Use different name for function composition
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@ -4,7 +4,6 @@ module Cat.Category.Monad where
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open import Agda.Primitive
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open import Data.Product
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open import Function renaming (_∘_ to _∘f_) using (_$_)
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open import Cubical
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open import Cubical.NType.Properties using (lemPropF ; lemSig ; lemSigP)
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@ -689,10 +688,11 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
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private
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ℓ = ℓa ⊔ ℓb
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module ℂ = Category ℂ
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open ℂ using (Object ; Arrow ; _∘_)
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open ℂ using (Object ; Arrow)
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open NaturalTransformation ℂ ℂ
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module M = Monoidal ℂ
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module K = Kleisli ℂ
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open import Function using (_∘_ ; _$_)
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module _ (omap : Omap ℂ ℂ) (pure : {X : Object} → Arrow X (omap X)) where
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open voe-2-3 ℂ omap pure
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@ -728,23 +728,23 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
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Kleisli→Monoidal = inverse Monoidal≃Kleisli
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forth : voe-2-3-1 → voe-2-3-2
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forth = voe-2-3-2-fromMonad ∘f Monoidal→Kleisli ∘f voe-2-3-1.toMonad
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forth = voe-2-3-2-fromMonad ∘ Monoidal→Kleisli ∘ voe-2-3-1.toMonad
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back : voe-2-3-2 → voe-2-3-1
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back = voe-2-3-1-fromMonad ∘f Kleisli→Monoidal ∘f voe-2-3-2.toMonad
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back = voe-2-3-1-fromMonad ∘ Kleisli→Monoidal ∘ voe-2-3-2.toMonad
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Voe-2-3-1-inverse = (toMonad ∘f fromMonad) ≡ Function.id
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Voe-2-3-1-inverse = (toMonad ∘ fromMonad) ≡ Function.id
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where
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fromMonad : (m : M.Monad) → voe-2-3.voe-2-3-1 ℂ (M.Monad.Romap m) (λ {X} → M.Monad.pureT m X)
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fromMonad = voe-2-3-1-fromMonad
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toMonad : ∀ {omap} {pure : {X : Object} → Arrow X (omap X)} → voe-2-3.voe-2-3-1 ℂ omap pure → M.Monad
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toMonad = voe-2-3.voe-2-3-1.toMonad
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-- voe-2-3-1-inverse : (voe-2-3.voe-2-3-1.toMonad ∘f voe-2-3-1-fromMonad) ≡ Function.id
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-- voe-2-3-1-inverse : (voe-2-3.voe-2-3-1.toMonad ∘ voe-2-3-1-fromMonad) ≡ Function.id
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voe-2-3-1-inverse : Voe-2-3-1-inverse
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voe-2-3-1-inverse = refl
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Voe-2-3-2-inverse = (toMonad ∘f fromMonad) ≡ Function.id
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Voe-2-3-2-inverse = (toMonad ∘ fromMonad) ≡ Function.id
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where
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fromMonad : (m : K.Monad) → voe-2-3.voe-2-3-2 ℂ (K.Monad.omap m) (K.Monad.pure m)
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fromMonad = voe-2-3-2-fromMonad
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@ -756,77 +756,77 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
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forthEq' : ∀ m → _ ≡ _
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forthEq' m = begin
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(forth ∘f back) m ≡⟨⟩
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(forth ∘ back) m ≡⟨⟩
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-- In full gory detail:
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( voe-2-3-2-fromMonad
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∘f Monoidal→Kleisli
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∘f voe-2-3.voe-2-3-1.toMonad
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∘f voe-2-3-1-fromMonad
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∘f Kleisli→Monoidal
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∘f voe-2-3.voe-2-3-2.toMonad
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∘ Monoidal→Kleisli
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∘ voe-2-3.voe-2-3-1.toMonad
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∘ voe-2-3-1-fromMonad
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∘ Kleisli→Monoidal
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∘ voe-2-3.voe-2-3-2.toMonad
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) m ≡⟨⟩ -- fromMonad and toMonad are inverses
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( voe-2-3-2-fromMonad
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∘f Monoidal→Kleisli
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∘f Kleisli→Monoidal
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∘f voe-2-3.voe-2-3-2.toMonad
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∘ Monoidal→Kleisli
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∘ Kleisli→Monoidal
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∘ voe-2-3.voe-2-3-2.toMonad
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) m ≡⟨ u ⟩
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-- Monoidal→Kleisli and Kleisli→Monoidal are inverses
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-- I should be able to prove this using congruence and `lem` below.
