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Merge remote-tracking branch 'Saizan/benchmark' into dev

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Frederik Hanghøj Iversen 2018-05-16 11:38:12 +02:00
parent 1c0b0d9db2 47c881ba2a
commit d4dc125fb0
6 changed files with 147 additions and 146 deletions
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139
src/Cat/Categories/Fun.agda
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@ -1,6 +1,7 @@
{-# OPTIONS --allow-unsolved-metas --cubical --caching #-} {-# OPTIONS --allow-unsolved-metas --cubical --caching #-}
module Cat.Categories.Fun where module Cat.Categories.Fun where
open import Cat.Prelude open import Cat.Prelude
open import Cat.Equivalence open import Cat.Equivalence
open import Cat.Category open import Cat.Category
@ -52,14 +53,14 @@ module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : C
lem : coe (pp {C}) 𝔻.identity ≡ f→g lem : coe (pp {C}) 𝔻.identity ≡ f→g
lem = trans (𝔻.9-1-9-right {b = Functor.omap F C} 𝔻.identity p*) 𝔻.rightIdentity lem = trans (𝔻.9-1-9-right {b = Functor.omap F C} 𝔻.identity p*) 𝔻.rightIdentity
idToNatTrans : NaturalTransformation F G -- idToNatTrans : NaturalTransformation F G
idToNatTrans = (λ C → coe pp 𝔻.identity) , λ f → begin -- idToNatTrans = (λ C → coe pp 𝔻.identity) , λ f → begin
coe pp 𝔻.identity 𝔻.<<< F.fmap f ≡⟨ cong (𝔻._<<< F.fmap f) lem ⟩ -- coe pp 𝔻.identity 𝔻.<<< F.fmap f ≡⟨ cong (𝔻._<<< F.fmap f) lem ⟩
-- Just need to show that f→g is a natural transformation -- -- Just need to show that f→g is a natural transformation
-- I know that it has an inverse; g→f -- -- I know that it has an inverse; g→f
f→g 𝔻.<<< F.fmap f ≡⟨ {!lem!} ⟩ -- f→g 𝔻.<<< F.fmap f ≡⟨ {!lem!} ⟩
G.fmap f 𝔻.<<< f→g ≡⟨ cong (G.fmap f 𝔻.<<<_) (sym lem) ⟩ -- G.fmap f 𝔻.<<< f→g ≡⟨ cong (G.fmap f 𝔻.<<<_) (sym lem) ⟩
G.fmap f 𝔻.<<< coe pp 𝔻.identity ∎ -- G.fmap f 𝔻.<<< coe pp 𝔻.identity ∎
module _ {A B : Functor ℂ 𝔻} where module _ {A B : Functor ℂ 𝔻} where
module A = Functor A module A = Functor A
@ -92,70 +93,70 @@ module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : C
U : (F : ℂ.Object → 𝔻.Object) → Set _ U : (F : ℂ.Object → 𝔻.Object) → Set _
U F = {A B : ℂ.Object} → ℂ [ A , B ] → 𝔻 [ F A , F B ] U F = {A B : ℂ.Object} → ℂ [ A , B ] → 𝔻 [ F A , F B ]
module _ -- module _
(omap : ℂ.Object → 𝔻.Object) -- (omap : ℂ.Object → 𝔻.Object)
(p : A.omap ≡ omap) -- (p : A.omap ≡ omap)
where -- where
D : Set _ -- D : Set _
D = ( fmap : U omap) -- D = ( fmap : U omap)
→ ( let -- → ( let
raw-B : RawFunctor ℂ 𝔻 -- raw-B : RawFunctor ℂ 𝔻
raw-B = record { omap = omap ; fmap = fmap } -- raw-B = record { omap = omap ; fmap = fmap }
) -- )
→ (isF-B' : IsFunctor ℂ 𝔻 raw-B) -- → (isF-B' : IsFunctor ℂ 𝔻 raw-B)
→ ( let -- → ( let
B' : Functor ℂ 𝔻 -- B' : Functor ℂ 𝔻
B' = record { raw = raw-B ; isFunctor = isF-B' } -- B' = record { raw = raw-B ; isFunctor = isF-B' }
) -- )
→ (iso' : A ≊ B') → PathP (λ i → U (p i)) A.fmap fmap -- → (iso' : A ≊ B') → PathP (λ i → U (p i)) A.fmap fmap
