Let me use a shorter example than the posted one:
original :: Int -> Int
original n
| n < 10 && n > 7 = 1 -- matches 8,9
| n < 12 && n > 5 = 2 -- matches 6,7,10,11
| n < 12 && n > 3 = 3 -- matches 4,5
| n < 13 && n > 0 = 4 -- matches 1,2,3,12
The code runs in GHCi as follows:
> map original [1..12]
[4,4,4,3,3,2,2,1,1,2,2,4]
Our aim is to “group” together the two branches requiring with n < 12, factoring this condition out. (This is not a huge gain in the original toy example, but it could be in more complex cases.)
We could naively think of splitting the code in two nested cases:
wrong1 :: Int -> Int
wrong1 n = case () of
_ | n < 10 && n > 7 -> 1
| n < 12 -> case () of
_ | n > 5 -> 2
| n > 3 -> 3
| n < 13 && n > 0 -> 4
Or, equivalently, using the MultiWayIf extension:
wrong2 :: Int -> Int
wrong2 n = if
| n < 10 && n > 7 -> 1
| n < 12 -> if | n > 5 -> 2
| n > 3 -> 3
| n < 13 && n > 0 -> 4
This however, leads to surprises:
> map wrong1 [1..12]
*** Exception: Non-exhaustive patterns in case
> map wrong2 [1..12]
*** Exception: Non-exhaustive guards in multi-way if
The issue is that when n is 1, the n < 12 branch is taken, the inner case is evaluated, and then no branch there considers 1. The original code simply tries the next branch, which handles it. However, wrong1,wrong2 are not backtracking to the outer case.
Please note that this is not a problem when you know that the outer case has non-overlapping conditions. In the code posted by the OP, this seems to be the case, so the wrong1,wrong2 approaches would work there (as shown by Jefffrey).
However, what about the general case, where there might be overlaps? Fortunately, Haskell is lazy, so it’s easy to roll our own control structures. For this, we can exploit the Maybe monad as follows:
correct :: Int -> Int
correct n = fromJust $ msum
[ guard (n < 10 && n > 7) >> return 1
, guard (n < 12) >> msum
[ guard (n > 5) >> return 2
, guard (n > 3) >> return 3 ]
, guard (n < 13 && n > 0) >> return 4 ]
It is a bit more verbose, but not by much. Writing code in this style is easier than it might look: a simple multiway conditional is written as
foo n = fromJust $ msum
[ guard boolean1 >> return value1
, guard boolean2 >> return value2
, ...
]
and, if you want a “nested” case, just replace any of the return value with a msum [ ... ].
Doing this ensures that we get the wanted backtracking. Indeed:
> map correct [1..12]
[4,4,4,3,3,2,2,1,1,2,2,4]
The trick here is that when a guard fails, it generates a Nothing value. The library function msum simply selects the first non-Nothing value in the list. So, even if every element in the inner list is Nothing, the outer msum will consider the next item in the outer list — backtracking, as wanted.