44 lines
1 KiB
Agda
44 lines
1 KiB
Agda
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
|