Lay out a strategy for showing the equivalence
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@ -4,6 +4,7 @@ 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)
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@ -553,7 +554,7 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
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Monoidal≃Kleisli : M.Monad ≃ K.Monad
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Monoidal≃Kleisli = forth , eqv
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module voe-2-3 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
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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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@ -562,7 +563,7 @@ module voe-2-3 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
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module M = Monoidal ℂ
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module K = Kleisli ℂ
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module _ (omap : Omap ℂ ℂ) (pure : {X : Object} → Arrow X (omap X)) where
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module voe-2-3 (omap : Omap ℂ ℂ) (pure : {X : Object} → Arrow X (omap X)) where
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record voe-2-3-1 : Set ℓ where
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open M
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@ -613,8 +614,8 @@ module voe-2-3 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
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field
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isMnd : IsMonad rawMnd
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mnd : Monad
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mnd = record
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toMonad : Monad
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toMonad = record
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{ raw = rawMnd
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; isMonad = isMnd
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}
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@ -635,8 +636,8 @@ module voe-2-3 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
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field
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isMnd : IsMonad rawMnd
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mnd : Monad
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mnd = record
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toMonad : Monad
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toMonad = record
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{ raw = rawMnd
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; isMonad = isMnd
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}
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@ -654,27 +655,128 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
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forth
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: {omap : Omap ℂ ℂ} {pure : {X : Object} → Arrow X (omap X)}
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→ voe-2-3-1 omap pure → M.Monad
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forth = voe-2-3-1.mnd
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forth {omap} {pure} m = voe-2-3-1.toMonad omap pure m
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back : (m : M.Monad) → voe-2-3-1 (M.Monad.Romap m) (λ {X} → M.Monad.pureT m X)
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back m = record
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{ fmap = Functor.func→ R
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; RisFunctor = Functor.isFunctor R
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; pureN = pureN
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; join = λ {X} → joinT X
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; joinN = joinN
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; isMnd = M.Monad.isMonad m
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}
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where
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raw = M.Monad.raw m
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R = M.RawMonad.R raw
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pureT = M.RawMonad.pureT raw
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pureN = M.RawMonad.pureN raw
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joinT = M.RawMonad.joinT raw
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joinN = M.RawMonad.joinN raw
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voe-2-3-1-fromMonad : (m : M.Monad) → voe-2-3-1 (M.Monad.Romap m) (λ {X} → M.Monad.pureT m X)
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voe-2-3-1-fromMonad m = record
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{ fmap = Functor.func→ R
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; RisFunctor = Functor.isFunctor R
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; pureN = pureN
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; join = λ {X} → joinT X
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; joinN = joinN
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; isMnd = M.Monad.isMonad m
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}
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where
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raw = M.Monad.raw m
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R = M.RawMonad.R raw
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pureT = M.RawMonad.pureT raw
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pureN = M.RawMonad.pureN raw
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joinT = M.RawMonad.joinT raw
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joinN = M.RawMonad.joinN raw
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-- Unfortunately the two above definitions don't really give rise to a
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-- bijection - at least not directly. Q: What to put in the indices for
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-- `voe-2-3-1`?
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equiv-2-3-1 : voe-2-3-1 {!!} {!!} ≃ M.Monad
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equiv-2-3-1 = {!!}
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voe-2-3-2-fromMonad : (m : K.Monad) → voe-2-3-2 (K.Monad.omap m) (K.Monad.pure m)
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voe-2-3-2-fromMonad m = record
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{ bind = K.Monad.bind m
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; isMnd = K.Monad.isMonad m
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}
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-- Unfortunately the two above definitions don't really give rise to a
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-- bijection - at least not directly. Q: What to put in the indices for
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-- `voe-2-3-1`?
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equiv-2-3-1 : voe-2-3-1 {!!} {!!} ≃ M.Monad
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equiv-2-3-1 = {!!}
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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 NaturalTransformation ℂ ℂ
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module M = Monoidal ℂ
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module K = Kleisli ℂ
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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 = ℂ} omap pure
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-- Idea:
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-- We want to prove
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--
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-- voe-2-3-1 ≃ voe-2-3-2
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--
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-- By using the equivalence we have already constructed.
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--
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-- We can construct `forth` by composing `forth0`, `forth1` and `forth2`:
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--
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-- forth0 : voe-2-3-1 → M.Monad
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--
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-- Where the we will naturally pick `omap` and `pure` as the corresponding
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-- fields in M.Monad
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--
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-- `forth1` will be the equivalence we have already constructed.
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--
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-- forth1 : M.Monad ≃ K.Monad
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--
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-- `forth2` is the straight-forward isomporphism:
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--
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-- forth1 : K.Monad → voe-2-3-2
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--
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-- NB! This may not be so straightforward since the index of `voe-2-3-2` is
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-- given before `K.Monad`.
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private
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Monoidal→Kleisli : M.Monad → K.Monad
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Monoidal→Kleisli = proj₁ Monoidal≃Kleisli
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Kleisli→Monoidal : K.Monad → M.Monad
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Kleisli→Monoidal = reverse 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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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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forthEq : ∀ m → (forth ∘f back) m ≡ m
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forthEq m = begin
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(forth ∘f 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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) 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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) 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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) 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 = begin
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(back ∘f 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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) 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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) 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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) m ≡⟨ {!!} ⟩ -- fromMonad and toMonad are inverses
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m ∎
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voe-isEquiv : isEquiv voe-2-3-1 voe-2-3-2 forth
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voe-isEquiv = gradLemma forth back forthEq backEq
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equiv-2-3 : voe-2-3-1 ≃ voe-2-3-2
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equiv-2-3 = forth , voe-isEquiv
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