Writer capability

An instance should fulfill the following laws. At this point these laws are not definitive, see https://github.com/haskell/mtl/issues/5.

listen @t (pure a) = pure (a, mempty)

listen @t (tell @t w) = tell @t w >> pure (w, w)

listen @t (m >>= k) = listen @t m >>= \(a, w1) -> listen @t (k a) >>= \(b, w2) -> pure (b, w1 `mappend` w2)

pass @t (tell @t w >> pure (a, f)) = tell @t (f w) >> pure a

writer @t (a, w) = tell @t w >> pure a

A note on the HasStream super class.

HasStream offers one yield method with the same signature as tell. Many people's intuition, however, wouldn't connect the two: yielding tosses the value down some black-box chute, while telling grows and accumulation via the monoid. The connection is since the chute is opaque, the tosser cannot rule out there being such an accumulation at the chutes other end.

Formally, we reach the same conclusion. HasStream has no laws, indicating the user can make no assumptions beyond the signature of yield. HasWriter, with tell defined as yield, is thus always compatable regardless of whatever additional methods it provides and laws by which it abides.

writer @tag (a, w) lifts a pure writer action (a, w) to a monadic action in an arbitrary monad m with capability HasWriter.

Appends w to the output of the writer capability tag and returns the value a.

tell @tag w appends w to the output of the writer capability tag.

listen @tag m executes the action m and returns the output of m in the writer capability tag along with result of m. Appends the output of m to the output of the writer capability tag.

pass @tag m executes the action m. Assuming m returns (a, f) and appends w to the output of the writer capability tag. pass @tag m instead appends w' = f w to the output and returns a.

Accumulate streamed values with their own monoid.