为什么此"with"块会破坏此功能的整体? [英] Why does this 'with' block spoil the totality of this function?

问题描述

我正在尝试通过自然数来计算奇偶校验和一半的底数:

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))

我尝试使用parity的明显实现:

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)

此类型检查并按预期工作.但是，如果我尝试将parity标记为total，则Idris开始抱怨:

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

我在parity中看到的唯一with块是从parity (S (S n))到parity n的递归调用的块，但是显然，这是有充分根据的，因为n在结构上小于S (S n) .

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).

如何使Idris相信parity是总数?

How do I convince Idris that parity is total?

推荐答案

对我来说这似乎是个错误，因为以下基于case的解决方案通过了总计检查器:

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)

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