-- D : Set _ -- -- D : Set _
-- D = PathP (λ i → U (p i)) A.fmap fmap -- -- D = PathP (λ i → U (p i)) A.fmap fmap
-- eeq : (λ f → A.fmap f) ≡ fmap -- -- eeq : (λ f → A.fmap f) ≡ fmap
-- eeq = funExtImp (λ A → funExtImp (λ B → funExt (λ f → isofmap {A} {B} f))) -- -- eeq = funExtImp (λ A → funExtImp (λ B → funExt (λ f → isofmap {A} {B} f)))
-- where -- -- where
-- module _ {X : ℂ.Object} {Y : ℂ.Object} (f : ℂ [ X , Y ]) where -- -- module _ {X : ℂ.Object} {Y : ℂ.Object} (f : ℂ [ X , Y ]) where
-- isofmap : A.fmap f ≡ fmap f -- -- isofmap : A.fmap f ≡ fmap f
-- isofmap = {!ap!} -- -- isofmap = {!ap!}
d : D A.omap refl -- d : D A.omap refl
d = res -- d = res
where -- where
module _ -- module _
( fmap : U A.omap ) -- ( fmap : U A.omap )
( let -- ( let
raw-B : RawFunctor ℂ 𝔻 -- raw-B : RawFunctor ℂ 𝔻
raw-B = record { omap = A.omap ; fmap = fmap } -- raw-B = record { omap = A.omap ; fmap = fmap }
) -- )
( isF-B' : IsFunctor ℂ 𝔻 raw-B ) -- ( isF-B' : IsFunctor ℂ 𝔻 raw-B )
( let -- ( let
B' : Functor ℂ 𝔻 -- B' : Functor ℂ 𝔻
B' = record { raw = raw-B ; isFunctor = isF-B' } -- B' = record { raw = raw-B ; isFunctor = isF-B' }
) -- )
( iso' : A ≊ B' ) -- ( iso' : A ≊ B' )
where -- where
module _ {X Y : ℂ.Object} (f : ℂ [ X , Y ]) where -- module _ {X Y : ℂ.Object} (f : ℂ [ X , Y ]) where
step : {!!} 𝔻.≊ {!!} -- step : {!!} 𝔻.≊ {!!}
step = {!!} -- step = {!!}
resres : A.fmap {X} {Y} f ≡ fmap {X} {Y} f -- resres : A.fmap {X} {Y} f ≡ fmap {X} {Y} f
resres = {!!} -- resres = {!!}
res : PathP (λ i → U A.omap) A.fmap fmap -- res : PathP (λ i → U A.omap) A.fmap fmap
res i {X} {Y} f = resres f i -- res i {X} {Y} f = resres f i
fmapEq : PathP (λ i → U (omapEq i)) A.fmap B.fmap -- fmapEq : PathP (λ i → U (omapEq i)) A.fmap B.fmap
fmapEq = pathJ D d B.omap omapEq B.fmap B.isFunctor iso -- fmapEq = pathJ D d B.omap omapEq B.fmap B.isFunctor iso
rawEq : A.raw ≡ B.raw -- rawEq : A.raw ≡ B.raw
rawEq i = record { omap = omapEq i ; fmap = fmapEq i } -- rawEq i = record { omap = omapEq i ; fmap = fmapEq i }
private -- private
f : (A ≡ B) → (A ≊ B) -- f : (A ≡ B) → (A ≊ B)
f p = idToNatTrans p , idToNatTrans (sym p) , NaturalTransformation≡ A A (funExt (λ C → {!!})) , {!!} -- f p = idToNatTrans p , idToNatTrans (sym p) , NaturalTransformation≡ A A (funExt (λ C → {!!})) , {!!}
g : (A ≊ B) → (A ≡ B) -- g : (A ≊ B) → (A ≡ B)
g = Functor≡ ∘ rawEq -- g = Functor≡ ∘ rawEq
inv : AreInverses f g -- inv : AreInverses f g
inv = {!funExt λ p → ?!} , {!!} -- inv = {!funExt λ p → ?!} , {!!}
postulate
iso : (A ≡ B) ≅ (A ≊ B) iso : (A ≡ B) ≅ (A ≊ B)
iso = f , g , inv -- iso = f , g , inv
univ : (A ≡ B) ≃ (A ≊ B) univ : (A ≡ B) ≃ (A ≊ B)
univ = fromIsomorphism _ _ iso univ = fromIsomorphism _ _ iso

2
src/Cat/Category.agda
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@ -44,7 +44,7 @@ open Cat.Equivalence
-- about these. The laws defined are the types the propositions - not the -- about these. The laws defined are the types the propositions - not the
-- witnesses to them! -- witnesses to them!
