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This commit is contained in:
Michael Zhang 2024-09-18 06:38:08 -05:00
parent 2f4c87ea1d
commit 79e8a2300a

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@ -22,7 +22,7 @@ open import Cubical.Foundations.Isomorphism
open import Cubical.Core.Primitives
open import Cubical.HITs.Susp.Base
open import Cubical.HITs.S1.Base
open import Data.Bool.Base
open import Data.Bool.Base hiding (_∧_; __)
```
</details>
@ -196,17 +196,39 @@ Between the second and third steps, I used functoriality of the $\mathsf{ap}$ op
```
fg (loop i) k =
let
leftFace = λ j → compPath-filler (λ i → f (merid false i)) (λ j → f (merid true (~ j))) j i
u = λ j → λ where
(i = i0) → base
(i = i1) → f (merid true (~ j))
(k = i0) → leftFace j
(k = i0) → compPath-filler
(λ i → f (merid false i))
(λ j → f (merid true (~ j))) j i
(k = i1) → loop i
in hcomp u (f (merid false i))
```
```
gf : (b : Susp Bool) → g (f b) ≡ b
gf b = {! !}
gf north = refl
gf south = merid true
-- Both merid true and merid false work here... why pick true?
gf (merid true i) j = merid true (i ∧ j)
```
For the last part, we are trying to prove:
`(merid false ∙ (λ i₁ → merid true (~ i₁))) i ≡ merid false i`
```
gf (merid false i) j =
let
u = λ k → λ where
(i = i0) → north
(i = i1) → merid true (j ~ k)
(j = i0) →
let u = λ k' → λ where
(i = i0) → north
(i = i1) → merid true (~ k')
in hfill u (inS (merid false i)) k
(j = i1) → merid false i
in hcomp u (merid false i)
```