water.hs

> import Control.Monad.Tardis
> :{
let levels hs = evalTardis (traverse 
      (\h -> do x <- min <$> getFuture <*> getPast -- Get the height of the enclosing walls
                modifyBackwards (`max` h)          -- Send the right max back to the past
                modifyForwards (`max` h)           -- Send the left max into the future
                pure (max 0 (x - h))               -- Calculate the height of the water itself
      ) hs) (0, 0)
:}
> sum $ levels [5,3,7,2,6,4,5,9,1,2]
14

This code has a slight subtlety in it that I wouldn't call a bug, strictly speaking, but is perhaps inhibitory to understanding the exact semantics of the Tardis Monad.

If we have a situation where the max height in the Future direction is less than the current height, then the modifyBackwards causes the getFuture on the previous line to return h, not the actual max height in the future direction. In that case, we know that x is either h, because there is a bigger number in the Past, or x is less than h, because the Past was a smaller number. Therefore, x - h <= 0, and we end up with the right answer in the end.

However, it's not quite what the code should be doing.