I recently found out that type holes combined with pattern matching on proofs provides a pretty nice Agda-like experience in Haskell. For example:
{-# LANGUAGE
DataKinds, PolyKinds, TypeFamilies,
UndecidableInstances, GADTs, TypeOperators #-}
data (==) :: k -> k -> * where
Refl :: x == x
sym :: a == b -> b == a
sym Refl = Refl
data Nat = Zero | Succ Nat
data SNat :: Nat -> * where
SZero :: SNat Zero
SSucc :: SNat n -> SNat (Succ n)
type family a + b where
Zero + b = b
Succ a + b = Succ (a + b)
addAssoc :: SNat a -> SNat b -> SNat c -> (a + (b + c)) == ((a + b) + c)
addAssoc SZero b c = Refl
addAssoc (SSucc a) b c = case addAssoc a b c of Refl -> Refl
addComm :: SNat a -> SNat b -> (a + b) == (b + a)
addComm SZero SZero = Refl
addComm (SSucc a) SZero = case addComm a SZero of Refl -> Refl
addComm SZero (SSucc b) = case addComm SZero b of Refl -> Refl
addComm sa@(SSucc a) sb@(SSucc b) =
case addComm a sb of
Refl -> case addComm b sa of
Refl -> case addComm a b of
Refl -> Refl
The really nice thing is that I can replace the right-hand sides of the Refl -> exp
constructions with a type hole, and my hole target types are updated with the proof, pretty much as with the rewrite
form in Agda.
However, sometimes the hole just fails to update:
(+.) :: SNat a -> SNat b -> SNat (a + b)
SZero +. b = b
SSucc a +. b = SSucc (a +. b)
infixl 5 +.
type family a * b where
Zero * b = Zero
Succ a * b = b + (a * b)
(*.) :: SNat a -> SNat b -> SNat (a * b)
SZero *. b = SZero
SSucc a *. b = b +. (a *. b)
infixl 6 *.
mulDistL :: SNat a -> SNat b -> SNat c -> (a * (b + c)) == ((a * b) + (a * c))
mulDistL SZero b c = Refl
mulDistL (SSucc a) b c =
case sym $ addAssoc b (a *. b) (c +. a *. c) of
-- At this point the target type is
-- ((b + c) + (n * (b + c))) == (b + ((n * b) + (c + (n * c))))
-- The next step would be to update the RHS of the equivalence:
Refl -> case addAssoc (a *. b) c (a *. c) of
Refl -> _ -- but the type of this hole remains unchanged...
Also, even though the target types do not necessarily line up inside the proof, if I paste in the whole thing from Agda it still checks fine:
mulDistL' :: SNat a -> SNat b -> SNat c -> (a * (b + c)) == ((a * b) + (a * c))
mulDistL' SZero b c = Refl
mulDistL' (SSucc a) b c = case
(sym $ addAssoc b (a *. b) (c +. a *. c),
addAssoc (a *. b) c (a *. c),
addComm (a *. b) c,
sym $ addAssoc c (a *. b) (a *. c),
addAssoc b c (a *. b +. a *. c),
mulDistL' a b c
) of (Refl, Refl, Refl, Refl, Refl, Refl) -> Refl
Do you have any ideas why this happens (or how I could do proof rewriting in a robust way)?
sym
calls inmulDistL'
and your code would still check. – kosmikus