no-eta-equality for monads speeds up Voevodsky
This commit is contained in:
Andrea Vezzosi
parent
c75a1d5d8b
commit
9f7a13b5da
|
|
@ -230,6 +230,7 @@ record IsMonad (raw : RawMonad) : Set ℓ where
|
|||
m ∎
|
||||
|
||||
record Monad : Set ℓ where
|
||||
no-eta-equality
|
||||
field
|
||||
raw : RawMonad
|
||||
isMonad : IsMonad raw
|
||||
|
|
|
|||
|
|
@ -123,6 +123,7 @@ record IsMonad (raw : RawMonad) : Set ℓ where
|
|||
∎
|
||||
|
||||
record Monad : Set ℓ where
|
||||
no-eta-equality
|
||||
field
|
||||
raw : RawMonad
|
||||
isMonad : IsMonad raw
|
||||
|
|
|
|||
|
|
@ -25,6 +25,7 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
|||
|
||||
module §2-3 (omap : Object → Object) (pure : {X : Object} → Arrow X (omap X)) where
|
||||
record §1 : Set ℓ where
|
||||
no-eta-equality
|
||||
open M
|
||||
|
||||
field
|
||||
|
|
@ -75,12 +76,11 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
|||
isMonad : IsMonad rawMnd
|
||||
|
||||
toMonad : Monad
|
||||
toMonad = record
|
||||
{ raw = rawMnd
|
||||
; isMonad = isMonad
|
||||
}
|
||||
toMonad .Monad.raw = rawMnd
|
||||
toMonad .Monad.isMonad = isMonad
|
||||
|
||||
record §2 : Set ℓ where
|
||||
no-eta-equality
|
||||
open K
|
||||
|
||||
field
|
||||
|
|
@ -97,28 +97,24 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
|||
isMonad : IsMonad rawMnd
|
||||
|
||||
toMonad : Monad
|
||||
toMonad = record
|
||||
{ raw = rawMnd
|
||||
; isMonad = isMonad
|
||||
}
|
||||
toMonad .Monad.raw = rawMnd
|
||||
toMonad .Monad.isMonad = isMonad
|
||||
|
||||
§1-fromMonad : (m : M.Monad) → §2-3.§1 (M.Monad.Romap m) (λ {X} → M.Monad.pureT m X)
|
||||
§1-fromMonad m = record
|
||||
{ fmap = Functor.fmap R
|
||||
; RisFunctor = Functor.isFunctor R
|
||||
; pureN = pureN
|
||||
; join = λ {X} → joinT X
|
||||
; joinN = joinN
|
||||
; isMonad = M.Monad.isMonad m
|
||||
}
|
||||
where
|
||||
module _ (m : M.Monad) where
|
||||
open M.Monad m
|
||||
|
||||
§1-fromMonad : §2-3.§1 (M.Monad.Romap m) (λ {X} → M.Monad.pureT m X)
|
||||
§1-fromMonad .§2-3.§1.fmap = Functor.fmap R
|
||||
§1-fromMonad .§2-3.§1.RisFunctor = Functor.isFunctor R
|
||||
§1-fromMonad .§2-3.§1.pureN = pureN
|
||||
§1-fromMonad .§2-3.§1.join {X} = joinT X
|
||||
§1-fromMonad .§2-3.§1.joinN = joinN
|
||||
§1-fromMonad .§2-3.§1.isMonad = M.Monad.isMonad m
|
||||
|
||||
|
||||
§2-fromMonad : (m : K.Monad) → §2-3.§2 (K.Monad.omap m) (K.Monad.pure m)
|
||||
§2-fromMonad m = record
|
||||
{ bind = K.Monad.bind m
|
||||
; isMonad = K.Monad.isMonad m
|
||||
}
|
||||
§2-fromMonad m .§2-3.§2.bind = K.Monad.bind m
|
||||
§2-fromMonad m .§2-3.§2.isMonad = K.Monad.isMonad m
|
||||
|
||||
-- | In the following we seek to transform the equivalence `Monoidal≃Kleisli`
|
||||
-- | to talk about voevodsky's construction.
|
||||
|
|
@ -145,65 +141,43 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
|||
back = §1-fromMonad ∘ Kleisli→Monoidal ∘ §2-3.§2.toMonad
|
||||
|
||||
forthEq : ∀ m → (forth ∘ back) m ≡ m
|
||||
forthEq m = begin
|
||||
(forth ∘ back) m ≡⟨⟩
|
||||
-- In full gory detail:
|
||||
( §2-fromMonad
|
||||
∘ Monoidal→Kleisli
|
||||
∘ §2-3.§1.toMonad
|
||||
∘ §1-fromMonad
|
||||
∘ Kleisli→Monoidal
|
||||
∘ §2-3.§2.toMonad
|
||||
) m ≡⟨⟩ -- fromMonad and toMonad are inverses
|
||||
( §2-fromMonad
|
||||
∘ Monoidal→Kleisli
|
||||
∘ Kleisli→Monoidal
|
||||
∘ §2-3.§2.toMonad
|
||||
) m ≡⟨ cong (λ φ → φ m) t ⟩
|
||||
-- Monoidal→Kleisli and Kleisli→Monoidal are inverses
|
||||
-- I should be able to prove this using congruence and `lem` below.
