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Tidy up proof a bit

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Frederik Hanghøj Iversen 2018-03-06 09:39:48 +01:00
parent 35419ad86e
commit 7647a452cd
1 changed files with 13 additions and 30 deletions
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43
src/Cat/Category/Monad.agda
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@ -72,54 +72,37 @@ module Monoidal {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
f ∎ f ∎
isDistributive : IsDistributive isDistributive : IsDistributive
isDistributive {X} {Y} {Z} g f = sym done isDistributive {X} {Y} {Z} g f = sym aux
where where
module R² = Functor F[ R ∘ R ] module R² = Functor F[ R ∘ R ]
distrib : ∀ {A B C D} {a : Arrow C D} {b : Arrow B C} {c : Arrow A B} distrib3 : ∀ {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 distrib3 {a = a} {b} {c} = begin
R.func→ (a ∘ b ∘ c) ≡⟨ distr ⟩ R.func→ (a ∘ b ∘ c) ≡⟨ R.isDistributive ⟩
R.func→ (a ∘ b) ∘ R.func→ c ≡⟨ cong (_∘ _) distr ⟩ R.func→ (a ∘ b) ∘ R.func→ c ≡⟨ cong (_∘ _) R.isDistributive ⟩
R.func→ a ∘ R.func→ b ∘ R.func→ c ∎ R.func→ a ∘ R.func→ b ∘ R.func→ c ∎
where aux = begin
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 : joinT Z ∘ R.func→ (joinT Z) ≡ joinT Z ∘ joinT (R.func* Z)
lemmm = isAssociative
lem4 : joinT (R.func* Z) ∘ R².func→ g ≡ R.func→ g ∘ joinT Y
lem4 = joinN g
done = begin
joinT Z ∘ R.func→ (joinT Z ∘ R.func→ g ∘ f) joinT Z ∘ R.func→ (joinT Z ∘ R.func→ g ∘ f)
≡⟨ cong (λ φ → joinT Z ∘ φ) distrib ⟩ ≡⟨ cong (λ φ → joinT Z ∘ φ) distrib3 ⟩
joinT Z ∘ (R.func→ (joinT Z) ∘ R.func→ (R.func→ g) ∘ R.func→ f) joinT Z ∘ (R.func→ (joinT Z) ∘ R.func→ (R.func→ g) ∘ R.func→ f)
≡⟨⟩ ≡⟨⟩
joinT Z ∘ (R.func→ (joinT Z) ∘ R².func→ g ∘ R.func→ f) joinT Z ∘ (R.func→ (joinT Z) ∘ R².func→ g ∘ R.func→ f)
≡⟨ cong (_∘_ (joinT Z)) (sym ℂ.isAssociative) ⟩ -- ●-solver? ≡⟨ cong (_∘_ (joinT Z)) (sym ℂ.isAssociative) ⟩
joinT Z ∘ (R.func→ (joinT Z) ∘ (R².func→ g ∘ R.func→ f)) joinT Z ∘ (R.func→ (joinT Z) ∘ (R².func→ g ∘ R.func→ f))
≡⟨ ℂ.isAssociative ⟩ ≡⟨ ℂ.isAssociative ⟩
(joinT Z ∘ R.func→ (joinT Z)) ∘ (R².func→ g ∘ R.func→ f) (joinT Z ∘ R.func→ (joinT Z)) ∘ (R².func→ g ∘ R.func→ f)
≡⟨ cong (λ φ → φ ∘ (R².func→ g ∘ R.func→ f)) isAssociative ⟩ ≡⟨ cong (λ φ → φ ∘ (R².func→ g ∘ R.func→ f)) isAssociative ⟩
(joinT Z ∘ joinT (R.func* Z)) ∘ (R².func→ g ∘ R.func→ f) (joinT Z ∘ joinT (R.func* Z)) ∘ (R².func→ g ∘ R.func→ f)
≡⟨ ℂ.isAssociative ⟩ -- ●-solver? ≡⟨ ℂ.isAssociative ⟩
joinT Z ∘ joinT (R.func* Z) ∘ R².func→ g ∘ R.func→ f joinT Z ∘ joinT (R.func* Z) ∘ R².func→ g ∘ R.func→ f
≡⟨⟩ -- ●-solver + lem4 ≡⟨⟩
((joinT Z ∘ joinT (R.func* Z)) ∘ R².func→ g) ∘ R.func→ f ((joinT Z ∘ joinT (R.func* Z)) ∘ R².func→ g) ∘ R.func→ f
≡⟨ cong (_∘ R.func→ f) (sym ℂ.isAssociative) ⟩ ≡⟨ cong (_∘ R.func→ f) (sym ℂ.isAssociative) ⟩
(joinT Z ∘ (joinT (R.func* Z) ∘ R².func→ g)) ∘ R.func→ f (joinT Z ∘ (joinT (R.func* Z) ∘ R².func→ g)) ∘ R.func→ f
≡⟨ cong (λ φ → φ ∘ R.func→ f) (cong (_∘_ (joinT Z)) lem4) ⟩ ≡⟨ cong (λ φ → φ ∘ R.func→ f) (cong (_∘_ (joinT Z)) (joinN g)) ⟩
(joinT Z ∘ (R.func→ g ∘ joinT Y)) ∘ R.func→ f ≡⟨ cong (_∘ R.func→ f) ℂ.isAssociative ⟩ (joinT Z ∘ (R.func→ g ∘ joinT Y)) ∘ R.func→ f
≡⟨ cong (_∘ R.func→ f) ℂ.isAssociative ⟩
joinT Z ∘ R.func→ g ∘ joinT Y ∘ R.func→ f joinT Z ∘ R.func→ g ∘ joinT Y ∘ R.func→ f
≡⟨ sym (Category.isAssociative ℂ) ⟩ ≡⟨ sym (Category.isAssociative ℂ) ⟩
joinT Z ∘ R.func→ g ∘ (joinT Y ∘ R.func→ f) joinT Z ∘ R.func→ g ∘ (joinT Y ∘ R.func→ f)