diff --git a/src/Data/NumIdr/Array/Array.idr b/src/Data/NumIdr/Array/Array.idr
index 4da2a90..8e2f4b7 100644
--- a/src/Data/NumIdr/Array/Array.idr
+++ b/src/Data/NumIdr/Array/Array.idr
@@ -2,6 +2,7 @@ module Data.NumIdr.Array.Array
 
 import Data.List1
 import Data.Vect
+import Data.Zippable
 import Data.NumIdr.PrimArray
 import Data.NumIdr.Array.Order
 import Data.NumIdr.Array.Coords
@@ -95,6 +96,12 @@ getAllCoords (Z :: s) = []
 getAllCoords (S d :: s) = [| forget (allFins d) :: getAllCoords s |]
 
 
+constant' : (s : Vect rk Nat) -> (ord : Order) -> a -> Array s a
+constant' s ord x = MkArray ord (calcStrides ord s) s (create (product s) (const x))
+
+constant : (s : Vect rk Nat) -> a -> Array s a
+constant s = constant' s COrder
+
 ||| Create an array given a vector of its elements. The elements of the vector
 ||| are arranged into the provided shape using the provided order.
 |||
@@ -150,7 +157,7 @@ array' s ord v = MkArray ord sts s (unsafeFromIns (product s) ins)
 ||| Construct an array using a structure of nested vectors.
 ||| To explicitly specify the shape and order of the array, use `array'`.
 export
-array : {s : _} -> Vects s a -> Array s a
+array : {s : Vect rk Nat} -> Vects s a -> Array s a
 array v = MkArray COrder (calcStrides COrder s) s (fromList $ collapse v)
 
 
@@ -170,7 +177,10 @@ indexRange : (rs : CoordsRange s) -> Array s a -> Array (newShape rs) a
 indexRange rs arr = let ord = getOrder arr
                         s = newShape {s = shape arr} $ rewrite sym $ shapeEq arr in rs
                         sts = calcStrides ord s
-                    in  believe_me $ MkArray ord sts s (unsafeFromIns (product s) $ map (\(is,is') => (getLocation' sts is', index (getLocation' (strides arr) is) (getPrim arr))) $ getCoordsList {s = shape arr} $ rewrite sym $ shapeEq arr in rs)
+                        -- TODO: Make this actually typecheck without resorting to this mess
+                    in  believe_me $ MkArray ord sts s (unsafeFromIns (product s) $
+                                                          map (\(is,is') => (getLocation' sts is', index (getLocation' (strides arr) is) (getPrim arr))) $
+                                                          getCoordsList {s = shape arr} $ rewrite sym $ shapeEq arr in rs)
 
 
 --------------------------------------------------------------------------------
@@ -197,6 +207,15 @@ reshape : (s' : Vect rk' Nat) -> Array {rk} s a ->
 reshape s' arr = reshape' s' (getOrder arr) arr
 
 
+||| Change the internal order of the array's elements.
+export
+reorder : Order -> Array s a -> Array s a
+reorder ord' arr = let s = shape arr
+                       sts = calcStrides ord' s
+                   in  rewrite shapeEq arr
+                   in  MkArray ord' sts _ (unsafeFromIns (product s) $
+                              map (\is => (getLocation' sts is, index (getLocation' (strides arr) is) (getPrim arr))) $
+                              getAllCoords' s)
 
 
 export
@@ -221,9 +240,101 @@ stack axis a b = let sA = shape a
                      sts = calcStrides COrder s
                      ins = map (mapFst $ getLocation' sts . toNats) (enumerate a)
                            ++ map (mapFst $ getLocation' sts . updateAt axis (+dA) . toNats) (enumerate b)
+                     -- TODO: prove that the type-level shape and `s` are equivalent
                  in  believe_me $ MkArray COrder sts s (unsafeFromIns (product s) ins)
 
