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

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Frederik Hanghøj Iversen 2018-03-30 12:23:29 +02:00
parent ba80fe96dc 34e633902f
commit 41b442c0d8
1 changed files with 9 additions and 36 deletions
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45
src/Cat/Category/Product.agda
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@ -154,67 +154,40 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
open RawCategory raw
propEqs : ∀ {X' : Object}{Y' : Object} (let X , xa , xb = X') (let Y , ya , yb = Y')
→ (xy : ℂ.Arrow X Y) → isProp (ℂ [ ya ∘ xy ] ≡ xa × ℂ [ yb ∘ xy ] ≡ xb)
propEqs xs = propSig (ℂ.arrowsAreSets _ _) (\ _ → ℂ.arrowsAreSets _ _)
isAssocitaive : IsAssociative
isAssocitaive
{A , a0 , a1}
{B , b0 , b1}
{C , c0 , c1}
{D , d0 , d1}
{ff@(f , f0 , f1)}
{gg@(g , g0 , g1)}
{hh@(h , h0 , h1)}
i
= s0 i , rl i , rr i
isAssocitaive {A'@(A , a0 , a1)} {B , _} {C , c0 , c1} {D'@(D , d0 , d1)} {ff@(f , f0 , f1)} {gg@(g , g0 , g1)} {hh@(h , h0 , h1)} i
= s0 i , lemPropF propEqs s0 {proj₂ l} {proj₂ r} i
where
l = hh ∘ (gg ∘ ff)
r = hh ∘ gg ∘ ff
-- s0 : h ℂ.∘ (g ℂ.∘ f) ≡ h ℂ.∘ g ℂ.∘ f
s0 : proj₁ l ≡ proj₁ r
s0 = ℂ.isAssociative {f = f} {g} {h}
prop0 : ∀ a → isProp (ℂ [ d0 ∘ a ] ≡ a0)
prop0 a = ℂ.arrowsAreSets (ℂ [ d0 ∘ a ]) a0
rl : (λ i → (ℂ [ d0 ∘ s0 i ]) ≡ a0) [ proj₁ (proj₂ l) ≡ proj₁ (proj₂ r) ]
rl = lemPropF prop0 s0
prop1 : ∀ a → isProp ((ℂ [ d1 ∘ a ]) ≡ a1)
prop1 a = ℂ.arrowsAreSets _ _
rr : (λ i → (ℂ [ d1 ∘ s0 i ]) ≡ a1) [ proj₂ (proj₂ l) ≡ proj₂ (proj₂ r) ]
rr = lemPropF prop1 s0
isIdentity : IsIdentity 𝟙
isIdentity {AA@(A , a0 , a1)} {BB@(B , b0 , b1)} {f , f0 , f1} = leftIdentity , rightIdentity
where
leftIdentity : 𝟙 ∘ (f , f0 , f1) ≡ (f , f0 , f1)
leftIdentity i = l i , rl i , rr i
leftIdentity i = l i , lemPropF propEqs l {proj₂ L} {proj₂ R} i
where
L = 𝟙 ∘ (f , f0 , f1)
R : Arrow AA BB
R = f , f0 , f1
l : proj₁ L ≡ proj₁ R
l = ℂ.leftIdentity
prop0 : ∀ a → isProp ((ℂ [ b0 ∘ a ]) ≡ a0)
prop0 a = ℂ.arrowsAreSets _ _
rl : (λ i → (ℂ [ b0 ∘ l i ]) ≡ a0) [ proj₁ (proj₂ L) ≡ proj₁ (proj₂ R) ]
rl = lemPropF prop0 l
prop1 : ∀ a → isProp (ℂ [ b1 ∘ a ] ≡ a1)
prop1 _ = ℂ.arrowsAreSets _ _
rr : (λ i → (ℂ [ b1 ∘ l i ]) ≡ a1) [ proj₂ (proj₂ L) ≡ proj₂ (proj₂ R) ]
rr = lemPropF prop1 l
rightIdentity : (f , f0 , f1) ∘ 𝟙 ≡ (f , f0 , f1)
rightIdentity i = l i , rl i , rr i
rightIdentity i = l i , lemPropF propEqs l {proj₂ L} {proj₂ R} i
where
L = (f , f0 , f1) ∘ 𝟙
R : Arrow AA BB
R = (f , f0 , f1)
l : ℂ [ f ∘ ℂ.𝟙 ] ≡ f
l = ℂ.rightIdentity
prop0 : ∀ a → isProp ((ℂ [ b0 ∘ a ]) ≡ a0)
prop0 _ = ℂ.arrowsAreSets _ _
rl : (λ i → (ℂ [ b0 ∘ l i ]) ≡ a0) [ proj₁ (proj₂ L) ≡ proj₁ (proj₂ R) ]
rl = lemPropF prop0 l
prop1 : ∀ a → isProp ((ℂ [ b1 ∘ a ]) ≡ a1)
prop1 _ = ℂ.arrowsAreSets _ _
rr : (λ i → (ℂ [ b1 ∘ l i ]) ≡ a1) [ proj₂ (proj₂ L) ≡ proj₂ (proj₂ R) ]
rr = lemPropF prop1 l
arrowsAreSets : ArrowsAreSets
arrowsAreSets {X , x0 , x1} {Y , y0 , y1} (f , f0 , f1) (g , g0 , g1) p q = {!!}