open import Data.Unit open import Data.Product open import Chapter1.Logic open import Chapter2.IDesc open import Chapter2.IDesc.Fixpoint module Chapter2.IDesc.InitialAlgebra {I : Set} (D : func I I) where Alg : (I → Set) → Set Alg X = ⟦ D ⟧func X ⇒ X module Fold {X : I → Set}(α : Alg X) where mutual fold : {i : I} → μ D i → X i fold ⟨ xs ⟩ = α (hyps (func.out D _) xs) hyps : (D' : IDesc I) → (xs : ⟦ D' ⟧ (μ D)) → ⟦ D' ⟧ X hyps `1 tt = tt hyps (`var i) xs = fold xs hyps (T `× T') (t , t') = hyps T t , hyps T' t' hyps (`σ n T) (k , xs) = k , hyps (T k) xs hyps (`Σ S T) (s , xs) = s , hyps (T s) xs hyps (`Π S T) f = λ s → hyps (T s) (f s) fold : {X : I → Set}(α : Alg X) → μ D ⇒ X fold = Fold.fold