 
 
 display : Show a => Array s a -> IO ()
 display = printLn . PrimArray.toList . getPrim
+
+
+--------------------------------------------------------------------------------
+-- Implementations
+--------------------------------------------------------------------------------
+
+
+-- Most of these implementations apply the operation pointwise. If there are
+-- multiple arrays involved with different orders, then all of the arrays are
+-- reordered to match one.
+
+
+export
+Functor (Array s) where
+  map f (MkArray ord sts s arr) = MkArray ord sts s (map f arr)
+
+export
+Zippable (Array s) where
+  zipWith f a b = rewrite shapeEq a
+                  in MkArray (getOrder a) (strides a) _ $
+                     if getOrder a == getOrder b
+                        then unsafeZipWith f (getPrim a) (getPrim b)
+                        else unsafeZipWith f (getPrim a) (getPrim $ reorder (getOrder a) b)
+
+  zipWith3 f a b c = rewrite shapeEq a
+                     in MkArray (getOrder a) (strides a) _ $
+                        if (getOrder a == getOrder b) && (getOrder b == getOrder c)
+                           then unsafeZipWith3 f (getPrim a) (getPrim b) (getPrim c)
+                           else unsafeZipWith3 f (getPrim a) (getPrim $ reorder (getOrder a) b)
+                                                             (getPrim $ reorder (getOrder a) c)
+
+  unzipWith f arr = rewrite shapeEq arr
+                    in case unzipWith f (getPrim arr) of
+                            (a, b) => (MkArray (getOrder arr) (strides arr) _ a,
+                                       MkArray (getOrder arr) (strides arr) _ b)
+
+  unzipWith3 f arr = rewrite shapeEq arr
+                     in case unzipWith3 f (getPrim arr) of
+                             (a, b, c) =>  (MkArray (getOrder arr) (strides arr) _ a,
+                                            MkArray (getOrder arr) (strides arr) _ b,
+                                            MkArray (getOrder arr) (strides arr) _ c)
+
+export
+{s : _} -> Applicative (Array s) where
+  pure = constant s
+  (<*>) = zipWith apply
+
+export
+{s : _} -> Monad (Array s) where
+  join arr = fromFunction s (\is => index is $ index is arr)
+
+
+-- Foldable and Traversable operate on the primitive array directly. This means
+-- that their operation is dependent on the order of the array.
+
+export
+Foldable (Array s) where
+  foldl f z = foldl f z . getPrim
+  foldr f z = foldr f z . getPrim
+  null arr = size arr == Z
+  toList = toList . getPrim
+
+export
+Traversable (Array s) where
+  traverse f (MkArray ord sts s arr) =
+    map (MkArray ord sts s) (traverse f arr)
+
+
+export
+Eq a => Eq (Array s a) where
+  a == b = if getOrder a == getOrder b
+              then unsafeEq (getPrim a) (getPrim b)
+              else unsafeEq (getPrim a) (getPrim $ reorder (getOrder a) b)
+
+export
+Semigroup a => Semigroup (Array s a) where
+  (<+>) = zipWith (<+>)
+
+export
+{s : _} -> Monoid a => Monoid (Array s a) where
+  neutral = constant s neutral
+
+
+-- the shape must be known at runtime due to `fromInteger`. If `fromInteger`
+-- were moved into its own interface, this constraint could be removed.
+{s : _} -> Num a => Num (Array s a) where
+  (+) = zipWith (+)
+  (*) = zipWith (*)
+
+  fromInteger = constant s . fromInteger
+
diff --git a/src/Data/NumIdr/Array/Coords.idr b/src/Data/NumIdr/Array/Coords.idr
index 8e0b108..2882269 100644
--- a/src/Data/NumIdr/Array/Coords.idr
+++ b/src/Data/NumIdr/Array/Coords.idr
@@ -42,7 +42,6 @@ namespace Coords
       mapWithIndex' f (x::xs) = f Z x :: mapWithIndex' (f . S) xs
 
