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Jules' dependent prism
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| module DepPrism where | |
| open import Prelude | |
| open import Data.Empty | |
| open import Data.Sum | |
| private variable | |
| ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Level | |
| X : 𝒰 ℓ₁ | |
| Y : 𝒰 ℓ₂ | |
| A : X → 𝒰 ℓ₃ | |
| B : Y → 𝒰 ℓ₄ | |
| lim : {B : Y → 𝒰 ℓ₄} {x : X} → A x ⊎ Y → 𝒰 ℓ₄ | |
| lim {ℓ₄} (inl _) = Lift ℓ₄ ⊥ | |
| lim {B} (inr y) = B y | |
| record DPrism (X : 𝒰 ℓ₁) (Y : 𝒰 ℓ₂) (A : X → 𝒰 ℓ₃) (B : Y → 𝒰 ℓ₄) : 𝒰 (ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃ ⊔ ℓ₄) where | |
| constructor dpr | |
| field | |
| fw : (x : X) → A x ⊎ Y | |
| bw : (x : X) → lim {A = A} {B = B} (fw x) → A x | |
| open DPrism | |
| bweq : (d : DPrism X Y A B) | |
| → (x : X) → (y : Y) → d .fw x ＝ inr y → B y → A x | |
| bweq {B} d x y e by with d .fw x | recall (d .fw) x | |
| ... | inl _ | ⟪ _ ⟫ = absurd (⊎-disjoint e) | |
| ... | inr _ | ⟪ eq ⟫ = d .bw x (subst lim (eq ⁻¹) (subst B (inr-inj e ⁻¹) by)) | |
| eqbw : (fw : (x : X) → A x ⊎ Y) | |
| → ((x : X) → (y : Y) → fw x ＝ inr y → B y → A x) | |
| → DPrism X Y A B | |
| eqbw {X} {A} {B} fw bw = dpr fw go | |
| where | |
| go : (x : X) → lim {A = A} {B = B} (fw x) → A x | |
| go x with fw x | recall fw x | |
| go x | inl _ | ⟪ eq ⟫ = λ () | |
| go x | inr y | ⟪ eq ⟫ = bw x y eq |
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| module DepPrism0 where | |
| open import Prelude | |
| open import Data.Empty | |
| open import Data.Sum | |
| private variable | |
| ℓ₁ ℓ₂ ℓₛ ℓₜ : Level | |
| X : 𝒰 ℓ₁ | |
| Y : 𝒰 ℓ₂ | |
| S : X → 𝒰 ℓₛ | |
| T : Y → 𝒰 ℓₜ | |
| lim : {T : Y → 𝒰 ℓₜ} {x : X} → S x ⊎ Y → 𝒰 ℓₜ | |
| lim {ℓₜ} (inl _) = Lift ℓₜ ⊥ | |
| lim {T} (inr y) = T y | |
| record @0 DPrism (X : 𝒰 ℓ₁) (Y : 𝒰 ℓ₂) (S : X → 𝒰 ℓₛ) (T : Y → 𝒰 ℓₜ) : 𝒰 (ℓ₁ ⊔ ℓ₂ ⊔ ℓₛ ⊔ ℓₜ) where | |
| constructor dpr | |
| field | |
| fw : (x : X) → S x ⊎ Y | |
| bw : (@0 x : X) → lim {S = S} {T = T} (fw x) → S x |
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| module DepPrismEq where | |
| open import Prelude | |
| open import Data.Empty | |
| open import Data.Sum | |
| private variable | |
| ℓ₁ ℓ₂ ℓₛ ℓₜ : Level | |
| record DPrism (X : 𝒰 ℓ₁) (Y : 𝒰 ℓ₂) (S : @0 X → 𝒰 ℓₛ) (T : @0 Y → 𝒰 ℓₜ) : 𝒰 (ℓ₁ ⊔ ℓ₂ ⊔ ℓₛ ⊔ ℓₜ) where | |
| constructor dpr | |
| field | |
| fw : (x : X) → S x ⊎ Y | |
| bw : (@0 x : X) → (@0 y : Y) → (@0 _ : fw x ＝ inr y) → T y → S x |
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| module DepPrism where | |
| open import Prelude | |
| open import Data.Empty | |
| open import Data.Sum | |
| private variable | |
| ℓ₁ ℓ₂ ℓₛ ℓₜ : Level | |
| X : 𝒰 ℓ₁ | |
| Y : 𝒰 ℓ₂ | |
| S : X → 𝒰 ℓₛ | |
| T : Y → 𝒰 ℓₜ | |
| lim : {T : Y → 𝒰 ℓₜ} {@0 x : X} → S x ⊎ Y → 𝒰 ℓₜ | |
| lim {ℓₜ} (inl _) = Lift ℓₜ ⊥ | |
| lim {T} (inr y) = T y | |
| record DPrism (X : 𝒰 ℓ₁) (Y : 𝒰 ℓ₂) (S : X → 𝒰 ℓₛ) (T : Y → 𝒰 ℓₜ) : 𝒰 (ℓ₁ ⊔ ℓ₂ ⊔ ℓₛ ⊔ ℓₜ) where | |
| constructor dpr | |
| field | |
| fw : (x : X) → S x ⊎ Y | |
| bw : (@0 x : X) → Erased (lim {S = λ y → S (y .erased)} {T = T} (fw x) → S x) |
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