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Move another proof to category definition

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Frederik Hanghøj Iversen 2018-02-26 20:36:39 +01:00
parent 101b2639e1
commit 1aaf81552c
1 changed files with 31 additions and 32 deletions
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63
src/Cat/Category/Monad.agda
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@ -48,12 +48,15 @@ module Monoidal {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
→ μ X ∘ η (R.func* X) ≡ 𝟙
× μ X ∘ R.func→ (η X) ≡ 𝟙
IsNatural' = ∀ {X Y f} → μ Y ∘ R.func→ f ∘ η X ≡ f
IsDistributive' = ∀ {X Y Z} {f : Arrow X (R.func* Y)} {g : Arrow Y (R.func* Z)}
→ μ Z ∘ R.func→ (μ Z ∘ R.func→ g ∘ f)
≡ μ Z ∘ R.func→ g ∘ (μ Y ∘ R.func→ f)
record IsMonad (raw : RawMonad) : Set ℓ where
open RawMonad raw public
field
isAssociative : IsAssociative
isInverse : IsInverse
isInverse : IsInverse
private
module R = Functor R
@ -68,6 +71,31 @@ module Monoidal {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
𝟙 ∘ f ≡⟨ proj₂ ℂ.isIdentity ⟩
f ∎
isDistributive' : IsDistributive'
isDistributive' {X} {Y} {Z} {f} {g} = 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→ (R.func→ g) ∘ R.func→ f) ≡⟨⟩
μ Z ∘ (R.func→ (μ Z) ∘ R².func→ g ∘ R.func→ f) ≡⟨ {!!} ⟩ -- ●-solver?
(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 ∘ μ (R.func* Z) ∘ R².func→ g ∘ R.func→ f ≡⟨ {!!} ⟩ -- ●-solver + lem4
μ Z ∘ R.func→ g ∘ μ Y ∘ R.func→ f ≡⟨ sym (Category.isAssociative ℂ) ⟩
μ Z ∘ R.func→ g ∘ (μ Y ∘ R.func→ f) ∎
where
module R² = Functor F[ R ∘ R ]
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 ∘ R.func→ b ∘ R.func→ c
distrib = {!!}
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 = {!!}
lemmm : μ Z ∘ R.func→ (μ Z) ≡ μ Z ∘ μ (R.func* Z)
lemmm = isAssociative
lem4 : μ (R.func* Z) ∘ R².func→ g ≡ R.func→ g ∘ μ Y
lem4 = μNat g
record Monad : Set ℓ where
field
raw : RawMonad
@ -212,7 +240,7 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
isIdentity : IsIdentity
isIdentity {X} = begin
bind pure ≡⟨⟩
bind pure ≡⟨⟩
bind (η X) ≡⟨⟩
μ X ∘ func→ R (η X) ≡⟨ proj₂ isInverse ⟩
𝟙 ∎
@ -232,38 +260,9 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
isDistributive : IsDistributive
isDistributive {X} {Y} {Z} g f = begin
bind g ∘ bind f ≡⟨⟩
μ Z ∘ R.func→ g ∘ (μ Y ∘ R.func→ f) ≡⟨ sym lem2 ⟩
μ Z ∘ R.func→ g ∘ (μ Y ∘ R.func→ f) ≡⟨ sym isDistributive' ⟩
μ Z ∘ R.func→ (μ Z ∘ R.func→ g ∘ f) ≡⟨⟩
μ Z ∘ R.func→ (bind g ∘ f) ∎
where
-- Proved it in reverse here... otherwise it could be neatly inlined.
lem2
: μ Z ∘ R.func→ (μ Z ∘ R.func→ g ∘ f)
≡ μ Z ∘ R.func→ g ∘ (μ Y ∘ R.func→ f)
lem2 = 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→ (R.func→ g) ∘ R.func→ f) ≡⟨⟩
μ Z ∘ (R.func→ (μ Z) ∘ RR.func→ g ∘ R.func→ f) ≡⟨ {!!} ⟩ -- ●-solver?
(μ Z ∘ R.func→ (μ Z)) ∘ (RR.func→ g ∘ R.func→ f) ≡⟨ cong (λ φ → φ ∘ (RR.func→ g ∘ R.func→ f)) lemmm ⟩
(μ Z ∘ μ (R.func* Z)) ∘ (RR.func→ g ∘ R.func→ f) ≡⟨ {!!} ⟩ -- ●-solver?
μ Z ∘ μ (R.func* Z) ∘ RR.func→ g ∘ R.func→ f ≡⟨ {!!} ⟩ -- ●-solver + lem4
μ Z ∘ R.func→ g ∘ μ Y ∘ R.func→ f ≡⟨ sym (Category.isAssociative ℂ) ⟩
μ Z ∘ R.func→ g ∘ (μ Y ∘ R.func→ f) ∎
where
module RR = Functor F[ R ∘ R ]
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 ∘ R.func→ b ∘ R.func→ c
distrib = {!!}
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 = {!!}
μN = proj₂ μNatTrans
lemmm : μ Z ∘ R.func→ (μ Z) ≡ μ Z ∘ μ (R.func* Z)
lemmm = isAssociative
lem4 : μ (R.func* Z) ∘ RR.func→ g ≡ R.func→ g ∘ μ Y
lem4 = μN g
module KI = K.IsMonad
forthIsMonad : K.IsMonad (forthRaw raw)