idristerminationtotality

Why does this 'with' block spoil the totality of this function?


I'm trying to compute parity together with the floor of the half, over natural numbers:

data IsEven : Nat -> Nat -> Type where
    Times2 : (n : Nat) -> IsEven (n + n) n

data IsOdd : Nat -> Nat -> Type where
    Times2Plus1 : (n : Nat) -> IsOdd (S (n + n)) n

parity : (n : Nat) -> Either (Exists (IsEven n)) (Exists (IsOdd n))

I tried going with the obvious implementation of parity:

parity Z = Left $ Evidence _ $ Times2 0
parity (S Z) = Right $ Evidence _ $ Times2Plus1 0
parity (S (S n)) with (parity n)
  parity (S (S (k + k))) | Left (Evidence _ (Times2 k)) =
      Left $ rewrite plusSuccRightSucc k k in Evidence _ $ Times2 (S k)
  parity (S (S (S ((k + k))))) | Right (Evidence _ (Times2Plus1 k)) =
      Right $ rewrite plusSuccRightSucc k k in Evidence _ $ Times2Plus1 (S k)

This typechecks and works as expected. However, if I try to mark parity as total, Idris starts complaining:

 parity is possibly not total due to: with block in parity

The only with block I see in parity is the one with the recursive call from parity (S (S n)) to parity n, but clearly that is well-founded, since n is structurally smaller than S (S n).

How do I convince Idris that parity is total?


Solution

  • It looks like a bug to me, because the following solution based on case passes the totality checker:

    total
    parity : (n : Nat) -> Either (Exists (IsEven n)) (Exists (IsOdd n))
    parity Z = Left $ Evidence _ $ Times2 0
    parity (S Z) = Right $ Evidence _ $ Times2Plus1 0
    parity (S (S k)) =
      case (parity k) of
        Left (Evidence k (Times2 k)) =>
          Left $ rewrite plusSuccRightSucc k k in Evidence _ $ Times2 (S k)
        Right (Evidence k (Times2Plus1 k)) =>
          Right $ rewrite plusSuccRightSucc k k in Evidence _ $ Times2Plus1 (S k)