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Use different name for function composition

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Frederik Hanghøj Iversen 2018-03-08 00:22:55 +01:00
parent 36cbe711fb
commit c8fef1d2b5
1 changed files with 44 additions and 44 deletions
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88
src/Cat/Category/Monad.agda
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@ -4,7 +4,6 @@ module Cat.Category.Monad where
open import Agda.Primitive open import Agda.Primitive
open import Data.Product open import Data.Product
open import Function renaming (_∘_ to _∘f_) using (_$_)
open import Cubical open import Cubical
open import Cubical.NType.Properties using (lemPropF ; lemSig ; lemSigP) open import Cubical.NType.Properties using (lemPropF ; lemSig ; lemSigP)
@ -689,10 +688,11 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
private private
ℓ = ℓa ⊔ ℓb ℓ = ℓa ⊔ ℓb
module ℂ = Category ℂ module ℂ = Category ℂ
open ℂ using (Object ; Arrow ; _∘_) open ℂ using (Object ; Arrow)
open NaturalTransformation ℂ ℂ open NaturalTransformation ℂ ℂ
module M = Monoidal ℂ module M = Monoidal ℂ
module K = Kleisli ℂ module K = Kleisli ℂ
open import Function using (_∘_ ; _$_)
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 omap pure
@ -728,23 +728,23 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
Kleisli→Monoidal = inverse Monoidal≃Kleisli Kleisli→Monoidal = inverse Monoidal≃Kleisli
forth : voe-2-3-1 → voe-2-3-2 forth : voe-2-3-1 → voe-2-3-2
forth = voe-2-3-2-fromMonad ∘f Monoidal→Kleisli ∘f voe-2-3-1.toMonad forth = voe-2-3-2-fromMonad ∘ Monoidal→Kleisli ∘ voe-2-3-1.toMonad
back : voe-2-3-2 → voe-2-3-1 back : voe-2-3-2 → voe-2-3-1
back = voe-2-3-1-fromMonad ∘f Kleisli→Monoidal ∘f voe-2-3-2.toMonad back = voe-2-3-1-fromMonad ∘ Kleisli→Monoidal ∘ voe-2-3-2.toMonad
Voe-2-3-1-inverse = (toMonad ∘f fromMonad) ≡ Function.id Voe-2-3-1-inverse = (toMonad ∘ fromMonad) ≡ Function.id
where where
fromMonad : (m : M.Monad) → voe-2-3.voe-2-3-1 ℂ (M.Monad.Romap m) (λ {X} → M.Monad.pureT m X) 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 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 : ∀ {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 toMonad = voe-2-3.voe-2-3-1.toMonad
-- voe-2-3-1-inverse : (voe-2-3.voe-2-3-1.toMonad ∘f voe-2-3-1-fromMonad) ≡ Function.id -- 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 : Voe-2-3-1-inverse
voe-2-3-1-inverse = refl voe-2-3-1-inverse = refl
Voe-2-3-2-inverse = (toMonad ∘f fromMonad) ≡ Function.id Voe-2-3-2-inverse = (toMonad ∘ fromMonad) ≡ Function.id
where where
fromMonad : (m : K.Monad) → voe-2-3.voe-2-3-2 ℂ (K.Monad.omap m) (K.Monad.pure m) 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 fromMonad = voe-2-3-2-fromMonad
@ -756,77 +756,77 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
forthEq' : ∀ m → _ ≡ _ forthEq' : ∀ m → _ ≡ _
forthEq' m = begin forthEq' m = begin
(forth ∘f back) m ≡⟨⟩ (forth ∘ back) m ≡⟨⟩
-- In full gory detail: -- In full gory detail:
( voe-2-3-2-fromMonad ( voe-2-3-2-fromMonad
∘f Monoidal→Kleisli ∘ Monoidal→Kleisli
