(&&&) :: a b c -> a b c' -> a b (c,c')
f &&& g = arr (\b -> (b,b)) >>> f *** g
+{-# RULES
+"compose/arr" forall f g .
+ arr f >>> arr g = arr (f >>> g)
+"first/arr" forall f .
+ first (arr f) = arr (first f)
+"second/arr" forall f .
+ second (arr f) = arr (second f)
+"product/arr" forall f g .
+ arr f *** arr g = arr (f *** g)
+"fanout/arr" forall f g .
+ arr f &&& arr g = arr (f &&& g)
+"compose/first" forall f g .
+ first f >>> first g = first (f >>> g)
+"compose/second" forall f g .
+ second f >>> second g = second (f >>> g)
+ #-}
+
-- Ordinary functions are arrows.
instance Arrow (->) where
where untag (Left x) = x
untag (Right y) = y
+{-# RULES
+"left/arr" forall f .
+ left (arr f) = arr (left f)
+"right/arr" forall f .
+ right (arr f) = arr (right f)
+"sum/arr" forall f g .
+ arr f +++ arr g = arr (f +++ g)
+"fanin/arr" forall f g .
+ arr f ||| arr g = arr (f ||| g)
+"compose/left" forall f g .
+ left f >>> left g = left (f >>> g)
+"compose/right" forall f g .
+ right f >>> right g = right (f >>> g)
+ #-}
+
instance ArrowChoice (->) where
left f = f +++ id
right f = id +++ f