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Indexed W-types: hlevel without univalence. #1172

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Nov 28, 2024
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Indexed W-types: hlevel without univalence. #1172

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11 changes: 5 additions & 6 deletions Cubical/Data/W/Indexed.agda
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Original file line number Diff line number Diff line change
Expand Up @@ -5,7 +5,6 @@ open import Cubical.Foundations.Function
open import Cubical.Foundations.Path
open import Cubical.Foundations.Isomorphism renaming (Iso to _≅_)
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Univalence
open import Cubical.Foundations.HLevels
open import Cubical.Functions.FunExtEquiv
open import Cubical.Data.Unit
Expand Down Expand Up @@ -55,8 +54,8 @@ module _ {X : Type ℓX} {S : X → Type ℓS} {P : ∀ x → S x → Type ℓP}
equivRepIW : (x : X) → IW S P inX x ≃ RepIW S P inX x
equivRepIW x = isoToEquiv (isoRepIW x)

pathRepIW : (x : X) → IW S P inX x ≡ RepIW S P inX x
pathRepIW x = ua (equivRepIW x)
--pathRepIW : (x : X) → IW S P inX x ≡ RepIW S P inX x
--pathRepIW x = ua (equivRepIW x)

isPropIW : (∀ x → isProp (S x)) → ∀ x → isProp (IW S P inX x)
isPropIW isPropS x (node s subtree) (node s' subtree') =
Expand Down Expand Up @@ -133,8 +132,8 @@ module IWPath {X : Type ℓX} {S : X → Type ℓS} {P : ∀ x → S x → Type
equivEncode : ∀ {x} (w w' : IW S P inX x) → (w ≡ w') ≃ Cover w w'
equivEncode w w' = isoToEquiv (isoEncode w w')

pathEncode : ∀ {x} (w w' : IW S P inX x) → (w ≡ w') ≡ Cover w w'
pathEncode w w' = ua (equivEncode w w')
--pathEncode : ∀ {x} (w w' : IW S P inX x) → (w ≡ w') ≡ Cover w w'
--pathEncode w w' = ua (equivEncode w w')

open IWPathTypes
open IWPath
Expand All @@ -143,7 +142,7 @@ isOfHLevelSuc-IW : {X : Type ℓX} {S : X → Type ℓS} {P : ∀ x → S x →
(n : HLevel) → (∀ x → isOfHLevel (suc n) (S x)) → ∀ x → isOfHLevel (suc n) (IW S P inX x)
isOfHLevelSuc-IW zero isHS x = isPropIW isHS x
isOfHLevelSuc-IW (suc n) isHS x w w' =
subst (isOfHLevel (suc n)) (λ i → pathEncode w w' (~ i))
isOfHLevelRetractFromIso (suc n) (isoEncode w w')
(isOfHLevelSuc-IW n
(λ (y , v , v') → isHS y (getShape v) (getShape v'))
(x , w , w')
Expand Down
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