cong (λ φ → φ ∘ (R².func→ g ∘ R.func→ f)) lemmm ⟩ ≡⟨⟩
(μ Z ∘ μ (R.func* Z)) ∘ (R².func→ g ∘ R.func→ f) ≡⟨ {!!} ⟩ -- ●-solver? μ Z ∘ (R.func→ (μ Z) ∘ R².func→ g ∘ R.func→ f)
μ Z ∘ μ (R.func* Z) ∘ R².func→ g ∘ R.func→ f ≡⟨ {!!} ⟩ -- ●-solver + lem4 ≡⟨ cong (_∘_ (μ Z)) (sym ℂ.isAssociative) ⟩ -- ●-solver?
μ Z ∘ R.func→ g ∘ μ Y ∘ R.func→ f ≡⟨ sym (Category.isAssociative ℂ) ⟩ μ Z ∘ (R.func→ (μ Z) ∘ (R².func→ g ∘ R.func→ f))
μ Z ∘ R.func→ g ∘ (μ Y ∘ R.func→ f) ∎ ≡⟨ ℂ.isAssociative ⟩
(μ Z ∘ R.func→ (μ Z)) ∘ (R².func→ g ∘ R.func→ f)
≡⟨ cong (λ φ → φ ∘ (R².func→ g ∘ R.func→ f)) isAssociative ⟩
(μ Z ∘ μ (R.func* Z)) ∘ (R².func→ g ∘ R.func→ f)
≡⟨ ℂ.isAssociative ⟩ -- ●-solver?
μ Z ∘ μ (R.func* Z) ∘ R².func→ g ∘ R.func→ f
≡⟨⟩ -- ●-solver + lem4
((μ Z ∘ μ (R.func* Z)) ∘ R².func→ g) ∘ R.func→ f
≡⟨ cong (_∘ R.func→ f) (sym ℂ.isAssociative) ⟩
(μ Z ∘ (μ (R.func* Z) ∘ R².func→ g)) ∘ R.func→ f
≡⟨ cong (λ φ → φ ∘ R.func→ f) (cong (_∘_ (μ Z)) lem4) ⟩
(μ Z ∘ (R.func→ g ∘ μ Y)) ∘ R.func→ f ≡⟨ cong (_∘ R.func→ f) ℂ.isAssociative ⟩
μ Z ∘ R.func→ g ∘ μ Y ∘ R.func→ f
≡⟨ sym (Category.isAssociative ℂ) ⟩
μ Z ∘ R.func→ g ∘ (μ Y ∘ R.func→ f)
∎
record Monad : Set ℓ where record Monad : Set ℓ where
field field