∘f voe-2-3.voe-2-3-1.toMonad ∘ voe-2-3.voe-2-3-1.toMonad
∘f voe-2-3-1-fromMonad ∘ voe-2-3-1-fromMonad
∘f Kleisli→Monoidal ∘ Kleisli→Monoidal
∘f 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
∘f Monoidal→Kleisli ∘ Monoidal→Kleisli
∘f Kleisli→Monoidal ∘ Kleisli→Monoidal
∘f 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-2-3-2-fromMonad
∘f voe-2-3.voe-2-3-2.toMonad ∘ voe-2-3.voe-2-3-2.toMonad
) m ≡⟨⟩ ) m ≡⟨⟩
( voe-2-3-2-fromMonad ( voe-2-3-2-fromMonad
∘f 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
m ∎ m ∎
where where
lem : Monoidal→Kleisli ∘f Kleisli→Monoidal ≡ Function.id lem : Monoidal→Kleisli ∘ Kleisli→Monoidal ≡ Function.id
lem = verso-recto Monoidal≃Kleisli lem = verso-recto Monoidal≃Kleisli
t : {ℓ : Level} {A B : Set ℓ} {a : _ → A} {b : B → _} t : {ℓ : Level} {A B : Set ℓ} {a : _ → A} {b : B → _}
→ a ∘f (Monoidal→Kleisli ∘f Kleisli→Monoidal) ∘f b ≡ a ∘f b → a ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ b ≡ a ∘ b
t {a = a} {b} = cong (λ φ → a ∘f φ ∘f b) lem t {a = a} {b} = cong (λ φ → a ∘ φ ∘ b) lem
u : {ℓ : Level} {A B : Set ℓ} {a : _ → A} {b : B → _} u : {ℓ : Level} {A B : Set ℓ} {a : _ → A} {b : B → _}
→ {m : _} → (a ∘f (Monoidal→Kleisli ∘f Kleisli→Monoidal) ∘f b) m ≡ (a ∘f 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 ∘f back) m ≡ m forthEq : ∀ m → (forth ∘ back) m ≡ m
forthEq m = begin forthEq m = begin
(forth ∘f back) m ≡⟨⟩ (forth ∘ back) m ≡⟨⟩
-- In full gory detail: -- In full gory detail:
( voe-2-3-2-fromMonad ( voe-2-3-2-fromMonad
∘f Monoidal→Kleisli ∘ Monoidal→Kleisli
∘f voe-2-3-1.toMonad ∘ voe-2-3-1.toMonad
∘f voe-2-3-1-fromMonad ∘ voe-2-3-1-fromMonad
∘f Kleisli→Monoidal ∘ Kleisli→Monoidal
∘f voe-2-3-2.toMonad ∘ 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
∘f Monoidal→Kleisli ∘ Monoidal→Kleisli
∘f Kleisli→Monoidal ∘ Kleisli→Monoidal
∘f voe-2-3-2.toMonad ∘ voe-2-3-2.toMonad
) m ≡⟨ {!!} ⟩ -- Monoidal→Kleisli and Kleisli→Monoidal are inverses ) m ≡⟨ {!!} ⟩ -- Monoidal→Kleisli and Kleisli→Monoidal are inverses
( voe-2-3-2-fromMonad ( voe-2-3-2-fromMonad
∘f voe-2-3-2.toMonad ∘ voe-2-3-2.toMonad
) m ≡⟨ {!!} ⟩ -- fromMonad and toMonad are inverses ) m ≡⟨ {!!} ⟩ -- fromMonad and toMonad are inverses
m ∎ m ∎
backEq : ∀ m → (back ∘f forth) m ≡ m backEq : ∀ m → (back ∘ forth) m ≡ m
backEq m = begin backEq m = begin
(back ∘f forth) m ≡⟨⟩ (back ∘ forth) m ≡⟨⟩
( voe-2-3-1-fromMonad ( voe-2-3-1-fromMonad
∘f Kleisli→Monoidal ∘ Kleisli→Monoidal
∘f voe-2-3-2.toMonad ∘ voe-2-3-2.toMonad
∘f voe-2-3-2-fromMonad ∘ voe-2-3-2-fromMonad
∘f Monoidal→Kleisli ∘ Monoidal→Kleisli
∘f voe-2-3-1.toMonad ∘ 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
∘f Kleisli→Monoidal ∘ Kleisli→Monoidal
∘f Monoidal→Kleisli ∘ Monoidal→Kleisli
∘f voe-2-3-1.toMonad ∘ voe-2-3-1.toMonad
) m ≡⟨ {!!} ⟩ -- Monoidal→Kleisli and Kleisli→Monoidal are inverses ) m ≡⟨ {!!} ⟩ -- Monoidal→Kleisli and Kleisli→Monoidal are inverses
( voe-2-3-1-fromMonad ( voe-2-3-1-fromMonad
∘f voe-2-3-1.toMonad ∘ voe-2-3-1.toMonad
) m ≡⟨ {!!} ⟩ -- fromMonad and toMonad are inverses ) m ≡⟨ {!!} ⟩ -- fromMonad and toMonad are inverses
m ∎ m ∎