-- widened borrow of a particular repeated amount 
-- i.e., for borrow2 2, the borrow is 22 = 100
borrow2 b | b == 1    = error "Cannot borrow 1: implies positional symbol zero disallowed"
          | otherwise = borrow' [] b
  where borrow' zs b (x:y:z:xs) | abs x >= b = zipUp ys zs
                                | otherwise  = borrow' (x:zs) b (y:z:xs)
                                  where ys     = r : y-x+r : z+q : xs
                                        (q, r) = x `quotRem` b
        zipUp = foldl (flip (:))

-- truncate the adic expansion to n digits, for a system where borrows are 2 wide and both b
truncadic b n = take n . (!! n) . iterate (borrow2 b)

--same but, produce a list of expansions, taking modulo m in between borrows
truncadicmod b m n = map (take n) . take n . iterate (map (`rem` m) . borrow2 2)

-- find moduli starting from s where truncations agree
-- caveat: the last of truncadicmod is not guaranteed to exhaust terms outside alphabet
-- i.e., for b = 2, the alphabet {-1, 0, 1}
findaccurate b n s xs = map fst $ filter ((==good) . snd) bads
  where good = truncadic b n xs
        bads = map (\m -> (,) m $ last $ truncadicmod b m n xs) [s..]

--given an amount of digits `n`, a 2-wide borrow `b`, and a canonical representation of `b`
--construct integer multiples of `b`
evens n b = ([0]:) . map (take n) . iterate addb
  --add b to an adic expansion
  where addb = (!! n ) . iterate (borrow2 b) . (b:) . tail

--first 200 digits of cendree-adic expansions of even numbers
adics  = evens 200 2 $ 0:0:cycle [1,-1]
adic2  = adics !! 1
adic4  = adics !! 2
adic4' = truncadic 2 200 $ (++repeat 0) $ map (*2) adic2

-- note: `truncadic 2 n $ 4:repeat 0` is the aggressive application of the carry to 4
-- but this could be done better since there are only 3 terms in the head, and no higher
-- series terms get in the way; the remainders would be emitted while the thunk continued