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Improvements to Univalence.agda #1237

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b7a7a97
improvements to Univalence module
anshwad10 Jul 27, 2025
3e17be3
simpler proof of `contrSinglEquiv`
anshwad10 Jul 27, 2025
3ed514c
Better `pathToEquiv`
anshwad10 Aug 16, 2025
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9 changes: 6 additions & 3 deletions Cubical/Core/Glue.agda
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Expand Up @@ -15,21 +15,24 @@ open import Cubical.Core.Primitives
open import Agda.Builtin.Cubical.Glue public
using ( isEquiv -- ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} (f : A → B) → Type (ℓ ⊔ ℓ')

; equiv-proof -- ∀ (y : B) → isContr (fiber f y)
; equiv-proof -- ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} {f : A → B} → isEquiv f
-- → (y : B) → isContr (fiber f y)

; _≃_ -- ∀ {ℓ ℓ'} (A : Type ℓ) (B : Type ℓ') → Type (ℓ ⊔ ℓ')

; equivFun -- ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} → A ≃ B → A → B

; equivProof -- ∀ {ℓ ℓ'} (T : Type ℓ) (A : Type ℓ') (w : T ≃ A) (a : A) φ →
-- Partial φ (fiber (equivFun w) a) → fiber (equivFun w) a
; equivProof -- ∀ {ℓ ℓ'} (T : Type ℓ) (A : Type ℓ') (w : T ≃ A) (a : A) φ
-- → Partial φ (fiber (equivFun w) a) → fiber (equivFun w) a

; primGlue -- ∀ {ℓ ℓ'} (A : Type ℓ) {φ : I} (T : Partial φ (Type ℓ'))
-- → (e : PartialP φ (λ o → T o ≃ A)) → Type ℓ'

; prim^unglue -- ∀ {ℓ ℓ'} {A : Type ℓ} {φ : I} {T : Partial φ (Type ℓ')}
-- → {e : PartialP φ (λ o → T o ≃ A)} → primGlue A T e → A

; pathToEquiv -- ∀ {ℓ} {A B : Type ℓ} → A ≡ B → A ≃ B

-- The ∀ operation on I. This is commented out as it is not currently used for anything
-- ; primFaceForall -- (I → I) → I
)
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13 changes: 13 additions & 0 deletions Cubical/Foundations/Equiv.agda
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Expand Up @@ -98,6 +98,19 @@ module _ {f : A → B} (equivF : isEquiv f) where
})
(commSqIsEq a i j)

commPathIsEq' : ∀ a → retIsEq (invIsEq a) ≡ cong invIsEq (secIsEq a)
commPathIsEq' a i j = hcomp (λ where
k (i = i0) → retIsEq (invIsEq (secIsEq a k)) j
k (i = i1) → invIsEq (secIsEq (secIsEq a k) j)
k (j = i0) → invIsEq (f (invIsEq (secIsEq a k)))
k (j = i1) → invIsEq (secIsEq a k)
) (hcomp (λ where
k (i = i0) → retIsEq (retIsEq (invIsEq a) (~ k)) (j ∨ ~ k)
k (i = i1) → invIsEq (commSqIsEq (invIsEq a) (~ k) j)
k (j = i0) → retIsEq (retIsEq (invIsEq a) (~ k)) (~ k ∧ ~ i)
k (j = i1) → retIsEq (invIsEq a) (~ k ∧ ~ i)
) (retIsEq (invIsEq a) (~ i)))

module _ (w : A ≃ B) where
invEq : B → A
invEq = invIsEq (snd w)
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