data Bool : Type where
     False : Bool
     True : Bool

not : Bool -> Bool
not False = True
not True = False

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)

ack : Nat -> Nat -> Nat
ack Z n = S n
ack (S k) Z = ack k (S Z)
ack (S j) (S k) = ack j (ack (S j) k)

foo : Nat -> Nat
foo Z = Z
foo (S Z) = (S Z)
foo (S (S k)) = foo (S k)

foo' : Nat -> Nat
foo' Z = Z
foo' (S Z) = (S Z)
foo' (S p@(S k)) = foo' p

data Bin : Type where
     EPS : Bin
     C0 : Bin -> Bin
     C1 : Bin -> Bin

foom : Bin -> Nat
foom EPS = Z
foom (C0 EPS) = Z
foom (C0 (C1 x)) = S (foom (C1 x))
foom (C0 (C0 x)) = foom (C0 x)
foom (C1 x) = S (foom x)

pfoom : Bin -> Nat
pfoom EPS = Z
pfoom (C0 EPS) = Z
pfoom (C0 (C1 x)) = S (pfoom (C0 x))
pfoom (C0 (C0 x)) = pfoom (C0 x)
pfoom (C1 x) = S (foom x)