[-- Attachment #1.1: Type: text/plain, Size: 1257 bytes --] I was experimenting cubical Agda with {-# OPTIONS --cubical --rewriting #-} I tried to define S^2, S^3 separately and found HIT worked fine. However when I was trying to write down Sphere n for general natural number n it simply won't allow me to do so. And the error message is "The target of a constructor must be the datatype applied to its parameters". So how do I define a dependent HIT? PtType : Set₁ PtType = Σ Set (λ A → A) Ω1 : PtType → PtType Ω1 A = ((A .snd) ≡ (A .snd)) , refl Ω : (n : ℕ) → PtType → PtType Ω 0 A = A Ω (suc n) A = Ω1 (Ω n A) data S² : Set where base : S² surf : ((Ω 2 (S² , base)) .fst) data S³ : Set where base : S³ cell : ((Ω 3 (S³ , base)) .fst) data Sphere : ℕ → Set where base : (n : ℕ) → Sphere n --cell : (n : ℕ) → ((Ω n ((Sphere n) , (base n))) .fst) -- Error! 222222222222 -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. For more options, visit https://groups.google.com/d/optout. [-- Attachment #1.2: Type: text/html, Size: 14981 bytes --]