`case` that refines arguments

问题

In this answer on a question about the totality checker, a workaround involving using case instead of with was recommended.

However, in situations where the result of the match refines the type of other arguments, this transformation is not straightforward to make.

For example, given the following definitions:

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

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

the following definition typechecks, but is not accepted as total:

foo1 : Nat -> Nat
foo1 n with (parity n)
  foo1 n@(k + k) | Left (Evidence _ (Times2 k)) = (n * n)
  foo1 n@(S (k + k)) | Right (Evidence _ (Times2Plus1 k)) = (2 * (k * n + k))

whereas the following one doesn't typecheck:

foo2 : Nat -> Nat
foo2 n = case parity n of
  Left (Evidence k (Times2 k)) => n * n
  Right (Evidence k (Times2Plus1 k)) => (2 * (k * n + k))

due to

Type mismatch between
      IsEven (k + k) k (Type of Times2 k)
and
      IsEven n k (Expected type)

I also tried expanding the with in foo1:

foo1' : (n : Nat) -> Either (Exists (IsEven n)) (Exists (IsOdd n)) -> Nat
foo1' n@(k + k) (Left (Evidence k (Times2 k))) = (n * n)
foo1' n@(S (k + k)) (Right (Evidence k (Times2Plus1 k))) = 2 * (k * n + k)

foo1 : Nat -> Nat
foo1 n = foo1' n (parity n)

but here, foo1' is also not accepted as total:

 foo1' is not total as there are missing cases

回答1:

One effective workaround I've found for this is to re-bind n in each branch, i.e. to write foo2 as

foo2 : Nat -> Nat
foo2 n = case parity n of
  Left (Evidence k (Times2 k)) => let n = k + k in n * n
  Right (Evidence k (Times2Plus1 k)) => let n = S (k + k) in (2 * (k * n + k))

Unfortunately, while this works for this simplified problem from my original question, it doesn't work when the types of the various branches differ; for example in

total Foo : Nat -> Type
Foo n = case parity n of
    Left (Evidence k (Times2 k)) => ()
    Right (Evidence k (Times2Plus1 k)) => Bool

foo : (n : Nat) -> Foo n
foo n = case parity n of
    Left (Evidence k (Times2 k)) => ()
    Right (Evidence k (Times2Plus1 k)) => True

which fails with

 Type mismatch between
         () (Type of ())
 and
         case parity (plus k k) of
           Left (Evidence k (Times2 k)) => ()
           Right (Evidence k (Times2Plus1 k)) => Bool (Expected type)

来源：https://stackoverflow.com/questions/46563713/case-that-refines-arguments