record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
no-eta-equality -- no-eta-equality
field field
Object : Set ℓa Object : Set ℓa
Arrow : Object → Object → Set ℓb Arrow : Object → Object → Set ℓb

1
src/Cat/Category/Monad/Kleisli.agda
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@ -230,6 +230,7 @@ record IsMonad (raw : RawMonad) : Set ℓ where
m ∎ m ∎
record Monad : Set ℓ where record Monad : Set ℓ where
no-eta-equality
field field
raw : RawMonad raw : RawMonad
isMonad : IsMonad raw isMonad : IsMonad raw

1
src/Cat/Category/Monad/Monoidal.agda
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@ -122,6 +122,7 @@ record IsMonad (raw : RawMonad) : Set ℓ where
R.fmap a <<< R.fmap b <<< R.fmap c ∎ R.fmap a <<< R.fmap b <<< R.fmap c ∎
record Monad : Set ℓ where record Monad : Set ℓ where
no-eta-equality
field field
raw : RawMonad raw : RawMonad
isMonad : IsMonad raw isMonad : IsMonad raw

147
src/Cat/Category/Monad/Voevodsky.agda
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@ -4,6 +4,7 @@ This module provides construction 2.3 in [voe]
{-# OPTIONS --cubical --caching #-} {-# OPTIONS --cubical --caching #-}
module Cat.Category.Monad.Voevodsky where module Cat.Category.Monad.Voevodsky where
open import Cat.Prelude open import Cat.Prelude
open import Cat.Category open import Cat.Category
@ -26,6 +27,7 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
module §2-3 (omap : Object → Object) (pure : {X : Object} → Arrow X (omap X)) where module §2-3 (omap : Object → Object) (pure : {X : Object} → Arrow X (omap X)) where
record §1 : Set ℓ where record §1 : Set ℓ where
no-eta-equality
open M open M
field field
@ -76,12 +78,11 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
isMonad : IsMonad rawMnd isMonad : IsMonad rawMnd
toMonad : Monad toMonad : Monad
toMonad = record toMonad .Monad.raw = rawMnd
{ raw = rawMnd toMonad .Monad.isMonad = isMonad
; isMonad = isMonad
}
record §2 : Set ℓ where record §2 : Set ℓ where
no-eta-equality
open K open K
field field
@ -98,28 +99,24 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
isMonad : IsMonad rawMnd isMonad : IsMonad rawMnd
toMonad : Monad toMonad : Monad
toMonad = record toMonad .Monad.raw = rawMnd
{ raw = rawMnd toMonad .Monad.isMonad = isMonad
; isMonad = isMonad
}
§1-fromMonad : (m : M.Monad) → §2-3.§1 (M.Monad.Romap m) (λ {X} → M.Monad.pureT m X) module _ (m : M.Monad) where
§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
open M.Monad m 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 : K.Monad) → §2-3.§2 (K.Monad.omap m) (K.Monad.pure m)
§2-fromMonad m = record §2-fromMonad m .§2-3.§2.bind = K.Monad.bind m
{ bind = K.Monad.bind m §2-fromMonad m .§2-3.§2.isMonad = K.Monad.isMonad m
; isMonad = K.Monad.isMonad m
}
-- | In the following we seek to transform the equivalence `Monoidal≃Kleisli` -- | In the following we seek to transform the equivalence `Monoidal≃Kleisli`
-- | to talk about voevodsky's construction. -- | to talk about voevodsky's construction.
