on b u x y runs the binary function b on the results of applying unary function u to two arguments x and y. From the opposite perspective, it transforms two inputs and combines the outputs.

((+) `on` f) x y = f x + f y

Typical usage: sortBy (compare `on` fst). Algebraic properties:

• (*) `on` id = (*) -- (if (*) ∉ {⊥, const
  ⊥})

• ((*) `on` f) `on` g = (*) `on` (f . g)

• flip on f . flip on g = flip on (g .
  f)

Like finally, but only performs the final action if there was an exception raised by the computation.

A more concise version of complement zeroBits.

>>> complement (zeroBits :: Word) == (oneBits :: Word)
True

>>> complement (oneBits :: Word) == (zeroBits :: Word)
True

Note

The constraint on oneBits is arguably too strong. However, as some types (such as Natural) have undefined complement, this is the only safe choice.