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Frederik Hanghøj Iversen 2018-03-08 00:36:38 +01:00
parent c8fef1d2b5
commit e43bee6d9f
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src/Cat/Category/Monad.agda
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@ -695,31 +695,7 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
open import Function using (_∘_ ; _$_) open import Function using (_∘_ ; _$_)
module _ (omap : Omap ℂ ℂ) (pure : {X : Object} → Arrow X (omap X)) where module _ (omap : Omap ℂ ℂ) (pure : {X : Object} → Arrow X (omap X)) where
open voe-2-3 ℂ omap pure open voe-2-3 ℂ
-- Idea:
-- We want to prove
--
-- voe-2-3-1 ≃ voe-2-3-2
--
-- By using the equivalence we have already constructed.
--
-- We can construct `forth` by composing `forth0`, `forth1` and `forth2`:
--
-- forth0 : voe-2-3-1 → M.Monad
--
-- Where the we will naturally pick `omap` and `pure` as the corresponding
-- fields in M.Monad
--
-- `forth1` will be the equivalence we have already constructed.
--
-- forth1 : M.Monad ≃ K.Monad
--
-- `forth2` is the straight-forward isomporphism:
--
-- forth1 : K.Monad → voe-2-3-2
--
-- NB! This may not be so straightforward since the index of `voe-2-3-2` is
-- given before `K.Monad`.
private private
Monoidal→Kleisli : M.Monad → K.Monad Monoidal→Kleisli : M.Monad → K.Monad
Monoidal→Kleisli = proj₁ Monoidal≃Kleisli Monoidal→Kleisli = proj₁ Monoidal≃Kleisli
@ -727,57 +703,36 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
Kleisli→Monoidal : K.Monad → M.Monad Kleisli→Monoidal : K.Monad → M.Monad
Kleisli→Monoidal = inverse Monoidal≃Kleisli Kleisli→Monoidal = inverse Monoidal≃Kleisli
forth : voe-2-3-1 → voe-2-3-2 forth : voe-2-3-1 omap pure → voe-2-3-2 omap pure
forth = voe-2-3-2-fromMonad ∘ Monoidal→Kleisli ∘ voe-2-3-1.toMonad forth = voe-2-3-2-fromMonad ∘ Monoidal→Kleisli ∘ voe-2-3.voe-2-3-1.toMonad
back : voe-2-3-2 → voe-2-3-1 back : voe-2-3-2 omap pure → voe-2-3-1 omap pure
back = voe-2-3-1-fromMonad ∘ Kleisli→Monoidal ∘ voe-2-3-2.toMonad back = voe-2-3-1-fromMonad ∘ Kleisli→Monoidal ∘ voe-2-3.voe-2-3-2.toMonad
Voe-2-3-1-inverse = (toMonad ∘ fromMonad) ≡ Function.id forthEq : ∀ m → _ ≡ _
where forthEq m = begin
fromMonad : (m : M.Monad) → voe-2-3.voe-2-3-1 ℂ (M.Monad.Romap m) (λ {X} → M.Monad.pureT m X)
fromMonad = voe-2-3-1-fromMonad
toMonad : ∀ {omap} {pure : {X : Object} → Arrow X (omap X)} → voe-2-3.voe-2-3-1 ℂ omap pure → M.Monad
toMonad = voe-2-3.voe-2-3-1.toMonad
-- voe-2-3-1-inverse : (voe-2-3.voe-2-3-1.toMonad ∘ voe-2-3-1-fromMonad) ≡ Function.id
voe-2-3-1-inverse : Voe-2-3-1-inverse
voe-2-3-1-inverse = refl
Voe-2-3-2-inverse = (toMonad ∘ fromMonad) ≡ Function.id
where
fromMonad : (m : K.Monad) → voe-2-3.voe-2-3-2 ℂ (K.Monad.omap m) (K.Monad.pure m)
fromMonad = voe-2-3-2-fromMonad
toMonad : ∀ {omap} {pure : {X : Object} → Arrow X (omap X)} → voe-2-3.voe-2-3-2 ℂ omap pure → K.Monad
toMonad = voe-2-3.voe-2-3-2.toMonad
voe-2-3-2-inverse : Voe-2-3-2-inverse
voe-2-3-2-inverse = refl
forthEq' : ∀ m → _ ≡ _
forthEq' m = begin
(forth ∘ back) m ≡⟨⟩ (forth ∘ back) m ≡⟨⟩
-- In full gory detail: -- In full gory detail:
( voe-2-3-2-fromMonad ( voe-2-3-2-fromMonad
∘ Monoidal→Kleisli ∘ Monoidal→Kleisli
∘ voe-2-3.voe-2-3-1.toMonad ∘ voe-2-3.voe-2-3-1.toMonad
∘ voe-2-3-1-fromMonad ∘ voe-2-3-1-fromMonad
∘ Kleisli→Monoidal ∘ Kleisli→Monoidal
∘ voe-2-3.voe-2-3-2.toMonad ∘ voe-2-3.voe-2-3-2.toMonad
) m ≡⟨⟩ -- fromMonad and toMonad are inverses ) m ≡⟨⟩ -- fromMonad and toMonad are inverses
( voe-2-3-2-fromMonad ( voe-2-3-2-fromMonad
∘ Monoidal→Kleisli ∘ Monoidal→Kleisli
∘ Kleisli→Monoidal ∘ Kleisli→Monoidal
∘ voe-2-3.voe-2-3-2.toMonad ∘ voe-2-3.voe-2-3-2.toMonad
) m ≡⟨ u ⟩ ) m ≡⟨ u ⟩
-- Monoidal→Kleisli and Kleisli→Monoidal are inverses -- Monoidal→Kleisli and Kleisli→Monoidal are inverses
-- I should be able to prove this using congruence and `lem` below. -- I should be able to prove this using congruence and `lem` below.
