no-eta-equality for monads speeds up Voevodsky
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Andrea Vezzosi
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@ -230,6 +230,7 @@ record IsMonad (raw : RawMonad) : Set ℓ where
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m ∎
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record Monad : Set ℓ where
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no-eta-equality
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field
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raw : RawMonad
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isMonad : IsMonad raw
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@ -123,6 +123,7 @@ record IsMonad (raw : RawMonad) : Set ℓ where
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∎
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record Monad : Set ℓ where
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no-eta-equality
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field
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raw : RawMonad
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isMonad : IsMonad raw
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@ -25,6 +25,7 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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module §2-3 (omap : Object → Object) (pure : {X : Object} → Arrow X (omap X)) where
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record §1 : Set ℓ where
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no-eta-equality
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open M
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field
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@ -75,12 +76,11 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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isMonad : IsMonad rawMnd
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toMonad : Monad
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toMonad = record
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{ raw = rawMnd
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; isMonad = isMonad
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}
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toMonad .Monad.raw = rawMnd
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toMonad .Monad.isMonad = isMonad
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record §2 : Set ℓ where
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no-eta-equality
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open K
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field
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@ -97,28 +97,24 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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isMonad : IsMonad rawMnd
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toMonad : Monad
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toMonad = record
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{ raw = rawMnd
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; isMonad = isMonad
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}
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toMonad .Monad.raw = rawMnd
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toMonad .Monad.isMonad = isMonad
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§1-fromMonad : (m : M.Monad) → §2-3.§1 (M.Monad.Romap m) (λ {X} → M.Monad.pureT m X)
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§1-fromMonad m = record
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{ fmap = Functor.fmap 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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; isMonad = M.Monad.isMonad m
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}
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where
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module _ (m : M.Monad) where
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open M.Monad m
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§1-fromMonad : §2-3.§1 (M.Monad.Romap m) (λ {X} → M.Monad.pureT m X)
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§1-fromMonad .§2-3.§1.fmap = Functor.fmap R
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§1-fromMonad .§2-3.§1.RisFunctor = Functor.isFunctor R
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§1-fromMonad .§2-3.§1.pureN = pureN
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§1-fromMonad .§2-3.§1.join {X} = joinT X
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§1-fromMonad .§2-3.§1.joinN = joinN
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§1-fromMonad .§2-3.§1.isMonad = M.Monad.isMonad m
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§2-fromMonad : (m : K.Monad) → §2-3.§2 (K.Monad.omap m) (K.Monad.pure m)
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§2-fromMonad m = record
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{ bind = K.Monad.bind m
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; isMonad = K.Monad.isMonad m
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}
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§2-fromMonad m .§2-3.§2.bind = K.Monad.bind m
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§2-fromMonad m .§2-3.§2.isMonad = K.Monad.isMonad m
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-- | In the following we seek to transform the equivalence `Monoidal≃Kleisli`
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-- | to talk about voevodsky's construction.
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@ -145,65 +141,43 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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back = §1-fromMonad ∘ Kleisli→Monoidal ∘ §2-3.§2.toMonad
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forthEq : ∀ m → (forth ∘ back) m ≡ m
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forthEq m = begin
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(forth ∘ back) m ≡⟨⟩
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-- In full gory detail:
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( §2-fromMonad
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∘ Monoidal→Kleisli
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∘ §2-3.§1.toMonad
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∘ §1-fromMonad
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∘ Kleisli→Monoidal
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∘ §2-3.§2.toMonad
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) m ≡⟨⟩ -- fromMonad and toMonad are inverses
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( §2-fromMonad
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∘ Monoidal→Kleisli
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∘ Kleisli→Monoidal
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∘ §2-3.§2.toMonad
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) m ≡⟨ cong (λ φ → φ m) t ⟩