|
||||
( §2-fromMonad
|
||||
∘ §2-3.§2.toMonad
|
||||
) m ≡⟨⟩
|
||||
( §2-fromMonad
|
||||
∘ §2-3.§2.toMonad
|
||||
) m ≡⟨⟩ -- fromMonad and toMonad are inverses
|
||||
m ∎
|
||||
forthEq m = trans
|
||||
(trans (cong-d (§2-fromMonad ∘ Monoidal→Kleisli) (lemmaz (Kleisli→Monoidal (§2-3.§2.toMonad m))))
|
||||
(cong-d (\ φ → §2-fromMonad (φ (§2-3.§2.toMonad m))) re-ve))
|
||||
lemma
|
||||
where
|
||||
t' : ((Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ §2-3.§2.toMonad {omap} {pure})
|
||||
≡ §2-3.§2.toMonad
|
||||
t' = cong (\ φ → φ ∘ §2-3.§2.toMonad) re-ve
|
||||
t : (§2-fromMonad ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ §2-3.§2.toMonad {omap} {pure})
|
||||
≡ (§2-fromMonad ∘ §2-3.§2.toMonad)
|
||||
t = cong-d (\ f → §2-fromMonad ∘ f) t'
|
||||
u : (§2-fromMonad ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ §2-3.§2.toMonad) m
|
||||
≡ (§2-fromMonad ∘ §2-3.§2.toMonad) m
|
||||
u = cong (\ φ → φ m) t
|
||||
lemma : (§2-fromMonad ∘ §2-3.§2.toMonad) m ≡ m
|
||||
§2-3.§2.bind (lemma i) = §2-3.§2.bind m
|
||||
§2-3.§2.isMonad (lemma i) = §2-3.§2.isMonad m
|
||||
lemmaz : ∀ m → §2-3.§1.toMonad (§1-fromMonad m) ≡ m
|
||||
M.Monad.raw (lemmaz m i) = M.Monad.raw m
|
||||
M.Monad.isMonad (lemmaz m i) = M.Monad.isMonad m
|
||||
|
||||
backEq : ∀ m → (back ∘ forth) m ≡ m
|
||||
backEq m = begin
|
||||
(back ∘ forth) m ≡⟨⟩
|
||||
( §1-fromMonad
|
||||
∘ Kleisli→Monoidal
|
||||
∘ §2-3.§2.toMonad
|
||||
∘ §2-fromMonad
|
||||
∘ Monoidal→Kleisli
|
||||
∘ §2-3.§1.toMonad
|
||||
) m ≡⟨⟩ -- fromMonad and toMonad are inverses
|
||||
( §1-fromMonad
|
||||
∘ Kleisli→Monoidal
|
||||
∘ Monoidal→Kleisli
|
||||
∘ §2-3.§1.toMonad
|
||||
) m ≡⟨ cong (λ φ → φ m) t ⟩ -- Monoidal→Kleisli and Kleisli→Monoidal are inverses
|
||||
( §1-fromMonad
|
||||
∘ §2-3.§1.toMonad
|
||||
) m ≡⟨⟩ -- fromMonad and toMonad are inverses
|
||||
m ∎
|
||||
where
|
||||
t : §1-fromMonad ∘ Kleisli→Monoidal ∘ Monoidal→Kleisli ∘ §2-3.§1.toMonad
|
||||
≡ §1-fromMonad ∘ §2-3.§1.toMonad
|
||||
-- Why does `re-ve` not satisfy this goal?
|
||||
t i m = §1-fromMonad (ve-re i (§2-3.§1.toMonad m))
|
||||
backEq m = trans (cong-d (§1-fromMonad ∘ Kleisli→Monoidal) (lemma (Monoidal→Kleisli (§2-3.§1.toMonad m))))
|
||||
(trans (cong-d (\ φ → §1-fromMonad (φ (§2-3.§1.toMonad m))) ve-re)
|
||||
lemmaz)
|
||||
where
|
||||
-- rhs = §1-fromMonad (Kleisli→Monoidal ((Monoidal→Kleisli (§2-3.§1.toMonad m))))
|
||||
-- foo : §1-fromMonad (Kleisli→Monoidal (§2-3.§2.toMonad (§2-fromMonad (Monoidal→Kleisli (§2-3.§1.toMonad m)))))
|
||||
-- ≡ §1-fromMonad (Kleisli→Monoidal ((Monoidal→Kleisli (§2-3.§1.toMonad m))))
|
||||
-- §2-3.§1.fmap (foo i) = §2-3.§1.fmap rhs
|
||||
-- §2-3.§1.join (foo i) = §2-3.§1.join rhs
|
||||
-- §2-3.§1.RisFunctor (foo i) = §2-3.§1.RisFunctor rhs
|
||||
-- §2-3.§1.pureN (foo i) = §2-3.§1.pureN rhs
|
||||
-- §2-3.§1.joinN (foo i) = §2-3.§1.joinN rhs
|
||||
-- §2-3.§1.isMonad (foo i) = §2-3.§1.isMonad rhs
|
||||
|
||||
lemmaz : §1-fromMonad (§2-3.§1.toMonad m) ≡ m
|
||||
§2-3.§1.fmap (lemmaz i) = §2-3.§1.fmap m
|
||||
§2-3.§1.join (lemmaz i) = §2-3.§1.join m
|
||||
§2-3.§1.RisFunctor (lemmaz i) = §2-3.§1.RisFunctor m
|
||||
§2-3.§1.pureN (lemmaz i) = §2-3.§1.pureN m
|
||||
§2-3.§1.joinN (lemmaz i) = §2-3.§1.joinN m
|
||||
§2-3.§1.isMonad (lemmaz i) = §2-3.§1.isMonad m
|
||||
lemma : ∀ m → §2-3.§2.toMonad (§2-fromMonad m) ≡ m
|
||||
K.Monad.raw (lemma m i) = K.Monad.raw m
|
||||
K.Monad.isMonad (lemma m i) = K.Monad.isMonad m
|
||||
|
||||
voe-isEquiv : isEquiv (§2-3.§1 omap pure) (§2-3.§2 omap pure) forth
|
||||
voe-isEquiv = gradLemma forth back forthEq backEq
|
||||
|
|
|
|||
Loading…
Reference in a new issue