[Haskell] Lemmas about type functions
ryani.spam at gmail.com
Fri Feb 15 18:05:44 EST 2008
I am pretty sure that this doesn't exist, but it's quite interesting.
I've submitted a feature request here:
On 2/15/08, Louis-Julien Guillemette <guillelj at iro.umontreal.ca> wrote:
> 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 ())
> 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
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