open import Axiom.Extensionality.Propositional

module _ (ext : ∀ {a b} → Extensionality a b) where

open import Prelude hiding (zero; suc)
open import Data.W
open import Data.Container
open import Data.Container.Relation.Unary.All ; open □

variable ℓ : Level


data Tag : Set where `zero `suc : Tag

Body : Tag → Set
Body t = case t of λ {`zero → ⊥ ; `suc → ⊤}

Repr : Container lzero lzero
Repr = Tag ▷ Body

Nat : Set
Nat = W Repr

Nat′ : Set
Nat′ = ⟦ Repr ⟧ Nat

zero : Nat
zero = sup (`zero , λ ())

suc : Nat → Nat
suc n = sup (`suc , const n)

elim : (P : Nat → Set ℓ) →
       (Z : P zero) →
       (S : ∀ n → P n → P (suc n)) →
       (n : Nat) → P n
elim P Z S = induction P λ {(tag , body)} →
  (case tag
   return (λ t → (n′ : Body t → Nat) →
                  □ Repr P (t , n′) →
                  P (sup (t , n′)))
   of λ where
     `zero → λ n′ _  → ≡.subst (λ n′ → P (sup (`zero , n′))) (ext λ ()) Z
     `suc  → λ n′ IH → S (n′ tt) (IH .proof tt)) body