Hi all, I've been using GHC's type families somewhat extensively in my type-preserving compiler (BTW, they work great), and quite often I come across the need to prove lemmas about those functions. As far as I understand there's currently no way to handle such lemmas purely at the type level, and I have to encode them as term-level functions. I wonder if I'm missing something, or otherwise if there are plans to provide some way to do this kind of type-level reasoning. Here's a simple example. I encode (de Bruijn) type contexts as lists of types of this form: (t0, (t1, (... , ()...))) I sometimes concatenate type contexts, and need a lemma stating that appending an empty context leaves it unchanged (ts ++ [] == ts). type family Cat ts0 ts type instance Cat () ts' = ts' type instance Cat (s, ts) ts' = (s, Cat ts ts') That is, I need to coerce: Exp (Cat ts ()) into: Exp ts The way I presently do it is through a term-level function that produces a witness that the two types are equivalent, like this: data Equiv s t where Equiv :: (s ~ t) => Equiv s t cat_nil :: EnvRep ts -> Equiv (Cat ts ()) ts cat_nil R0 = Equiv cat_nil (Rs _ ts_r) = case cat_nil ts_r of Equiv -> Equiv coerce :: EnvRep ts -> Exp (Cat ts ()) -> Exp ts coerce ts_r e = case cat_nil ts_r of Equiv -> e Louis