Add array document notes

packages/@ot-doc/document/src/lib/array-document.ts

@@ -0,0 +1,49 @@
+
+/**
+ * Array document
+ * Given an ordered set S, we can define an array document model.
+ * (As the set theory notations to express Array/List are not convenient to
+ * use, I'll switch to Haskell notation instead)
+ *    @8.1 T:
+ *      data E s = Ins Integer s
+ *               | Del Integer s
+ *      type T s = [E s]
+ *    @8.2 Inv:
+ *      inv :: Order s => [E s] -> [E s]
+ *      inv = fmap $ \case
+ *                    Ins n s -> Del n s
+ *                    Del n s -> Ins n s
+ *    @8.3 Comp:
+ *      comp :: Order s => [E s] -> [E s] -> Maybe [E s]
+ *      comp xs ys = combine (inverse xs) ys where
+ *        combine [] ys         = Just ys
+ *        combine xs []         = Just (inverse xs)
+ *        combine (x:xs) (y:ys) =
+ *          case (x, y) of
+ *            (Ins m u) (Del n v) -> if m == n
+ *                                    then if u == v then comp xs ys
+ *                                                   else Nothing
+ *                                    else combine xs (x:y:ys)
+ *            (Del m u) (Ins n v) -> if m == n
+ *                                    then if u == v then comp xs ys
+ *                                                   else Nothing
+ *                                    else combine xs (x:y:ys)
+ *            otherwise           -> combine xs (x:y:ys)
+ *    @8.4 Tran:
+ *      tran :: Order s => [E s] -> [E s] -> Maybe [E s]
+ *      tran = liftTran tranElem where
+ *        tranElem :: E s -> E s -> Maybe [E s]
+ *        tranElem (Ins m x) (Ins n y)
+ *          | m < n     = Ins m x
+ *          | m > n     = Ins (m + 1) x
+ *          | x < y     = Ins m x
+ *          | otherwise = Ins (m + 1) x
+ *        tranElem (Del m x) (Ins n y)
+ *          | m < n     = Del m x
+ *          | otherwise = Del (m + 1) x
+ *        tranElem (Ins m x) (Del n y)
+ *          | m <= n    = Ins m x
+ *          | otherwise = Ins (m - 1) x
+ *        tranElem (Del m x) (Del n y)
+ *          | 
+ */