( voe-2-3-2-fromMonad
∘ voe-2-3.voe-2-3-2.toMonad
) m ≡⟨⟩
( voe-2-3-2-fromMonad ( voe-2-3-2-fromMonad
∘ voe-2-3.voe-2-3-2.toMonad ∘ voe-2-3.voe-2-3-2.toMonad
) m ≡⟨⟩ ) m ≡⟨⟩ -- fromMonad and toMonad are inverses
( voe-2-3-2-fromMonad
∘ voe-2-3.voe-2-3-2.toMonad
) m ≡⟨⟩ -- fromMonad and toMonad are inverses
m ∎ m ∎
where where
lem : Monoidal→Kleisli ∘ Kleisli→Monoidal ≡ Function.id lem : Monoidal→Kleisli ∘ Kleisli→Monoidal ≡ Function.id
@ -789,49 +744,30 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
→ {m : _} → (a ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ b) m ≡ (a ∘ b) m → {m : _} → (a ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ b) m ≡ (a ∘ b) m
u {m = m} = cong (λ φ → φ m) t u {m = m} = cong (λ φ → φ m) t
forthEq : ∀ m → (forth ∘ back) m ≡ m
forthEq m = begin
(forth ∘ back) m ≡⟨⟩
-- In full gory detail:
( voe-2-3-2-fromMonad
∘ Monoidal→Kleisli
∘ voe-2-3-1.toMonad
∘ voe-2-3-1-fromMonad
∘ Kleisli→Monoidal
∘ voe-2-3-2.toMonad
) m ≡⟨ {!!} ⟩ -- fromMonad and toMonad are inverses
( voe-2-3-2-fromMonad
∘ Monoidal→Kleisli
∘ Kleisli→Monoidal
∘ voe-2-3-2.toMonad
) m ≡⟨ {!!} ⟩ -- Monoidal→Kleisli and Kleisli→Monoidal are inverses
( voe-2-3-2-fromMonad
∘ voe-2-3-2.toMonad
) m ≡⟨ {!!} ⟩ -- fromMonad and toMonad are inverses
m ∎
backEq : ∀ m → (back ∘ forth) m ≡ m backEq : ∀ m → (back ∘ forth) m ≡ m
backEq m = begin backEq m = begin
(back ∘ forth) m ≡⟨⟩ (back ∘ forth) m ≡⟨⟩
( voe-2-3-1-fromMonad ( voe-2-3-1-fromMonad
∘ Kleisli→Monoidal ∘ Kleisli→Monoidal
∘ voe-2-3-2.toMonad ∘ voe-2-3.voe-2-3-2.toMonad
∘ voe-2-3-2-fromMonad ∘ voe-2-3-2-fromMonad
∘ Monoidal→Kleisli ∘ Monoidal→Kleisli
∘ voe-2-3-1.toMonad ∘ voe-2-3.voe-2-3-1.toMonad
) m ≡⟨ {!!} ⟩ -- fromMonad and toMonad are inverses ) m ≡⟨⟩ -- fromMonad and toMonad are inverses
( voe-2-3-1-fromMonad ( voe-2-3-1-fromMonad
∘ Kleisli→Monoidal ∘ Kleisli→Monoidal
∘ Monoidal→Kleisli ∘ Monoidal→Kleisli
∘ voe-2-3-1.toMonad ∘ voe-2-3.voe-2-3-1.toMonad
) m ≡⟨ {!!} ⟩ -- Monoidal→Kleisli and Kleisli→Monoidal are inverses ) m ≡⟨ cong (λ φ → φ m) t ⟩ -- Monoidal→Kleisli and Kleisli→Monoidal are inverses
( voe-2-3-1-fromMonad ( voe-2-3-1-fromMonad
∘ voe-2-3-1.toMonad ∘ voe-2-3.voe-2-3-1.toMonad
) m ≡⟨ {!!} ⟩ -- fromMonad and toMonad are inverses ) m ≡⟨⟩ -- fromMonad and toMonad are inverses
m ∎ m ∎
where
t = cong (λ φ → voe-2-3-1-fromMonad ∘ φ ∘ voe-2-3.voe-2-3-1.toMonad) (recto-verso Monoidal≃Kleisli)
voe-isEquiv : isEquiv voe-2-3-1 voe-2-3-2 forth voe-isEquiv : isEquiv (voe-2-3-1 omap pure) (voe-2-3-2 omap pure) forth
voe-isEquiv = gradLemma forth back forthEq backEq voe-isEquiv = gradLemma forth back forthEq backEq
equiv-2-3 : voe-2-3-1 ≃ voe-2-3-2 equiv-2-3 : voe-2-3-1 omap pure ≃ voe-2-3-2 omap pure
equiv-2-3 = forth , voe-isEquiv equiv-2-3 = forth , voe-isEquiv