Hi Henning, Am Mittwoch, den 03.10.2012, 19:52 +0200 schrieb Henning Thielemann:
I wondered whether there is a brilliant typing technique that makes Data.Map.! a total function. That is, is it possible to give (!) a type, such that m!k expects a proof that the key k is actually present in the dictionary m? How can I provide the proof that k is in m?
I think it is possible to do this using the same trick that ST is using, i.e. rank 2 types. The problem is that this code is likely to be unwieldy to use, as the variable needs to change with every change to the map – but I guess if you create your map once and then use it in various places without modification, this can actually work. So we need to tag values (maps and keys) with variables. We could use Edward Kmett’s tagged library, but for now lets just roll our own (full code in the right order attached): newtype Tagged a x = Tag { unTag :: x } retag :: Tagged t1 x -> Tagged t2 x retag (Tag x) = (Tag x) The idea is that if you have a "Tagged a (Map k v)" and a "Tagged a k", where the a’s are the same variable, then you have a guarantee that the key is in the map. In the code we are writing now we can easily break that, e.g. using retag, but hopefully not via the functions that we are going to export in the end. The lookup function just ensures that the tags agree, if all goes well the "fromJust" will never fail: lookup :: Ord k => Tagged t k -> Tagged t (Map k v) -> v lookup (Tag k) (Tag m) = fromJust $ M.lookup k m We want a function that takes an existing map and tags it. As with most of the functions that follow, we use continuation passing style with a rank-2-function to ensure that no assumptions can be made about the tag variable at all. fromMap :: Map k v -> (forall t. Tagged t (Map k v) -> c) -> c fromMap m f = (f (Tag m)) The user of this code has no tagged keys of this type yet, so he cannot use lookup yet. Let’s allow him to add an entry: insert :: Ord k => k -> v -> Tagged t1 (Map k v) -> (forall t2. Tagged t2 (Map k v) -> Tagged t2 k -> (Tagged t1 k -> Tagged t2 k) -> c) -> c insert k v (Tag m) f = f (Tag (M.insert k v m)) (Tag k) retag Besides the usual arguments (key, value, map), it takes a continuation. This receives the updated map with a /new/ tag – after all, our knowledge about it has changed. We also return the tagged key; this is the proof (or the certificate) that the key is in the map that can be passed to lookup. And finally we pass a function that takes a proof „k is in the original map“ and turns it into a proof „k is in the update map“. We can also allow the programmer to check if a key is present, and obtain a proof if that is the case: check :: Ord k => k -> Tagged t1 (Map k v) -> (forall t2. Tagged t2 (Map k v) -> Maybe (Tagged t2 k) -> (Tagged t1 k -> Tagged t2 k) -> c) -> c check k (Tag m) f = f (Tag m) (if M.member k m then Just (Tag k) else Nothing) retag This would be useful when a non-empty map is converted with fromMap. The type of this function could be varied, e.g. by not creating a new tag at all if the type is not present. Finally we need to select the functions that are safe to use for the export list: module SafeMap (Tagged, unTag, fromMap, insert, lookup, check) More functions need to be added, e.g. an adjust (which luckily would not require a continuation-style type). Here is some code that uses the library to implement fromList (it is nice to see that this is possible given the primitives above): {-# LANGUAGE Rank2Types #-} import Data.Map (Map, empty) import SafeMap fromList :: Ord k => [(k,v)] -> (forall t. Tagged t (Map k v) -> [Tagged t k] -> c) -> c fromList [] f = fromMap empty $ \m -> f m [] fromList ((k,v):l) f = fromList l $ \m tks -> insert k v m $ \m' tk rt -> f m' (tk : map rt tks) And here is it in use: *Main> fromList [(1,1),(2,2)] (curry show) "(Tag {unTag = fromList [(1,1),(2,2)]},[Tag {unTag = 1},Tag {unTag = 2}])" The "map rt" will make this quadratic, as GHC does not detect if you map an (operationally speaking) identity over a list. Hopefully that will be fixed eventually: http://hackage.haskell.org/trac/ghc/ticket/2110 Is this anything like what you wanted? Greetings, Joachim -- Joachim "nomeata" Breitner mail@joachim-breitner.de | nomeata@debian.org | GPG: 0x4743206C xmpp: nomeata@joachim-breitner.de | http://www.joachim-breitner.de/