-- <pre>


plus:: Nat ~> Nat ~> Nat
{plus Z m} = m
{plus (S n) m} = S {plus n m}


plusZ :: Nat' n -> Equal {plus n Z} n
plusZ Z = Eq
plusZ (S m) = Eq
  where theorem indHyp = plusZ m

plusAssoc :: Nat' n -> Equal {plus {plus n b} c} {plus n {plus b c}}
plusAssoc n = undefined

times:: Nat ~> Nat ~> Nat
{times Z m} = Z
{times (S n) m} = {plus m {times n m}}


timesZ:: Nat' n -> Equal {times n Z} Z
timesZ Z = Eq
timesZ (S n) = Eq
  where theorem ih = timesZ n

distr:: Nat' x -> Equal {times {plus x y} z} {plus {times x z} {times y z}}
distr Z = Eq
distr (S n) = Eq
  where theorem ih = distr n
        theorem plusAssoc

oneTimes:: Nat' n -> Equal {times 1t n} n
oneTimes Z = Eq
oneTimes (S m) = Eq
  where theorem ih = plusZ m


-- These functions are homework for Next week
-- See Programming in Omega. Exercise 27, page 52


{-
plusS :: Nat' n -> Equal {plus n (S m)} (S{plus n m})
plusCommutes :: Nat' n -> Nat' m -> Equal {plus n m} {plus m n}
plusAssoc :: Nat' n -> Equal {plus {plus n b} c} {plus n {plus b c}}
plusNorm :: Nat' x -> Equal {plus x {plus y z}} {plus y {plus x z}}
-}
-- </pre>