open import IDesc.IDesc
open import Orn.Ornament
module FunOrn.Lift.Fold
{I I⁺ : Set }
{D : func I I}
{u : I⁺ → I}
(o : orn D u u)
where
open import Data.Product
open import Logic.Logic
open import IDesc.InitialAlgebra
open import Orn.Reornament
open import FunOrn.Functions
open import FunOrn.FunOrnament
open import FunOrn.Patch
liftAlg : ∀{T} → Alg D (λ _ → ⟦ T ⟧Type) → FunctionOrn T → Set
liftAlg α T⁺ = Alg ⌈ o ⌉D (λ ix → Patch (fold D α (proj₂ ix)) T⁺)
lift-fold : {i : I}{i⁺ : u ⁻¹ i}
{T : Type }{T⁺ : FunctionOrn T}
(α : Alg D (λ _ → ⟦ T ⟧Type))
(β : liftAlg α T⁺) →
Patch (fold D α) (μ⁺ o [ i⁺ ]→ T⁺)
lift-fold {i⁺ = inv i⁺} α β =
λ x x⁺⁺ → fold ⌈ o ⌉D (λ {ix} ih → β {ix} ih) x⁺⁺