GADT的详尽检查失败了

Question

Consider the following code:

data (:+:) f g a = Inl (f a) | Inr (g a)

data A
data B

data Foo l where
  Foo :: Foo A

data Bar l where
  Bar :: Bar B

type Sig = Foo :+: Bar

fun :: Sig B -> Int
fun (Inr Bar) = 1

Even though fun is an exhaustive match, when compiling with -Wall, GHC complains about a missing case. However, if I add another constructor:

data (:+:) f g a = Inl (f a) | Inr (g a)

data A
data B

data Foo l where
  Foo :: Foo A
  Baz :: Foo B

data Bar l where
  Bar :: Bar B

type Sig = Foo :+: Bar

fun :: Sig B -> Int
fun (Inr Bar) = 1
fun (Inl Baz) = 2

Then GHC correctly detects that fun is total.

I am using code similar to this in my work, and would like GHC to raise warnings if I have missed cases, and not raise warnings if I don't. Why is GHC complaining on the first program, and how can I get the first sample to compile without warnings without adding spurious constructors or cases?

Answer 1

The problem actually reported is:

Warning: Pattern match(es) are non-exhaustive
         In an equation for `fun': Patterns not matched: Inl _

Which is true. You provide a case for the Inr constructor, but not the Inl constructor.

What you're hoping is that since there's no way to provide a value of type Sig B that uses the Inl constructor (it would need an argument of type Foo B , but the only constructor for Foo is of type Foo A ), that ghc will notice that you don't need to handle the Inl constructor.

The trouble is that due to bottom every type is inhabited. There are values of type Sig B that use the Inl constructor; there are even non-bottom values. They must contain bottom, but they are not themselves bottom. So it is possible for the program to be evaluating a call to fun that fails to match; that's what ghc is warning about.

So to fix that you need to change fun to something like this:

fun :: Sig B -> Int
fun (Inr Bar) = 1
fun (Inl foo) = error "whoops"

But now of course if you later add Baz :: Foo B this function is a time bomb waiting to happen. It's be nice for ghc to warn about that , but the only way to make that happen is to pattern match foo against a currently-exhaustive set of patterns. Unfortunately there are no valid patterns you can put there! foo is known to be of type Foo B , which is only inhabited by bottom, and you can't write a pattern for bottom.

But you could pass it to a function that accepts an argument of polymorphic type Foo a . That function could then match against all the currently-existing Foo constructors, so that you'll get a warning if you later add one. Something like this:

fun :: Sig B -> Int
fun (Inr Bar) = 1
fun (Inl foo) = errorFoo foo
    where 
        errorFoo :: Foo a -> b
        errorFoo Foo = error "whoops"

Now You've properly handled all the constructors of :+: in fun , the "impossible" case simply errors out if it ever actually occurs and if you ever add Baz :: Foo B you get a warning about a non-exhaustive pattern in errorFoo , which is at least directing you to look at fun because it's defined in an attached where .

On the downside, when you add unrelated constructors to Foo (say more of type Foo A ) you'll have to add more cases to errorFoo , and that could be unfun (though easy and mechanical) if you've got lots of functions applying this pattern.

Answer 2

I'm sorry to tell you this, but your first example is not quite as exhaustive as you think it is:

∀x. x ⊢ fun (Inl (undefined :: Foo B))
*** Exception: Test.hs:48:1-17: Non-exhaustive patterns in function fun

Annoying, yes, but them's the breaks. ⊥ is why we can't have nice things. :[

Answer 3

As already mentioned. The case you are not handling is Inl _|_ , which is not itself _|_ , and thus must be handled.

Luckily there is a perfectly nice way to handle this:

data (:+:) f g a = Inl (f a) | Inr (g a)

data A
data B

data Foo l where
  Foo :: Foo A
  Baz :: Foo B

data Bar l where
  Bar :: Bar B

type Sig = Foo :+: Bar

fun :: Sig B -> Int
fun (Inr Bar) = 1
fun (Inl x) = case x of {}

Now if you do add in the Baz :: Foo B constructor, you will appropriately get:

    Pattern match(es) are non-exhaustive
    In a case alternative: Patterns not matched: Baz
   |
21 | fun (Inl x) = case x of {}
   |               ^^^^

Thus you can appropriately change the code to something like your second example to properly handle the new case you have created.