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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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( §2-fromMonad
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∘ §2-3.§2.toMonad
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) m ≡⟨⟩
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( §2-fromMonad
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∘ §2-3.§2.toMonad
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) m ≡⟨⟩ -- fromMonad and toMonad are inverses
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m ∎
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forthEq m = trans
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(trans (cong-d (§2-fromMonad ∘ Monoidal→Kleisli) (lemmaz (Kleisli→Monoidal (§2-3.§2.toMonad m))))
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(cong-d (\ φ → §2-fromMonad (φ (§2-3.§2.toMonad m))) re-ve))
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lemma
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where
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t' : ((Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ §2-3.§2.toMonad {omap} {pure})
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≡ §2-3.§2.toMonad
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t' = cong (\ φ → φ ∘ §2-3.§2.toMonad) re-ve
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t : (§2-fromMonad ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ §2-3.§2.toMonad {omap} {pure})
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≡ (§2-fromMonad ∘ §2-3.§2.toMonad)
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t = cong-d (\ f → §2-fromMonad ∘ f) t'
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u : (§2-fromMonad ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ §2-3.§2.toMonad) m
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≡ (§2-fromMonad ∘ §2-3.§2.toMonad) m
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u = cong (\ φ → φ m) t
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lemma : (§2-fromMonad ∘ §2-3.§2.toMonad) m ≡ m
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§2-3.§2.bind (lemma i) = §2-3.§2.bind m
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§2-3.§2.isMonad (lemma i) = §2-3.§2.isMonad m
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lemmaz : ∀ m → §2-3.§1.toMonad (§1-fromMonad m) ≡ m
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M.Monad.raw (lemmaz m i) = M.Monad.raw m
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M.Monad.isMonad (lemmaz m i) = M.Monad.isMonad m
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backEq : ∀ m → (back ∘ forth) m ≡ m
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backEq m = begin
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(back ∘ forth) m ≡⟨⟩
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( §1-fromMonad
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∘ Kleisli→Monoidal
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∘ §2-3.§2.toMonad
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∘ §2-fromMonad
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∘ Monoidal→Kleisli
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∘ §2-3.§1.toMonad
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) m ≡⟨⟩ -- fromMonad and toMonad are inverses
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( §1-fromMonad
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∘ Kleisli→Monoidal
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∘ Monoidal→Kleisli
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∘ §2-3.§1.toMonad
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) m ≡⟨ cong (λ φ → φ m) t ⟩ -- Monoidal→Kleisli and Kleisli→Monoidal are inverses
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( §1-fromMonad
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∘ §2-3.§1.toMonad
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) m ≡⟨⟩ -- fromMonad and toMonad are inverses
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m ∎
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backEq m = trans (cong-d (§1-fromMonad ∘ Kleisli→Monoidal) (lemma (Monoidal→Kleisli (§2-3.§1.toMonad m))))
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(trans (cong-d (\ φ → §1-fromMonad (φ (§2-3.§1.toMonad m))) ve-re)
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lemmaz)
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where
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t : §1-fromMonad ∘ Kleisli→Monoidal ∘ Monoidal→Kleisli ∘ §2-3.§1.toMonad
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≡ §1-fromMonad ∘ §2-3.§1.toMonad
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-- Why does `re-ve` not satisfy this goal?
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t i m = §1-fromMonad (ve-re i (§2-3.§1.toMonad m))
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-- rhs = §1-fromMonad (Kleisli→Monoidal ((Monoidal→Kleisli (§2-3.§1.toMonad m))))
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-- foo : §1-fromMonad (Kleisli→Monoidal (§2-3.§2.toMonad (§2-fromMonad (Monoidal→Kleisli (§2-3.§1.toMonad m)))))
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-- ≡ §1-fromMonad (Kleisli→Monoidal ((Monoidal→Kleisli (§2-3.§1.toMonad m))))
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-- §2-3.§1.fmap (foo i) = §2-3.§1.fmap rhs
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-- §2-3.§1.join (foo i) = §2-3.§1.join rhs
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-- §2-3.§1.RisFunctor (foo i) = §2-3.§1.RisFunctor rhs
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-- §2-3.§1.pureN (foo i) = §2-3.§1.pureN rhs
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-- §2-3.§1.joinN (foo i) = §2-3.§1.joinN rhs
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-- §2-3.§1.isMonad (foo i) = §2-3.§1.isMonad rhs
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lemmaz : §1-fromMonad (§2-3.§1.toMonad m) ≡ m
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§2-3.§1.fmap (lemmaz i) = §2-3.§1.fmap m
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§2-3.§1.join (lemmaz i) = §2-3.§1.join m
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§2-3.§1.RisFunctor (lemmaz i) = §2-3.§1.RisFunctor m
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§2-3.§1.pureN (lemmaz i) = §2-3.§1.pureN m
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§2-3.§1.joinN (lemmaz i) = §2-3.§1.joinN m
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§2-3.§1.isMonad (lemmaz i) = §2-3.§1.isMonad m
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lemma : ∀ m → §2-3.§2.toMonad (§2-fromMonad m) ≡ m
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K.Monad.raw (lemma m i) = K.Monad.raw m
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K.Monad.isMonad (lemma m i) = K.Monad.isMonad m
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voe-isEquiv : isEquiv (§2-3.§1 omap pure) (§2-3.§2 omap pure) forth
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voe-isEquiv = gradLemma forth back forthEq backEq
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