Discreteness and H-Levels of W-types

[TOTBWF]

Description

This PR proves that a W-type W A B is discrete when A is discrete and B x is Listable for every x. I've also added
a characterisation of equality types of W-types. These should make arguments about decidable equality a bit easier to make for inductive types: instead of doing a giant $O(n^2)$ case bash, we can construct an equivalence with a W-type instead. Some automation on this front would be useful, but I've spent enough time shaving this yak already :)

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[Lavenza]

Pull request preview

Changed pages

• Data.Wellfounded.W (+324, -4)
• 1Lab.Type.Pi (+7, -0)

[ncfavier]

Cool!

[plt-amy]

I have this much simpler identity system construction sitting around:

W-is-hlevel {ℓ} {ℓ'} {A = A} {B = B} n ahl = identity-system→hlevel n W-ids W-tr where
  W-code : W A B → W A B → Type (ℓ ⊔ ℓ')
  W-code (sup x f) (sup y g) = Σ[ p ∈ x ≡ᵢ y ] (∀ i → W-code (f i) (g (substᵢ B p i)))

  W-code-refl : ∀ x → W-code x x
  W-code-refl (sup x f) = reflᵢ , λ a → W-code-refl (f a)

  W-decode : ∀ x y → W-code x y → x ≡ y
  W-decode (sup x f) (sup y g) (reflᵢ , p) = ap (sup x) (funext λ a → W-decode _ _ (p a))

  W-coh : ∀ x y (p : W-code x y) → PathP (λ i → W-code x (W-decode x y p i)) (W-code-refl x) p
  W-coh (sup x f) (sup y g) (reflᵢ , p) = Σ-pathp refl (funextP λ x → W-coh (f x) (g x) (p x))

  W-ids : is-identity-system W-code W-code-refl
  W-ids .to-path = W-decode _ _
  W-ids .to-path-over = W-coh _ _

  W-tr : ∀ x y → is-hlevel (W-code x y) n
  W-tr (sup x f) (sup y g) = Σ-is-hlevel n (≡ᵢ-is-hlevel' ahl x y) λ x → Π-is-hlevel n λ i → W-tr _ _

[TOTBWF]

I think there is value in having both, but mostly because I think the proof of Path≃Code is cute 🙂

We should probably provide a version of your identity system as an extensionality transformer though.