 
-  export
   index : Coords s -> Vects s a -> a
   index []      x = x
   index (i::is) v = index is $ index i v
diff --git a/src/Data/NumIdr/Array/Order.idr b/src/Data/NumIdr/Array/Order.idr
index 827ca78..c277771 100644
--- a/src/Data/NumIdr/Array/Order.idr
+++ b/src/Data/NumIdr/Array/Order.idr
@@ -21,6 +21,15 @@ data Order : Type where
   FOrder : Order
 
 
+public export
+Eq Order where
+  COrder == COrder = True
+  FOrder == FOrder = True
+  COrder == FOrder = False
+  FOrder == COrder = False
+
+
+
 scanr : (el -> res -> res) -> res -> Vect len el -> Vect (S len) res
 scanr _ q0 []      = [q0]
 scanr f q0 (x::xs) = f x (head qs) :: qs
diff --git a/src/Data/NumIdr/PrimArray.idr b/src/Data/NumIdr/PrimArray.idr
index 3477d4a..a033aff 100644
--- a/src/Data/NumIdr/PrimArray.idr
+++ b/src/Data/NumIdr/PrimArray.idr
@@ -1,5 +1,7 @@
 module Data.NumIdr.PrimArray
 
+import Data.Nat
+import Data.IORef
 import Data.IOArray.Prims
 
 %default total
@@ -92,3 +94,56 @@ export
 map : (a -> b) -> PrimArray a -> PrimArray b
 map f arr = create (length arr) (\n => f $ index n arr)
 
+
+export
+unsafeZipWith : (a -> b -> c) -> PrimArray a -> PrimArray b -> PrimArray c
+unsafeZipWith f a b = create (length a) (\n => f (index n a) (index n b))
+
+export
+unsafeZipWith3 : (a -> b -> c -> d) ->
+                 PrimArray a -> PrimArray b -> PrimArray c -> PrimArray d
+unsafeZipWith3 f a b c = create (length a) (\n => f (index n a) (index n b) (index n c))
+
+export
+unzipWith : (a -> (b, c)) -> PrimArray a -> (PrimArray b, PrimArray c)
+unzipWith f arr = (map (fst . f) arr, map (snd . f) arr)
+
+export
+unzipWith3 : (a -> (b, c, d)) -> PrimArray a -> (PrimArray b, PrimArray c, PrimArray d)
+unzipWith3 f arr = (map ((\(x,_,_) => x) . f) arr,
+                          map ((\(_,y,_) => y) . f) arr,
+                          map ((\(_,_,z) => z) . f) arr)
+
+
+export
+foldl : (b -> a -> b) -> b -> PrimArray a -> b
+foldl f z (MkPrimArray size arr) = unsafePerformIO $ do
+    ref <- newIORef z
+    for_ [0..pred size] $ \n => do
+        x <- readIORef ref
+        y <- arrayDataGet n arr
+        writeIORef ref (f x y)
+    readIORef ref
+
+export
+foldr : (a -> b -> b) -> b -> PrimArray a -> b
+foldr f z (MkPrimArray size arr) = unsafePerformIO $ do
+    ref <- newIORef z
+    for_ [pred size..0] $ \n => do
+         x <- arrayDataGet n arr
+         y <- readIORef ref
+         writeIORef ref (f x y)
+    readIORef ref
+
+export
+traverse : Applicative f => (a -> f b) -> PrimArray a -> f (PrimArray b)
+traverse f = map fromList . traverse f . toList
+
+
+||| Compares two primitive arrays for equal elements. This function assumes
+||| the arrays same the same length; it must not be used in any other case.
+export
+unsafeEq : Eq a => PrimArray a -> PrimArray a -> Bool
+unsafeEq a b = unsafePerformIO $
+                 map (concat @{All}) $ for [0..pred (arraySize a)] $
+                 \n => (==) <$> arrayDataGet n (content a) <*> arrayDataGet n (content b)