(Moving this discussion to glasgow-users. It's just not appropriate on the cafe.)
I am no longer a novice, and yet would still have a hard time making any use of the laws as written in constructing instances. Instead, I'd ignore the laws and write a natural intuitive instance, and it would invariably work.
Seems my approach is very similar to Viktor's. My (very informal) understanding of the Laws looks nothing like the docos. I regard Foldable structures as merely more efficient ways to hold a List. Then I expect 'moral equivalences':
toList . fromList ~=~ id -- going via the Foldable structure fromList . toList ~=~ id toList ~=~ foldr (:) []
But those aren't equalities. 'moral equivalence' means the Lists have the same elements, not necessarily in the same order; the structures have the same elements but possibly in a different arrangement -- that is, in the `Tree` example, there might be `Empty` scattered about, and elements held variously in `Leaf`s vs `Node`s. So more accurately:
fromList . toList . fromList === fromList -- i.e. there's a 'canonical' arrangement toList . fromList . toList === toList -- i.e. there's a 'canonical' List ordering
(That triple-journey business is a similar style to defining Lattice pseudocomplements https://en.wikipedia.org/wiki/Pseudocomplement#Properties -- if I can chuck in some math theory.) I'd expect all other methods to be one of: `reduceStuff === reduceStuff . toList` or `mapStuff === fromList . mapStuff . toList`. But! there's no method `fromList` in Foldable. Why not?/please explain. (Are there Foldable structures which we can't load from a List? At least assuming the List is finite.) `fromList` is the first thing I write after declaring the datatype, so I can easily load up some test data. There is one example `fromList` in the doco. Is that not generalisable? `foldMap Leaf` would be brutal, but should work? `foldMap singleton` ? (But there's no method `singleton`.)