open nat prod

inductive ifin : ℕ → Type := -- inductively defined fin-type
| fz : Π n, ifin (succ n)
| fs : Π {n}, ifin n → ifin (succ n)

open ifin

definition foo {N : Type} : Π{n : ℕ}, N → ifin n → (N × ifin n)
| (succ k) n (fz k) := sorry
| (succ k) n (fs x) := sorry

definition bar {N : Type} : Π{n : ℕ}, (N × ifin n) → (N × ifin n)
| ⌞succ k⌟ (n, fz k) := sorry
| ⌞succ k⌟ (n, fs x) := sorry

definition bar2 {N : Type} : Π{n : ℕ}, (N × ifin n) → (N × ifin n)
| (succ k) (n, fz k) := sorry
| (succ k) (n, fs x) := sorry