@ -147,64 +144,64 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
forthEq : ∀ m → (forth ∘ back) m ≡ m forthEq : ∀ m → (forth ∘ back) m ≡ m
forthEq m = begin forthEq m = begin
(forth ∘ back) m ≡⟨⟩ §2-fromMonad
-- In full gory detail: (Monoidal→Kleisli
( §2-fromMonad (§2-3.§1.toMonad
∘ Monoidal→Kleisli (§1-fromMonad (Kleisli→Monoidal (§2-3.§2.toMonad m)))))
∘ §2-3.§1.toMonad ≡⟨ cong-d (§2-fromMonad ∘ Monoidal→Kleisli) (lemmaz (Kleisli→Monoidal (§2-3.§2.toMonad m))) ⟩
∘ §1-fromMonad §2-fromMonad
∘ Kleisli→Monoidal ((Monoidal→Kleisli ∘ Kleisli→Monoidal)
∘ §2-3.§2.toMonad (§2-3.§2.toMonad m))
) m ≡⟨⟩ -- fromMonad and toMonad are inverses ≡⟨ (cong-d (\ φ → §2-fromMonad (φ (§2-3.§2.toMonad m))) re-ve) ⟩
( §2-fromMonad (§2-fromMonad ∘ §2-3.§2.toMonad) m
∘ Monoidal→Kleisli ≡⟨ lemma ⟩
∘ Kleisli→Monoidal m ∎
∘ §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 ∎
where where
t' : ((Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ §2-3.§2.toMonad {omap} {pure}) lemma : (§2-fromMonad ∘ §2-3.§2.toMonad) m ≡ m
≡ §2-3.§2.toMonad §2-3.§2.bind (lemma i) = §2-3.§2.bind m
t' = cong (\ φ → φ ∘ §2-3.§2.toMonad) re-ve §2-3.§2.isMonad (lemma i) = §2-3.§2.isMonad m
t : (§2-fromMonad ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ §2-3.§2.toMonad {omap} {pure}) lemmaz : ∀ m → §2-3.§1.toMonad (§1-fromMonad m) ≡ m
≡ (§2-fromMonad ∘ §2-3.§2.toMonad) M.Monad.raw (lemmaz m i) = M.Monad.raw m
t = cong-d (\ f → §2-fromMonad ∘ f) t' M.Monad.isMonad (lemmaz m i) = M.Monad.isMonad m
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
backEq : ∀ m → (back ∘ forth) m ≡ m backEq : ∀ m → (back ∘ forth) m ≡ m
backEq m = begin backEq m = begin
(back ∘ forth) m ≡⟨⟩ §1-fromMonad
( §1-fromMonad (Kleisli→Monoidal
∘ Kleisli→Monoidal (§2-3.§2.toMonad
∘ §2-3.§2.toMonad (§2-fromMonad (Monoidal→Kleisli (§2-3.§1.toMonad m)))))
∘ §2-fromMonad ≡⟨ cong-d (§1-fromMonad ∘ Kleisli→Monoidal) (lemma (Monoidal→Kleisli (§2-3.§1.toMonad m))) ⟩
∘ Monoidal→Kleisli §1-fromMonad
∘ §2-3.§1.toMonad ((Kleisli→Monoidal ∘ Monoidal→Kleisli)
) m ≡⟨⟩ -- fromMonad and toMonad are inverses (§2-3.§1.toMonad m))
( §1-fromMonad ≡⟨ (cong-d (\ φ → §1-fromMonad (φ (§2-3.§1.toMonad m))) ve-re) ⟩
∘ Kleisli→Monoidal §1-fromMonad (§2-3.§1.toMonad m)
∘ Monoidal→Kleisli ≡⟨ lemmaz ⟩
∘ §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 ∎ m ∎
where where
t : §1-fromMonad ∘ Kleisli→Monoidal ∘ Monoidal→Kleisli ∘ §2-3.§1.toMonad -- having eta equality on causes roughly the same work as checking this proof of foo,
≡ §1-fromMonad ∘ §2-3.§1.toMonad -- which is quite expensive because it ends up reducing complex terms.
-- Why does `re-ve` not satisfy this goal?
t i m = §1-fromMonad (ve-re i (§2-3.§1.toMonad m)) -- 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 : isEquiv (§2-3.§1 omap pure) (§2-3.§2 omap pure) forth
voe-isEquiv = gradLemma forth back forthEq backEq voe-isEquiv = gradLemma forth back forthEq backEq

3
src/Cat/Category/Product.agda
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@ -1,6 +1,7 @@
{-# OPTIONS --cubical --caching #-} {-# OPTIONS --cubical --caching #-}
module Cat.Category.Product where module Cat.Category.Product where
open import Cat.Prelude as P hiding (_×_ ; fst ; snd) open import Cat.Prelude as P hiding (_×_ ; fst ; snd)
open import Cat.Equivalence open import Cat.Equivalence
@ -11,7 +12,7 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
module _ (A B : Object) where module _ (A B : Object) where
record RawProduct : Set (ℓa ⊔ ℓb) where record RawProduct : Set (ℓa ⊔ ℓb) where
no-eta-equality -- no-eta-equality
field field
object : Object object : Object
fst : ℂ [ object , A ] fst : ℂ [ object , A ]