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Change what is needed

This commit is contained in:
Frederik Hanghøj Iversen 2018-04-04 12:01:29 +02:00
parent f66d180ec3
commit 84f88ac2ae
1 changed files with 13 additions and 7 deletions
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20
src/Cat/Category.agda
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@ -137,8 +137,11 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
-- It may be that we need something weaker than this, in that there
-- may be some other lemmas available to us.
-- For instance, `need0` should be available to us when we prove `need1`.
need0 : ∀ Y → A ≡ Y
need1 : (f : Arrow A A) → identity ≡ f
need0 : (s : Σ Object (A ≅_)) → (open Σ s renaming (proj₁ to Y) using ()) → A ≡ Y
need2 : (iso : A ≅ A)
→ (open Σ iso renaming (proj₁ to f ; proj₂ to iso-f))
→ (open Σ iso-f renaming (proj₁ to f~ ; proj₂ to areInv))
→ (identity , identity) ≡ (f , f~)
c : Σ Object (A ≅_)
c = A , idIso A
@ -146,7 +149,7 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
module _ (y : Σ Object (A ≅_)) where
open Σ y renaming (proj₁ to Y ; proj₂ to isoY)
q : A ≡ Y
q = need0 Y
q = need0 y
-- Some error with primComp
isoAY : A ≅ Y
@ -162,20 +165,23 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
where
open Σ A≅Y renaming (proj₁ to f ; proj₂ to iso-f)
open Σ iso-f renaming (proj₁ to f~ ; proj₂ to areInv)
aaa : (identity , identity) ≡ (f , f~)
aaa = need2 A≅Y
a0 : identity ≡ f
a0 = need1 f
a0 i = fst (aaa i)
a1 : identity ≡ f~
a1 = need1 f~
a1 i = snd (aaa i)
-- we do have this!
-- I just need to rearrange the proofs a bit.
postulate
prop : ∀ {A B} (fg : Arrow A B × Arrow B A) → isProp (IsInverseOf (fst fg) (snd fg))
a2 : PathP (λ i → IsInverseOf (a0 i) (a1 i)) isIdentity areInv
a2 = lemPropF prop λ i → a0 i , a1 i
a2 = lemPropF prop aaa
d* : D Y q
d* = pathJ D d Y q
p : (A , idIso A) ≡ (Y , isoY)
p i = need0 Y i , lem i
p i = q i , lem i
univ-lem : isContr (Σ Object (A ≅_))
univ-lem = c , p