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( voe-2-3-2-fromMonad
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∘f voe-2-3.voe-2-3-2.toMonad
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∘ voe-2-3.voe-2-3-2.toMonad
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) m ≡⟨⟩
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( voe-2-3-2-fromMonad
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∘f voe-2-3.voe-2-3-2.toMonad
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∘ voe-2-3.voe-2-3-2.toMonad
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) m ≡⟨⟩ -- fromMonad and toMonad are inverses
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m ∎
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where
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lem : Monoidal→Kleisli ∘f Kleisli→Monoidal ≡ Function.id
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lem : Monoidal→Kleisli ∘ Kleisli→Monoidal ≡ Function.id
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lem = verso-recto Monoidal≃Kleisli
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t : {ℓ : Level} {A B : Set ℓ} {a : _ → A} {b : B → _}
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→ a ∘f (Monoidal→Kleisli ∘f Kleisli→Monoidal) ∘f b ≡ a ∘f b
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t {a = a} {b} = cong (λ φ → a ∘f φ ∘f b) lem
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→ a ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ b ≡ a ∘ b
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t {a = a} {b} = cong (λ φ → a ∘ φ ∘ b) lem
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u : {ℓ : Level} {A B : Set ℓ} {a : _ → A} {b : B → _}
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→ {m : _} → (a ∘f (Monoidal→Kleisli ∘f Kleisli→Monoidal) ∘f b) m ≡ (a ∘f b) m
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→ {m : _} → (a ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ b) m ≡ (a ∘ b) m
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u {m = m} = cong (λ φ → φ m) t
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forthEq : ∀ m → (forth ∘f back) m ≡ m
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forthEq : ∀ m → (forth ∘ back) m ≡ m
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forthEq m = begin
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(forth ∘f back) m ≡⟨⟩
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(forth ∘ back) m ≡⟨⟩
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-- In full gory detail:
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( voe-2-3-2-fromMonad
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∘f Monoidal→Kleisli
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∘f voe-2-3-1.toMonad
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∘f voe-2-3-1-fromMonad
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∘f Kleisli→Monoidal
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∘f voe-2-3-2.toMonad
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∘ Monoidal→Kleisli
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∘ voe-2-3-1.toMonad
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∘ voe-2-3-1-fromMonad
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∘ Kleisli→Monoidal
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∘ voe-2-3-2.toMonad
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) m ≡⟨ {!!} ⟩ -- fromMonad and toMonad are inverses
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( voe-2-3-2-fromMonad
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∘f Monoidal→Kleisli
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∘f Kleisli→Monoidal
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∘f voe-2-3-2.toMonad
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∘ Monoidal→Kleisli
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∘ Kleisli→Monoidal
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∘ voe-2-3-2.toMonad
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) m ≡⟨ {!!} ⟩ -- Monoidal→Kleisli and Kleisli→Monoidal are inverses
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( voe-2-3-2-fromMonad
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∘f voe-2-3-2.toMonad
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∘ voe-2-3-2.toMonad
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) m ≡⟨ {!!} ⟩ -- fromMonad and toMonad are inverses
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m ∎
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backEq : ∀ m → (back ∘f forth) m ≡ m
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backEq : ∀ m → (back ∘ forth) m ≡ m
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backEq m = begin
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(back ∘f forth) m ≡⟨⟩
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(back ∘ forth) m ≡⟨⟩
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( voe-2-3-1-fromMonad
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∘f Kleisli→Monoidal
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∘f voe-2-3-2.toMonad
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∘f voe-2-3-2-fromMonad
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∘f Monoidal→Kleisli
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∘f voe-2-3-1.toMonad
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∘ Kleisli→Monoidal
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∘ voe-2-3-2.toMonad
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∘ voe-2-3-2-fromMonad
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∘ Monoidal→Kleisli
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∘ voe-2-3-1.toMonad
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) m ≡⟨ {!!} ⟩ -- fromMonad and toMonad are inverses
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( voe-2-3-1-fromMonad
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∘f Kleisli→Monoidal
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∘f Monoidal→Kleisli
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∘f voe-2-3-1.toMonad
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∘ Kleisli→Monoidal
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∘ Monoidal→Kleisli
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∘ voe-2-3-1.toMonad
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) m ≡⟨ {!!} ⟩ -- Monoidal→Kleisli and Kleisli→Monoidal are inverses
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( voe-2-3-1-fromMonad
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∘f voe-2-3-1.toMonad
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∘ voe-2-3-1.toMonad
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) m ≡⟨ {!!} ⟩ -- fromMonad and toMonad are inverses
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m ∎
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