data Nat : Type where
     Z : Nat
     S : Nat -> Nat

plus : Nat -> Nat -> Nat
plus Z $y = y
plus (S $k) $y = S (plus k y)

data Parity : Nat -> Type where
     Even : (n : Nat) -> Parity (plus n n)
     Odd : (n : Nat) -> Parity (S (plus n n))

half : (n : Nat) -> Parity n -> Nat
half (plus $n $n) (Even $n) = n
half (S (plus $n $m)) (Odd $n) = n