3

I am trying to implement a version of Braun Tree in Haskell, which requires me to prove this fact about Nats

If n :~: m, then n + 1 :~: m + 1

However, I am failing to fill the undefined below. I tried Refl, but the type system complains.

import GHC.TypeLits
import Data.Type.Equality

sucCong :: ((p :: Nat) :~: (q :: Nat)) -> (p + 1 :: Nat) :~: (q + 1 :: Nat)
sucCong proof = undefined

How do I prove this?

Agnishom Chattopadhyay
  • 1,971
  • 14
  • 32

1 Answers1

4

Pattern-match on the proof to make its statement available to the constraint solver. Then there is only one constructor for the type (:~:) so there's not much choice in the rest of the definition anyway.

sucCong :: (p :~: q) -> (p+1) :~: (q+1)
sucCong Refl = Refl
Li-yao Xia
  • 31,896
  • 2
  • 33
  • 56
  • 2
    So many functions involving `(:~:)` end up having that exact same implementation... – Carl Jul 11 '18 at 18:35